Research ArticleBooms and Busts in the Oil Market Identifying SpeculativeBubbles Using a Continuous-Time Dynamic System

Kaizhi Yu and Yun Zhang

School of Statistics Southwestern University of Finance and Economics Chengdu 611130 China

Correspondence should be addressed to Yun Zhang zyunsmailswufeeducn

Received 6 September 2020 Revised 23 October 2020 Accepted 19 December 2020 Published 7 January 2021

Academic Editor Shouwei Li

Copyright copy 2021 Kaizhi Yu and Yun Zhang -is is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

-e sharp changes in oil prices since 2004 featured a nonlinear data-generating mechanism which displayed bubble-like behaviorA popular view is that such a salient pattern cannot be explained by shifts in economic fundamentals but was driven by speculativebubbles as a consequence of the increased financialization of oil future markets Testing this hypothesis however is challengingsince the fundamental component of the oil price is unobservable -is paper attempts to isolate the contribution of speculativebubbles and fundamentals to the evolution of oil prices by providing a stylized model of commodity pricing Motivated by ourtheoretical model we adopt a continuous-time model with a random and time-varying persistence parameter to empiricallyinvestigate the presence of speculative bubbles in daily oil future prices over the period April 1983 to June 2020 We do not findany evidence in favor of speculative bubbles although we indeed find that oil prices exhibit episodes of unstable behaviorafter 2004

1 Introduction

Figure 1 displays two leading crude oil prices from 1986 to2020 using daily data As is evident from the figure onestriking characteristic of the oil prices has been the pro-longed buildups and dramatic collapses over the past twodecades -e most remarkable spike in the data series oc-curred from 2004 to mid-2008 followed by a rapid decline-e price of oil experienced another episode of slump be-tween 2014 and 2016 In April 2020 it had plummeted to anall-time low falling into the negative territory Such salientempirical patterns have led many observers to attribute thesesteep price changes to the presence of speculative bubbles inthe oil market [1ndash3] One of their reasons is that the massiveoil price fluctuations have coincided with the increasedfinancialization of oil future markets since the early 2000s(for a comprehensive review of financialization of com-modity markets see Cheng and Xiong [4]) A similar ar-gument is also made in Tang and Xiong [5] Can thewidespread claim be substantiated

In this paper we exploit the insights provided by atheoretical model of rational expectation commodity pricingin conjunction with empirical study to investigate the ex-istence and duration of speculative bubbles in the volatile oilprices To achieve this goal we develop a reduced-formmodel of demand and supply for separating the contributionof speculative bubbles and traditional economic funda-mentals to oil price movements On this basis our theo-retical formulation decomposes the oil price into twocomponents the fundamental component and the unob-served bubble component In particular the model impliesthat bubbles if they do exist should manifest explosivedynamic features in prices From an empirical perspectivethis appealing property reduces bubble detection to exam-ining evidence for stochastic explosive behavior in priceseries To proceed we use a continuous-time dynamicsystem to analyze daily oil prices over the period April 41983 through to June 30 2020 -is novel approach pro-vides a recursive test procedure that delivers a mechanismfor real-time detecting various forms of unstable and ex-plosive behavior and date-stamping the origination and

HindawiComplexityVolume 2021 Article ID 8883416 19 pageshttpsdoiorg10115520218883416

termination of market exuberance Overall our methodol-ogy provides no evidence in support of speculative bubblesin oil prices over the sample period examined despite thefact that the series displays periods of instability dynamicsafter 2004

-is paper is related to different strands of the literature-eoretically rational expectation bubble models are welldocumented in long literature studying asset pricing [6ndash9]Diba and Grossman [7] first pointed out under a standardasset-pricing framework with rational risk-neutral agentsthat the detection of explosive (submartingale) behavior canbe reflective of a bubble-like phenomenon -e similar ar-gument has also been put forward by Pavlidis et al [10]Martınez-Garcıa and Grossman [11] and Shi et al [12]amongmany others specifically in the context of the stylizedasset-pricing model for housing Our paper is not the first tolook at whether bubbles are a key characteristic of thechanges in oil prices In fact a growing body of empiricalresearch has extensively applied the supremum augmentedDickeyndashFuller (SADF) test of Phillips et al [13] (PWYhereafter) and generalized supremum augmented Dick-eyndashFuller (GSADF) test proposed by Phillips et al [14 15](PSY hereafter) to examine the possible presence of spec-ulative bubbles [16ndash18] -e general idea of the SADF andGSADF methods involves a recursive evolving algorithmcoupled with right-tailed unit root tests In contrast to ourconclusion these studies provide some evidence of specu-lative bubbles Finally this paper also speaks to the extantliterature that studies the role of speculation in the fluctu-ations of oil prices based on structural vector autore-gressions (SVARs) -ese studies find strong evidence thathistorical oil price movements had suffered little if anyspeculation effects [19ndash22]

Our paper makes several contributions First buildingupon prior analysis by Campbell and Shiller [23] andPindyck [24] we contribute to the literature by refining themodel of rational commodity pricing From a theoreticalperspective the price evolution of storable commoditiessuch as crude oil does not directly fit in the theory of classicalrational bubbles whereby the fundamental value is

approximated by a stream of expected future dividendpayments In order to determine the fundamental level of oilprices our model makes the link between the latent con-venience yield and a reduced-form model of dynamic de-mand and supply Second compared to active literaturewhich relies on a recursive method implemented in a mildlyexplosive discrete time model we employ a novel econo-metric approach for detecting bubbles Inspired by the workof Tao et al [25] the present paper estimates a continuous-time model a model that allows for a random persistenceparameter and its endogeneity -e advantages of extendinga discrete time model to continuous time are threefold Firstthe persistence parameter as is well known is not deter-mined by the choice of data frequency in the continuoustime framework [26] Consequently the empirical resultsabout the explosive behavior in continuous-time models areless sensitive to sampling frequency than that in a discretetime setting Second our continuous system formulationsreadily accommodate an initial condition and drift effectsgiving a limit theory without nuisance parameters Bycontrast these additional unknown parameters appear indiscrete time models thereby complicating inference -irdcombined with high-frequency data the continuous-timemodel enables us to gain substantial power in identifyingbubbles that collapse periodically especially when these bub-bles are short-lived To the best of our knowledge this paper isthe first to use the above methodologies to detect any potentialextreme behavior in the oil market -ird unlike the SVARliterature our goal is not to precisely identify the sources ofprice shocks but rather to see whether speculation plays acentral role in the price of oil In this sense this paper offers acomplementary approach Our findings also add furthersupport to a fundamental-driven explanation for the recenthigh price volatility in the oil market

-e remainder of the paper is organized as follows Afterreviewing the relevant literature in the next section we lay out asimple theoretical framework in Section 3 Section 4 introducesthe econometric methodology and describes the bubble de-tection strategy in more detail Section 5 describes the dataemployed in the empirical analysis while in Section 6 wepresent the main empirical results and evaluate the sensitivityof our findings Section 7 discusses and compares the resultswith other studies Section 8 contains some concluding re-marks and draws out substantive policy implications

2 Literature Review

Over the past decades and particularly after the 2007-2008global financial crisis the rapid surge and subsequent col-lapse of asset prices more loosely called as speculativebubbles have reawakened interests in econometric tests ofbubbles and have enlivened an emerging string of literatureaimed at date-stamping strategies Speculative bubbles havea long-standing history in asset markets Well-known his-torical episodes comprise the 17th-century ldquoTulip Maniardquoand the Mississippi Bubble of 1718ndash1720 Such speculativebehavior and crashes in the financial sector can triggerserious economic recessions such as the great stock marketcrash episode prior to the Great Depression or the recent

86 88 90 92 94 96 98 00 02 04 06 08 10 12 14 16 18 20

020

ndash20ndash40

406080

100120140

WTI spot priceBrent spot price

Figure 1 Time series of daily spot prices for crude oil WTI andBrent

2 Complexity

subprime mortgage crisis followed by the global economicshock -ere is a general consensus that speculative bubblesare defined as the divergence of an assetrsquos price from itsintrinsic value [27]

-e classic rational bubble is the standard theoreticalframework to model the presence and persistence of speculativebubbles with seminal papers by Blanchard and Watson [6]Tirole [28 29] Flood and Hodrick [30] and Froot and Obstfeld[31] In this workhorse model of bubbles it is economicallyrational for investors to buy overpriced securities today that areunjustified by fundamental factors as long as they believe that theselling price becomes high tomorrow In spite of the central roleof rational bubbles in the theoretical literature empirical evi-dence in support of their existence remains elusive -e chal-lenges in empirically testing for the potential bubbles lie ininvestigating their explosivity and dating their occurrence Earlyempirical studies have designed and adopted various time seriesmethods to capture exuberance in asset and commodity mar-kets-is includes variance bound tests proposed by Shiller [32]and LeRoy and Porter [33] two-step tests initiated byWest [34]and integration and cointegration tests developed by Diba andGrossman [7 35] as well as intrinsic bubbles considered byFroot andObstfeld [31] Furthermore Hommand Breitung [36]applied Chow and CUSUM-type tests -e list is by no meansthorough and exhaustive Gurkaynak [37] provided a detailedsurvey of different econometric tests on asset price bubblesMostnotable among these studies is the wide use of traditional unitroot tests eg augmented DickeyndashFuller Although popularlyapplied standard unit root tests have been criticized by Evans[38] As Evans noted when exploring to detect periodicallycollapsing bubbles the unit root-based tests have extremely lowpower because the observed series may appear more like astationary process than like an explosive process

-e recent literature has proposed several other testmethodologies to deal with this weakness -e two mostprominent test procedures of which are the SADF putforward by PWY and its extension the GSADF of PSY Inorder to handle the effect of major downturns in the datatrajectories on the testing power the methods of PWY andPSY implement the forward recursive algorithm and therecursive evolving algorithm respectively Both of themestimate augmented DickeyndashFuller regression equationsusing subsamples of available data An attractive feature ofthese two methods is that they allow dating the originationand termination of explosive periods in real time -e PWYand PSY techniques have gained increasing popularity in awide variety of markets Phillips and Yu [16] are the first todirectly study the bubble characteristics in the crude oil priceseries from January 1999 to January 2009 based on themethod of PWY -ey found statistical evidence of thepresence of a short price bubble between March and July2008 Tsvetanov et al [17] Figuerola-Ferretti et al [18] andCaspi et al [39] employed the GSADF test of PSY usingdifferent samples and reported multiple bubble episodes ofoil price explosivity during the period under examination

Instead of taking the approaches of PWY and PSY anumber of studies have implemented some alternativemethodologies towards speculative or rational bubbles in crudeoil markets Markov regime-switching regression models

provide amethod which assumes that the bubble component isin one of several regimes with positive probability Shi andArora [40] are the first to apply this approach to analyze thebubble issue in oil price dynamics -e authors used threedifferent regime-switching models and reported favorableevidence for the claim of Phillips and Yu [16] that a short-livedbubble existed in 2008 Lammerding et al [41] offered aBayesian Markov switching approach to distinguish a stablephase in one regime and an explosive phase in another regime-eir two Markov-regimes test procedure uncovers significantevidence in support of the existence of bubbles in the oil pricesMore recently Pavlidis et al [42] developed two novel ap-proaches for bubble detection by exploiting differences be-tween spot and forward (or futures) prices Pavlidis et al [43]applied them to crude oil markets and found no speculativebubbles in oil prices -e main advantage of their approach isthat the need for approximating fundamentals of the oil pricecan be obviated by using the information incorporated inmarket expectations However the method is tied to an ap-propriatemeasure ofmarket expectations Altogether it is clearthat extant empirical research on speculative bubbles in the oilmarket has yielded mixed results

3 Theoretical Model

Speculative bubbles in crude oil markets can be defined assystematic deviates from the fundamental price of oil One ofthe most widely recognized models allowing for bubbles is therational expectation asset-pricingmodel-us our theoreticalframework closely relates to that of Diba and Grossman [35]Gurkaynak [37] and Pavlidis and Vasilopoulos [44] How-ever we extend the stylized asset-pricing model to explainwhy prices tend to show excess volatility in storable com-modity markets Consider that rational infinitely-lived in-vestors obtain utility from individual consumption in astandard endowment economy -e representative investorrsquosmaximization problem is

maxEt 1113944

infin

i0βi

u Ct+i( 1113857⎧⎨

⎩

⎫⎬

⎭ (1)

where Et denotes the rational expectation operator condi-tional on all the information available at time t Ct+i is theconsumption quantity at period t + i and β stands for thediscount rate of future consumption which is restricted totake values in (0 1) such that the time preference of an agentis always positive Consistent with the standard assumptionsin economics the instantaneous utility function u(middot) isconsidered to be concave continuously differentiable andstrictly increasing in Ct+i

At each time period the investor subjects to the budgetconstraint

Ct+i lewt+i + Pt+i + ψt+i( 1113857xt minus Pt+ixt+i (2)

where xt is the storable asset Pt represents the price of acommodity in the unit of the consumption good and wt refersto an initial endowment that can be immediately consumed orphysically held to gain the stream of current and expected

Complexity 3

discounted future benefits called the convenience yield ψt Inthis case the convenience yield is analogous to the payoff(dividend) on a stock -e concept of convenience yield is wellknown and has been widely used in analyzing commoditymarkets Specifically it measures a premium that accrues toinventory holders stemming from possible alternative inven-tory functions Forward-looking investors usually establishinventories to smooth production requirements avoid stock-outs and accommodate sales scheduling [45ndash48] Changes indesired inventories reflect the shifts in the agentsrsquo expectationsabout current and future market conditions As such ourmodel links the latent convenience yield to a reduced-formdynamic model of demand and supply

-e first-order condition for the representative investorrsquosutility optimization problem specified by (1) and (2) yields

Et βuprime Ct+i( 1113857 Pt+i + ψt+i1113858 11138591113864 1113865 Et uprime Ct+iminus 1( 1113857Pt+iminus 11113864 1113865 (3)

Intuitively equation (3) states that for the optimal timepath of x an investor cannot increase expected utility byselling or buying an asset at time t and performing a reversetransaction at time t + 1 For commodity pricing purposesutility is often assumed to be linear that implies risk neu-trality and constant marginal utility In this setup equation(3) can be simply written as

βEt Pt+i + ψt+i( 1113857 Et Pt+iminus 1( 1113857 (4)

Assuming further the presence of a risk-free bondavailable in clear financial markets with a time-invariant netinterest rate R standard no arbitrage condition indicates

Pt 1

1 + REt Pt+1 + ψt+1( 1113857 (5)

Solving this first-order stochastic difference (equation(5)) by forward iteration derives the general solution

Pt 1113944infin

i1

1(1 + R)

iEt ψt+1( 1113857 + lim

i⟶infinEt

1(1 + R)

iPt+i1113888 1113889 (6)

-e last equation has decomposed the price into twoparts -e first term of the right-hand side is regarded as amarket fundamental component which depends on thediscounted sum of expected future convenience yields -esecond term is commonly referred to as a periodicallycollapsing speculative bubble component Imposing thetransversality condition

limi⟶infin

Et

1(1 + R)

iPt+i1113888 1113889 0 (7)

rules out the existence of a bubble component -e price Pt

reduces to the fundamental price which will be representedby P

ft whereas if the transversality condition is not justified

there are infinitely many solutions to equation (5) -ey takethe more general form

Pt Pft + Bt (8)

where Bt is a rational bubble component that has to satisfythe submartingale property

Et Bt+1( 1113857 (1 + R)Bt (9)

If a bubble is present in the commodity market equation(9) requires the bubble to grow in expectation geometricallyat the rate of R in order for any rational investor willing tohold the asset Investors may drive the price far higher thanthe fundamental value as long as they believe that they cancompensate for the extra payment Bt through selling theasset to other market participants at an even higher price inthe future In other words commodity price bubbles couldbe a self-fulfilling prophecy Taken together equations (8)and (9) imply that Bt will display explosive dynamics as wellas the spot pricePt irrespective of whether the fundamentalcomponent P

ft is or is not itself nonexplosive

As illustrated in the following the contemporaneousfuture price at time t with maturity at t + T FTt will also beexplosive Let F1t denote the future price for delivery at t + 1To avoid arbitrage ψt must satisfy

ψt (1 + R)Pt minus F1t (10)

Given this condition and equation (5) it follows that F1t

is an unbiased predictor of the expected future spot price

F1t Et Pt+1( 1113857 (11)

Using equations (8) and (9) we obtain the followingexpression for the future price

FTt Et Pf

t+T1113872 1113873 + Et Bt+T( 1113857 Et Pf

t+T1113872 1113873 +(1 + R)T

Bt

(12)

We can easily see that the future price is a linear functionof the explosive process -is statistical property has mo-tivated our econometric tests for identifying explosivecharacteristics in prices

4 Econometric Methodology

In this section we give a brief description of bubble de-tectionmethods used in our empirical study so as to facilitateease interpretation of the results We first describe a con-tinuous-time model with randomized persistence and then astatistical inference procedure in the continuous systemFinally we discuss the model in the endogeneity setting

41 Model Specification Following Tao et al [25] we beginour analysis with a variant version of the Orn-steinndashUhlenbeck process given by

dx(t) x(t)μd(t) + σdWε(t) x(0) x0 (13)

where Wε(t) is a standard Brownian motion with constantvariance 1 also termed a standard Wiener process and μ isthe crucial persistence parameter that is meant to govern thedynamic behavior of x(t) When μlt 0 the process x(t) isstationary when μ 0 it is nonstationary and when μgt 0 itmanifests explosive characteristics In (13) μ is assumed tobe time-invariant which may not be corroborated by long-run data Hence the model studied in the paper takes theform of

4 Complexity

dx(t) x(t) μd(t) + 1113957σWμ(t)1113960 1113961 + σdWε(t) x(0) x0

(14)

where Wμ(t) and Wε(t) are two standard Brownianmotionsand the initial value x0 is viewed to be independent of Wμ(t)

and Wε(t) For 1113957σ2 ne 0 we see that random shocks to the driftterm have been incorporated into model (14) At this mo-ment we explore the situation where Wμ(t) and Wε(t) are

independent noise processes and subsequently allow for theendogenous case by relaxing the independence assumption

Solving model (14) yields the strong solution [49]

dx(t) exp 1113957σWμ(t) + μ minus12

1113957σ21113874 1113875t1113874 1113875x(0) + V(t) (15)

where

V(t) σ 1113946t

0exp 1113957σ Wμ(t) minus Wμ(s)1113872 1113873 + μ minus

12

1113957σ21113874 1113875(t minus s)1113874 1113875dWε(t)

sim MN 0 σ2 1113946t

0e21113957σ Wμ(t)minus Wμ(s)( 1113857+2 μminus (12)1113957σ

2( 1113857(tminus s)ds1113888 1113889

(16)

such that

E V(t)2

1113966 1113967 σ2E 1113946t

0e21113957σ Wμ(t)minus Wμ(s)( 1113857+2 μminus (12)1113957σ

2( 1113857(tminus s)ds1113896 1113897

σ2

2e2 μ+(12)1113957σ

2( 1113857t

minus 1μ +(12)1113957σ21113872 1113873

(17)

Under μ + (12)1113957σ2 gt 0 E V(t)21113966 1113967 is expected to growexponentially

-e corresponding discrete-time representation of (14) isderived directly from the strong solution and is of the form

xtΔ exp 1113957σ WμtΔ minus Wμ(tminus 1)Δ1113960 1113961 + μ minus12

1113957σ21113874 1113875Δ1113882 1113883x(tminus 1)Δ

+ σ 1113946tΔ

(tminus 1)Δexp 1113957σ WμtΔ minus Wμs1113872 1113873 + μ minus

12

1113957σ21113874 1113875(tΔ minus s)1113882 1113883dWεs

(18)

where Δ is the sampling interval over time span T andt 1 TΔ With respect to the random coefficientautoregression model [50] the autoregressive coefficient isdefined by

ρtΔ exp 1113957σ WμtΔ minus Wμ(tminus 1)Δ1113960 1113961 + μ minus12

1113957σ21113874 1113875Δ1113882 1113883 (19)

Here we introduce some notations to simplify thediscrete-time system Let

ϕ ≔ μ minus12

1113957σ2

κ ≔ μ +12

1113957σ2

utΔ ≔WμtΔ minus Wμ(tminus 1)Δ

Δ

radic simN(0 1)

ρtΔ ≔ exp 1113957σ WμtΔ minus Wμ(tminus 1)Δ1113960 1113961 + μ minus12

1113957σ21113874 1113875Δ1113882 1113883 exp ϕΔ + 1113957σΔ

radicutΔ1113966 1113967

ηtΔ ≔ 1113946tΔ

(tminus 1)Δexp 1113957σ WμtΔ minus Wμs1113872 1113873 + μ minus

12

1113957σ21113874 1113875(tΔ minus s)1113882 1113883dWεssimN 0e2κΔ

minus 12κ

1113888 1113889

(20)

Complexity 5

-e discrete-time representation (18) can then be re-written as

xtΔ exp 1113957σΔ

radicutΔ + ϕΔ1113966 1113967x(tminus 1)Δ + σηtΔ ρtΔx(tminus 1)Δ + σηtΔ

(21)

Assuming 1113957σ2 gt 0 models (14) and (21) have the followingstationarity properties (1) for (μ + 1113957σ22)lt 0 the process isasymptotically covariance stationary sinceμlt 0 means thatboth E(ρtΔ) eμΔ lt 1 and E(ρ2tΔ) exp(2μΔ+1113957σ2Δ) exp(2κΔ)lt 1 (2) for (μ + 1113957σ22)lt 0 the process re-mains asymptotically stationary but is not asymptoticallycovariance stationary any longer Practically speaking wehave to impose the condition of κ (μ + 1113957σ22)lt 0 toguarantee the covariance stationary Whenκ (μ + 1113957σ22)lt 0 it follows from (19) that the variance ofV(t) converges to minus σ22κltinfin as t⟶infinWhen κ 0 thevariance is equal to σ2t and diverges as t⟶infin (3) for (μ +1113957σ22)gt 0 and μlt 0 the process is still asymptotically sta-tionary but is not covariance stationary with evident per-sistence in the trajectory (4) when μ 0 ϕ (μ minus 1113957σ22)lt 0ensures the asymptotic stationarity but leads to infinitesecond moments withκ (μ + 1113957σ22)gt 0 -e process ex-hibits apparent unstable behavior and larger variation (5)when μgt 0 and μ minus 1113957σ22lt 0 the process continues to becovariance nonstationary Contrast to the traditional ex-plosive AR (1) model the model retains asymptoticallystationary In this instance bubble-like phenomenon shouldbe clearly observed in actual data (6) if μ minus 1113957σ22gt 0 themodel is no longer asymptotically stationary and the ex-plosive growth behavior is now obvious

As shown in Tao et al [25] when 1113957σ2 0 model (21)collapses to a simple autoregression with a constant coef-ficient It explains why a popular way to look for bubbles is totest ϕ 0 against ϕgt 0 via standard or recursive right-tailedunit root tests [15ndash18 27] For the purpose of the presentpaper we only examine three price behavior features that aredivided into three categories

(1) Unstable κ μ + 1113957σ22gt 0 Here the model is as-ymptotically covariance nonstationary and applies tocapture unstable behavior

(2) Locally explosive μgt 0 -e model in this case hasthe ability for generating both explosive and col-lapsing behavior

(3) Explosive ϕ μ minus 1113957σ22gt 0 Under the circumstancethe model is not asymptotically stationary and iscapable of generating explosive behavior

Based on the above typology explosive embodies locallyexplosive while locally explosive embodies unstable

42 Inference Scheme -is section is concerned with puttingthe continuous-time model specified above into the appli-cation of discretely sampled data We utilize a novel two-stage approach proposed by Phillips and Yu [51] to estimateμ 1113957σ2 and σ2 In the first stage realized volatility is employedto provide a regression model for estimating the diffusionparameters 1113957σ2 and σ2 In the second stage we maximize thein-fill likelihood function to estimate the drift parameter μ

Suppose data are observed in N equally spaced blocksover the time interval [0 T] with sampling interval h -eoverall time span of the data is T so that N Th If eachblock has M observations the total number of observationsis M times N TΔ where Δ is the sampling frequency Todevelop asymptotics we assume that as Δ⟶ 0M times N⟶infin According to the algorithm of Phillips andYu [51] the estimators based on realized variance (RV) andrealized quarticity (RQ) are denoted by

1113954θ argminθisinΘ

QΔ(θ) (22)

where 1113954θ ≔ (1113957σ2 σ2)prime and QΔ(θ) Δ1113936Nn1 (logRVnminus log

[x]nh(nminus 1)h + (12)s2n)2s2nwith s2n max

2Δ(RQ2

nRV2n)

11139681113864

2M

radic

RVn 1113944M

i1x(nminus 1)h+iΔ minus x(nminus 1)h+(iminus 1)Δ1113960 1113961

2 n 1 2 N

RQn 13Δ

1113944

M

i1x(nminus 1)h+iΔ minus x(nminus 1)h+(iminus 1)Δ1113960 1113961

4 n 1 2 N

(23)

In the second stage the approximate logarithmic in-filllikelihood function is maximized with regard to μ (repre-senting the resulting estimator as 1113954μ)

1113954μ argmaxμ

1MN

log ℓALF(μ) (24)

where

ℓALF(μ) 1113944MtimesN

t1

μx(tminus 1)Δ11139541113957σ2x2

(tminus 1)Δ +1113954σ2

xtΔ minus x(tminus 1)Δ1113872 1113873 minusΔ2

1113944

MtimesN

t1

μx(tminus 1)Δ11139541113957σ2x2

(tminus 1)Δ +1113954σ2

(25)

One important goal in the present empirical work is tocheck the presence of randomness in the persistenceparameter of model (14) which is equivalent to test the

null hypothesis 1113957σ2 0 To achieve this aim we resort tothe locally best invariant test (LBI test) statistic [52]namely

6 Complexity

1113957ZN ≔1113936

MtimesNt1 1113957ε2tΔ minus (1MN)1113936

MtimesNt1 1113957ε2tΔ1113872 1113873

21113874 11138751113957x

2(tminus 1)Δ

(1MN)1113936MtimesNt1 1113957ε4tΔ minus (1MN)1113936

MtimesNt1 1113957ε2tΔ1113872 1113873

21113969

(1MN)1113936MtimesNt1 1113957x

4(tminus 1)Δ minus (1MN)1113936

MtimesNt1 1113957x

2(tminus 1)Δ1113872 1113873

21113969 (26)

where 1113957x2tΔ (xtΔ

1 + x2

tΔ

1113969) 1113957εtΔ xtΔ minus 1113957ρx(tminus 1)Δ and

1113957ρ (1113936MtimesNt1 1113957x(tminus 1)Δx(tminus 1)Δ)

minus 11113936MtimesNt1 1113957x(tminus 1)Δx(tminus 1)Δ -us as

N⟶infin MNminus 12 1113957ZN⟶L N(0 1) under H0 1113957σ2 0 andMNminus (12) 1113957ZN⟶L N(0 1) under H0 1113957σ2 0

Relying on asymptotics with in-fill (as the sampling fre-quency tends to zeroΔ⟶ 0) and long time-span asymptoticsTao et al [25] developed new limit theory for the continuous-time system where the persistence parameter is random -easymptotic properties of the diffusion parameter and drift pa-rameter estimates can be used to construct test statistics for threetypes of unstableexplosive behavior which are

tκ 1Δ

1113944

MtimesN

t1

x2(tminus 1)Δ

11139541113957σ2x2(tminus 1)Δ + σ2

⎛⎝ ⎞⎠

12

1113954βκ minus β0κ1113872 1113873⟶L

N(0 1)

tμ 1Δ

1113944

MtimesN

t1

x2(tminus 1)Δ

11139541113957σ2x2(tminus 1)Δ + 1113954σ2

⎛⎝ ⎞⎠

12

1113954ρ minus ρ0( 1113857⟶L

N(0 1)

tϕ 1Δ

1113944

MtimesN

t1

x2(tminus 1)Δ

11139541113957σ2x2(tminus 1)Δ + 1113954σ2

⎛⎝ ⎞⎠

12

1113954βϕ minus β0ϕ1113872 1113873⟶L

N(0 1)

(27)

where

1113954βκ exp 1113954μ +12

11139541113957σ21113874 1113875Δ1113882 1113883

1113954βϕ exp 1113954μ minus12

11139541113957σ21113874 1113875Δ1113882 1113883

β0κ exp μ0 +12μ201113874 1113875Δ1113882 1113883

β0ϕ exp μ0 minus12μ201113874 1113875Δ1113882 1113883

1113954ρ 1 +1113936

MtimesNt1 x(tminus 1)Δ xtΔ minus x(tminus 1)Δ1113872 1113873 1113954

1113957σ2x2(tminus 1)Δ +

1113954σ21113874 1113875

1113936MtimesNt1 x

2(tminus 1)Δ

11139541113957σ2x2

(tminus 1)Δ +1113954σ21113874 1113875

ρ0 1

(28)

As can be seen this test procedure does not directlyverify whether or not κ (μ + 1113957σ22) 0 or μ 0 orϕ (μ minus 1113957σ22) 0 but transforms the null hypotheses toβk 1 or ρ 1 or βϕ 1 -ese three t-test statistics areaccompanied by a date-stamping strategy that permits us toidentify the exact origination and termination periods ofprice exuberance Given this instrument estimates of thesedates are as follows

1113954ribun inf

sisin 1113954r(iminus 1)e

un 11113858 1113859s tκ(s)gtZα1113864 1113865

1113954rieun inf

sisin 1113954r(iminus 1)b

un 11113858 1113859s tκ(s)ltZα1113864 1113865

1113954rible inf

sisin 1113954r(iminus 1)e

le 11113858 1113859s tμ(s)gtZα1113966 1113967

1113954rieun inf

sisin 1113954r(iminus 1)b

le 11113858 1113859s tμ(s)ltZα1113966 1113967

1113954ribe inf

sisin 1113954r(iminus 1)e

e 11113858 1113859s tϕ(s)gtZα1113966 1113967

1113954rieun inf

sisin 1113954r(iminus 1)b

e 11113858 1113859s tϕ(s)ltZα1113966 1113967

(29)

where Zα is the right-tailed critical value of the standardnormal distribution corresponding to a significance level ofα 1113954rib

un1113954rible1113954r

ibe denote the beginnings on the dates of the ith

extreme price behavior period and similarly 1113954rieun1113954r

iele1113954r

iee

denote the ends on the dates of the ith extreme price be-havior period

43 Extend the Model with Endogeneity Now we are con-cerned with the endogeneity case where Wμ(t) is not in-dependent of Wε(t) To see this let us suppose that data aregenerated from the following stochastic differential equationsystem

dx(t) x(t)d1113957B(t) + σdB(t) x(0) x0

d1113957B(t) μ dt + 1113957σWμ(t)

dB(t) σdWε(t)

(30)

where (Wμ(t) Wε(t)) is a two-dimensional Brownianmotion with covariance parameter η As Follmer [53] notedthe strong solution obtained from this continuous systemwould be of the explicit form

dx(t) exp 1113957σWμ(t) + μ minus12

1113957σ21113874 1113875t1113874 1113875x(0) + Z(t) (31)

where

Z(t) σ 1113946t

0exp 1113957σ Wμ(t) minus Wμ(s)1113872 1113873 + μ minus

12

1113957σ21113874 1113875(t minus s)1113874 1113875dWε(t)

minus ω1113946t

0exp 1113957σ Wμ(t) minus Wμ(s)1113872 1113873 + μ minus

12

1113957σ21113874 1113875(t minus s)1113874 1113875ds

V(t) minus P(t)

(32)

Notice that when endogeneity is included in model (14)the driver process becomes Z(t) as a substitute for V(t)Z(t) consists of a component V(t) and another oneconnected with the covariance parameter of (1113957B tB) ω and

Complexity 7

variance of V(t) -e strong solution gives the exact dis-crete-time form of model (31)

xtΔ exp 1113957σΔ

radicuΔ + μ minus

12

1113957σ21113874 1113875Δ1113882 1113883x(tminus 1)Δ + ZtΔ ρtΔ x(tminus 1)Δ + ZtΔ

+ σ 1113946tΔ

(tminus 1)Δexp 1113957σ WμtΔ minus Wμs1113872 1113873 + μ minus

12

1113957σ21113874 1113875(tΔ minus s)1113882 1113883dWεs

(33)

where uiidsim

t N(0 1) and

ZtΔ σ 1113946tΔ

(tminus 1)Δexp 1113957σ WμtΔ minus Wμs1113872 1113873 + μ minus

12

1113957σ21113874 1113875(tΔ minus s)1113882 1113883dWεs

minus ω1113946tΔ

(tminus 1)Δexp 1113957σ WμtΔ minus Wμs1113872 11138731113872 1113873 + μ minus

12

1113957σ21113874 1113875(tΔ minus s)1113874 1113875ds

(34)

Somewhat surprisingly Tao et al [25] indicated thatendogeneity would not affect the sample path features ofmodel (31) Our analysis therefore is the same as before

(1) Unstable κ μ + 1113957σ22gt 0(2) Locally explosive μgt 0(3) Explosive ϕ μ minus 1113957σ22gt 0

We can estimate 1113954θlowast ≔ ( 11139541113957σ2 1113954η 1113954σ2)prime in the same fashion asbefore -e likelihood ratio test (LR test) has been proposedby Tao et al [25] to test the null hypothesis of endogeneityinterest

LR Δminus 1Q

rΔ minus Q

urΔ( 1113857simχ2(1) underH0 η0 0 (35)

Tao et al [25] also developed the asymptotic theory for 1113954μand 1113954ρ ≔ exp(1113954μΔ) Based on their limit properties the es-timates of 1113954μ and 1113954ρ continue to follow asymptotic normaldistributions -is good quality enables us to convenientlyadopt the aforementioned testing procedures after making asmall adjustment to the variance of the limit distribution

5 Data

For our main empirical analysis we apply daily WTI crudeoil prices for the front-month (one-month) future contracton the New York Mercantile Exchange (NYMEX) -e dataare retrieved from the Energy Information Administration(EIA) Our sample spans the period from April 4 1983 toJune 30 2020 and comprises a total of 9352 daily obser-vations -e beginning and end dates of the sample periodare dictated by the availability of the data

It is well known that WIT and Brent are grades of in-ternationally traded crude oil that are widely used asbenchmarks in oil pricing globally -e reason for this paperrelying on the WTI price of crude oil is twofold On the onehand oil speculation is often blamed on people who tradefutures in the US market WTI is the primary benchmark

for oil extracted and consumed in the US-eWTI crude oilfuture market is also the worldrsquos most liquid and largestcrude oil future market Nearly 12 million contracts trade aday with more than 2 million in open interest On the otherhand WTI future contracts have the longest available his-tory dating back to 1983 giving us a large sample size to gainpowers in detecting short-lived bubbles

Our sample period under investigation includes severalmajor exogenous events that to a large extent are reckonedto have influenced oil markets -ese contain the collapse ofOPEC in 1986 the Persian Gulf War of 199091 the Asianfinancial crisis of 199798 the terrorist attacks on September11 2001 the Venezuelan crisis and Iraq war of 200203 theincreased financialization of oil future markets since 2004the OPEC oil supply announcements in 2008 and 2014 the2019 novel coronavirus disease (COVID-19) and moststrikingly the 200708 global financial crisis Figure 2 plotsthe time series trajectories of the WTI nominal front-monthfuture price and the corresponding real price A look at thefigure illustrates that the oil prices grew steadily from thestarting of the sample until the early 2000s -e price seriessince then begun to surge and the sharp increase in the priceof oil continued until July 2008 As shown in Figure 2 itcollapsed to about 30 dollars by the end of 2008 Between2014 and 2016 the price of oil also experienced a crash Morerecently WTI crude price even went subzero for the firsttime in history A common view is that the steep rise anddecline in the price of oil in the 2000s could be attributed tospeculation

6 Empirical Results

61 Main Results We start the empirical illustration withbaseline model (14) expanding it to the endogeneity ap-plication later In order to formally examine the existenceand duration of unstable and explosive behavior in the crudeoil market we run our tests on the daily price data

8 Complexity

Following the recommendation of PSY the minimumwindow size set for the recursive test procedure is the first 5years of the entire sample Figure 3 plots the test empiricaloutcomes along with their corresponding 95 critical valuesequences Also plotted are the trajectories of test statisticsfor the random coefficient and the price data Some inter-esting results are shown in this figure

Looking at the recursive test results reported in the lastpanel of Figure 3 we observe that over the initial period ofobservation the null hypothesis of a time-invariant autor-egressive coefficient cannot be rejected at the 5 statisticalsignificance level -e test statistic experiences a dramaticincrease in the late 1980s and exceeds the 95 critical valuein November 1990 Overall the observed pattern of timevariation rises and becomes much stronger as the timeperiod expands -e evidence that the LBI test statistics forthe majority of samples are greater than the 95 criticalvalue suggests that the data-generating process is wellcharacterized by the model with time-varying coefficientsOne point worth noting is that the sequence falls in thesecond half of 2008 and slumps in early 2020 -e styled factseems to indicate that the crude oil market has undergonedrastic changes Interestingly these located dates coincidewith the 200708 global financial crisis and the COVID-19pandemic

We now turn to our focus on different types of unstableand explosive behavior-e shaded areas in green in Figure 3are the identified periods where the t-test statistic crosses its95 critical value As is evident from the first panel theprocedure detects two unstable episodes -e first unstableepisode is relatively short and only lasts two months(2005M08-M09) -e second unstable episode appears inJanuary 2006 and continues until March 2020 -ese un-stable periods might be associated with the financializationof oil future markets from 2004 It can be seen in the secondpanel of Figure 3 that the detected episodes of locally ex-plosive (submartingale) behavior are clustered in 2007-20082009ndash2014 and 2018 As stated above since local explo-siveness implies instability and displays more extreme be-havior it should lead to a much shorter duration of

exuberance than that of instability -e third panel ofFigure 3 reflects that the t-test statistic is below the 95critical value throughout the sample period -is points outthat explosive behavior in crude oil price series does notexist Table 1 summarizes time horizons for the three kindsof unstable and explosive behavior identified by ourapproach

To shed light on a more realistic scenario of data-gen-erating process we implement the extreme sample pathbehavior test by extending the model to the endogeneitycase Figure 4 plots the empirical test results together with95 critical values A comparison of the LR test statisticswith the critical value sequence provides novel evidence insupport of the presence of endogeneity during the wholesample period -us it is believed that endogeneity is es-sential to the oil price formation process Furthermore thelast panel of Figure 4 highlights that the quadratic varianceof the price process is related to the test statistic forendogeneity -is finding is in accordance with the fact thatwe have exploited the realized variance series to obtain theLR test statistics In this sense Figure 5 displays differencesin the realized variance estimates which are captured by thetest statistic for endogeneity employing different methodsSimilar to Figure 3 we again identify episodes of unstabledynamics and fail to detect explosive behavior when con-sidering the model with endogeneity In contrast the sta-tistic sequence for local explosiveness exhibits a differentpattern Previous evidence in favor of locally explosive be-havior disappeared Table 2 summarizes time horizons forthe three forms of unstable and explosive behavior -eseresults suggest that endogeneity feedbacks play a significantrole in investigating the existence of explosive price be-havior -e conclusion that emerges is that there are nospeculative bubbles in the crude oil market

In principal our findings are in accordance with theexplanations made by Kilian and Murphy [19] Baumeisterand Hamilton [21] and Knittel and Pindyck [54] that thevariations in crude oil prices are driven by economicfundamentals

62 Comparison to Alternative Strategies Rich literature hasstudied speculative bubbles in the crude oil market using thetechniques developed by PWY and then extended by PSY Abrief description of the SADF methodology (used in PWY)and the GSADF methodology (used in PSY) can be found inAppendix It is therefore helpful to compare results foralternative methods as the detection strategy in the presentpaper is quite different from them Figure 6 plots thebackward ADF and backward SADF statistic sequences andthe associated 95 critical value sequences for oil futureprices 95 percent critical values are taken from 2000 MonteCarlo simulations with the actual sample size -e initialwindow size is set to 5 years A glance at Figure 6 reveals thatexplosive behavior occurred mainly in two occasions usingPWY and PSY dating strategies -ey are before and during2007-2008 global recession We also observe that PWYdetects several short-lived episodes between 2011 and 2014but PSY fails to do so For daily oil future prices over April 4

0

20

ndash20

ndash40

40

60

80

100

120

140

84 86 88 90 92 94 96 98 00 02 04 06 08 10 12 14 16 18 20

Nominal WTI front-month future pricesReal WTI front-month future prices

Figure 2 Real and nominal prices of WTI front-month futurecontracts

Complexity 9

1983 to June 30 2020 time horizons for explosive behavioridentified by PWY and PSY are summarized in Table 3 Inline with previous studies we could conclude from bothPWY and PSY tests that there is evidence of bubbles in thecrude oil market data However as shown in Gurkaynak [37]and Pavlidis et al [43] explosive dynamics in asset prices perse do not constitute evidence either in support of or againstspeculative bubbles since such explosive behavior may

actually reflect explosive market fundamentals or bothfactors As aforementioned our random coefficient autor-egressive model framework has considered endogeneityfeedbacks that to some extent alleviate this problemMoreover the evidence in Figures 3 and 4 suggests theabsence of explosive dynamics in the crude oil market -efact permits us to safely ignore the distinction betweenbubbles or explosive fundamentals

1988

-03

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

ndash4

ndash2

0

2

4

6

8

0

20

40

60

80

100

120

140

t-statistic of k95 critical value of N (0 1)Crude oil future prices

Instability detection based on the time-varyingcoefficient model

(a)

Local explosiveness detection based on the time-varyingcoefficient model

1988

-03

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

t-statistic of μ95 critical value of N (0 1)Crude oil future prices

ndash4

ndash2

0

2

4

6

8

0

20

40

60

80

100

120

140

(b)

Explosiveness detection based on the time-varyingcoefficient model

1988

-03

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

t-statistic of ϕ95 critical value of N (0 1)Crude oil future prices

ndash4

ndash2

0

2

4

6

8

0

20

40

60

80

100

120

140

(c)

LBItest statistic for testing time-varying coefficient19

88-0

3

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

LBItest95 critical value of N (0 1)

ndash5

ndash10

0

5

10

15

20

(d)

Figure 3 Date-stamping unstable and explosive episodes in oil prices with the time-varying coefficient model assuming no endogeneity

Table 1 Time horizons of unstable and explosive periods identified by models assuming no endogeneity

Unstable Locally explosive ExplosiveAug 2005-Sep 2005 Dec 2007ndashSep 2008 mdash

Jan 2006ndashMar 2020 Jun 2009ndashOct 2014 mdashJun 2018ndashSep 2018 mdash

10 Complexity

63 Robustness of the Results In the following we providetwo robustness checks to assess whether our previousconclusions of the paper still hold

First in our benchmark model our use of front-monthfuture price instead of other future prices at different ma-turities is justified by its being the most liquid future contractwith the highest open interest To explore the sensitivity ofour results to the future contracts at different maturities werepeat the analysis described above using the prices forlonger-term contracts (ie 2 3 and 4 months-ahead fu-tures) It should be pointed out that the WTI future contractusually expires on the third business day before the 25thcalendar day of the contract month proceeding the deliverymonth

Second becauseWTI crude oil is by tradition traded innominal US dollars per barrel it might be of concern that

inflation and other monetary variables may have realeffects on oil price It is therefore of interest to use realrather than nominal price In fact recent work documentsthat the use of either price is inconsequential in practice[55] Although we mainly focus on nominal prices forrobustness check we shall report the results for realprices To convert nominal daily series into daily realseries we use linearly interpolated monthly ConsumerPrice Index (CPI) Monthly CPI series for the UnitedStates is obtained from the websites of the Federal ReserveBank of St Louis

Further evidence presented in Figures 7ndash10 demon-strates that our results are remarkably robust to thesemodifications and the conclusions remain qualitativelyunchanged

ndash4

ndash2

0

2

4

6

8

0

20

40

60

80

100

120

140

1988

-03

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

Instability detection based on the time-varyingcoefficient model

t-statistic of k95 critical value of N (0 1)Crude oil future prices

(a)

ndash4

ndash2

0

2

4

6

8

0

20

40

60

80

100

120

140

1988

-03

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

t-statistic of μ95 critical value of N (0 1)Crude oil future prices

Local explosiveness detection based on the time-varyingcoefficient model

(b)

ndash4

ndash2

0

2

4

6

8

0

20

40

60

80

100

120

140

1988

-03

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

Explosiveness detection based on the time-varyingcoefficient model

t-statistic of ϕ95 critical value of N (0 1)Crude oil future prices

(c)

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

LR-statistic95 critical values of x2 (1)Realized variance

700

600

500

400

300

200

100

0

ndash100

350

300

250

200

150

100

50

0

Test for endogeneity in time-varyingcoefficient model

(d)

Figure 4 Date-stamping unstable and explosive episodes in oil prices with the time-varying coefficient model with endogeneity

Complexity 11

2

3

4

5

6

7

8

0

50

100

150

200

250

300

350

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

LLF0

LLF1

Realized variance of crude oil future prices

Figure 5 Realized variance and likelihood values for random coefficient regression

Table 2 Time horizons of unstable and explosive periods detected by models with endogeneity

Unstable Locally explosive ExplosiveMay 2008ndashJul 2008 mdash mdashDec 2010ndashOct 2014 mdash mdashNov 2016ndashApr 2017 mdash mdashJul 2017ndashFeb 2020 mdash mdash

ndash4

ndash2

0

2

4

6

8

0

20

40

60

80

100

120

140

1988

-03

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

Explosiveness detection using SADF methodology

BADF sequence95 critical valuesCrude oil future prices

(a)

ndash4

ndash2

0

2

4

6

8

0

20

40

60

80

100

120

140

1988

-03

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

Explosiveness detection using GSADF methodology

BSADF sequence95 critical valuesCrude oil future prices

(b)

Figure 6 Date-stamping explosive episodes in oil prices with the time-invariant coefficient model

12 Complexity

Table 3 Time horizons of explosive behavior identified by SADF and GSADF methodologies

SADF GSADFAug 2005ndashAug 2006 Oct 2011ndashApr 2012 Sep 1990-Oct 1990Jun 2007ndashOct 2008 Aug 2012-Sep 2012 Feb 1991-Mar 1991Jun 2009-Jul 2009 Jul 2013-Aug 2013 Oct 2004-Nov 2004Feb 2011ndashApr 2011 Feb 2014-Mar 2014 Feb 2005-Mar 2005

Jul 2011ndashAug 2012 Jun 2014-Jul 2014May 2005ndashJan 2006Mar 2006ndashAug 2006Jul 2007ndashAug 2008

ndash4

ndash2

0

2

4

6

8

0

10

20

30

40

50

60

70

1988

-03

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

95 critical value of N (0 1)Real oil future prices

t-statistic of μ~

Local explosiveness detection based on the time-varyingcoefficient model

(a)

ndash4

ndash2

0

2

4

6

8

0

10

20

30

40

50

60

70

1988

-03

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

95 critical value of N (0 1)Real oil future prices

t-statistic of ϕ

Explosiveness detection based on the time-varyingcoefficient model

(b)

Figure 7 Date-stamping explosive episodes in real oil prices with the time-varying coefficient model with endogeneity

ndash4

ndash2

0

2

4

6

8

0

50

100

150

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

Local explosiveness detection based on the time-varyingcoefficient model

t-statistic of μ95 critical value of N (0 1)2-month crude oil future prices

(a)

ndash4

ndash2

0

2

4

6

8

0

50

100

150

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

Local explosiveness detection based on the time-varyingcoefficient model

t-statistic of ϕ95 critical value of N (0 1)2-month crude oil future prices

(b)

Figure 8 Date-stamping explosive episodes in 2-month future oil prices with the time-varying coefficient model with endogeneity

Complexity 13

7 Discussion

-e oil price rally since the early 2000s has attracted greatinterest in the sources of oil price shocks Some causal ob-servers have stressed that financial speculation has become asignificant driver of the dynamics of oil prices A large body ofliterature emerged that has sought to investigate its existenceIn this paper we employ a novel approach for testing forspeculative bubbles in oil prices In synthesis our empiricalfindings indicate that there is less evidence of exuberance in

the oil future markets Our results are very robust in severalscenarios We argue that the booms and busts in the oil pricemovements during the sample period are driven primarily byeconomic fundamentals rather than speculation

Overall our results confirm the main findings obtainedby structural VAR-based studies including Kilian andMurphy [19] Kilian and Lee [20] Baumeister and Hamilton[21] and Zhou [22] In the past decade the use of structuralVAR models to understand the evolution of the oil price hasbecome common -e traditional SVAR literature

ndash4

ndash2

0

2

4

6

8

0

50

100

15019

88-0

3

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

Local explosiveness detection based on the time-varyingcoefficient model

t-statistic of μ95 critical value of N (0 1)3-month crude oil future prices

(a)

ndash4

ndash2

0

2

4

6

8

0

50

100

150

1988

-03

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

Explosiveness detection based on the time-varyingcoefficient model

t-statistic of ϕ95 critical value of N (0 1)3-month crude oil future prices

(b)

Figure 9 Date-stamping explosive episodes in 3-month future oil prices with the time-varying coefficient model with endogeneity

ndash4

ndash2

0

2

4

6

8

0

50

100

150

Local explosiveness detection based on the time-varyingcoefficient model

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

t-statistic of μ95 critical value of N (0 1)4-month crude oil future prices

(a)

ndash4

ndash2

0

2

4

6

8

0

50

100

150

Explosiveness detection based on the time-varyingcoefficient model

t-statistic of ϕ95 critical value of N (0 1)4-month crude oil futures prices

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

(b)

Figure 10 Date-stamping explosive episodes in 4-month future oil prices with the time-varying coefficient model with endogeneity

14 Complexity

documents that the fundamental drivers of oil price fluc-tuations have been dominated essentially by supply anddemand shocks see eg Kilian [56] Lippi and Nobili [57]Kilian andMurphy [58] and Baumeister and Peersman [59]Kilian and Murphy [19] focused on the role speculation byaugmenting the SVAR framework of their earlier paper byglobal oil inventory data -ey specified a four-variablemodel for the global oil market (the percentage change inglobal oil production the real price of oil a suitable proxy ofcyclical variation in global real economic activity andchanges in global crude oil inventories) -is setting andtheir sign-identified scheme allow them to separately explainvariation in these data as four different types of shocks (1) ashock to the amount of oil pumped out of the ground (ldquooilsupply shockrdquo) (2) an oil flow demand shock (3) an oil-specific oil demand shock and especially (4) a speculative(or inventory) demand shock -eir core finding is that thespike in the real price of crude oil prices during 2003ndash2008was caused by unexpected increase in global oil consump-tion Using two alternative proxies for global above-groundcrude oil inventories Kilian and Lee [20] reestimated themodel of Kilian and Murphy [19] and ruled out the role ofspeculation in explaining the recent surge of the real oil priceon the basis of their estimation results Also Zhou [22]assessed the sensitivity of the findings in Kilian and Murphy[19] to the introduction of various state-of-the-art oil marketmethods She shows that the core conclusions reached byKilian and Murphy [19] are highly robust to some refine-ments corrections and extension of the KilianndashMurphymodel Baumeister and Hamilton [21] recently contendedthat prior information plays a key role in any SVAR modelsand traditional identification approaches can be generalizedto a special case of a Bayesian perspective -ey revisited theimportance of supply and demand shocks in the oil marketand demonstrated that speculative demand shock turns outto be only a small factor in accounting for most historical oilprice shocks

For comparison purpose we have also applied the testsof PWY and PSY which are the most popular econometrictechniques in the literature on speculative bubbles in the oilmarkets If we were to follow the test procedures of PWY andPSY our conclusions are also broadly consistent with therich literature that supports the existence of speculativebubbles in oil prices However these cointegration-basedbubble tests suffer from the joint hypothesis problem ldquoEverytest of a bubble is a joint test of the presence of the bubble inthe data and the validity of the model applied by theeconometricianrdquo (see Giglio et al [60]) More specifically themain drawback of such indirect model-dependent tests ofbubbles typically refers to the difficulty in distinguishingbetween the unknown true model for market fundamentalsand the presence of bubbles (see for example Gurkaynak[37] Pavlidis et al [42] Pavlidis et al [43]) Given thefundamental value of oil price is unobservable the literaturehas resorted to observe economic and financial variableswhich are used to estimate the intrinsic values For exampleboth Phillips and Yu [16] and Caspi et al [39] simplynormalized crude oil prices by US oil inventory data as thefundamental value -is measure is questionable since their

method cannot sufficiently capture the information onmarket fundamentals and neglect the major driversmdashlikeglobal economic activitymdashalso affecting the fundamentalvalue of crude oil -is issue of misspecified fundamentals(either in terms of model misspecification or omitted var-iables or both of these reasons) may make researchers fail toarrive at a conclusion from evidence of explosive pricesalone To make matters worse even though we have cap-tured the true market fundamentals we may still fail todetect bubbles if the explosive behavior in the price of oil juststems from the explosive fundamentals

Compared to earlier studies as briefly mentioned aboveour methodology has the main advantage that by consid-ering the dependence between the random coefficient andsystem shocks we have incorporated endogeneity into thecontinuous-time random coefficient setting and thus we donot need to prespecify the fundamental component of the oilprice -e test results for endogeneity suggest that endo-geneity is important in the data-generating process Inparticular our empirical analysis also finds that there are noexplosive episodes in crude oil markets-is implies that it isnot necessary to require the specification of a fundamentalvalue of the oil price Like Pavlidis et al [43] their methodsand ours have in common that the problem of the well-known joint hypothesis has to be some extent been ef-fectively ameliorated Although our approach is quite dis-tinct we draw the same substantive conclusion thatspeculative bubbles can be ruled out as a key determinant ofthe price of crude oil in the 2000s reaffirming the con-clusions of Kilian and Murphy [19] Baumeister andHamilton [21] Zhou [22] and Fattouh et al [61]

8 Conclusions

Repeated surges and crashes in crude oil prices during lastdecades have sparked a heated debate on their underlyingcauses among academics commentors and market regu-lators One of the central questions concerns whetherfinancialization of oil future markets creates a speculativebubble that subsequently collapses Answering this questionis difficult because one has to rely upon strong assumptionsabout the data-generating mechanism for the fundamentalvalue of the oil price In this paper we present a stylizedmodel of asset pricing for a commodity to address this issueOur theoretical framework disentangles the effects ofspeculative bubbles from the traditional fundamental effectson oil price dynamics We use this setting to illustrate thatbubbles if they are present should demonstrate explosivebehavior in prices We then provide a novel approach forbubble detection that covers various explosive scenarios andfind no evidence of speculative bubbles in oil markets evenduring episodes where the most popular SADF and GSADFmethodologies identify a speculative bubble

Our conclusions have pertinent implications for policymakers Given the historically important role of oil prices inthe real economy one of the serious concerns in policycircles over the past decade has been whether speculativetrading activities in oil future markets are responsible for thelarge price swings For example in their 2008 testimonies to

Complexity 15

the US Senate Commerce Committee Michael Masters andGeorge Soros posited that speculators had fueled price in-creases since 2003 and had caused a part of a ldquosuperbubblerdquoin 2007 and 2008 resulting in significantly higher energycosts for consumers -is widespread bubble view domi-nated policy discussions and efforts to limit speculativetrading in the future market quickly followed As a responseto the excessive speculation pressures on prices theDoddndashFrank Wall Street Reform and Consumer ProtectionAct was enacted by the US Congress in 2010 As a directpolicy implication our results highlight that imposingstricter regulations on trading in oil derivative marketscannot be expected to lower the oil price given the fact thatthe synchronized boom and bust cycles witnessed in the past20 years are mainly due to economic fundamentals whichappear likely to resurge with the business cycle -e aboveanalysis implies that there is no need to enforce additionalregulatory limits against financial arbitrage in the com-modity market

Appendix

The SADF and GSADF Methodologies

Appendix describes the SADF and GSADF testing proce-dures used to detect speculative bubbles in Section 62 For agenetic time series process xt consider the following rolling-window type augmented DickeyndashFuller (ADF) regressionequation

xt ar1r2+ μr1r2

xtminus 1 + 1113944

J

j1ϕj

r1r2Δxtminus j + εt εt sim NID 0 δ2r1r2

1113874 1113875

(A1)

where Δxtminus j with j 1 J are lagged first differences ofxt incorporated to accommodate potential serial correla-tion ar1 r2

μr1 r2 and ϕj

r1 r2 with j 1 J are regression

coefficients the subscripts r1 and r2 refer to fractions of thetotal sample size (ofT observations) that set the start and endpoints of a subsample period and NID represents inde-pendent and normal distribution -e ADF test statisticcorresponding to the unit root null hypothesisH0 μr1 r2

0 is simply the t-ratio

ADFr2r1

1113954μr1 r2

se 1113954μr1 r21113872 1113873

(A2)

and the right-side alternative hypothesis of explosive be-havior is H0 μr1r2

gt 0 We can obtain the standard ADF teststatistic by running regression (A1) on the whole sampleie by specifying r1 0 and r2 1 A large amount of re-search has demonstrated that the standard ADF test ADF10has extremely low power in detecting periodically collapsingbubbles

To address the problem PSW proposed a recursiveregression technique which involves estimating (A1)employing a forward expanding sample Specifically theyused subsamples of the available data to recursively calculatea sequence of ADF statistics denoted by ADFr2

0 -e startpoint of the sample period is fixed at r1 0 while the endpoint r2 increases from r0 (the minimum window size) toone (the full sample period) -e SADF statistic called thesupremum of ADFr2

0 is defined by

SADF r0( 1113857 supr2isin r0 1[ ]

ADFr20

(A3)

and has the following limit distribution under the nullhypothesis

supr2isin r0 1[ ]

1113938r2

0 WdW

1113938r2

0 W21113872 1113873(12)

(A4)

where W is the standard Brownian motion Testing thepresence of explosive behavior entails comparing the ADFr2

0statistic with the right-side critical values obtained throughMonte Carlo simulations

More recently PSY showed that when the sample perioddisplays more than one boom-bust bubble the SADF testprocedure may perform poorly and fail to detect or con-sistently date-stamp alternative bubbles -is deficiencymotivates PSY to formulate an extension of the SADF testthe GSADF test which covers more subsamples of the datathan the SADF by allowing both the end point r2 and thestart point r1 to vary As a result the GSADF test has moreflexibility and gains substantial power making this testconsistent with multiple boom-bust bubbles -e GSADFtest statistic is defined as

GSADF r0( 1113857 supr2isin r0 1[ ]r1isin 0r2minus r0[ ]

ADFr2r1 (A5)

with the limit distribution

supr2isin r0 1[ ]r1isin 0r2minus r0[ ]

(12)rw W r2( 11138572

minus W r1( 11138572

minus rw1113960 1113961 minus 1113938r2

r1W(r)dr W r2( 1113857 minus W r1( 11138571113858 1113859

r12w rw 1113938

r2

r1W(r)2dr minus 1113938

r2

r1W(r)dr1113876 1113877

21113896 1113897

12

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎭

(A6)

where rw r2 minus r1 is the expanding window size Like theSADF test comparison of the GASDF statistic with the right-

tailed critical values from its limit distribution enables us to testfor the unit root hypothesis against explosive behavior

16 Complexity

If the null hypothesis of a unit root in xt is rejectedthen in the second stage the SADF and GSADF tests canidentify the exact episodes during which the xt seriesexhibits explosive dynamics To locate the emergence andconclusion of explosiveness the SADF methodologymatches the sequence of the recursive backward ADF(BADF) test statistics against the right-tailed criticalvalues of the limit distribution of the standard Dick-eyndashFuller test statistic Suppose that re is the originationdate and rf is the finish date of the bubble in the data-ese estimates of dates can be constructed from

1113954re infr2isin r0 1[ ]

r2 ADFr20 gt cv

αr2

1113966 1113967

1113954rf infr2isin 1113954re11113858 1113859

r2 ADFr20 lt cv

αr2

1113966 1113967(A7)

where cvαr2 is 100(1 minus α) the right-tailed critical value of theADF statistic corresponding to the chosen significance levelof α For the GSADF methodology the date-stampingstrategy in this case is based on the backward SADF(BSADF) statistic

BSADFr2r0( 1113857 sup

r1isin 0r2minus r0[ ]ADFr2

r1

(A8)

Likewise the origination date of the bubble is defined asthe first observation that the BSADF statistic is greater thanits critical value

1113954re infr2isin r0 1[ ]

r2 BSADFr2r0( 1113857gt cv

βT

r21113966 1113967 (A9)

and the finish date as the first date after which the BSADFdrops back below its critical value

1113954rf infr2isin 1113954re11113858 1113859

r2 BSADFr2r0( 1113857lt cvβT

r21113966 1113967 (A10)

where cvβTr2 is 100(1 minus βT) the right-tailed critical value of

the SADF based on Tr2 and the corresponding significancelevel is βT

Data Availability

-e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

-e authors declare no conflicts of interest

Acknowledgments

-e authors gratefully acknowledge the financial supportfrom the National Social Science Foundation of China underGrant no 18BTJ039

References

[1] M W Masters Testimony before the US Senate Committee onHomeland Security and Governmental Affairs 2008 httpswwwhsgacsenategovimomediadoc052008Masterspdfattempt2

[2] M W Masters Testimony before the Commodity FuturesTrading Commission 2010 httpscftcgovsitesdefaultfilesidcgroupspublicnewsroomdocumentsfilemetalmarkets032510_masterspdf

[3] J P Kennedy ldquo-e high cost of gambling on oilrdquo Section A p23 New York Times New York NY USANew York Times2012

[4] I-H Cheng and W Xiong ldquoFinancialization of commoditymarketsrdquo Annual Review of Financial Economics vol 6 no 1pp 419ndash441 2014

[5] K Tang and W Xiong ldquoIndex investment and the financi-alization of commoditiesrdquo Financial Analysts Journal vol 68no 6 pp 54ndash74 2012

[6] O J Blanchard and M W Watson ldquoBubbles rational ex-pectations and financial marketsrdquo in Crisis in the Economicand Financial Structure P Wachtel Ed Lexington BooksLexington MA USA pp 295ndash315 1982

[7] B T Diba and H I Grossman ldquoOn the inception of rationalbubblesrdquo e Quarterly Journal of Economics vol 102 no 3pp 697ndash700 1987

[8] G W Evans ldquo-e fragility of sunspots and bubblesrdquo Journalof Monetary Economics vol 23 no 2 pp 297ndash317 1989

[9] G W Evans and S Honkapohja ldquoOn the robustness ofbubbles in linear RE modelsrdquo International Economic Reviewvol 33 no 1 pp 1ndash14 1992

[10] E Pavlidis A Yusupova I Paya et al ldquoEpisodes of exu-berance in housing markets in search of the smoking gunrdquoe Journal of Real Estate Finance and Economics vol 53no 4 pp 419ndash449 2016

[11] E Martınez-Garcıa and V Grossman ldquoExplosive dynamics inhouse prices An exploration of financial market spillovers inhousing markets around the worldrdquo Journal of InternationalMoney and Finance vol 101 p 102103 2020

[12] S Shi A Rahman and B Z Wang ldquoAustralian housingmarket booms fundamentals or speculationlowastrdquo EconomicRecord 2020 In press

[13] P C B Phillips Y Wu and J Yu ldquoExplosive behavior in the1990s NASDAQ when did exuberance escalate asset val-ueslowastrdquo International Economic Review vol 52 no 1pp 201ndash226 2011

[14] P C B Phillips S Shi and J Yu ldquoTesting for multiplebubbles historical episodes of exuberance and collapse in theSampp 500rdquo International Economic Review vol 56 no 4pp 1043ndash1078 2015

[15] P C B Phillips S Shi and J Yu ldquoTesting for multiplebubbles limit theory of real-time detectorsrdquo InternationalEconomic Review vol 56 no 4 pp 1079ndash1134 2015

[16] P C B Phillips and J Yu ldquoDating the timeline of financialbubbles during the subprime crisisrdquo Quantitative Economicsvol 2 no 3 pp 455ndash491 2011

[17] D Tsvetanov J Coakley and N Kellard ldquoBubbling over thebehaviour of oil futures along the yield curverdquo Journal ofEmpirical Finance vol 38 no Part B pp 516ndash533 2016

[18] I Figuerola-Ferretti J R Mccrorie and I ParaskevopoulosldquoMild explosivity in recent crude oil pricesrdquo Energy Eco-nomics vol 87 p 104387 2020

[19] L Kilian and D P Murphy ldquo-e role of inventories andspeculative trading in the global market for crude oilrdquo Journalof Applied Econometrics vol 29 no 3 pp 454ndash478 2014

Complexity 17

subprime mortgage crisis followed by the global economicshock -ere is a general consensus that speculative bubblesare defined as the divergence of an assetrsquos price from itsintrinsic value [27]

-e classic rational bubble is the standard theoreticalframework to model the presence and persistence of speculativebubbles with seminal papers by Blanchard and Watson [6]Tirole [28 29] Flood and Hodrick [30] and Froot and Obstfeld[31] In this workhorse model of bubbles it is economicallyrational for investors to buy overpriced securities today that areunjustified by fundamental factors as long as they believe that theselling price becomes high tomorrow In spite of the central roleof rational bubbles in the theoretical literature empirical evi-dence in support of their existence remains elusive -e chal-lenges in empirically testing for the potential bubbles lie ininvestigating their explosivity and dating their occurrence Earlyempirical studies have designed and adopted various time seriesmethods to capture exuberance in asset and commodity mar-kets-is includes variance bound tests proposed by Shiller [32]and LeRoy and Porter [33] two-step tests initiated byWest [34]and integration and cointegration tests developed by Diba andGrossman [7 35] as well as intrinsic bubbles considered byFroot andObstfeld [31] Furthermore Hommand Breitung [36]applied Chow and CUSUM-type tests -e list is by no meansthorough and exhaustive Gurkaynak [37] provided a detailedsurvey of different econometric tests on asset price bubblesMostnotable among these studies is the wide use of traditional unitroot tests eg augmented DickeyndashFuller Although popularlyapplied standard unit root tests have been criticized by Evans[38] As Evans noted when exploring to detect periodicallycollapsing bubbles the unit root-based tests have extremely lowpower because the observed series may appear more like astationary process than like an explosive process

-e recent literature has proposed several other testmethodologies to deal with this weakness -e two mostprominent test procedures of which are the SADF putforward by PWY and its extension the GSADF of PSY Inorder to handle the effect of major downturns in the datatrajectories on the testing power the methods of PWY andPSY implement the forward recursive algorithm and therecursive evolving algorithm respectively Both of themestimate augmented DickeyndashFuller regression equationsusing subsamples of available data An attractive feature ofthese two methods is that they allow dating the originationand termination of explosive periods in real time -e PWYand PSY techniques have gained increasing popularity in awide variety of markets Phillips and Yu [16] are the first todirectly study the bubble characteristics in the crude oil priceseries from January 1999 to January 2009 based on themethod of PWY -ey found statistical evidence of thepresence of a short price bubble between March and July2008 Tsvetanov et al [17] Figuerola-Ferretti et al [18] andCaspi et al [39] employed the GSADF test of PSY usingdifferent samples and reported multiple bubble episodes ofoil price explosivity during the period under examination

Instead of taking the approaches of PWY and PSY anumber of studies have implemented some alternativemethodologies towards speculative or rational bubbles in crudeoil markets Markov regime-switching regression models

provide amethod which assumes that the bubble component isin one of several regimes with positive probability Shi andArora [40] are the first to apply this approach to analyze thebubble issue in oil price dynamics -e authors used threedifferent regime-switching models and reported favorableevidence for the claim of Phillips and Yu [16] that a short-livedbubble existed in 2008 Lammerding et al [41] offered aBayesian Markov switching approach to distinguish a stablephase in one regime and an explosive phase in another regime-eir two Markov-regimes test procedure uncovers significantevidence in support of the existence of bubbles in the oil pricesMore recently Pavlidis et al [42] developed two novel ap-proaches for bubble detection by exploiting differences be-tween spot and forward (or futures) prices Pavlidis et al [43]applied them to crude oil markets and found no speculativebubbles in oil prices -e main advantage of their approach isthat the need for approximating fundamentals of the oil pricecan be obviated by using the information incorporated inmarket expectations However the method is tied to an ap-propriatemeasure ofmarket expectations Altogether it is clearthat extant empirical research on speculative bubbles in the oilmarket has yielded mixed results

3 Theoretical Model

Speculative bubbles in crude oil markets can be defined assystematic deviates from the fundamental price of oil One ofthe most widely recognized models allowing for bubbles is therational expectation asset-pricingmodel-us our theoreticalframework closely relates to that of Diba and Grossman [35]Gurkaynak [37] and Pavlidis and Vasilopoulos [44] How-ever we extend the stylized asset-pricing model to explainwhy prices tend to show excess volatility in storable com-modity markets Consider that rational infinitely-lived in-vestors obtain utility from individual consumption in astandard endowment economy -e representative investorrsquosmaximization problem is

maxEt 1113944

infin

i0βi

u Ct+i( 1113857⎧⎨

⎩

⎫⎬

⎭ (1)

where Et denotes the rational expectation operator condi-tional on all the information available at time t Ct+i is theconsumption quantity at period t + i and β stands for thediscount rate of future consumption which is restricted totake values in (0 1) such that the time preference of an agentis always positive Consistent with the standard assumptionsin economics the instantaneous utility function u(middot) isconsidered to be concave continuously differentiable andstrictly increasing in Ct+i

At each time period the investor subjects to the budgetconstraint

Ct+i lewt+i + Pt+i + ψt+i( 1113857xt minus Pt+ixt+i (2)

where xt is the storable asset Pt represents the price of acommodity in the unit of the consumption good and wt refersto an initial endowment that can be immediately consumed orphysically held to gain the stream of current and expected

Complexity 3

discounted future benefits called the convenience yield ψt Inthis case the convenience yield is analogous to the payoff(dividend) on a stock -e concept of convenience yield is wellknown and has been widely used in analyzing commoditymarkets Specifically it measures a premium that accrues toinventory holders stemming from possible alternative inven-tory functions Forward-looking investors usually establishinventories to smooth production requirements avoid stock-outs and accommodate sales scheduling [45ndash48] Changes indesired inventories reflect the shifts in the agentsrsquo expectationsabout current and future market conditions As such ourmodel links the latent convenience yield to a reduced-formdynamic model of demand and supply

-e first-order condition for the representative investorrsquosutility optimization problem specified by (1) and (2) yields

Et βuprime Ct+i( 1113857 Pt+i + ψt+i1113858 11138591113864 1113865 Et uprime Ct+iminus 1( 1113857Pt+iminus 11113864 1113865 (3)

Intuitively equation (3) states that for the optimal timepath of x an investor cannot increase expected utility byselling or buying an asset at time t and performing a reversetransaction at time t + 1 For commodity pricing purposesutility is often assumed to be linear that implies risk neu-trality and constant marginal utility In this setup equation(3) can be simply written as

βEt Pt+i + ψt+i( 1113857 Et Pt+iminus 1( 1113857 (4)

Assuming further the presence of a risk-free bondavailable in clear financial markets with a time-invariant netinterest rate R standard no arbitrage condition indicates

Pt 1

1 + REt Pt+1 + ψt+1( 1113857 (5)

Solving this first-order stochastic difference (equation(5)) by forward iteration derives the general solution

Pt 1113944infin

i1

1(1 + R)

iEt ψt+1( 1113857 + lim

i⟶infinEt

1(1 + R)

iPt+i1113888 1113889 (6)

-e last equation has decomposed the price into twoparts -e first term of the right-hand side is regarded as amarket fundamental component which depends on thediscounted sum of expected future convenience yields -esecond term is commonly referred to as a periodicallycollapsing speculative bubble component Imposing thetransversality condition

limi⟶infin

Et

1(1 + R)

iPt+i1113888 1113889 0 (7)

rules out the existence of a bubble component -e price Pt

reduces to the fundamental price which will be representedby P

ft whereas if the transversality condition is not justified

there are infinitely many solutions to equation (5) -ey takethe more general form

Pt Pft + Bt (8)

where Bt is a rational bubble component that has to satisfythe submartingale property

Et Bt+1( 1113857 (1 + R)Bt (9)

If a bubble is present in the commodity market equation(9) requires the bubble to grow in expectation geometricallyat the rate of R in order for any rational investor willing tohold the asset Investors may drive the price far higher thanthe fundamental value as long as they believe that they cancompensate for the extra payment Bt through selling theasset to other market participants at an even higher price inthe future In other words commodity price bubbles couldbe a self-fulfilling prophecy Taken together equations (8)and (9) imply that Bt will display explosive dynamics as wellas the spot pricePt irrespective of whether the fundamentalcomponent P

ft is or is not itself nonexplosive

As illustrated in the following the contemporaneousfuture price at time t with maturity at t + T FTt will also beexplosive Let F1t denote the future price for delivery at t + 1To avoid arbitrage ψt must satisfy

ψt (1 + R)Pt minus F1t (10)

Given this condition and equation (5) it follows that F1t

is an unbiased predictor of the expected future spot price

F1t Et Pt+1( 1113857 (11)

Using equations (8) and (9) we obtain the followingexpression for the future price

FTt Et Pf

t+T1113872 1113873 + Et Bt+T( 1113857 Et Pf

t+T1113872 1113873 +(1 + R)T

Bt

(12)

We can easily see that the future price is a linear functionof the explosive process -is statistical property has mo-tivated our econometric tests for identifying explosivecharacteristics in prices

4 Econometric Methodology

In this section we give a brief description of bubble de-tectionmethods used in our empirical study so as to facilitateease interpretation of the results We first describe a con-tinuous-time model with randomized persistence and then astatistical inference procedure in the continuous systemFinally we discuss the model in the endogeneity setting

41 Model Specification Following Tao et al [25] we beginour analysis with a variant version of the Orn-steinndashUhlenbeck process given by

dx(t) x(t)μd(t) + σdWε(t) x(0) x0 (13)

where Wε(t) is a standard Brownian motion with constantvariance 1 also termed a standard Wiener process and μ isthe crucial persistence parameter that is meant to govern thedynamic behavior of x(t) When μlt 0 the process x(t) isstationary when μ 0 it is nonstationary and when μgt 0 itmanifests explosive characteristics In (13) μ is assumed tobe time-invariant which may not be corroborated by long-run data Hence the model studied in the paper takes theform of

4 Complexity

dx(t) x(t) μd(t) + 1113957σWμ(t)1113960 1113961 + σdWε(t) x(0) x0

(14)

where Wμ(t) and Wε(t) are two standard Brownianmotionsand the initial value x0 is viewed to be independent of Wμ(t)

and Wε(t) For 1113957σ2 ne 0 we see that random shocks to the driftterm have been incorporated into model (14) At this mo-ment we explore the situation where Wμ(t) and Wε(t) are

independent noise processes and subsequently allow for theendogenous case by relaxing the independence assumption

Solving model (14) yields the strong solution [49]

dx(t) exp 1113957σWμ(t) + μ minus12

1113957σ21113874 1113875t1113874 1113875x(0) + V(t) (15)

where

V(t) σ 1113946t

0exp 1113957σ Wμ(t) minus Wμ(s)1113872 1113873 + μ minus

12

1113957σ21113874 1113875(t minus s)1113874 1113875dWε(t)

sim MN 0 σ2 1113946t

0e21113957σ Wμ(t)minus Wμ(s)( 1113857+2 μminus (12)1113957σ

2( 1113857(tminus s)ds1113888 1113889

(16)

such that

E V(t)2

1113966 1113967 σ2E 1113946t

0e21113957σ Wμ(t)minus Wμ(s)( 1113857+2 μminus (12)1113957σ

2( 1113857(tminus s)ds1113896 1113897

σ2

2e2 μ+(12)1113957σ

2( 1113857t

minus 1μ +(12)1113957σ21113872 1113873

(17)

Under μ + (12)1113957σ2 gt 0 E V(t)21113966 1113967 is expected to growexponentially

-e corresponding discrete-time representation of (14) isderived directly from the strong solution and is of the form

xtΔ exp 1113957σ WμtΔ minus Wμ(tminus 1)Δ1113960 1113961 + μ minus12

1113957σ21113874 1113875Δ1113882 1113883x(tminus 1)Δ

+ σ 1113946tΔ

(tminus 1)Δexp 1113957σ WμtΔ minus Wμs1113872 1113873 + μ minus

12

1113957σ21113874 1113875(tΔ minus s)1113882 1113883dWεs

(18)

where Δ is the sampling interval over time span T andt 1 TΔ With respect to the random coefficientautoregression model [50] the autoregressive coefficient isdefined by

ρtΔ exp 1113957σ WμtΔ minus Wμ(tminus 1)Δ1113960 1113961 + μ minus12

1113957σ21113874 1113875Δ1113882 1113883 (19)

Here we introduce some notations to simplify thediscrete-time system Let

ϕ ≔ μ minus12

1113957σ2

κ ≔ μ +12

1113957σ2

utΔ ≔WμtΔ minus Wμ(tminus 1)Δ

Δ

radic simN(0 1)

ρtΔ ≔ exp 1113957σ WμtΔ minus Wμ(tminus 1)Δ1113960 1113961 + μ minus12

1113957σ21113874 1113875Δ1113882 1113883 exp ϕΔ + 1113957σΔ

radicutΔ1113966 1113967

ηtΔ ≔ 1113946tΔ

(tminus 1)Δexp 1113957σ WμtΔ minus Wμs1113872 1113873 + μ minus

12

1113957σ21113874 1113875(tΔ minus s)1113882 1113883dWεssimN 0e2κΔ

minus 12κ

1113888 1113889

(20)

Complexity 5

-e discrete-time representation (18) can then be re-written as

xtΔ exp 1113957σΔ

radicutΔ + ϕΔ1113966 1113967x(tminus 1)Δ + σηtΔ ρtΔx(tminus 1)Δ + σηtΔ

(21)

Assuming 1113957σ2 gt 0 models (14) and (21) have the followingstationarity properties (1) for (μ + 1113957σ22)lt 0 the process isasymptotically covariance stationary sinceμlt 0 means thatboth E(ρtΔ) eμΔ lt 1 and E(ρ2tΔ) exp(2μΔ+1113957σ2Δ) exp(2κΔ)lt 1 (2) for (μ + 1113957σ22)lt 0 the process re-mains asymptotically stationary but is not asymptoticallycovariance stationary any longer Practically speaking wehave to impose the condition of κ (μ + 1113957σ22)lt 0 toguarantee the covariance stationary Whenκ (μ + 1113957σ22)lt 0 it follows from (19) that the variance ofV(t) converges to minus σ22κltinfin as t⟶infinWhen κ 0 thevariance is equal to σ2t and diverges as t⟶infin (3) for (μ +1113957σ22)gt 0 and μlt 0 the process is still asymptotically sta-tionary but is not covariance stationary with evident per-sistence in the trajectory (4) when μ 0 ϕ (μ minus 1113957σ22)lt 0ensures the asymptotic stationarity but leads to infinitesecond moments withκ (μ + 1113957σ22)gt 0 -e process ex-hibits apparent unstable behavior and larger variation (5)when μgt 0 and μ minus 1113957σ22lt 0 the process continues to becovariance nonstationary Contrast to the traditional ex-plosive AR (1) model the model retains asymptoticallystationary In this instance bubble-like phenomenon shouldbe clearly observed in actual data (6) if μ minus 1113957σ22gt 0 themodel is no longer asymptotically stationary and the ex-plosive growth behavior is now obvious

As shown in Tao et al [25] when 1113957σ2 0 model (21)collapses to a simple autoregression with a constant coef-ficient It explains why a popular way to look for bubbles is totest ϕ 0 against ϕgt 0 via standard or recursive right-tailedunit root tests [15ndash18 27] For the purpose of the presentpaper we only examine three price behavior features that aredivided into three categories

(1) Unstable κ μ + 1113957σ22gt 0 Here the model is as-ymptotically covariance nonstationary and applies tocapture unstable behavior

(2) Locally explosive μgt 0 -e model in this case hasthe ability for generating both explosive and col-lapsing behavior

(3) Explosive ϕ μ minus 1113957σ22gt 0 Under the circumstancethe model is not asymptotically stationary and iscapable of generating explosive behavior

Based on the above typology explosive embodies locallyexplosive while locally explosive embodies unstable

42 Inference Scheme -is section is concerned with puttingthe continuous-time model specified above into the appli-cation of discretely sampled data We utilize a novel two-stage approach proposed by Phillips and Yu [51] to estimateμ 1113957σ2 and σ2 In the first stage realized volatility is employedto provide a regression model for estimating the diffusionparameters 1113957σ2 and σ2 In the second stage we maximize thein-fill likelihood function to estimate the drift parameter μ

Suppose data are observed in N equally spaced blocksover the time interval [0 T] with sampling interval h -eoverall time span of the data is T so that N Th If eachblock has M observations the total number of observationsis M times N TΔ where Δ is the sampling frequency Todevelop asymptotics we assume that as Δ⟶ 0M times N⟶infin According to the algorithm of Phillips andYu [51] the estimators based on realized variance (RV) andrealized quarticity (RQ) are denoted by

1113954θ argminθisinΘ

QΔ(θ) (22)

where 1113954θ ≔ (1113957σ2 σ2)prime and QΔ(θ) Δ1113936Nn1 (logRVnminus log

[x]nh(nminus 1)h + (12)s2n)2s2nwith s2n max

2Δ(RQ2

nRV2n)

11139681113864

2M

radic

RVn 1113944M

i1x(nminus 1)h+iΔ minus x(nminus 1)h+(iminus 1)Δ1113960 1113961

2 n 1 2 N

RQn 13Δ

1113944

M

i1x(nminus 1)h+iΔ minus x(nminus 1)h+(iminus 1)Δ1113960 1113961

4 n 1 2 N

(23)

In the second stage the approximate logarithmic in-filllikelihood function is maximized with regard to μ (repre-senting the resulting estimator as 1113954μ)

1113954μ argmaxμ

1MN

log ℓALF(μ) (24)

where

ℓALF(μ) 1113944MtimesN

t1

μx(tminus 1)Δ11139541113957σ2x2

(tminus 1)Δ +1113954σ2

xtΔ minus x(tminus 1)Δ1113872 1113873 minusΔ2

1113944

MtimesN

t1

μx(tminus 1)Δ11139541113957σ2x2

(tminus 1)Δ +1113954σ2

(25)

One important goal in the present empirical work is tocheck the presence of randomness in the persistenceparameter of model (14) which is equivalent to test the

null hypothesis 1113957σ2 0 To achieve this aim we resort tothe locally best invariant test (LBI test) statistic [52]namely

6 Complexity

1113957ZN ≔1113936

MtimesNt1 1113957ε2tΔ minus (1MN)1113936

MtimesNt1 1113957ε2tΔ1113872 1113873

21113874 11138751113957x

2(tminus 1)Δ

(1MN)1113936MtimesNt1 1113957ε4tΔ minus (1MN)1113936

MtimesNt1 1113957ε2tΔ1113872 1113873

21113969

(1MN)1113936MtimesNt1 1113957x

4(tminus 1)Δ minus (1MN)1113936

MtimesNt1 1113957x

2(tminus 1)Δ1113872 1113873

21113969 (26)

where 1113957x2tΔ (xtΔ

1 + x2

tΔ

1113969) 1113957εtΔ xtΔ minus 1113957ρx(tminus 1)Δ and

1113957ρ (1113936MtimesNt1 1113957x(tminus 1)Δx(tminus 1)Δ)

minus 11113936MtimesNt1 1113957x(tminus 1)Δx(tminus 1)Δ -us as

N⟶infin MNminus 12 1113957ZN⟶L N(0 1) under H0 1113957σ2 0 andMNminus (12) 1113957ZN⟶L N(0 1) under H0 1113957σ2 0

Relying on asymptotics with in-fill (as the sampling fre-quency tends to zeroΔ⟶ 0) and long time-span asymptoticsTao et al [25] developed new limit theory for the continuous-time system where the persistence parameter is random -easymptotic properties of the diffusion parameter and drift pa-rameter estimates can be used to construct test statistics for threetypes of unstableexplosive behavior which are

tκ 1Δ

1113944

MtimesN

t1

x2(tminus 1)Δ

11139541113957σ2x2(tminus 1)Δ + σ2

⎛⎝ ⎞⎠

12

1113954βκ minus β0κ1113872 1113873⟶L

N(0 1)

tμ 1Δ

1113944

MtimesN

t1

x2(tminus 1)Δ

11139541113957σ2x2(tminus 1)Δ + 1113954σ2

⎛⎝ ⎞⎠

12

1113954ρ minus ρ0( 1113857⟶L

N(0 1)

tϕ 1Δ

1113944

MtimesN

t1

x2(tminus 1)Δ

11139541113957σ2x2(tminus 1)Δ + 1113954σ2

⎛⎝ ⎞⎠

12

1113954βϕ minus β0ϕ1113872 1113873⟶L

N(0 1)

(27)

where

1113954βκ exp 1113954μ +12

11139541113957σ21113874 1113875Δ1113882 1113883

1113954βϕ exp 1113954μ minus12

11139541113957σ21113874 1113875Δ1113882 1113883

β0κ exp μ0 +12μ201113874 1113875Δ1113882 1113883

β0ϕ exp μ0 minus12μ201113874 1113875Δ1113882 1113883

1113954ρ 1 +1113936

MtimesNt1 x(tminus 1)Δ xtΔ minus x(tminus 1)Δ1113872 1113873 1113954

1113957σ2x2(tminus 1)Δ +

1113954σ21113874 1113875

1113936MtimesNt1 x

2(tminus 1)Δ

11139541113957σ2x2

(tminus 1)Δ +1113954σ21113874 1113875

ρ0 1

(28)

As can be seen this test procedure does not directlyverify whether or not κ (μ + 1113957σ22) 0 or μ 0 orϕ (μ minus 1113957σ22) 0 but transforms the null hypotheses toβk 1 or ρ 1 or βϕ 1 -ese three t-test statistics areaccompanied by a date-stamping strategy that permits us toidentify the exact origination and termination periods ofprice exuberance Given this instrument estimates of thesedates are as follows

1113954ribun inf

sisin 1113954r(iminus 1)e

un 11113858 1113859s tκ(s)gtZα1113864 1113865

1113954rieun inf

sisin 1113954r(iminus 1)b

un 11113858 1113859s tκ(s)ltZα1113864 1113865

1113954rible inf

sisin 1113954r(iminus 1)e

le 11113858 1113859s tμ(s)gtZα1113966 1113967

1113954rieun inf

sisin 1113954r(iminus 1)b

le 11113858 1113859s tμ(s)ltZα1113966 1113967

1113954ribe inf

sisin 1113954r(iminus 1)e

e 11113858 1113859s tϕ(s)gtZα1113966 1113967

1113954rieun inf

sisin 1113954r(iminus 1)b

e 11113858 1113859s tϕ(s)ltZα1113966 1113967

(29)

where Zα is the right-tailed critical value of the standardnormal distribution corresponding to a significance level ofα 1113954rib

un1113954rible1113954r

ibe denote the beginnings on the dates of the ith

extreme price behavior period and similarly 1113954rieun1113954r

iele1113954r

iee

denote the ends on the dates of the ith extreme price be-havior period

43 Extend the Model with Endogeneity Now we are con-cerned with the endogeneity case where Wμ(t) is not in-dependent of Wε(t) To see this let us suppose that data aregenerated from the following stochastic differential equationsystem

dx(t) x(t)d1113957B(t) + σdB(t) x(0) x0

d1113957B(t) μ dt + 1113957σWμ(t)

dB(t) σdWε(t)

(30)

where (Wμ(t) Wε(t)) is a two-dimensional Brownianmotion with covariance parameter η As Follmer [53] notedthe strong solution obtained from this continuous systemwould be of the explicit form

dx(t) exp 1113957σWμ(t) + μ minus12

1113957σ21113874 1113875t1113874 1113875x(0) + Z(t) (31)

where

Z(t) σ 1113946t

0exp 1113957σ Wμ(t) minus Wμ(s)1113872 1113873 + μ minus

12

1113957σ21113874 1113875(t minus s)1113874 1113875dWε(t)

minus ω1113946t

0exp 1113957σ Wμ(t) minus Wμ(s)1113872 1113873 + μ minus

12

1113957σ21113874 1113875(t minus s)1113874 1113875ds

V(t) minus P(t)

(32)

Notice that when endogeneity is included in model (14)the driver process becomes Z(t) as a substitute for V(t)Z(t) consists of a component V(t) and another oneconnected with the covariance parameter of (1113957B tB) ω and

Complexity 7

variance of V(t) -e strong solution gives the exact dis-crete-time form of model (31)

xtΔ exp 1113957σΔ

radicuΔ + μ minus

12

1113957σ21113874 1113875Δ1113882 1113883x(tminus 1)Δ + ZtΔ ρtΔ x(tminus 1)Δ + ZtΔ

+ σ 1113946tΔ

(tminus 1)Δexp 1113957σ WμtΔ minus Wμs1113872 1113873 + μ minus

12

1113957σ21113874 1113875(tΔ minus s)1113882 1113883dWεs

(33)

where uiidsim

t N(0 1) and

ZtΔ σ 1113946tΔ

(tminus 1)Δexp 1113957σ WμtΔ minus Wμs1113872 1113873 + μ minus

12

1113957σ21113874 1113875(tΔ minus s)1113882 1113883dWεs

minus ω1113946tΔ

(tminus 1)Δexp 1113957σ WμtΔ minus Wμs1113872 11138731113872 1113873 + μ minus

12

1113957σ21113874 1113875(tΔ minus s)1113874 1113875ds

(34)

Somewhat surprisingly Tao et al [25] indicated thatendogeneity would not affect the sample path features ofmodel (31) Our analysis therefore is the same as before

(1) Unstable κ μ + 1113957σ22gt 0(2) Locally explosive μgt 0(3) Explosive ϕ μ minus 1113957σ22gt 0

We can estimate 1113954θlowast ≔ ( 11139541113957σ2 1113954η 1113954σ2)prime in the same fashion asbefore -e likelihood ratio test (LR test) has been proposedby Tao et al [25] to test the null hypothesis of endogeneityinterest

LR Δminus 1Q

rΔ minus Q

urΔ( 1113857simχ2(1) underH0 η0 0 (35)

Tao et al [25] also developed the asymptotic theory for 1113954μand 1113954ρ ≔ exp(1113954μΔ) Based on their limit properties the es-timates of 1113954μ and 1113954ρ continue to follow asymptotic normaldistributions -is good quality enables us to convenientlyadopt the aforementioned testing procedures after making asmall adjustment to the variance of the limit distribution

5 Data

For our main empirical analysis we apply daily WTI crudeoil prices for the front-month (one-month) future contracton the New York Mercantile Exchange (NYMEX) -e dataare retrieved from the Energy Information Administration(EIA) Our sample spans the period from April 4 1983 toJune 30 2020 and comprises a total of 9352 daily obser-vations -e beginning and end dates of the sample periodare dictated by the availability of the data

It is well known that WIT and Brent are grades of in-ternationally traded crude oil that are widely used asbenchmarks in oil pricing globally -e reason for this paperrelying on the WTI price of crude oil is twofold On the onehand oil speculation is often blamed on people who tradefutures in the US market WTI is the primary benchmark

for oil extracted and consumed in the US-eWTI crude oilfuture market is also the worldrsquos most liquid and largestcrude oil future market Nearly 12 million contracts trade aday with more than 2 million in open interest On the otherhand WTI future contracts have the longest available his-tory dating back to 1983 giving us a large sample size to gainpowers in detecting short-lived bubbles

Our sample period under investigation includes severalmajor exogenous events that to a large extent are reckonedto have influenced oil markets -ese contain the collapse ofOPEC in 1986 the Persian Gulf War of 199091 the Asianfinancial crisis of 199798 the terrorist attacks on September11 2001 the Venezuelan crisis and Iraq war of 200203 theincreased financialization of oil future markets since 2004the OPEC oil supply announcements in 2008 and 2014 the2019 novel coronavirus disease (COVID-19) and moststrikingly the 200708 global financial crisis Figure 2 plotsthe time series trajectories of the WTI nominal front-monthfuture price and the corresponding real price A look at thefigure illustrates that the oil prices grew steadily from thestarting of the sample until the early 2000s -e price seriessince then begun to surge and the sharp increase in the priceof oil continued until July 2008 As shown in Figure 2 itcollapsed to about 30 dollars by the end of 2008 Between2014 and 2016 the price of oil also experienced a crash Morerecently WTI crude price even went subzero for the firsttime in history A common view is that the steep rise anddecline in the price of oil in the 2000s could be attributed tospeculation

6 Empirical Results

61 Main Results We start the empirical illustration withbaseline model (14) expanding it to the endogeneity ap-plication later In order to formally examine the existenceand duration of unstable and explosive behavior in the crudeoil market we run our tests on the daily price data

8 Complexity

Following the recommendation of PSY the minimumwindow size set for the recursive test procedure is the first 5years of the entire sample Figure 3 plots the test empiricaloutcomes along with their corresponding 95 critical valuesequences Also plotted are the trajectories of test statisticsfor the random coefficient and the price data Some inter-esting results are shown in this figure

Looking at the recursive test results reported in the lastpanel of Figure 3 we observe that over the initial period ofobservation the null hypothesis of a time-invariant autor-egressive coefficient cannot be rejected at the 5 statisticalsignificance level -e test statistic experiences a dramaticincrease in the late 1980s and exceeds the 95 critical valuein November 1990 Overall the observed pattern of timevariation rises and becomes much stronger as the timeperiod expands -e evidence that the LBI test statistics forthe majority of samples are greater than the 95 criticalvalue suggests that the data-generating process is wellcharacterized by the model with time-varying coefficientsOne point worth noting is that the sequence falls in thesecond half of 2008 and slumps in early 2020 -e styled factseems to indicate that the crude oil market has undergonedrastic changes Interestingly these located dates coincidewith the 200708 global financial crisis and the COVID-19pandemic

We now turn to our focus on different types of unstableand explosive behavior-e shaded areas in green in Figure 3are the identified periods where the t-test statistic crosses its95 critical value As is evident from the first panel theprocedure detects two unstable episodes -e first unstableepisode is relatively short and only lasts two months(2005M08-M09) -e second unstable episode appears inJanuary 2006 and continues until March 2020 -ese un-stable periods might be associated with the financializationof oil future markets from 2004 It can be seen in the secondpanel of Figure 3 that the detected episodes of locally ex-plosive (submartingale) behavior are clustered in 2007-20082009ndash2014 and 2018 As stated above since local explo-siveness implies instability and displays more extreme be-havior it should lead to a much shorter duration of

exuberance than that of instability -e third panel ofFigure 3 reflects that the t-test statistic is below the 95critical value throughout the sample period -is points outthat explosive behavior in crude oil price series does notexist Table 1 summarizes time horizons for the three kindsof unstable and explosive behavior identified by ourapproach

To shed light on a more realistic scenario of data-gen-erating process we implement the extreme sample pathbehavior test by extending the model to the endogeneitycase Figure 4 plots the empirical test results together with95 critical values A comparison of the LR test statisticswith the critical value sequence provides novel evidence insupport of the presence of endogeneity during the wholesample period -us it is believed that endogeneity is es-sential to the oil price formation process Furthermore thelast panel of Figure 4 highlights that the quadratic varianceof the price process is related to the test statistic forendogeneity -is finding is in accordance with the fact thatwe have exploited the realized variance series to obtain theLR test statistics In this sense Figure 5 displays differencesin the realized variance estimates which are captured by thetest statistic for endogeneity employing different methodsSimilar to Figure 3 we again identify episodes of unstabledynamics and fail to detect explosive behavior when con-sidering the model with endogeneity In contrast the sta-tistic sequence for local explosiveness exhibits a differentpattern Previous evidence in favor of locally explosive be-havior disappeared Table 2 summarizes time horizons forthe three forms of unstable and explosive behavior -eseresults suggest that endogeneity feedbacks play a significantrole in investigating the existence of explosive price be-havior -e conclusion that emerges is that there are nospeculative bubbles in the crude oil market

In principal our findings are in accordance with theexplanations made by Kilian and Murphy [19] Baumeisterand Hamilton [21] and Knittel and Pindyck [54] that thevariations in crude oil prices are driven by economicfundamentals

62 Comparison to Alternative Strategies Rich literature hasstudied speculative bubbles in the crude oil market using thetechniques developed by PWY and then extended by PSY Abrief description of the SADF methodology (used in PWY)and the GSADF methodology (used in PSY) can be found inAppendix It is therefore helpful to compare results foralternative methods as the detection strategy in the presentpaper is quite different from them Figure 6 plots thebackward ADF and backward SADF statistic sequences andthe associated 95 critical value sequences for oil futureprices 95 percent critical values are taken from 2000 MonteCarlo simulations with the actual sample size -e initialwindow size is set to 5 years A glance at Figure 6 reveals thatexplosive behavior occurred mainly in two occasions usingPWY and PSY dating strategies -ey are before and during2007-2008 global recession We also observe that PWYdetects several short-lived episodes between 2011 and 2014but PSY fails to do so For daily oil future prices over April 4

0

20

ndash20

ndash40

40

60

80

100

120

140

84 86 88 90 92 94 96 98 00 02 04 06 08 10 12 14 16 18 20

Nominal WTI front-month future pricesReal WTI front-month future prices

Figure 2 Real and nominal prices of WTI front-month futurecontracts

Complexity 9

1983 to June 30 2020 time horizons for explosive behavioridentified by PWY and PSY are summarized in Table 3 Inline with previous studies we could conclude from bothPWY and PSY tests that there is evidence of bubbles in thecrude oil market data However as shown in Gurkaynak [37]and Pavlidis et al [43] explosive dynamics in asset prices perse do not constitute evidence either in support of or againstspeculative bubbles since such explosive behavior may

actually reflect explosive market fundamentals or bothfactors As aforementioned our random coefficient autor-egressive model framework has considered endogeneityfeedbacks that to some extent alleviate this problemMoreover the evidence in Figures 3 and 4 suggests theabsence of explosive dynamics in the crude oil market -efact permits us to safely ignore the distinction betweenbubbles or explosive fundamentals

1988

-03

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

ndash4

ndash2

0

2

4

6

8

0

20

40

60

80

100

120

140

t-statistic of k95 critical value of N (0 1)Crude oil future prices

Instability detection based on the time-varyingcoefficient model

(a)

Local explosiveness detection based on the time-varyingcoefficient model

1988

-03

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

t-statistic of μ95 critical value of N (0 1)Crude oil future prices

ndash4

ndash2

0

2

4

6

8

0

20

40

60

80

100

120

140

(b)

Explosiveness detection based on the time-varyingcoefficient model

1988

-03

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

t-statistic of ϕ95 critical value of N (0 1)Crude oil future prices

ndash4

ndash2

0

2

4

6

8

0

20

40

60

80

100

120

140

(c)

LBItest statistic for testing time-varying coefficient19

88-0

3

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

LBItest95 critical value of N (0 1)

ndash5

ndash10

0

5

10

15

20

(d)

Figure 3 Date-stamping unstable and explosive episodes in oil prices with the time-varying coefficient model assuming no endogeneity

Table 1 Time horizons of unstable and explosive periods identified by models assuming no endogeneity

Unstable Locally explosive ExplosiveAug 2005-Sep 2005 Dec 2007ndashSep 2008 mdash

Jan 2006ndashMar 2020 Jun 2009ndashOct 2014 mdashJun 2018ndashSep 2018 mdash

10 Complexity

63 Robustness of the Results In the following we providetwo robustness checks to assess whether our previousconclusions of the paper still hold

First in our benchmark model our use of front-monthfuture price instead of other future prices at different ma-turities is justified by its being the most liquid future contractwith the highest open interest To explore the sensitivity ofour results to the future contracts at different maturities werepeat the analysis described above using the prices forlonger-term contracts (ie 2 3 and 4 months-ahead fu-tures) It should be pointed out that the WTI future contractusually expires on the third business day before the 25thcalendar day of the contract month proceeding the deliverymonth

Second becauseWTI crude oil is by tradition traded innominal US dollars per barrel it might be of concern that

inflation and other monetary variables may have realeffects on oil price It is therefore of interest to use realrather than nominal price In fact recent work documentsthat the use of either price is inconsequential in practice[55] Although we mainly focus on nominal prices forrobustness check we shall report the results for realprices To convert nominal daily series into daily realseries we use linearly interpolated monthly ConsumerPrice Index (CPI) Monthly CPI series for the UnitedStates is obtained from the websites of the Federal ReserveBank of St Louis

Further evidence presented in Figures 7ndash10 demon-strates that our results are remarkably robust to thesemodifications and the conclusions remain qualitativelyunchanged

ndash4

ndash2

0

2

4

6

8

0

20

40

60

80

100

120

140

1988

-03

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

Instability detection based on the time-varyingcoefficient model

t-statistic of k95 critical value of N (0 1)Crude oil future prices

(a)

ndash4

ndash2

0

2

4

6

8

0

20

40

60

80

100

120

140

1988

-03

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

t-statistic of μ95 critical value of N (0 1)Crude oil future prices

Local explosiveness detection based on the time-varyingcoefficient model

(b)

ndash4

ndash2

0

2

4

6

8

0

20

40

60

80

100

120

140

1988

-03

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

Explosiveness detection based on the time-varyingcoefficient model

t-statistic of ϕ95 critical value of N (0 1)Crude oil future prices

(c)

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

LR-statistic95 critical values of x2 (1)Realized variance

700

600

500

400

300

200

100

0

ndash100

350

300

250

200

150

100

50

0

Test for endogeneity in time-varyingcoefficient model

(d)

Figure 4 Date-stamping unstable and explosive episodes in oil prices with the time-varying coefficient model with endogeneity

Complexity 11

2

3

4

5

6

7

8

0

50

100

150

200

250

300

350

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

LLF0

LLF1

Realized variance of crude oil future prices

Figure 5 Realized variance and likelihood values for random coefficient regression

Table 2 Time horizons of unstable and explosive periods detected by models with endogeneity

Unstable Locally explosive ExplosiveMay 2008ndashJul 2008 mdash mdashDec 2010ndashOct 2014 mdash mdashNov 2016ndashApr 2017 mdash mdashJul 2017ndashFeb 2020 mdash mdash

ndash4

ndash2

0

2

4

6

8

0

20

40

60

80

100

120

140

1988

-03

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

Explosiveness detection using SADF methodology

BADF sequence95 critical valuesCrude oil future prices

(a)

ndash4

ndash2

0

2

4

6

8

0

20

40

60

80

100

120

140

1988

-03

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

Explosiveness detection using GSADF methodology

BSADF sequence95 critical valuesCrude oil future prices

(b)

Figure 6 Date-stamping explosive episodes in oil prices with the time-invariant coefficient model

12 Complexity

Table 3 Time horizons of explosive behavior identified by SADF and GSADF methodologies

SADF GSADFAug 2005ndashAug 2006 Oct 2011ndashApr 2012 Sep 1990-Oct 1990Jun 2007ndashOct 2008 Aug 2012-Sep 2012 Feb 1991-Mar 1991Jun 2009-Jul 2009 Jul 2013-Aug 2013 Oct 2004-Nov 2004Feb 2011ndashApr 2011 Feb 2014-Mar 2014 Feb 2005-Mar 2005

Jul 2011ndashAug 2012 Jun 2014-Jul 2014May 2005ndashJan 2006Mar 2006ndashAug 2006Jul 2007ndashAug 2008

ndash4

ndash2

0

2

4

6

8

0

10

20

30

40

50

60

70

1988

-03

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

95 critical value of N (0 1)Real oil future prices

t-statistic of μ~

Local explosiveness detection based on the time-varyingcoefficient model

(a)

ndash4

ndash2

0

2

4

6

8

0

10

20

30

40

50

60

70

1988

-03

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

95 critical value of N (0 1)Real oil future prices

t-statistic of ϕ

Explosiveness detection based on the time-varyingcoefficient model

(b)

Figure 7 Date-stamping explosive episodes in real oil prices with the time-varying coefficient model with endogeneity

ndash4

ndash2

0

2

4

6

8

0

50

100

150

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

Local explosiveness detection based on the time-varyingcoefficient model

t-statistic of μ95 critical value of N (0 1)2-month crude oil future prices

(a)

ndash4

ndash2

0

2

4

6

8

0

50

100

150

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

Local explosiveness detection based on the time-varyingcoefficient model

t-statistic of ϕ95 critical value of N (0 1)2-month crude oil future prices

(b)

Figure 8 Date-stamping explosive episodes in 2-month future oil prices with the time-varying coefficient model with endogeneity

Complexity 13

7 Discussion

-e oil price rally since the early 2000s has attracted greatinterest in the sources of oil price shocks Some causal ob-servers have stressed that financial speculation has become asignificant driver of the dynamics of oil prices A large body ofliterature emerged that has sought to investigate its existenceIn this paper we employ a novel approach for testing forspeculative bubbles in oil prices In synthesis our empiricalfindings indicate that there is less evidence of exuberance in

the oil future markets Our results are very robust in severalscenarios We argue that the booms and busts in the oil pricemovements during the sample period are driven primarily byeconomic fundamentals rather than speculation

Overall our results confirm the main findings obtainedby structural VAR-based studies including Kilian andMurphy [19] Kilian and Lee [20] Baumeister and Hamilton[21] and Zhou [22] In the past decade the use of structuralVAR models to understand the evolution of the oil price hasbecome common -e traditional SVAR literature

ndash4

ndash2

0

2

4

6

8

0

50

100

15019

88-0

3

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

Local explosiveness detection based on the time-varyingcoefficient model

t-statistic of μ95 critical value of N (0 1)3-month crude oil future prices

(a)

ndash4

ndash2

0

2

4

6

8

0

50

100

150

1988

-03

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

Explosiveness detection based on the time-varyingcoefficient model

t-statistic of ϕ95 critical value of N (0 1)3-month crude oil future prices

(b)

Figure 9 Date-stamping explosive episodes in 3-month future oil prices with the time-varying coefficient model with endogeneity

ndash4

ndash2

0

2

4

6

8

0

50

100

150

Local explosiveness detection based on the time-varyingcoefficient model

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

t-statistic of μ95 critical value of N (0 1)4-month crude oil future prices

(a)

ndash4

ndash2

0

2

4

6

8

0

50

100

150

Explosiveness detection based on the time-varyingcoefficient model

t-statistic of ϕ95 critical value of N (0 1)4-month crude oil futures prices

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

(b)

Figure 10 Date-stamping explosive episodes in 4-month future oil prices with the time-varying coefficient model with endogeneity

14 Complexity

documents that the fundamental drivers of oil price fluc-tuations have been dominated essentially by supply anddemand shocks see eg Kilian [56] Lippi and Nobili [57]Kilian andMurphy [58] and Baumeister and Peersman [59]Kilian and Murphy [19] focused on the role speculation byaugmenting the SVAR framework of their earlier paper byglobal oil inventory data -ey specified a four-variablemodel for the global oil market (the percentage change inglobal oil production the real price of oil a suitable proxy ofcyclical variation in global real economic activity andchanges in global crude oil inventories) -is setting andtheir sign-identified scheme allow them to separately explainvariation in these data as four different types of shocks (1) ashock to the amount of oil pumped out of the ground (ldquooilsupply shockrdquo) (2) an oil flow demand shock (3) an oil-specific oil demand shock and especially (4) a speculative(or inventory) demand shock -eir core finding is that thespike in the real price of crude oil prices during 2003ndash2008was caused by unexpected increase in global oil consump-tion Using two alternative proxies for global above-groundcrude oil inventories Kilian and Lee [20] reestimated themodel of Kilian and Murphy [19] and ruled out the role ofspeculation in explaining the recent surge of the real oil priceon the basis of their estimation results Also Zhou [22]assessed the sensitivity of the findings in Kilian and Murphy[19] to the introduction of various state-of-the-art oil marketmethods She shows that the core conclusions reached byKilian and Murphy [19] are highly robust to some refine-ments corrections and extension of the KilianndashMurphymodel Baumeister and Hamilton [21] recently contendedthat prior information plays a key role in any SVAR modelsand traditional identification approaches can be generalizedto a special case of a Bayesian perspective -ey revisited theimportance of supply and demand shocks in the oil marketand demonstrated that speculative demand shock turns outto be only a small factor in accounting for most historical oilprice shocks

For comparison purpose we have also applied the testsof PWY and PSY which are the most popular econometrictechniques in the literature on speculative bubbles in the oilmarkets If we were to follow the test procedures of PWY andPSY our conclusions are also broadly consistent with therich literature that supports the existence of speculativebubbles in oil prices However these cointegration-basedbubble tests suffer from the joint hypothesis problem ldquoEverytest of a bubble is a joint test of the presence of the bubble inthe data and the validity of the model applied by theeconometricianrdquo (see Giglio et al [60]) More specifically themain drawback of such indirect model-dependent tests ofbubbles typically refers to the difficulty in distinguishingbetween the unknown true model for market fundamentalsand the presence of bubbles (see for example Gurkaynak[37] Pavlidis et al [42] Pavlidis et al [43]) Given thefundamental value of oil price is unobservable the literaturehas resorted to observe economic and financial variableswhich are used to estimate the intrinsic values For exampleboth Phillips and Yu [16] and Caspi et al [39] simplynormalized crude oil prices by US oil inventory data as thefundamental value -is measure is questionable since their

method cannot sufficiently capture the information onmarket fundamentals and neglect the major driversmdashlikeglobal economic activitymdashalso affecting the fundamentalvalue of crude oil -is issue of misspecified fundamentals(either in terms of model misspecification or omitted var-iables or both of these reasons) may make researchers fail toarrive at a conclusion from evidence of explosive pricesalone To make matters worse even though we have cap-tured the true market fundamentals we may still fail todetect bubbles if the explosive behavior in the price of oil juststems from the explosive fundamentals

Compared to earlier studies as briefly mentioned aboveour methodology has the main advantage that by consid-ering the dependence between the random coefficient andsystem shocks we have incorporated endogeneity into thecontinuous-time random coefficient setting and thus we donot need to prespecify the fundamental component of the oilprice -e test results for endogeneity suggest that endo-geneity is important in the data-generating process Inparticular our empirical analysis also finds that there are noexplosive episodes in crude oil markets-is implies that it isnot necessary to require the specification of a fundamentalvalue of the oil price Like Pavlidis et al [43] their methodsand ours have in common that the problem of the well-known joint hypothesis has to be some extent been ef-fectively ameliorated Although our approach is quite dis-tinct we draw the same substantive conclusion thatspeculative bubbles can be ruled out as a key determinant ofthe price of crude oil in the 2000s reaffirming the con-clusions of Kilian and Murphy [19] Baumeister andHamilton [21] Zhou [22] and Fattouh et al [61]

8 Conclusions

Repeated surges and crashes in crude oil prices during lastdecades have sparked a heated debate on their underlyingcauses among academics commentors and market regu-lators One of the central questions concerns whetherfinancialization of oil future markets creates a speculativebubble that subsequently collapses Answering this questionis difficult because one has to rely upon strong assumptionsabout the data-generating mechanism for the fundamentalvalue of the oil price In this paper we present a stylizedmodel of asset pricing for a commodity to address this issueOur theoretical framework disentangles the effects ofspeculative bubbles from the traditional fundamental effectson oil price dynamics We use this setting to illustrate thatbubbles if they are present should demonstrate explosivebehavior in prices We then provide a novel approach forbubble detection that covers various explosive scenarios andfind no evidence of speculative bubbles in oil markets evenduring episodes where the most popular SADF and GSADFmethodologies identify a speculative bubble

Our conclusions have pertinent implications for policymakers Given the historically important role of oil prices inthe real economy one of the serious concerns in policycircles over the past decade has been whether speculativetrading activities in oil future markets are responsible for thelarge price swings For example in their 2008 testimonies to

Complexity 15

the US Senate Commerce Committee Michael Masters andGeorge Soros posited that speculators had fueled price in-creases since 2003 and had caused a part of a ldquosuperbubblerdquoin 2007 and 2008 resulting in significantly higher energycosts for consumers -is widespread bubble view domi-nated policy discussions and efforts to limit speculativetrading in the future market quickly followed As a responseto the excessive speculation pressures on prices theDoddndashFrank Wall Street Reform and Consumer ProtectionAct was enacted by the US Congress in 2010 As a directpolicy implication our results highlight that imposingstricter regulations on trading in oil derivative marketscannot be expected to lower the oil price given the fact thatthe synchronized boom and bust cycles witnessed in the past20 years are mainly due to economic fundamentals whichappear likely to resurge with the business cycle -e aboveanalysis implies that there is no need to enforce additionalregulatory limits against financial arbitrage in the com-modity market

Appendix

The SADF and GSADF Methodologies

Appendix describes the SADF and GSADF testing proce-dures used to detect speculative bubbles in Section 62 For agenetic time series process xt consider the following rolling-window type augmented DickeyndashFuller (ADF) regressionequation

xt ar1r2+ μr1r2

xtminus 1 + 1113944

J

j1ϕj

r1r2Δxtminus j + εt εt sim NID 0 δ2r1r2

1113874 1113875

(A1)

where Δxtminus j with j 1 J are lagged first differences ofxt incorporated to accommodate potential serial correla-tion ar1 r2

μr1 r2 and ϕj

r1 r2 with j 1 J are regression

coefficients the subscripts r1 and r2 refer to fractions of thetotal sample size (ofT observations) that set the start and endpoints of a subsample period and NID represents inde-pendent and normal distribution -e ADF test statisticcorresponding to the unit root null hypothesisH0 μr1 r2

0 is simply the t-ratio

ADFr2r1

1113954μr1 r2

se 1113954μr1 r21113872 1113873

(A2)

and the right-side alternative hypothesis of explosive be-havior is H0 μr1r2

gt 0 We can obtain the standard ADF teststatistic by running regression (A1) on the whole sampleie by specifying r1 0 and r2 1 A large amount of re-search has demonstrated that the standard ADF test ADF10has extremely low power in detecting periodically collapsingbubbles

To address the problem PSW proposed a recursiveregression technique which involves estimating (A1)employing a forward expanding sample Specifically theyused subsamples of the available data to recursively calculatea sequence of ADF statistics denoted by ADFr2

0 -e startpoint of the sample period is fixed at r1 0 while the endpoint r2 increases from r0 (the minimum window size) toone (the full sample period) -e SADF statistic called thesupremum of ADFr2

0 is defined by

SADF r0( 1113857 supr2isin r0 1[ ]

ADFr20

(A3)

and has the following limit distribution under the nullhypothesis

supr2isin r0 1[ ]

1113938r2

0 WdW

1113938r2

0 W21113872 1113873(12)

(A4)

where W is the standard Brownian motion Testing thepresence of explosive behavior entails comparing the ADFr2

0statistic with the right-side critical values obtained throughMonte Carlo simulations

More recently PSY showed that when the sample perioddisplays more than one boom-bust bubble the SADF testprocedure may perform poorly and fail to detect or con-sistently date-stamp alternative bubbles -is deficiencymotivates PSY to formulate an extension of the SADF testthe GSADF test which covers more subsamples of the datathan the SADF by allowing both the end point r2 and thestart point r1 to vary As a result the GSADF test has moreflexibility and gains substantial power making this testconsistent with multiple boom-bust bubbles -e GSADFtest statistic is defined as

GSADF r0( 1113857 supr2isin r0 1[ ]r1isin 0r2minus r0[ ]

ADFr2r1 (A5)

with the limit distribution

supr2isin r0 1[ ]r1isin 0r2minus r0[ ]

(12)rw W r2( 11138572

minus W r1( 11138572

minus rw1113960 1113961 minus 1113938r2

r1W(r)dr W r2( 1113857 minus W r1( 11138571113858 1113859

r12w rw 1113938

r2

r1W(r)2dr minus 1113938

r2

r1W(r)dr1113876 1113877

21113896 1113897

12

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎭

(A6)

where rw r2 minus r1 is the expanding window size Like theSADF test comparison of the GASDF statistic with the right-

tailed critical values from its limit distribution enables us to testfor the unit root hypothesis against explosive behavior

16 Complexity

If the null hypothesis of a unit root in xt is rejectedthen in the second stage the SADF and GSADF tests canidentify the exact episodes during which the xt seriesexhibits explosive dynamics To locate the emergence andconclusion of explosiveness the SADF methodologymatches the sequence of the recursive backward ADF(BADF) test statistics against the right-tailed criticalvalues of the limit distribution of the standard Dick-eyndashFuller test statistic Suppose that re is the originationdate and rf is the finish date of the bubble in the data-ese estimates of dates can be constructed from

1113954re infr2isin r0 1[ ]

r2 ADFr20 gt cv

αr2

1113966 1113967

1113954rf infr2isin 1113954re11113858 1113859

r2 ADFr20 lt cv

αr2

1113966 1113967(A7)

where cvαr2 is 100(1 minus α) the right-tailed critical value of theADF statistic corresponding to the chosen significance levelof α For the GSADF methodology the date-stampingstrategy in this case is based on the backward SADF(BSADF) statistic

BSADFr2r0( 1113857 sup

r1isin 0r2minus r0[ ]ADFr2

r1

(A8)

Likewise the origination date of the bubble is defined asthe first observation that the BSADF statistic is greater thanits critical value

1113954re infr2isin r0 1[ ]

r2 BSADFr2r0( 1113857gt cv

βT

r21113966 1113967 (A9)

and the finish date as the first date after which the BSADFdrops back below its critical value

1113954rf infr2isin 1113954re11113858 1113859

r2 BSADFr2r0( 1113857lt cvβT

r21113966 1113967 (A10)

where cvβTr2 is 100(1 minus βT) the right-tailed critical value of

the SADF based on Tr2 and the corresponding significancelevel is βT

Data Availability

-e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

-e authors declare no conflicts of interest

Acknowledgments

-e authors gratefully acknowledge the financial supportfrom the National Social Science Foundation of China underGrant no 18BTJ039

References

[1] M W Masters Testimony before the US Senate Committee onHomeland Security and Governmental Affairs 2008 httpswwwhsgacsenategovimomediadoc052008Masterspdfattempt2

[2] M W Masters Testimony before the Commodity FuturesTrading Commission 2010 httpscftcgovsitesdefaultfilesidcgroupspublicnewsroomdocumentsfilemetalmarkets032510_masterspdf

[3] J P Kennedy ldquo-e high cost of gambling on oilrdquo Section A p23 New York Times New York NY USANew York Times2012

[4] I-H Cheng and W Xiong ldquoFinancialization of commoditymarketsrdquo Annual Review of Financial Economics vol 6 no 1pp 419ndash441 2014

[5] K Tang and W Xiong ldquoIndex investment and the financi-alization of commoditiesrdquo Financial Analysts Journal vol 68no 6 pp 54ndash74 2012

[6] O J Blanchard and M W Watson ldquoBubbles rational ex-pectations and financial marketsrdquo in Crisis in the Economicand Financial Structure P Wachtel Ed Lexington BooksLexington MA USA pp 295ndash315 1982

[7] B T Diba and H I Grossman ldquoOn the inception of rationalbubblesrdquo e Quarterly Journal of Economics vol 102 no 3pp 697ndash700 1987

[8] G W Evans ldquo-e fragility of sunspots and bubblesrdquo Journalof Monetary Economics vol 23 no 2 pp 297ndash317 1989

[9] G W Evans and S Honkapohja ldquoOn the robustness ofbubbles in linear RE modelsrdquo International Economic Reviewvol 33 no 1 pp 1ndash14 1992

[10] E Pavlidis A Yusupova I Paya et al ldquoEpisodes of exu-berance in housing markets in search of the smoking gunrdquoe Journal of Real Estate Finance and Economics vol 53no 4 pp 419ndash449 2016

[11] E Martınez-Garcıa and V Grossman ldquoExplosive dynamics inhouse prices An exploration of financial market spillovers inhousing markets around the worldrdquo Journal of InternationalMoney and Finance vol 101 p 102103 2020

[12] S Shi A Rahman and B Z Wang ldquoAustralian housingmarket booms fundamentals or speculationlowastrdquo EconomicRecord 2020 In press

[13] P C B Phillips Y Wu and J Yu ldquoExplosive behavior in the1990s NASDAQ when did exuberance escalate asset val-ueslowastrdquo International Economic Review vol 52 no 1pp 201ndash226 2011

[14] P C B Phillips S Shi and J Yu ldquoTesting for multiplebubbles historical episodes of exuberance and collapse in theSampp 500rdquo International Economic Review vol 56 no 4pp 1043ndash1078 2015

[15] P C B Phillips S Shi and J Yu ldquoTesting for multiplebubbles limit theory of real-time detectorsrdquo InternationalEconomic Review vol 56 no 4 pp 1079ndash1134 2015

[16] P C B Phillips and J Yu ldquoDating the timeline of financialbubbles during the subprime crisisrdquo Quantitative Economicsvol 2 no 3 pp 455ndash491 2011

[17] D Tsvetanov J Coakley and N Kellard ldquoBubbling over thebehaviour of oil futures along the yield curverdquo Journal ofEmpirical Finance vol 38 no Part B pp 516ndash533 2016

[18] I Figuerola-Ferretti J R Mccrorie and I ParaskevopoulosldquoMild explosivity in recent crude oil pricesrdquo Energy Eco-nomics vol 87 p 104387 2020

[19] L Kilian and D P Murphy ldquo-e role of inventories andspeculative trading in the global market for crude oilrdquo Journalof Applied Econometrics vol 29 no 3 pp 454ndash478 2014

Complexity 17

discounted future benefits called the convenience yield ψt Inthis case the convenience yield is analogous to the payoff(dividend) on a stock -e concept of convenience yield is wellknown and has been widely used in analyzing commoditymarkets Specifically it measures a premium that accrues toinventory holders stemming from possible alternative inven-tory functions Forward-looking investors usually establishinventories to smooth production requirements avoid stock-outs and accommodate sales scheduling [45ndash48] Changes indesired inventories reflect the shifts in the agentsrsquo expectationsabout current and future market conditions As such ourmodel links the latent convenience yield to a reduced-formdynamic model of demand and supply

-e first-order condition for the representative investorrsquosutility optimization problem specified by (1) and (2) yields

Et βuprime Ct+i( 1113857 Pt+i + ψt+i1113858 11138591113864 1113865 Et uprime Ct+iminus 1( 1113857Pt+iminus 11113864 1113865 (3)

Intuitively equation (3) states that for the optimal timepath of x an investor cannot increase expected utility byselling or buying an asset at time t and performing a reversetransaction at time t + 1 For commodity pricing purposesutility is often assumed to be linear that implies risk neu-trality and constant marginal utility In this setup equation(3) can be simply written as

βEt Pt+i + ψt+i( 1113857 Et Pt+iminus 1( 1113857 (4)

Assuming further the presence of a risk-free bondavailable in clear financial markets with a time-invariant netinterest rate R standard no arbitrage condition indicates

Pt 1

1 + REt Pt+1 + ψt+1( 1113857 (5)

Solving this first-order stochastic difference (equation(5)) by forward iteration derives the general solution

Pt 1113944infin

i1

1(1 + R)

iEt ψt+1( 1113857 + lim

i⟶infinEt

1(1 + R)

iPt+i1113888 1113889 (6)

-e last equation has decomposed the price into twoparts -e first term of the right-hand side is regarded as amarket fundamental component which depends on thediscounted sum of expected future convenience yields -esecond term is commonly referred to as a periodicallycollapsing speculative bubble component Imposing thetransversality condition

limi⟶infin

Et

1(1 + R)

iPt+i1113888 1113889 0 (7)

rules out the existence of a bubble component -e price Pt

reduces to the fundamental price which will be representedby P

ft whereas if the transversality condition is not justified

there are infinitely many solutions to equation (5) -ey takethe more general form

Pt Pft + Bt (8)

where Bt is a rational bubble component that has to satisfythe submartingale property

Et Bt+1( 1113857 (1 + R)Bt (9)

If a bubble is present in the commodity market equation(9) requires the bubble to grow in expectation geometricallyat the rate of R in order for any rational investor willing tohold the asset Investors may drive the price far higher thanthe fundamental value as long as they believe that they cancompensate for the extra payment Bt through selling theasset to other market participants at an even higher price inthe future In other words commodity price bubbles couldbe a self-fulfilling prophecy Taken together equations (8)and (9) imply that Bt will display explosive dynamics as wellas the spot pricePt irrespective of whether the fundamentalcomponent P

ft is or is not itself nonexplosive

As illustrated in the following the contemporaneousfuture price at time t with maturity at t + T FTt will also beexplosive Let F1t denote the future price for delivery at t + 1To avoid arbitrage ψt must satisfy

ψt (1 + R)Pt minus F1t (10)

Given this condition and equation (5) it follows that F1t

is an unbiased predictor of the expected future spot price

F1t Et Pt+1( 1113857 (11)

Using equations (8) and (9) we obtain the followingexpression for the future price

FTt Et Pf

t+T1113872 1113873 + Et Bt+T( 1113857 Et Pf

t+T1113872 1113873 +(1 + R)T

Bt

(12)

We can easily see that the future price is a linear functionof the explosive process -is statistical property has mo-tivated our econometric tests for identifying explosivecharacteristics in prices

4 Econometric Methodology

In this section we give a brief description of bubble de-tectionmethods used in our empirical study so as to facilitateease interpretation of the results We first describe a con-tinuous-time model with randomized persistence and then astatistical inference procedure in the continuous systemFinally we discuss the model in the endogeneity setting

41 Model Specification Following Tao et al [25] we beginour analysis with a variant version of the Orn-steinndashUhlenbeck process given by

dx(t) x(t)μd(t) + σdWε(t) x(0) x0 (13)

where Wε(t) is a standard Brownian motion with constantvariance 1 also termed a standard Wiener process and μ isthe crucial persistence parameter that is meant to govern thedynamic behavior of x(t) When μlt 0 the process x(t) isstationary when μ 0 it is nonstationary and when μgt 0 itmanifests explosive characteristics In (13) μ is assumed tobe time-invariant which may not be corroborated by long-run data Hence the model studied in the paper takes theform of

4 Complexity

dx(t) x(t) μd(t) + 1113957σWμ(t)1113960 1113961 + σdWε(t) x(0) x0

(14)

where Wμ(t) and Wε(t) are two standard Brownianmotionsand the initial value x0 is viewed to be independent of Wμ(t)

and Wε(t) For 1113957σ2 ne 0 we see that random shocks to the driftterm have been incorporated into model (14) At this mo-ment we explore the situation where Wμ(t) and Wε(t) are

independent noise processes and subsequently allow for theendogenous case by relaxing the independence assumption

Solving model (14) yields the strong solution [49]

dx(t) exp 1113957σWμ(t) + μ minus12

1113957σ21113874 1113875t1113874 1113875x(0) + V(t) (15)

where

V(t) σ 1113946t

0exp 1113957σ Wμ(t) minus Wμ(s)1113872 1113873 + μ minus

12

1113957σ21113874 1113875(t minus s)1113874 1113875dWε(t)

sim MN 0 σ2 1113946t

0e21113957σ Wμ(t)minus Wμ(s)( 1113857+2 μminus (12)1113957σ

2( 1113857(tminus s)ds1113888 1113889

(16)

such that

E V(t)2

1113966 1113967 σ2E 1113946t

0e21113957σ Wμ(t)minus Wμ(s)( 1113857+2 μminus (12)1113957σ

2( 1113857(tminus s)ds1113896 1113897

σ2

2e2 μ+(12)1113957σ

2( 1113857t

minus 1μ +(12)1113957σ21113872 1113873

(17)

Under μ + (12)1113957σ2 gt 0 E V(t)21113966 1113967 is expected to growexponentially

-e corresponding discrete-time representation of (14) isderived directly from the strong solution and is of the form

xtΔ exp 1113957σ WμtΔ minus Wμ(tminus 1)Δ1113960 1113961 + μ minus12

1113957σ21113874 1113875Δ1113882 1113883x(tminus 1)Δ

+ σ 1113946tΔ

(tminus 1)Δexp 1113957σ WμtΔ minus Wμs1113872 1113873 + μ minus

12

1113957σ21113874 1113875(tΔ minus s)1113882 1113883dWεs

(18)

where Δ is the sampling interval over time span T andt 1 TΔ With respect to the random coefficientautoregression model [50] the autoregressive coefficient isdefined by

ρtΔ exp 1113957σ WμtΔ minus Wμ(tminus 1)Δ1113960 1113961 + μ minus12

1113957σ21113874 1113875Δ1113882 1113883 (19)

Here we introduce some notations to simplify thediscrete-time system Let

ϕ ≔ μ minus12

1113957σ2

κ ≔ μ +12

1113957σ2

utΔ ≔WμtΔ minus Wμ(tminus 1)Δ

Δ

radic simN(0 1)

ρtΔ ≔ exp 1113957σ WμtΔ minus Wμ(tminus 1)Δ1113960 1113961 + μ minus12

1113957σ21113874 1113875Δ1113882 1113883 exp ϕΔ + 1113957σΔ

radicutΔ1113966 1113967

ηtΔ ≔ 1113946tΔ

(tminus 1)Δexp 1113957σ WμtΔ minus Wμs1113872 1113873 + μ minus

12

1113957σ21113874 1113875(tΔ minus s)1113882 1113883dWεssimN 0e2κΔ

minus 12κ

1113888 1113889

(20)

Complexity 5

-e discrete-time representation (18) can then be re-written as

xtΔ exp 1113957σΔ

radicutΔ + ϕΔ1113966 1113967x(tminus 1)Δ + σηtΔ ρtΔx(tminus 1)Δ + σηtΔ

(21)

Assuming 1113957σ2 gt 0 models (14) and (21) have the followingstationarity properties (1) for (μ + 1113957σ22)lt 0 the process isasymptotically covariance stationary sinceμlt 0 means thatboth E(ρtΔ) eμΔ lt 1 and E(ρ2tΔ) exp(2μΔ+1113957σ2Δ) exp(2κΔ)lt 1 (2) for (μ + 1113957σ22)lt 0 the process re-mains asymptotically stationary but is not asymptoticallycovariance stationary any longer Practically speaking wehave to impose the condition of κ (μ + 1113957σ22)lt 0 toguarantee the covariance stationary Whenκ (μ + 1113957σ22)lt 0 it follows from (19) that the variance ofV(t) converges to minus σ22κltinfin as t⟶infinWhen κ 0 thevariance is equal to σ2t and diverges as t⟶infin (3) for (μ +1113957σ22)gt 0 and μlt 0 the process is still asymptotically sta-tionary but is not covariance stationary with evident per-sistence in the trajectory (4) when μ 0 ϕ (μ minus 1113957σ22)lt 0ensures the asymptotic stationarity but leads to infinitesecond moments withκ (μ + 1113957σ22)gt 0 -e process ex-hibits apparent unstable behavior and larger variation (5)when μgt 0 and μ minus 1113957σ22lt 0 the process continues to becovariance nonstationary Contrast to the traditional ex-plosive AR (1) model the model retains asymptoticallystationary In this instance bubble-like phenomenon shouldbe clearly observed in actual data (6) if μ minus 1113957σ22gt 0 themodel is no longer asymptotically stationary and the ex-plosive growth behavior is now obvious

As shown in Tao et al [25] when 1113957σ2 0 model (21)collapses to a simple autoregression with a constant coef-ficient It explains why a popular way to look for bubbles is totest ϕ 0 against ϕgt 0 via standard or recursive right-tailedunit root tests [15ndash18 27] For the purpose of the presentpaper we only examine three price behavior features that aredivided into three categories

(1) Unstable κ μ + 1113957σ22gt 0 Here the model is as-ymptotically covariance nonstationary and applies tocapture unstable behavior

(2) Locally explosive μgt 0 -e model in this case hasthe ability for generating both explosive and col-lapsing behavior

(3) Explosive ϕ μ minus 1113957σ22gt 0 Under the circumstancethe model is not asymptotically stationary and iscapable of generating explosive behavior

Based on the above typology explosive embodies locallyexplosive while locally explosive embodies unstable

42 Inference Scheme -is section is concerned with puttingthe continuous-time model specified above into the appli-cation of discretely sampled data We utilize a novel two-stage approach proposed by Phillips and Yu [51] to estimateμ 1113957σ2 and σ2 In the first stage realized volatility is employedto provide a regression model for estimating the diffusionparameters 1113957σ2 and σ2 In the second stage we maximize thein-fill likelihood function to estimate the drift parameter μ

Suppose data are observed in N equally spaced blocksover the time interval [0 T] with sampling interval h -eoverall time span of the data is T so that N Th If eachblock has M observations the total number of observationsis M times N TΔ where Δ is the sampling frequency Todevelop asymptotics we assume that as Δ⟶ 0M times N⟶infin According to the algorithm of Phillips andYu [51] the estimators based on realized variance (RV) andrealized quarticity (RQ) are denoted by

1113954θ argminθisinΘ

QΔ(θ) (22)

where 1113954θ ≔ (1113957σ2 σ2)prime and QΔ(θ) Δ1113936Nn1 (logRVnminus log

[x]nh(nminus 1)h + (12)s2n)2s2nwith s2n max

2Δ(RQ2

nRV2n)

11139681113864

2M

radic

RVn 1113944M

i1x(nminus 1)h+iΔ minus x(nminus 1)h+(iminus 1)Δ1113960 1113961

2 n 1 2 N

RQn 13Δ

1113944

M

i1x(nminus 1)h+iΔ minus x(nminus 1)h+(iminus 1)Δ1113960 1113961

4 n 1 2 N

(23)

In the second stage the approximate logarithmic in-filllikelihood function is maximized with regard to μ (repre-senting the resulting estimator as 1113954μ)

1113954μ argmaxμ

1MN

log ℓALF(μ) (24)

where

ℓALF(μ) 1113944MtimesN

t1

μx(tminus 1)Δ11139541113957σ2x2

(tminus 1)Δ +1113954σ2

xtΔ minus x(tminus 1)Δ1113872 1113873 minusΔ2

1113944

MtimesN

t1

μx(tminus 1)Δ11139541113957σ2x2

(tminus 1)Δ +1113954σ2

(25)

One important goal in the present empirical work is tocheck the presence of randomness in the persistenceparameter of model (14) which is equivalent to test the

null hypothesis 1113957σ2 0 To achieve this aim we resort tothe locally best invariant test (LBI test) statistic [52]namely

6 Complexity

1113957ZN ≔1113936

MtimesNt1 1113957ε2tΔ minus (1MN)1113936

MtimesNt1 1113957ε2tΔ1113872 1113873

21113874 11138751113957x

2(tminus 1)Δ

(1MN)1113936MtimesNt1 1113957ε4tΔ minus (1MN)1113936

MtimesNt1 1113957ε2tΔ1113872 1113873

21113969

(1MN)1113936MtimesNt1 1113957x

4(tminus 1)Δ minus (1MN)1113936

MtimesNt1 1113957x

2(tminus 1)Δ1113872 1113873

21113969 (26)

where 1113957x2tΔ (xtΔ

1 + x2

tΔ

1113969) 1113957εtΔ xtΔ minus 1113957ρx(tminus 1)Δ and

1113957ρ (1113936MtimesNt1 1113957x(tminus 1)Δx(tminus 1)Δ)

minus 11113936MtimesNt1 1113957x(tminus 1)Δx(tminus 1)Δ -us as

N⟶infin MNminus 12 1113957ZN⟶L N(0 1) under H0 1113957σ2 0 andMNminus (12) 1113957ZN⟶L N(0 1) under H0 1113957σ2 0

Relying on asymptotics with in-fill (as the sampling fre-quency tends to zeroΔ⟶ 0) and long time-span asymptoticsTao et al [25] developed new limit theory for the continuous-time system where the persistence parameter is random -easymptotic properties of the diffusion parameter and drift pa-rameter estimates can be used to construct test statistics for threetypes of unstableexplosive behavior which are

tκ 1Δ

1113944

MtimesN

t1

x2(tminus 1)Δ

11139541113957σ2x2(tminus 1)Δ + σ2

⎛⎝ ⎞⎠

12

1113954βκ minus β0κ1113872 1113873⟶L

N(0 1)

tμ 1Δ

1113944

MtimesN

t1

x2(tminus 1)Δ

11139541113957σ2x2(tminus 1)Δ + 1113954σ2

⎛⎝ ⎞⎠

12

1113954ρ minus ρ0( 1113857⟶L

N(0 1)

tϕ 1Δ

1113944

MtimesN

t1

x2(tminus 1)Δ

11139541113957σ2x2(tminus 1)Δ + 1113954σ2

⎛⎝ ⎞⎠

12

1113954βϕ minus β0ϕ1113872 1113873⟶L

N(0 1)

(27)

where

1113954βκ exp 1113954μ +12

11139541113957σ21113874 1113875Δ1113882 1113883

1113954βϕ exp 1113954μ minus12

11139541113957σ21113874 1113875Δ1113882 1113883

β0κ exp μ0 +12μ201113874 1113875Δ1113882 1113883

β0ϕ exp μ0 minus12μ201113874 1113875Δ1113882 1113883

1113954ρ 1 +1113936

MtimesNt1 x(tminus 1)Δ xtΔ minus x(tminus 1)Δ1113872 1113873 1113954

1113957σ2x2(tminus 1)Δ +

1113954σ21113874 1113875

1113936MtimesNt1 x

2(tminus 1)Δ

11139541113957σ2x2

(tminus 1)Δ +1113954σ21113874 1113875

ρ0 1

(28)

As can be seen this test procedure does not directlyverify whether or not κ (μ + 1113957σ22) 0 or μ 0 orϕ (μ minus 1113957σ22) 0 but transforms the null hypotheses toβk 1 or ρ 1 or βϕ 1 -ese three t-test statistics areaccompanied by a date-stamping strategy that permits us toidentify the exact origination and termination periods ofprice exuberance Given this instrument estimates of thesedates are as follows

1113954ribun inf

sisin 1113954r(iminus 1)e

un 11113858 1113859s tκ(s)gtZα1113864 1113865

1113954rieun inf

sisin 1113954r(iminus 1)b

un 11113858 1113859s tκ(s)ltZα1113864 1113865

1113954rible inf

sisin 1113954r(iminus 1)e

le 11113858 1113859s tμ(s)gtZα1113966 1113967

1113954rieun inf

sisin 1113954r(iminus 1)b

le 11113858 1113859s tμ(s)ltZα1113966 1113967

1113954ribe inf

sisin 1113954r(iminus 1)e

e 11113858 1113859s tϕ(s)gtZα1113966 1113967

1113954rieun inf

sisin 1113954r(iminus 1)b

e 11113858 1113859s tϕ(s)ltZα1113966 1113967

(29)

where Zα is the right-tailed critical value of the standardnormal distribution corresponding to a significance level ofα 1113954rib

un1113954rible1113954r

ibe denote the beginnings on the dates of the ith

extreme price behavior period and similarly 1113954rieun1113954r

iele1113954r

iee

denote the ends on the dates of the ith extreme price be-havior period

43 Extend the Model with Endogeneity Now we are con-cerned with the endogeneity case where Wμ(t) is not in-dependent of Wε(t) To see this let us suppose that data aregenerated from the following stochastic differential equationsystem

dx(t) x(t)d1113957B(t) + σdB(t) x(0) x0

d1113957B(t) μ dt + 1113957σWμ(t)

dB(t) σdWε(t)

(30)

where (Wμ(t) Wε(t)) is a two-dimensional Brownianmotion with covariance parameter η As Follmer [53] notedthe strong solution obtained from this continuous systemwould be of the explicit form

dx(t) exp 1113957σWμ(t) + μ minus12

1113957σ21113874 1113875t1113874 1113875x(0) + Z(t) (31)

where

Z(t) σ 1113946t

0exp 1113957σ Wμ(t) minus Wμ(s)1113872 1113873 + μ minus

12

1113957σ21113874 1113875(t minus s)1113874 1113875dWε(t)

minus ω1113946t

0exp 1113957σ Wμ(t) minus Wμ(s)1113872 1113873 + μ minus

12

1113957σ21113874 1113875(t minus s)1113874 1113875ds

V(t) minus P(t)

(32)

Notice that when endogeneity is included in model (14)the driver process becomes Z(t) as a substitute for V(t)Z(t) consists of a component V(t) and another oneconnected with the covariance parameter of (1113957B tB) ω and

Complexity 7

variance of V(t) -e strong solution gives the exact dis-crete-time form of model (31)

xtΔ exp 1113957σΔ

radicuΔ + μ minus

12

1113957σ21113874 1113875Δ1113882 1113883x(tminus 1)Δ + ZtΔ ρtΔ x(tminus 1)Δ + ZtΔ

+ σ 1113946tΔ

(tminus 1)Δexp 1113957σ WμtΔ minus Wμs1113872 1113873 + μ minus

12

1113957σ21113874 1113875(tΔ minus s)1113882 1113883dWεs

(33)

where uiidsim

t N(0 1) and

ZtΔ σ 1113946tΔ

(tminus 1)Δexp 1113957σ WμtΔ minus Wμs1113872 1113873 + μ minus

12

1113957σ21113874 1113875(tΔ minus s)1113882 1113883dWεs

minus ω1113946tΔ

(tminus 1)Δexp 1113957σ WμtΔ minus Wμs1113872 11138731113872 1113873 + μ minus

12

1113957σ21113874 1113875(tΔ minus s)1113874 1113875ds

(34)

Somewhat surprisingly Tao et al [25] indicated thatendogeneity would not affect the sample path features ofmodel (31) Our analysis therefore is the same as before

(1) Unstable κ μ + 1113957σ22gt 0(2) Locally explosive μgt 0(3) Explosive ϕ μ minus 1113957σ22gt 0

We can estimate 1113954θlowast ≔ ( 11139541113957σ2 1113954η 1113954σ2)prime in the same fashion asbefore -e likelihood ratio test (LR test) has been proposedby Tao et al [25] to test the null hypothesis of endogeneityinterest

LR Δminus 1Q

rΔ minus Q

urΔ( 1113857simχ2(1) underH0 η0 0 (35)

Tao et al [25] also developed the asymptotic theory for 1113954μand 1113954ρ ≔ exp(1113954μΔ) Based on their limit properties the es-timates of 1113954μ and 1113954ρ continue to follow asymptotic normaldistributions -is good quality enables us to convenientlyadopt the aforementioned testing procedures after making asmall adjustment to the variance of the limit distribution

5 Data

For our main empirical analysis we apply daily WTI crudeoil prices for the front-month (one-month) future contracton the New York Mercantile Exchange (NYMEX) -e dataare retrieved from the Energy Information Administration(EIA) Our sample spans the period from April 4 1983 toJune 30 2020 and comprises a total of 9352 daily obser-vations -e beginning and end dates of the sample periodare dictated by the availability of the data

It is well known that WIT and Brent are grades of in-ternationally traded crude oil that are widely used asbenchmarks in oil pricing globally -e reason for this paperrelying on the WTI price of crude oil is twofold On the onehand oil speculation is often blamed on people who tradefutures in the US market WTI is the primary benchmark

for oil extracted and consumed in the US-eWTI crude oilfuture market is also the worldrsquos most liquid and largestcrude oil future market Nearly 12 million contracts trade aday with more than 2 million in open interest On the otherhand WTI future contracts have the longest available his-tory dating back to 1983 giving us a large sample size to gainpowers in detecting short-lived bubbles

Our sample period under investigation includes severalmajor exogenous events that to a large extent are reckonedto have influenced oil markets -ese contain the collapse ofOPEC in 1986 the Persian Gulf War of 199091 the Asianfinancial crisis of 199798 the terrorist attacks on September11 2001 the Venezuelan crisis and Iraq war of 200203 theincreased financialization of oil future markets since 2004the OPEC oil supply announcements in 2008 and 2014 the2019 novel coronavirus disease (COVID-19) and moststrikingly the 200708 global financial crisis Figure 2 plotsthe time series trajectories of the WTI nominal front-monthfuture price and the corresponding real price A look at thefigure illustrates that the oil prices grew steadily from thestarting of the sample until the early 2000s -e price seriessince then begun to surge and the sharp increase in the priceof oil continued until July 2008 As shown in Figure 2 itcollapsed to about 30 dollars by the end of 2008 Between2014 and 2016 the price of oil also experienced a crash Morerecently WTI crude price even went subzero for the firsttime in history A common view is that the steep rise anddecline in the price of oil in the 2000s could be attributed tospeculation

6 Empirical Results

61 Main Results We start the empirical illustration withbaseline model (14) expanding it to the endogeneity ap-plication later In order to formally examine the existenceand duration of unstable and explosive behavior in the crudeoil market we run our tests on the daily price data

8 Complexity

Following the recommendation of PSY the minimumwindow size set for the recursive test procedure is the first 5years of the entire sample Figure 3 plots the test empiricaloutcomes along with their corresponding 95 critical valuesequences Also plotted are the trajectories of test statisticsfor the random coefficient and the price data Some inter-esting results are shown in this figure

Looking at the recursive test results reported in the lastpanel of Figure 3 we observe that over the initial period ofobservation the null hypothesis of a time-invariant autor-egressive coefficient cannot be rejected at the 5 statisticalsignificance level -e test statistic experiences a dramaticincrease in the late 1980s and exceeds the 95 critical valuein November 1990 Overall the observed pattern of timevariation rises and becomes much stronger as the timeperiod expands -e evidence that the LBI test statistics forthe majority of samples are greater than the 95 criticalvalue suggests that the data-generating process is wellcharacterized by the model with time-varying coefficientsOne point worth noting is that the sequence falls in thesecond half of 2008 and slumps in early 2020 -e styled factseems to indicate that the crude oil market has undergonedrastic changes Interestingly these located dates coincidewith the 200708 global financial crisis and the COVID-19pandemic

We now turn to our focus on different types of unstableand explosive behavior-e shaded areas in green in Figure 3are the identified periods where the t-test statistic crosses its95 critical value As is evident from the first panel theprocedure detects two unstable episodes -e first unstableepisode is relatively short and only lasts two months(2005M08-M09) -e second unstable episode appears inJanuary 2006 and continues until March 2020 -ese un-stable periods might be associated with the financializationof oil future markets from 2004 It can be seen in the secondpanel of Figure 3 that the detected episodes of locally ex-plosive (submartingale) behavior are clustered in 2007-20082009ndash2014 and 2018 As stated above since local explo-siveness implies instability and displays more extreme be-havior it should lead to a much shorter duration of

exuberance than that of instability -e third panel ofFigure 3 reflects that the t-test statistic is below the 95critical value throughout the sample period -is points outthat explosive behavior in crude oil price series does notexist Table 1 summarizes time horizons for the three kindsof unstable and explosive behavior identified by ourapproach

To shed light on a more realistic scenario of data-gen-erating process we implement the extreme sample pathbehavior test by extending the model to the endogeneitycase Figure 4 plots the empirical test results together with95 critical values A comparison of the LR test statisticswith the critical value sequence provides novel evidence insupport of the presence of endogeneity during the wholesample period -us it is believed that endogeneity is es-sential to the oil price formation process Furthermore thelast panel of Figure 4 highlights that the quadratic varianceof the price process is related to the test statistic forendogeneity -is finding is in accordance with the fact thatwe have exploited the realized variance series to obtain theLR test statistics In this sense Figure 5 displays differencesin the realized variance estimates which are captured by thetest statistic for endogeneity employing different methodsSimilar to Figure 3 we again identify episodes of unstabledynamics and fail to detect explosive behavior when con-sidering the model with endogeneity In contrast the sta-tistic sequence for local explosiveness exhibits a differentpattern Previous evidence in favor of locally explosive be-havior disappeared Table 2 summarizes time horizons forthe three forms of unstable and explosive behavior -eseresults suggest that endogeneity feedbacks play a significantrole in investigating the existence of explosive price be-havior -e conclusion that emerges is that there are nospeculative bubbles in the crude oil market

In principal our findings are in accordance with theexplanations made by Kilian and Murphy [19] Baumeisterand Hamilton [21] and Knittel and Pindyck [54] that thevariations in crude oil prices are driven by economicfundamentals

62 Comparison to Alternative Strategies Rich literature hasstudied speculative bubbles in the crude oil market using thetechniques developed by PWY and then extended by PSY Abrief description of the SADF methodology (used in PWY)and the GSADF methodology (used in PSY) can be found inAppendix It is therefore helpful to compare results foralternative methods as the detection strategy in the presentpaper is quite different from them Figure 6 plots thebackward ADF and backward SADF statistic sequences andthe associated 95 critical value sequences for oil futureprices 95 percent critical values are taken from 2000 MonteCarlo simulations with the actual sample size -e initialwindow size is set to 5 years A glance at Figure 6 reveals thatexplosive behavior occurred mainly in two occasions usingPWY and PSY dating strategies -ey are before and during2007-2008 global recession We also observe that PWYdetects several short-lived episodes between 2011 and 2014but PSY fails to do so For daily oil future prices over April 4

0

20

ndash20

ndash40

40

60

80

100

120

140

84 86 88 90 92 94 96 98 00 02 04 06 08 10 12 14 16 18 20

Nominal WTI front-month future pricesReal WTI front-month future prices

Figure 2 Real and nominal prices of WTI front-month futurecontracts

Complexity 9

1983 to June 30 2020 time horizons for explosive behavioridentified by PWY and PSY are summarized in Table 3 Inline with previous studies we could conclude from bothPWY and PSY tests that there is evidence of bubbles in thecrude oil market data However as shown in Gurkaynak [37]and Pavlidis et al [43] explosive dynamics in asset prices perse do not constitute evidence either in support of or againstspeculative bubbles since such explosive behavior may

actually reflect explosive market fundamentals or bothfactors As aforementioned our random coefficient autor-egressive model framework has considered endogeneityfeedbacks that to some extent alleviate this problemMoreover the evidence in Figures 3 and 4 suggests theabsence of explosive dynamics in the crude oil market -efact permits us to safely ignore the distinction betweenbubbles or explosive fundamentals

1988

-03

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

ndash4

ndash2

0

2

4

6

8

0

20

40

60

80

100

120

140

t-statistic of k95 critical value of N (0 1)Crude oil future prices

Instability detection based on the time-varyingcoefficient model

(a)

Local explosiveness detection based on the time-varyingcoefficient model

1988

-03

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

t-statistic of μ95 critical value of N (0 1)Crude oil future prices

ndash4

ndash2

0

2

4

6

8

0

20

40

60

80

100

120

140

(b)

Explosiveness detection based on the time-varyingcoefficient model

1988

-03

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

t-statistic of ϕ95 critical value of N (0 1)Crude oil future prices

ndash4

ndash2

0

2

4

6

8

0

20

40

60

80

100

120

140

(c)

LBItest statistic for testing time-varying coefficient19

88-0

3

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

LBItest95 critical value of N (0 1)

ndash5

ndash10

0

5

10

15

20

(d)

Figure 3 Date-stamping unstable and explosive episodes in oil prices with the time-varying coefficient model assuming no endogeneity

Table 1 Time horizons of unstable and explosive periods identified by models assuming no endogeneity

Unstable Locally explosive ExplosiveAug 2005-Sep 2005 Dec 2007ndashSep 2008 mdash

Jan 2006ndashMar 2020 Jun 2009ndashOct 2014 mdashJun 2018ndashSep 2018 mdash

10 Complexity

63 Robustness of the Results In the following we providetwo robustness checks to assess whether our previousconclusions of the paper still hold

First in our benchmark model our use of front-monthfuture price instead of other future prices at different ma-turities is justified by its being the most liquid future contractwith the highest open interest To explore the sensitivity ofour results to the future contracts at different maturities werepeat the analysis described above using the prices forlonger-term contracts (ie 2 3 and 4 months-ahead fu-tures) It should be pointed out that the WTI future contractusually expires on the third business day before the 25thcalendar day of the contract month proceeding the deliverymonth

Second becauseWTI crude oil is by tradition traded innominal US dollars per barrel it might be of concern that

inflation and other monetary variables may have realeffects on oil price It is therefore of interest to use realrather than nominal price In fact recent work documentsthat the use of either price is inconsequential in practice[55] Although we mainly focus on nominal prices forrobustness check we shall report the results for realprices To convert nominal daily series into daily realseries we use linearly interpolated monthly ConsumerPrice Index (CPI) Monthly CPI series for the UnitedStates is obtained from the websites of the Federal ReserveBank of St Louis

Further evidence presented in Figures 7ndash10 demon-strates that our results are remarkably robust to thesemodifications and the conclusions remain qualitativelyunchanged

ndash4

ndash2

0

2

4

6

8

0

20

40

60

80

100

120

140

1988

-03

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

Instability detection based on the time-varyingcoefficient model

t-statistic of k95 critical value of N (0 1)Crude oil future prices

(a)

ndash4

ndash2

0

2

4

6

8

0

20

40

60

80

100

120

140

1988

-03

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

t-statistic of μ95 critical value of N (0 1)Crude oil future prices

Local explosiveness detection based on the time-varyingcoefficient model

(b)

ndash4

ndash2

0

2

4

6

8

0

20

40

60

80

100

120

140

1988

-03

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

Explosiveness detection based on the time-varyingcoefficient model

t-statistic of ϕ95 critical value of N (0 1)Crude oil future prices

(c)

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

LR-statistic95 critical values of x2 (1)Realized variance

700

600

500

400

300

200

100

0

ndash100

350

300

250

200

150

100

50

0

Test for endogeneity in time-varyingcoefficient model

(d)

Figure 4 Date-stamping unstable and explosive episodes in oil prices with the time-varying coefficient model with endogeneity

Complexity 11

2

3

4

5

6

7

8

0

50

100

150

200

250

300

350

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

LLF0

LLF1

Realized variance of crude oil future prices

Figure 5 Realized variance and likelihood values for random coefficient regression

Table 2 Time horizons of unstable and explosive periods detected by models with endogeneity

Unstable Locally explosive ExplosiveMay 2008ndashJul 2008 mdash mdashDec 2010ndashOct 2014 mdash mdashNov 2016ndashApr 2017 mdash mdashJul 2017ndashFeb 2020 mdash mdash

ndash4

ndash2

0

2

4

6

8

0

20

40

60

80

100

120

140

1988

-03

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

Explosiveness detection using SADF methodology

BADF sequence95 critical valuesCrude oil future prices

(a)

ndash4

ndash2

0

2

4

6

8

0

20

40

60

80

100

120

140

1988

-03

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

Explosiveness detection using GSADF methodology

BSADF sequence95 critical valuesCrude oil future prices

(b)

Figure 6 Date-stamping explosive episodes in oil prices with the time-invariant coefficient model

12 Complexity

Table 3 Time horizons of explosive behavior identified by SADF and GSADF methodologies

SADF GSADFAug 2005ndashAug 2006 Oct 2011ndashApr 2012 Sep 1990-Oct 1990Jun 2007ndashOct 2008 Aug 2012-Sep 2012 Feb 1991-Mar 1991Jun 2009-Jul 2009 Jul 2013-Aug 2013 Oct 2004-Nov 2004Feb 2011ndashApr 2011 Feb 2014-Mar 2014 Feb 2005-Mar 2005

Jul 2011ndashAug 2012 Jun 2014-Jul 2014May 2005ndashJan 2006Mar 2006ndashAug 2006Jul 2007ndashAug 2008

ndash4

ndash2

0

2

4

6

8

0

10

20

30

40

50

60

70

1988

-03

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

95 critical value of N (0 1)Real oil future prices

t-statistic of μ~

Local explosiveness detection based on the time-varyingcoefficient model

(a)

ndash4

ndash2

0

2

4

6

8

0

10

20

30

40

50

60

70

1988

-03

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

95 critical value of N (0 1)Real oil future prices

t-statistic of ϕ

Explosiveness detection based on the time-varyingcoefficient model

(b)

Figure 7 Date-stamping explosive episodes in real oil prices with the time-varying coefficient model with endogeneity

ndash4

ndash2

0

2

4

6

8

0

50

100

150

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

Local explosiveness detection based on the time-varyingcoefficient model

t-statistic of μ95 critical value of N (0 1)2-month crude oil future prices

(a)

ndash4

ndash2

0

2

4

6

8

0

50

100

150

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

Local explosiveness detection based on the time-varyingcoefficient model

t-statistic of ϕ95 critical value of N (0 1)2-month crude oil future prices

(b)

Figure 8 Date-stamping explosive episodes in 2-month future oil prices with the time-varying coefficient model with endogeneity

Complexity 13

7 Discussion

-e oil price rally since the early 2000s has attracted greatinterest in the sources of oil price shocks Some causal ob-servers have stressed that financial speculation has become asignificant driver of the dynamics of oil prices A large body ofliterature emerged that has sought to investigate its existenceIn this paper we employ a novel approach for testing forspeculative bubbles in oil prices In synthesis our empiricalfindings indicate that there is less evidence of exuberance in

the oil future markets Our results are very robust in severalscenarios We argue that the booms and busts in the oil pricemovements during the sample period are driven primarily byeconomic fundamentals rather than speculation

Overall our results confirm the main findings obtainedby structural VAR-based studies including Kilian andMurphy [19] Kilian and Lee [20] Baumeister and Hamilton[21] and Zhou [22] In the past decade the use of structuralVAR models to understand the evolution of the oil price hasbecome common -e traditional SVAR literature

ndash4

ndash2

0

2

4

6

8

0

50

100

15019

88-0

3

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

Local explosiveness detection based on the time-varyingcoefficient model

t-statistic of μ95 critical value of N (0 1)3-month crude oil future prices

(a)

ndash4

ndash2

0

2

4

6

8

0

50

100

150

1988

-03

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

Explosiveness detection based on the time-varyingcoefficient model

t-statistic of ϕ95 critical value of N (0 1)3-month crude oil future prices

(b)

Figure 9 Date-stamping explosive episodes in 3-month future oil prices with the time-varying coefficient model with endogeneity

ndash4

ndash2

0

2

4

6

8

0

50

100

150

Local explosiveness detection based on the time-varyingcoefficient model

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

t-statistic of μ95 critical value of N (0 1)4-month crude oil future prices

(a)

ndash4

ndash2

0

2

4

6

8

0

50

100

150

Explosiveness detection based on the time-varyingcoefficient model

t-statistic of ϕ95 critical value of N (0 1)4-month crude oil futures prices

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

(b)

Figure 10 Date-stamping explosive episodes in 4-month future oil prices with the time-varying coefficient model with endogeneity

14 Complexity

documents that the fundamental drivers of oil price fluc-tuations have been dominated essentially by supply anddemand shocks see eg Kilian [56] Lippi and Nobili [57]Kilian andMurphy [58] and Baumeister and Peersman [59]Kilian and Murphy [19] focused on the role speculation byaugmenting the SVAR framework of their earlier paper byglobal oil inventory data -ey specified a four-variablemodel for the global oil market (the percentage change inglobal oil production the real price of oil a suitable proxy ofcyclical variation in global real economic activity andchanges in global crude oil inventories) -is setting andtheir sign-identified scheme allow them to separately explainvariation in these data as four different types of shocks (1) ashock to the amount of oil pumped out of the ground (ldquooilsupply shockrdquo) (2) an oil flow demand shock (3) an oil-specific oil demand shock and especially (4) a speculative(or inventory) demand shock -eir core finding is that thespike in the real price of crude oil prices during 2003ndash2008was caused by unexpected increase in global oil consump-tion Using two alternative proxies for global above-groundcrude oil inventories Kilian and Lee [20] reestimated themodel of Kilian and Murphy [19] and ruled out the role ofspeculation in explaining the recent surge of the real oil priceon the basis of their estimation results Also Zhou [22]assessed the sensitivity of the findings in Kilian and Murphy[19] to the introduction of various state-of-the-art oil marketmethods She shows that the core conclusions reached byKilian and Murphy [19] are highly robust to some refine-ments corrections and extension of the KilianndashMurphymodel Baumeister and Hamilton [21] recently contendedthat prior information plays a key role in any SVAR modelsand traditional identification approaches can be generalizedto a special case of a Bayesian perspective -ey revisited theimportance of supply and demand shocks in the oil marketand demonstrated that speculative demand shock turns outto be only a small factor in accounting for most historical oilprice shocks

For comparison purpose we have also applied the testsof PWY and PSY which are the most popular econometrictechniques in the literature on speculative bubbles in the oilmarkets If we were to follow the test procedures of PWY andPSY our conclusions are also broadly consistent with therich literature that supports the existence of speculativebubbles in oil prices However these cointegration-basedbubble tests suffer from the joint hypothesis problem ldquoEverytest of a bubble is a joint test of the presence of the bubble inthe data and the validity of the model applied by theeconometricianrdquo (see Giglio et al [60]) More specifically themain drawback of such indirect model-dependent tests ofbubbles typically refers to the difficulty in distinguishingbetween the unknown true model for market fundamentalsand the presence of bubbles (see for example Gurkaynak[37] Pavlidis et al [42] Pavlidis et al [43]) Given thefundamental value of oil price is unobservable the literaturehas resorted to observe economic and financial variableswhich are used to estimate the intrinsic values For exampleboth Phillips and Yu [16] and Caspi et al [39] simplynormalized crude oil prices by US oil inventory data as thefundamental value -is measure is questionable since their

method cannot sufficiently capture the information onmarket fundamentals and neglect the major driversmdashlikeglobal economic activitymdashalso affecting the fundamentalvalue of crude oil -is issue of misspecified fundamentals(either in terms of model misspecification or omitted var-iables or both of these reasons) may make researchers fail toarrive at a conclusion from evidence of explosive pricesalone To make matters worse even though we have cap-tured the true market fundamentals we may still fail todetect bubbles if the explosive behavior in the price of oil juststems from the explosive fundamentals

Compared to earlier studies as briefly mentioned aboveour methodology has the main advantage that by consid-ering the dependence between the random coefficient andsystem shocks we have incorporated endogeneity into thecontinuous-time random coefficient setting and thus we donot need to prespecify the fundamental component of the oilprice -e test results for endogeneity suggest that endo-geneity is important in the data-generating process Inparticular our empirical analysis also finds that there are noexplosive episodes in crude oil markets-is implies that it isnot necessary to require the specification of a fundamentalvalue of the oil price Like Pavlidis et al [43] their methodsand ours have in common that the problem of the well-known joint hypothesis has to be some extent been ef-fectively ameliorated Although our approach is quite dis-tinct we draw the same substantive conclusion thatspeculative bubbles can be ruled out as a key determinant ofthe price of crude oil in the 2000s reaffirming the con-clusions of Kilian and Murphy [19] Baumeister andHamilton [21] Zhou [22] and Fattouh et al [61]

8 Conclusions

Repeated surges and crashes in crude oil prices during lastdecades have sparked a heated debate on their underlyingcauses among academics commentors and market regu-lators One of the central questions concerns whetherfinancialization of oil future markets creates a speculativebubble that subsequently collapses Answering this questionis difficult because one has to rely upon strong assumptionsabout the data-generating mechanism for the fundamentalvalue of the oil price In this paper we present a stylizedmodel of asset pricing for a commodity to address this issueOur theoretical framework disentangles the effects ofspeculative bubbles from the traditional fundamental effectson oil price dynamics We use this setting to illustrate thatbubbles if they are present should demonstrate explosivebehavior in prices We then provide a novel approach forbubble detection that covers various explosive scenarios andfind no evidence of speculative bubbles in oil markets evenduring episodes where the most popular SADF and GSADFmethodologies identify a speculative bubble

Our conclusions have pertinent implications for policymakers Given the historically important role of oil prices inthe real economy one of the serious concerns in policycircles over the past decade has been whether speculativetrading activities in oil future markets are responsible for thelarge price swings For example in their 2008 testimonies to

Complexity 15

the US Senate Commerce Committee Michael Masters andGeorge Soros posited that speculators had fueled price in-creases since 2003 and had caused a part of a ldquosuperbubblerdquoin 2007 and 2008 resulting in significantly higher energycosts for consumers -is widespread bubble view domi-nated policy discussions and efforts to limit speculativetrading in the future market quickly followed As a responseto the excessive speculation pressures on prices theDoddndashFrank Wall Street Reform and Consumer ProtectionAct was enacted by the US Congress in 2010 As a directpolicy implication our results highlight that imposingstricter regulations on trading in oil derivative marketscannot be expected to lower the oil price given the fact thatthe synchronized boom and bust cycles witnessed in the past20 years are mainly due to economic fundamentals whichappear likely to resurge with the business cycle -e aboveanalysis implies that there is no need to enforce additionalregulatory limits against financial arbitrage in the com-modity market

Appendix

The SADF and GSADF Methodologies

Appendix describes the SADF and GSADF testing proce-dures used to detect speculative bubbles in Section 62 For agenetic time series process xt consider the following rolling-window type augmented DickeyndashFuller (ADF) regressionequation

xt ar1r2+ μr1r2

xtminus 1 + 1113944

J

j1ϕj

r1r2Δxtminus j + εt εt sim NID 0 δ2r1r2

1113874 1113875

(A1)

where Δxtminus j with j 1 J are lagged first differences ofxt incorporated to accommodate potential serial correla-tion ar1 r2

μr1 r2 and ϕj

r1 r2 with j 1 J are regression

coefficients the subscripts r1 and r2 refer to fractions of thetotal sample size (ofT observations) that set the start and endpoints of a subsample period and NID represents inde-pendent and normal distribution -e ADF test statisticcorresponding to the unit root null hypothesisH0 μr1 r2

0 is simply the t-ratio

ADFr2r1

1113954μr1 r2

se 1113954μr1 r21113872 1113873

(A2)

and the right-side alternative hypothesis of explosive be-havior is H0 μr1r2

gt 0 We can obtain the standard ADF teststatistic by running regression (A1) on the whole sampleie by specifying r1 0 and r2 1 A large amount of re-search has demonstrated that the standard ADF test ADF10has extremely low power in detecting periodically collapsingbubbles

To address the problem PSW proposed a recursiveregression technique which involves estimating (A1)employing a forward expanding sample Specifically theyused subsamples of the available data to recursively calculatea sequence of ADF statistics denoted by ADFr2

0 -e startpoint of the sample period is fixed at r1 0 while the endpoint r2 increases from r0 (the minimum window size) toone (the full sample period) -e SADF statistic called thesupremum of ADFr2

0 is defined by

SADF r0( 1113857 supr2isin r0 1[ ]

ADFr20

(A3)

and has the following limit distribution under the nullhypothesis

supr2isin r0 1[ ]

1113938r2

0 WdW

1113938r2

0 W21113872 1113873(12)

(A4)

where W is the standard Brownian motion Testing thepresence of explosive behavior entails comparing the ADFr2

0statistic with the right-side critical values obtained throughMonte Carlo simulations

More recently PSY showed that when the sample perioddisplays more than one boom-bust bubble the SADF testprocedure may perform poorly and fail to detect or con-sistently date-stamp alternative bubbles -is deficiencymotivates PSY to formulate an extension of the SADF testthe GSADF test which covers more subsamples of the datathan the SADF by allowing both the end point r2 and thestart point r1 to vary As a result the GSADF test has moreflexibility and gains substantial power making this testconsistent with multiple boom-bust bubbles -e GSADFtest statistic is defined as

GSADF r0( 1113857 supr2isin r0 1[ ]r1isin 0r2minus r0[ ]

ADFr2r1 (A5)

with the limit distribution

supr2isin r0 1[ ]r1isin 0r2minus r0[ ]

(12)rw W r2( 11138572

minus W r1( 11138572

minus rw1113960 1113961 minus 1113938r2

r1W(r)dr W r2( 1113857 minus W r1( 11138571113858 1113859

r12w rw 1113938

r2

r1W(r)2dr minus 1113938

r2

r1W(r)dr1113876 1113877

21113896 1113897

12

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎭

(A6)

where rw r2 minus r1 is the expanding window size Like theSADF test comparison of the GASDF statistic with the right-

tailed critical values from its limit distribution enables us to testfor the unit root hypothesis against explosive behavior

16 Complexity

If the null hypothesis of a unit root in xt is rejectedthen in the second stage the SADF and GSADF tests canidentify the exact episodes during which the xt seriesexhibits explosive dynamics To locate the emergence andconclusion of explosiveness the SADF methodologymatches the sequence of the recursive backward ADF(BADF) test statistics against the right-tailed criticalvalues of the limit distribution of the standard Dick-eyndashFuller test statistic Suppose that re is the originationdate and rf is the finish date of the bubble in the data-ese estimates of dates can be constructed from

1113954re infr2isin r0 1[ ]

r2 ADFr20 gt cv

αr2

1113966 1113967

1113954rf infr2isin 1113954re11113858 1113859

r2 ADFr20 lt cv

αr2

1113966 1113967(A7)

where cvαr2 is 100(1 minus α) the right-tailed critical value of theADF statistic corresponding to the chosen significance levelof α For the GSADF methodology the date-stampingstrategy in this case is based on the backward SADF(BSADF) statistic

BSADFr2r0( 1113857 sup

r1isin 0r2minus r0[ ]ADFr2

r1

(A8)

Likewise the origination date of the bubble is defined asthe first observation that the BSADF statistic is greater thanits critical value

1113954re infr2isin r0 1[ ]

r2 BSADFr2r0( 1113857gt cv

βT

r21113966 1113967 (A9)

and the finish date as the first date after which the BSADFdrops back below its critical value

1113954rf infr2isin 1113954re11113858 1113859

r2 BSADFr2r0( 1113857lt cvβT

r21113966 1113967 (A10)

where cvβTr2 is 100(1 minus βT) the right-tailed critical value of

the SADF based on Tr2 and the corresponding significancelevel is βT

Data Availability

-e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

-e authors declare no conflicts of interest

Acknowledgments

-e authors gratefully acknowledge the financial supportfrom the National Social Science Foundation of China underGrant no 18BTJ039

References

[1] M W Masters Testimony before the US Senate Committee onHomeland Security and Governmental Affairs 2008 httpswwwhsgacsenategovimomediadoc052008Masterspdfattempt2

[2] M W Masters Testimony before the Commodity FuturesTrading Commission 2010 httpscftcgovsitesdefaultfilesidcgroupspublicnewsroomdocumentsfilemetalmarkets032510_masterspdf

[3] J P Kennedy ldquo-e high cost of gambling on oilrdquo Section A p23 New York Times New York NY USANew York Times2012

[4] I-H Cheng and W Xiong ldquoFinancialization of commoditymarketsrdquo Annual Review of Financial Economics vol 6 no 1pp 419ndash441 2014

[5] K Tang and W Xiong ldquoIndex investment and the financi-alization of commoditiesrdquo Financial Analysts Journal vol 68no 6 pp 54ndash74 2012

[6] O J Blanchard and M W Watson ldquoBubbles rational ex-pectations and financial marketsrdquo in Crisis in the Economicand Financial Structure P Wachtel Ed Lexington BooksLexington MA USA pp 295ndash315 1982

[7] B T Diba and H I Grossman ldquoOn the inception of rationalbubblesrdquo e Quarterly Journal of Economics vol 102 no 3pp 697ndash700 1987

[8] G W Evans ldquo-e fragility of sunspots and bubblesrdquo Journalof Monetary Economics vol 23 no 2 pp 297ndash317 1989

[9] G W Evans and S Honkapohja ldquoOn the robustness ofbubbles in linear RE modelsrdquo International Economic Reviewvol 33 no 1 pp 1ndash14 1992

[10] E Pavlidis A Yusupova I Paya et al ldquoEpisodes of exu-berance in housing markets in search of the smoking gunrdquoe Journal of Real Estate Finance and Economics vol 53no 4 pp 419ndash449 2016

[11] E Martınez-Garcıa and V Grossman ldquoExplosive dynamics inhouse prices An exploration of financial market spillovers inhousing markets around the worldrdquo Journal of InternationalMoney and Finance vol 101 p 102103 2020

[12] S Shi A Rahman and B Z Wang ldquoAustralian housingmarket booms fundamentals or speculationlowastrdquo EconomicRecord 2020 In press

[13] P C B Phillips Y Wu and J Yu ldquoExplosive behavior in the1990s NASDAQ when did exuberance escalate asset val-ueslowastrdquo International Economic Review vol 52 no 1pp 201ndash226 2011

[14] P C B Phillips S Shi and J Yu ldquoTesting for multiplebubbles historical episodes of exuberance and collapse in theSampp 500rdquo International Economic Review vol 56 no 4pp 1043ndash1078 2015

[15] P C B Phillips S Shi and J Yu ldquoTesting for multiplebubbles limit theory of real-time detectorsrdquo InternationalEconomic Review vol 56 no 4 pp 1079ndash1134 2015

[16] P C B Phillips and J Yu ldquoDating the timeline of financialbubbles during the subprime crisisrdquo Quantitative Economicsvol 2 no 3 pp 455ndash491 2011

[17] D Tsvetanov J Coakley and N Kellard ldquoBubbling over thebehaviour of oil futures along the yield curverdquo Journal ofEmpirical Finance vol 38 no Part B pp 516ndash533 2016

[18] I Figuerola-Ferretti J R Mccrorie and I ParaskevopoulosldquoMild explosivity in recent crude oil pricesrdquo Energy Eco-nomics vol 87 p 104387 2020

[19] L Kilian and D P Murphy ldquo-e role of inventories andspeculative trading in the global market for crude oilrdquo Journalof Applied Econometrics vol 29 no 3 pp 454ndash478 2014

Complexity 17

dx(t) x(t) μd(t) + 1113957σWμ(t)1113960 1113961 + σdWε(t) x(0) x0

(14)

where Wμ(t) and Wε(t) are two standard Brownianmotionsand the initial value x0 is viewed to be independent of Wμ(t)

and Wε(t) For 1113957σ2 ne 0 we see that random shocks to the driftterm have been incorporated into model (14) At this mo-ment we explore the situation where Wμ(t) and Wε(t) are

independent noise processes and subsequently allow for theendogenous case by relaxing the independence assumption

Solving model (14) yields the strong solution [49]

dx(t) exp 1113957σWμ(t) + μ minus12

1113957σ21113874 1113875t1113874 1113875x(0) + V(t) (15)

where

V(t) σ 1113946t

0exp 1113957σ Wμ(t) minus Wμ(s)1113872 1113873 + μ minus

12

1113957σ21113874 1113875(t minus s)1113874 1113875dWε(t)

sim MN 0 σ2 1113946t

0e21113957σ Wμ(t)minus Wμ(s)( 1113857+2 μminus (12)1113957σ

2( 1113857(tminus s)ds1113888 1113889

(16)

such that

E V(t)2

1113966 1113967 σ2E 1113946t

0e21113957σ Wμ(t)minus Wμ(s)( 1113857+2 μminus (12)1113957σ

2( 1113857(tminus s)ds1113896 1113897

σ2

2e2 μ+(12)1113957σ

2( 1113857t

minus 1μ +(12)1113957σ21113872 1113873

(17)

Under μ + (12)1113957σ2 gt 0 E V(t)21113966 1113967 is expected to growexponentially

-e corresponding discrete-time representation of (14) isderived directly from the strong solution and is of the form

xtΔ exp 1113957σ WμtΔ minus Wμ(tminus 1)Δ1113960 1113961 + μ minus12

1113957σ21113874 1113875Δ1113882 1113883x(tminus 1)Δ

+ σ 1113946tΔ

(tminus 1)Δexp 1113957σ WμtΔ minus Wμs1113872 1113873 + μ minus

12

1113957σ21113874 1113875(tΔ minus s)1113882 1113883dWεs

(18)

where Δ is the sampling interval over time span T andt 1 TΔ With respect to the random coefficientautoregression model [50] the autoregressive coefficient isdefined by

ρtΔ exp 1113957σ WμtΔ minus Wμ(tminus 1)Δ1113960 1113961 + μ minus12

1113957σ21113874 1113875Δ1113882 1113883 (19)

Here we introduce some notations to simplify thediscrete-time system Let

ϕ ≔ μ minus12

1113957σ2

κ ≔ μ +12

1113957σ2

utΔ ≔WμtΔ minus Wμ(tminus 1)Δ

Δ

radic simN(0 1)

ρtΔ ≔ exp 1113957σ WμtΔ minus Wμ(tminus 1)Δ1113960 1113961 + μ minus12

1113957σ21113874 1113875Δ1113882 1113883 exp ϕΔ + 1113957σΔ

radicutΔ1113966 1113967

ηtΔ ≔ 1113946tΔ

(tminus 1)Δexp 1113957σ WμtΔ minus Wμs1113872 1113873 + μ minus

12

1113957σ21113874 1113875(tΔ minus s)1113882 1113883dWεssimN 0e2κΔ

minus 12κ

1113888 1113889

(20)

Complexity 5

-e discrete-time representation (18) can then be re-written as

xtΔ exp 1113957σΔ

radicutΔ + ϕΔ1113966 1113967x(tminus 1)Δ + σηtΔ ρtΔx(tminus 1)Δ + σηtΔ

(21)

Assuming 1113957σ2 gt 0 models (14) and (21) have the followingstationarity properties (1) for (μ + 1113957σ22)lt 0 the process isasymptotically covariance stationary sinceμlt 0 means thatboth E(ρtΔ) eμΔ lt 1 and E(ρ2tΔ) exp(2μΔ+1113957σ2Δ) exp(2κΔ)lt 1 (2) for (μ + 1113957σ22)lt 0 the process re-mains asymptotically stationary but is not asymptoticallycovariance stationary any longer Practically speaking wehave to impose the condition of κ (μ + 1113957σ22)lt 0 toguarantee the covariance stationary Whenκ (μ + 1113957σ22)lt 0 it follows from (19) that the variance ofV(t) converges to minus σ22κltinfin as t⟶infinWhen κ 0 thevariance is equal to σ2t and diverges as t⟶infin (3) for (μ +1113957σ22)gt 0 and μlt 0 the process is still asymptotically sta-tionary but is not covariance stationary with evident per-sistence in the trajectory (4) when μ 0 ϕ (μ minus 1113957σ22)lt 0ensures the asymptotic stationarity but leads to infinitesecond moments withκ (μ + 1113957σ22)gt 0 -e process ex-hibits apparent unstable behavior and larger variation (5)when μgt 0 and μ minus 1113957σ22lt 0 the process continues to becovariance nonstationary Contrast to the traditional ex-plosive AR (1) model the model retains asymptoticallystationary In this instance bubble-like phenomenon shouldbe clearly observed in actual data (6) if μ minus 1113957σ22gt 0 themodel is no longer asymptotically stationary and the ex-plosive growth behavior is now obvious

As shown in Tao et al [25] when 1113957σ2 0 model (21)collapses to a simple autoregression with a constant coef-ficient It explains why a popular way to look for bubbles is totest ϕ 0 against ϕgt 0 via standard or recursive right-tailedunit root tests [15ndash18 27] For the purpose of the presentpaper we only examine three price behavior features that aredivided into three categories

(1) Unstable κ μ + 1113957σ22gt 0 Here the model is as-ymptotically covariance nonstationary and applies tocapture unstable behavior

(2) Locally explosive μgt 0 -e model in this case hasthe ability for generating both explosive and col-lapsing behavior

(3) Explosive ϕ μ minus 1113957σ22gt 0 Under the circumstancethe model is not asymptotically stationary and iscapable of generating explosive behavior

Based on the above typology explosive embodies locallyexplosive while locally explosive embodies unstable

42 Inference Scheme -is section is concerned with puttingthe continuous-time model specified above into the appli-cation of discretely sampled data We utilize a novel two-stage approach proposed by Phillips and Yu [51] to estimateμ 1113957σ2 and σ2 In the first stage realized volatility is employedto provide a regression model for estimating the diffusionparameters 1113957σ2 and σ2 In the second stage we maximize thein-fill likelihood function to estimate the drift parameter μ

Suppose data are observed in N equally spaced blocksover the time interval [0 T] with sampling interval h -eoverall time span of the data is T so that N Th If eachblock has M observations the total number of observationsis M times N TΔ where Δ is the sampling frequency Todevelop asymptotics we assume that as Δ⟶ 0M times N⟶infin According to the algorithm of Phillips andYu [51] the estimators based on realized variance (RV) andrealized quarticity (RQ) are denoted by

1113954θ argminθisinΘ

QΔ(θ) (22)

where 1113954θ ≔ (1113957σ2 σ2)prime and QΔ(θ) Δ1113936Nn1 (logRVnminus log

[x]nh(nminus 1)h + (12)s2n)2s2nwith s2n max

2Δ(RQ2

nRV2n)

11139681113864

2M

radic

RVn 1113944M

i1x(nminus 1)h+iΔ minus x(nminus 1)h+(iminus 1)Δ1113960 1113961

2 n 1 2 N

RQn 13Δ

1113944

M

i1x(nminus 1)h+iΔ minus x(nminus 1)h+(iminus 1)Δ1113960 1113961

4 n 1 2 N

(23)

In the second stage the approximate logarithmic in-filllikelihood function is maximized with regard to μ (repre-senting the resulting estimator as 1113954μ)

1113954μ argmaxμ

1MN

log ℓALF(μ) (24)

where

ℓALF(μ) 1113944MtimesN

t1

μx(tminus 1)Δ11139541113957σ2x2

(tminus 1)Δ +1113954σ2

xtΔ minus x(tminus 1)Δ1113872 1113873 minusΔ2

1113944

MtimesN

t1

μx(tminus 1)Δ11139541113957σ2x2

(tminus 1)Δ +1113954σ2

(25)

One important goal in the present empirical work is tocheck the presence of randomness in the persistenceparameter of model (14) which is equivalent to test the

null hypothesis 1113957σ2 0 To achieve this aim we resort tothe locally best invariant test (LBI test) statistic [52]namely

6 Complexity

1113957ZN ≔1113936

MtimesNt1 1113957ε2tΔ minus (1MN)1113936

MtimesNt1 1113957ε2tΔ1113872 1113873

21113874 11138751113957x

2(tminus 1)Δ

(1MN)1113936MtimesNt1 1113957ε4tΔ minus (1MN)1113936

MtimesNt1 1113957ε2tΔ1113872 1113873

21113969

(1MN)1113936MtimesNt1 1113957x

4(tminus 1)Δ minus (1MN)1113936

MtimesNt1 1113957x

2(tminus 1)Δ1113872 1113873

21113969 (26)

where 1113957x2tΔ (xtΔ

1 + x2

tΔ

1113969) 1113957εtΔ xtΔ minus 1113957ρx(tminus 1)Δ and

1113957ρ (1113936MtimesNt1 1113957x(tminus 1)Δx(tminus 1)Δ)

minus 11113936MtimesNt1 1113957x(tminus 1)Δx(tminus 1)Δ -us as

N⟶infin MNminus 12 1113957ZN⟶L N(0 1) under H0 1113957σ2 0 andMNminus (12) 1113957ZN⟶L N(0 1) under H0 1113957σ2 0

Relying on asymptotics with in-fill (as the sampling fre-quency tends to zeroΔ⟶ 0) and long time-span asymptoticsTao et al [25] developed new limit theory for the continuous-time system where the persistence parameter is random -easymptotic properties of the diffusion parameter and drift pa-rameter estimates can be used to construct test statistics for threetypes of unstableexplosive behavior which are

tκ 1Δ

1113944

MtimesN

t1

x2(tminus 1)Δ

11139541113957σ2x2(tminus 1)Δ + σ2

⎛⎝ ⎞⎠

12

1113954βκ minus β0κ1113872 1113873⟶L

N(0 1)

tμ 1Δ

1113944

MtimesN

t1

x2(tminus 1)Δ

11139541113957σ2x2(tminus 1)Δ + 1113954σ2

⎛⎝ ⎞⎠

12

1113954ρ minus ρ0( 1113857⟶L

N(0 1)

tϕ 1Δ

1113944

MtimesN

t1

x2(tminus 1)Δ

11139541113957σ2x2(tminus 1)Δ + 1113954σ2

⎛⎝ ⎞⎠

12

1113954βϕ minus β0ϕ1113872 1113873⟶L

N(0 1)

(27)

where

1113954βκ exp 1113954μ +12

11139541113957σ21113874 1113875Δ1113882 1113883

1113954βϕ exp 1113954μ minus12

11139541113957σ21113874 1113875Δ1113882 1113883

β0κ exp μ0 +12μ201113874 1113875Δ1113882 1113883

β0ϕ exp μ0 minus12μ201113874 1113875Δ1113882 1113883

1113954ρ 1 +1113936

MtimesNt1 x(tminus 1)Δ xtΔ minus x(tminus 1)Δ1113872 1113873 1113954

1113957σ2x2(tminus 1)Δ +

1113954σ21113874 1113875

1113936MtimesNt1 x

2(tminus 1)Δ

11139541113957σ2x2

(tminus 1)Δ +1113954σ21113874 1113875

ρ0 1

(28)

As can be seen this test procedure does not directlyverify whether or not κ (μ + 1113957σ22) 0 or μ 0 orϕ (μ minus 1113957σ22) 0 but transforms the null hypotheses toβk 1 or ρ 1 or βϕ 1 -ese three t-test statistics areaccompanied by a date-stamping strategy that permits us toidentify the exact origination and termination periods ofprice exuberance Given this instrument estimates of thesedates are as follows

1113954ribun inf

sisin 1113954r(iminus 1)e

un 11113858 1113859s tκ(s)gtZα1113864 1113865

1113954rieun inf

sisin 1113954r(iminus 1)b

un 11113858 1113859s tκ(s)ltZα1113864 1113865

1113954rible inf

sisin 1113954r(iminus 1)e

le 11113858 1113859s tμ(s)gtZα1113966 1113967

1113954rieun inf

sisin 1113954r(iminus 1)b

le 11113858 1113859s tμ(s)ltZα1113966 1113967

1113954ribe inf

sisin 1113954r(iminus 1)e

e 11113858 1113859s tϕ(s)gtZα1113966 1113967

1113954rieun inf

sisin 1113954r(iminus 1)b

e 11113858 1113859s tϕ(s)ltZα1113966 1113967

(29)

where Zα is the right-tailed critical value of the standardnormal distribution corresponding to a significance level ofα 1113954rib

un1113954rible1113954r

ibe denote the beginnings on the dates of the ith

extreme price behavior period and similarly 1113954rieun1113954r

iele1113954r

iee

denote the ends on the dates of the ith extreme price be-havior period

43 Extend the Model with Endogeneity Now we are con-cerned with the endogeneity case where Wμ(t) is not in-dependent of Wε(t) To see this let us suppose that data aregenerated from the following stochastic differential equationsystem

dx(t) x(t)d1113957B(t) + σdB(t) x(0) x0

d1113957B(t) μ dt + 1113957σWμ(t)

dB(t) σdWε(t)

(30)

where (Wμ(t) Wε(t)) is a two-dimensional Brownianmotion with covariance parameter η As Follmer [53] notedthe strong solution obtained from this continuous systemwould be of the explicit form

dx(t) exp 1113957σWμ(t) + μ minus12

1113957σ21113874 1113875t1113874 1113875x(0) + Z(t) (31)

where

Z(t) σ 1113946t

0exp 1113957σ Wμ(t) minus Wμ(s)1113872 1113873 + μ minus

12

1113957σ21113874 1113875(t minus s)1113874 1113875dWε(t)

minus ω1113946t

0exp 1113957σ Wμ(t) minus Wμ(s)1113872 1113873 + μ minus

12

1113957σ21113874 1113875(t minus s)1113874 1113875ds

V(t) minus P(t)

(32)

Notice that when endogeneity is included in model (14)the driver process becomes Z(t) as a substitute for V(t)Z(t) consists of a component V(t) and another oneconnected with the covariance parameter of (1113957B tB) ω and

Complexity 7

variance of V(t) -e strong solution gives the exact dis-crete-time form of model (31)

xtΔ exp 1113957σΔ

radicuΔ + μ minus

12

1113957σ21113874 1113875Δ1113882 1113883x(tminus 1)Δ + ZtΔ ρtΔ x(tminus 1)Δ + ZtΔ

+ σ 1113946tΔ

(tminus 1)Δexp 1113957σ WμtΔ minus Wμs1113872 1113873 + μ minus

12

1113957σ21113874 1113875(tΔ minus s)1113882 1113883dWεs

(33)

where uiidsim

t N(0 1) and

ZtΔ σ 1113946tΔ

(tminus 1)Δexp 1113957σ WμtΔ minus Wμs1113872 1113873 + μ minus

12

1113957σ21113874 1113875(tΔ minus s)1113882 1113883dWεs

minus ω1113946tΔ

(tminus 1)Δexp 1113957σ WμtΔ minus Wμs1113872 11138731113872 1113873 + μ minus

12

1113957σ21113874 1113875(tΔ minus s)1113874 1113875ds

(34)

Somewhat surprisingly Tao et al [25] indicated thatendogeneity would not affect the sample path features ofmodel (31) Our analysis therefore is the same as before

(1) Unstable κ μ + 1113957σ22gt 0(2) Locally explosive μgt 0(3) Explosive ϕ μ minus 1113957σ22gt 0

We can estimate 1113954θlowast ≔ ( 11139541113957σ2 1113954η 1113954σ2)prime in the same fashion asbefore -e likelihood ratio test (LR test) has been proposedby Tao et al [25] to test the null hypothesis of endogeneityinterest

LR Δminus 1Q

rΔ minus Q

urΔ( 1113857simχ2(1) underH0 η0 0 (35)

Tao et al [25] also developed the asymptotic theory for 1113954μand 1113954ρ ≔ exp(1113954μΔ) Based on their limit properties the es-timates of 1113954μ and 1113954ρ continue to follow asymptotic normaldistributions -is good quality enables us to convenientlyadopt the aforementioned testing procedures after making asmall adjustment to the variance of the limit distribution

5 Data

For our main empirical analysis we apply daily WTI crudeoil prices for the front-month (one-month) future contracton the New York Mercantile Exchange (NYMEX) -e dataare retrieved from the Energy Information Administration(EIA) Our sample spans the period from April 4 1983 toJune 30 2020 and comprises a total of 9352 daily obser-vations -e beginning and end dates of the sample periodare dictated by the availability of the data

It is well known that WIT and Brent are grades of in-ternationally traded crude oil that are widely used asbenchmarks in oil pricing globally -e reason for this paperrelying on the WTI price of crude oil is twofold On the onehand oil speculation is often blamed on people who tradefutures in the US market WTI is the primary benchmark

for oil extracted and consumed in the US-eWTI crude oilfuture market is also the worldrsquos most liquid and largestcrude oil future market Nearly 12 million contracts trade aday with more than 2 million in open interest On the otherhand WTI future contracts have the longest available his-tory dating back to 1983 giving us a large sample size to gainpowers in detecting short-lived bubbles

Our sample period under investigation includes severalmajor exogenous events that to a large extent are reckonedto have influenced oil markets -ese contain the collapse ofOPEC in 1986 the Persian Gulf War of 199091 the Asianfinancial crisis of 199798 the terrorist attacks on September11 2001 the Venezuelan crisis and Iraq war of 200203 theincreased financialization of oil future markets since 2004the OPEC oil supply announcements in 2008 and 2014 the2019 novel coronavirus disease (COVID-19) and moststrikingly the 200708 global financial crisis Figure 2 plotsthe time series trajectories of the WTI nominal front-monthfuture price and the corresponding real price A look at thefigure illustrates that the oil prices grew steadily from thestarting of the sample until the early 2000s -e price seriessince then begun to surge and the sharp increase in the priceof oil continued until July 2008 As shown in Figure 2 itcollapsed to about 30 dollars by the end of 2008 Between2014 and 2016 the price of oil also experienced a crash Morerecently WTI crude price even went subzero for the firsttime in history A common view is that the steep rise anddecline in the price of oil in the 2000s could be attributed tospeculation

6 Empirical Results

61 Main Results We start the empirical illustration withbaseline model (14) expanding it to the endogeneity ap-plication later In order to formally examine the existenceand duration of unstable and explosive behavior in the crudeoil market we run our tests on the daily price data

8 Complexity

Following the recommendation of PSY the minimumwindow size set for the recursive test procedure is the first 5years of the entire sample Figure 3 plots the test empiricaloutcomes along with their corresponding 95 critical valuesequences Also plotted are the trajectories of test statisticsfor the random coefficient and the price data Some inter-esting results are shown in this figure

Looking at the recursive test results reported in the lastpanel of Figure 3 we observe that over the initial period ofobservation the null hypothesis of a time-invariant autor-egressive coefficient cannot be rejected at the 5 statisticalsignificance level -e test statistic experiences a dramaticincrease in the late 1980s and exceeds the 95 critical valuein November 1990 Overall the observed pattern of timevariation rises and becomes much stronger as the timeperiod expands -e evidence that the LBI test statistics forthe majority of samples are greater than the 95 criticalvalue suggests that the data-generating process is wellcharacterized by the model with time-varying coefficientsOne point worth noting is that the sequence falls in thesecond half of 2008 and slumps in early 2020 -e styled factseems to indicate that the crude oil market has undergonedrastic changes Interestingly these located dates coincidewith the 200708 global financial crisis and the COVID-19pandemic

We now turn to our focus on different types of unstableand explosive behavior-e shaded areas in green in Figure 3are the identified periods where the t-test statistic crosses its95 critical value As is evident from the first panel theprocedure detects two unstable episodes -e first unstableepisode is relatively short and only lasts two months(2005M08-M09) -e second unstable episode appears inJanuary 2006 and continues until March 2020 -ese un-stable periods might be associated with the financializationof oil future markets from 2004 It can be seen in the secondpanel of Figure 3 that the detected episodes of locally ex-plosive (submartingale) behavior are clustered in 2007-20082009ndash2014 and 2018 As stated above since local explo-siveness implies instability and displays more extreme be-havior it should lead to a much shorter duration of

exuberance than that of instability -e third panel ofFigure 3 reflects that the t-test statistic is below the 95critical value throughout the sample period -is points outthat explosive behavior in crude oil price series does notexist Table 1 summarizes time horizons for the three kindsof unstable and explosive behavior identified by ourapproach

To shed light on a more realistic scenario of data-gen-erating process we implement the extreme sample pathbehavior test by extending the model to the endogeneitycase Figure 4 plots the empirical test results together with95 critical values A comparison of the LR test statisticswith the critical value sequence provides novel evidence insupport of the presence of endogeneity during the wholesample period -us it is believed that endogeneity is es-sential to the oil price formation process Furthermore thelast panel of Figure 4 highlights that the quadratic varianceof the price process is related to the test statistic forendogeneity -is finding is in accordance with the fact thatwe have exploited the realized variance series to obtain theLR test statistics In this sense Figure 5 displays differencesin the realized variance estimates which are captured by thetest statistic for endogeneity employing different methodsSimilar to Figure 3 we again identify episodes of unstabledynamics and fail to detect explosive behavior when con-sidering the model with endogeneity In contrast the sta-tistic sequence for local explosiveness exhibits a differentpattern Previous evidence in favor of locally explosive be-havior disappeared Table 2 summarizes time horizons forthe three forms of unstable and explosive behavior -eseresults suggest that endogeneity feedbacks play a significantrole in investigating the existence of explosive price be-havior -e conclusion that emerges is that there are nospeculative bubbles in the crude oil market

In principal our findings are in accordance with theexplanations made by Kilian and Murphy [19] Baumeisterand Hamilton [21] and Knittel and Pindyck [54] that thevariations in crude oil prices are driven by economicfundamentals

62 Comparison to Alternative Strategies Rich literature hasstudied speculative bubbles in the crude oil market using thetechniques developed by PWY and then extended by PSY Abrief description of the SADF methodology (used in PWY)and the GSADF methodology (used in PSY) can be found inAppendix It is therefore helpful to compare results foralternative methods as the detection strategy in the presentpaper is quite different from them Figure 6 plots thebackward ADF and backward SADF statistic sequences andthe associated 95 critical value sequences for oil futureprices 95 percent critical values are taken from 2000 MonteCarlo simulations with the actual sample size -e initialwindow size is set to 5 years A glance at Figure 6 reveals thatexplosive behavior occurred mainly in two occasions usingPWY and PSY dating strategies -ey are before and during2007-2008 global recession We also observe that PWYdetects several short-lived episodes between 2011 and 2014but PSY fails to do so For daily oil future prices over April 4

0

20

ndash20

ndash40

40

60

80

100

120

140

84 86 88 90 92 94 96 98 00 02 04 06 08 10 12 14 16 18 20

Nominal WTI front-month future pricesReal WTI front-month future prices

Figure 2 Real and nominal prices of WTI front-month futurecontracts

Complexity 9

1983 to June 30 2020 time horizons for explosive behavioridentified by PWY and PSY are summarized in Table 3 Inline with previous studies we could conclude from bothPWY and PSY tests that there is evidence of bubbles in thecrude oil market data However as shown in Gurkaynak [37]and Pavlidis et al [43] explosive dynamics in asset prices perse do not constitute evidence either in support of or againstspeculative bubbles since such explosive behavior may

actually reflect explosive market fundamentals or bothfactors As aforementioned our random coefficient autor-egressive model framework has considered endogeneityfeedbacks that to some extent alleviate this problemMoreover the evidence in Figures 3 and 4 suggests theabsence of explosive dynamics in the crude oil market -efact permits us to safely ignore the distinction betweenbubbles or explosive fundamentals

1988

-03

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

ndash4

ndash2

0

2

4

6

8

0

20

40

60

80

100

120

140

t-statistic of k95 critical value of N (0 1)Crude oil future prices

Instability detection based on the time-varyingcoefficient model

(a)

Local explosiveness detection based on the time-varyingcoefficient model

1988

-03

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

t-statistic of μ95 critical value of N (0 1)Crude oil future prices

ndash4

ndash2

0

2

4

6

8

0

20

40

60

80

100

120

140

(b)

Explosiveness detection based on the time-varyingcoefficient model

1988

-03

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

t-statistic of ϕ95 critical value of N (0 1)Crude oil future prices

ndash4

ndash2

0

2

4

6

8

0

20

40

60

80

100

120

140

(c)

LBItest statistic for testing time-varying coefficient19

88-0

3

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

LBItest95 critical value of N (0 1)

ndash5

ndash10

0

5

10

15

20

(d)

Figure 3 Date-stamping unstable and explosive episodes in oil prices with the time-varying coefficient model assuming no endogeneity

Table 1 Time horizons of unstable and explosive periods identified by models assuming no endogeneity

Unstable Locally explosive ExplosiveAug 2005-Sep 2005 Dec 2007ndashSep 2008 mdash

Jan 2006ndashMar 2020 Jun 2009ndashOct 2014 mdashJun 2018ndashSep 2018 mdash

10 Complexity

63 Robustness of the Results In the following we providetwo robustness checks to assess whether our previousconclusions of the paper still hold

First in our benchmark model our use of front-monthfuture price instead of other future prices at different ma-turities is justified by its being the most liquid future contractwith the highest open interest To explore the sensitivity ofour results to the future contracts at different maturities werepeat the analysis described above using the prices forlonger-term contracts (ie 2 3 and 4 months-ahead fu-tures) It should be pointed out that the WTI future contractusually expires on the third business day before the 25thcalendar day of the contract month proceeding the deliverymonth

Second becauseWTI crude oil is by tradition traded innominal US dollars per barrel it might be of concern that

inflation and other monetary variables may have realeffects on oil price It is therefore of interest to use realrather than nominal price In fact recent work documentsthat the use of either price is inconsequential in practice[55] Although we mainly focus on nominal prices forrobustness check we shall report the results for realprices To convert nominal daily series into daily realseries we use linearly interpolated monthly ConsumerPrice Index (CPI) Monthly CPI series for the UnitedStates is obtained from the websites of the Federal ReserveBank of St Louis

Further evidence presented in Figures 7ndash10 demon-strates that our results are remarkably robust to thesemodifications and the conclusions remain qualitativelyunchanged

ndash4

ndash2

0

2

4

6

8

0

20

40

60

80

100

120

140

1988

-03

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

Instability detection based on the time-varyingcoefficient model

t-statistic of k95 critical value of N (0 1)Crude oil future prices

(a)

ndash4

ndash2

0

2

4

6

8

0

20

40

60

80

100

120

140

1988

-03

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

t-statistic of μ95 critical value of N (0 1)Crude oil future prices

Local explosiveness detection based on the time-varyingcoefficient model

(b)

ndash4

ndash2

0

2

4

6

8

0

20

40

60

80

100

120

140

1988

-03

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

Explosiveness detection based on the time-varyingcoefficient model

t-statistic of ϕ95 critical value of N (0 1)Crude oil future prices

(c)

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

LR-statistic95 critical values of x2 (1)Realized variance

700

600

500

400

300

200

100

0

ndash100

350

300

250

200

150

100

50

0

Test for endogeneity in time-varyingcoefficient model

(d)

Figure 4 Date-stamping unstable and explosive episodes in oil prices with the time-varying coefficient model with endogeneity

Complexity 11

2

3

4

5

6

7

8

0

50

100

150

200

250

300

350

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

LLF0

LLF1

Realized variance of crude oil future prices

Figure 5 Realized variance and likelihood values for random coefficient regression

Table 2 Time horizons of unstable and explosive periods detected by models with endogeneity

Unstable Locally explosive ExplosiveMay 2008ndashJul 2008 mdash mdashDec 2010ndashOct 2014 mdash mdashNov 2016ndashApr 2017 mdash mdashJul 2017ndashFeb 2020 mdash mdash

ndash4

ndash2

0

2

4

6

8

0

20

40

60

80

100

120

140

1988

-03

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

Explosiveness detection using SADF methodology

BADF sequence95 critical valuesCrude oil future prices

(a)

ndash4

ndash2

0

2

4

6

8

0

20

40

60

80

100

120

140

1988

-03

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

Explosiveness detection using GSADF methodology

BSADF sequence95 critical valuesCrude oil future prices

(b)

Figure 6 Date-stamping explosive episodes in oil prices with the time-invariant coefficient model

12 Complexity

Table 3 Time horizons of explosive behavior identified by SADF and GSADF methodologies

SADF GSADFAug 2005ndashAug 2006 Oct 2011ndashApr 2012 Sep 1990-Oct 1990Jun 2007ndashOct 2008 Aug 2012-Sep 2012 Feb 1991-Mar 1991Jun 2009-Jul 2009 Jul 2013-Aug 2013 Oct 2004-Nov 2004Feb 2011ndashApr 2011 Feb 2014-Mar 2014 Feb 2005-Mar 2005

Jul 2011ndashAug 2012 Jun 2014-Jul 2014May 2005ndashJan 2006Mar 2006ndashAug 2006Jul 2007ndashAug 2008

ndash4

ndash2

0

2

4

6

8

0

10

20

30

40

50

60

70

1988

-03

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

95 critical value of N (0 1)Real oil future prices

t-statistic of μ~

Local explosiveness detection based on the time-varyingcoefficient model

(a)

ndash4

ndash2

0

2

4

6

8

0

10

20

30

40

50

60

70

1988

-03

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

95 critical value of N (0 1)Real oil future prices

t-statistic of ϕ

Explosiveness detection based on the time-varyingcoefficient model

(b)

Figure 7 Date-stamping explosive episodes in real oil prices with the time-varying coefficient model with endogeneity

ndash4

ndash2

0

2

4

6

8

0

50

100

150

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

Local explosiveness detection based on the time-varyingcoefficient model

t-statistic of μ95 critical value of N (0 1)2-month crude oil future prices

(a)

ndash4

ndash2

0

2

4

6

8

0

50

100

150

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

Local explosiveness detection based on the time-varyingcoefficient model

t-statistic of ϕ95 critical value of N (0 1)2-month crude oil future prices

(b)

Figure 8 Date-stamping explosive episodes in 2-month future oil prices with the time-varying coefficient model with endogeneity

Complexity 13

7 Discussion

-e oil price rally since the early 2000s has attracted greatinterest in the sources of oil price shocks Some causal ob-servers have stressed that financial speculation has become asignificant driver of the dynamics of oil prices A large body ofliterature emerged that has sought to investigate its existenceIn this paper we employ a novel approach for testing forspeculative bubbles in oil prices In synthesis our empiricalfindings indicate that there is less evidence of exuberance in

the oil future markets Our results are very robust in severalscenarios We argue that the booms and busts in the oil pricemovements during the sample period are driven primarily byeconomic fundamentals rather than speculation

Overall our results confirm the main findings obtainedby structural VAR-based studies including Kilian andMurphy [19] Kilian and Lee [20] Baumeister and Hamilton[21] and Zhou [22] In the past decade the use of structuralVAR models to understand the evolution of the oil price hasbecome common -e traditional SVAR literature

ndash4

ndash2

0

2

4

6

8

0

50

100

15019

88-0

3

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

Local explosiveness detection based on the time-varyingcoefficient model

t-statistic of μ95 critical value of N (0 1)3-month crude oil future prices

(a)

ndash4

ndash2

0

2

4

6

8

0

50

100

150

1988

-03

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

Explosiveness detection based on the time-varyingcoefficient model

t-statistic of ϕ95 critical value of N (0 1)3-month crude oil future prices

(b)

Figure 9 Date-stamping explosive episodes in 3-month future oil prices with the time-varying coefficient model with endogeneity

ndash4

ndash2

0

2

4

6

8

0

50

100

150

Local explosiveness detection based on the time-varyingcoefficient model

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

t-statistic of μ95 critical value of N (0 1)4-month crude oil future prices

(a)

ndash4

ndash2

0

2

4

6

8

0

50

100

150

Explosiveness detection based on the time-varyingcoefficient model

t-statistic of ϕ95 critical value of N (0 1)4-month crude oil futures prices

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

(b)

Figure 10 Date-stamping explosive episodes in 4-month future oil prices with the time-varying coefficient model with endogeneity

14 Complexity

documents that the fundamental drivers of oil price fluc-tuations have been dominated essentially by supply anddemand shocks see eg Kilian [56] Lippi and Nobili [57]Kilian andMurphy [58] and Baumeister and Peersman [59]Kilian and Murphy [19] focused on the role speculation byaugmenting the SVAR framework of their earlier paper byglobal oil inventory data -ey specified a four-variablemodel for the global oil market (the percentage change inglobal oil production the real price of oil a suitable proxy ofcyclical variation in global real economic activity andchanges in global crude oil inventories) -is setting andtheir sign-identified scheme allow them to separately explainvariation in these data as four different types of shocks (1) ashock to the amount of oil pumped out of the ground (ldquooilsupply shockrdquo) (2) an oil flow demand shock (3) an oil-specific oil demand shock and especially (4) a speculative(or inventory) demand shock -eir core finding is that thespike in the real price of crude oil prices during 2003ndash2008was caused by unexpected increase in global oil consump-tion Using two alternative proxies for global above-groundcrude oil inventories Kilian and Lee [20] reestimated themodel of Kilian and Murphy [19] and ruled out the role ofspeculation in explaining the recent surge of the real oil priceon the basis of their estimation results Also Zhou [22]assessed the sensitivity of the findings in Kilian and Murphy[19] to the introduction of various state-of-the-art oil marketmethods She shows that the core conclusions reached byKilian and Murphy [19] are highly robust to some refine-ments corrections and extension of the KilianndashMurphymodel Baumeister and Hamilton [21] recently contendedthat prior information plays a key role in any SVAR modelsand traditional identification approaches can be generalizedto a special case of a Bayesian perspective -ey revisited theimportance of supply and demand shocks in the oil marketand demonstrated that speculative demand shock turns outto be only a small factor in accounting for most historical oilprice shocks

For comparison purpose we have also applied the testsof PWY and PSY which are the most popular econometrictechniques in the literature on speculative bubbles in the oilmarkets If we were to follow the test procedures of PWY andPSY our conclusions are also broadly consistent with therich literature that supports the existence of speculativebubbles in oil prices However these cointegration-basedbubble tests suffer from the joint hypothesis problem ldquoEverytest of a bubble is a joint test of the presence of the bubble inthe data and the validity of the model applied by theeconometricianrdquo (see Giglio et al [60]) More specifically themain drawback of such indirect model-dependent tests ofbubbles typically refers to the difficulty in distinguishingbetween the unknown true model for market fundamentalsand the presence of bubbles (see for example Gurkaynak[37] Pavlidis et al [42] Pavlidis et al [43]) Given thefundamental value of oil price is unobservable the literaturehas resorted to observe economic and financial variableswhich are used to estimate the intrinsic values For exampleboth Phillips and Yu [16] and Caspi et al [39] simplynormalized crude oil prices by US oil inventory data as thefundamental value -is measure is questionable since their

method cannot sufficiently capture the information onmarket fundamentals and neglect the major driversmdashlikeglobal economic activitymdashalso affecting the fundamentalvalue of crude oil -is issue of misspecified fundamentals(either in terms of model misspecification or omitted var-iables or both of these reasons) may make researchers fail toarrive at a conclusion from evidence of explosive pricesalone To make matters worse even though we have cap-tured the true market fundamentals we may still fail todetect bubbles if the explosive behavior in the price of oil juststems from the explosive fundamentals

Compared to earlier studies as briefly mentioned aboveour methodology has the main advantage that by consid-ering the dependence between the random coefficient andsystem shocks we have incorporated endogeneity into thecontinuous-time random coefficient setting and thus we donot need to prespecify the fundamental component of the oilprice -e test results for endogeneity suggest that endo-geneity is important in the data-generating process Inparticular our empirical analysis also finds that there are noexplosive episodes in crude oil markets-is implies that it isnot necessary to require the specification of a fundamentalvalue of the oil price Like Pavlidis et al [43] their methodsand ours have in common that the problem of the well-known joint hypothesis has to be some extent been ef-fectively ameliorated Although our approach is quite dis-tinct we draw the same substantive conclusion thatspeculative bubbles can be ruled out as a key determinant ofthe price of crude oil in the 2000s reaffirming the con-clusions of Kilian and Murphy [19] Baumeister andHamilton [21] Zhou [22] and Fattouh et al [61]

8 Conclusions

Repeated surges and crashes in crude oil prices during lastdecades have sparked a heated debate on their underlyingcauses among academics commentors and market regu-lators One of the central questions concerns whetherfinancialization of oil future markets creates a speculativebubble that subsequently collapses Answering this questionis difficult because one has to rely upon strong assumptionsabout the data-generating mechanism for the fundamentalvalue of the oil price In this paper we present a stylizedmodel of asset pricing for a commodity to address this issueOur theoretical framework disentangles the effects ofspeculative bubbles from the traditional fundamental effectson oil price dynamics We use this setting to illustrate thatbubbles if they are present should demonstrate explosivebehavior in prices We then provide a novel approach forbubble detection that covers various explosive scenarios andfind no evidence of speculative bubbles in oil markets evenduring episodes where the most popular SADF and GSADFmethodologies identify a speculative bubble

Our conclusions have pertinent implications for policymakers Given the historically important role of oil prices inthe real economy one of the serious concerns in policycircles over the past decade has been whether speculativetrading activities in oil future markets are responsible for thelarge price swings For example in their 2008 testimonies to

Complexity 15

the US Senate Commerce Committee Michael Masters andGeorge Soros posited that speculators had fueled price in-creases since 2003 and had caused a part of a ldquosuperbubblerdquoin 2007 and 2008 resulting in significantly higher energycosts for consumers -is widespread bubble view domi-nated policy discussions and efforts to limit speculativetrading in the future market quickly followed As a responseto the excessive speculation pressures on prices theDoddndashFrank Wall Street Reform and Consumer ProtectionAct was enacted by the US Congress in 2010 As a directpolicy implication our results highlight that imposingstricter regulations on trading in oil derivative marketscannot be expected to lower the oil price given the fact thatthe synchronized boom and bust cycles witnessed in the past20 years are mainly due to economic fundamentals whichappear likely to resurge with the business cycle -e aboveanalysis implies that there is no need to enforce additionalregulatory limits against financial arbitrage in the com-modity market

Appendix

The SADF and GSADF Methodologies

Appendix describes the SADF and GSADF testing proce-dures used to detect speculative bubbles in Section 62 For agenetic time series process xt consider the following rolling-window type augmented DickeyndashFuller (ADF) regressionequation

xt ar1r2+ μr1r2

xtminus 1 + 1113944

J

j1ϕj

r1r2Δxtminus j + εt εt sim NID 0 δ2r1r2

1113874 1113875

(A1)

where Δxtminus j with j 1 J are lagged first differences ofxt incorporated to accommodate potential serial correla-tion ar1 r2

μr1 r2 and ϕj

r1 r2 with j 1 J are regression

coefficients the subscripts r1 and r2 refer to fractions of thetotal sample size (ofT observations) that set the start and endpoints of a subsample period and NID represents inde-pendent and normal distribution -e ADF test statisticcorresponding to the unit root null hypothesisH0 μr1 r2

0 is simply the t-ratio

ADFr2r1

1113954μr1 r2

se 1113954μr1 r21113872 1113873

(A2)

and the right-side alternative hypothesis of explosive be-havior is H0 μr1r2

gt 0 We can obtain the standard ADF teststatistic by running regression (A1) on the whole sampleie by specifying r1 0 and r2 1 A large amount of re-search has demonstrated that the standard ADF test ADF10has extremely low power in detecting periodically collapsingbubbles

To address the problem PSW proposed a recursiveregression technique which involves estimating (A1)employing a forward expanding sample Specifically theyused subsamples of the available data to recursively calculatea sequence of ADF statistics denoted by ADFr2

0 -e startpoint of the sample period is fixed at r1 0 while the endpoint r2 increases from r0 (the minimum window size) toone (the full sample period) -e SADF statistic called thesupremum of ADFr2

0 is defined by

SADF r0( 1113857 supr2isin r0 1[ ]

ADFr20

(A3)

and has the following limit distribution under the nullhypothesis

supr2isin r0 1[ ]

1113938r2

0 WdW

1113938r2

0 W21113872 1113873(12)

(A4)

where W is the standard Brownian motion Testing thepresence of explosive behavior entails comparing the ADFr2

0statistic with the right-side critical values obtained throughMonte Carlo simulations

More recently PSY showed that when the sample perioddisplays more than one boom-bust bubble the SADF testprocedure may perform poorly and fail to detect or con-sistently date-stamp alternative bubbles -is deficiencymotivates PSY to formulate an extension of the SADF testthe GSADF test which covers more subsamples of the datathan the SADF by allowing both the end point r2 and thestart point r1 to vary As a result the GSADF test has moreflexibility and gains substantial power making this testconsistent with multiple boom-bust bubbles -e GSADFtest statistic is defined as

GSADF r0( 1113857 supr2isin r0 1[ ]r1isin 0r2minus r0[ ]

ADFr2r1 (A5)

with the limit distribution

supr2isin r0 1[ ]r1isin 0r2minus r0[ ]

(12)rw W r2( 11138572

minus W r1( 11138572

minus rw1113960 1113961 minus 1113938r2

r1W(r)dr W r2( 1113857 minus W r1( 11138571113858 1113859

r12w rw 1113938

r2

r1W(r)2dr minus 1113938

r2

r1W(r)dr1113876 1113877

21113896 1113897

12

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎭

(A6)

where rw r2 minus r1 is the expanding window size Like theSADF test comparison of the GASDF statistic with the right-

tailed critical values from its limit distribution enables us to testfor the unit root hypothesis against explosive behavior

16 Complexity

If the null hypothesis of a unit root in xt is rejectedthen in the second stage the SADF and GSADF tests canidentify the exact episodes during which the xt seriesexhibits explosive dynamics To locate the emergence andconclusion of explosiveness the SADF methodologymatches the sequence of the recursive backward ADF(BADF) test statistics against the right-tailed criticalvalues of the limit distribution of the standard Dick-eyndashFuller test statistic Suppose that re is the originationdate and rf is the finish date of the bubble in the data-ese estimates of dates can be constructed from

1113954re infr2isin r0 1[ ]

r2 ADFr20 gt cv

αr2

1113966 1113967

1113954rf infr2isin 1113954re11113858 1113859

r2 ADFr20 lt cv

αr2

1113966 1113967(A7)

where cvαr2 is 100(1 minus α) the right-tailed critical value of theADF statistic corresponding to the chosen significance levelof α For the GSADF methodology the date-stampingstrategy in this case is based on the backward SADF(BSADF) statistic

BSADFr2r0( 1113857 sup

r1isin 0r2minus r0[ ]ADFr2

r1

(A8)

Likewise the origination date of the bubble is defined asthe first observation that the BSADF statistic is greater thanits critical value

1113954re infr2isin r0 1[ ]

r2 BSADFr2r0( 1113857gt cv

βT

r21113966 1113967 (A9)

and the finish date as the first date after which the BSADFdrops back below its critical value

1113954rf infr2isin 1113954re11113858 1113859

r2 BSADFr2r0( 1113857lt cvβT

r21113966 1113967 (A10)

where cvβTr2 is 100(1 minus βT) the right-tailed critical value of

the SADF based on Tr2 and the corresponding significancelevel is βT

Data Availability

-e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

-e authors declare no conflicts of interest

Acknowledgments

-e authors gratefully acknowledge the financial supportfrom the National Social Science Foundation of China underGrant no 18BTJ039

References

[1] M W Masters Testimony before the US Senate Committee onHomeland Security and Governmental Affairs 2008 httpswwwhsgacsenategovimomediadoc052008Masterspdfattempt2

[2] M W Masters Testimony before the Commodity FuturesTrading Commission 2010 httpscftcgovsitesdefaultfilesidcgroupspublicnewsroomdocumentsfilemetalmarkets032510_masterspdf

[3] J P Kennedy ldquo-e high cost of gambling on oilrdquo Section A p23 New York Times New York NY USANew York Times2012

[4] I-H Cheng and W Xiong ldquoFinancialization of commoditymarketsrdquo Annual Review of Financial Economics vol 6 no 1pp 419ndash441 2014

[5] K Tang and W Xiong ldquoIndex investment and the financi-alization of commoditiesrdquo Financial Analysts Journal vol 68no 6 pp 54ndash74 2012

[6] O J Blanchard and M W Watson ldquoBubbles rational ex-pectations and financial marketsrdquo in Crisis in the Economicand Financial Structure P Wachtel Ed Lexington BooksLexington MA USA pp 295ndash315 1982

[7] B T Diba and H I Grossman ldquoOn the inception of rationalbubblesrdquo e Quarterly Journal of Economics vol 102 no 3pp 697ndash700 1987

[8] G W Evans ldquo-e fragility of sunspots and bubblesrdquo Journalof Monetary Economics vol 23 no 2 pp 297ndash317 1989

[9] G W Evans and S Honkapohja ldquoOn the robustness ofbubbles in linear RE modelsrdquo International Economic Reviewvol 33 no 1 pp 1ndash14 1992

[10] E Pavlidis A Yusupova I Paya et al ldquoEpisodes of exu-berance in housing markets in search of the smoking gunrdquoe Journal of Real Estate Finance and Economics vol 53no 4 pp 419ndash449 2016

[11] E Martınez-Garcıa and V Grossman ldquoExplosive dynamics inhouse prices An exploration of financial market spillovers inhousing markets around the worldrdquo Journal of InternationalMoney and Finance vol 101 p 102103 2020

[12] S Shi A Rahman and B Z Wang ldquoAustralian housingmarket booms fundamentals or speculationlowastrdquo EconomicRecord 2020 In press

[13] P C B Phillips Y Wu and J Yu ldquoExplosive behavior in the1990s NASDAQ when did exuberance escalate asset val-ueslowastrdquo International Economic Review vol 52 no 1pp 201ndash226 2011

[14] P C B Phillips S Shi and J Yu ldquoTesting for multiplebubbles historical episodes of exuberance and collapse in theSampp 500rdquo International Economic Review vol 56 no 4pp 1043ndash1078 2015

[15] P C B Phillips S Shi and J Yu ldquoTesting for multiplebubbles limit theory of real-time detectorsrdquo InternationalEconomic Review vol 56 no 4 pp 1079ndash1134 2015

[16] P C B Phillips and J Yu ldquoDating the timeline of financialbubbles during the subprime crisisrdquo Quantitative Economicsvol 2 no 3 pp 455ndash491 2011

[17] D Tsvetanov J Coakley and N Kellard ldquoBubbling over thebehaviour of oil futures along the yield curverdquo Journal ofEmpirical Finance vol 38 no Part B pp 516ndash533 2016

[18] I Figuerola-Ferretti J R Mccrorie and I ParaskevopoulosldquoMild explosivity in recent crude oil pricesrdquo Energy Eco-nomics vol 87 p 104387 2020

[19] L Kilian and D P Murphy ldquo-e role of inventories andspeculative trading in the global market for crude oilrdquo Journalof Applied Econometrics vol 29 no 3 pp 454ndash478 2014

Complexity 17

-e discrete-time representation (18) can then be re-written as

xtΔ exp 1113957σΔ

radicutΔ + ϕΔ1113966 1113967x(tminus 1)Δ + σηtΔ ρtΔx(tminus 1)Δ + σηtΔ

(21)

Assuming 1113957σ2 gt 0 models (14) and (21) have the followingstationarity properties (1) for (μ + 1113957σ22)lt 0 the process isasymptotically covariance stationary sinceμlt 0 means thatboth E(ρtΔ) eμΔ lt 1 and E(ρ2tΔ) exp(2μΔ+1113957σ2Δ) exp(2κΔ)lt 1 (2) for (μ + 1113957σ22)lt 0 the process re-mains asymptotically stationary but is not asymptoticallycovariance stationary any longer Practically speaking wehave to impose the condition of κ (μ + 1113957σ22)lt 0 toguarantee the covariance stationary Whenκ (μ + 1113957σ22)lt 0 it follows from (19) that the variance ofV(t) converges to minus σ22κltinfin as t⟶infinWhen κ 0 thevariance is equal to σ2t and diverges as t⟶infin (3) for (μ +1113957σ22)gt 0 and μlt 0 the process is still asymptotically sta-tionary but is not covariance stationary with evident per-sistence in the trajectory (4) when μ 0 ϕ (μ minus 1113957σ22)lt 0ensures the asymptotic stationarity but leads to infinitesecond moments withκ (μ + 1113957σ22)gt 0 -e process ex-hibits apparent unstable behavior and larger variation (5)when μgt 0 and μ minus 1113957σ22lt 0 the process continues to becovariance nonstationary Contrast to the traditional ex-plosive AR (1) model the model retains asymptoticallystationary In this instance bubble-like phenomenon shouldbe clearly observed in actual data (6) if μ minus 1113957σ22gt 0 themodel is no longer asymptotically stationary and the ex-plosive growth behavior is now obvious

As shown in Tao et al [25] when 1113957σ2 0 model (21)collapses to a simple autoregression with a constant coef-ficient It explains why a popular way to look for bubbles is totest ϕ 0 against ϕgt 0 via standard or recursive right-tailedunit root tests [15ndash18 27] For the purpose of the presentpaper we only examine three price behavior features that aredivided into three categories

(1) Unstable κ μ + 1113957σ22gt 0 Here the model is as-ymptotically covariance nonstationary and applies tocapture unstable behavior

(2) Locally explosive μgt 0 -e model in this case hasthe ability for generating both explosive and col-lapsing behavior

(3) Explosive ϕ μ minus 1113957σ22gt 0 Under the circumstancethe model is not asymptotically stationary and iscapable of generating explosive behavior

Based on the above typology explosive embodies locallyexplosive while locally explosive embodies unstable

42 Inference Scheme -is section is concerned with puttingthe continuous-time model specified above into the appli-cation of discretely sampled data We utilize a novel two-stage approach proposed by Phillips and Yu [51] to estimateμ 1113957σ2 and σ2 In the first stage realized volatility is employedto provide a regression model for estimating the diffusionparameters 1113957σ2 and σ2 In the second stage we maximize thein-fill likelihood function to estimate the drift parameter μ

Suppose data are observed in N equally spaced blocksover the time interval [0 T] with sampling interval h -eoverall time span of the data is T so that N Th If eachblock has M observations the total number of observationsis M times N TΔ where Δ is the sampling frequency Todevelop asymptotics we assume that as Δ⟶ 0M times N⟶infin According to the algorithm of Phillips andYu [51] the estimators based on realized variance (RV) andrealized quarticity (RQ) are denoted by

1113954θ argminθisinΘ

QΔ(θ) (22)

where 1113954θ ≔ (1113957σ2 σ2)prime and QΔ(θ) Δ1113936Nn1 (logRVnminus log

[x]nh(nminus 1)h + (12)s2n)2s2nwith s2n max

2Δ(RQ2

nRV2n)

11139681113864

2M

radic

RVn 1113944M

i1x(nminus 1)h+iΔ minus x(nminus 1)h+(iminus 1)Δ1113960 1113961

2 n 1 2 N

RQn 13Δ

1113944

M

i1x(nminus 1)h+iΔ minus x(nminus 1)h+(iminus 1)Δ1113960 1113961

4 n 1 2 N

(23)

In the second stage the approximate logarithmic in-filllikelihood function is maximized with regard to μ (repre-senting the resulting estimator as 1113954μ)

1113954μ argmaxμ

1MN

log ℓALF(μ) (24)

where

ℓALF(μ) 1113944MtimesN

t1

μx(tminus 1)Δ11139541113957σ2x2

(tminus 1)Δ +1113954σ2

xtΔ minus x(tminus 1)Δ1113872 1113873 minusΔ2

1113944

MtimesN

t1

μx(tminus 1)Δ11139541113957σ2x2

(tminus 1)Δ +1113954σ2

(25)

One important goal in the present empirical work is tocheck the presence of randomness in the persistenceparameter of model (14) which is equivalent to test the

null hypothesis 1113957σ2 0 To achieve this aim we resort tothe locally best invariant test (LBI test) statistic [52]namely

6 Complexity

1113957ZN ≔1113936

MtimesNt1 1113957ε2tΔ minus (1MN)1113936

MtimesNt1 1113957ε2tΔ1113872 1113873

21113874 11138751113957x

2(tminus 1)Δ

(1MN)1113936MtimesNt1 1113957ε4tΔ minus (1MN)1113936

MtimesNt1 1113957ε2tΔ1113872 1113873

21113969

(1MN)1113936MtimesNt1 1113957x

4(tminus 1)Δ minus (1MN)1113936

MtimesNt1 1113957x

2(tminus 1)Δ1113872 1113873

21113969 (26)

where 1113957x2tΔ (xtΔ

1 + x2

tΔ

1113969) 1113957εtΔ xtΔ minus 1113957ρx(tminus 1)Δ and

1113957ρ (1113936MtimesNt1 1113957x(tminus 1)Δx(tminus 1)Δ)

minus 11113936MtimesNt1 1113957x(tminus 1)Δx(tminus 1)Δ -us as

N⟶infin MNminus 12 1113957ZN⟶L N(0 1) under H0 1113957σ2 0 andMNminus (12) 1113957ZN⟶L N(0 1) under H0 1113957σ2 0

Relying on asymptotics with in-fill (as the sampling fre-quency tends to zeroΔ⟶ 0) and long time-span asymptoticsTao et al [25] developed new limit theory for the continuous-time system where the persistence parameter is random -easymptotic properties of the diffusion parameter and drift pa-rameter estimates can be used to construct test statistics for threetypes of unstableexplosive behavior which are

tκ 1Δ

1113944

MtimesN

t1

x2(tminus 1)Δ

11139541113957σ2x2(tminus 1)Δ + σ2

⎛⎝ ⎞⎠

12

1113954βκ minus β0κ1113872 1113873⟶L

N(0 1)

tμ 1Δ

1113944

MtimesN

t1

x2(tminus 1)Δ

11139541113957σ2x2(tminus 1)Δ + 1113954σ2

⎛⎝ ⎞⎠

12

1113954ρ minus ρ0( 1113857⟶L

N(0 1)

tϕ 1Δ

1113944

MtimesN

t1

x2(tminus 1)Δ

11139541113957σ2x2(tminus 1)Δ + 1113954σ2

⎛⎝ ⎞⎠

12

1113954βϕ minus β0ϕ1113872 1113873⟶L

N(0 1)

(27)

where

1113954βκ exp 1113954μ +12

11139541113957σ21113874 1113875Δ1113882 1113883

1113954βϕ exp 1113954μ minus12

11139541113957σ21113874 1113875Δ1113882 1113883

β0κ exp μ0 +12μ201113874 1113875Δ1113882 1113883

β0ϕ exp μ0 minus12μ201113874 1113875Δ1113882 1113883

1113954ρ 1 +1113936

MtimesNt1 x(tminus 1)Δ xtΔ minus x(tminus 1)Δ1113872 1113873 1113954

1113957σ2x2(tminus 1)Δ +

1113954σ21113874 1113875

1113936MtimesNt1 x

2(tminus 1)Δ

11139541113957σ2x2

(tminus 1)Δ +1113954σ21113874 1113875

ρ0 1

(28)

As can be seen this test procedure does not directlyverify whether or not κ (μ + 1113957σ22) 0 or μ 0 orϕ (μ minus 1113957σ22) 0 but transforms the null hypotheses toβk 1 or ρ 1 or βϕ 1 -ese three t-test statistics areaccompanied by a date-stamping strategy that permits us toidentify the exact origination and termination periods ofprice exuberance Given this instrument estimates of thesedates are as follows

1113954ribun inf

sisin 1113954r(iminus 1)e

un 11113858 1113859s tκ(s)gtZα1113864 1113865

1113954rieun inf

sisin 1113954r(iminus 1)b

un 11113858 1113859s tκ(s)ltZα1113864 1113865

1113954rible inf

sisin 1113954r(iminus 1)e

le 11113858 1113859s tμ(s)gtZα1113966 1113967

1113954rieun inf

sisin 1113954r(iminus 1)b

le 11113858 1113859s tμ(s)ltZα1113966 1113967

1113954ribe inf

sisin 1113954r(iminus 1)e

e 11113858 1113859s tϕ(s)gtZα1113966 1113967

1113954rieun inf

sisin 1113954r(iminus 1)b

e 11113858 1113859s tϕ(s)ltZα1113966 1113967

(29)

where Zα is the right-tailed critical value of the standardnormal distribution corresponding to a significance level ofα 1113954rib

un1113954rible1113954r

ibe denote the beginnings on the dates of the ith

extreme price behavior period and similarly 1113954rieun1113954r

iele1113954r

iee

denote the ends on the dates of the ith extreme price be-havior period

43 Extend the Model with Endogeneity Now we are con-cerned with the endogeneity case where Wμ(t) is not in-dependent of Wε(t) To see this let us suppose that data aregenerated from the following stochastic differential equationsystem

dx(t) x(t)d1113957B(t) + σdB(t) x(0) x0

d1113957B(t) μ dt + 1113957σWμ(t)

dB(t) σdWε(t)

(30)

where (Wμ(t) Wε(t)) is a two-dimensional Brownianmotion with covariance parameter η As Follmer [53] notedthe strong solution obtained from this continuous systemwould be of the explicit form

dx(t) exp 1113957σWμ(t) + μ minus12

1113957σ21113874 1113875t1113874 1113875x(0) + Z(t) (31)

where

Z(t) σ 1113946t

0exp 1113957σ Wμ(t) minus Wμ(s)1113872 1113873 + μ minus

12

1113957σ21113874 1113875(t minus s)1113874 1113875dWε(t)

minus ω1113946t

0exp 1113957σ Wμ(t) minus Wμ(s)1113872 1113873 + μ minus

12

1113957σ21113874 1113875(t minus s)1113874 1113875ds

V(t) minus P(t)

(32)

Notice that when endogeneity is included in model (14)the driver process becomes Z(t) as a substitute for V(t)Z(t) consists of a component V(t) and another oneconnected with the covariance parameter of (1113957B tB) ω and

Complexity 7

variance of V(t) -e strong solution gives the exact dis-crete-time form of model (31)

xtΔ exp 1113957σΔ

radicuΔ + μ minus

12

1113957σ21113874 1113875Δ1113882 1113883x(tminus 1)Δ + ZtΔ ρtΔ x(tminus 1)Δ + ZtΔ

+ σ 1113946tΔ

(tminus 1)Δexp 1113957σ WμtΔ minus Wμs1113872 1113873 + μ minus

12

1113957σ21113874 1113875(tΔ minus s)1113882 1113883dWεs

(33)

where uiidsim

t N(0 1) and

ZtΔ σ 1113946tΔ

(tminus 1)Δexp 1113957σ WμtΔ minus Wμs1113872 1113873 + μ minus

12

1113957σ21113874 1113875(tΔ minus s)1113882 1113883dWεs

minus ω1113946tΔ

(tminus 1)Δexp 1113957σ WμtΔ minus Wμs1113872 11138731113872 1113873 + μ minus

12

1113957σ21113874 1113875(tΔ minus s)1113874 1113875ds

(34)

Somewhat surprisingly Tao et al [25] indicated thatendogeneity would not affect the sample path features ofmodel (31) Our analysis therefore is the same as before

(1) Unstable κ μ + 1113957σ22gt 0(2) Locally explosive μgt 0(3) Explosive ϕ μ minus 1113957σ22gt 0

We can estimate 1113954θlowast ≔ ( 11139541113957σ2 1113954η 1113954σ2)prime in the same fashion asbefore -e likelihood ratio test (LR test) has been proposedby Tao et al [25] to test the null hypothesis of endogeneityinterest

LR Δminus 1Q

rΔ minus Q

urΔ( 1113857simχ2(1) underH0 η0 0 (35)

Tao et al [25] also developed the asymptotic theory for 1113954μand 1113954ρ ≔ exp(1113954μΔ) Based on their limit properties the es-timates of 1113954μ and 1113954ρ continue to follow asymptotic normaldistributions -is good quality enables us to convenientlyadopt the aforementioned testing procedures after making asmall adjustment to the variance of the limit distribution

5 Data

For our main empirical analysis we apply daily WTI crudeoil prices for the front-month (one-month) future contracton the New York Mercantile Exchange (NYMEX) -e dataare retrieved from the Energy Information Administration(EIA) Our sample spans the period from April 4 1983 toJune 30 2020 and comprises a total of 9352 daily obser-vations -e beginning and end dates of the sample periodare dictated by the availability of the data

It is well known that WIT and Brent are grades of in-ternationally traded crude oil that are widely used asbenchmarks in oil pricing globally -e reason for this paperrelying on the WTI price of crude oil is twofold On the onehand oil speculation is often blamed on people who tradefutures in the US market WTI is the primary benchmark

for oil extracted and consumed in the US-eWTI crude oilfuture market is also the worldrsquos most liquid and largestcrude oil future market Nearly 12 million contracts trade aday with more than 2 million in open interest On the otherhand WTI future contracts have the longest available his-tory dating back to 1983 giving us a large sample size to gainpowers in detecting short-lived bubbles

Our sample period under investigation includes severalmajor exogenous events that to a large extent are reckonedto have influenced oil markets -ese contain the collapse ofOPEC in 1986 the Persian Gulf War of 199091 the Asianfinancial crisis of 199798 the terrorist attacks on September11 2001 the Venezuelan crisis and Iraq war of 200203 theincreased financialization of oil future markets since 2004the OPEC oil supply announcements in 2008 and 2014 the2019 novel coronavirus disease (COVID-19) and moststrikingly the 200708 global financial crisis Figure 2 plotsthe time series trajectories of the WTI nominal front-monthfuture price and the corresponding real price A look at thefigure illustrates that the oil prices grew steadily from thestarting of the sample until the early 2000s -e price seriessince then begun to surge and the sharp increase in the priceof oil continued until July 2008 As shown in Figure 2 itcollapsed to about 30 dollars by the end of 2008 Between2014 and 2016 the price of oil also experienced a crash Morerecently WTI crude price even went subzero for the firsttime in history A common view is that the steep rise anddecline in the price of oil in the 2000s could be attributed tospeculation

6 Empirical Results

61 Main Results We start the empirical illustration withbaseline model (14) expanding it to the endogeneity ap-plication later In order to formally examine the existenceand duration of unstable and explosive behavior in the crudeoil market we run our tests on the daily price data

8 Complexity

Following the recommendation of PSY the minimumwindow size set for the recursive test procedure is the first 5years of the entire sample Figure 3 plots the test empiricaloutcomes along with their corresponding 95 critical valuesequences Also plotted are the trajectories of test statisticsfor the random coefficient and the price data Some inter-esting results are shown in this figure

Looking at the recursive test results reported in the lastpanel of Figure 3 we observe that over the initial period ofobservation the null hypothesis of a time-invariant autor-egressive coefficient cannot be rejected at the 5 statisticalsignificance level -e test statistic experiences a dramaticincrease in the late 1980s and exceeds the 95 critical valuein November 1990 Overall the observed pattern of timevariation rises and becomes much stronger as the timeperiod expands -e evidence that the LBI test statistics forthe majority of samples are greater than the 95 criticalvalue suggests that the data-generating process is wellcharacterized by the model with time-varying coefficientsOne point worth noting is that the sequence falls in thesecond half of 2008 and slumps in early 2020 -e styled factseems to indicate that the crude oil market has undergonedrastic changes Interestingly these located dates coincidewith the 200708 global financial crisis and the COVID-19pandemic

We now turn to our focus on different types of unstableand explosive behavior-e shaded areas in green in Figure 3are the identified periods where the t-test statistic crosses its95 critical value As is evident from the first panel theprocedure detects two unstable episodes -e first unstableepisode is relatively short and only lasts two months(2005M08-M09) -e second unstable episode appears inJanuary 2006 and continues until March 2020 -ese un-stable periods might be associated with the financializationof oil future markets from 2004 It can be seen in the secondpanel of Figure 3 that the detected episodes of locally ex-plosive (submartingale) behavior are clustered in 2007-20082009ndash2014 and 2018 As stated above since local explo-siveness implies instability and displays more extreme be-havior it should lead to a much shorter duration of

exuberance than that of instability -e third panel ofFigure 3 reflects that the t-test statistic is below the 95critical value throughout the sample period -is points outthat explosive behavior in crude oil price series does notexist Table 1 summarizes time horizons for the three kindsof unstable and explosive behavior identified by ourapproach

To shed light on a more realistic scenario of data-gen-erating process we implement the extreme sample pathbehavior test by extending the model to the endogeneitycase Figure 4 plots the empirical test results together with95 critical values A comparison of the LR test statisticswith the critical value sequence provides novel evidence insupport of the presence of endogeneity during the wholesample period -us it is believed that endogeneity is es-sential to the oil price formation process Furthermore thelast panel of Figure 4 highlights that the quadratic varianceof the price process is related to the test statistic forendogeneity -is finding is in accordance with the fact thatwe have exploited the realized variance series to obtain theLR test statistics In this sense Figure 5 displays differencesin the realized variance estimates which are captured by thetest statistic for endogeneity employing different methodsSimilar to Figure 3 we again identify episodes of unstabledynamics and fail to detect explosive behavior when con-sidering the model with endogeneity In contrast the sta-tistic sequence for local explosiveness exhibits a differentpattern Previous evidence in favor of locally explosive be-havior disappeared Table 2 summarizes time horizons forthe three forms of unstable and explosive behavior -eseresults suggest that endogeneity feedbacks play a significantrole in investigating the existence of explosive price be-havior -e conclusion that emerges is that there are nospeculative bubbles in the crude oil market

In principal our findings are in accordance with theexplanations made by Kilian and Murphy [19] Baumeisterand Hamilton [21] and Knittel and Pindyck [54] that thevariations in crude oil prices are driven by economicfundamentals

62 Comparison to Alternative Strategies Rich literature hasstudied speculative bubbles in the crude oil market using thetechniques developed by PWY and then extended by PSY Abrief description of the SADF methodology (used in PWY)and the GSADF methodology (used in PSY) can be found inAppendix It is therefore helpful to compare results foralternative methods as the detection strategy in the presentpaper is quite different from them Figure 6 plots thebackward ADF and backward SADF statistic sequences andthe associated 95 critical value sequences for oil futureprices 95 percent critical values are taken from 2000 MonteCarlo simulations with the actual sample size -e initialwindow size is set to 5 years A glance at Figure 6 reveals thatexplosive behavior occurred mainly in two occasions usingPWY and PSY dating strategies -ey are before and during2007-2008 global recession We also observe that PWYdetects several short-lived episodes between 2011 and 2014but PSY fails to do so For daily oil future prices over April 4

0

20

ndash20

ndash40

40

60

80

100

120

140

84 86 88 90 92 94 96 98 00 02 04 06 08 10 12 14 16 18 20

Nominal WTI front-month future pricesReal WTI front-month future prices

Figure 2 Real and nominal prices of WTI front-month futurecontracts

Complexity 9

1983 to June 30 2020 time horizons for explosive behavioridentified by PWY and PSY are summarized in Table 3 Inline with previous studies we could conclude from bothPWY and PSY tests that there is evidence of bubbles in thecrude oil market data However as shown in Gurkaynak [37]and Pavlidis et al [43] explosive dynamics in asset prices perse do not constitute evidence either in support of or againstspeculative bubbles since such explosive behavior may

actually reflect explosive market fundamentals or bothfactors As aforementioned our random coefficient autor-egressive model framework has considered endogeneityfeedbacks that to some extent alleviate this problemMoreover the evidence in Figures 3 and 4 suggests theabsence of explosive dynamics in the crude oil market -efact permits us to safely ignore the distinction betweenbubbles or explosive fundamentals

1988

-03

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

ndash4

ndash2

0

2

4

6

8

0

20

40

60

80

100

120

140

t-statistic of k95 critical value of N (0 1)Crude oil future prices

Instability detection based on the time-varyingcoefficient model

(a)

Local explosiveness detection based on the time-varyingcoefficient model

1988

-03

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

t-statistic of μ95 critical value of N (0 1)Crude oil future prices

ndash4

ndash2

0

2

4

6

8

0

20

40

60

80

100

120

140

(b)

Explosiveness detection based on the time-varyingcoefficient model

1988

-03

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

t-statistic of ϕ95 critical value of N (0 1)Crude oil future prices

ndash4

ndash2

0

2

4

6

8

0

20

40

60

80

100

120

140

(c)

LBItest statistic for testing time-varying coefficient19

88-0

3

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

LBItest95 critical value of N (0 1)

ndash5

ndash10

0

5

10

15

20

(d)

Figure 3 Date-stamping unstable and explosive episodes in oil prices with the time-varying coefficient model assuming no endogeneity

Table 1 Time horizons of unstable and explosive periods identified by models assuming no endogeneity

Unstable Locally explosive ExplosiveAug 2005-Sep 2005 Dec 2007ndashSep 2008 mdash

Jan 2006ndashMar 2020 Jun 2009ndashOct 2014 mdashJun 2018ndashSep 2018 mdash

10 Complexity

63 Robustness of the Results In the following we providetwo robustness checks to assess whether our previousconclusions of the paper still hold

First in our benchmark model our use of front-monthfuture price instead of other future prices at different ma-turities is justified by its being the most liquid future contractwith the highest open interest To explore the sensitivity ofour results to the future contracts at different maturities werepeat the analysis described above using the prices forlonger-term contracts (ie 2 3 and 4 months-ahead fu-tures) It should be pointed out that the WTI future contractusually expires on the third business day before the 25thcalendar day of the contract month proceeding the deliverymonth

Second becauseWTI crude oil is by tradition traded innominal US dollars per barrel it might be of concern that

inflation and other monetary variables may have realeffects on oil price It is therefore of interest to use realrather than nominal price In fact recent work documentsthat the use of either price is inconsequential in practice[55] Although we mainly focus on nominal prices forrobustness check we shall report the results for realprices To convert nominal daily series into daily realseries we use linearly interpolated monthly ConsumerPrice Index (CPI) Monthly CPI series for the UnitedStates is obtained from the websites of the Federal ReserveBank of St Louis

Further evidence presented in Figures 7ndash10 demon-strates that our results are remarkably robust to thesemodifications and the conclusions remain qualitativelyunchanged

ndash4

ndash2

0

2

4

6

8

0

20

40

60

80

100

120

140

1988

-03

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

Instability detection based on the time-varyingcoefficient model

t-statistic of k95 critical value of N (0 1)Crude oil future prices

(a)

ndash4

ndash2

0

2

4

6

8

0

20

40

60

80

100

120

140

1988

-03

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

t-statistic of μ95 critical value of N (0 1)Crude oil future prices

Local explosiveness detection based on the time-varyingcoefficient model

(b)

ndash4

ndash2

0

2

4

6

8

0

20

40

60

80

100

120

140

1988

-03

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

Explosiveness detection based on the time-varyingcoefficient model

t-statistic of ϕ95 critical value of N (0 1)Crude oil future prices

(c)

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

LR-statistic95 critical values of x2 (1)Realized variance

700

600

500

400

300

200

100

0

ndash100

350

300

250

200

150

100

50

0

Test for endogeneity in time-varyingcoefficient model

(d)

Figure 4 Date-stamping unstable and explosive episodes in oil prices with the time-varying coefficient model with endogeneity

Complexity 11

2

3

4

5

6

7

8

0

50

100

150

200

250

300

350

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

LLF0

LLF1

Realized variance of crude oil future prices

Figure 5 Realized variance and likelihood values for random coefficient regression

Table 2 Time horizons of unstable and explosive periods detected by models with endogeneity

Unstable Locally explosive ExplosiveMay 2008ndashJul 2008 mdash mdashDec 2010ndashOct 2014 mdash mdashNov 2016ndashApr 2017 mdash mdashJul 2017ndashFeb 2020 mdash mdash

ndash4

ndash2

0

2

4

6

8

0

20

40

60

80

100

120

140

1988

-03

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

Explosiveness detection using SADF methodology

BADF sequence95 critical valuesCrude oil future prices

(a)

ndash4

ndash2

0

2

4

6

8

0

20

40

60

80

100

120

140

1988

-03

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

Explosiveness detection using GSADF methodology

BSADF sequence95 critical valuesCrude oil future prices

(b)

Figure 6 Date-stamping explosive episodes in oil prices with the time-invariant coefficient model

12 Complexity

Table 3 Time horizons of explosive behavior identified by SADF and GSADF methodologies

SADF GSADFAug 2005ndashAug 2006 Oct 2011ndashApr 2012 Sep 1990-Oct 1990Jun 2007ndashOct 2008 Aug 2012-Sep 2012 Feb 1991-Mar 1991Jun 2009-Jul 2009 Jul 2013-Aug 2013 Oct 2004-Nov 2004Feb 2011ndashApr 2011 Feb 2014-Mar 2014 Feb 2005-Mar 2005

Jul 2011ndashAug 2012 Jun 2014-Jul 2014May 2005ndashJan 2006Mar 2006ndashAug 2006Jul 2007ndashAug 2008

ndash4

ndash2

0

2

4

6

8

0

10

20

30

40

50

60

70

1988

-03

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

95 critical value of N (0 1)Real oil future prices

t-statistic of μ~

Local explosiveness detection based on the time-varyingcoefficient model

(a)

ndash4

ndash2

0

2

4

6

8

0

10

20

30

40

50

60

70

1988

-03

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

95 critical value of N (0 1)Real oil future prices

t-statistic of ϕ

Explosiveness detection based on the time-varyingcoefficient model

(b)

Figure 7 Date-stamping explosive episodes in real oil prices with the time-varying coefficient model with endogeneity

ndash4

ndash2

0

2

4

6

8

0

50

100

150

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

Local explosiveness detection based on the time-varyingcoefficient model

t-statistic of μ95 critical value of N (0 1)2-month crude oil future prices

(a)

ndash4

ndash2

0

2

4

6

8

0

50

100

150

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

Local explosiveness detection based on the time-varyingcoefficient model

t-statistic of ϕ95 critical value of N (0 1)2-month crude oil future prices

(b)

Figure 8 Date-stamping explosive episodes in 2-month future oil prices with the time-varying coefficient model with endogeneity

Complexity 13

7 Discussion

-e oil price rally since the early 2000s has attracted greatinterest in the sources of oil price shocks Some causal ob-servers have stressed that financial speculation has become asignificant driver of the dynamics of oil prices A large body ofliterature emerged that has sought to investigate its existenceIn this paper we employ a novel approach for testing forspeculative bubbles in oil prices In synthesis our empiricalfindings indicate that there is less evidence of exuberance in

the oil future markets Our results are very robust in severalscenarios We argue that the booms and busts in the oil pricemovements during the sample period are driven primarily byeconomic fundamentals rather than speculation

Overall our results confirm the main findings obtainedby structural VAR-based studies including Kilian andMurphy [19] Kilian and Lee [20] Baumeister and Hamilton[21] and Zhou [22] In the past decade the use of structuralVAR models to understand the evolution of the oil price hasbecome common -e traditional SVAR literature

ndash4

ndash2

0

2

4

6

8

0

50

100

15019

88-0

3

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

Local explosiveness detection based on the time-varyingcoefficient model

t-statistic of μ95 critical value of N (0 1)3-month crude oil future prices

(a)

ndash4

ndash2

0

2

4

6

8

0

50

100

150

1988

-03

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

Explosiveness detection based on the time-varyingcoefficient model

t-statistic of ϕ95 critical value of N (0 1)3-month crude oil future prices

(b)

Figure 9 Date-stamping explosive episodes in 3-month future oil prices with the time-varying coefficient model with endogeneity

ndash4

ndash2

0

2

4

6

8

0

50

100

150

Local explosiveness detection based on the time-varyingcoefficient model

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

t-statistic of μ95 critical value of N (0 1)4-month crude oil future prices

(a)

ndash4

ndash2

0

2

4

6

8

0

50

100

150

Explosiveness detection based on the time-varyingcoefficient model

t-statistic of ϕ95 critical value of N (0 1)4-month crude oil futures prices

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

(b)

Figure 10 Date-stamping explosive episodes in 4-month future oil prices with the time-varying coefficient model with endogeneity

14 Complexity

documents that the fundamental drivers of oil price fluc-tuations have been dominated essentially by supply anddemand shocks see eg Kilian [56] Lippi and Nobili [57]Kilian andMurphy [58] and Baumeister and Peersman [59]Kilian and Murphy [19] focused on the role speculation byaugmenting the SVAR framework of their earlier paper byglobal oil inventory data -ey specified a four-variablemodel for the global oil market (the percentage change inglobal oil production the real price of oil a suitable proxy ofcyclical variation in global real economic activity andchanges in global crude oil inventories) -is setting andtheir sign-identified scheme allow them to separately explainvariation in these data as four different types of shocks (1) ashock to the amount of oil pumped out of the ground (ldquooilsupply shockrdquo) (2) an oil flow demand shock (3) an oil-specific oil demand shock and especially (4) a speculative(or inventory) demand shock -eir core finding is that thespike in the real price of crude oil prices during 2003ndash2008was caused by unexpected increase in global oil consump-tion Using two alternative proxies for global above-groundcrude oil inventories Kilian and Lee [20] reestimated themodel of Kilian and Murphy [19] and ruled out the role ofspeculation in explaining the recent surge of the real oil priceon the basis of their estimation results Also Zhou [22]assessed the sensitivity of the findings in Kilian and Murphy[19] to the introduction of various state-of-the-art oil marketmethods She shows that the core conclusions reached byKilian and Murphy [19] are highly robust to some refine-ments corrections and extension of the KilianndashMurphymodel Baumeister and Hamilton [21] recently contendedthat prior information plays a key role in any SVAR modelsand traditional identification approaches can be generalizedto a special case of a Bayesian perspective -ey revisited theimportance of supply and demand shocks in the oil marketand demonstrated that speculative demand shock turns outto be only a small factor in accounting for most historical oilprice shocks

For comparison purpose we have also applied the testsof PWY and PSY which are the most popular econometrictechniques in the literature on speculative bubbles in the oilmarkets If we were to follow the test procedures of PWY andPSY our conclusions are also broadly consistent with therich literature that supports the existence of speculativebubbles in oil prices However these cointegration-basedbubble tests suffer from the joint hypothesis problem ldquoEverytest of a bubble is a joint test of the presence of the bubble inthe data and the validity of the model applied by theeconometricianrdquo (see Giglio et al [60]) More specifically themain drawback of such indirect model-dependent tests ofbubbles typically refers to the difficulty in distinguishingbetween the unknown true model for market fundamentalsand the presence of bubbles (see for example Gurkaynak[37] Pavlidis et al [42] Pavlidis et al [43]) Given thefundamental value of oil price is unobservable the literaturehas resorted to observe economic and financial variableswhich are used to estimate the intrinsic values For exampleboth Phillips and Yu [16] and Caspi et al [39] simplynormalized crude oil prices by US oil inventory data as thefundamental value -is measure is questionable since their

method cannot sufficiently capture the information onmarket fundamentals and neglect the major driversmdashlikeglobal economic activitymdashalso affecting the fundamentalvalue of crude oil -is issue of misspecified fundamentals(either in terms of model misspecification or omitted var-iables or both of these reasons) may make researchers fail toarrive at a conclusion from evidence of explosive pricesalone To make matters worse even though we have cap-tured the true market fundamentals we may still fail todetect bubbles if the explosive behavior in the price of oil juststems from the explosive fundamentals

Compared to earlier studies as briefly mentioned aboveour methodology has the main advantage that by consid-ering the dependence between the random coefficient andsystem shocks we have incorporated endogeneity into thecontinuous-time random coefficient setting and thus we donot need to prespecify the fundamental component of the oilprice -e test results for endogeneity suggest that endo-geneity is important in the data-generating process Inparticular our empirical analysis also finds that there are noexplosive episodes in crude oil markets-is implies that it isnot necessary to require the specification of a fundamentalvalue of the oil price Like Pavlidis et al [43] their methodsand ours have in common that the problem of the well-known joint hypothesis has to be some extent been ef-fectively ameliorated Although our approach is quite dis-tinct we draw the same substantive conclusion thatspeculative bubbles can be ruled out as a key determinant ofthe price of crude oil in the 2000s reaffirming the con-clusions of Kilian and Murphy [19] Baumeister andHamilton [21] Zhou [22] and Fattouh et al [61]

8 Conclusions

Repeated surges and crashes in crude oil prices during lastdecades have sparked a heated debate on their underlyingcauses among academics commentors and market regu-lators One of the central questions concerns whetherfinancialization of oil future markets creates a speculativebubble that subsequently collapses Answering this questionis difficult because one has to rely upon strong assumptionsabout the data-generating mechanism for the fundamentalvalue of the oil price In this paper we present a stylizedmodel of asset pricing for a commodity to address this issueOur theoretical framework disentangles the effects ofspeculative bubbles from the traditional fundamental effectson oil price dynamics We use this setting to illustrate thatbubbles if they are present should demonstrate explosivebehavior in prices We then provide a novel approach forbubble detection that covers various explosive scenarios andfind no evidence of speculative bubbles in oil markets evenduring episodes where the most popular SADF and GSADFmethodologies identify a speculative bubble

Our conclusions have pertinent implications for policymakers Given the historically important role of oil prices inthe real economy one of the serious concerns in policycircles over the past decade has been whether speculativetrading activities in oil future markets are responsible for thelarge price swings For example in their 2008 testimonies to

Complexity 15

the US Senate Commerce Committee Michael Masters andGeorge Soros posited that speculators had fueled price in-creases since 2003 and had caused a part of a ldquosuperbubblerdquoin 2007 and 2008 resulting in significantly higher energycosts for consumers -is widespread bubble view domi-nated policy discussions and efforts to limit speculativetrading in the future market quickly followed As a responseto the excessive speculation pressures on prices theDoddndashFrank Wall Street Reform and Consumer ProtectionAct was enacted by the US Congress in 2010 As a directpolicy implication our results highlight that imposingstricter regulations on trading in oil derivative marketscannot be expected to lower the oil price given the fact thatthe synchronized boom and bust cycles witnessed in the past20 years are mainly due to economic fundamentals whichappear likely to resurge with the business cycle -e aboveanalysis implies that there is no need to enforce additionalregulatory limits against financial arbitrage in the com-modity market

Appendix

The SADF and GSADF Methodologies

Appendix describes the SADF and GSADF testing proce-dures used to detect speculative bubbles in Section 62 For agenetic time series process xt consider the following rolling-window type augmented DickeyndashFuller (ADF) regressionequation

xt ar1r2+ μr1r2

xtminus 1 + 1113944

J

j1ϕj

r1r2Δxtminus j + εt εt sim NID 0 δ2r1r2

1113874 1113875

(A1)

where Δxtminus j with j 1 J are lagged first differences ofxt incorporated to accommodate potential serial correla-tion ar1 r2

μr1 r2 and ϕj

r1 r2 with j 1 J are regression

coefficients the subscripts r1 and r2 refer to fractions of thetotal sample size (ofT observations) that set the start and endpoints of a subsample period and NID represents inde-pendent and normal distribution -e ADF test statisticcorresponding to the unit root null hypothesisH0 μr1 r2

0 is simply the t-ratio

ADFr2r1

1113954μr1 r2

se 1113954μr1 r21113872 1113873

(A2)

and the right-side alternative hypothesis of explosive be-havior is H0 μr1r2

gt 0 We can obtain the standard ADF teststatistic by running regression (A1) on the whole sampleie by specifying r1 0 and r2 1 A large amount of re-search has demonstrated that the standard ADF test ADF10has extremely low power in detecting periodically collapsingbubbles

To address the problem PSW proposed a recursiveregression technique which involves estimating (A1)employing a forward expanding sample Specifically theyused subsamples of the available data to recursively calculatea sequence of ADF statistics denoted by ADFr2

0 -e startpoint of the sample period is fixed at r1 0 while the endpoint r2 increases from r0 (the minimum window size) toone (the full sample period) -e SADF statistic called thesupremum of ADFr2

0 is defined by

SADF r0( 1113857 supr2isin r0 1[ ]

ADFr20

(A3)

and has the following limit distribution under the nullhypothesis

supr2isin r0 1[ ]

1113938r2

0 WdW

1113938r2

0 W21113872 1113873(12)

(A4)

where W is the standard Brownian motion Testing thepresence of explosive behavior entails comparing the ADFr2

0statistic with the right-side critical values obtained throughMonte Carlo simulations

More recently PSY showed that when the sample perioddisplays more than one boom-bust bubble the SADF testprocedure may perform poorly and fail to detect or con-sistently date-stamp alternative bubbles -is deficiencymotivates PSY to formulate an extension of the SADF testthe GSADF test which covers more subsamples of the datathan the SADF by allowing both the end point r2 and thestart point r1 to vary As a result the GSADF test has moreflexibility and gains substantial power making this testconsistent with multiple boom-bust bubbles -e GSADFtest statistic is defined as

GSADF r0( 1113857 supr2isin r0 1[ ]r1isin 0r2minus r0[ ]

ADFr2r1 (A5)

with the limit distribution

supr2isin r0 1[ ]r1isin 0r2minus r0[ ]

(12)rw W r2( 11138572

minus W r1( 11138572

minus rw1113960 1113961 minus 1113938r2

r1W(r)dr W r2( 1113857 minus W r1( 11138571113858 1113859

r12w rw 1113938

r2

r1W(r)2dr minus 1113938

r2

r1W(r)dr1113876 1113877

21113896 1113897

12

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎭

(A6)

where rw r2 minus r1 is the expanding window size Like theSADF test comparison of the GASDF statistic with the right-

tailed critical values from its limit distribution enables us to testfor the unit root hypothesis against explosive behavior

16 Complexity

If the null hypothesis of a unit root in xt is rejectedthen in the second stage the SADF and GSADF tests canidentify the exact episodes during which the xt seriesexhibits explosive dynamics To locate the emergence andconclusion of explosiveness the SADF methodologymatches the sequence of the recursive backward ADF(BADF) test statistics against the right-tailed criticalvalues of the limit distribution of the standard Dick-eyndashFuller test statistic Suppose that re is the originationdate and rf is the finish date of the bubble in the data-ese estimates of dates can be constructed from

1113954re infr2isin r0 1[ ]

r2 ADFr20 gt cv

αr2

1113966 1113967

1113954rf infr2isin 1113954re11113858 1113859

r2 ADFr20 lt cv

αr2

1113966 1113967(A7)

where cvαr2 is 100(1 minus α) the right-tailed critical value of theADF statistic corresponding to the chosen significance levelof α For the GSADF methodology the date-stampingstrategy in this case is based on the backward SADF(BSADF) statistic

BSADFr2r0( 1113857 sup

r1isin 0r2minus r0[ ]ADFr2

r1

(A8)

Likewise the origination date of the bubble is defined asthe first observation that the BSADF statistic is greater thanits critical value

1113954re infr2isin r0 1[ ]

r2 BSADFr2r0( 1113857gt cv

βT

r21113966 1113967 (A9)

and the finish date as the first date after which the BSADFdrops back below its critical value

1113954rf infr2isin 1113954re11113858 1113859

r2 BSADFr2r0( 1113857lt cvβT

r21113966 1113967 (A10)

where cvβTr2 is 100(1 minus βT) the right-tailed critical value of

the SADF based on Tr2 and the corresponding significancelevel is βT

Data Availability

-e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

-e authors declare no conflicts of interest

Acknowledgments

-e authors gratefully acknowledge the financial supportfrom the National Social Science Foundation of China underGrant no 18BTJ039

References

[1] M W Masters Testimony before the US Senate Committee onHomeland Security and Governmental Affairs 2008 httpswwwhsgacsenategovimomediadoc052008Masterspdfattempt2

[2] M W Masters Testimony before the Commodity FuturesTrading Commission 2010 httpscftcgovsitesdefaultfilesidcgroupspublicnewsroomdocumentsfilemetalmarkets032510_masterspdf

[3] J P Kennedy ldquo-e high cost of gambling on oilrdquo Section A p23 New York Times New York NY USANew York Times2012

[4] I-H Cheng and W Xiong ldquoFinancialization of commoditymarketsrdquo Annual Review of Financial Economics vol 6 no 1pp 419ndash441 2014

[5] K Tang and W Xiong ldquoIndex investment and the financi-alization of commoditiesrdquo Financial Analysts Journal vol 68no 6 pp 54ndash74 2012

[6] O J Blanchard and M W Watson ldquoBubbles rational ex-pectations and financial marketsrdquo in Crisis in the Economicand Financial Structure P Wachtel Ed Lexington BooksLexington MA USA pp 295ndash315 1982

[7] B T Diba and H I Grossman ldquoOn the inception of rationalbubblesrdquo e Quarterly Journal of Economics vol 102 no 3pp 697ndash700 1987

[8] G W Evans ldquo-e fragility of sunspots and bubblesrdquo Journalof Monetary Economics vol 23 no 2 pp 297ndash317 1989

[9] G W Evans and S Honkapohja ldquoOn the robustness ofbubbles in linear RE modelsrdquo International Economic Reviewvol 33 no 1 pp 1ndash14 1992

[10] E Pavlidis A Yusupova I Paya et al ldquoEpisodes of exu-berance in housing markets in search of the smoking gunrdquoe Journal of Real Estate Finance and Economics vol 53no 4 pp 419ndash449 2016

[11] E Martınez-Garcıa and V Grossman ldquoExplosive dynamics inhouse prices An exploration of financial market spillovers inhousing markets around the worldrdquo Journal of InternationalMoney and Finance vol 101 p 102103 2020

[12] S Shi A Rahman and B Z Wang ldquoAustralian housingmarket booms fundamentals or speculationlowastrdquo EconomicRecord 2020 In press

[13] P C B Phillips Y Wu and J Yu ldquoExplosive behavior in the1990s NASDAQ when did exuberance escalate asset val-ueslowastrdquo International Economic Review vol 52 no 1pp 201ndash226 2011

[14] P C B Phillips S Shi and J Yu ldquoTesting for multiplebubbles historical episodes of exuberance and collapse in theSampp 500rdquo International Economic Review vol 56 no 4pp 1043ndash1078 2015

[15] P C B Phillips S Shi and J Yu ldquoTesting for multiplebubbles limit theory of real-time detectorsrdquo InternationalEconomic Review vol 56 no 4 pp 1079ndash1134 2015

[16] P C B Phillips and J Yu ldquoDating the timeline of financialbubbles during the subprime crisisrdquo Quantitative Economicsvol 2 no 3 pp 455ndash491 2011

[17] D Tsvetanov J Coakley and N Kellard ldquoBubbling over thebehaviour of oil futures along the yield curverdquo Journal ofEmpirical Finance vol 38 no Part B pp 516ndash533 2016

[18] I Figuerola-Ferretti J R Mccrorie and I ParaskevopoulosldquoMild explosivity in recent crude oil pricesrdquo Energy Eco-nomics vol 87 p 104387 2020

[19] L Kilian and D P Murphy ldquo-e role of inventories andspeculative trading in the global market for crude oilrdquo Journalof Applied Econometrics vol 29 no 3 pp 454ndash478 2014

Complexity 17

1113957ZN ≔1113936

MtimesNt1 1113957ε2tΔ minus (1MN)1113936

MtimesNt1 1113957ε2tΔ1113872 1113873

21113874 11138751113957x

2(tminus 1)Δ

(1MN)1113936MtimesNt1 1113957ε4tΔ minus (1MN)1113936

MtimesNt1 1113957ε2tΔ1113872 1113873

21113969

(1MN)1113936MtimesNt1 1113957x

4(tminus 1)Δ minus (1MN)1113936

MtimesNt1 1113957x

2(tminus 1)Δ1113872 1113873

21113969 (26)

where 1113957x2tΔ (xtΔ

1 + x2

tΔ

1113969) 1113957εtΔ xtΔ minus 1113957ρx(tminus 1)Δ and

1113957ρ (1113936MtimesNt1 1113957x(tminus 1)Δx(tminus 1)Δ)

minus 11113936MtimesNt1 1113957x(tminus 1)Δx(tminus 1)Δ -us as

N⟶infin MNminus 12 1113957ZN⟶L N(0 1) under H0 1113957σ2 0 andMNminus (12) 1113957ZN⟶L N(0 1) under H0 1113957σ2 0

Relying on asymptotics with in-fill (as the sampling fre-quency tends to zeroΔ⟶ 0) and long time-span asymptoticsTao et al [25] developed new limit theory for the continuous-time system where the persistence parameter is random -easymptotic properties of the diffusion parameter and drift pa-rameter estimates can be used to construct test statistics for threetypes of unstableexplosive behavior which are

tκ 1Δ

1113944

MtimesN

t1

x2(tminus 1)Δ

11139541113957σ2x2(tminus 1)Δ + σ2

⎛⎝ ⎞⎠

12

1113954βκ minus β0κ1113872 1113873⟶L

N(0 1)

tμ 1Δ

1113944

MtimesN

t1

x2(tminus 1)Δ

11139541113957σ2x2(tminus 1)Δ + 1113954σ2

⎛⎝ ⎞⎠

12

1113954ρ minus ρ0( 1113857⟶L

N(0 1)

tϕ 1Δ

1113944

MtimesN

t1

x2(tminus 1)Δ

11139541113957σ2x2(tminus 1)Δ + 1113954σ2

⎛⎝ ⎞⎠

12

1113954βϕ minus β0ϕ1113872 1113873⟶L

N(0 1)

(27)

where

1113954βκ exp 1113954μ +12

11139541113957σ21113874 1113875Δ1113882 1113883

1113954βϕ exp 1113954μ minus12

11139541113957σ21113874 1113875Δ1113882 1113883

β0κ exp μ0 +12μ201113874 1113875Δ1113882 1113883

β0ϕ exp μ0 minus12μ201113874 1113875Δ1113882 1113883

1113954ρ 1 +1113936

MtimesNt1 x(tminus 1)Δ xtΔ minus x(tminus 1)Δ1113872 1113873 1113954

1113957σ2x2(tminus 1)Δ +

1113954σ21113874 1113875

1113936MtimesNt1 x

2(tminus 1)Δ

11139541113957σ2x2

(tminus 1)Δ +1113954σ21113874 1113875

ρ0 1

(28)

As can be seen this test procedure does not directlyverify whether or not κ (μ + 1113957σ22) 0 or μ 0 orϕ (μ minus 1113957σ22) 0 but transforms the null hypotheses toβk 1 or ρ 1 or βϕ 1 -ese three t-test statistics areaccompanied by a date-stamping strategy that permits us toidentify the exact origination and termination periods ofprice exuberance Given this instrument estimates of thesedates are as follows

1113954ribun inf

sisin 1113954r(iminus 1)e

un 11113858 1113859s tκ(s)gtZα1113864 1113865

1113954rieun inf

sisin 1113954r(iminus 1)b

un 11113858 1113859s tκ(s)ltZα1113864 1113865

1113954rible inf

sisin 1113954r(iminus 1)e

le 11113858 1113859s tμ(s)gtZα1113966 1113967

1113954rieun inf

sisin 1113954r(iminus 1)b

le 11113858 1113859s tμ(s)ltZα1113966 1113967

1113954ribe inf

sisin 1113954r(iminus 1)e

e 11113858 1113859s tϕ(s)gtZα1113966 1113967

1113954rieun inf

sisin 1113954r(iminus 1)b

e 11113858 1113859s tϕ(s)ltZα1113966 1113967

(29)

where Zα is the right-tailed critical value of the standardnormal distribution corresponding to a significance level ofα 1113954rib

un1113954rible1113954r

ibe denote the beginnings on the dates of the ith

extreme price behavior period and similarly 1113954rieun1113954r

iele1113954r

iee

denote the ends on the dates of the ith extreme price be-havior period

43 Extend the Model with Endogeneity Now we are con-cerned with the endogeneity case where Wμ(t) is not in-dependent of Wε(t) To see this let us suppose that data aregenerated from the following stochastic differential equationsystem

dx(t) x(t)d1113957B(t) + σdB(t) x(0) x0

d1113957B(t) μ dt + 1113957σWμ(t)

dB(t) σdWε(t)

(30)

where (Wμ(t) Wε(t)) is a two-dimensional Brownianmotion with covariance parameter η As Follmer [53] notedthe strong solution obtained from this continuous systemwould be of the explicit form

dx(t) exp 1113957σWμ(t) + μ minus12

1113957σ21113874 1113875t1113874 1113875x(0) + Z(t) (31)

where

Z(t) σ 1113946t

0exp 1113957σ Wμ(t) minus Wμ(s)1113872 1113873 + μ minus

12

1113957σ21113874 1113875(t minus s)1113874 1113875dWε(t)

minus ω1113946t

0exp 1113957σ Wμ(t) minus Wμ(s)1113872 1113873 + μ minus

12

1113957σ21113874 1113875(t minus s)1113874 1113875ds

V(t) minus P(t)

(32)

Notice that when endogeneity is included in model (14)the driver process becomes Z(t) as a substitute for V(t)Z(t) consists of a component V(t) and another oneconnected with the covariance parameter of (1113957B tB) ω and

Complexity 7

variance of V(t) -e strong solution gives the exact dis-crete-time form of model (31)

xtΔ exp 1113957σΔ

radicuΔ + μ minus

12

1113957σ21113874 1113875Δ1113882 1113883x(tminus 1)Δ + ZtΔ ρtΔ x(tminus 1)Δ + ZtΔ

+ σ 1113946tΔ

(tminus 1)Δexp 1113957σ WμtΔ minus Wμs1113872 1113873 + μ minus

12

1113957σ21113874 1113875(tΔ minus s)1113882 1113883dWεs

(33)

where uiidsim

t N(0 1) and

ZtΔ σ 1113946tΔ

(tminus 1)Δexp 1113957σ WμtΔ minus Wμs1113872 1113873 + μ minus

12

1113957σ21113874 1113875(tΔ minus s)1113882 1113883dWεs

minus ω1113946tΔ

(tminus 1)Δexp 1113957σ WμtΔ minus Wμs1113872 11138731113872 1113873 + μ minus

12

1113957σ21113874 1113875(tΔ minus s)1113874 1113875ds

(34)

Somewhat surprisingly Tao et al [25] indicated thatendogeneity would not affect the sample path features ofmodel (31) Our analysis therefore is the same as before

(1) Unstable κ μ + 1113957σ22gt 0(2) Locally explosive μgt 0(3) Explosive ϕ μ minus 1113957σ22gt 0

We can estimate 1113954θlowast ≔ ( 11139541113957σ2 1113954η 1113954σ2)prime in the same fashion asbefore -e likelihood ratio test (LR test) has been proposedby Tao et al [25] to test the null hypothesis of endogeneityinterest

LR Δminus 1Q

rΔ minus Q

urΔ( 1113857simχ2(1) underH0 η0 0 (35)

Tao et al [25] also developed the asymptotic theory for 1113954μand 1113954ρ ≔ exp(1113954μΔ) Based on their limit properties the es-timates of 1113954μ and 1113954ρ continue to follow asymptotic normaldistributions -is good quality enables us to convenientlyadopt the aforementioned testing procedures after making asmall adjustment to the variance of the limit distribution

5 Data

For our main empirical analysis we apply daily WTI crudeoil prices for the front-month (one-month) future contracton the New York Mercantile Exchange (NYMEX) -e dataare retrieved from the Energy Information Administration(EIA) Our sample spans the period from April 4 1983 toJune 30 2020 and comprises a total of 9352 daily obser-vations -e beginning and end dates of the sample periodare dictated by the availability of the data

It is well known that WIT and Brent are grades of in-ternationally traded crude oil that are widely used asbenchmarks in oil pricing globally -e reason for this paperrelying on the WTI price of crude oil is twofold On the onehand oil speculation is often blamed on people who tradefutures in the US market WTI is the primary benchmark

for oil extracted and consumed in the US-eWTI crude oilfuture market is also the worldrsquos most liquid and largestcrude oil future market Nearly 12 million contracts trade aday with more than 2 million in open interest On the otherhand WTI future contracts have the longest available his-tory dating back to 1983 giving us a large sample size to gainpowers in detecting short-lived bubbles

Our sample period under investigation includes severalmajor exogenous events that to a large extent are reckonedto have influenced oil markets -ese contain the collapse ofOPEC in 1986 the Persian Gulf War of 199091 the Asianfinancial crisis of 199798 the terrorist attacks on September11 2001 the Venezuelan crisis and Iraq war of 200203 theincreased financialization of oil future markets since 2004the OPEC oil supply announcements in 2008 and 2014 the2019 novel coronavirus disease (COVID-19) and moststrikingly the 200708 global financial crisis Figure 2 plotsthe time series trajectories of the WTI nominal front-monthfuture price and the corresponding real price A look at thefigure illustrates that the oil prices grew steadily from thestarting of the sample until the early 2000s -e price seriessince then begun to surge and the sharp increase in the priceof oil continued until July 2008 As shown in Figure 2 itcollapsed to about 30 dollars by the end of 2008 Between2014 and 2016 the price of oil also experienced a crash Morerecently WTI crude price even went subzero for the firsttime in history A common view is that the steep rise anddecline in the price of oil in the 2000s could be attributed tospeculation

6 Empirical Results

61 Main Results We start the empirical illustration withbaseline model (14) expanding it to the endogeneity ap-plication later In order to formally examine the existenceand duration of unstable and explosive behavior in the crudeoil market we run our tests on the daily price data

8 Complexity

Following the recommendation of PSY the minimumwindow size set for the recursive test procedure is the first 5years of the entire sample Figure 3 plots the test empiricaloutcomes along with their corresponding 95 critical valuesequences Also plotted are the trajectories of test statisticsfor the random coefficient and the price data Some inter-esting results are shown in this figure

Looking at the recursive test results reported in the lastpanel of Figure 3 we observe that over the initial period ofobservation the null hypothesis of a time-invariant autor-egressive coefficient cannot be rejected at the 5 statisticalsignificance level -e test statistic experiences a dramaticincrease in the late 1980s and exceeds the 95 critical valuein November 1990 Overall the observed pattern of timevariation rises and becomes much stronger as the timeperiod expands -e evidence that the LBI test statistics forthe majority of samples are greater than the 95 criticalvalue suggests that the data-generating process is wellcharacterized by the model with time-varying coefficientsOne point worth noting is that the sequence falls in thesecond half of 2008 and slumps in early 2020 -e styled factseems to indicate that the crude oil market has undergonedrastic changes Interestingly these located dates coincidewith the 200708 global financial crisis and the COVID-19pandemic

We now turn to our focus on different types of unstableand explosive behavior-e shaded areas in green in Figure 3are the identified periods where the t-test statistic crosses its95 critical value As is evident from the first panel theprocedure detects two unstable episodes -e first unstableepisode is relatively short and only lasts two months(2005M08-M09) -e second unstable episode appears inJanuary 2006 and continues until March 2020 -ese un-stable periods might be associated with the financializationof oil future markets from 2004 It can be seen in the secondpanel of Figure 3 that the detected episodes of locally ex-plosive (submartingale) behavior are clustered in 2007-20082009ndash2014 and 2018 As stated above since local explo-siveness implies instability and displays more extreme be-havior it should lead to a much shorter duration of

exuberance than that of instability -e third panel ofFigure 3 reflects that the t-test statistic is below the 95critical value throughout the sample period -is points outthat explosive behavior in crude oil price series does notexist Table 1 summarizes time horizons for the three kindsof unstable and explosive behavior identified by ourapproach

To shed light on a more realistic scenario of data-gen-erating process we implement the extreme sample pathbehavior test by extending the model to the endogeneitycase Figure 4 plots the empirical test results together with95 critical values A comparison of the LR test statisticswith the critical value sequence provides novel evidence insupport of the presence of endogeneity during the wholesample period -us it is believed that endogeneity is es-sential to the oil price formation process Furthermore thelast panel of Figure 4 highlights that the quadratic varianceof the price process is related to the test statistic forendogeneity -is finding is in accordance with the fact thatwe have exploited the realized variance series to obtain theLR test statistics In this sense Figure 5 displays differencesin the realized variance estimates which are captured by thetest statistic for endogeneity employing different methodsSimilar to Figure 3 we again identify episodes of unstabledynamics and fail to detect explosive behavior when con-sidering the model with endogeneity In contrast the sta-tistic sequence for local explosiveness exhibits a differentpattern Previous evidence in favor of locally explosive be-havior disappeared Table 2 summarizes time horizons forthe three forms of unstable and explosive behavior -eseresults suggest that endogeneity feedbacks play a significantrole in investigating the existence of explosive price be-havior -e conclusion that emerges is that there are nospeculative bubbles in the crude oil market

In principal our findings are in accordance with theexplanations made by Kilian and Murphy [19] Baumeisterand Hamilton [21] and Knittel and Pindyck [54] that thevariations in crude oil prices are driven by economicfundamentals

62 Comparison to Alternative Strategies Rich literature hasstudied speculative bubbles in the crude oil market using thetechniques developed by PWY and then extended by PSY Abrief description of the SADF methodology (used in PWY)and the GSADF methodology (used in PSY) can be found inAppendix It is therefore helpful to compare results foralternative methods as the detection strategy in the presentpaper is quite different from them Figure 6 plots thebackward ADF and backward SADF statistic sequences andthe associated 95 critical value sequences for oil futureprices 95 percent critical values are taken from 2000 MonteCarlo simulations with the actual sample size -e initialwindow size is set to 5 years A glance at Figure 6 reveals thatexplosive behavior occurred mainly in two occasions usingPWY and PSY dating strategies -ey are before and during2007-2008 global recession We also observe that PWYdetects several short-lived episodes between 2011 and 2014but PSY fails to do so For daily oil future prices over April 4

0

20

ndash20

ndash40

40

60

80

100

120

140

84 86 88 90 92 94 96 98 00 02 04 06 08 10 12 14 16 18 20

Nominal WTI front-month future pricesReal WTI front-month future prices

Figure 2 Real and nominal prices of WTI front-month futurecontracts

Complexity 9

1983 to June 30 2020 time horizons for explosive behavioridentified by PWY and PSY are summarized in Table 3 Inline with previous studies we could conclude from bothPWY and PSY tests that there is evidence of bubbles in thecrude oil market data However as shown in Gurkaynak [37]and Pavlidis et al [43] explosive dynamics in asset prices perse do not constitute evidence either in support of or againstspeculative bubbles since such explosive behavior may

actually reflect explosive market fundamentals or bothfactors As aforementioned our random coefficient autor-egressive model framework has considered endogeneityfeedbacks that to some extent alleviate this problemMoreover the evidence in Figures 3 and 4 suggests theabsence of explosive dynamics in the crude oil market -efact permits us to safely ignore the distinction betweenbubbles or explosive fundamentals

1988

-03

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

ndash4

ndash2

0

2

4

6

8

0

20

40

60

80

100

120

140

t-statistic of k95 critical value of N (0 1)Crude oil future prices

Instability detection based on the time-varyingcoefficient model

(a)

Local explosiveness detection based on the time-varyingcoefficient model

1988

-03

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

t-statistic of μ95 critical value of N (0 1)Crude oil future prices

ndash4

ndash2

0

2

4

6

8

0

20

40

60

80

100

120

140

(b)

Explosiveness detection based on the time-varyingcoefficient model

1988

-03

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

t-statistic of ϕ95 critical value of N (0 1)Crude oil future prices

ndash4

ndash2

0

2

4

6

8

0

20

40

60

80

100

120

140

(c)

LBItest statistic for testing time-varying coefficient19

88-0

3

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

LBItest95 critical value of N (0 1)

ndash5

ndash10

0

5

10

15

20

(d)

Figure 3 Date-stamping unstable and explosive episodes in oil prices with the time-varying coefficient model assuming no endogeneity

Table 1 Time horizons of unstable and explosive periods identified by models assuming no endogeneity

Unstable Locally explosive ExplosiveAug 2005-Sep 2005 Dec 2007ndashSep 2008 mdash

Jan 2006ndashMar 2020 Jun 2009ndashOct 2014 mdashJun 2018ndashSep 2018 mdash

10 Complexity

63 Robustness of the Results In the following we providetwo robustness checks to assess whether our previousconclusions of the paper still hold

First in our benchmark model our use of front-monthfuture price instead of other future prices at different ma-turities is justified by its being the most liquid future contractwith the highest open interest To explore the sensitivity ofour results to the future contracts at different maturities werepeat the analysis described above using the prices forlonger-term contracts (ie 2 3 and 4 months-ahead fu-tures) It should be pointed out that the WTI future contractusually expires on the third business day before the 25thcalendar day of the contract month proceeding the deliverymonth

Second becauseWTI crude oil is by tradition traded innominal US dollars per barrel it might be of concern that

inflation and other monetary variables may have realeffects on oil price It is therefore of interest to use realrather than nominal price In fact recent work documentsthat the use of either price is inconsequential in practice[55] Although we mainly focus on nominal prices forrobustness check we shall report the results for realprices To convert nominal daily series into daily realseries we use linearly interpolated monthly ConsumerPrice Index (CPI) Monthly CPI series for the UnitedStates is obtained from the websites of the Federal ReserveBank of St Louis

Further evidence presented in Figures 7ndash10 demon-strates that our results are remarkably robust to thesemodifications and the conclusions remain qualitativelyunchanged

ndash4

ndash2

0

2

4

6

8

0

20

40

60

80

100

120

140

1988

-03

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

Instability detection based on the time-varyingcoefficient model

t-statistic of k95 critical value of N (0 1)Crude oil future prices

(a)

ndash4

ndash2

0

2

4

6

8

0

20

40

60

80

100

120

140

1988

-03

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

t-statistic of μ95 critical value of N (0 1)Crude oil future prices

Local explosiveness detection based on the time-varyingcoefficient model

(b)

ndash4

ndash2

0

2

4

6

8

0

20

40

60

80

100

120

140

1988

-03

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

Explosiveness detection based on the time-varyingcoefficient model

t-statistic of ϕ95 critical value of N (0 1)Crude oil future prices

(c)

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

LR-statistic95 critical values of x2 (1)Realized variance

700

600

500

400

300

200

100

0

ndash100

350

300

250

200

150

100

50

0

Test for endogeneity in time-varyingcoefficient model

(d)

Figure 4 Date-stamping unstable and explosive episodes in oil prices with the time-varying coefficient model with endogeneity

Complexity 11

2

3

4

5

6

7

8

0

50

100

150

200

250

300

350

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

LLF0

LLF1

Realized variance of crude oil future prices

Figure 5 Realized variance and likelihood values for random coefficient regression

Table 2 Time horizons of unstable and explosive periods detected by models with endogeneity

Unstable Locally explosive ExplosiveMay 2008ndashJul 2008 mdash mdashDec 2010ndashOct 2014 mdash mdashNov 2016ndashApr 2017 mdash mdashJul 2017ndashFeb 2020 mdash mdash

ndash4

ndash2

0

2

4

6

8

0

20

40

60

80

100

120

140

1988

-03

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

Explosiveness detection using SADF methodology

BADF sequence95 critical valuesCrude oil future prices

(a)

ndash4

ndash2

0

2

4

6

8

0

20

40

60

80

100

120

140

1988

-03

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

Explosiveness detection using GSADF methodology

BSADF sequence95 critical valuesCrude oil future prices

(b)

Figure 6 Date-stamping explosive episodes in oil prices with the time-invariant coefficient model

12 Complexity

Table 3 Time horizons of explosive behavior identified by SADF and GSADF methodologies

SADF GSADFAug 2005ndashAug 2006 Oct 2011ndashApr 2012 Sep 1990-Oct 1990Jun 2007ndashOct 2008 Aug 2012-Sep 2012 Feb 1991-Mar 1991Jun 2009-Jul 2009 Jul 2013-Aug 2013 Oct 2004-Nov 2004Feb 2011ndashApr 2011 Feb 2014-Mar 2014 Feb 2005-Mar 2005

Jul 2011ndashAug 2012 Jun 2014-Jul 2014May 2005ndashJan 2006Mar 2006ndashAug 2006Jul 2007ndashAug 2008

ndash4

ndash2

0

2

4

6

8

0

10

20

30

40

50

60

70

1988

-03

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

95 critical value of N (0 1)Real oil future prices

t-statistic of μ~

Local explosiveness detection based on the time-varyingcoefficient model

(a)

ndash4

ndash2

0

2

4

6

8

0

10

20

30

40

50

60

70

1988

-03

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

95 critical value of N (0 1)Real oil future prices

t-statistic of ϕ

Explosiveness detection based on the time-varyingcoefficient model

(b)

Figure 7 Date-stamping explosive episodes in real oil prices with the time-varying coefficient model with endogeneity

ndash4

ndash2

0

2

4

6

8

0

50

100

150

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

Local explosiveness detection based on the time-varyingcoefficient model

t-statistic of μ95 critical value of N (0 1)2-month crude oil future prices

(a)

ndash4

ndash2

0

2

4

6

8

0

50

100

150

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

Local explosiveness detection based on the time-varyingcoefficient model

t-statistic of ϕ95 critical value of N (0 1)2-month crude oil future prices

(b)

Figure 8 Date-stamping explosive episodes in 2-month future oil prices with the time-varying coefficient model with endogeneity

Complexity 13

7 Discussion

-e oil price rally since the early 2000s has attracted greatinterest in the sources of oil price shocks Some causal ob-servers have stressed that financial speculation has become asignificant driver of the dynamics of oil prices A large body ofliterature emerged that has sought to investigate its existenceIn this paper we employ a novel approach for testing forspeculative bubbles in oil prices In synthesis our empiricalfindings indicate that there is less evidence of exuberance in

the oil future markets Our results are very robust in severalscenarios We argue that the booms and busts in the oil pricemovements during the sample period are driven primarily byeconomic fundamentals rather than speculation

Overall our results confirm the main findings obtainedby structural VAR-based studies including Kilian andMurphy [19] Kilian and Lee [20] Baumeister and Hamilton[21] and Zhou [22] In the past decade the use of structuralVAR models to understand the evolution of the oil price hasbecome common -e traditional SVAR literature

ndash4

ndash2

0

2

4

6

8

0

50

100

15019

88-0

3

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

Local explosiveness detection based on the time-varyingcoefficient model

t-statistic of μ95 critical value of N (0 1)3-month crude oil future prices

(a)

ndash4

ndash2

0

2

4

6

8

0

50

100

150

1988

-03

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

Explosiveness detection based on the time-varyingcoefficient model

t-statistic of ϕ95 critical value of N (0 1)3-month crude oil future prices

(b)

Figure 9 Date-stamping explosive episodes in 3-month future oil prices with the time-varying coefficient model with endogeneity

ndash4

ndash2

0

2

4

6

8

0

50

100

150

Local explosiveness detection based on the time-varyingcoefficient model

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

t-statistic of μ95 critical value of N (0 1)4-month crude oil future prices

(a)

ndash4

ndash2

0

2

4

6

8

0

50

100

150

Explosiveness detection based on the time-varyingcoefficient model

t-statistic of ϕ95 critical value of N (0 1)4-month crude oil futures prices

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

(b)

Figure 10 Date-stamping explosive episodes in 4-month future oil prices with the time-varying coefficient model with endogeneity

14 Complexity

documents that the fundamental drivers of oil price fluc-tuations have been dominated essentially by supply anddemand shocks see eg Kilian [56] Lippi and Nobili [57]Kilian andMurphy [58] and Baumeister and Peersman [59]Kilian and Murphy [19] focused on the role speculation byaugmenting the SVAR framework of their earlier paper byglobal oil inventory data -ey specified a four-variablemodel for the global oil market (the percentage change inglobal oil production the real price of oil a suitable proxy ofcyclical variation in global real economic activity andchanges in global crude oil inventories) -is setting andtheir sign-identified scheme allow them to separately explainvariation in these data as four different types of shocks (1) ashock to the amount of oil pumped out of the ground (ldquooilsupply shockrdquo) (2) an oil flow demand shock (3) an oil-specific oil demand shock and especially (4) a speculative(or inventory) demand shock -eir core finding is that thespike in the real price of crude oil prices during 2003ndash2008was caused by unexpected increase in global oil consump-tion Using two alternative proxies for global above-groundcrude oil inventories Kilian and Lee [20] reestimated themodel of Kilian and Murphy [19] and ruled out the role ofspeculation in explaining the recent surge of the real oil priceon the basis of their estimation results Also Zhou [22]assessed the sensitivity of the findings in Kilian and Murphy[19] to the introduction of various state-of-the-art oil marketmethods She shows that the core conclusions reached byKilian and Murphy [19] are highly robust to some refine-ments corrections and extension of the KilianndashMurphymodel Baumeister and Hamilton [21] recently contendedthat prior information plays a key role in any SVAR modelsand traditional identification approaches can be generalizedto a special case of a Bayesian perspective -ey revisited theimportance of supply and demand shocks in the oil marketand demonstrated that speculative demand shock turns outto be only a small factor in accounting for most historical oilprice shocks

For comparison purpose we have also applied the testsof PWY and PSY which are the most popular econometrictechniques in the literature on speculative bubbles in the oilmarkets If we were to follow the test procedures of PWY andPSY our conclusions are also broadly consistent with therich literature that supports the existence of speculativebubbles in oil prices However these cointegration-basedbubble tests suffer from the joint hypothesis problem ldquoEverytest of a bubble is a joint test of the presence of the bubble inthe data and the validity of the model applied by theeconometricianrdquo (see Giglio et al [60]) More specifically themain drawback of such indirect model-dependent tests ofbubbles typically refers to the difficulty in distinguishingbetween the unknown true model for market fundamentalsand the presence of bubbles (see for example Gurkaynak[37] Pavlidis et al [42] Pavlidis et al [43]) Given thefundamental value of oil price is unobservable the literaturehas resorted to observe economic and financial variableswhich are used to estimate the intrinsic values For exampleboth Phillips and Yu [16] and Caspi et al [39] simplynormalized crude oil prices by US oil inventory data as thefundamental value -is measure is questionable since their

method cannot sufficiently capture the information onmarket fundamentals and neglect the major driversmdashlikeglobal economic activitymdashalso affecting the fundamentalvalue of crude oil -is issue of misspecified fundamentals(either in terms of model misspecification or omitted var-iables or both of these reasons) may make researchers fail toarrive at a conclusion from evidence of explosive pricesalone To make matters worse even though we have cap-tured the true market fundamentals we may still fail todetect bubbles if the explosive behavior in the price of oil juststems from the explosive fundamentals

Compared to earlier studies as briefly mentioned aboveour methodology has the main advantage that by consid-ering the dependence between the random coefficient andsystem shocks we have incorporated endogeneity into thecontinuous-time random coefficient setting and thus we donot need to prespecify the fundamental component of the oilprice -e test results for endogeneity suggest that endo-geneity is important in the data-generating process Inparticular our empirical analysis also finds that there are noexplosive episodes in crude oil markets-is implies that it isnot necessary to require the specification of a fundamentalvalue of the oil price Like Pavlidis et al [43] their methodsand ours have in common that the problem of the well-known joint hypothesis has to be some extent been ef-fectively ameliorated Although our approach is quite dis-tinct we draw the same substantive conclusion thatspeculative bubbles can be ruled out as a key determinant ofthe price of crude oil in the 2000s reaffirming the con-clusions of Kilian and Murphy [19] Baumeister andHamilton [21] Zhou [22] and Fattouh et al [61]

8 Conclusions

Repeated surges and crashes in crude oil prices during lastdecades have sparked a heated debate on their underlyingcauses among academics commentors and market regu-lators One of the central questions concerns whetherfinancialization of oil future markets creates a speculativebubble that subsequently collapses Answering this questionis difficult because one has to rely upon strong assumptionsabout the data-generating mechanism for the fundamentalvalue of the oil price In this paper we present a stylizedmodel of asset pricing for a commodity to address this issueOur theoretical framework disentangles the effects ofspeculative bubbles from the traditional fundamental effectson oil price dynamics We use this setting to illustrate thatbubbles if they are present should demonstrate explosivebehavior in prices We then provide a novel approach forbubble detection that covers various explosive scenarios andfind no evidence of speculative bubbles in oil markets evenduring episodes where the most popular SADF and GSADFmethodologies identify a speculative bubble

Our conclusions have pertinent implications for policymakers Given the historically important role of oil prices inthe real economy one of the serious concerns in policycircles over the past decade has been whether speculativetrading activities in oil future markets are responsible for thelarge price swings For example in their 2008 testimonies to

Complexity 15

the US Senate Commerce Committee Michael Masters andGeorge Soros posited that speculators had fueled price in-creases since 2003 and had caused a part of a ldquosuperbubblerdquoin 2007 and 2008 resulting in significantly higher energycosts for consumers -is widespread bubble view domi-nated policy discussions and efforts to limit speculativetrading in the future market quickly followed As a responseto the excessive speculation pressures on prices theDoddndashFrank Wall Street Reform and Consumer ProtectionAct was enacted by the US Congress in 2010 As a directpolicy implication our results highlight that imposingstricter regulations on trading in oil derivative marketscannot be expected to lower the oil price given the fact thatthe synchronized boom and bust cycles witnessed in the past20 years are mainly due to economic fundamentals whichappear likely to resurge with the business cycle -e aboveanalysis implies that there is no need to enforce additionalregulatory limits against financial arbitrage in the com-modity market

Appendix

The SADF and GSADF Methodologies

Appendix describes the SADF and GSADF testing proce-dures used to detect speculative bubbles in Section 62 For agenetic time series process xt consider the following rolling-window type augmented DickeyndashFuller (ADF) regressionequation

xt ar1r2+ μr1r2

xtminus 1 + 1113944

J

j1ϕj

r1r2Δxtminus j + εt εt sim NID 0 δ2r1r2

1113874 1113875

(A1)

where Δxtminus j with j 1 J are lagged first differences ofxt incorporated to accommodate potential serial correla-tion ar1 r2

μr1 r2 and ϕj

r1 r2 with j 1 J are regression

coefficients the subscripts r1 and r2 refer to fractions of thetotal sample size (ofT observations) that set the start and endpoints of a subsample period and NID represents inde-pendent and normal distribution -e ADF test statisticcorresponding to the unit root null hypothesisH0 μr1 r2

0 is simply the t-ratio

ADFr2r1

1113954μr1 r2

se 1113954μr1 r21113872 1113873

(A2)

and the right-side alternative hypothesis of explosive be-havior is H0 μr1r2

gt 0 We can obtain the standard ADF teststatistic by running regression (A1) on the whole sampleie by specifying r1 0 and r2 1 A large amount of re-search has demonstrated that the standard ADF test ADF10has extremely low power in detecting periodically collapsingbubbles

To address the problem PSW proposed a recursiveregression technique which involves estimating (A1)employing a forward expanding sample Specifically theyused subsamples of the available data to recursively calculatea sequence of ADF statistics denoted by ADFr2

0 -e startpoint of the sample period is fixed at r1 0 while the endpoint r2 increases from r0 (the minimum window size) toone (the full sample period) -e SADF statistic called thesupremum of ADFr2

0 is defined by

SADF r0( 1113857 supr2isin r0 1[ ]

ADFr20

(A3)

and has the following limit distribution under the nullhypothesis

supr2isin r0 1[ ]

1113938r2

0 WdW

1113938r2

0 W21113872 1113873(12)

(A4)

where W is the standard Brownian motion Testing thepresence of explosive behavior entails comparing the ADFr2

0statistic with the right-side critical values obtained throughMonte Carlo simulations

More recently PSY showed that when the sample perioddisplays more than one boom-bust bubble the SADF testprocedure may perform poorly and fail to detect or con-sistently date-stamp alternative bubbles -is deficiencymotivates PSY to formulate an extension of the SADF testthe GSADF test which covers more subsamples of the datathan the SADF by allowing both the end point r2 and thestart point r1 to vary As a result the GSADF test has moreflexibility and gains substantial power making this testconsistent with multiple boom-bust bubbles -e GSADFtest statistic is defined as

GSADF r0( 1113857 supr2isin r0 1[ ]r1isin 0r2minus r0[ ]

ADFr2r1 (A5)

with the limit distribution

supr2isin r0 1[ ]r1isin 0r2minus r0[ ]

(12)rw W r2( 11138572

minus W r1( 11138572

minus rw1113960 1113961 minus 1113938r2

r1W(r)dr W r2( 1113857 minus W r1( 11138571113858 1113859

r12w rw 1113938

r2

r1W(r)2dr minus 1113938

r2

r1W(r)dr1113876 1113877

21113896 1113897

12

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎭

(A6)

where rw r2 minus r1 is the expanding window size Like theSADF test comparison of the GASDF statistic with the right-

tailed critical values from its limit distribution enables us to testfor the unit root hypothesis against explosive behavior

16 Complexity

If the null hypothesis of a unit root in xt is rejectedthen in the second stage the SADF and GSADF tests canidentify the exact episodes during which the xt seriesexhibits explosive dynamics To locate the emergence andconclusion of explosiveness the SADF methodologymatches the sequence of the recursive backward ADF(BADF) test statistics against the right-tailed criticalvalues of the limit distribution of the standard Dick-eyndashFuller test statistic Suppose that re is the originationdate and rf is the finish date of the bubble in the data-ese estimates of dates can be constructed from

1113954re infr2isin r0 1[ ]

r2 ADFr20 gt cv

αr2

1113966 1113967

1113954rf infr2isin 1113954re11113858 1113859

r2 ADFr20 lt cv

αr2

1113966 1113967(A7)

where cvαr2 is 100(1 minus α) the right-tailed critical value of theADF statistic corresponding to the chosen significance levelof α For the GSADF methodology the date-stampingstrategy in this case is based on the backward SADF(BSADF) statistic

BSADFr2r0( 1113857 sup

r1isin 0r2minus r0[ ]ADFr2

r1

(A8)

Likewise the origination date of the bubble is defined asthe first observation that the BSADF statistic is greater thanits critical value

1113954re infr2isin r0 1[ ]

r2 BSADFr2r0( 1113857gt cv

βT

r21113966 1113967 (A9)

and the finish date as the first date after which the BSADFdrops back below its critical value

1113954rf infr2isin 1113954re11113858 1113859

r2 BSADFr2r0( 1113857lt cvβT

r21113966 1113967 (A10)

where cvβTr2 is 100(1 minus βT) the right-tailed critical value of

the SADF based on Tr2 and the corresponding significancelevel is βT

Data Availability

-e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

-e authors declare no conflicts of interest

Acknowledgments

-e authors gratefully acknowledge the financial supportfrom the National Social Science Foundation of China underGrant no 18BTJ039

References

[1] M W Masters Testimony before the US Senate Committee onHomeland Security and Governmental Affairs 2008 httpswwwhsgacsenategovimomediadoc052008Masterspdfattempt2

[2] M W Masters Testimony before the Commodity FuturesTrading Commission 2010 httpscftcgovsitesdefaultfilesidcgroupspublicnewsroomdocumentsfilemetalmarkets032510_masterspdf

[3] J P Kennedy ldquo-e high cost of gambling on oilrdquo Section A p23 New York Times New York NY USANew York Times2012

[4] I-H Cheng and W Xiong ldquoFinancialization of commoditymarketsrdquo Annual Review of Financial Economics vol 6 no 1pp 419ndash441 2014

[5] K Tang and W Xiong ldquoIndex investment and the financi-alization of commoditiesrdquo Financial Analysts Journal vol 68no 6 pp 54ndash74 2012

[6] O J Blanchard and M W Watson ldquoBubbles rational ex-pectations and financial marketsrdquo in Crisis in the Economicand Financial Structure P Wachtel Ed Lexington BooksLexington MA USA pp 295ndash315 1982

[7] B T Diba and H I Grossman ldquoOn the inception of rationalbubblesrdquo e Quarterly Journal of Economics vol 102 no 3pp 697ndash700 1987

[8] G W Evans ldquo-e fragility of sunspots and bubblesrdquo Journalof Monetary Economics vol 23 no 2 pp 297ndash317 1989

[9] G W Evans and S Honkapohja ldquoOn the robustness ofbubbles in linear RE modelsrdquo International Economic Reviewvol 33 no 1 pp 1ndash14 1992

[10] E Pavlidis A Yusupova I Paya et al ldquoEpisodes of exu-berance in housing markets in search of the smoking gunrdquoe Journal of Real Estate Finance and Economics vol 53no 4 pp 419ndash449 2016

[11] E Martınez-Garcıa and V Grossman ldquoExplosive dynamics inhouse prices An exploration of financial market spillovers inhousing markets around the worldrdquo Journal of InternationalMoney and Finance vol 101 p 102103 2020

[12] S Shi A Rahman and B Z Wang ldquoAustralian housingmarket booms fundamentals or speculationlowastrdquo EconomicRecord 2020 In press

[13] P C B Phillips Y Wu and J Yu ldquoExplosive behavior in the1990s NASDAQ when did exuberance escalate asset val-ueslowastrdquo International Economic Review vol 52 no 1pp 201ndash226 2011

[14] P C B Phillips S Shi and J Yu ldquoTesting for multiplebubbles historical episodes of exuberance and collapse in theSampp 500rdquo International Economic Review vol 56 no 4pp 1043ndash1078 2015

[15] P C B Phillips S Shi and J Yu ldquoTesting for multiplebubbles limit theory of real-time detectorsrdquo InternationalEconomic Review vol 56 no 4 pp 1079ndash1134 2015

[16] P C B Phillips and J Yu ldquoDating the timeline of financialbubbles during the subprime crisisrdquo Quantitative Economicsvol 2 no 3 pp 455ndash491 2011

[17] D Tsvetanov J Coakley and N Kellard ldquoBubbling over thebehaviour of oil futures along the yield curverdquo Journal ofEmpirical Finance vol 38 no Part B pp 516ndash533 2016

[18] I Figuerola-Ferretti J R Mccrorie and I ParaskevopoulosldquoMild explosivity in recent crude oil pricesrdquo Energy Eco-nomics vol 87 p 104387 2020

[19] L Kilian and D P Murphy ldquo-e role of inventories andspeculative trading in the global market for crude oilrdquo Journalof Applied Econometrics vol 29 no 3 pp 454ndash478 2014

Complexity 17

variance of V(t) -e strong solution gives the exact dis-crete-time form of model (31)

xtΔ exp 1113957σΔ

radicuΔ + μ minus

12

1113957σ21113874 1113875Δ1113882 1113883x(tminus 1)Δ + ZtΔ ρtΔ x(tminus 1)Δ + ZtΔ

+ σ 1113946tΔ

(tminus 1)Δexp 1113957σ WμtΔ minus Wμs1113872 1113873 + μ minus

12

1113957σ21113874 1113875(tΔ minus s)1113882 1113883dWεs

(33)

where uiidsim

t N(0 1) and

ZtΔ σ 1113946tΔ

(tminus 1)Δexp 1113957σ WμtΔ minus Wμs1113872 1113873 + μ minus

12

1113957σ21113874 1113875(tΔ minus s)1113882 1113883dWεs

minus ω1113946tΔ

(tminus 1)Δexp 1113957σ WμtΔ minus Wμs1113872 11138731113872 1113873 + μ minus

12

1113957σ21113874 1113875(tΔ minus s)1113874 1113875ds

(34)

Somewhat surprisingly Tao et al [25] indicated thatendogeneity would not affect the sample path features ofmodel (31) Our analysis therefore is the same as before

(1) Unstable κ μ + 1113957σ22gt 0(2) Locally explosive μgt 0(3) Explosive ϕ μ minus 1113957σ22gt 0

We can estimate 1113954θlowast ≔ ( 11139541113957σ2 1113954η 1113954σ2)prime in the same fashion asbefore -e likelihood ratio test (LR test) has been proposedby Tao et al [25] to test the null hypothesis of endogeneityinterest

LR Δminus 1Q

rΔ minus Q

urΔ( 1113857simχ2(1) underH0 η0 0 (35)

Tao et al [25] also developed the asymptotic theory for 1113954μand 1113954ρ ≔ exp(1113954μΔ) Based on their limit properties the es-timates of 1113954μ and 1113954ρ continue to follow asymptotic normaldistributions -is good quality enables us to convenientlyadopt the aforementioned testing procedures after making asmall adjustment to the variance of the limit distribution

5 Data

For our main empirical analysis we apply daily WTI crudeoil prices for the front-month (one-month) future contracton the New York Mercantile Exchange (NYMEX) -e dataare retrieved from the Energy Information Administration(EIA) Our sample spans the period from April 4 1983 toJune 30 2020 and comprises a total of 9352 daily obser-vations -e beginning and end dates of the sample periodare dictated by the availability of the data

It is well known that WIT and Brent are grades of in-ternationally traded crude oil that are widely used asbenchmarks in oil pricing globally -e reason for this paperrelying on the WTI price of crude oil is twofold On the onehand oil speculation is often blamed on people who tradefutures in the US market WTI is the primary benchmark

for oil extracted and consumed in the US-eWTI crude oilfuture market is also the worldrsquos most liquid and largestcrude oil future market Nearly 12 million contracts trade aday with more than 2 million in open interest On the otherhand WTI future contracts have the longest available his-tory dating back to 1983 giving us a large sample size to gainpowers in detecting short-lived bubbles

Our sample period under investigation includes severalmajor exogenous events that to a large extent are reckonedto have influenced oil markets -ese contain the collapse ofOPEC in 1986 the Persian Gulf War of 199091 the Asianfinancial crisis of 199798 the terrorist attacks on September11 2001 the Venezuelan crisis and Iraq war of 200203 theincreased financialization of oil future markets since 2004the OPEC oil supply announcements in 2008 and 2014 the2019 novel coronavirus disease (COVID-19) and moststrikingly the 200708 global financial crisis Figure 2 plotsthe time series trajectories of the WTI nominal front-monthfuture price and the corresponding real price A look at thefigure illustrates that the oil prices grew steadily from thestarting of the sample until the early 2000s -e price seriessince then begun to surge and the sharp increase in the priceof oil continued until July 2008 As shown in Figure 2 itcollapsed to about 30 dollars by the end of 2008 Between2014 and 2016 the price of oil also experienced a crash Morerecently WTI crude price even went subzero for the firsttime in history A common view is that the steep rise anddecline in the price of oil in the 2000s could be attributed tospeculation

6 Empirical Results

61 Main Results We start the empirical illustration withbaseline model (14) expanding it to the endogeneity ap-plication later In order to formally examine the existenceand duration of unstable and explosive behavior in the crudeoil market we run our tests on the daily price data

8 Complexity

Following the recommendation of PSY the minimumwindow size set for the recursive test procedure is the first 5years of the entire sample Figure 3 plots the test empiricaloutcomes along with their corresponding 95 critical valuesequences Also plotted are the trajectories of test statisticsfor the random coefficient and the price data Some inter-esting results are shown in this figure

Looking at the recursive test results reported in the lastpanel of Figure 3 we observe that over the initial period ofobservation the null hypothesis of a time-invariant autor-egressive coefficient cannot be rejected at the 5 statisticalsignificance level -e test statistic experiences a dramaticincrease in the late 1980s and exceeds the 95 critical valuein November 1990 Overall the observed pattern of timevariation rises and becomes much stronger as the timeperiod expands -e evidence that the LBI test statistics forthe majority of samples are greater than the 95 criticalvalue suggests that the data-generating process is wellcharacterized by the model with time-varying coefficientsOne point worth noting is that the sequence falls in thesecond half of 2008 and slumps in early 2020 -e styled factseems to indicate that the crude oil market has undergonedrastic changes Interestingly these located dates coincidewith the 200708 global financial crisis and the COVID-19pandemic

We now turn to our focus on different types of unstableand explosive behavior-e shaded areas in green in Figure 3are the identified periods where the t-test statistic crosses its95 critical value As is evident from the first panel theprocedure detects two unstable episodes -e first unstableepisode is relatively short and only lasts two months(2005M08-M09) -e second unstable episode appears inJanuary 2006 and continues until March 2020 -ese un-stable periods might be associated with the financializationof oil future markets from 2004 It can be seen in the secondpanel of Figure 3 that the detected episodes of locally ex-plosive (submartingale) behavior are clustered in 2007-20082009ndash2014 and 2018 As stated above since local explo-siveness implies instability and displays more extreme be-havior it should lead to a much shorter duration of

exuberance than that of instability -e third panel ofFigure 3 reflects that the t-test statistic is below the 95critical value throughout the sample period -is points outthat explosive behavior in crude oil price series does notexist Table 1 summarizes time horizons for the three kindsof unstable and explosive behavior identified by ourapproach

To shed light on a more realistic scenario of data-gen-erating process we implement the extreme sample pathbehavior test by extending the model to the endogeneitycase Figure 4 plots the empirical test results together with95 critical values A comparison of the LR test statisticswith the critical value sequence provides novel evidence insupport of the presence of endogeneity during the wholesample period -us it is believed that endogeneity is es-sential to the oil price formation process Furthermore thelast panel of Figure 4 highlights that the quadratic varianceof the price process is related to the test statistic forendogeneity -is finding is in accordance with the fact thatwe have exploited the realized variance series to obtain theLR test statistics In this sense Figure 5 displays differencesin the realized variance estimates which are captured by thetest statistic for endogeneity employing different methodsSimilar to Figure 3 we again identify episodes of unstabledynamics and fail to detect explosive behavior when con-sidering the model with endogeneity In contrast the sta-tistic sequence for local explosiveness exhibits a differentpattern Previous evidence in favor of locally explosive be-havior disappeared Table 2 summarizes time horizons forthe three forms of unstable and explosive behavior -eseresults suggest that endogeneity feedbacks play a significantrole in investigating the existence of explosive price be-havior -e conclusion that emerges is that there are nospeculative bubbles in the crude oil market

In principal our findings are in accordance with theexplanations made by Kilian and Murphy [19] Baumeisterand Hamilton [21] and Knittel and Pindyck [54] that thevariations in crude oil prices are driven by economicfundamentals

62 Comparison to Alternative Strategies Rich literature hasstudied speculative bubbles in the crude oil market using thetechniques developed by PWY and then extended by PSY Abrief description of the SADF methodology (used in PWY)and the GSADF methodology (used in PSY) can be found inAppendix It is therefore helpful to compare results foralternative methods as the detection strategy in the presentpaper is quite different from them Figure 6 plots thebackward ADF and backward SADF statistic sequences andthe associated 95 critical value sequences for oil futureprices 95 percent critical values are taken from 2000 MonteCarlo simulations with the actual sample size -e initialwindow size is set to 5 years A glance at Figure 6 reveals thatexplosive behavior occurred mainly in two occasions usingPWY and PSY dating strategies -ey are before and during2007-2008 global recession We also observe that PWYdetects several short-lived episodes between 2011 and 2014but PSY fails to do so For daily oil future prices over April 4

0

20

ndash20

ndash40

40

60

80

100

120

140

84 86 88 90 92 94 96 98 00 02 04 06 08 10 12 14 16 18 20

Nominal WTI front-month future pricesReal WTI front-month future prices

Figure 2 Real and nominal prices of WTI front-month futurecontracts

Complexity 9

1983 to June 30 2020 time horizons for explosive behavioridentified by PWY and PSY are summarized in Table 3 Inline with previous studies we could conclude from bothPWY and PSY tests that there is evidence of bubbles in thecrude oil market data However as shown in Gurkaynak [37]and Pavlidis et al [43] explosive dynamics in asset prices perse do not constitute evidence either in support of or againstspeculative bubbles since such explosive behavior may

actually reflect explosive market fundamentals or bothfactors As aforementioned our random coefficient autor-egressive model framework has considered endogeneityfeedbacks that to some extent alleviate this problemMoreover the evidence in Figures 3 and 4 suggests theabsence of explosive dynamics in the crude oil market -efact permits us to safely ignore the distinction betweenbubbles or explosive fundamentals

1988

-03

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

ndash4

ndash2

0

2

4

6

8

0

20

40

60

80

100

120

140

t-statistic of k95 critical value of N (0 1)Crude oil future prices

Instability detection based on the time-varyingcoefficient model

(a)

Local explosiveness detection based on the time-varyingcoefficient model

1988

-03

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

t-statistic of μ95 critical value of N (0 1)Crude oil future prices

ndash4

ndash2

0

2

4

6

8

0

20

40

60

80

100

120

140

(b)

Explosiveness detection based on the time-varyingcoefficient model

1988

-03

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

t-statistic of ϕ95 critical value of N (0 1)Crude oil future prices

ndash4

ndash2

0

2

4

6

8

0

20

40

60

80

100

120

140

(c)

LBItest statistic for testing time-varying coefficient19

88-0

3

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

LBItest95 critical value of N (0 1)

ndash5

ndash10

0

5

10

15

20

(d)

Figure 3 Date-stamping unstable and explosive episodes in oil prices with the time-varying coefficient model assuming no endogeneity

Table 1 Time horizons of unstable and explosive periods identified by models assuming no endogeneity

Unstable Locally explosive ExplosiveAug 2005-Sep 2005 Dec 2007ndashSep 2008 mdash

Jan 2006ndashMar 2020 Jun 2009ndashOct 2014 mdashJun 2018ndashSep 2018 mdash

10 Complexity

63 Robustness of the Results In the following we providetwo robustness checks to assess whether our previousconclusions of the paper still hold

First in our benchmark model our use of front-monthfuture price instead of other future prices at different ma-turities is justified by its being the most liquid future contractwith the highest open interest To explore the sensitivity ofour results to the future contracts at different maturities werepeat the analysis described above using the prices forlonger-term contracts (ie 2 3 and 4 months-ahead fu-tures) It should be pointed out that the WTI future contractusually expires on the third business day before the 25thcalendar day of the contract month proceeding the deliverymonth

Second becauseWTI crude oil is by tradition traded innominal US dollars per barrel it might be of concern that

inflation and other monetary variables may have realeffects on oil price It is therefore of interest to use realrather than nominal price In fact recent work documentsthat the use of either price is inconsequential in practice[55] Although we mainly focus on nominal prices forrobustness check we shall report the results for realprices To convert nominal daily series into daily realseries we use linearly interpolated monthly ConsumerPrice Index (CPI) Monthly CPI series for the UnitedStates is obtained from the websites of the Federal ReserveBank of St Louis

Further evidence presented in Figures 7ndash10 demon-strates that our results are remarkably robust to thesemodifications and the conclusions remain qualitativelyunchanged

ndash4

ndash2

0

2

4

6

8

0

20

40

60

80

100

120

140

1988

-03

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

Instability detection based on the time-varyingcoefficient model

t-statistic of k95 critical value of N (0 1)Crude oil future prices

(a)

ndash4

ndash2

0

2

4

6

8

0

20

40

60

80

100

120

140

1988

-03

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

t-statistic of μ95 critical value of N (0 1)Crude oil future prices

Local explosiveness detection based on the time-varyingcoefficient model

(b)

ndash4

ndash2

0

2

4

6

8

0

20

40

60

80

100

120

140

1988

-03

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

Explosiveness detection based on the time-varyingcoefficient model

t-statistic of ϕ95 critical value of N (0 1)Crude oil future prices

(c)

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

LR-statistic95 critical values of x2 (1)Realized variance

700

600

500

400

300

200

100

0

ndash100

350

300

250

200

150

100

50

0

Test for endogeneity in time-varyingcoefficient model

(d)

Figure 4 Date-stamping unstable and explosive episodes in oil prices with the time-varying coefficient model with endogeneity

Complexity 11

2

3

4

5

6

7

8

0

50

100

150

200

250

300

350

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

LLF0

LLF1

Realized variance of crude oil future prices

Figure 5 Realized variance and likelihood values for random coefficient regression

Table 2 Time horizons of unstable and explosive periods detected by models with endogeneity

Unstable Locally explosive ExplosiveMay 2008ndashJul 2008 mdash mdashDec 2010ndashOct 2014 mdash mdashNov 2016ndashApr 2017 mdash mdashJul 2017ndashFeb 2020 mdash mdash

ndash4

ndash2

0

2

4

6

8

0

20

40

60

80

100

120

140

1988

-03

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

Explosiveness detection using SADF methodology

BADF sequence95 critical valuesCrude oil future prices

(a)

ndash4

ndash2

0

2

4

6

8

0

20

40

60

80

100

120

140

1988

-03

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

Explosiveness detection using GSADF methodology

BSADF sequence95 critical valuesCrude oil future prices

(b)

Figure 6 Date-stamping explosive episodes in oil prices with the time-invariant coefficient model

12 Complexity

Table 3 Time horizons of explosive behavior identified by SADF and GSADF methodologies

SADF GSADFAug 2005ndashAug 2006 Oct 2011ndashApr 2012 Sep 1990-Oct 1990Jun 2007ndashOct 2008 Aug 2012-Sep 2012 Feb 1991-Mar 1991Jun 2009-Jul 2009 Jul 2013-Aug 2013 Oct 2004-Nov 2004Feb 2011ndashApr 2011 Feb 2014-Mar 2014 Feb 2005-Mar 2005

Jul 2011ndashAug 2012 Jun 2014-Jul 2014May 2005ndashJan 2006Mar 2006ndashAug 2006Jul 2007ndashAug 2008

ndash4

ndash2

0

2

4

6

8

0

10

20

30

40

50

60

70

1988

-03

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

95 critical value of N (0 1)Real oil future prices

t-statistic of μ~

Local explosiveness detection based on the time-varyingcoefficient model

(a)

ndash4

ndash2

0

2

4

6

8

0

10

20

30

40

50

60

70

1988

-03

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

95 critical value of N (0 1)Real oil future prices

t-statistic of ϕ

Explosiveness detection based on the time-varyingcoefficient model

(b)

Figure 7 Date-stamping explosive episodes in real oil prices with the time-varying coefficient model with endogeneity

ndash4

ndash2

0

2

4

6

8

0

50

100

150

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

Local explosiveness detection based on the time-varyingcoefficient model

t-statistic of μ95 critical value of N (0 1)2-month crude oil future prices

(a)

ndash4

ndash2

0

2

4

6

8

0

50

100

150

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

Local explosiveness detection based on the time-varyingcoefficient model

t-statistic of ϕ95 critical value of N (0 1)2-month crude oil future prices

(b)

Figure 8 Date-stamping explosive episodes in 2-month future oil prices with the time-varying coefficient model with endogeneity

Complexity 13

7 Discussion

-e oil price rally since the early 2000s has attracted greatinterest in the sources of oil price shocks Some causal ob-servers have stressed that financial speculation has become asignificant driver of the dynamics of oil prices A large body ofliterature emerged that has sought to investigate its existenceIn this paper we employ a novel approach for testing forspeculative bubbles in oil prices In synthesis our empiricalfindings indicate that there is less evidence of exuberance in

the oil future markets Our results are very robust in severalscenarios We argue that the booms and busts in the oil pricemovements during the sample period are driven primarily byeconomic fundamentals rather than speculation

Overall our results confirm the main findings obtainedby structural VAR-based studies including Kilian andMurphy [19] Kilian and Lee [20] Baumeister and Hamilton[21] and Zhou [22] In the past decade the use of structuralVAR models to understand the evolution of the oil price hasbecome common -e traditional SVAR literature

ndash4

ndash2

0

2

4

6

8

0

50

100

15019

88-0

3

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

Local explosiveness detection based on the time-varyingcoefficient model

t-statistic of μ95 critical value of N (0 1)3-month crude oil future prices

(a)

ndash4

ndash2

0

2

4

6

8

0

50

100

150

1988

-03

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

Explosiveness detection based on the time-varyingcoefficient model

t-statistic of ϕ95 critical value of N (0 1)3-month crude oil future prices

(b)

Figure 9 Date-stamping explosive episodes in 3-month future oil prices with the time-varying coefficient model with endogeneity

ndash4

ndash2

0

2

4

6

8

0

50

100

150

Local explosiveness detection based on the time-varyingcoefficient model

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

t-statistic of μ95 critical value of N (0 1)4-month crude oil future prices

(a)

ndash4

ndash2

0

2

4

6

8

0

50

100

150

Explosiveness detection based on the time-varyingcoefficient model

t-statistic of ϕ95 critical value of N (0 1)4-month crude oil futures prices

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

(b)

Figure 10 Date-stamping explosive episodes in 4-month future oil prices with the time-varying coefficient model with endogeneity

14 Complexity

documents that the fundamental drivers of oil price fluc-tuations have been dominated essentially by supply anddemand shocks see eg Kilian [56] Lippi and Nobili [57]Kilian andMurphy [58] and Baumeister and Peersman [59]Kilian and Murphy [19] focused on the role speculation byaugmenting the SVAR framework of their earlier paper byglobal oil inventory data -ey specified a four-variablemodel for the global oil market (the percentage change inglobal oil production the real price of oil a suitable proxy ofcyclical variation in global real economic activity andchanges in global crude oil inventories) -is setting andtheir sign-identified scheme allow them to separately explainvariation in these data as four different types of shocks (1) ashock to the amount of oil pumped out of the ground (ldquooilsupply shockrdquo) (2) an oil flow demand shock (3) an oil-specific oil demand shock and especially (4) a speculative(or inventory) demand shock -eir core finding is that thespike in the real price of crude oil prices during 2003ndash2008was caused by unexpected increase in global oil consump-tion Using two alternative proxies for global above-groundcrude oil inventories Kilian and Lee [20] reestimated themodel of Kilian and Murphy [19] and ruled out the role ofspeculation in explaining the recent surge of the real oil priceon the basis of their estimation results Also Zhou [22]assessed the sensitivity of the findings in Kilian and Murphy[19] to the introduction of various state-of-the-art oil marketmethods She shows that the core conclusions reached byKilian and Murphy [19] are highly robust to some refine-ments corrections and extension of the KilianndashMurphymodel Baumeister and Hamilton [21] recently contendedthat prior information plays a key role in any SVAR modelsand traditional identification approaches can be generalizedto a special case of a Bayesian perspective -ey revisited theimportance of supply and demand shocks in the oil marketand demonstrated that speculative demand shock turns outto be only a small factor in accounting for most historical oilprice shocks

For comparison purpose we have also applied the testsof PWY and PSY which are the most popular econometrictechniques in the literature on speculative bubbles in the oilmarkets If we were to follow the test procedures of PWY andPSY our conclusions are also broadly consistent with therich literature that supports the existence of speculativebubbles in oil prices However these cointegration-basedbubble tests suffer from the joint hypothesis problem ldquoEverytest of a bubble is a joint test of the presence of the bubble inthe data and the validity of the model applied by theeconometricianrdquo (see Giglio et al [60]) More specifically themain drawback of such indirect model-dependent tests ofbubbles typically refers to the difficulty in distinguishingbetween the unknown true model for market fundamentalsand the presence of bubbles (see for example Gurkaynak[37] Pavlidis et al [42] Pavlidis et al [43]) Given thefundamental value of oil price is unobservable the literaturehas resorted to observe economic and financial variableswhich are used to estimate the intrinsic values For exampleboth Phillips and Yu [16] and Caspi et al [39] simplynormalized crude oil prices by US oil inventory data as thefundamental value -is measure is questionable since their

method cannot sufficiently capture the information onmarket fundamentals and neglect the major driversmdashlikeglobal economic activitymdashalso affecting the fundamentalvalue of crude oil -is issue of misspecified fundamentals(either in terms of model misspecification or omitted var-iables or both of these reasons) may make researchers fail toarrive at a conclusion from evidence of explosive pricesalone To make matters worse even though we have cap-tured the true market fundamentals we may still fail todetect bubbles if the explosive behavior in the price of oil juststems from the explosive fundamentals

Compared to earlier studies as briefly mentioned aboveour methodology has the main advantage that by consid-ering the dependence between the random coefficient andsystem shocks we have incorporated endogeneity into thecontinuous-time random coefficient setting and thus we donot need to prespecify the fundamental component of the oilprice -e test results for endogeneity suggest that endo-geneity is important in the data-generating process Inparticular our empirical analysis also finds that there are noexplosive episodes in crude oil markets-is implies that it isnot necessary to require the specification of a fundamentalvalue of the oil price Like Pavlidis et al [43] their methodsand ours have in common that the problem of the well-known joint hypothesis has to be some extent been ef-fectively ameliorated Although our approach is quite dis-tinct we draw the same substantive conclusion thatspeculative bubbles can be ruled out as a key determinant ofthe price of crude oil in the 2000s reaffirming the con-clusions of Kilian and Murphy [19] Baumeister andHamilton [21] Zhou [22] and Fattouh et al [61]

8 Conclusions

Repeated surges and crashes in crude oil prices during lastdecades have sparked a heated debate on their underlyingcauses among academics commentors and market regu-lators One of the central questions concerns whetherfinancialization of oil future markets creates a speculativebubble that subsequently collapses Answering this questionis difficult because one has to rely upon strong assumptionsabout the data-generating mechanism for the fundamentalvalue of the oil price In this paper we present a stylizedmodel of asset pricing for a commodity to address this issueOur theoretical framework disentangles the effects ofspeculative bubbles from the traditional fundamental effectson oil price dynamics We use this setting to illustrate thatbubbles if they are present should demonstrate explosivebehavior in prices We then provide a novel approach forbubble detection that covers various explosive scenarios andfind no evidence of speculative bubbles in oil markets evenduring episodes where the most popular SADF and GSADFmethodologies identify a speculative bubble

Our conclusions have pertinent implications for policymakers Given the historically important role of oil prices inthe real economy one of the serious concerns in policycircles over the past decade has been whether speculativetrading activities in oil future markets are responsible for thelarge price swings For example in their 2008 testimonies to

Complexity 15

the US Senate Commerce Committee Michael Masters andGeorge Soros posited that speculators had fueled price in-creases since 2003 and had caused a part of a ldquosuperbubblerdquoin 2007 and 2008 resulting in significantly higher energycosts for consumers -is widespread bubble view domi-nated policy discussions and efforts to limit speculativetrading in the future market quickly followed As a responseto the excessive speculation pressures on prices theDoddndashFrank Wall Street Reform and Consumer ProtectionAct was enacted by the US Congress in 2010 As a directpolicy implication our results highlight that imposingstricter regulations on trading in oil derivative marketscannot be expected to lower the oil price given the fact thatthe synchronized boom and bust cycles witnessed in the past20 years are mainly due to economic fundamentals whichappear likely to resurge with the business cycle -e aboveanalysis implies that there is no need to enforce additionalregulatory limits against financial arbitrage in the com-modity market

Appendix

The SADF and GSADF Methodologies

Appendix describes the SADF and GSADF testing proce-dures used to detect speculative bubbles in Section 62 For agenetic time series process xt consider the following rolling-window type augmented DickeyndashFuller (ADF) regressionequation

xt ar1r2+ μr1r2

xtminus 1 + 1113944

J

j1ϕj

r1r2Δxtminus j + εt εt sim NID 0 δ2r1r2

1113874 1113875

(A1)

where Δxtminus j with j 1 J are lagged first differences ofxt incorporated to accommodate potential serial correla-tion ar1 r2

μr1 r2 and ϕj

r1 r2 with j 1 J are regression

coefficients the subscripts r1 and r2 refer to fractions of thetotal sample size (ofT observations) that set the start and endpoints of a subsample period and NID represents inde-pendent and normal distribution -e ADF test statisticcorresponding to the unit root null hypothesisH0 μr1 r2

0 is simply the t-ratio

ADFr2r1

1113954μr1 r2

se 1113954μr1 r21113872 1113873

(A2)

and the right-side alternative hypothesis of explosive be-havior is H0 μr1r2

gt 0 We can obtain the standard ADF teststatistic by running regression (A1) on the whole sampleie by specifying r1 0 and r2 1 A large amount of re-search has demonstrated that the standard ADF test ADF10has extremely low power in detecting periodically collapsingbubbles

To address the problem PSW proposed a recursiveregression technique which involves estimating (A1)employing a forward expanding sample Specifically theyused subsamples of the available data to recursively calculatea sequence of ADF statistics denoted by ADFr2

0 -e startpoint of the sample period is fixed at r1 0 while the endpoint r2 increases from r0 (the minimum window size) toone (the full sample period) -e SADF statistic called thesupremum of ADFr2

0 is defined by

SADF r0( 1113857 supr2isin r0 1[ ]

ADFr20

(A3)

and has the following limit distribution under the nullhypothesis

supr2isin r0 1[ ]

1113938r2

0 WdW

1113938r2

0 W21113872 1113873(12)

(A4)

where W is the standard Brownian motion Testing thepresence of explosive behavior entails comparing the ADFr2

0statistic with the right-side critical values obtained throughMonte Carlo simulations

More recently PSY showed that when the sample perioddisplays more than one boom-bust bubble the SADF testprocedure may perform poorly and fail to detect or con-sistently date-stamp alternative bubbles -is deficiencymotivates PSY to formulate an extension of the SADF testthe GSADF test which covers more subsamples of the datathan the SADF by allowing both the end point r2 and thestart point r1 to vary As a result the GSADF test has moreflexibility and gains substantial power making this testconsistent with multiple boom-bust bubbles -e GSADFtest statistic is defined as

GSADF r0( 1113857 supr2isin r0 1[ ]r1isin 0r2minus r0[ ]

ADFr2r1 (A5)

with the limit distribution

supr2isin r0 1[ ]r1isin 0r2minus r0[ ]

(12)rw W r2( 11138572

minus W r1( 11138572

minus rw1113960 1113961 minus 1113938r2

r1W(r)dr W r2( 1113857 minus W r1( 11138571113858 1113859

r12w rw 1113938

r2

r1W(r)2dr minus 1113938

r2

r1W(r)dr1113876 1113877

21113896 1113897

12

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎭

(A6)

where rw r2 minus r1 is the expanding window size Like theSADF test comparison of the GASDF statistic with the right-

tailed critical values from its limit distribution enables us to testfor the unit root hypothesis against explosive behavior

16 Complexity

If the null hypothesis of a unit root in xt is rejectedthen in the second stage the SADF and GSADF tests canidentify the exact episodes during which the xt seriesexhibits explosive dynamics To locate the emergence andconclusion of explosiveness the SADF methodologymatches the sequence of the recursive backward ADF(BADF) test statistics against the right-tailed criticalvalues of the limit distribution of the standard Dick-eyndashFuller test statistic Suppose that re is the originationdate and rf is the finish date of the bubble in the data-ese estimates of dates can be constructed from

1113954re infr2isin r0 1[ ]

r2 ADFr20 gt cv

αr2

1113966 1113967

1113954rf infr2isin 1113954re11113858 1113859

r2 ADFr20 lt cv

αr2

1113966 1113967(A7)

where cvαr2 is 100(1 minus α) the right-tailed critical value of theADF statistic corresponding to the chosen significance levelof α For the GSADF methodology the date-stampingstrategy in this case is based on the backward SADF(BSADF) statistic

BSADFr2r0( 1113857 sup

r1isin 0r2minus r0[ ]ADFr2

r1

(A8)

Likewise the origination date of the bubble is defined asthe first observation that the BSADF statistic is greater thanits critical value

1113954re infr2isin r0 1[ ]

r2 BSADFr2r0( 1113857gt cv

βT

r21113966 1113967 (A9)

and the finish date as the first date after which the BSADFdrops back below its critical value

1113954rf infr2isin 1113954re11113858 1113859

r2 BSADFr2r0( 1113857lt cvβT

r21113966 1113967 (A10)

where cvβTr2 is 100(1 minus βT) the right-tailed critical value of

the SADF based on Tr2 and the corresponding significancelevel is βT

Data Availability

-e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

-e authors declare no conflicts of interest

Acknowledgments

-e authors gratefully acknowledge the financial supportfrom the National Social Science Foundation of China underGrant no 18BTJ039

References

[1] M W Masters Testimony before the US Senate Committee onHomeland Security and Governmental Affairs 2008 httpswwwhsgacsenategovimomediadoc052008Masterspdfattempt2

[2] M W Masters Testimony before the Commodity FuturesTrading Commission 2010 httpscftcgovsitesdefaultfilesidcgroupspublicnewsroomdocumentsfilemetalmarkets032510_masterspdf

[3] J P Kennedy ldquo-e high cost of gambling on oilrdquo Section A p23 New York Times New York NY USANew York Times2012

[4] I-H Cheng and W Xiong ldquoFinancialization of commoditymarketsrdquo Annual Review of Financial Economics vol 6 no 1pp 419ndash441 2014

[5] K Tang and W Xiong ldquoIndex investment and the financi-alization of commoditiesrdquo Financial Analysts Journal vol 68no 6 pp 54ndash74 2012

[6] O J Blanchard and M W Watson ldquoBubbles rational ex-pectations and financial marketsrdquo in Crisis in the Economicand Financial Structure P Wachtel Ed Lexington BooksLexington MA USA pp 295ndash315 1982

[7] B T Diba and H I Grossman ldquoOn the inception of rationalbubblesrdquo e Quarterly Journal of Economics vol 102 no 3pp 697ndash700 1987

[8] G W Evans ldquo-e fragility of sunspots and bubblesrdquo Journalof Monetary Economics vol 23 no 2 pp 297ndash317 1989

[9] G W Evans and S Honkapohja ldquoOn the robustness ofbubbles in linear RE modelsrdquo International Economic Reviewvol 33 no 1 pp 1ndash14 1992

[10] E Pavlidis A Yusupova I Paya et al ldquoEpisodes of exu-berance in housing markets in search of the smoking gunrdquoe Journal of Real Estate Finance and Economics vol 53no 4 pp 419ndash449 2016

[11] E Martınez-Garcıa and V Grossman ldquoExplosive dynamics inhouse prices An exploration of financial market spillovers inhousing markets around the worldrdquo Journal of InternationalMoney and Finance vol 101 p 102103 2020

[12] S Shi A Rahman and B Z Wang ldquoAustralian housingmarket booms fundamentals or speculationlowastrdquo EconomicRecord 2020 In press

[13] P C B Phillips Y Wu and J Yu ldquoExplosive behavior in the1990s NASDAQ when did exuberance escalate asset val-ueslowastrdquo International Economic Review vol 52 no 1pp 201ndash226 2011

[14] P C B Phillips S Shi and J Yu ldquoTesting for multiplebubbles historical episodes of exuberance and collapse in theSampp 500rdquo International Economic Review vol 56 no 4pp 1043ndash1078 2015

[15] P C B Phillips S Shi and J Yu ldquoTesting for multiplebubbles limit theory of real-time detectorsrdquo InternationalEconomic Review vol 56 no 4 pp 1079ndash1134 2015

[16] P C B Phillips and J Yu ldquoDating the timeline of financialbubbles during the subprime crisisrdquo Quantitative Economicsvol 2 no 3 pp 455ndash491 2011

[17] D Tsvetanov J Coakley and N Kellard ldquoBubbling over thebehaviour of oil futures along the yield curverdquo Journal ofEmpirical Finance vol 38 no Part B pp 516ndash533 2016

[18] I Figuerola-Ferretti J R Mccrorie and I ParaskevopoulosldquoMild explosivity in recent crude oil pricesrdquo Energy Eco-nomics vol 87 p 104387 2020

[19] L Kilian and D P Murphy ldquo-e role of inventories andspeculative trading in the global market for crude oilrdquo Journalof Applied Econometrics vol 29 no 3 pp 454ndash478 2014

Complexity 17

Following the recommendation of PSY the minimumwindow size set for the recursive test procedure is the first 5years of the entire sample Figure 3 plots the test empiricaloutcomes along with their corresponding 95 critical valuesequences Also plotted are the trajectories of test statisticsfor the random coefficient and the price data Some inter-esting results are shown in this figure

Looking at the recursive test results reported in the lastpanel of Figure 3 we observe that over the initial period ofobservation the null hypothesis of a time-invariant autor-egressive coefficient cannot be rejected at the 5 statisticalsignificance level -e test statistic experiences a dramaticincrease in the late 1980s and exceeds the 95 critical valuein November 1990 Overall the observed pattern of timevariation rises and becomes much stronger as the timeperiod expands -e evidence that the LBI test statistics forthe majority of samples are greater than the 95 criticalvalue suggests that the data-generating process is wellcharacterized by the model with time-varying coefficientsOne point worth noting is that the sequence falls in thesecond half of 2008 and slumps in early 2020 -e styled factseems to indicate that the crude oil market has undergonedrastic changes Interestingly these located dates coincidewith the 200708 global financial crisis and the COVID-19pandemic

We now turn to our focus on different types of unstableand explosive behavior-e shaded areas in green in Figure 3are the identified periods where the t-test statistic crosses its95 critical value As is evident from the first panel theprocedure detects two unstable episodes -e first unstableepisode is relatively short and only lasts two months(2005M08-M09) -e second unstable episode appears inJanuary 2006 and continues until March 2020 -ese un-stable periods might be associated with the financializationof oil future markets from 2004 It can be seen in the secondpanel of Figure 3 that the detected episodes of locally ex-plosive (submartingale) behavior are clustered in 2007-20082009ndash2014 and 2018 As stated above since local explo-siveness implies instability and displays more extreme be-havior it should lead to a much shorter duration of

exuberance than that of instability -e third panel ofFigure 3 reflects that the t-test statistic is below the 95critical value throughout the sample period -is points outthat explosive behavior in crude oil price series does notexist Table 1 summarizes time horizons for the three kindsof unstable and explosive behavior identified by ourapproach

To shed light on a more realistic scenario of data-gen-erating process we implement the extreme sample pathbehavior test by extending the model to the endogeneitycase Figure 4 plots the empirical test results together with95 critical values A comparison of the LR test statisticswith the critical value sequence provides novel evidence insupport of the presence of endogeneity during the wholesample period -us it is believed that endogeneity is es-sential to the oil price formation process Furthermore thelast panel of Figure 4 highlights that the quadratic varianceof the price process is related to the test statistic forendogeneity -is finding is in accordance with the fact thatwe have exploited the realized variance series to obtain theLR test statistics In this sense Figure 5 displays differencesin the realized variance estimates which are captured by thetest statistic for endogeneity employing different methodsSimilar to Figure 3 we again identify episodes of unstabledynamics and fail to detect explosive behavior when con-sidering the model with endogeneity In contrast the sta-tistic sequence for local explosiveness exhibits a differentpattern Previous evidence in favor of locally explosive be-havior disappeared Table 2 summarizes time horizons forthe three forms of unstable and explosive behavior -eseresults suggest that endogeneity feedbacks play a significantrole in investigating the existence of explosive price be-havior -e conclusion that emerges is that there are nospeculative bubbles in the crude oil market

In principal our findings are in accordance with theexplanations made by Kilian and Murphy [19] Baumeisterand Hamilton [21] and Knittel and Pindyck [54] that thevariations in crude oil prices are driven by economicfundamentals

62 Comparison to Alternative Strategies Rich literature hasstudied speculative bubbles in the crude oil market using thetechniques developed by PWY and then extended by PSY Abrief description of the SADF methodology (used in PWY)and the GSADF methodology (used in PSY) can be found inAppendix It is therefore helpful to compare results foralternative methods as the detection strategy in the presentpaper is quite different from them Figure 6 plots thebackward ADF and backward SADF statistic sequences andthe associated 95 critical value sequences for oil futureprices 95 percent critical values are taken from 2000 MonteCarlo simulations with the actual sample size -e initialwindow size is set to 5 years A glance at Figure 6 reveals thatexplosive behavior occurred mainly in two occasions usingPWY and PSY dating strategies -ey are before and during2007-2008 global recession We also observe that PWYdetects several short-lived episodes between 2011 and 2014but PSY fails to do so For daily oil future prices over April 4

0

20

ndash20

ndash40

40

60

80

100

120

140

84 86 88 90 92 94 96 98 00 02 04 06 08 10 12 14 16 18 20

Nominal WTI front-month future pricesReal WTI front-month future prices

Figure 2 Real and nominal prices of WTI front-month futurecontracts

Complexity 9

1983 to June 30 2020 time horizons for explosive behavioridentified by PWY and PSY are summarized in Table 3 Inline with previous studies we could conclude from bothPWY and PSY tests that there is evidence of bubbles in thecrude oil market data However as shown in Gurkaynak [37]and Pavlidis et al [43] explosive dynamics in asset prices perse do not constitute evidence either in support of or againstspeculative bubbles since such explosive behavior may

actually reflect explosive market fundamentals or bothfactors As aforementioned our random coefficient autor-egressive model framework has considered endogeneityfeedbacks that to some extent alleviate this problemMoreover the evidence in Figures 3 and 4 suggests theabsence of explosive dynamics in the crude oil market -efact permits us to safely ignore the distinction betweenbubbles or explosive fundamentals

1988

-03

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

ndash4

ndash2

0

2

4

6

8

0

20

40

60

80

100

120

140

t-statistic of k95 critical value of N (0 1)Crude oil future prices

Instability detection based on the time-varyingcoefficient model

(a)

Local explosiveness detection based on the time-varyingcoefficient model

1988

-03

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

t-statistic of μ95 critical value of N (0 1)Crude oil future prices

ndash4

ndash2

0

2

4

6

8

0

20

40

60

80

100

120

140

(b)

Explosiveness detection based on the time-varyingcoefficient model

1988

-03

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

t-statistic of ϕ95 critical value of N (0 1)Crude oil future prices

ndash4

ndash2

0

2

4

6

8

0

20

40

60

80

100

120

140

(c)

LBItest statistic for testing time-varying coefficient19

88-0

3

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

LBItest95 critical value of N (0 1)

ndash5

ndash10

0

5

10

15

20

(d)

Figure 3 Date-stamping unstable and explosive episodes in oil prices with the time-varying coefficient model assuming no endogeneity

Table 1 Time horizons of unstable and explosive periods identified by models assuming no endogeneity

Unstable Locally explosive ExplosiveAug 2005-Sep 2005 Dec 2007ndashSep 2008 mdash

Jan 2006ndashMar 2020 Jun 2009ndashOct 2014 mdashJun 2018ndashSep 2018 mdash

10 Complexity

63 Robustness of the Results In the following we providetwo robustness checks to assess whether our previousconclusions of the paper still hold

First in our benchmark model our use of front-monthfuture price instead of other future prices at different ma-turities is justified by its being the most liquid future contractwith the highest open interest To explore the sensitivity ofour results to the future contracts at different maturities werepeat the analysis described above using the prices forlonger-term contracts (ie 2 3 and 4 months-ahead fu-tures) It should be pointed out that the WTI future contractusually expires on the third business day before the 25thcalendar day of the contract month proceeding the deliverymonth

Second becauseWTI crude oil is by tradition traded innominal US dollars per barrel it might be of concern that

inflation and other monetary variables may have realeffects on oil price It is therefore of interest to use realrather than nominal price In fact recent work documentsthat the use of either price is inconsequential in practice[55] Although we mainly focus on nominal prices forrobustness check we shall report the results for realprices To convert nominal daily series into daily realseries we use linearly interpolated monthly ConsumerPrice Index (CPI) Monthly CPI series for the UnitedStates is obtained from the websites of the Federal ReserveBank of St Louis

Further evidence presented in Figures 7ndash10 demon-strates that our results are remarkably robust to thesemodifications and the conclusions remain qualitativelyunchanged

ndash4

ndash2

0

2

4

6

8

0

20

40

60

80

100

120

140

1988

-03

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

Instability detection based on the time-varyingcoefficient model

t-statistic of k95 critical value of N (0 1)Crude oil future prices

(a)

ndash4

ndash2

0

2

4

6

8

0

20

40

60

80

100

120

140

1988

-03

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

t-statistic of μ95 critical value of N (0 1)Crude oil future prices

Local explosiveness detection based on the time-varyingcoefficient model

(b)

ndash4

ndash2

0

2

4

6

8

0

20

40

60

80

100

120

140

1988

-03

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

Explosiveness detection based on the time-varyingcoefficient model

t-statistic of ϕ95 critical value of N (0 1)Crude oil future prices

(c)

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

LR-statistic95 critical values of x2 (1)Realized variance

700

600

500

400

300

200

100

0

ndash100

350

300

250

200

150

100

50

0

Test for endogeneity in time-varyingcoefficient model

(d)

Figure 4 Date-stamping unstable and explosive episodes in oil prices with the time-varying coefficient model with endogeneity

Complexity 11

2

3

4

5

6

7

8

0

50

100

150

200

250

300

350

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

LLF0

LLF1

Realized variance of crude oil future prices

Figure 5 Realized variance and likelihood values for random coefficient regression

Table 2 Time horizons of unstable and explosive periods detected by models with endogeneity

Unstable Locally explosive ExplosiveMay 2008ndashJul 2008 mdash mdashDec 2010ndashOct 2014 mdash mdashNov 2016ndashApr 2017 mdash mdashJul 2017ndashFeb 2020 mdash mdash

ndash4

ndash2

0

2

4

6

8

0

20

40

60

80

100

120

140

1988

-03

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

Explosiveness detection using SADF methodology

BADF sequence95 critical valuesCrude oil future prices

(a)

ndash4

ndash2

0

2

4

6

8

0

20

40

60

80

100

120

140

1988

-03

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

Explosiveness detection using GSADF methodology

BSADF sequence95 critical valuesCrude oil future prices

(b)

Figure 6 Date-stamping explosive episodes in oil prices with the time-invariant coefficient model

12 Complexity

Table 3 Time horizons of explosive behavior identified by SADF and GSADF methodologies

SADF GSADFAug 2005ndashAug 2006 Oct 2011ndashApr 2012 Sep 1990-Oct 1990Jun 2007ndashOct 2008 Aug 2012-Sep 2012 Feb 1991-Mar 1991Jun 2009-Jul 2009 Jul 2013-Aug 2013 Oct 2004-Nov 2004Feb 2011ndashApr 2011 Feb 2014-Mar 2014 Feb 2005-Mar 2005

Jul 2011ndashAug 2012 Jun 2014-Jul 2014May 2005ndashJan 2006Mar 2006ndashAug 2006Jul 2007ndashAug 2008

ndash4

ndash2

0

2

4

6

8

0

10

20

30

40

50

60

70

1988

-03

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

95 critical value of N (0 1)Real oil future prices

t-statistic of μ~

Local explosiveness detection based on the time-varyingcoefficient model

(a)

ndash4

ndash2

0

2

4

6

8

0

10

20

30

40

50

60

70

1988

-03

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

95 critical value of N (0 1)Real oil future prices

t-statistic of ϕ

Explosiveness detection based on the time-varyingcoefficient model

(b)

Figure 7 Date-stamping explosive episodes in real oil prices with the time-varying coefficient model with endogeneity

ndash4

ndash2

0

2

4

6

8

0

50

100

150

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

Local explosiveness detection based on the time-varyingcoefficient model

t-statistic of μ95 critical value of N (0 1)2-month crude oil future prices

(a)

ndash4

ndash2

0

2

4

6

8

0

50

100

150

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

Local explosiveness detection based on the time-varyingcoefficient model

t-statistic of ϕ95 critical value of N (0 1)2-month crude oil future prices

(b)

Figure 8 Date-stamping explosive episodes in 2-month future oil prices with the time-varying coefficient model with endogeneity

Complexity 13

7 Discussion

-e oil price rally since the early 2000s has attracted greatinterest in the sources of oil price shocks Some causal ob-servers have stressed that financial speculation has become asignificant driver of the dynamics of oil prices A large body ofliterature emerged that has sought to investigate its existenceIn this paper we employ a novel approach for testing forspeculative bubbles in oil prices In synthesis our empiricalfindings indicate that there is less evidence of exuberance in

the oil future markets Our results are very robust in severalscenarios We argue that the booms and busts in the oil pricemovements during the sample period are driven primarily byeconomic fundamentals rather than speculation

Overall our results confirm the main findings obtainedby structural VAR-based studies including Kilian andMurphy [19] Kilian and Lee [20] Baumeister and Hamilton[21] and Zhou [22] In the past decade the use of structuralVAR models to understand the evolution of the oil price hasbecome common -e traditional SVAR literature

ndash4

ndash2

0

2

4

6

8

0

50

100

15019

88-0

3

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

Local explosiveness detection based on the time-varyingcoefficient model

t-statistic of μ95 critical value of N (0 1)3-month crude oil future prices

(a)

ndash4

ndash2

0

2

4

6

8

0

50

100

150

1988

-03

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

Explosiveness detection based on the time-varyingcoefficient model

t-statistic of ϕ95 critical value of N (0 1)3-month crude oil future prices

(b)

Figure 9 Date-stamping explosive episodes in 3-month future oil prices with the time-varying coefficient model with endogeneity

ndash4

ndash2

0

2

4

6

8

0

50

100

150

Local explosiveness detection based on the time-varyingcoefficient model

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

t-statistic of μ95 critical value of N (0 1)4-month crude oil future prices

(a)

ndash4

ndash2

0

2

4

6

8

0

50

100

150

Explosiveness detection based on the time-varyingcoefficient model

t-statistic of ϕ95 critical value of N (0 1)4-month crude oil futures prices

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

(b)

Figure 10 Date-stamping explosive episodes in 4-month future oil prices with the time-varying coefficient model with endogeneity

14 Complexity

documents that the fundamental drivers of oil price fluc-tuations have been dominated essentially by supply anddemand shocks see eg Kilian [56] Lippi and Nobili [57]Kilian andMurphy [58] and Baumeister and Peersman [59]Kilian and Murphy [19] focused on the role speculation byaugmenting the SVAR framework of their earlier paper byglobal oil inventory data -ey specified a four-variablemodel for the global oil market (the percentage change inglobal oil production the real price of oil a suitable proxy ofcyclical variation in global real economic activity andchanges in global crude oil inventories) -is setting andtheir sign-identified scheme allow them to separately explainvariation in these data as four different types of shocks (1) ashock to the amount of oil pumped out of the ground (ldquooilsupply shockrdquo) (2) an oil flow demand shock (3) an oil-specific oil demand shock and especially (4) a speculative(or inventory) demand shock -eir core finding is that thespike in the real price of crude oil prices during 2003ndash2008was caused by unexpected increase in global oil consump-tion Using two alternative proxies for global above-groundcrude oil inventories Kilian and Lee [20] reestimated themodel of Kilian and Murphy [19] and ruled out the role ofspeculation in explaining the recent surge of the real oil priceon the basis of their estimation results Also Zhou [22]assessed the sensitivity of the findings in Kilian and Murphy[19] to the introduction of various state-of-the-art oil marketmethods She shows that the core conclusions reached byKilian and Murphy [19] are highly robust to some refine-ments corrections and extension of the KilianndashMurphymodel Baumeister and Hamilton [21] recently contendedthat prior information plays a key role in any SVAR modelsand traditional identification approaches can be generalizedto a special case of a Bayesian perspective -ey revisited theimportance of supply and demand shocks in the oil marketand demonstrated that speculative demand shock turns outto be only a small factor in accounting for most historical oilprice shocks

For comparison purpose we have also applied the testsof PWY and PSY which are the most popular econometrictechniques in the literature on speculative bubbles in the oilmarkets If we were to follow the test procedures of PWY andPSY our conclusions are also broadly consistent with therich literature that supports the existence of speculativebubbles in oil prices However these cointegration-basedbubble tests suffer from the joint hypothesis problem ldquoEverytest of a bubble is a joint test of the presence of the bubble inthe data and the validity of the model applied by theeconometricianrdquo (see Giglio et al [60]) More specifically themain drawback of such indirect model-dependent tests ofbubbles typically refers to the difficulty in distinguishingbetween the unknown true model for market fundamentalsand the presence of bubbles (see for example Gurkaynak[37] Pavlidis et al [42] Pavlidis et al [43]) Given thefundamental value of oil price is unobservable the literaturehas resorted to observe economic and financial variableswhich are used to estimate the intrinsic values For exampleboth Phillips and Yu [16] and Caspi et al [39] simplynormalized crude oil prices by US oil inventory data as thefundamental value -is measure is questionable since their

method cannot sufficiently capture the information onmarket fundamentals and neglect the major driversmdashlikeglobal economic activitymdashalso affecting the fundamentalvalue of crude oil -is issue of misspecified fundamentals(either in terms of model misspecification or omitted var-iables or both of these reasons) may make researchers fail toarrive at a conclusion from evidence of explosive pricesalone To make matters worse even though we have cap-tured the true market fundamentals we may still fail todetect bubbles if the explosive behavior in the price of oil juststems from the explosive fundamentals

Compared to earlier studies as briefly mentioned aboveour methodology has the main advantage that by consid-ering the dependence between the random coefficient andsystem shocks we have incorporated endogeneity into thecontinuous-time random coefficient setting and thus we donot need to prespecify the fundamental component of the oilprice -e test results for endogeneity suggest that endo-geneity is important in the data-generating process Inparticular our empirical analysis also finds that there are noexplosive episodes in crude oil markets-is implies that it isnot necessary to require the specification of a fundamentalvalue of the oil price Like Pavlidis et al [43] their methodsand ours have in common that the problem of the well-known joint hypothesis has to be some extent been ef-fectively ameliorated Although our approach is quite dis-tinct we draw the same substantive conclusion thatspeculative bubbles can be ruled out as a key determinant ofthe price of crude oil in the 2000s reaffirming the con-clusions of Kilian and Murphy [19] Baumeister andHamilton [21] Zhou [22] and Fattouh et al [61]

8 Conclusions

Repeated surges and crashes in crude oil prices during lastdecades have sparked a heated debate on their underlyingcauses among academics commentors and market regu-lators One of the central questions concerns whetherfinancialization of oil future markets creates a speculativebubble that subsequently collapses Answering this questionis difficult because one has to rely upon strong assumptionsabout the data-generating mechanism for the fundamentalvalue of the oil price In this paper we present a stylizedmodel of asset pricing for a commodity to address this issueOur theoretical framework disentangles the effects ofspeculative bubbles from the traditional fundamental effectson oil price dynamics We use this setting to illustrate thatbubbles if they are present should demonstrate explosivebehavior in prices We then provide a novel approach forbubble detection that covers various explosive scenarios andfind no evidence of speculative bubbles in oil markets evenduring episodes where the most popular SADF and GSADFmethodologies identify a speculative bubble

Our conclusions have pertinent implications for policymakers Given the historically important role of oil prices inthe real economy one of the serious concerns in policycircles over the past decade has been whether speculativetrading activities in oil future markets are responsible for thelarge price swings For example in their 2008 testimonies to

Complexity 15

the US Senate Commerce Committee Michael Masters andGeorge Soros posited that speculators had fueled price in-creases since 2003 and had caused a part of a ldquosuperbubblerdquoin 2007 and 2008 resulting in significantly higher energycosts for consumers -is widespread bubble view domi-nated policy discussions and efforts to limit speculativetrading in the future market quickly followed As a responseto the excessive speculation pressures on prices theDoddndashFrank Wall Street Reform and Consumer ProtectionAct was enacted by the US Congress in 2010 As a directpolicy implication our results highlight that imposingstricter regulations on trading in oil derivative marketscannot be expected to lower the oil price given the fact thatthe synchronized boom and bust cycles witnessed in the past20 years are mainly due to economic fundamentals whichappear likely to resurge with the business cycle -e aboveanalysis implies that there is no need to enforce additionalregulatory limits against financial arbitrage in the com-modity market

Appendix

The SADF and GSADF Methodologies

Appendix describes the SADF and GSADF testing proce-dures used to detect speculative bubbles in Section 62 For agenetic time series process xt consider the following rolling-window type augmented DickeyndashFuller (ADF) regressionequation

xt ar1r2+ μr1r2

xtminus 1 + 1113944

J

j1ϕj

r1r2Δxtminus j + εt εt sim NID 0 δ2r1r2

1113874 1113875

(A1)

where Δxtminus j with j 1 J are lagged first differences ofxt incorporated to accommodate potential serial correla-tion ar1 r2

μr1 r2 and ϕj

r1 r2 with j 1 J are regression

coefficients the subscripts r1 and r2 refer to fractions of thetotal sample size (ofT observations) that set the start and endpoints of a subsample period and NID represents inde-pendent and normal distribution -e ADF test statisticcorresponding to the unit root null hypothesisH0 μr1 r2

0 is simply the t-ratio

ADFr2r1

1113954μr1 r2

se 1113954μr1 r21113872 1113873

(A2)

and the right-side alternative hypothesis of explosive be-havior is H0 μr1r2

gt 0 We can obtain the standard ADF teststatistic by running regression (A1) on the whole sampleie by specifying r1 0 and r2 1 A large amount of re-search has demonstrated that the standard ADF test ADF10has extremely low power in detecting periodically collapsingbubbles

To address the problem PSW proposed a recursiveregression technique which involves estimating (A1)employing a forward expanding sample Specifically theyused subsamples of the available data to recursively calculatea sequence of ADF statistics denoted by ADFr2

0 -e startpoint of the sample period is fixed at r1 0 while the endpoint r2 increases from r0 (the minimum window size) toone (the full sample period) -e SADF statistic called thesupremum of ADFr2

0 is defined by

SADF r0( 1113857 supr2isin r0 1[ ]

ADFr20

(A3)

and has the following limit distribution under the nullhypothesis

supr2isin r0 1[ ]

1113938r2

0 WdW

1113938r2

0 W21113872 1113873(12)

(A4)

where W is the standard Brownian motion Testing thepresence of explosive behavior entails comparing the ADFr2

0statistic with the right-side critical values obtained throughMonte Carlo simulations

More recently PSY showed that when the sample perioddisplays more than one boom-bust bubble the SADF testprocedure may perform poorly and fail to detect or con-sistently date-stamp alternative bubbles -is deficiencymotivates PSY to formulate an extension of the SADF testthe GSADF test which covers more subsamples of the datathan the SADF by allowing both the end point r2 and thestart point r1 to vary As a result the GSADF test has moreflexibility and gains substantial power making this testconsistent with multiple boom-bust bubbles -e GSADFtest statistic is defined as

GSADF r0( 1113857 supr2isin r0 1[ ]r1isin 0r2minus r0[ ]

ADFr2r1 (A5)

with the limit distribution

supr2isin r0 1[ ]r1isin 0r2minus r0[ ]

(12)rw W r2( 11138572

minus W r1( 11138572

minus rw1113960 1113961 minus 1113938r2

r1W(r)dr W r2( 1113857 minus W r1( 11138571113858 1113859

r12w rw 1113938

r2

r1W(r)2dr minus 1113938

r2

r1W(r)dr1113876 1113877

21113896 1113897

12

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎭

(A6)

where rw r2 minus r1 is the expanding window size Like theSADF test comparison of the GASDF statistic with the right-

tailed critical values from its limit distribution enables us to testfor the unit root hypothesis against explosive behavior

16 Complexity

If the null hypothesis of a unit root in xt is rejectedthen in the second stage the SADF and GSADF tests canidentify the exact episodes during which the xt seriesexhibits explosive dynamics To locate the emergence andconclusion of explosiveness the SADF methodologymatches the sequence of the recursive backward ADF(BADF) test statistics against the right-tailed criticalvalues of the limit distribution of the standard Dick-eyndashFuller test statistic Suppose that re is the originationdate and rf is the finish date of the bubble in the data-ese estimates of dates can be constructed from

1113954re infr2isin r0 1[ ]

r2 ADFr20 gt cv

αr2

1113966 1113967

1113954rf infr2isin 1113954re11113858 1113859

r2 ADFr20 lt cv

αr2

1113966 1113967(A7)

where cvαr2 is 100(1 minus α) the right-tailed critical value of theADF statistic corresponding to the chosen significance levelof α For the GSADF methodology the date-stampingstrategy in this case is based on the backward SADF(BSADF) statistic

BSADFr2r0( 1113857 sup

r1isin 0r2minus r0[ ]ADFr2

r1

(A8)

Likewise the origination date of the bubble is defined asthe first observation that the BSADF statistic is greater thanits critical value

1113954re infr2isin r0 1[ ]

r2 BSADFr2r0( 1113857gt cv

βT

r21113966 1113967 (A9)

and the finish date as the first date after which the BSADFdrops back below its critical value

1113954rf infr2isin 1113954re11113858 1113859

r2 BSADFr2r0( 1113857lt cvβT

r21113966 1113967 (A10)

where cvβTr2 is 100(1 minus βT) the right-tailed critical value of

the SADF based on Tr2 and the corresponding significancelevel is βT

Data Availability

-e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

-e authors declare no conflicts of interest

Acknowledgments

-e authors gratefully acknowledge the financial supportfrom the National Social Science Foundation of China underGrant no 18BTJ039

References

[1] M W Masters Testimony before the US Senate Committee onHomeland Security and Governmental Affairs 2008 httpswwwhsgacsenategovimomediadoc052008Masterspdfattempt2

[2] M W Masters Testimony before the Commodity FuturesTrading Commission 2010 httpscftcgovsitesdefaultfilesidcgroupspublicnewsroomdocumentsfilemetalmarkets032510_masterspdf

[3] J P Kennedy ldquo-e high cost of gambling on oilrdquo Section A p23 New York Times New York NY USANew York Times2012

[4] I-H Cheng and W Xiong ldquoFinancialization of commoditymarketsrdquo Annual Review of Financial Economics vol 6 no 1pp 419ndash441 2014

[5] K Tang and W Xiong ldquoIndex investment and the financi-alization of commoditiesrdquo Financial Analysts Journal vol 68no 6 pp 54ndash74 2012

[6] O J Blanchard and M W Watson ldquoBubbles rational ex-pectations and financial marketsrdquo in Crisis in the Economicand Financial Structure P Wachtel Ed Lexington BooksLexington MA USA pp 295ndash315 1982

[7] B T Diba and H I Grossman ldquoOn the inception of rationalbubblesrdquo e Quarterly Journal of Economics vol 102 no 3pp 697ndash700 1987

[8] G W Evans ldquo-e fragility of sunspots and bubblesrdquo Journalof Monetary Economics vol 23 no 2 pp 297ndash317 1989

[9] G W Evans and S Honkapohja ldquoOn the robustness ofbubbles in linear RE modelsrdquo International Economic Reviewvol 33 no 1 pp 1ndash14 1992

[10] E Pavlidis A Yusupova I Paya et al ldquoEpisodes of exu-berance in housing markets in search of the smoking gunrdquoe Journal of Real Estate Finance and Economics vol 53no 4 pp 419ndash449 2016

[11] E Martınez-Garcıa and V Grossman ldquoExplosive dynamics inhouse prices An exploration of financial market spillovers inhousing markets around the worldrdquo Journal of InternationalMoney and Finance vol 101 p 102103 2020

[12] S Shi A Rahman and B Z Wang ldquoAustralian housingmarket booms fundamentals or speculationlowastrdquo EconomicRecord 2020 In press

[13] P C B Phillips Y Wu and J Yu ldquoExplosive behavior in the1990s NASDAQ when did exuberance escalate asset val-ueslowastrdquo International Economic Review vol 52 no 1pp 201ndash226 2011

[14] P C B Phillips S Shi and J Yu ldquoTesting for multiplebubbles historical episodes of exuberance and collapse in theSampp 500rdquo International Economic Review vol 56 no 4pp 1043ndash1078 2015

[15] P C B Phillips S Shi and J Yu ldquoTesting for multiplebubbles limit theory of real-time detectorsrdquo InternationalEconomic Review vol 56 no 4 pp 1079ndash1134 2015

[16] P C B Phillips and J Yu ldquoDating the timeline of financialbubbles during the subprime crisisrdquo Quantitative Economicsvol 2 no 3 pp 455ndash491 2011

[17] D Tsvetanov J Coakley and N Kellard ldquoBubbling over thebehaviour of oil futures along the yield curverdquo Journal ofEmpirical Finance vol 38 no Part B pp 516ndash533 2016

[18] I Figuerola-Ferretti J R Mccrorie and I ParaskevopoulosldquoMild explosivity in recent crude oil pricesrdquo Energy Eco-nomics vol 87 p 104387 2020

[19] L Kilian and D P Murphy ldquo-e role of inventories andspeculative trading in the global market for crude oilrdquo Journalof Applied Econometrics vol 29 no 3 pp 454ndash478 2014

Complexity 17

1983 to June 30 2020 time horizons for explosive behavioridentified by PWY and PSY are summarized in Table 3 Inline with previous studies we could conclude from bothPWY and PSY tests that there is evidence of bubbles in thecrude oil market data However as shown in Gurkaynak [37]and Pavlidis et al [43] explosive dynamics in asset prices perse do not constitute evidence either in support of or againstspeculative bubbles since such explosive behavior may

actually reflect explosive market fundamentals or bothfactors As aforementioned our random coefficient autor-egressive model framework has considered endogeneityfeedbacks that to some extent alleviate this problemMoreover the evidence in Figures 3 and 4 suggests theabsence of explosive dynamics in the crude oil market -efact permits us to safely ignore the distinction betweenbubbles or explosive fundamentals

1988

-03

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

ndash4

ndash2

0

2

4

6

8

0

20

40

60

80

100

120

140

t-statistic of k95 critical value of N (0 1)Crude oil future prices

Instability detection based on the time-varyingcoefficient model

(a)

Local explosiveness detection based on the time-varyingcoefficient model

1988

-03

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

t-statistic of μ95 critical value of N (0 1)Crude oil future prices

ndash4

ndash2

0

2

4

6

8

0

20

40

60

80

100

120

140

(b)

Explosiveness detection based on the time-varyingcoefficient model

1988

-03

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

t-statistic of ϕ95 critical value of N (0 1)Crude oil future prices

ndash4

ndash2

0

2

4

6

8

0

20

40

60

80

100

120

140

(c)

LBItest statistic for testing time-varying coefficient19

88-0

3

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

LBItest95 critical value of N (0 1)

ndash5

ndash10

0

5

10

15

20

(d)

Figure 3 Date-stamping unstable and explosive episodes in oil prices with the time-varying coefficient model assuming no endogeneity

Table 1 Time horizons of unstable and explosive periods identified by models assuming no endogeneity

Unstable Locally explosive ExplosiveAug 2005-Sep 2005 Dec 2007ndashSep 2008 mdash

Jan 2006ndashMar 2020 Jun 2009ndashOct 2014 mdashJun 2018ndashSep 2018 mdash

10 Complexity

63 Robustness of the Results In the following we providetwo robustness checks to assess whether our previousconclusions of the paper still hold

First in our benchmark model our use of front-monthfuture price instead of other future prices at different ma-turities is justified by its being the most liquid future contractwith the highest open interest To explore the sensitivity ofour results to the future contracts at different maturities werepeat the analysis described above using the prices forlonger-term contracts (ie 2 3 and 4 months-ahead fu-tures) It should be pointed out that the WTI future contractusually expires on the third business day before the 25thcalendar day of the contract month proceeding the deliverymonth

Second becauseWTI crude oil is by tradition traded innominal US dollars per barrel it might be of concern that

inflation and other monetary variables may have realeffects on oil price It is therefore of interest to use realrather than nominal price In fact recent work documentsthat the use of either price is inconsequential in practice[55] Although we mainly focus on nominal prices forrobustness check we shall report the results for realprices To convert nominal daily series into daily realseries we use linearly interpolated monthly ConsumerPrice Index (CPI) Monthly CPI series for the UnitedStates is obtained from the websites of the Federal ReserveBank of St Louis

Further evidence presented in Figures 7ndash10 demon-strates that our results are remarkably robust to thesemodifications and the conclusions remain qualitativelyunchanged

ndash4

ndash2

0

2

4

6

8

0

20

40

60

80

100

120

140

1988

-03

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

Instability detection based on the time-varyingcoefficient model

t-statistic of k95 critical value of N (0 1)Crude oil future prices

(a)

ndash4

ndash2

0

2

4

6

8

0

20

40

60

80

100

120

140

1988

-03

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

t-statistic of μ95 critical value of N (0 1)Crude oil future prices

Local explosiveness detection based on the time-varyingcoefficient model

(b)

ndash4

ndash2

0

2

4

6

8

0

20

40

60

80

100

120

140

1988

-03

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

Explosiveness detection based on the time-varyingcoefficient model

t-statistic of ϕ95 critical value of N (0 1)Crude oil future prices

(c)

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

LR-statistic95 critical values of x2 (1)Realized variance

700

600

500

400

300

200

100

0

ndash100

350

300

250

200

150

100

50

0

Test for endogeneity in time-varyingcoefficient model

(d)

Figure 4 Date-stamping unstable and explosive episodes in oil prices with the time-varying coefficient model with endogeneity

Complexity 11

2

3

4

5

6

7

8

0

50

100

150

200

250

300

350

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

LLF0

LLF1

Realized variance of crude oil future prices

Figure 5 Realized variance and likelihood values for random coefficient regression

Table 2 Time horizons of unstable and explosive periods detected by models with endogeneity

Unstable Locally explosive ExplosiveMay 2008ndashJul 2008 mdash mdashDec 2010ndashOct 2014 mdash mdashNov 2016ndashApr 2017 mdash mdashJul 2017ndashFeb 2020 mdash mdash

ndash4

ndash2

0

2

4

6

8

0

20

40

60

80

100

120

140

1988

-03

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

Explosiveness detection using SADF methodology

BADF sequence95 critical valuesCrude oil future prices

(a)

ndash4

ndash2

0

2

4

6

8

0

20

40

60

80

100

120

140

1988

-03

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

Explosiveness detection using GSADF methodology

BSADF sequence95 critical valuesCrude oil future prices

(b)

Figure 6 Date-stamping explosive episodes in oil prices with the time-invariant coefficient model

12 Complexity

Table 3 Time horizons of explosive behavior identified by SADF and GSADF methodologies

SADF GSADFAug 2005ndashAug 2006 Oct 2011ndashApr 2012 Sep 1990-Oct 1990Jun 2007ndashOct 2008 Aug 2012-Sep 2012 Feb 1991-Mar 1991Jun 2009-Jul 2009 Jul 2013-Aug 2013 Oct 2004-Nov 2004Feb 2011ndashApr 2011 Feb 2014-Mar 2014 Feb 2005-Mar 2005

Jul 2011ndashAug 2012 Jun 2014-Jul 2014May 2005ndashJan 2006Mar 2006ndashAug 2006Jul 2007ndashAug 2008

ndash4

ndash2

0

2

4

6

8

0

10

20

30

40

50

60

70

1988

-03

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

95 critical value of N (0 1)Real oil future prices

t-statistic of μ~

Local explosiveness detection based on the time-varyingcoefficient model

(a)

ndash4

ndash2

0

2

4

6

8

0

10

20

30

40

50

60

70

1988

-03

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

95 critical value of N (0 1)Real oil future prices

t-statistic of ϕ

Explosiveness detection based on the time-varyingcoefficient model

(b)

Figure 7 Date-stamping explosive episodes in real oil prices with the time-varying coefficient model with endogeneity

ndash4

ndash2

0

2

4

6

8

0

50

100

150

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

Local explosiveness detection based on the time-varyingcoefficient model

t-statistic of μ95 critical value of N (0 1)2-month crude oil future prices

(a)

ndash4

ndash2

0

2

4

6

8

0

50

100

150

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

Local explosiveness detection based on the time-varyingcoefficient model

t-statistic of ϕ95 critical value of N (0 1)2-month crude oil future prices

(b)

Figure 8 Date-stamping explosive episodes in 2-month future oil prices with the time-varying coefficient model with endogeneity

Complexity 13

7 Discussion

-e oil price rally since the early 2000s has attracted greatinterest in the sources of oil price shocks Some causal ob-servers have stressed that financial speculation has become asignificant driver of the dynamics of oil prices A large body ofliterature emerged that has sought to investigate its existenceIn this paper we employ a novel approach for testing forspeculative bubbles in oil prices In synthesis our empiricalfindings indicate that there is less evidence of exuberance in

the oil future markets Our results are very robust in severalscenarios We argue that the booms and busts in the oil pricemovements during the sample period are driven primarily byeconomic fundamentals rather than speculation

Overall our results confirm the main findings obtainedby structural VAR-based studies including Kilian andMurphy [19] Kilian and Lee [20] Baumeister and Hamilton[21] and Zhou [22] In the past decade the use of structuralVAR models to understand the evolution of the oil price hasbecome common -e traditional SVAR literature

ndash4

ndash2

0

2

4

6

8

0

50

100

15019

88-0

3

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

Local explosiveness detection based on the time-varyingcoefficient model

t-statistic of μ95 critical value of N (0 1)3-month crude oil future prices

(a)

ndash4

ndash2

0

2

4

6

8

0

50

100

150

1988

-03

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

Explosiveness detection based on the time-varyingcoefficient model

t-statistic of ϕ95 critical value of N (0 1)3-month crude oil future prices

(b)

Figure 9 Date-stamping explosive episodes in 3-month future oil prices with the time-varying coefficient model with endogeneity

ndash4

ndash2

0

2

4

6

8

0

50

100

150

Local explosiveness detection based on the time-varyingcoefficient model

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

t-statistic of μ95 critical value of N (0 1)4-month crude oil future prices

(a)

ndash4

ndash2

0

2

4

6

8

0

50

100

150

Explosiveness detection based on the time-varyingcoefficient model

t-statistic of ϕ95 critical value of N (0 1)4-month crude oil futures prices

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

(b)

Figure 10 Date-stamping explosive episodes in 4-month future oil prices with the time-varying coefficient model with endogeneity

14 Complexity

documents that the fundamental drivers of oil price fluc-tuations have been dominated essentially by supply anddemand shocks see eg Kilian [56] Lippi and Nobili [57]Kilian andMurphy [58] and Baumeister and Peersman [59]Kilian and Murphy [19] focused on the role speculation byaugmenting the SVAR framework of their earlier paper byglobal oil inventory data -ey specified a four-variablemodel for the global oil market (the percentage change inglobal oil production the real price of oil a suitable proxy ofcyclical variation in global real economic activity andchanges in global crude oil inventories) -is setting andtheir sign-identified scheme allow them to separately explainvariation in these data as four different types of shocks (1) ashock to the amount of oil pumped out of the ground (ldquooilsupply shockrdquo) (2) an oil flow demand shock (3) an oil-specific oil demand shock and especially (4) a speculative(or inventory) demand shock -eir core finding is that thespike in the real price of crude oil prices during 2003ndash2008was caused by unexpected increase in global oil consump-tion Using two alternative proxies for global above-groundcrude oil inventories Kilian and Lee [20] reestimated themodel of Kilian and Murphy [19] and ruled out the role ofspeculation in explaining the recent surge of the real oil priceon the basis of their estimation results Also Zhou [22]assessed the sensitivity of the findings in Kilian and Murphy[19] to the introduction of various state-of-the-art oil marketmethods She shows that the core conclusions reached byKilian and Murphy [19] are highly robust to some refine-ments corrections and extension of the KilianndashMurphymodel Baumeister and Hamilton [21] recently contendedthat prior information plays a key role in any SVAR modelsand traditional identification approaches can be generalizedto a special case of a Bayesian perspective -ey revisited theimportance of supply and demand shocks in the oil marketand demonstrated that speculative demand shock turns outto be only a small factor in accounting for most historical oilprice shocks

For comparison purpose we have also applied the testsof PWY and PSY which are the most popular econometrictechniques in the literature on speculative bubbles in the oilmarkets If we were to follow the test procedures of PWY andPSY our conclusions are also broadly consistent with therich literature that supports the existence of speculativebubbles in oil prices However these cointegration-basedbubble tests suffer from the joint hypothesis problem ldquoEverytest of a bubble is a joint test of the presence of the bubble inthe data and the validity of the model applied by theeconometricianrdquo (see Giglio et al [60]) More specifically themain drawback of such indirect model-dependent tests ofbubbles typically refers to the difficulty in distinguishingbetween the unknown true model for market fundamentalsand the presence of bubbles (see for example Gurkaynak[37] Pavlidis et al [42] Pavlidis et al [43]) Given thefundamental value of oil price is unobservable the literaturehas resorted to observe economic and financial variableswhich are used to estimate the intrinsic values For exampleboth Phillips and Yu [16] and Caspi et al [39] simplynormalized crude oil prices by US oil inventory data as thefundamental value -is measure is questionable since their

method cannot sufficiently capture the information onmarket fundamentals and neglect the major driversmdashlikeglobal economic activitymdashalso affecting the fundamentalvalue of crude oil -is issue of misspecified fundamentals(either in terms of model misspecification or omitted var-iables or both of these reasons) may make researchers fail toarrive at a conclusion from evidence of explosive pricesalone To make matters worse even though we have cap-tured the true market fundamentals we may still fail todetect bubbles if the explosive behavior in the price of oil juststems from the explosive fundamentals

Compared to earlier studies as briefly mentioned aboveour methodology has the main advantage that by consid-ering the dependence between the random coefficient andsystem shocks we have incorporated endogeneity into thecontinuous-time random coefficient setting and thus we donot need to prespecify the fundamental component of the oilprice -e test results for endogeneity suggest that endo-geneity is important in the data-generating process Inparticular our empirical analysis also finds that there are noexplosive episodes in crude oil markets-is implies that it isnot necessary to require the specification of a fundamentalvalue of the oil price Like Pavlidis et al [43] their methodsand ours have in common that the problem of the well-known joint hypothesis has to be some extent been ef-fectively ameliorated Although our approach is quite dis-tinct we draw the same substantive conclusion thatspeculative bubbles can be ruled out as a key determinant ofthe price of crude oil in the 2000s reaffirming the con-clusions of Kilian and Murphy [19] Baumeister andHamilton [21] Zhou [22] and Fattouh et al [61]

8 Conclusions

Repeated surges and crashes in crude oil prices during lastdecades have sparked a heated debate on their underlyingcauses among academics commentors and market regu-lators One of the central questions concerns whetherfinancialization of oil future markets creates a speculativebubble that subsequently collapses Answering this questionis difficult because one has to rely upon strong assumptionsabout the data-generating mechanism for the fundamentalvalue of the oil price In this paper we present a stylizedmodel of asset pricing for a commodity to address this issueOur theoretical framework disentangles the effects ofspeculative bubbles from the traditional fundamental effectson oil price dynamics We use this setting to illustrate thatbubbles if they are present should demonstrate explosivebehavior in prices We then provide a novel approach forbubble detection that covers various explosive scenarios andfind no evidence of speculative bubbles in oil markets evenduring episodes where the most popular SADF and GSADFmethodologies identify a speculative bubble

Our conclusions have pertinent implications for policymakers Given the historically important role of oil prices inthe real economy one of the serious concerns in policycircles over the past decade has been whether speculativetrading activities in oil future markets are responsible for thelarge price swings For example in their 2008 testimonies to

Complexity 15

the US Senate Commerce Committee Michael Masters andGeorge Soros posited that speculators had fueled price in-creases since 2003 and had caused a part of a ldquosuperbubblerdquoin 2007 and 2008 resulting in significantly higher energycosts for consumers -is widespread bubble view domi-nated policy discussions and efforts to limit speculativetrading in the future market quickly followed As a responseto the excessive speculation pressures on prices theDoddndashFrank Wall Street Reform and Consumer ProtectionAct was enacted by the US Congress in 2010 As a directpolicy implication our results highlight that imposingstricter regulations on trading in oil derivative marketscannot be expected to lower the oil price given the fact thatthe synchronized boom and bust cycles witnessed in the past20 years are mainly due to economic fundamentals whichappear likely to resurge with the business cycle -e aboveanalysis implies that there is no need to enforce additionalregulatory limits against financial arbitrage in the com-modity market

Appendix

The SADF and GSADF Methodologies

Appendix describes the SADF and GSADF testing proce-dures used to detect speculative bubbles in Section 62 For agenetic time series process xt consider the following rolling-window type augmented DickeyndashFuller (ADF) regressionequation

xt ar1r2+ μr1r2

xtminus 1 + 1113944

J

j1ϕj

r1r2Δxtminus j + εt εt sim NID 0 δ2r1r2

1113874 1113875

(A1)

where Δxtminus j with j 1 J are lagged first differences ofxt incorporated to accommodate potential serial correla-tion ar1 r2

μr1 r2 and ϕj

r1 r2 with j 1 J are regression

coefficients the subscripts r1 and r2 refer to fractions of thetotal sample size (ofT observations) that set the start and endpoints of a subsample period and NID represents inde-pendent and normal distribution -e ADF test statisticcorresponding to the unit root null hypothesisH0 μr1 r2

0 is simply the t-ratio

ADFr2r1

1113954μr1 r2

se 1113954μr1 r21113872 1113873

(A2)

and the right-side alternative hypothesis of explosive be-havior is H0 μr1r2

gt 0 We can obtain the standard ADF teststatistic by running regression (A1) on the whole sampleie by specifying r1 0 and r2 1 A large amount of re-search has demonstrated that the standard ADF test ADF10has extremely low power in detecting periodically collapsingbubbles

To address the problem PSW proposed a recursiveregression technique which involves estimating (A1)employing a forward expanding sample Specifically theyused subsamples of the available data to recursively calculatea sequence of ADF statistics denoted by ADFr2

0 -e startpoint of the sample period is fixed at r1 0 while the endpoint r2 increases from r0 (the minimum window size) toone (the full sample period) -e SADF statistic called thesupremum of ADFr2

0 is defined by

SADF r0( 1113857 supr2isin r0 1[ ]

ADFr20

(A3)

and has the following limit distribution under the nullhypothesis

supr2isin r0 1[ ]

1113938r2

0 WdW

1113938r2

0 W21113872 1113873(12)

(A4)

where W is the standard Brownian motion Testing thepresence of explosive behavior entails comparing the ADFr2

0statistic with the right-side critical values obtained throughMonte Carlo simulations

More recently PSY showed that when the sample perioddisplays more than one boom-bust bubble the SADF testprocedure may perform poorly and fail to detect or con-sistently date-stamp alternative bubbles -is deficiencymotivates PSY to formulate an extension of the SADF testthe GSADF test which covers more subsamples of the datathan the SADF by allowing both the end point r2 and thestart point r1 to vary As a result the GSADF test has moreflexibility and gains substantial power making this testconsistent with multiple boom-bust bubbles -e GSADFtest statistic is defined as

GSADF r0( 1113857 supr2isin r0 1[ ]r1isin 0r2minus r0[ ]

ADFr2r1 (A5)

with the limit distribution

supr2isin r0 1[ ]r1isin 0r2minus r0[ ]

(12)rw W r2( 11138572

minus W r1( 11138572

minus rw1113960 1113961 minus 1113938r2

r1W(r)dr W r2( 1113857 minus W r1( 11138571113858 1113859

r12w rw 1113938

r2

r1W(r)2dr minus 1113938

r2

r1W(r)dr1113876 1113877

21113896 1113897

12

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎭

(A6)

where rw r2 minus r1 is the expanding window size Like theSADF test comparison of the GASDF statistic with the right-

tailed critical values from its limit distribution enables us to testfor the unit root hypothesis against explosive behavior

16 Complexity

If the null hypothesis of a unit root in xt is rejectedthen in the second stage the SADF and GSADF tests canidentify the exact episodes during which the xt seriesexhibits explosive dynamics To locate the emergence andconclusion of explosiveness the SADF methodologymatches the sequence of the recursive backward ADF(BADF) test statistics against the right-tailed criticalvalues of the limit distribution of the standard Dick-eyndashFuller test statistic Suppose that re is the originationdate and rf is the finish date of the bubble in the data-ese estimates of dates can be constructed from

1113954re infr2isin r0 1[ ]

r2 ADFr20 gt cv

αr2

1113966 1113967

1113954rf infr2isin 1113954re11113858 1113859

r2 ADFr20 lt cv

αr2

1113966 1113967(A7)

where cvαr2 is 100(1 minus α) the right-tailed critical value of theADF statistic corresponding to the chosen significance levelof α For the GSADF methodology the date-stampingstrategy in this case is based on the backward SADF(BSADF) statistic

BSADFr2r0( 1113857 sup

r1isin 0r2minus r0[ ]ADFr2

r1

(A8)

Likewise the origination date of the bubble is defined asthe first observation that the BSADF statistic is greater thanits critical value

1113954re infr2isin r0 1[ ]

r2 BSADFr2r0( 1113857gt cv

βT

r21113966 1113967 (A9)

and the finish date as the first date after which the BSADFdrops back below its critical value

1113954rf infr2isin 1113954re11113858 1113859

r2 BSADFr2r0( 1113857lt cvβT

r21113966 1113967 (A10)

where cvβTr2 is 100(1 minus βT) the right-tailed critical value of

the SADF based on Tr2 and the corresponding significancelevel is βT

Data Availability

-e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

-e authors declare no conflicts of interest

Acknowledgments

-e authors gratefully acknowledge the financial supportfrom the National Social Science Foundation of China underGrant no 18BTJ039

References

[1] M W Masters Testimony before the US Senate Committee onHomeland Security and Governmental Affairs 2008 httpswwwhsgacsenategovimomediadoc052008Masterspdfattempt2

[2] M W Masters Testimony before the Commodity FuturesTrading Commission 2010 httpscftcgovsitesdefaultfilesidcgroupspublicnewsroomdocumentsfilemetalmarkets032510_masterspdf

[3] J P Kennedy ldquo-e high cost of gambling on oilrdquo Section A p23 New York Times New York NY USANew York Times2012

[4] I-H Cheng and W Xiong ldquoFinancialization of commoditymarketsrdquo Annual Review of Financial Economics vol 6 no 1pp 419ndash441 2014

[5] K Tang and W Xiong ldquoIndex investment and the financi-alization of commoditiesrdquo Financial Analysts Journal vol 68no 6 pp 54ndash74 2012

[6] O J Blanchard and M W Watson ldquoBubbles rational ex-pectations and financial marketsrdquo in Crisis in the Economicand Financial Structure P Wachtel Ed Lexington BooksLexington MA USA pp 295ndash315 1982

[7] B T Diba and H I Grossman ldquoOn the inception of rationalbubblesrdquo e Quarterly Journal of Economics vol 102 no 3pp 697ndash700 1987

[8] G W Evans ldquo-e fragility of sunspots and bubblesrdquo Journalof Monetary Economics vol 23 no 2 pp 297ndash317 1989

[9] G W Evans and S Honkapohja ldquoOn the robustness ofbubbles in linear RE modelsrdquo International Economic Reviewvol 33 no 1 pp 1ndash14 1992

[10] E Pavlidis A Yusupova I Paya et al ldquoEpisodes of exu-berance in housing markets in search of the smoking gunrdquoe Journal of Real Estate Finance and Economics vol 53no 4 pp 419ndash449 2016

[11] E Martınez-Garcıa and V Grossman ldquoExplosive dynamics inhouse prices An exploration of financial market spillovers inhousing markets around the worldrdquo Journal of InternationalMoney and Finance vol 101 p 102103 2020

[12] S Shi A Rahman and B Z Wang ldquoAustralian housingmarket booms fundamentals or speculationlowastrdquo EconomicRecord 2020 In press

[13] P C B Phillips Y Wu and J Yu ldquoExplosive behavior in the1990s NASDAQ when did exuberance escalate asset val-ueslowastrdquo International Economic Review vol 52 no 1pp 201ndash226 2011

[14] P C B Phillips S Shi and J Yu ldquoTesting for multiplebubbles historical episodes of exuberance and collapse in theSampp 500rdquo International Economic Review vol 56 no 4pp 1043ndash1078 2015

[15] P C B Phillips S Shi and J Yu ldquoTesting for multiplebubbles limit theory of real-time detectorsrdquo InternationalEconomic Review vol 56 no 4 pp 1079ndash1134 2015

[16] P C B Phillips and J Yu ldquoDating the timeline of financialbubbles during the subprime crisisrdquo Quantitative Economicsvol 2 no 3 pp 455ndash491 2011

[17] D Tsvetanov J Coakley and N Kellard ldquoBubbling over thebehaviour of oil futures along the yield curverdquo Journal ofEmpirical Finance vol 38 no Part B pp 516ndash533 2016

[18] I Figuerola-Ferretti J R Mccrorie and I ParaskevopoulosldquoMild explosivity in recent crude oil pricesrdquo Energy Eco-nomics vol 87 p 104387 2020

[19] L Kilian and D P Murphy ldquo-e role of inventories andspeculative trading in the global market for crude oilrdquo Journalof Applied Econometrics vol 29 no 3 pp 454ndash478 2014

Complexity 17

63 Robustness of the Results In the following we providetwo robustness checks to assess whether our previousconclusions of the paper still hold

First in our benchmark model our use of front-monthfuture price instead of other future prices at different ma-turities is justified by its being the most liquid future contractwith the highest open interest To explore the sensitivity ofour results to the future contracts at different maturities werepeat the analysis described above using the prices forlonger-term contracts (ie 2 3 and 4 months-ahead fu-tures) It should be pointed out that the WTI future contractusually expires on the third business day before the 25thcalendar day of the contract month proceeding the deliverymonth

Second becauseWTI crude oil is by tradition traded innominal US dollars per barrel it might be of concern that

inflation and other monetary variables may have realeffects on oil price It is therefore of interest to use realrather than nominal price In fact recent work documentsthat the use of either price is inconsequential in practice[55] Although we mainly focus on nominal prices forrobustness check we shall report the results for realprices To convert nominal daily series into daily realseries we use linearly interpolated monthly ConsumerPrice Index (CPI) Monthly CPI series for the UnitedStates is obtained from the websites of the Federal ReserveBank of St Louis

Further evidence presented in Figures 7ndash10 demon-strates that our results are remarkably robust to thesemodifications and the conclusions remain qualitativelyunchanged

ndash4

ndash2

0

2

4

6

8

0

20

40

60

80

100

120

140

1988

-03

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

Instability detection based on the time-varyingcoefficient model

t-statistic of k95 critical value of N (0 1)Crude oil future prices

(a)

ndash4

ndash2

0

2

4

6

8

0

20

40

60

80

100

120

140

1988

-03

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

t-statistic of μ95 critical value of N (0 1)Crude oil future prices

Local explosiveness detection based on the time-varyingcoefficient model

(b)

ndash4

ndash2

0

2

4

6

8

0

20

40

60

80

100

120

140

1988

-03

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

Explosiveness detection based on the time-varyingcoefficient model

t-statistic of ϕ95 critical value of N (0 1)Crude oil future prices

(c)

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

LR-statistic95 critical values of x2 (1)Realized variance

700

600

500

400

300

200

100

0

ndash100

350

300

250

200

150

100

50

0

Test for endogeneity in time-varyingcoefficient model

(d)

Figure 4 Date-stamping unstable and explosive episodes in oil prices with the time-varying coefficient model with endogeneity

Complexity 11

2

3

4

5

6

7

8

0

50

100

150

200

250

300

350

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

LLF0

LLF1

Realized variance of crude oil future prices

Figure 5 Realized variance and likelihood values for random coefficient regression

Table 2 Time horizons of unstable and explosive periods detected by models with endogeneity

Unstable Locally explosive ExplosiveMay 2008ndashJul 2008 mdash mdashDec 2010ndashOct 2014 mdash mdashNov 2016ndashApr 2017 mdash mdashJul 2017ndashFeb 2020 mdash mdash

ndash4

ndash2

0

2

4

6

8

0

20

40

60

80

100

120

140

1988

-03

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

Explosiveness detection using SADF methodology

BADF sequence95 critical valuesCrude oil future prices

(a)

ndash4

ndash2

0

2

4

6

8

0

20

40

60

80

100

120

140

1988

-03

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

Explosiveness detection using GSADF methodology

BSADF sequence95 critical valuesCrude oil future prices

(b)

Figure 6 Date-stamping explosive episodes in oil prices with the time-invariant coefficient model

12 Complexity

Table 3 Time horizons of explosive behavior identified by SADF and GSADF methodologies

SADF GSADFAug 2005ndashAug 2006 Oct 2011ndashApr 2012 Sep 1990-Oct 1990Jun 2007ndashOct 2008 Aug 2012-Sep 2012 Feb 1991-Mar 1991Jun 2009-Jul 2009 Jul 2013-Aug 2013 Oct 2004-Nov 2004Feb 2011ndashApr 2011 Feb 2014-Mar 2014 Feb 2005-Mar 2005

Jul 2011ndashAug 2012 Jun 2014-Jul 2014May 2005ndashJan 2006Mar 2006ndashAug 2006Jul 2007ndashAug 2008

ndash4

ndash2

0

2

4

6

8

0

10

20

30

40

50

60

70

1988

-03

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

95 critical value of N (0 1)Real oil future prices

t-statistic of μ~

Local explosiveness detection based on the time-varyingcoefficient model

(a)

ndash4

ndash2

0

2

4

6

8

0

10

20

30

40

50

60

70

1988

-03

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

95 critical value of N (0 1)Real oil future prices

t-statistic of ϕ

Explosiveness detection based on the time-varyingcoefficient model

(b)

Figure 7 Date-stamping explosive episodes in real oil prices with the time-varying coefficient model with endogeneity

ndash4

ndash2

0

2

4

6

8

0

50

100

150

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

Local explosiveness detection based on the time-varyingcoefficient model

t-statistic of μ95 critical value of N (0 1)2-month crude oil future prices

(a)

ndash4

ndash2

0

2

4

6

8

0

50

100

150

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

Local explosiveness detection based on the time-varyingcoefficient model

t-statistic of ϕ95 critical value of N (0 1)2-month crude oil future prices

(b)

Figure 8 Date-stamping explosive episodes in 2-month future oil prices with the time-varying coefficient model with endogeneity

Complexity 13

7 Discussion

-e oil price rally since the early 2000s has attracted greatinterest in the sources of oil price shocks Some causal ob-servers have stressed that financial speculation has become asignificant driver of the dynamics of oil prices A large body ofliterature emerged that has sought to investigate its existenceIn this paper we employ a novel approach for testing forspeculative bubbles in oil prices In synthesis our empiricalfindings indicate that there is less evidence of exuberance in

the oil future markets Our results are very robust in severalscenarios We argue that the booms and busts in the oil pricemovements during the sample period are driven primarily byeconomic fundamentals rather than speculation

Overall our results confirm the main findings obtainedby structural VAR-based studies including Kilian andMurphy [19] Kilian and Lee [20] Baumeister and Hamilton[21] and Zhou [22] In the past decade the use of structuralVAR models to understand the evolution of the oil price hasbecome common -e traditional SVAR literature

ndash4

ndash2

0

2

4

6

8

0

50

100

15019

88-0

3

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

Local explosiveness detection based on the time-varyingcoefficient model

t-statistic of μ95 critical value of N (0 1)3-month crude oil future prices

(a)

ndash4

ndash2

0

2

4

6

8

0

50

100

150

1988

-03

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

Explosiveness detection based on the time-varyingcoefficient model

t-statistic of ϕ95 critical value of N (0 1)3-month crude oil future prices

(b)

Figure 9 Date-stamping explosive episodes in 3-month future oil prices with the time-varying coefficient model with endogeneity

ndash4

ndash2

0

2

4

6

8

0

50

100

150

Local explosiveness detection based on the time-varyingcoefficient model

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

t-statistic of μ95 critical value of N (0 1)4-month crude oil future prices

(a)

ndash4

ndash2

0

2

4

6

8

0

50

100

150

Explosiveness detection based on the time-varyingcoefficient model

t-statistic of ϕ95 critical value of N (0 1)4-month crude oil futures prices

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

(b)

Figure 10 Date-stamping explosive episodes in 4-month future oil prices with the time-varying coefficient model with endogeneity

14 Complexity

documents that the fundamental drivers of oil price fluc-tuations have been dominated essentially by supply anddemand shocks see eg Kilian [56] Lippi and Nobili [57]Kilian andMurphy [58] and Baumeister and Peersman [59]Kilian and Murphy [19] focused on the role speculation byaugmenting the SVAR framework of their earlier paper byglobal oil inventory data -ey specified a four-variablemodel for the global oil market (the percentage change inglobal oil production the real price of oil a suitable proxy ofcyclical variation in global real economic activity andchanges in global crude oil inventories) -is setting andtheir sign-identified scheme allow them to separately explainvariation in these data as four different types of shocks (1) ashock to the amount of oil pumped out of the ground (ldquooilsupply shockrdquo) (2) an oil flow demand shock (3) an oil-specific oil demand shock and especially (4) a speculative(or inventory) demand shock -eir core finding is that thespike in the real price of crude oil prices during 2003ndash2008was caused by unexpected increase in global oil consump-tion Using two alternative proxies for global above-groundcrude oil inventories Kilian and Lee [20] reestimated themodel of Kilian and Murphy [19] and ruled out the role ofspeculation in explaining the recent surge of the real oil priceon the basis of their estimation results Also Zhou [22]assessed the sensitivity of the findings in Kilian and Murphy[19] to the introduction of various state-of-the-art oil marketmethods She shows that the core conclusions reached byKilian and Murphy [19] are highly robust to some refine-ments corrections and extension of the KilianndashMurphymodel Baumeister and Hamilton [21] recently contendedthat prior information plays a key role in any SVAR modelsand traditional identification approaches can be generalizedto a special case of a Bayesian perspective -ey revisited theimportance of supply and demand shocks in the oil marketand demonstrated that speculative demand shock turns outto be only a small factor in accounting for most historical oilprice shocks

For comparison purpose we have also applied the testsof PWY and PSY which are the most popular econometrictechniques in the literature on speculative bubbles in the oilmarkets If we were to follow the test procedures of PWY andPSY our conclusions are also broadly consistent with therich literature that supports the existence of speculativebubbles in oil prices However these cointegration-basedbubble tests suffer from the joint hypothesis problem ldquoEverytest of a bubble is a joint test of the presence of the bubble inthe data and the validity of the model applied by theeconometricianrdquo (see Giglio et al [60]) More specifically themain drawback of such indirect model-dependent tests ofbubbles typically refers to the difficulty in distinguishingbetween the unknown true model for market fundamentalsand the presence of bubbles (see for example Gurkaynak[37] Pavlidis et al [42] Pavlidis et al [43]) Given thefundamental value of oil price is unobservable the literaturehas resorted to observe economic and financial variableswhich are used to estimate the intrinsic values For exampleboth Phillips and Yu [16] and Caspi et al [39] simplynormalized crude oil prices by US oil inventory data as thefundamental value -is measure is questionable since their

method cannot sufficiently capture the information onmarket fundamentals and neglect the major driversmdashlikeglobal economic activitymdashalso affecting the fundamentalvalue of crude oil -is issue of misspecified fundamentals(either in terms of model misspecification or omitted var-iables or both of these reasons) may make researchers fail toarrive at a conclusion from evidence of explosive pricesalone To make matters worse even though we have cap-tured the true market fundamentals we may still fail todetect bubbles if the explosive behavior in the price of oil juststems from the explosive fundamentals

Compared to earlier studies as briefly mentioned aboveour methodology has the main advantage that by consid-ering the dependence between the random coefficient andsystem shocks we have incorporated endogeneity into thecontinuous-time random coefficient setting and thus we donot need to prespecify the fundamental component of the oilprice -e test results for endogeneity suggest that endo-geneity is important in the data-generating process Inparticular our empirical analysis also finds that there are noexplosive episodes in crude oil markets-is implies that it isnot necessary to require the specification of a fundamentalvalue of the oil price Like Pavlidis et al [43] their methodsand ours have in common that the problem of the well-known joint hypothesis has to be some extent been ef-fectively ameliorated Although our approach is quite dis-tinct we draw the same substantive conclusion thatspeculative bubbles can be ruled out as a key determinant ofthe price of crude oil in the 2000s reaffirming the con-clusions of Kilian and Murphy [19] Baumeister andHamilton [21] Zhou [22] and Fattouh et al [61]

8 Conclusions

Repeated surges and crashes in crude oil prices during lastdecades have sparked a heated debate on their underlyingcauses among academics commentors and market regu-lators One of the central questions concerns whetherfinancialization of oil future markets creates a speculativebubble that subsequently collapses Answering this questionis difficult because one has to rely upon strong assumptionsabout the data-generating mechanism for the fundamentalvalue of the oil price In this paper we present a stylizedmodel of asset pricing for a commodity to address this issueOur theoretical framework disentangles the effects ofspeculative bubbles from the traditional fundamental effectson oil price dynamics We use this setting to illustrate thatbubbles if they are present should demonstrate explosivebehavior in prices We then provide a novel approach forbubble detection that covers various explosive scenarios andfind no evidence of speculative bubbles in oil markets evenduring episodes where the most popular SADF and GSADFmethodologies identify a speculative bubble

Our conclusions have pertinent implications for policymakers Given the historically important role of oil prices inthe real economy one of the serious concerns in policycircles over the past decade has been whether speculativetrading activities in oil future markets are responsible for thelarge price swings For example in their 2008 testimonies to

Complexity 15

the US Senate Commerce Committee Michael Masters andGeorge Soros posited that speculators had fueled price in-creases since 2003 and had caused a part of a ldquosuperbubblerdquoin 2007 and 2008 resulting in significantly higher energycosts for consumers -is widespread bubble view domi-nated policy discussions and efforts to limit speculativetrading in the future market quickly followed As a responseto the excessive speculation pressures on prices theDoddndashFrank Wall Street Reform and Consumer ProtectionAct was enacted by the US Congress in 2010 As a directpolicy implication our results highlight that imposingstricter regulations on trading in oil derivative marketscannot be expected to lower the oil price given the fact thatthe synchronized boom and bust cycles witnessed in the past20 years are mainly due to economic fundamentals whichappear likely to resurge with the business cycle -e aboveanalysis implies that there is no need to enforce additionalregulatory limits against financial arbitrage in the com-modity market

Appendix

The SADF and GSADF Methodologies

Appendix describes the SADF and GSADF testing proce-dures used to detect speculative bubbles in Section 62 For agenetic time series process xt consider the following rolling-window type augmented DickeyndashFuller (ADF) regressionequation

xt ar1r2+ μr1r2

xtminus 1 + 1113944

J

j1ϕj

r1r2Δxtminus j + εt εt sim NID 0 δ2r1r2

1113874 1113875

(A1)

where Δxtminus j with j 1 J are lagged first differences ofxt incorporated to accommodate potential serial correla-tion ar1 r2

μr1 r2 and ϕj

r1 r2 with j 1 J are regression

coefficients the subscripts r1 and r2 refer to fractions of thetotal sample size (ofT observations) that set the start and endpoints of a subsample period and NID represents inde-pendent and normal distribution -e ADF test statisticcorresponding to the unit root null hypothesisH0 μr1 r2

0 is simply the t-ratio

ADFr2r1

1113954μr1 r2

se 1113954μr1 r21113872 1113873

(A2)

and the right-side alternative hypothesis of explosive be-havior is H0 μr1r2

gt 0 We can obtain the standard ADF teststatistic by running regression (A1) on the whole sampleie by specifying r1 0 and r2 1 A large amount of re-search has demonstrated that the standard ADF test ADF10has extremely low power in detecting periodically collapsingbubbles

To address the problem PSW proposed a recursiveregression technique which involves estimating (A1)employing a forward expanding sample Specifically theyused subsamples of the available data to recursively calculatea sequence of ADF statistics denoted by ADFr2

0 -e startpoint of the sample period is fixed at r1 0 while the endpoint r2 increases from r0 (the minimum window size) toone (the full sample period) -e SADF statistic called thesupremum of ADFr2

0 is defined by

SADF r0( 1113857 supr2isin r0 1[ ]

ADFr20

(A3)

and has the following limit distribution under the nullhypothesis

supr2isin r0 1[ ]

1113938r2

0 WdW

1113938r2

0 W21113872 1113873(12)

(A4)

where W is the standard Brownian motion Testing thepresence of explosive behavior entails comparing the ADFr2

0statistic with the right-side critical values obtained throughMonte Carlo simulations

More recently PSY showed that when the sample perioddisplays more than one boom-bust bubble the SADF testprocedure may perform poorly and fail to detect or con-sistently date-stamp alternative bubbles -is deficiencymotivates PSY to formulate an extension of the SADF testthe GSADF test which covers more subsamples of the datathan the SADF by allowing both the end point r2 and thestart point r1 to vary As a result the GSADF test has moreflexibility and gains substantial power making this testconsistent with multiple boom-bust bubbles -e GSADFtest statistic is defined as

GSADF r0( 1113857 supr2isin r0 1[ ]r1isin 0r2minus r0[ ]

ADFr2r1 (A5)

with the limit distribution

supr2isin r0 1[ ]r1isin 0r2minus r0[ ]

(12)rw W r2( 11138572

minus W r1( 11138572

minus rw1113960 1113961 minus 1113938r2

r1W(r)dr W r2( 1113857 minus W r1( 11138571113858 1113859

r12w rw 1113938

r2

r1W(r)2dr minus 1113938

r2

r1W(r)dr1113876 1113877

21113896 1113897

12

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎭

(A6)

where rw r2 minus r1 is the expanding window size Like theSADF test comparison of the GASDF statistic with the right-

tailed critical values from its limit distribution enables us to testfor the unit root hypothesis against explosive behavior

16 Complexity

If the null hypothesis of a unit root in xt is rejectedthen in the second stage the SADF and GSADF tests canidentify the exact episodes during which the xt seriesexhibits explosive dynamics To locate the emergence andconclusion of explosiveness the SADF methodologymatches the sequence of the recursive backward ADF(BADF) test statistics against the right-tailed criticalvalues of the limit distribution of the standard Dick-eyndashFuller test statistic Suppose that re is the originationdate and rf is the finish date of the bubble in the data-ese estimates of dates can be constructed from

1113954re infr2isin r0 1[ ]

r2 ADFr20 gt cv

αr2

1113966 1113967

1113954rf infr2isin 1113954re11113858 1113859

r2 ADFr20 lt cv

αr2

1113966 1113967(A7)

where cvαr2 is 100(1 minus α) the right-tailed critical value of theADF statistic corresponding to the chosen significance levelof α For the GSADF methodology the date-stampingstrategy in this case is based on the backward SADF(BSADF) statistic

BSADFr2r0( 1113857 sup

r1isin 0r2minus r0[ ]ADFr2

r1

(A8)

Likewise the origination date of the bubble is defined asthe first observation that the BSADF statistic is greater thanits critical value

1113954re infr2isin r0 1[ ]

r2 BSADFr2r0( 1113857gt cv

βT

r21113966 1113967 (A9)

and the finish date as the first date after which the BSADFdrops back below its critical value

1113954rf infr2isin 1113954re11113858 1113859

r2 BSADFr2r0( 1113857lt cvβT

r21113966 1113967 (A10)

where cvβTr2 is 100(1 minus βT) the right-tailed critical value of

the SADF based on Tr2 and the corresponding significancelevel is βT

Data Availability

-e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

-e authors declare no conflicts of interest

Acknowledgments

-e authors gratefully acknowledge the financial supportfrom the National Social Science Foundation of China underGrant no 18BTJ039

References

[1] M W Masters Testimony before the US Senate Committee onHomeland Security and Governmental Affairs 2008 httpswwwhsgacsenategovimomediadoc052008Masterspdfattempt2

[2] M W Masters Testimony before the Commodity FuturesTrading Commission 2010 httpscftcgovsitesdefaultfilesidcgroupspublicnewsroomdocumentsfilemetalmarkets032510_masterspdf

[3] J P Kennedy ldquo-e high cost of gambling on oilrdquo Section A p23 New York Times New York NY USANew York Times2012

[4] I-H Cheng and W Xiong ldquoFinancialization of commoditymarketsrdquo Annual Review of Financial Economics vol 6 no 1pp 419ndash441 2014

[5] K Tang and W Xiong ldquoIndex investment and the financi-alization of commoditiesrdquo Financial Analysts Journal vol 68no 6 pp 54ndash74 2012

[6] O J Blanchard and M W Watson ldquoBubbles rational ex-pectations and financial marketsrdquo in Crisis in the Economicand Financial Structure P Wachtel Ed Lexington BooksLexington MA USA pp 295ndash315 1982

[7] B T Diba and H I Grossman ldquoOn the inception of rationalbubblesrdquo e Quarterly Journal of Economics vol 102 no 3pp 697ndash700 1987

[8] G W Evans ldquo-e fragility of sunspots and bubblesrdquo Journalof Monetary Economics vol 23 no 2 pp 297ndash317 1989

[9] G W Evans and S Honkapohja ldquoOn the robustness ofbubbles in linear RE modelsrdquo International Economic Reviewvol 33 no 1 pp 1ndash14 1992

[10] E Pavlidis A Yusupova I Paya et al ldquoEpisodes of exu-berance in housing markets in search of the smoking gunrdquoe Journal of Real Estate Finance and Economics vol 53no 4 pp 419ndash449 2016

[11] E Martınez-Garcıa and V Grossman ldquoExplosive dynamics inhouse prices An exploration of financial market spillovers inhousing markets around the worldrdquo Journal of InternationalMoney and Finance vol 101 p 102103 2020

[12] S Shi A Rahman and B Z Wang ldquoAustralian housingmarket booms fundamentals or speculationlowastrdquo EconomicRecord 2020 In press

[13] P C B Phillips Y Wu and J Yu ldquoExplosive behavior in the1990s NASDAQ when did exuberance escalate asset val-ueslowastrdquo International Economic Review vol 52 no 1pp 201ndash226 2011

[14] P C B Phillips S Shi and J Yu ldquoTesting for multiplebubbles historical episodes of exuberance and collapse in theSampp 500rdquo International Economic Review vol 56 no 4pp 1043ndash1078 2015

[15] P C B Phillips S Shi and J Yu ldquoTesting for multiplebubbles limit theory of real-time detectorsrdquo InternationalEconomic Review vol 56 no 4 pp 1079ndash1134 2015

[16] P C B Phillips and J Yu ldquoDating the timeline of financialbubbles during the subprime crisisrdquo Quantitative Economicsvol 2 no 3 pp 455ndash491 2011

[17] D Tsvetanov J Coakley and N Kellard ldquoBubbling over thebehaviour of oil futures along the yield curverdquo Journal ofEmpirical Finance vol 38 no Part B pp 516ndash533 2016

[18] I Figuerola-Ferretti J R Mccrorie and I ParaskevopoulosldquoMild explosivity in recent crude oil pricesrdquo Energy Eco-nomics vol 87 p 104387 2020

[19] L Kilian and D P Murphy ldquo-e role of inventories andspeculative trading in the global market for crude oilrdquo Journalof Applied Econometrics vol 29 no 3 pp 454ndash478 2014

Complexity 17

2

3

4

5

6

7

8

0

50

100

150

200

250

300

350

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

LLF0

LLF1

Realized variance of crude oil future prices

Figure 5 Realized variance and likelihood values for random coefficient regression

Table 2 Time horizons of unstable and explosive periods detected by models with endogeneity

Unstable Locally explosive ExplosiveMay 2008ndashJul 2008 mdash mdashDec 2010ndashOct 2014 mdash mdashNov 2016ndashApr 2017 mdash mdashJul 2017ndashFeb 2020 mdash mdash

ndash4

ndash2

0

2

4

6

8

0

20

40

60

80

100

120

140

1988

-03

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

Explosiveness detection using SADF methodology

BADF sequence95 critical valuesCrude oil future prices

(a)

ndash4

ndash2

0

2

4

6

8

0

20

40

60

80

100

120

140

1988

-03

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

Explosiveness detection using GSADF methodology

BSADF sequence95 critical valuesCrude oil future prices

(b)

Figure 6 Date-stamping explosive episodes in oil prices with the time-invariant coefficient model

12 Complexity

Table 3 Time horizons of explosive behavior identified by SADF and GSADF methodologies

SADF GSADFAug 2005ndashAug 2006 Oct 2011ndashApr 2012 Sep 1990-Oct 1990Jun 2007ndashOct 2008 Aug 2012-Sep 2012 Feb 1991-Mar 1991Jun 2009-Jul 2009 Jul 2013-Aug 2013 Oct 2004-Nov 2004Feb 2011ndashApr 2011 Feb 2014-Mar 2014 Feb 2005-Mar 2005

Jul 2011ndashAug 2012 Jun 2014-Jul 2014May 2005ndashJan 2006Mar 2006ndashAug 2006Jul 2007ndashAug 2008

ndash4

ndash2

0

2

4

6

8

0

10

20

30

40

50

60

70

1988

-03

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

95 critical value of N (0 1)Real oil future prices

t-statistic of μ~

Local explosiveness detection based on the time-varyingcoefficient model

(a)

ndash4

ndash2

0

2

4

6

8

0

10

20

30

40

50

60

70

1988

-03

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

95 critical value of N (0 1)Real oil future prices

t-statistic of ϕ

Explosiveness detection based on the time-varyingcoefficient model

(b)

Figure 7 Date-stamping explosive episodes in real oil prices with the time-varying coefficient model with endogeneity

ndash4

ndash2

0

2

4

6

8

0

50

100

150

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

Local explosiveness detection based on the time-varyingcoefficient model

t-statistic of μ95 critical value of N (0 1)2-month crude oil future prices

(a)

ndash4

ndash2

0

2

4

6

8

0

50

100

150

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

Local explosiveness detection based on the time-varyingcoefficient model

t-statistic of ϕ95 critical value of N (0 1)2-month crude oil future prices

(b)

Figure 8 Date-stamping explosive episodes in 2-month future oil prices with the time-varying coefficient model with endogeneity

Complexity 13

7 Discussion

-e oil price rally since the early 2000s has attracted greatinterest in the sources of oil price shocks Some causal ob-servers have stressed that financial speculation has become asignificant driver of the dynamics of oil prices A large body ofliterature emerged that has sought to investigate its existenceIn this paper we employ a novel approach for testing forspeculative bubbles in oil prices In synthesis our empiricalfindings indicate that there is less evidence of exuberance in

the oil future markets Our results are very robust in severalscenarios We argue that the booms and busts in the oil pricemovements during the sample period are driven primarily byeconomic fundamentals rather than speculation

Overall our results confirm the main findings obtainedby structural VAR-based studies including Kilian andMurphy [19] Kilian and Lee [20] Baumeister and Hamilton[21] and Zhou [22] In the past decade the use of structuralVAR models to understand the evolution of the oil price hasbecome common -e traditional SVAR literature

ndash4

ndash2

0

2

4

6

8

0

50

100

15019

88-0

3

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

Local explosiveness detection based on the time-varyingcoefficient model

t-statistic of μ95 critical value of N (0 1)3-month crude oil future prices

(a)

ndash4

ndash2

0

2

4

6

8

0

50

100

150

1988

-03

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

Explosiveness detection based on the time-varyingcoefficient model

t-statistic of ϕ95 critical value of N (0 1)3-month crude oil future prices

(b)

Figure 9 Date-stamping explosive episodes in 3-month future oil prices with the time-varying coefficient model with endogeneity

ndash4

ndash2

0

2

4

6

8

0

50

100

150

Local explosiveness detection based on the time-varyingcoefficient model

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

t-statistic of μ95 critical value of N (0 1)4-month crude oil future prices

(a)

ndash4

ndash2

0

2

4

6

8

0

50

100

150

Explosiveness detection based on the time-varyingcoefficient model

t-statistic of ϕ95 critical value of N (0 1)4-month crude oil futures prices

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

(b)

Figure 10 Date-stamping explosive episodes in 4-month future oil prices with the time-varying coefficient model with endogeneity

14 Complexity

documents that the fundamental drivers of oil price fluc-tuations have been dominated essentially by supply anddemand shocks see eg Kilian [56] Lippi and Nobili [57]Kilian andMurphy [58] and Baumeister and Peersman [59]Kilian and Murphy [19] focused on the role speculation byaugmenting the SVAR framework of their earlier paper byglobal oil inventory data -ey specified a four-variablemodel for the global oil market (the percentage change inglobal oil production the real price of oil a suitable proxy ofcyclical variation in global real economic activity andchanges in global crude oil inventories) -is setting andtheir sign-identified scheme allow them to separately explainvariation in these data as four different types of shocks (1) ashock to the amount of oil pumped out of the ground (ldquooilsupply shockrdquo) (2) an oil flow demand shock (3) an oil-specific oil demand shock and especially (4) a speculative(or inventory) demand shock -eir core finding is that thespike in the real price of crude oil prices during 2003ndash2008was caused by unexpected increase in global oil consump-tion Using two alternative proxies for global above-groundcrude oil inventories Kilian and Lee [20] reestimated themodel of Kilian and Murphy [19] and ruled out the role ofspeculation in explaining the recent surge of the real oil priceon the basis of their estimation results Also Zhou [22]assessed the sensitivity of the findings in Kilian and Murphy[19] to the introduction of various state-of-the-art oil marketmethods She shows that the core conclusions reached byKilian and Murphy [19] are highly robust to some refine-ments corrections and extension of the KilianndashMurphymodel Baumeister and Hamilton [21] recently contendedthat prior information plays a key role in any SVAR modelsand traditional identification approaches can be generalizedto a special case of a Bayesian perspective -ey revisited theimportance of supply and demand shocks in the oil marketand demonstrated that speculative demand shock turns outto be only a small factor in accounting for most historical oilprice shocks

For comparison purpose we have also applied the testsof PWY and PSY which are the most popular econometrictechniques in the literature on speculative bubbles in the oilmarkets If we were to follow the test procedures of PWY andPSY our conclusions are also broadly consistent with therich literature that supports the existence of speculativebubbles in oil prices However these cointegration-basedbubble tests suffer from the joint hypothesis problem ldquoEverytest of a bubble is a joint test of the presence of the bubble inthe data and the validity of the model applied by theeconometricianrdquo (see Giglio et al [60]) More specifically themain drawback of such indirect model-dependent tests ofbubbles typically refers to the difficulty in distinguishingbetween the unknown true model for market fundamentalsand the presence of bubbles (see for example Gurkaynak[37] Pavlidis et al [42] Pavlidis et al [43]) Given thefundamental value of oil price is unobservable the literaturehas resorted to observe economic and financial variableswhich are used to estimate the intrinsic values For exampleboth Phillips and Yu [16] and Caspi et al [39] simplynormalized crude oil prices by US oil inventory data as thefundamental value -is measure is questionable since their

method cannot sufficiently capture the information onmarket fundamentals and neglect the major driversmdashlikeglobal economic activitymdashalso affecting the fundamentalvalue of crude oil -is issue of misspecified fundamentals(either in terms of model misspecification or omitted var-iables or both of these reasons) may make researchers fail toarrive at a conclusion from evidence of explosive pricesalone To make matters worse even though we have cap-tured the true market fundamentals we may still fail todetect bubbles if the explosive behavior in the price of oil juststems from the explosive fundamentals

Compared to earlier studies as briefly mentioned aboveour methodology has the main advantage that by consid-ering the dependence between the random coefficient andsystem shocks we have incorporated endogeneity into thecontinuous-time random coefficient setting and thus we donot need to prespecify the fundamental component of the oilprice -e test results for endogeneity suggest that endo-geneity is important in the data-generating process Inparticular our empirical analysis also finds that there are noexplosive episodes in crude oil markets-is implies that it isnot necessary to require the specification of a fundamentalvalue of the oil price Like Pavlidis et al [43] their methodsand ours have in common that the problem of the well-known joint hypothesis has to be some extent been ef-fectively ameliorated Although our approach is quite dis-tinct we draw the same substantive conclusion thatspeculative bubbles can be ruled out as a key determinant ofthe price of crude oil in the 2000s reaffirming the con-clusions of Kilian and Murphy [19] Baumeister andHamilton [21] Zhou [22] and Fattouh et al [61]

8 Conclusions

Repeated surges and crashes in crude oil prices during lastdecades have sparked a heated debate on their underlyingcauses among academics commentors and market regu-lators One of the central questions concerns whetherfinancialization of oil future markets creates a speculativebubble that subsequently collapses Answering this questionis difficult because one has to rely upon strong assumptionsabout the data-generating mechanism for the fundamentalvalue of the oil price In this paper we present a stylizedmodel of asset pricing for a commodity to address this issueOur theoretical framework disentangles the effects ofspeculative bubbles from the traditional fundamental effectson oil price dynamics We use this setting to illustrate thatbubbles if they are present should demonstrate explosivebehavior in prices We then provide a novel approach forbubble detection that covers various explosive scenarios andfind no evidence of speculative bubbles in oil markets evenduring episodes where the most popular SADF and GSADFmethodologies identify a speculative bubble

Our conclusions have pertinent implications for policymakers Given the historically important role of oil prices inthe real economy one of the serious concerns in policycircles over the past decade has been whether speculativetrading activities in oil future markets are responsible for thelarge price swings For example in their 2008 testimonies to

Complexity 15

the US Senate Commerce Committee Michael Masters andGeorge Soros posited that speculators had fueled price in-creases since 2003 and had caused a part of a ldquosuperbubblerdquoin 2007 and 2008 resulting in significantly higher energycosts for consumers -is widespread bubble view domi-nated policy discussions and efforts to limit speculativetrading in the future market quickly followed As a responseto the excessive speculation pressures on prices theDoddndashFrank Wall Street Reform and Consumer ProtectionAct was enacted by the US Congress in 2010 As a directpolicy implication our results highlight that imposingstricter regulations on trading in oil derivative marketscannot be expected to lower the oil price given the fact thatthe synchronized boom and bust cycles witnessed in the past20 years are mainly due to economic fundamentals whichappear likely to resurge with the business cycle -e aboveanalysis implies that there is no need to enforce additionalregulatory limits against financial arbitrage in the com-modity market

Appendix

The SADF and GSADF Methodologies

Appendix describes the SADF and GSADF testing proce-dures used to detect speculative bubbles in Section 62 For agenetic time series process xt consider the following rolling-window type augmented DickeyndashFuller (ADF) regressionequation

xt ar1r2+ μr1r2

xtminus 1 + 1113944

J

j1ϕj

r1r2Δxtminus j + εt εt sim NID 0 δ2r1r2

1113874 1113875

(A1)

where Δxtminus j with j 1 J are lagged first differences ofxt incorporated to accommodate potential serial correla-tion ar1 r2

μr1 r2 and ϕj

r1 r2 with j 1 J are regression

coefficients the subscripts r1 and r2 refer to fractions of thetotal sample size (ofT observations) that set the start and endpoints of a subsample period and NID represents inde-pendent and normal distribution -e ADF test statisticcorresponding to the unit root null hypothesisH0 μr1 r2

0 is simply the t-ratio

ADFr2r1

1113954μr1 r2

se 1113954μr1 r21113872 1113873

(A2)

and the right-side alternative hypothesis of explosive be-havior is H0 μr1r2

gt 0 We can obtain the standard ADF teststatistic by running regression (A1) on the whole sampleie by specifying r1 0 and r2 1 A large amount of re-search has demonstrated that the standard ADF test ADF10has extremely low power in detecting periodically collapsingbubbles

To address the problem PSW proposed a recursiveregression technique which involves estimating (A1)employing a forward expanding sample Specifically theyused subsamples of the available data to recursively calculatea sequence of ADF statistics denoted by ADFr2

0 -e startpoint of the sample period is fixed at r1 0 while the endpoint r2 increases from r0 (the minimum window size) toone (the full sample period) -e SADF statistic called thesupremum of ADFr2

0 is defined by

SADF r0( 1113857 supr2isin r0 1[ ]

ADFr20

(A3)

and has the following limit distribution under the nullhypothesis

supr2isin r0 1[ ]

1113938r2

0 WdW

1113938r2

0 W21113872 1113873(12)

(A4)

where W is the standard Brownian motion Testing thepresence of explosive behavior entails comparing the ADFr2

0statistic with the right-side critical values obtained throughMonte Carlo simulations

More recently PSY showed that when the sample perioddisplays more than one boom-bust bubble the SADF testprocedure may perform poorly and fail to detect or con-sistently date-stamp alternative bubbles -is deficiencymotivates PSY to formulate an extension of the SADF testthe GSADF test which covers more subsamples of the datathan the SADF by allowing both the end point r2 and thestart point r1 to vary As a result the GSADF test has moreflexibility and gains substantial power making this testconsistent with multiple boom-bust bubbles -e GSADFtest statistic is defined as

GSADF r0( 1113857 supr2isin r0 1[ ]r1isin 0r2minus r0[ ]

ADFr2r1 (A5)

with the limit distribution

supr2isin r0 1[ ]r1isin 0r2minus r0[ ]

(12)rw W r2( 11138572

minus W r1( 11138572

minus rw1113960 1113961 minus 1113938r2

r1W(r)dr W r2( 1113857 minus W r1( 11138571113858 1113859

r12w rw 1113938

r2

r1W(r)2dr minus 1113938

r2

r1W(r)dr1113876 1113877

21113896 1113897

12

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎭

(A6)

where rw r2 minus r1 is the expanding window size Like theSADF test comparison of the GASDF statistic with the right-

tailed critical values from its limit distribution enables us to testfor the unit root hypothesis against explosive behavior

16 Complexity

If the null hypothesis of a unit root in xt is rejectedthen in the second stage the SADF and GSADF tests canidentify the exact episodes during which the xt seriesexhibits explosive dynamics To locate the emergence andconclusion of explosiveness the SADF methodologymatches the sequence of the recursive backward ADF(BADF) test statistics against the right-tailed criticalvalues of the limit distribution of the standard Dick-eyndashFuller test statistic Suppose that re is the originationdate and rf is the finish date of the bubble in the data-ese estimates of dates can be constructed from

1113954re infr2isin r0 1[ ]

r2 ADFr20 gt cv

αr2

1113966 1113967

1113954rf infr2isin 1113954re11113858 1113859

r2 ADFr20 lt cv

αr2

1113966 1113967(A7)

where cvαr2 is 100(1 minus α) the right-tailed critical value of theADF statistic corresponding to the chosen significance levelof α For the GSADF methodology the date-stampingstrategy in this case is based on the backward SADF(BSADF) statistic

BSADFr2r0( 1113857 sup

r1isin 0r2minus r0[ ]ADFr2

r1

(A8)

Likewise the origination date of the bubble is defined asthe first observation that the BSADF statistic is greater thanits critical value

1113954re infr2isin r0 1[ ]

r2 BSADFr2r0( 1113857gt cv

βT

r21113966 1113967 (A9)

and the finish date as the first date after which the BSADFdrops back below its critical value

1113954rf infr2isin 1113954re11113858 1113859

r2 BSADFr2r0( 1113857lt cvβT

r21113966 1113967 (A10)

where cvβTr2 is 100(1 minus βT) the right-tailed critical value of

the SADF based on Tr2 and the corresponding significancelevel is βT

Data Availability

-e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

-e authors declare no conflicts of interest

Acknowledgments

-e authors gratefully acknowledge the financial supportfrom the National Social Science Foundation of China underGrant no 18BTJ039

References

[1] M W Masters Testimony before the US Senate Committee onHomeland Security and Governmental Affairs 2008 httpswwwhsgacsenategovimomediadoc052008Masterspdfattempt2

[2] M W Masters Testimony before the Commodity FuturesTrading Commission 2010 httpscftcgovsitesdefaultfilesidcgroupspublicnewsroomdocumentsfilemetalmarkets032510_masterspdf

[3] J P Kennedy ldquo-e high cost of gambling on oilrdquo Section A p23 New York Times New York NY USANew York Times2012

[4] I-H Cheng and W Xiong ldquoFinancialization of commoditymarketsrdquo Annual Review of Financial Economics vol 6 no 1pp 419ndash441 2014

[5] K Tang and W Xiong ldquoIndex investment and the financi-alization of commoditiesrdquo Financial Analysts Journal vol 68no 6 pp 54ndash74 2012

[6] O J Blanchard and M W Watson ldquoBubbles rational ex-pectations and financial marketsrdquo in Crisis in the Economicand Financial Structure P Wachtel Ed Lexington BooksLexington MA USA pp 295ndash315 1982

[7] B T Diba and H I Grossman ldquoOn the inception of rationalbubblesrdquo e Quarterly Journal of Economics vol 102 no 3pp 697ndash700 1987

[8] G W Evans ldquo-e fragility of sunspots and bubblesrdquo Journalof Monetary Economics vol 23 no 2 pp 297ndash317 1989

[9] G W Evans and S Honkapohja ldquoOn the robustness ofbubbles in linear RE modelsrdquo International Economic Reviewvol 33 no 1 pp 1ndash14 1992

[10] E Pavlidis A Yusupova I Paya et al ldquoEpisodes of exu-berance in housing markets in search of the smoking gunrdquoe Journal of Real Estate Finance and Economics vol 53no 4 pp 419ndash449 2016

[11] E Martınez-Garcıa and V Grossman ldquoExplosive dynamics inhouse prices An exploration of financial market spillovers inhousing markets around the worldrdquo Journal of InternationalMoney and Finance vol 101 p 102103 2020

[12] S Shi A Rahman and B Z Wang ldquoAustralian housingmarket booms fundamentals or speculationlowastrdquo EconomicRecord 2020 In press

[13] P C B Phillips Y Wu and J Yu ldquoExplosive behavior in the1990s NASDAQ when did exuberance escalate asset val-ueslowastrdquo International Economic Review vol 52 no 1pp 201ndash226 2011

[14] P C B Phillips S Shi and J Yu ldquoTesting for multiplebubbles historical episodes of exuberance and collapse in theSampp 500rdquo International Economic Review vol 56 no 4pp 1043ndash1078 2015

[15] P C B Phillips S Shi and J Yu ldquoTesting for multiplebubbles limit theory of real-time detectorsrdquo InternationalEconomic Review vol 56 no 4 pp 1079ndash1134 2015

[16] P C B Phillips and J Yu ldquoDating the timeline of financialbubbles during the subprime crisisrdquo Quantitative Economicsvol 2 no 3 pp 455ndash491 2011

[17] D Tsvetanov J Coakley and N Kellard ldquoBubbling over thebehaviour of oil futures along the yield curverdquo Journal ofEmpirical Finance vol 38 no Part B pp 516ndash533 2016

[18] I Figuerola-Ferretti J R Mccrorie and I ParaskevopoulosldquoMild explosivity in recent crude oil pricesrdquo Energy Eco-nomics vol 87 p 104387 2020

[19] L Kilian and D P Murphy ldquo-e role of inventories andspeculative trading in the global market for crude oilrdquo Journalof Applied Econometrics vol 29 no 3 pp 454ndash478 2014

Complexity 17

Table 3 Time horizons of explosive behavior identified by SADF and GSADF methodologies

SADF GSADFAug 2005ndashAug 2006 Oct 2011ndashApr 2012 Sep 1990-Oct 1990Jun 2007ndashOct 2008 Aug 2012-Sep 2012 Feb 1991-Mar 1991Jun 2009-Jul 2009 Jul 2013-Aug 2013 Oct 2004-Nov 2004Feb 2011ndashApr 2011 Feb 2014-Mar 2014 Feb 2005-Mar 2005

Jul 2011ndashAug 2012 Jun 2014-Jul 2014May 2005ndashJan 2006Mar 2006ndashAug 2006Jul 2007ndashAug 2008

ndash4

ndash2

0

2

4

6

8

0

10

20

30

40

50

60

70

1988

-03

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

95 critical value of N (0 1)Real oil future prices

t-statistic of μ~

Local explosiveness detection based on the time-varyingcoefficient model

(a)

ndash4

ndash2

0

2

4

6

8

0

10

20

30

40

50

60

70

1988

-03

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

95 critical value of N (0 1)Real oil future prices

t-statistic of ϕ

Explosiveness detection based on the time-varyingcoefficient model

(b)

Figure 7 Date-stamping explosive episodes in real oil prices with the time-varying coefficient model with endogeneity

ndash4

ndash2

0

2

4

6

8

0

50

100

150

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

Local explosiveness detection based on the time-varyingcoefficient model

t-statistic of μ95 critical value of N (0 1)2-month crude oil future prices

(a)

ndash4

ndash2

0

2

4

6

8

0

50

100

150

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

Local explosiveness detection based on the time-varyingcoefficient model

t-statistic of ϕ95 critical value of N (0 1)2-month crude oil future prices

(b)

Figure 8 Date-stamping explosive episodes in 2-month future oil prices with the time-varying coefficient model with endogeneity

Complexity 13

7 Discussion

-e oil price rally since the early 2000s has attracted greatinterest in the sources of oil price shocks Some causal ob-servers have stressed that financial speculation has become asignificant driver of the dynamics of oil prices A large body ofliterature emerged that has sought to investigate its existenceIn this paper we employ a novel approach for testing forspeculative bubbles in oil prices In synthesis our empiricalfindings indicate that there is less evidence of exuberance in

the oil future markets Our results are very robust in severalscenarios We argue that the booms and busts in the oil pricemovements during the sample period are driven primarily byeconomic fundamentals rather than speculation

Overall our results confirm the main findings obtainedby structural VAR-based studies including Kilian andMurphy [19] Kilian and Lee [20] Baumeister and Hamilton[21] and Zhou [22] In the past decade the use of structuralVAR models to understand the evolution of the oil price hasbecome common -e traditional SVAR literature

ndash4

ndash2

0

2

4

6

8

0

50

100

15019

88-0

3

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

Local explosiveness detection based on the time-varyingcoefficient model

t-statistic of μ95 critical value of N (0 1)3-month crude oil future prices

(a)

ndash4

ndash2

0

2

4

6

8

0

50

100

150

1988

-03

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

Explosiveness detection based on the time-varyingcoefficient model

t-statistic of ϕ95 critical value of N (0 1)3-month crude oil future prices

(b)

Figure 9 Date-stamping explosive episodes in 3-month future oil prices with the time-varying coefficient model with endogeneity

ndash4

ndash2

0

2

4

6

8

0

50

100

150

Local explosiveness detection based on the time-varyingcoefficient model

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

t-statistic of μ95 critical value of N (0 1)4-month crude oil future prices

(a)

ndash4

ndash2

0

2

4

6

8

0

50

100

150

Explosiveness detection based on the time-varyingcoefficient model

t-statistic of ϕ95 critical value of N (0 1)4-month crude oil futures prices

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

(b)

Figure 10 Date-stamping explosive episodes in 4-month future oil prices with the time-varying coefficient model with endogeneity

14 Complexity

documents that the fundamental drivers of oil price fluc-tuations have been dominated essentially by supply anddemand shocks see eg Kilian [56] Lippi and Nobili [57]Kilian andMurphy [58] and Baumeister and Peersman [59]Kilian and Murphy [19] focused on the role speculation byaugmenting the SVAR framework of their earlier paper byglobal oil inventory data -ey specified a four-variablemodel for the global oil market (the percentage change inglobal oil production the real price of oil a suitable proxy ofcyclical variation in global real economic activity andchanges in global crude oil inventories) -is setting andtheir sign-identified scheme allow them to separately explainvariation in these data as four different types of shocks (1) ashock to the amount of oil pumped out of the ground (ldquooilsupply shockrdquo) (2) an oil flow demand shock (3) an oil-specific oil demand shock and especially (4) a speculative(or inventory) demand shock -eir core finding is that thespike in the real price of crude oil prices during 2003ndash2008was caused by unexpected increase in global oil consump-tion Using two alternative proxies for global above-groundcrude oil inventories Kilian and Lee [20] reestimated themodel of Kilian and Murphy [19] and ruled out the role ofspeculation in explaining the recent surge of the real oil priceon the basis of their estimation results Also Zhou [22]assessed the sensitivity of the findings in Kilian and Murphy[19] to the introduction of various state-of-the-art oil marketmethods She shows that the core conclusions reached byKilian and Murphy [19] are highly robust to some refine-ments corrections and extension of the KilianndashMurphymodel Baumeister and Hamilton [21] recently contendedthat prior information plays a key role in any SVAR modelsand traditional identification approaches can be generalizedto a special case of a Bayesian perspective -ey revisited theimportance of supply and demand shocks in the oil marketand demonstrated that speculative demand shock turns outto be only a small factor in accounting for most historical oilprice shocks

For comparison purpose we have also applied the testsof PWY and PSY which are the most popular econometrictechniques in the literature on speculative bubbles in the oilmarkets If we were to follow the test procedures of PWY andPSY our conclusions are also broadly consistent with therich literature that supports the existence of speculativebubbles in oil prices However these cointegration-basedbubble tests suffer from the joint hypothesis problem ldquoEverytest of a bubble is a joint test of the presence of the bubble inthe data and the validity of the model applied by theeconometricianrdquo (see Giglio et al [60]) More specifically themain drawback of such indirect model-dependent tests ofbubbles typically refers to the difficulty in distinguishingbetween the unknown true model for market fundamentalsand the presence of bubbles (see for example Gurkaynak[37] Pavlidis et al [42] Pavlidis et al [43]) Given thefundamental value of oil price is unobservable the literaturehas resorted to observe economic and financial variableswhich are used to estimate the intrinsic values For exampleboth Phillips and Yu [16] and Caspi et al [39] simplynormalized crude oil prices by US oil inventory data as thefundamental value -is measure is questionable since their

method cannot sufficiently capture the information onmarket fundamentals and neglect the major driversmdashlikeglobal economic activitymdashalso affecting the fundamentalvalue of crude oil -is issue of misspecified fundamentals(either in terms of model misspecification or omitted var-iables or both of these reasons) may make researchers fail toarrive at a conclusion from evidence of explosive pricesalone To make matters worse even though we have cap-tured the true market fundamentals we may still fail todetect bubbles if the explosive behavior in the price of oil juststems from the explosive fundamentals

Compared to earlier studies as briefly mentioned aboveour methodology has the main advantage that by consid-ering the dependence between the random coefficient andsystem shocks we have incorporated endogeneity into thecontinuous-time random coefficient setting and thus we donot need to prespecify the fundamental component of the oilprice -e test results for endogeneity suggest that endo-geneity is important in the data-generating process Inparticular our empirical analysis also finds that there are noexplosive episodes in crude oil markets-is implies that it isnot necessary to require the specification of a fundamentalvalue of the oil price Like Pavlidis et al [43] their methodsand ours have in common that the problem of the well-known joint hypothesis has to be some extent been ef-fectively ameliorated Although our approach is quite dis-tinct we draw the same substantive conclusion thatspeculative bubbles can be ruled out as a key determinant ofthe price of crude oil in the 2000s reaffirming the con-clusions of Kilian and Murphy [19] Baumeister andHamilton [21] Zhou [22] and Fattouh et al [61]

8 Conclusions

Repeated surges and crashes in crude oil prices during lastdecades have sparked a heated debate on their underlyingcauses among academics commentors and market regu-lators One of the central questions concerns whetherfinancialization of oil future markets creates a speculativebubble that subsequently collapses Answering this questionis difficult because one has to rely upon strong assumptionsabout the data-generating mechanism for the fundamentalvalue of the oil price In this paper we present a stylizedmodel of asset pricing for a commodity to address this issueOur theoretical framework disentangles the effects ofspeculative bubbles from the traditional fundamental effectson oil price dynamics We use this setting to illustrate thatbubbles if they are present should demonstrate explosivebehavior in prices We then provide a novel approach forbubble detection that covers various explosive scenarios andfind no evidence of speculative bubbles in oil markets evenduring episodes where the most popular SADF and GSADFmethodologies identify a speculative bubble

Our conclusions have pertinent implications for policymakers Given the historically important role of oil prices inthe real economy one of the serious concerns in policycircles over the past decade has been whether speculativetrading activities in oil future markets are responsible for thelarge price swings For example in their 2008 testimonies to

Complexity 15

the US Senate Commerce Committee Michael Masters andGeorge Soros posited that speculators had fueled price in-creases since 2003 and had caused a part of a ldquosuperbubblerdquoin 2007 and 2008 resulting in significantly higher energycosts for consumers -is widespread bubble view domi-nated policy discussions and efforts to limit speculativetrading in the future market quickly followed As a responseto the excessive speculation pressures on prices theDoddndashFrank Wall Street Reform and Consumer ProtectionAct was enacted by the US Congress in 2010 As a directpolicy implication our results highlight that imposingstricter regulations on trading in oil derivative marketscannot be expected to lower the oil price given the fact thatthe synchronized boom and bust cycles witnessed in the past20 years are mainly due to economic fundamentals whichappear likely to resurge with the business cycle -e aboveanalysis implies that there is no need to enforce additionalregulatory limits against financial arbitrage in the com-modity market

Appendix

The SADF and GSADF Methodologies

Appendix describes the SADF and GSADF testing proce-dures used to detect speculative bubbles in Section 62 For agenetic time series process xt consider the following rolling-window type augmented DickeyndashFuller (ADF) regressionequation

xt ar1r2+ μr1r2

xtminus 1 + 1113944

J

j1ϕj

r1r2Δxtminus j + εt εt sim NID 0 δ2r1r2

1113874 1113875

(A1)

where Δxtminus j with j 1 J are lagged first differences ofxt incorporated to accommodate potential serial correla-tion ar1 r2

μr1 r2 and ϕj

r1 r2 with j 1 J are regression

coefficients the subscripts r1 and r2 refer to fractions of thetotal sample size (ofT observations) that set the start and endpoints of a subsample period and NID represents inde-pendent and normal distribution -e ADF test statisticcorresponding to the unit root null hypothesisH0 μr1 r2

0 is simply the t-ratio

ADFr2r1

1113954μr1 r2

se 1113954μr1 r21113872 1113873

(A2)

and the right-side alternative hypothesis of explosive be-havior is H0 μr1r2

gt 0 We can obtain the standard ADF teststatistic by running regression (A1) on the whole sampleie by specifying r1 0 and r2 1 A large amount of re-search has demonstrated that the standard ADF test ADF10has extremely low power in detecting periodically collapsingbubbles

To address the problem PSW proposed a recursiveregression technique which involves estimating (A1)employing a forward expanding sample Specifically theyused subsamples of the available data to recursively calculatea sequence of ADF statistics denoted by ADFr2

0 -e startpoint of the sample period is fixed at r1 0 while the endpoint r2 increases from r0 (the minimum window size) toone (the full sample period) -e SADF statistic called thesupremum of ADFr2

0 is defined by

SADF r0( 1113857 supr2isin r0 1[ ]

ADFr20

(A3)

and has the following limit distribution under the nullhypothesis

supr2isin r0 1[ ]

1113938r2

0 WdW

1113938r2

0 W21113872 1113873(12)

(A4)

where W is the standard Brownian motion Testing thepresence of explosive behavior entails comparing the ADFr2

0statistic with the right-side critical values obtained throughMonte Carlo simulations

More recently PSY showed that when the sample perioddisplays more than one boom-bust bubble the SADF testprocedure may perform poorly and fail to detect or con-sistently date-stamp alternative bubbles -is deficiencymotivates PSY to formulate an extension of the SADF testthe GSADF test which covers more subsamples of the datathan the SADF by allowing both the end point r2 and thestart point r1 to vary As a result the GSADF test has moreflexibility and gains substantial power making this testconsistent with multiple boom-bust bubbles -e GSADFtest statistic is defined as

GSADF r0( 1113857 supr2isin r0 1[ ]r1isin 0r2minus r0[ ]

ADFr2r1 (A5)

with the limit distribution

supr2isin r0 1[ ]r1isin 0r2minus r0[ ]

(12)rw W r2( 11138572

minus W r1( 11138572

minus rw1113960 1113961 minus 1113938r2

r1W(r)dr W r2( 1113857 minus W r1( 11138571113858 1113859

r12w rw 1113938

r2

r1W(r)2dr minus 1113938

r2

r1W(r)dr1113876 1113877

21113896 1113897

12

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎭

(A6)

where rw r2 minus r1 is the expanding window size Like theSADF test comparison of the GASDF statistic with the right-

tailed critical values from its limit distribution enables us to testfor the unit root hypothesis against explosive behavior

16 Complexity

If the null hypothesis of a unit root in xt is rejectedthen in the second stage the SADF and GSADF tests canidentify the exact episodes during which the xt seriesexhibits explosive dynamics To locate the emergence andconclusion of explosiveness the SADF methodologymatches the sequence of the recursive backward ADF(BADF) test statistics against the right-tailed criticalvalues of the limit distribution of the standard Dick-eyndashFuller test statistic Suppose that re is the originationdate and rf is the finish date of the bubble in the data-ese estimates of dates can be constructed from

1113954re infr2isin r0 1[ ]

r2 ADFr20 gt cv

αr2

1113966 1113967

1113954rf infr2isin 1113954re11113858 1113859

r2 ADFr20 lt cv

αr2

1113966 1113967(A7)

where cvαr2 is 100(1 minus α) the right-tailed critical value of theADF statistic corresponding to the chosen significance levelof α For the GSADF methodology the date-stampingstrategy in this case is based on the backward SADF(BSADF) statistic

BSADFr2r0( 1113857 sup

r1isin 0r2minus r0[ ]ADFr2

r1

(A8)

Likewise the origination date of the bubble is defined asthe first observation that the BSADF statistic is greater thanits critical value

1113954re infr2isin r0 1[ ]

r2 BSADFr2r0( 1113857gt cv

βT

r21113966 1113967 (A9)

and the finish date as the first date after which the BSADFdrops back below its critical value

1113954rf infr2isin 1113954re11113858 1113859

r2 BSADFr2r0( 1113857lt cvβT

r21113966 1113967 (A10)

where cvβTr2 is 100(1 minus βT) the right-tailed critical value of

the SADF based on Tr2 and the corresponding significancelevel is βT

Data Availability

-e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

-e authors declare no conflicts of interest

Acknowledgments

-e authors gratefully acknowledge the financial supportfrom the National Social Science Foundation of China underGrant no 18BTJ039

References

[1] M W Masters Testimony before the US Senate Committee onHomeland Security and Governmental Affairs 2008 httpswwwhsgacsenategovimomediadoc052008Masterspdfattempt2

[2] M W Masters Testimony before the Commodity FuturesTrading Commission 2010 httpscftcgovsitesdefaultfilesidcgroupspublicnewsroomdocumentsfilemetalmarkets032510_masterspdf

[3] J P Kennedy ldquo-e high cost of gambling on oilrdquo Section A p23 New York Times New York NY USANew York Times2012

[4] I-H Cheng and W Xiong ldquoFinancialization of commoditymarketsrdquo Annual Review of Financial Economics vol 6 no 1pp 419ndash441 2014

[5] K Tang and W Xiong ldquoIndex investment and the financi-alization of commoditiesrdquo Financial Analysts Journal vol 68no 6 pp 54ndash74 2012

[6] O J Blanchard and M W Watson ldquoBubbles rational ex-pectations and financial marketsrdquo in Crisis in the Economicand Financial Structure P Wachtel Ed Lexington BooksLexington MA USA pp 295ndash315 1982

[7] B T Diba and H I Grossman ldquoOn the inception of rationalbubblesrdquo e Quarterly Journal of Economics vol 102 no 3pp 697ndash700 1987

[8] G W Evans ldquo-e fragility of sunspots and bubblesrdquo Journalof Monetary Economics vol 23 no 2 pp 297ndash317 1989

[9] G W Evans and S Honkapohja ldquoOn the robustness ofbubbles in linear RE modelsrdquo International Economic Reviewvol 33 no 1 pp 1ndash14 1992

[10] E Pavlidis A Yusupova I Paya et al ldquoEpisodes of exu-berance in housing markets in search of the smoking gunrdquoe Journal of Real Estate Finance and Economics vol 53no 4 pp 419ndash449 2016

[11] E Martınez-Garcıa and V Grossman ldquoExplosive dynamics inhouse prices An exploration of financial market spillovers inhousing markets around the worldrdquo Journal of InternationalMoney and Finance vol 101 p 102103 2020

[12] S Shi A Rahman and B Z Wang ldquoAustralian housingmarket booms fundamentals or speculationlowastrdquo EconomicRecord 2020 In press

[13] P C B Phillips Y Wu and J Yu ldquoExplosive behavior in the1990s NASDAQ when did exuberance escalate asset val-ueslowastrdquo International Economic Review vol 52 no 1pp 201ndash226 2011

[14] P C B Phillips S Shi and J Yu ldquoTesting for multiplebubbles historical episodes of exuberance and collapse in theSampp 500rdquo International Economic Review vol 56 no 4pp 1043ndash1078 2015

[15] P C B Phillips S Shi and J Yu ldquoTesting for multiplebubbles limit theory of real-time detectorsrdquo InternationalEconomic Review vol 56 no 4 pp 1079ndash1134 2015

[16] P C B Phillips and J Yu ldquoDating the timeline of financialbubbles during the subprime crisisrdquo Quantitative Economicsvol 2 no 3 pp 455ndash491 2011

[17] D Tsvetanov J Coakley and N Kellard ldquoBubbling over thebehaviour of oil futures along the yield curverdquo Journal ofEmpirical Finance vol 38 no Part B pp 516ndash533 2016

[18] I Figuerola-Ferretti J R Mccrorie and I ParaskevopoulosldquoMild explosivity in recent crude oil pricesrdquo Energy Eco-nomics vol 87 p 104387 2020

[19] L Kilian and D P Murphy ldquo-e role of inventories andspeculative trading in the global market for crude oilrdquo Journalof Applied Econometrics vol 29 no 3 pp 454ndash478 2014

Complexity 17

7 Discussion

-e oil price rally since the early 2000s has attracted greatinterest in the sources of oil price shocks Some causal ob-servers have stressed that financial speculation has become asignificant driver of the dynamics of oil prices A large body ofliterature emerged that has sought to investigate its existenceIn this paper we employ a novel approach for testing forspeculative bubbles in oil prices In synthesis our empiricalfindings indicate that there is less evidence of exuberance in

the oil future markets Our results are very robust in severalscenarios We argue that the booms and busts in the oil pricemovements during the sample period are driven primarily byeconomic fundamentals rather than speculation

Overall our results confirm the main findings obtainedby structural VAR-based studies including Kilian andMurphy [19] Kilian and Lee [20] Baumeister and Hamilton[21] and Zhou [22] In the past decade the use of structuralVAR models to understand the evolution of the oil price hasbecome common -e traditional SVAR literature

ndash4

ndash2

0

2

4

6

8

0

50

100

15019

88-0

3

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

Local explosiveness detection based on the time-varyingcoefficient model

t-statistic of μ95 critical value of N (0 1)3-month crude oil future prices

(a)

ndash4

ndash2

0

2

4

6

8

0

50

100

150

1988

-03

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

Explosiveness detection based on the time-varyingcoefficient model

t-statistic of ϕ95 critical value of N (0 1)3-month crude oil future prices

(b)

Figure 9 Date-stamping explosive episodes in 3-month future oil prices with the time-varying coefficient model with endogeneity

ndash4

ndash2

0

2

4

6

8

0

50

100

150

Local explosiveness detection based on the time-varyingcoefficient model

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

t-statistic of μ95 critical value of N (0 1)4-month crude oil future prices

(a)

ndash4

ndash2

0

2

4

6

8

0

50

100

150

Explosiveness detection based on the time-varyingcoefficient model

t-statistic of ϕ95 critical value of N (0 1)4-month crude oil futures prices

1991

-09

1995

-04

1998

-11

2002

-05

2005

-12

2009

-07

2013

-01

2016

-08

2020

-03

(b)

Figure 10 Date-stamping explosive episodes in 4-month future oil prices with the time-varying coefficient model with endogeneity

14 Complexity

documents that the fundamental drivers of oil price fluc-tuations have been dominated essentially by supply anddemand shocks see eg Kilian [56] Lippi and Nobili [57]Kilian andMurphy [58] and Baumeister and Peersman [59]Kilian and Murphy [19] focused on the role speculation byaugmenting the SVAR framework of their earlier paper byglobal oil inventory data -ey specified a four-variablemodel for the global oil market (the percentage change inglobal oil production the real price of oil a suitable proxy ofcyclical variation in global real economic activity andchanges in global crude oil inventories) -is setting andtheir sign-identified scheme allow them to separately explainvariation in these data as four different types of shocks (1) ashock to the amount of oil pumped out of the ground (ldquooilsupply shockrdquo) (2) an oil flow demand shock (3) an oil-specific oil demand shock and especially (4) a speculative(or inventory) demand shock -eir core finding is that thespike in the real price of crude oil prices during 2003ndash2008was caused by unexpected increase in global oil consump-tion Using two alternative proxies for global above-groundcrude oil inventories Kilian and Lee [20] reestimated themodel of Kilian and Murphy [19] and ruled out the role ofspeculation in explaining the recent surge of the real oil priceon the basis of their estimation results Also Zhou [22]assessed the sensitivity of the findings in Kilian and Murphy[19] to the introduction of various state-of-the-art oil marketmethods She shows that the core conclusions reached byKilian and Murphy [19] are highly robust to some refine-ments corrections and extension of the KilianndashMurphymodel Baumeister and Hamilton [21] recently contendedthat prior information plays a key role in any SVAR modelsand traditional identification approaches can be generalizedto a special case of a Bayesian perspective -ey revisited theimportance of supply and demand shocks in the oil marketand demonstrated that speculative demand shock turns outto be only a small factor in accounting for most historical oilprice shocks

For comparison purpose we have also applied the testsof PWY and PSY which are the most popular econometrictechniques in the literature on speculative bubbles in the oilmarkets If we were to follow the test procedures of PWY andPSY our conclusions are also broadly consistent with therich literature that supports the existence of speculativebubbles in oil prices However these cointegration-basedbubble tests suffer from the joint hypothesis problem ldquoEverytest of a bubble is a joint test of the presence of the bubble inthe data and the validity of the model applied by theeconometricianrdquo (see Giglio et al [60]) More specifically themain drawback of such indirect model-dependent tests ofbubbles typically refers to the difficulty in distinguishingbetween the unknown true model for market fundamentalsand the presence of bubbles (see for example Gurkaynak[37] Pavlidis et al [42] Pavlidis et al [43]) Given thefundamental value of oil price is unobservable the literaturehas resorted to observe economic and financial variableswhich are used to estimate the intrinsic values For exampleboth Phillips and Yu [16] and Caspi et al [39] simplynormalized crude oil prices by US oil inventory data as thefundamental value -is measure is questionable since their

method cannot sufficiently capture the information onmarket fundamentals and neglect the major driversmdashlikeglobal economic activitymdashalso affecting the fundamentalvalue of crude oil -is issue of misspecified fundamentals(either in terms of model misspecification or omitted var-iables or both of these reasons) may make researchers fail toarrive at a conclusion from evidence of explosive pricesalone To make matters worse even though we have cap-tured the true market fundamentals we may still fail todetect bubbles if the explosive behavior in the price of oil juststems from the explosive fundamentals

Compared to earlier studies as briefly mentioned aboveour methodology has the main advantage that by consid-ering the dependence between the random coefficient andsystem shocks we have incorporated endogeneity into thecontinuous-time random coefficient setting and thus we donot need to prespecify the fundamental component of the oilprice -e test results for endogeneity suggest that endo-geneity is important in the data-generating process Inparticular our empirical analysis also finds that there are noexplosive episodes in crude oil markets-is implies that it isnot necessary to require the specification of a fundamentalvalue of the oil price Like Pavlidis et al [43] their methodsand ours have in common that the problem of the well-known joint hypothesis has to be some extent been ef-fectively ameliorated Although our approach is quite dis-tinct we draw the same substantive conclusion thatspeculative bubbles can be ruled out as a key determinant ofthe price of crude oil in the 2000s reaffirming the con-clusions of Kilian and Murphy [19] Baumeister andHamilton [21] Zhou [22] and Fattouh et al [61]

8 Conclusions

Repeated surges and crashes in crude oil prices during lastdecades have sparked a heated debate on their underlyingcauses among academics commentors and market regu-lators One of the central questions concerns whetherfinancialization of oil future markets creates a speculativebubble that subsequently collapses Answering this questionis difficult because one has to rely upon strong assumptionsabout the data-generating mechanism for the fundamentalvalue of the oil price In this paper we present a stylizedmodel of asset pricing for a commodity to address this issueOur theoretical framework disentangles the effects ofspeculative bubbles from the traditional fundamental effectson oil price dynamics We use this setting to illustrate thatbubbles if they are present should demonstrate explosivebehavior in prices We then provide a novel approach forbubble detection that covers various explosive scenarios andfind no evidence of speculative bubbles in oil markets evenduring episodes where the most popular SADF and GSADFmethodologies identify a speculative bubble

Our conclusions have pertinent implications for policymakers Given the historically important role of oil prices inthe real economy one of the serious concerns in policycircles over the past decade has been whether speculativetrading activities in oil future markets are responsible for thelarge price swings For example in their 2008 testimonies to

Complexity 15

the US Senate Commerce Committee Michael Masters andGeorge Soros posited that speculators had fueled price in-creases since 2003 and had caused a part of a ldquosuperbubblerdquoin 2007 and 2008 resulting in significantly higher energycosts for consumers -is widespread bubble view domi-nated policy discussions and efforts to limit speculativetrading in the future market quickly followed As a responseto the excessive speculation pressures on prices theDoddndashFrank Wall Street Reform and Consumer ProtectionAct was enacted by the US Congress in 2010 As a directpolicy implication our results highlight that imposingstricter regulations on trading in oil derivative marketscannot be expected to lower the oil price given the fact thatthe synchronized boom and bust cycles witnessed in the past20 years are mainly due to economic fundamentals whichappear likely to resurge with the business cycle -e aboveanalysis implies that there is no need to enforce additionalregulatory limits against financial arbitrage in the com-modity market

Appendix

The SADF and GSADF Methodologies

Appendix describes the SADF and GSADF testing proce-dures used to detect speculative bubbles in Section 62 For agenetic time series process xt consider the following rolling-window type augmented DickeyndashFuller (ADF) regressionequation

xt ar1r2+ μr1r2

xtminus 1 + 1113944

J

j1ϕj

r1r2Δxtminus j + εt εt sim NID 0 δ2r1r2

1113874 1113875

(A1)

where Δxtminus j with j 1 J are lagged first differences ofxt incorporated to accommodate potential serial correla-tion ar1 r2

μr1 r2 and ϕj

r1 r2 with j 1 J are regression

coefficients the subscripts r1 and r2 refer to fractions of thetotal sample size (ofT observations) that set the start and endpoints of a subsample period and NID represents inde-pendent and normal distribution -e ADF test statisticcorresponding to the unit root null hypothesisH0 μr1 r2

0 is simply the t-ratio

ADFr2r1

1113954μr1 r2

se 1113954μr1 r21113872 1113873

(A2)

and the right-side alternative hypothesis of explosive be-havior is H0 μr1r2

gt 0 We can obtain the standard ADF teststatistic by running regression (A1) on the whole sampleie by specifying r1 0 and r2 1 A large amount of re-search has demonstrated that the standard ADF test ADF10has extremely low power in detecting periodically collapsingbubbles

To address the problem PSW proposed a recursiveregression technique which involves estimating (A1)employing a forward expanding sample Specifically theyused subsamples of the available data to recursively calculatea sequence of ADF statistics denoted by ADFr2

0 -e startpoint of the sample period is fixed at r1 0 while the endpoint r2 increases from r0 (the minimum window size) toone (the full sample period) -e SADF statistic called thesupremum of ADFr2

0 is defined by

SADF r0( 1113857 supr2isin r0 1[ ]

ADFr20

(A3)

and has the following limit distribution under the nullhypothesis

supr2isin r0 1[ ]

1113938r2

0 WdW

1113938r2

0 W21113872 1113873(12)

(A4)

where W is the standard Brownian motion Testing thepresence of explosive behavior entails comparing the ADFr2

0statistic with the right-side critical values obtained throughMonte Carlo simulations

More recently PSY showed that when the sample perioddisplays more than one boom-bust bubble the SADF testprocedure may perform poorly and fail to detect or con-sistently date-stamp alternative bubbles -is deficiencymotivates PSY to formulate an extension of the SADF testthe GSADF test which covers more subsamples of the datathan the SADF by allowing both the end point r2 and thestart point r1 to vary As a result the GSADF test has moreflexibility and gains substantial power making this testconsistent with multiple boom-bust bubbles -e GSADFtest statistic is defined as

GSADF r0( 1113857 supr2isin r0 1[ ]r1isin 0r2minus r0[ ]

ADFr2r1 (A5)

with the limit distribution

supr2isin r0 1[ ]r1isin 0r2minus r0[ ]

(12)rw W r2( 11138572

minus W r1( 11138572

minus rw1113960 1113961 minus 1113938r2

r1W(r)dr W r2( 1113857 minus W r1( 11138571113858 1113859

r12w rw 1113938

r2

r1W(r)2dr minus 1113938

r2

r1W(r)dr1113876 1113877

21113896 1113897

12

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎭

(A6)

where rw r2 minus r1 is the expanding window size Like theSADF test comparison of the GASDF statistic with the right-

tailed critical values from its limit distribution enables us to testfor the unit root hypothesis against explosive behavior

16 Complexity

If the null hypothesis of a unit root in xt is rejectedthen in the second stage the SADF and GSADF tests canidentify the exact episodes during which the xt seriesexhibits explosive dynamics To locate the emergence andconclusion of explosiveness the SADF methodologymatches the sequence of the recursive backward ADF(BADF) test statistics against the right-tailed criticalvalues of the limit distribution of the standard Dick-eyndashFuller test statistic Suppose that re is the originationdate and rf is the finish date of the bubble in the data-ese estimates of dates can be constructed from

1113954re infr2isin r0 1[ ]

r2 ADFr20 gt cv

αr2

1113966 1113967

1113954rf infr2isin 1113954re11113858 1113859

r2 ADFr20 lt cv

αr2

1113966 1113967(A7)

where cvαr2 is 100(1 minus α) the right-tailed critical value of theADF statistic corresponding to the chosen significance levelof α For the GSADF methodology the date-stampingstrategy in this case is based on the backward SADF(BSADF) statistic

BSADFr2r0( 1113857 sup

r1isin 0r2minus r0[ ]ADFr2

r1

(A8)

Likewise the origination date of the bubble is defined asthe first observation that the BSADF statistic is greater thanits critical value

1113954re infr2isin r0 1[ ]

r2 BSADFr2r0( 1113857gt cv

βT

r21113966 1113967 (A9)

and the finish date as the first date after which the BSADFdrops back below its critical value

1113954rf infr2isin 1113954re11113858 1113859

r2 BSADFr2r0( 1113857lt cvβT

r21113966 1113967 (A10)

where cvβTr2 is 100(1 minus βT) the right-tailed critical value of

the SADF based on Tr2 and the corresponding significancelevel is βT

Data Availability

-e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

-e authors declare no conflicts of interest

Acknowledgments

-e authors gratefully acknowledge the financial supportfrom the National Social Science Foundation of China underGrant no 18BTJ039

References

[1] M W Masters Testimony before the US Senate Committee onHomeland Security and Governmental Affairs 2008 httpswwwhsgacsenategovimomediadoc052008Masterspdfattempt2

[2] M W Masters Testimony before the Commodity FuturesTrading Commission 2010 httpscftcgovsitesdefaultfilesidcgroupspublicnewsroomdocumentsfilemetalmarkets032510_masterspdf

[3] J P Kennedy ldquo-e high cost of gambling on oilrdquo Section A p23 New York Times New York NY USANew York Times2012

[4] I-H Cheng and W Xiong ldquoFinancialization of commoditymarketsrdquo Annual Review of Financial Economics vol 6 no 1pp 419ndash441 2014

[5] K Tang and W Xiong ldquoIndex investment and the financi-alization of commoditiesrdquo Financial Analysts Journal vol 68no 6 pp 54ndash74 2012

[6] O J Blanchard and M W Watson ldquoBubbles rational ex-pectations and financial marketsrdquo in Crisis in the Economicand Financial Structure P Wachtel Ed Lexington BooksLexington MA USA pp 295ndash315 1982

[7] B T Diba and H I Grossman ldquoOn the inception of rationalbubblesrdquo e Quarterly Journal of Economics vol 102 no 3pp 697ndash700 1987

[8] G W Evans ldquo-e fragility of sunspots and bubblesrdquo Journalof Monetary Economics vol 23 no 2 pp 297ndash317 1989

[9] G W Evans and S Honkapohja ldquoOn the robustness ofbubbles in linear RE modelsrdquo International Economic Reviewvol 33 no 1 pp 1ndash14 1992

[10] E Pavlidis A Yusupova I Paya et al ldquoEpisodes of exu-berance in housing markets in search of the smoking gunrdquoe Journal of Real Estate Finance and Economics vol 53no 4 pp 419ndash449 2016

[11] E Martınez-Garcıa and V Grossman ldquoExplosive dynamics inhouse prices An exploration of financial market spillovers inhousing markets around the worldrdquo Journal of InternationalMoney and Finance vol 101 p 102103 2020

[12] S Shi A Rahman and B Z Wang ldquoAustralian housingmarket booms fundamentals or speculationlowastrdquo EconomicRecord 2020 In press

[13] P C B Phillips Y Wu and J Yu ldquoExplosive behavior in the1990s NASDAQ when did exuberance escalate asset val-ueslowastrdquo International Economic Review vol 52 no 1pp 201ndash226 2011

[14] P C B Phillips S Shi and J Yu ldquoTesting for multiplebubbles historical episodes of exuberance and collapse in theSampp 500rdquo International Economic Review vol 56 no 4pp 1043ndash1078 2015

[15] P C B Phillips S Shi and J Yu ldquoTesting for multiplebubbles limit theory of real-time detectorsrdquo InternationalEconomic Review vol 56 no 4 pp 1079ndash1134 2015

[16] P C B Phillips and J Yu ldquoDating the timeline of financialbubbles during the subprime crisisrdquo Quantitative Economicsvol 2 no 3 pp 455ndash491 2011

[17] D Tsvetanov J Coakley and N Kellard ldquoBubbling over thebehaviour of oil futures along the yield curverdquo Journal ofEmpirical Finance vol 38 no Part B pp 516ndash533 2016

[18] I Figuerola-Ferretti J R Mccrorie and I ParaskevopoulosldquoMild explosivity in recent crude oil pricesrdquo Energy Eco-nomics vol 87 p 104387 2020

[19] L Kilian and D P Murphy ldquo-e role of inventories andspeculative trading in the global market for crude oilrdquo Journalof Applied Econometrics vol 29 no 3 pp 454ndash478 2014

Complexity 17

documents that the fundamental drivers of oil price fluc-tuations have been dominated essentially by supply anddemand shocks see eg Kilian [56] Lippi and Nobili [57]Kilian andMurphy [58] and Baumeister and Peersman [59]Kilian and Murphy [19] focused on the role speculation byaugmenting the SVAR framework of their earlier paper byglobal oil inventory data -ey specified a four-variablemodel for the global oil market (the percentage change inglobal oil production the real price of oil a suitable proxy ofcyclical variation in global real economic activity andchanges in global crude oil inventories) -is setting andtheir sign-identified scheme allow them to separately explainvariation in these data as four different types of shocks (1) ashock to the amount of oil pumped out of the ground (ldquooilsupply shockrdquo) (2) an oil flow demand shock (3) an oil-specific oil demand shock and especially (4) a speculative(or inventory) demand shock -eir core finding is that thespike in the real price of crude oil prices during 2003ndash2008was caused by unexpected increase in global oil consump-tion Using two alternative proxies for global above-groundcrude oil inventories Kilian and Lee [20] reestimated themodel of Kilian and Murphy [19] and ruled out the role ofspeculation in explaining the recent surge of the real oil priceon the basis of their estimation results Also Zhou [22]assessed the sensitivity of the findings in Kilian and Murphy[19] to the introduction of various state-of-the-art oil marketmethods She shows that the core conclusions reached byKilian and Murphy [19] are highly robust to some refine-ments corrections and extension of the KilianndashMurphymodel Baumeister and Hamilton [21] recently contendedthat prior information plays a key role in any SVAR modelsand traditional identification approaches can be generalizedto a special case of a Bayesian perspective -ey revisited theimportance of supply and demand shocks in the oil marketand demonstrated that speculative demand shock turns outto be only a small factor in accounting for most historical oilprice shocks

For comparison purpose we have also applied the testsof PWY and PSY which are the most popular econometrictechniques in the literature on speculative bubbles in the oilmarkets If we were to follow the test procedures of PWY andPSY our conclusions are also broadly consistent with therich literature that supports the existence of speculativebubbles in oil prices However these cointegration-basedbubble tests suffer from the joint hypothesis problem ldquoEverytest of a bubble is a joint test of the presence of the bubble inthe data and the validity of the model applied by theeconometricianrdquo (see Giglio et al [60]) More specifically themain drawback of such indirect model-dependent tests ofbubbles typically refers to the difficulty in distinguishingbetween the unknown true model for market fundamentalsand the presence of bubbles (see for example Gurkaynak[37] Pavlidis et al [42] Pavlidis et al [43]) Given thefundamental value of oil price is unobservable the literaturehas resorted to observe economic and financial variableswhich are used to estimate the intrinsic values For exampleboth Phillips and Yu [16] and Caspi et al [39] simplynormalized crude oil prices by US oil inventory data as thefundamental value -is measure is questionable since their

method cannot sufficiently capture the information onmarket fundamentals and neglect the major driversmdashlikeglobal economic activitymdashalso affecting the fundamentalvalue of crude oil -is issue of misspecified fundamentals(either in terms of model misspecification or omitted var-iables or both of these reasons) may make researchers fail toarrive at a conclusion from evidence of explosive pricesalone To make matters worse even though we have cap-tured the true market fundamentals we may still fail todetect bubbles if the explosive behavior in the price of oil juststems from the explosive fundamentals

Compared to earlier studies as briefly mentioned aboveour methodology has the main advantage that by consid-ering the dependence between the random coefficient andsystem shocks we have incorporated endogeneity into thecontinuous-time random coefficient setting and thus we donot need to prespecify the fundamental component of the oilprice -e test results for endogeneity suggest that endo-geneity is important in the data-generating process Inparticular our empirical analysis also finds that there are noexplosive episodes in crude oil markets-is implies that it isnot necessary to require the specification of a fundamentalvalue of the oil price Like Pavlidis et al [43] their methodsand ours have in common that the problem of the well-known joint hypothesis has to be some extent been ef-fectively ameliorated Although our approach is quite dis-tinct we draw the same substantive conclusion thatspeculative bubbles can be ruled out as a key determinant ofthe price of crude oil in the 2000s reaffirming the con-clusions of Kilian and Murphy [19] Baumeister andHamilton [21] Zhou [22] and Fattouh et al [61]

8 Conclusions

Repeated surges and crashes in crude oil prices during lastdecades have sparked a heated debate on their underlyingcauses among academics commentors and market regu-lators One of the central questions concerns whetherfinancialization of oil future markets creates a speculativebubble that subsequently collapses Answering this questionis difficult because one has to rely upon strong assumptionsabout the data-generating mechanism for the fundamentalvalue of the oil price In this paper we present a stylizedmodel of asset pricing for a commodity to address this issueOur theoretical framework disentangles the effects ofspeculative bubbles from the traditional fundamental effectson oil price dynamics We use this setting to illustrate thatbubbles if they are present should demonstrate explosivebehavior in prices We then provide a novel approach forbubble detection that covers various explosive scenarios andfind no evidence of speculative bubbles in oil markets evenduring episodes where the most popular SADF and GSADFmethodologies identify a speculative bubble

Our conclusions have pertinent implications for policymakers Given the historically important role of oil prices inthe real economy one of the serious concerns in policycircles over the past decade has been whether speculativetrading activities in oil future markets are responsible for thelarge price swings For example in their 2008 testimonies to

Complexity 15

the US Senate Commerce Committee Michael Masters andGeorge Soros posited that speculators had fueled price in-creases since 2003 and had caused a part of a ldquosuperbubblerdquoin 2007 and 2008 resulting in significantly higher energycosts for consumers -is widespread bubble view domi-nated policy discussions and efforts to limit speculativetrading in the future market quickly followed As a responseto the excessive speculation pressures on prices theDoddndashFrank Wall Street Reform and Consumer ProtectionAct was enacted by the US Congress in 2010 As a directpolicy implication our results highlight that imposingstricter regulations on trading in oil derivative marketscannot be expected to lower the oil price given the fact thatthe synchronized boom and bust cycles witnessed in the past20 years are mainly due to economic fundamentals whichappear likely to resurge with the business cycle -e aboveanalysis implies that there is no need to enforce additionalregulatory limits against financial arbitrage in the com-modity market

Appendix

The SADF and GSADF Methodologies

Appendix describes the SADF and GSADF testing proce-dures used to detect speculative bubbles in Section 62 For agenetic time series process xt consider the following rolling-window type augmented DickeyndashFuller (ADF) regressionequation

xt ar1r2+ μr1r2

xtminus 1 + 1113944

J

j1ϕj

r1r2Δxtminus j + εt εt sim NID 0 δ2r1r2

1113874 1113875

(A1)

where Δxtminus j with j 1 J are lagged first differences ofxt incorporated to accommodate potential serial correla-tion ar1 r2

μr1 r2 and ϕj

r1 r2 with j 1 J are regression

coefficients the subscripts r1 and r2 refer to fractions of thetotal sample size (ofT observations) that set the start and endpoints of a subsample period and NID represents inde-pendent and normal distribution -e ADF test statisticcorresponding to the unit root null hypothesisH0 μr1 r2

0 is simply the t-ratio

ADFr2r1

1113954μr1 r2

se 1113954μr1 r21113872 1113873

(A2)

and the right-side alternative hypothesis of explosive be-havior is H0 μr1r2

gt 0 We can obtain the standard ADF teststatistic by running regression (A1) on the whole sampleie by specifying r1 0 and r2 1 A large amount of re-search has demonstrated that the standard ADF test ADF10has extremely low power in detecting periodically collapsingbubbles

To address the problem PSW proposed a recursiveregression technique which involves estimating (A1)employing a forward expanding sample Specifically theyused subsamples of the available data to recursively calculatea sequence of ADF statistics denoted by ADFr2

0 -e startpoint of the sample period is fixed at r1 0 while the endpoint r2 increases from r0 (the minimum window size) toone (the full sample period) -e SADF statistic called thesupremum of ADFr2

0 is defined by

SADF r0( 1113857 supr2isin r0 1[ ]

ADFr20

(A3)

and has the following limit distribution under the nullhypothesis

supr2isin r0 1[ ]

1113938r2

0 WdW

1113938r2

0 W21113872 1113873(12)

(A4)

where W is the standard Brownian motion Testing thepresence of explosive behavior entails comparing the ADFr2

0statistic with the right-side critical values obtained throughMonte Carlo simulations

More recently PSY showed that when the sample perioddisplays more than one boom-bust bubble the SADF testprocedure may perform poorly and fail to detect or con-sistently date-stamp alternative bubbles -is deficiencymotivates PSY to formulate an extension of the SADF testthe GSADF test which covers more subsamples of the datathan the SADF by allowing both the end point r2 and thestart point r1 to vary As a result the GSADF test has moreflexibility and gains substantial power making this testconsistent with multiple boom-bust bubbles -e GSADFtest statistic is defined as

GSADF r0( 1113857 supr2isin r0 1[ ]r1isin 0r2minus r0[ ]

ADFr2r1 (A5)

with the limit distribution

supr2isin r0 1[ ]r1isin 0r2minus r0[ ]

(12)rw W r2( 11138572

minus W r1( 11138572

minus rw1113960 1113961 minus 1113938r2

r1W(r)dr W r2( 1113857 minus W r1( 11138571113858 1113859

r12w rw 1113938

r2

r1W(r)2dr minus 1113938

r2

r1W(r)dr1113876 1113877

21113896 1113897

12

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎭

(A6)

where rw r2 minus r1 is the expanding window size Like theSADF test comparison of the GASDF statistic with the right-

tailed critical values from its limit distribution enables us to testfor the unit root hypothesis against explosive behavior

16 Complexity

If the null hypothesis of a unit root in xt is rejectedthen in the second stage the SADF and GSADF tests canidentify the exact episodes during which the xt seriesexhibits explosive dynamics To locate the emergence andconclusion of explosiveness the SADF methodologymatches the sequence of the recursive backward ADF(BADF) test statistics against the right-tailed criticalvalues of the limit distribution of the standard Dick-eyndashFuller test statistic Suppose that re is the originationdate and rf is the finish date of the bubble in the data-ese estimates of dates can be constructed from

1113954re infr2isin r0 1[ ]

r2 ADFr20 gt cv

αr2

1113966 1113967

1113954rf infr2isin 1113954re11113858 1113859

r2 ADFr20 lt cv

αr2

1113966 1113967(A7)

where cvαr2 is 100(1 minus α) the right-tailed critical value of theADF statistic corresponding to the chosen significance levelof α For the GSADF methodology the date-stampingstrategy in this case is based on the backward SADF(BSADF) statistic

BSADFr2r0( 1113857 sup

r1isin 0r2minus r0[ ]ADFr2

r1

(A8)

Likewise the origination date of the bubble is defined asthe first observation that the BSADF statistic is greater thanits critical value

1113954re infr2isin r0 1[ ]

r2 BSADFr2r0( 1113857gt cv

βT

r21113966 1113967 (A9)

and the finish date as the first date after which the BSADFdrops back below its critical value

1113954rf infr2isin 1113954re11113858 1113859

r2 BSADFr2r0( 1113857lt cvβT

r21113966 1113967 (A10)

where cvβTr2 is 100(1 minus βT) the right-tailed critical value of

the SADF based on Tr2 and the corresponding significancelevel is βT

Data Availability

-e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

-e authors declare no conflicts of interest

Acknowledgments

-e authors gratefully acknowledge the financial supportfrom the National Social Science Foundation of China underGrant no 18BTJ039

References

[1] M W Masters Testimony before the US Senate Committee onHomeland Security and Governmental Affairs 2008 httpswwwhsgacsenategovimomediadoc052008Masterspdfattempt2

[2] M W Masters Testimony before the Commodity FuturesTrading Commission 2010 httpscftcgovsitesdefaultfilesidcgroupspublicnewsroomdocumentsfilemetalmarkets032510_masterspdf

[3] J P Kennedy ldquo-e high cost of gambling on oilrdquo Section A p23 New York Times New York NY USANew York Times2012

[4] I-H Cheng and W Xiong ldquoFinancialization of commoditymarketsrdquo Annual Review of Financial Economics vol 6 no 1pp 419ndash441 2014

[5] K Tang and W Xiong ldquoIndex investment and the financi-alization of commoditiesrdquo Financial Analysts Journal vol 68no 6 pp 54ndash74 2012

[6] O J Blanchard and M W Watson ldquoBubbles rational ex-pectations and financial marketsrdquo in Crisis in the Economicand Financial Structure P Wachtel Ed Lexington BooksLexington MA USA pp 295ndash315 1982

[7] B T Diba and H I Grossman ldquoOn the inception of rationalbubblesrdquo e Quarterly Journal of Economics vol 102 no 3pp 697ndash700 1987

[8] G W Evans ldquo-e fragility of sunspots and bubblesrdquo Journalof Monetary Economics vol 23 no 2 pp 297ndash317 1989

[9] G W Evans and S Honkapohja ldquoOn the robustness ofbubbles in linear RE modelsrdquo International Economic Reviewvol 33 no 1 pp 1ndash14 1992

[10] E Pavlidis A Yusupova I Paya et al ldquoEpisodes of exu-berance in housing markets in search of the smoking gunrdquoe Journal of Real Estate Finance and Economics vol 53no 4 pp 419ndash449 2016

[11] E Martınez-Garcıa and V Grossman ldquoExplosive dynamics inhouse prices An exploration of financial market spillovers inhousing markets around the worldrdquo Journal of InternationalMoney and Finance vol 101 p 102103 2020

[12] S Shi A Rahman and B Z Wang ldquoAustralian housingmarket booms fundamentals or speculationlowastrdquo EconomicRecord 2020 In press

[13] P C B Phillips Y Wu and J Yu ldquoExplosive behavior in the1990s NASDAQ when did exuberance escalate asset val-ueslowastrdquo International Economic Review vol 52 no 1pp 201ndash226 2011

[14] P C B Phillips S Shi and J Yu ldquoTesting for multiplebubbles historical episodes of exuberance and collapse in theSampp 500rdquo International Economic Review vol 56 no 4pp 1043ndash1078 2015

[15] P C B Phillips S Shi and J Yu ldquoTesting for multiplebubbles limit theory of real-time detectorsrdquo InternationalEconomic Review vol 56 no 4 pp 1079ndash1134 2015

[16] P C B Phillips and J Yu ldquoDating the timeline of financialbubbles during the subprime crisisrdquo Quantitative Economicsvol 2 no 3 pp 455ndash491 2011

[17] D Tsvetanov J Coakley and N Kellard ldquoBubbling over thebehaviour of oil futures along the yield curverdquo Journal ofEmpirical Finance vol 38 no Part B pp 516ndash533 2016

[18] I Figuerola-Ferretti J R Mccrorie and I ParaskevopoulosldquoMild explosivity in recent crude oil pricesrdquo Energy Eco-nomics vol 87 p 104387 2020

[19] L Kilian and D P Murphy ldquo-e role of inventories andspeculative trading in the global market for crude oilrdquo Journalof Applied Econometrics vol 29 no 3 pp 454ndash478 2014

Complexity 17

the US Senate Commerce Committee Michael Masters andGeorge Soros posited that speculators had fueled price in-creases since 2003 and had caused a part of a ldquosuperbubblerdquoin 2007 and 2008 resulting in significantly higher energycosts for consumers -is widespread bubble view domi-nated policy discussions and efforts to limit speculativetrading in the future market quickly followed As a responseto the excessive speculation pressures on prices theDoddndashFrank Wall Street Reform and Consumer ProtectionAct was enacted by the US Congress in 2010 As a directpolicy implication our results highlight that imposingstricter regulations on trading in oil derivative marketscannot be expected to lower the oil price given the fact thatthe synchronized boom and bust cycles witnessed in the past20 years are mainly due to economic fundamentals whichappear likely to resurge with the business cycle -e aboveanalysis implies that there is no need to enforce additionalregulatory limits against financial arbitrage in the com-modity market

Appendix

The SADF and GSADF Methodologies

Appendix describes the SADF and GSADF testing proce-dures used to detect speculative bubbles in Section 62 For agenetic time series process xt consider the following rolling-window type augmented DickeyndashFuller (ADF) regressionequation

xt ar1r2+ μr1r2

xtminus 1 + 1113944

J

j1ϕj

r1r2Δxtminus j + εt εt sim NID 0 δ2r1r2

1113874 1113875

(A1)

where Δxtminus j with j 1 J are lagged first differences ofxt incorporated to accommodate potential serial correla-tion ar1 r2

μr1 r2 and ϕj

r1 r2 with j 1 J are regression

coefficients the subscripts r1 and r2 refer to fractions of thetotal sample size (ofT observations) that set the start and endpoints of a subsample period and NID represents inde-pendent and normal distribution -e ADF test statisticcorresponding to the unit root null hypothesisH0 μr1 r2

0 is simply the t-ratio

ADFr2r1

1113954μr1 r2

se 1113954μr1 r21113872 1113873

(A2)

and the right-side alternative hypothesis of explosive be-havior is H0 μr1r2

gt 0 We can obtain the standard ADF teststatistic by running regression (A1) on the whole sampleie by specifying r1 0 and r2 1 A large amount of re-search has demonstrated that the standard ADF test ADF10has extremely low power in detecting periodically collapsingbubbles

To address the problem PSW proposed a recursiveregression technique which involves estimating (A1)employing a forward expanding sample Specifically theyused subsamples of the available data to recursively calculatea sequence of ADF statistics denoted by ADFr2

0 -e startpoint of the sample period is fixed at r1 0 while the endpoint r2 increases from r0 (the minimum window size) toone (the full sample period) -e SADF statistic called thesupremum of ADFr2

0 is defined by

SADF r0( 1113857 supr2isin r0 1[ ]

ADFr20

(A3)

and has the following limit distribution under the nullhypothesis

supr2isin r0 1[ ]

1113938r2

0 WdW

1113938r2

0 W21113872 1113873(12)

(A4)

where W is the standard Brownian motion Testing thepresence of explosive behavior entails comparing the ADFr2

0statistic with the right-side critical values obtained throughMonte Carlo simulations

More recently PSY showed that when the sample perioddisplays more than one boom-bust bubble the SADF testprocedure may perform poorly and fail to detect or con-sistently date-stamp alternative bubbles -is deficiencymotivates PSY to formulate an extension of the SADF testthe GSADF test which covers more subsamples of the datathan the SADF by allowing both the end point r2 and thestart point r1 to vary As a result the GSADF test has moreflexibility and gains substantial power making this testconsistent with multiple boom-bust bubbles -e GSADFtest statistic is defined as

GSADF r0( 1113857 supr2isin r0 1[ ]r1isin 0r2minus r0[ ]

ADFr2r1 (A5)

with the limit distribution

supr2isin r0 1[ ]r1isin 0r2minus r0[ ]

(12)rw W r2( 11138572

minus W r1( 11138572

minus rw1113960 1113961 minus 1113938r2

r1W(r)dr W r2( 1113857 minus W r1( 11138571113858 1113859

r12w rw 1113938

r2

r1W(r)2dr minus 1113938

r2

r1W(r)dr1113876 1113877

21113896 1113897

12

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎭

(A6)

where rw r2 minus r1 is the expanding window size Like theSADF test comparison of the GASDF statistic with the right-

tailed critical values from its limit distribution enables us to testfor the unit root hypothesis against explosive behavior

16 Complexity

If the null hypothesis of a unit root in xt is rejectedthen in the second stage the SADF and GSADF tests canidentify the exact episodes during which the xt seriesexhibits explosive dynamics To locate the emergence andconclusion of explosiveness the SADF methodologymatches the sequence of the recursive backward ADF(BADF) test statistics against the right-tailed criticalvalues of the limit distribution of the standard Dick-eyndashFuller test statistic Suppose that re is the originationdate and rf is the finish date of the bubble in the data-ese estimates of dates can be constructed from

1113954re infr2isin r0 1[ ]

r2 ADFr20 gt cv

αr2

1113966 1113967

1113954rf infr2isin 1113954re11113858 1113859

r2 ADFr20 lt cv

αr2

1113966 1113967(A7)

where cvαr2 is 100(1 minus α) the right-tailed critical value of theADF statistic corresponding to the chosen significance levelof α For the GSADF methodology the date-stampingstrategy in this case is based on the backward SADF(BSADF) statistic

BSADFr2r0( 1113857 sup

r1isin 0r2minus r0[ ]ADFr2

r1

(A8)

Likewise the origination date of the bubble is defined asthe first observation that the BSADF statistic is greater thanits critical value

1113954re infr2isin r0 1[ ]

r2 BSADFr2r0( 1113857gt cv

βT

r21113966 1113967 (A9)

and the finish date as the first date after which the BSADFdrops back below its critical value

1113954rf infr2isin 1113954re11113858 1113859

r2 BSADFr2r0( 1113857lt cvβT

r21113966 1113967 (A10)

where cvβTr2 is 100(1 minus βT) the right-tailed critical value of

the SADF based on Tr2 and the corresponding significancelevel is βT

Data Availability

-e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

-e authors declare no conflicts of interest

Acknowledgments

-e authors gratefully acknowledge the financial supportfrom the National Social Science Foundation of China underGrant no 18BTJ039

References

[1] M W Masters Testimony before the US Senate Committee onHomeland Security and Governmental Affairs 2008 httpswwwhsgacsenategovimomediadoc052008Masterspdfattempt2

[2] M W Masters Testimony before the Commodity FuturesTrading Commission 2010 httpscftcgovsitesdefaultfilesidcgroupspublicnewsroomdocumentsfilemetalmarkets032510_masterspdf

[3] J P Kennedy ldquo-e high cost of gambling on oilrdquo Section A p23 New York Times New York NY USANew York Times2012

[4] I-H Cheng and W Xiong ldquoFinancialization of commoditymarketsrdquo Annual Review of Financial Economics vol 6 no 1pp 419ndash441 2014

[5] K Tang and W Xiong ldquoIndex investment and the financi-alization of commoditiesrdquo Financial Analysts Journal vol 68no 6 pp 54ndash74 2012

[6] O J Blanchard and M W Watson ldquoBubbles rational ex-pectations and financial marketsrdquo in Crisis in the Economicand Financial Structure P Wachtel Ed Lexington BooksLexington MA USA pp 295ndash315 1982

[7] B T Diba and H I Grossman ldquoOn the inception of rationalbubblesrdquo e Quarterly Journal of Economics vol 102 no 3pp 697ndash700 1987

[8] G W Evans ldquo-e fragility of sunspots and bubblesrdquo Journalof Monetary Economics vol 23 no 2 pp 297ndash317 1989

[9] G W Evans and S Honkapohja ldquoOn the robustness ofbubbles in linear RE modelsrdquo International Economic Reviewvol 33 no 1 pp 1ndash14 1992

[10] E Pavlidis A Yusupova I Paya et al ldquoEpisodes of exu-berance in housing markets in search of the smoking gunrdquoe Journal of Real Estate Finance and Economics vol 53no 4 pp 419ndash449 2016

[11] E Martınez-Garcıa and V Grossman ldquoExplosive dynamics inhouse prices An exploration of financial market spillovers inhousing markets around the worldrdquo Journal of InternationalMoney and Finance vol 101 p 102103 2020

[12] S Shi A Rahman and B Z Wang ldquoAustralian housingmarket booms fundamentals or speculationlowastrdquo EconomicRecord 2020 In press

[13] P C B Phillips Y Wu and J Yu ldquoExplosive behavior in the1990s NASDAQ when did exuberance escalate asset val-ueslowastrdquo International Economic Review vol 52 no 1pp 201ndash226 2011

[14] P C B Phillips S Shi and J Yu ldquoTesting for multiplebubbles historical episodes of exuberance and collapse in theSampp 500rdquo International Economic Review vol 56 no 4pp 1043ndash1078 2015

[15] P C B Phillips S Shi and J Yu ldquoTesting for multiplebubbles limit theory of real-time detectorsrdquo InternationalEconomic Review vol 56 no 4 pp 1079ndash1134 2015

[16] P C B Phillips and J Yu ldquoDating the timeline of financialbubbles during the subprime crisisrdquo Quantitative Economicsvol 2 no 3 pp 455ndash491 2011

[17] D Tsvetanov J Coakley and N Kellard ldquoBubbling over thebehaviour of oil futures along the yield curverdquo Journal ofEmpirical Finance vol 38 no Part B pp 516ndash533 2016

[18] I Figuerola-Ferretti J R Mccrorie and I ParaskevopoulosldquoMild explosivity in recent crude oil pricesrdquo Energy Eco-nomics vol 87 p 104387 2020

[19] L Kilian and D P Murphy ldquo-e role of inventories andspeculative trading in the global market for crude oilrdquo Journalof Applied Econometrics vol 29 no 3 pp 454ndash478 2014

Complexity 17

If the null hypothesis of a unit root in xt is rejectedthen in the second stage the SADF and GSADF tests canidentify the exact episodes during which the xt seriesexhibits explosive dynamics To locate the emergence andconclusion of explosiveness the SADF methodologymatches the sequence of the recursive backward ADF(BADF) test statistics against the right-tailed criticalvalues of the limit distribution of the standard Dick-eyndashFuller test statistic Suppose that re is the originationdate and rf is the finish date of the bubble in the data-ese estimates of dates can be constructed from

1113954re infr2isin r0 1[ ]

r2 ADFr20 gt cv

αr2

1113966 1113967

1113954rf infr2isin 1113954re11113858 1113859

r2 ADFr20 lt cv

αr2

1113966 1113967(A7)

where cvαr2 is 100(1 minus α) the right-tailed critical value of theADF statistic corresponding to the chosen significance levelof α For the GSADF methodology the date-stampingstrategy in this case is based on the backward SADF(BSADF) statistic

BSADFr2r0( 1113857 sup

r1isin 0r2minus r0[ ]ADFr2

r1

(A8)

Likewise the origination date of the bubble is defined asthe first observation that the BSADF statistic is greater thanits critical value

1113954re infr2isin r0 1[ ]

r2 BSADFr2r0( 1113857gt cv

βT

r21113966 1113967 (A9)

and the finish date as the first date after which the BSADFdrops back below its critical value

1113954rf infr2isin 1113954re11113858 1113859

r2 BSADFr2r0( 1113857lt cvβT

r21113966 1113967 (A10)

where cvβTr2 is 100(1 minus βT) the right-tailed critical value of

the SADF based on Tr2 and the corresponding significancelevel is βT

Data Availability

-e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

-e authors declare no conflicts of interest

Acknowledgments

-e authors gratefully acknowledge the financial supportfrom the National Social Science Foundation of China underGrant no 18BTJ039

References

[1] M W Masters Testimony before the US Senate Committee onHomeland Security and Governmental Affairs 2008 httpswwwhsgacsenategovimomediadoc052008Masterspdfattempt2

[2] M W Masters Testimony before the Commodity FuturesTrading Commission 2010 httpscftcgovsitesdefaultfilesidcgroupspublicnewsroomdocumentsfilemetalmarkets032510_masterspdf

[3] J P Kennedy ldquo-e high cost of gambling on oilrdquo Section A p23 New York Times New York NY USANew York Times2012

[4] I-H Cheng and W Xiong ldquoFinancialization of commoditymarketsrdquo Annual Review of Financial Economics vol 6 no 1pp 419ndash441 2014

[5] K Tang and W Xiong ldquoIndex investment and the financi-alization of commoditiesrdquo Financial Analysts Journal vol 68no 6 pp 54ndash74 2012

[6] O J Blanchard and M W Watson ldquoBubbles rational ex-pectations and financial marketsrdquo in Crisis in the Economicand Financial Structure P Wachtel Ed Lexington BooksLexington MA USA pp 295ndash315 1982

[7] B T Diba and H I Grossman ldquoOn the inception of rationalbubblesrdquo e Quarterly Journal of Economics vol 102 no 3pp 697ndash700 1987

[8] G W Evans ldquo-e fragility of sunspots and bubblesrdquo Journalof Monetary Economics vol 23 no 2 pp 297ndash317 1989

[9] G W Evans and S Honkapohja ldquoOn the robustness ofbubbles in linear RE modelsrdquo International Economic Reviewvol 33 no 1 pp 1ndash14 1992

[10] E Pavlidis A Yusupova I Paya et al ldquoEpisodes of exu-berance in housing markets in search of the smoking gunrdquoe Journal of Real Estate Finance and Economics vol 53no 4 pp 419ndash449 2016

[11] E Martınez-Garcıa and V Grossman ldquoExplosive dynamics inhouse prices An exploration of financial market spillovers inhousing markets around the worldrdquo Journal of InternationalMoney and Finance vol 101 p 102103 2020

[12] S Shi A Rahman and B Z Wang ldquoAustralian housingmarket booms fundamentals or speculationlowastrdquo EconomicRecord 2020 In press

[13] P C B Phillips Y Wu and J Yu ldquoExplosive behavior in the1990s NASDAQ when did exuberance escalate asset val-ueslowastrdquo International Economic Review vol 52 no 1pp 201ndash226 2011

[14] P C B Phillips S Shi and J Yu ldquoTesting for multiplebubbles historical episodes of exuberance and collapse in theSampp 500rdquo International Economic Review vol 56 no 4pp 1043ndash1078 2015

[15] P C B Phillips S Shi and J Yu ldquoTesting for multiplebubbles limit theory of real-time detectorsrdquo InternationalEconomic Review vol 56 no 4 pp 1079ndash1134 2015

[16] P C B Phillips and J Yu ldquoDating the timeline of financialbubbles during the subprime crisisrdquo Quantitative Economicsvol 2 no 3 pp 455ndash491 2011

[17] D Tsvetanov J Coakley and N Kellard ldquoBubbling over thebehaviour of oil futures along the yield curverdquo Journal ofEmpirical Finance vol 38 no Part B pp 516ndash533 2016

[18] I Figuerola-Ferretti J R Mccrorie and I ParaskevopoulosldquoMild explosivity in recent crude oil pricesrdquo Energy Eco-nomics vol 87 p 104387 2020

[19] L Kilian and D P Murphy ldquo-e role of inventories andspeculative trading in the global market for crude oilrdquo Journalof Applied Econometrics vol 29 no 3 pp 454ndash478 2014

Complexity 17

[20] L Kilian and T K Lee ldquoQuantifying the speculative com-ponent in the real price of oil the role of global oil inven-toriesrdquo Journal of International Money and Finance vol 42pp 71ndash87 2014

[21] C Baumeister and J D Hamilton ldquoStructural interpretationof vector autoregressions with incomplete identificationrevisiting the role of oil supply and demand shocksrdquoAmerican Economic Review vol 109 no 5 pp 1873ndash19102019

[22] X Zhou ldquoRefining the workhorse oil market modelrdquo Journalof Applied Econometrics vol 35 no 1 pp 130ndash140 2020

[23] J Y Campbell and R J Shiller ldquo-e dividend-price ratio andexpectations of future dividends and discount factorsrdquo Reviewof Financial Studies vol 1 no 3 pp 195ndash228 1988

[24] R S Pindyck ldquo-e present value model of rational com-modity pricingrdquo e Economic Journal vol 103 no 418pp 511ndash530 1993

[25] Y Tao P C B Phillips and J Yu ldquoRandom coefficientcontinuous systems testing for extreme sample path behav-iorrdquo Journal of Econometrics vol 209 no 2 pp 208ndash2372019

[26] X Wang and J Yu ldquoDouble asymptotics for explosive con-tinuous time modelsrdquo Journal of Econometrics vol 193 no 1pp 35ndash53 2016

[27] J E Stiglitz ldquoSymposium on bubblesrdquo Journal of EconomicPerspectives vol 4 no 2 pp 3ndash18 1990

[28] J Tirole ldquoOn the possibility of speculation under rationalexpectationsrdquo Econometrica vol 50 no 5 pp 1163ndash11811982

[29] J Tirole ldquoAsset bubbles and overlapping generationsrdquoEconometrica vol 53 no 6 pp 1499ndash1528 1985

[30] R P Flood and R J Hodrick ldquoOn testing for speculativebubblesrdquo Journal of Economic Perspectives vol 4 no 2pp 85ndash101 1990

[31] K A Froot and M Obstfeld ldquoIntrinsic bubbles the case ofstock pricesrdquo American Economic Review vol 81 no 5pp 1189ndash1214 1991

[32] R J Shiller ldquoDo stock prices move too much to be justified bysubsequent changes in dividendsrdquo American Economic Re-view vol 71 no 3 pp 421ndash436 1981

[33] S F LeRoy and R D Porter ldquo-e present-value relation testsbased on implied variance boundsrdquo Econometrica vol 49no 3 pp 555ndash574 1981

[34] K D West ldquoA specification test for speculative bubblesrdquoeQuarterly Journal of Economics vol 102 no 3 pp 553ndash5801987

[35] B T Diba andH I Grossman ldquo-e theory of rational bubblesin stock pricesrdquo e Economic Journal vol 98 no 392pp 746ndash754 1988

[36] U Homm and J Breitung ldquoTesting for speculative bubbles instock markets a comparison of alternative methodsrdquo Journalof Financial Econometrics vol 10 no 1 pp 198ndash231 2012

[37] R S Gurkaynak ldquoEconometric tests of asset price bubblestaking stockrdquo Journal of Economic Surveys vol 22 no 1pp 166ndash186 2008

[38] G W Evans ldquoPitfalls in testing for explosive bubbles in assetpricesrdquo American Economic Review vol 81 no 4 pp 922ndash9301991

[39] I Caspi N Katzke and R Gupta ldquoDate stamping historicalperiods of oil price explosivity 1876ndash2014rdquo Energy Eco-nomics vol 70 pp 582ndash587 2018

[40] S Shi and V Arora ldquoAn application of models of speculativebehaviour to oil pricesrdquo Economics Letters vol 115 no 3pp 469ndash472 2012

[41] M Lammerding P Stephan M Trede and B WilflingldquoSpeculative bubbles in recent oil price dynamics evidencefrom a bayesian markov-switching state-space approachrdquoEnergy Economics vol 36 pp 491ndash502 2013

[42] E G Pavlidis I Paya and D A Peel ldquoTesting for speculativebubbles using spot and forward pricesrdquo International Eco-nomic Review vol 58 no 4 pp 1191ndash1226 2017

[43] E G Pavlidis I Paya and D A Peel ldquoUsing market ex-pectations to test for speculative bubbles in the crude oilmarketrdquo Journal of Money Credit and Banking vol 50 no 5pp 833ndash856 2018

[44] E G Pavlidis and K Vasilopoulos ldquoSpeculative bubbles insegmented markets evidence from Chinese cross-listedstocksrdquo Journal of International Money and Finance vol 109p 102222 2020

[45] R S Pindyck ldquo-e dynamics of commodity spot and futuresmarkets a primerrdquo Energy Journal vol 22 no 3 pp 1ndash29 2001

[46] R Alquist and L Kilian ldquoWhat do we learn from the price ofcrude oil futuresrdquo Journal of Applied Econometrics vol 25no 4 pp 539ndash573 2010

[47] R Alquist and O Gervais ldquo-e role of financial speculation indriving the price of crude oilrdquo Energy Journal vol 34pp 35ndash54 2013

[48] R Alquist H B Gregory and A Diez de los Rios ldquoWhat doesthe convenience yield curve tell us about the crude oilmarketrdquo Bank of Canada Ottawa Canada Working PaperNo 2014-42 2014

[49] H Follmer andM Schweizer ldquoA microeconomic approach todiffusion models for stock pricesrdquo Mathematical Financevol 3 no 1 pp 1ndash23 1993

[50] D F Nicholls and B G Quinn ldquo-e estimation of randomcoefficient autoregressive models Irdquo Journal of Time SeriesAnalysis vol 1 no 1 pp 37ndash46 1980

[51] P C B Phillips and J Yu ldquoA two-stage realized volatilityapproach to estimation of diffusion processes with discretedatardquo Journal of Econometrics vol 150 no 2 pp 139ndash1502009

[52] S Lee ldquoCoefficient constancy test in a random coefficientautoregressive modelrdquo Journal of Statistical Planning andInference vol 74 no 1 pp 93ndash101 1998

[53] H Follmer ldquoStock price fluctuation as a diffusion in a randomenvironmentrdquo Philosophical Transactions of the Royal SocietyA vol 347 no 1684 pp 471ndash483 1994

[54] C R Knittel and R S Pindyck ldquo-e simple economics ofcommodity price speculationrdquo American Economic JournalMacroeconomics vol 8 no 2 pp 85ndash110 2016

[55] M J Lombardi and I V Robays ldquoDo financial investorsdestabilize the oil pricerdquo European Central Bank FrankfurtGermany Working Paper Series 1346 2011

[56] L Kilian ldquoNot all oil price shocks are alike disentanglingdemand and supply shocks in the crude oil marketrdquoAmericanEconomic Review vol 99 no 3 pp 1053ndash1069 2009

[57] F Lippi and A Nobili ldquoOil and the macroeconomy aquantitative structural analysisrdquo Journal of the EuropeanEconomic Association vol 10 no 5 pp 1059ndash1083 2012

[58] L Kilian and D P Murphy ldquoWhy agnostic sign restrictionsare not enough understanding the dynamics of oil marketVAR modelsrdquo Journal of the European Economic Associationvol 10 no 5 pp 1166ndash1188 2012

[59] C Baumeister and G Peersman ldquoTime-varying effects of oilsupply shocks on the US economyrdquo American EconomicJournal Macroeconomics vol 5 no 4 pp 1ndash28 2013

18 Complexity

[60] S Giglio M Maggiori and J Stroebel ldquoNo-bubble conditionmodel-free tests in housing marketsrdquo Econometrica vol 84no 3 pp 1047ndash1091 2016

[61] B Fattouh L Kilian and L Mahadeva ldquo-e role of specu-lation in oil markets What have we learned so farrdquo EnergyJournal vol 34 no 3 pp 7ndash33 2013

Complexity 19

[60] S Giglio M Maggiori and J Stroebel ldquoNo-bubble conditionmodel-free tests in housing marketsrdquo Econometrica vol 84no 3 pp 1047ndash1091 2016

[61] B Fattouh L Kilian and L Mahadeva ldquo-e role of specu-lation in oil markets What have we learned so farrdquo EnergyJournal vol 34 no 3 pp 7ndash33 2013

Complexity 19