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1
“NONPARAMETRIC NONLINEAR CAUSALITY TESTING WITH
STEPWISE MULTIVARIATE FILTERING”
Stelios D. Bekiros 1
European University Institute, Via Delle Fontanelle 10, 50014
San Domenico di Fiesole (FI), Italy
Cees G.H Diks
Center for Nonlinear Dynamics in Economics and Finance (CeNDEF)
Department of Quantitative Economics, University of Amsterdam, Roetersstraat 11,
1018 WB Amsterdam, The Netherlands
Abstract
The present study investigates the linear and nonlinear causal linkages in exchange, equity and derivatives markets. Specifically, in case of exchange markets, among six currencies denoted relative to United States dollar (USD), namely EUR, GBP, JPY, CHF, AUD and CAD. The prime motivation for choosing these exchange rates comes from the fact that they are the most liquid and widely traded, covering about 90% of total FX trading worldwide. The data spans two periods (PI: 3/20/1991 – 3/20/1997, PII: 3/20/2003 – 3/20/2007) before and after the structural break of the Asian financial crisis, which set a platform for departure for causality testing. In case of derivatives markets, between daily spot and futures prices for maturities of 1, 2, 3 and 4 months of West Texas Intermediate (WTI) crude oil. The data cover two periods October 1991-October 1999 and November 1999-October 2007, with the latter being significantly more turbulent. Finally in case of stock markets, among NIKKEI 255, FTSE100 and NYSE. The data spans four periods (PI: 3/20/1991 – 3/20/2007, PII: 3/20/1991 – 3/20/1997, PIII: 3/20/2003 – 3/20/2007, PIV: 3/20/2000 – 3/22/2004), namely the total period, a pre- and post-bubble period and an in-bubble period. The structural break point is again the Asian financial crisis.
We apply a new nonparametric test for Granger non-causality as well as the conventional linear Granger test on the return time series after controlling for cointegration. In addition to the traditional pairwise analysis, we test for causality while correcting for the effects of the other variables in a multi-variate formulation. To check if any of the observed causality is strictly nonlinear in nature, we also examine the nonlinear causal relationships of VAR or VECM filtered residuals. Finally, we investigate the hypothesis of nonlinear non-causality after controlling for conditional heteroskedasticity in the data using a GARCH-BEKK model. Our approach allows the entire variance-covariance structure of the time series interrelationship to be incorporated in order to explicitly capture the volatility spillover mechanism. In case of exchange markets, whilst the nonparametric test statistics are smaller in some cases, significant nonlinear causal linkages persisted even after GARCH filtering during both the pre- and post-Asian crisis period. This indicates that currency returns may exhibit asymmetries and statistically significant higher-order moments. Similar
1 E-mail addresses: [email protected] , [email protected] (corresponding author); Tel.: +39-055-4685-698.
This study was conducted at the Center for Nonlinear Dynamics in Economics and Finance (CeNDEF), Department of Quantitative Economics, University of Amsterdam.
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results are presented for stock markets. Finally, in oil derivatives markets whereas the linear causal relationships disappear after VECM cointegration filtering, nonlinear causal linkages in some cases persist even after GARCH filtering in both periods. This indicates that spot and futures returns may exhibit asymmetries and statistically significant higher-order moments. Moreover, the results imply that if nonlinear effects are accounted for, neither market leads or lags the other consistently, videlicet the pattern of leads and lags changes over time.
JEL classification: C14; C51; F31; Q40
Keywords: Nonparametric nonlinear causality; VAR/VECM filtering; Cointegration; GARCH-BEKK filtering; foreign exchange; oil futures; stock markets.
3
A. Foreign Exchange Markets
1A. Introduction
In the nineties the gradual abolition of capital controls and trade barriers provided
the foundation for liberalized and deregulated emerging financial markets. This less
restrictive environment created a systematic interrelationship between and within the stock
and currency markets. Specifically, foreign exchange markets have grown fiscally and
monetarily (i.e. by achieving lower inflation and/or interest rate differentials) more similar,
which generally led to lower exchange rate volatility and caused asymmetry in reactions
toward macroeconomic developments to significantly decrease (Laopodis, 1998). This growing
similarity may also reflect a temporary, or long-term, causal relationship between the major
currency markets. A rich empirical literature exists on the propagation mechanism
(spillovers) of US currency volatility across other foreign exchange markets and on “stylized
facts” like leptokurtosis and volatility clustering. These studies focus on the investigation of
the stochastic behavior of the US dollar, mostly employing the autoregressive conditional
heteroskedastic (ARCH) methodology of Engle (1982) (Engle and Bollerslev, 1986; Boothe
and Glassman, 1987; Hsieh, 1989; Baillie and Bollerslev, 1989, 1990; Engle et al., 1990). The
nature of the volatility transmission mechanism as well as the degree of price information
efficiency was investigated in the beginning of the higher integration of foreign exchange
rates vis-à-vis the US dollar (Hogan and Sharpe, 1984; Ito and Roley, 1987). Some empirical
evidence by Koutmos and Booth (1995) and Laopodis (1997) suggests that the size and/or
sign of an innovation in US exchange rates may seriously affect the extent of dependence and
spillovers across markets. Given the status of the USD as the anchor currency, it should be
interesting to examine its volatility transfers and more general the nature of causal linkages
with other major currencies. If these exist, it would suggest that on a global scale, news
originating in a specific market is fully and efficiently transmitted to other foreign markets,
thereby providing support to the “meteor shower” notion coined by Engle et al. (1990). This
term applies to a situation where volatility originating in one market flows over other
4
markets, as opposed to the term “heat wave” also introduced by Engle et al. (1990) which
suggests that volatility would continue in the originating market the next day or that it is
country-specific.
The nature of causality in currency markets, i.e. linear or nonlinear is also a matter
for investigation. Ever since the influential work of Meese and Rogoff (1983) in which they
examined the failure of some linear exchange rate models, several more recent studies have
provided further evidence of the empirical failure of the linear models. The theoretical
extension of the linear exchange rate framework to nonlinear models has been growing in the
literature. According to Ma and Kanas (2000) these nonlinear extensions include the concept
of bubbles with self-fulfilling expectations (Blanchard and Watson, 1982; Froot and Obstfeld,
1991), target zone models (Krugman, 1991), models of micro-foundation of trading behaviour
(Krugman and Miller, 1993), nonlinear monetary policies (Flood and Isard, 1989), and noise
trading (Shiller, 1984; Kyle, 1985; Black, 1986; Frankel and Froot, 1986; Summers, 1986; De
Long et al., 1990). Empirical studies have mainly tested for nonlinearities due to target zones,
and have failed to support such nonlinearities (Flood et al., 1991; Lindberg and Soderlind,
1994). It still remains an open question whether these or other types of nonlinear
interdependencies across currency markets exist.
The recent empirical evidence is invariably based on the linear Granger causality
test (Granger, 1969). The conventional approach of testing for Granger causality is to assume
a parametric, linear time series model for the conditional mean. This approach is appealing,
since the test reduces to determining whether the lags of one variable enter into the equation
for another variable, although it requires the linearity assumption. Moreover, tests based on
residuals will be sensitive only to causality in the conditional mean while covariables may
influence the conditional distribution of the response in nonlinear ways. Additionally, Baek
and Brock (1992) noted that parametric linear Granger causality tests have low power
against certain nonlinear alternatives. In view of this, nonparametric techniques are
appealing because they place direct emphasis on prediction without imposing a linear
5
functional form. Various nonparametric causality tests have been proposed in the literature.
The test by Hiemstra and Jones (1994) which is a modified version of the Baek and Brock
(1992) test is regarded as a test for a nonlinear dynamic relationship. This test can detect the
nonlinear Granger-causal relationship between variables by testing whether the past values
influence present and future values. However, Diks et. al. (2006) demonstrate that the
relationship tested by Hiemstra and Jones test is not generally compatible with Granger
causality, leading to the possibility of spurious rejections of the null hypothesis. As an
alternative we developed a new test statistic that overcomes these limitations.
The aim of the current paper is to test for the existence of both linear and nonlinear
causal relationships among six currencies denoted relative to United States dollar (USD),
namely Euro (EUR), Great Britain Pound (GBP), Japanese Yen (JPY), Swiss Frank (CHF),
Australian Dollar (AUD) and Canadian Dollar (CAD). The prime motivation for choosing
these exchange rates (also known as “FX majors”) comes from the fact that they are the most
liquid and widely traded exchange rates in the world. Trades involving “majors” make up
about 90% of total Forex trading worldwide. The data cover a pre- and a post-Asian crisis
period. The Asian crisis started with a 15 - 20% devaluation of Thailand’s Bath which took
place on July 2, 1997. Subsequently it was followed by devaluations of the Philippine Peso,
the Malaysian Ringgit, the Indonesian Rupiah and the Singaporean Dollar. In addition, the
currencies of South Korea and Taiwan suffered. Further in October, 1997 the Hong Kong
stock market collapsed with a 40% loss. In January 1998, the currencies of most South-East
Asian countries regained parts of the earlier losses. In that context, it is worth investigating
whether the time period after the 1997 Asian financial crisis may have changed the direction
and strength of the causal relationships among the currencies under study.
In the present study we apply a three-step empirical framework for examining
dynamic relationships among foreign exchange markets. First, we explore linear and
nonlinear dynamic linkages between exchange rates, applying both a parametric Granger
causality test and the new nonparametric causality test. Then, after filtering return series
6
pairwise, as well as in a six-variate formulation for linear Vector AutoRegressive (VAR)
structure, the series of residuals are examined pairwise by the nonparametric causality test.
This step ensures that any remaining causality is strictly nonlinear in nature, as the VAR
model has already purged the residuals of linear causality. Finally, in the last step, we
investigate the hypothesis of nonlinear non-causality after controlling for conditional
heteroskedasticity in the data using a GARCH-BEKK model again, both pairwise and in a
six-variate representation. Our approach allows the entire variance-covariance structure of
the currency interrelationship to be incorporated. The empirical methodology employed with
the multivariate GARCH-BEKK model can help not only to understand the short-run
movements but also explicitly capture the volatility spillover mechanism. The method’s
advantage rests with its ability to examine all markets concurrently and paired, assuming
that spillovers are realizations of a process of international news affecting the examined
markets. Improved knowledge of the direction and nature of causality and interdependency
between the currency markets and consequently the degree of their integration will expand
the information set available to international portfolio managers, multinational corporations,
and policymakers for decision-making.
The remainder of the 1st part of the paper is organized as follows. Section 2A briefly
reviews the linear Granger causality framework and provides a description of the new
nonparametric test for nonlinear Granger causality. Section 3A describes the data used and
Section 4A presents the results. Section 5A concludes with suggestions for future research.
2A. The Nonparametric Causality Test
Granger (1969) non-causality has turned out to be a useful notion for characterizing
dependence relations between time series in economics and econometrics. Assume
that { }≥, ; 1t tX Y t are two scalar-valued strictly stationary time series. Intuitively { }tX is a
strictly Granger cause of { }tY if past and current values of X contain additional information
7
on future values of Y that is not contained only in the past and current tY values. Let ,X tF and
,Y tF denote the information sets consisting of past observations of tX and tY up to and including
time t, and let ‘~’ denote equivalence in distribution. Then { }tX is a Granger cause of { }tY if,
for ≥ 1k :
( ) ( )+ +1 , ,,..., ,t t k X t Y tY Y F F ( )+ +1 ,,...,t t k X tY Y F (1)
In practice = 1k is used most often, in which case testing for Granger non-causality amounts
to comparing the one-step-ahead conditional distribution of { }tY with and without past and
current observed values of { }tX . A conventional approach of testing for Granger causality
among stationary time series is to assume a parametric, linear, time series model for the
conditional mean ( )( )+1 , ,,t X t Y tE Y F F . Then, causality can be tested by comparing the residuals
of a fitted autoregressive model of tY with those obtained by regressing tY on past values of
both { }tX and { }tY (Granger, 1969). Now, assume delay vectors
( )− +=ℓ
ℓ 1 ,...,X
t Xt tX XX and ( )− +=ℓ
ℓ 1 ,...,Y
t Yt tY YY , ( )≥ℓ ℓ, 1X Y . In practice the null hypothesis that past
observations of ℓ X
tX contain no additional information (beyond that in
ℓY
tY ) about +1tY is tested,
i.e.:
( )+ +ℓ ℓ ℓ
0 1 1: ; ~X Y Y
t t t t tH Y YX Y Y (2)
For a strictly stationary bivariate time series Eq. (2) comes down to a statement about the
invariant distribution of the ( + +ℓ ℓ 1X Y )-dimensional vector ( )= ℓ ℓ, ,X X
t tt tZW X Y where += 1t tZ Y .
To keep the notation compact, and to bring about the fact that the null hypothesis is a
statement about the invariant distribution of ( )ℓ ℓ, ,X X
t t tZX Y we drop the time index and also
= =ℓ ℓ 1X Y is assumed. Hence, under the null, the conditional distribution of Z given (X, Y) =
(x, y) is the same as that of Z given Y = y. Further, Eq. (2) can be restated in terms of ratios
8
of joint distributions. Specifically, the joint probability density function , , ( , , )X Y Zf x y z and its
marginals must satisfy the following relationship:
= ⋅, , , ,( , , ) ( , ) ( , )
( ) ( ) ( )X Y Z X Y Y Z
Y Y Y
f x y z f x y f y z
f y f y f y (3)
This explicitly states that X and Z are independent conditionally on Y = y for each fixed value
of y. Diks and Panchenko (2006) show that this reformulated H0 implies:
≡ − = , , , ,( , , ) ( ) ( , ) ( , ) 0X Y Z Y X Y Y Zq E f X Y Z f Y f X Y f Y Z (4)
Let ˆ ( )W if W denote a local density estimator of a dW - variate random vector W at Wi defined by
ε − −
≠= − ∑1 ,
ˆ ( ) (2 ) ( 1)Wd WW i n ijj j if W n I where ( )ε= − <W
ij i j nI I W W with ⋅( )I the indicator function and
ε n the bandwidth, depending on the sample size n. Given this estimator, the test statistic is
a scaled sample version of q in Eq. (4):
ε Ζ
−= ⋅ −
−∑ , , , ,
1 ˆ ˆ ˆ ˆ( ) ( ( , , ) ( ) ( , ) ( , ))( 2)
n n X Z Y i i i Y i X Y i i Y i i
i
nT f X Z Y f Y f X Y f Y Z
n n (5)
For = =ℓ ℓ 1X Y , if βε β−
= > < <
1 1 0,
4 3n Cn C then it is proven under strong mixing that the
test statistic in Eq. (5) satisfies:
( )ε −→
( )(0,1)
Dn n
n
T qn N
S (6)
where →D
denotes convergence in distribution and Sn is an estimator of the asymptotic
variance of ⋅( )nT . We implement a one-tailed version of the test, rejecting the null hypothesis
if the left-hand-side of Eq. (6) is too large.
9
3A. Data and preliminary analysis
The data consist of six time series of daily closing foreign exchange rates denoted
relative to United States dollar (USD), namely Euro (EUR), Great Britain Pound (GBP),
Japanese Yen (JPY), Swiss Frank (CHF), Australian Dollar (AUD) and Canadian Dollar
(CAD). The exact ratios represent EUR/USD, GBP/USD, USD/JPY, USD/CHF, AUD/USD
and USD/CAD respectively. These are the most liquid and widely traded currency pairs in
the world and make up about 90% of total Forex trading worldwide. The data cover two
periods, before and after a structural break, which sets a platform for departure for causality
tests. The first period PI spans from March 20, 1991 to March 20, 1997, denoting a pre-Asian
crisis period (1567 observations)2, and the second PII, a post-Asian crisis period, from March
20, 2003 to March 20, 2007 (1044 observations). Recall that the on-set of the Asian financial
crisis started with the devaluation of Thailand’s Bath which took place on July 2, 1997 and
followed by devaluations of the Philippine Peso, the Malaysian Ringgit, the Indonesian
Rupiah, the Singaporean Dollar and in October, 1997 the Hong Kong stock market collapsed
with a 40% loss. In January 1998, the currencies of South-East Asian countries began to
regain part of the earlier losses.
4A. Empirical results
The empirical methodology comprises three steps. In the first pre-filtering step, we
explore the linear and nonlinear dynamic linkages applying both a Granger causality test
and the nonparametric test on the raw log-differenced time series of the currencies. Then, we
implement both bi-variate and six-variate VAR filtering on the return series and the
residuals are examined pairwise by the test. Finally, we investigate the hypothesis of
nonlinear non-causality after controlling for conditional heteroskedasticity using a GARCH-
BEKK filter again pairwise as well as in a six-variate representation.
2 Euro values before 1999 refer to ECU (European Currency Unit). The ECU, a basket of EU currencies, ceased to
exist on 1 January 1999, when it was replaced by the Euro at a rate of 1:1. From that date, the currencies of the Euro-zone became sub-divisions of the Euro at irrevocably fixed rates of conversion.
10
Descriptive statistics and Augmented Dickey-Fuller (ADF) test for the logarithmic
levels and log-daily returns are reported in Tables 1A and 2A. The lag lengths which are
consistently zero in all cases were selected using the Schwartz Information Criterion (SIC).
All the variables appear to be nonstationary in log-levels and stationary in log-returns based
on the reported p-values. The results are reported in the corresponding columns of Tables 3A
and 4. In order to overcome the difficulty of presenting large tables with numbers we use the
following simplifying notation: “ ** ” indicating that the corresponding p-value of a particular
causality test is smaller than 1% and “ * ” that the corresponding p-value of a test is in the
range 1-5%; Directional causalities will be denoted by the functional representation →.
[ Insert Table 1A here ]
[ Insert Table 2A here ]
4A.1 Causality testing on raw returns
The linear Granger causality test is usually constructed in the context of a reduced-
form vector autoregression (VAR). Let tY the vector of endogenous variables and ℓ number of
lags. Then the VAR( ℓ ) model is given as follows:
ε−
=
= +∑ℓ
1t s t s t
s
Y A Y (7)
where [ ]=ℓ1 ,...,t t tY YY the ×ℓ 1 vector of endogenous variables, sA the ×ℓ ℓ parameter matrices
and ε t the residual vector, for which εε ε ε =
= = ≠
∑' ( ) 0 , ( )
0 t t s
t sE E
t s. Specifically, in case of
two stationary time series { }tX and { }tY the bivariate VAR model is given by:
ε
ε
= + +=
= + +
ℓ ℓ
ℓ ℓ
,
,
( ) ( ) 1,2,...,
( ) ( )t t t X t
t t t Y t
X A X B Yt N
Y C X D Y (8)
where ℓ ℓ ℓ( ), ( ), ( )A B C and ℓ( )D are all polynomials in the lag operator with all roots outside the
unit circle. The error terms are separate i.i.d. processes with zero mean and constant
11
variance. The test whether Y strictly Granger causes X is simply a test of the joint restriction
that all the coefficients of the lag polynomial ℓ( )B are zero, whilst similarly, a test of whether
X strictly Granger causes Y is a test regarding ℓ( )C . In each case, the null hypothesis of no
Granger causality is rejected if the exclusion restriction is rejected. If both ℓ( )B and ℓ( )C joint
tests for significance show that they are different from zero, the series are bi-causally related.
For each of the six raw return series linear causality testing was carried out using the
Granger’s test. The lag lengths of the VAR specification were selected using the Schwartz
Information Criterion (SIC), which are presented in Tables 3A, 4A in parentheses. For the
new test, in what follows we discuss results for lags = =ℓ ℓ 1X Y . To implement the test, the
constant C for the bandwidth ε n was set at 7.5, which is close to the value 8.0 for ARCH
processes suggested by Diks and Panchenko (2006). With the theoretical optimal rate
β = 2 7 this implies a bandwidth value of approximately 1, for both PI and PII. Selected
bandwidth values smaller (larger) than 1 resulted, in general, in larger (smaller) p-values.
The results presented in Tables 3A and 4A allow for the following observations: GBP,
JPY, CHF and AUD linearly cause EUR with small differences in PI and PII as well as
regarding the degree of statistical significance. None of the currencies Granger causes GBP.
Further, there is a strong causal relationship which affects JPY, CHF, AUD and CAD, with
JPY Granger causing the others. Finally, CHF presents a significant unidirectional linear
relationship CHF→CAD in PII. AUD and CAD appear to lack any causal relationship. We
next discuss the results for the nonparametric test. Interestingly starting form the latter
example, in period PI, there is now strong evidence of bi-directional nonlinear relationship
AUD↔CAD. Any of the previously detected linear causality among JPY, CHF, AUD and
CAD has vanished with the exception of a unidirectional nonlinear relationship JPY←CAD
in both periods. Again none of the currencies causes GBP. Only in PI, GBP causes CHF and
the same currency causes AUD in PII. Finally, strong nonlinear causality appears from
others currencies toward EUR but it no longer exists for AUD and CAD.
12
[ Insert Table 3A here ]
[ Insert Table 4A here ]
4A.2 Causality testing on VAR-filtered residuals
The results from the previous step suggest that there are some significant and
persistent linear and nonlinear causal linkages between the FX rates. However, even though
we found nonlinear causality, the new test should be reapplied to filtered VAR-residuals to
ensure that any causality found is strictly nonlinear in nature. The lag lengths of the VAR
specification were based on the Schwartz Information Criterion (SIC).
The application of the nonparametric test on the VAR residuals points roughly
towards the preservation of the results reported for the raw returns in the pairwise and six-
variate implementation. Comparing the summary results in Table 3A, it is interesting to see
that they show identical significant causal nonlinear relationships, except for the absence
now of causality GBP→EUR in PII and the emergence of the unidirectional causality
CHF→JPY in PI. In the six-variate representation (Table 4A) the causal relationship
JPY→EUR has vanished and the same applies to the unidirectional linkage CAD→JPY. The
nature and source of the detected nonlinearities are different from that of the linear Granger
causality test and may also imply a temporary, or long-term, causal relationship between the
currencies. For instance, exchange rate volatility might induce nonlinear causality. Given
the status of the USD as the anchor currency, an innovation in US stock markets or an
increase in the interest rates from the Federal Reserve System in the USA etc., may
seriously affect the extent of dependency and volatility spillovers across currencies. The
nature of the volatility transmission mechanism can be investigated after controlling for
conditional heteroskedasticity using a GARCH-BEKK model, pairwise and in a six-variate
representation. This approach allows the entire variance-covariance structure of the
currency interrelationship to be incorporated.
13
4A.3 Causality testing on GARCH-BEKK filtered VAR-residuals
The use of the new test on filtered data with a multivariate GARCH model enables
one to determine whether the posited model is sufficient to describe the relationship among
the series. If the statistical evidence of nonlinear Granger causality lies in the conditional
variances and covariances then it would be strongly reduced when the appropriate
multivariate GARCH model is fitted to the raw or linearly filtered data. However, failure to
accept the no-causality null hypothesis may also constitute evidence that the selected
multivariate GARCH model was incorrectly specified. This line of analysis is similar to the
use of the univariate BDS test on raw data and on GARCH models (Brock et al., 1996;
Brooks, 1996; Hsieh, 1989). Many GARCH models can be used for this purpose. In the
present study the GARCH-BEKK model of Engle and Kroner (1995) is used. The BEKK (p,q)
model is defined as:
ε ε− − −= =
′ ′ ′ ′= + +∑ ∑1 1
q p
tj j
jk t j t j jk jk t j jkH C C A A G H G , ε = 1/2t t tH v (9)
where jkC,A and jkG are (NxN) matrices and C is upper triangular. tH is the conditional
covariance matrix of { }ε t with ε −Φ 1| ~ (0 )t t t, H and −Φ 1t the information set at time t − 1. The
residuals are obtained by the whitening matrix transformation ε1/2 tH . Gourieroux (1997)
gives sufficient conditions for tA and tG in order to guarantee that tH is always positive
definite.
Tables 3A and 4A show results before and after GARCH-BEKK (1,1) filtering. The
order parameters were determined for the time series in terms of the minimal SIC. One
conclusion after the nonlinear causality testing is that in some cases the statistical
significance is weaker after filtering than before. These differences in statistical significance
indicate that the nonlinear causality is largely due to simple volatility effects. However, this
is not indicative of a general conclusion. Instead, significant nonlinear interdependencies
remain after the pairwise and six-variate GARCH-BEKK filtering revealing that volatility
14
effects and spillovers are probably not the only ones inducing nonlinear causality. This of
course does not apply to all the pairs of FX rates but some main results can be drawn for
specific relationships. These are also depicted graphically in Figure 1A where strong
causality (“ ** ”) is denoted by a “double arrow”. In particular, the pairwise nonlinear
causality reveals the unidirectional linkages JPY→EUR, CHF→EUR, GBP→EUR and
AUD→CAD in PI, while in PII, JPY→EUR and CHF→EUR remain and AUD→ EUR is
added. Thus, there is strong evidence of the influence of the aforementioned currencies on
EUR though some in pre- and others in post-crisis period. This is perhaps an after-effect of
the independent and robust Euro zone behavior against the USD (Bénassy-Quéré et. al.,
2000; Wang et al., 2005). A potential increase/decrease in the US dollar volatility affects the
Euro zone currencies less than (and with a significant delay) the USD closest dependent
economies of Canada and Australia.
Now, incorporating the effect of all the currencies in a six-variate GARCH-BEKK
framework, the “whitened” residuals present different causal relationships than before.
Specifically, in PI two bidirectional linkages exist, namely GBP↔EUR and EUR↔CAD and
two unidirectional relationships CHF→EUR, CAD→JPY. In PII, besides the causal linkage
of GBP→EUR, the influence of CAD is strongly evident in forming the unidirectional
relationships, i.e., CAD→AUD, CAD→CHF, CAD→EUR and CAD→GBP. A possible
explanation is that Canadian market acting as a proxy for the neighboring US stock, bond or
currency market reacts faster to any innovation or volatility jumps (Yang, 2005; Bessler et.
al., 2003; Eun, 1989). Finally, in all results, third moment causality may be also a significant
factor of the remaining interdependence.
5A. Conclusions for currency markets
In the first part of the study we investigated the existence of linear and nonlinear
causal relationships among the most liquid and widely traded currencies in the world
denoted relative to United States dollar (USD), namely Euro (EUR), Great Britain Pound
15
(GBP), Japanese Yen (JPY), Swiss Frank (CHF), Australian Dollar (AUD) and Canadian
Dollar (CAD). Several interesting conclusions have already emerged from this study. In
particular, it was shown that almost all FX markets considered here have become more
internationally integrated after the Asian financial crisis. Additionally, whilst the linear
causal relationships detected on the returns have disappeared after proper filtering,
nonlinear causal linkages in some cases emerged and more importantly persisted even after
GARCH filtering during both the pre- and post-Asian financial crisis period. For instance,
there is strong evidence of the influence of the other currencies on EUR and that Canadian
currency market substituting for US financial market nonlinearly causes the other currency
markets beyond the conventional spillover effect. These results, apart from offering a much
better understanding of the dynamic linear and nonlinear relationships underlying the major
currency markets, may have important implications for market efficiency. For instance, they
may be useful in future research to quantify the process of financial integration or may
influence the greater predictability of these markets.
An interesting subject for future research is the nature and source of the nonlinear
causal linkages. As was shown, volatility effects might partly induce nonlinear causality. The
fitted GARCH-BEKK models account for a large part of the nonlinearity in daily exchange
rates, but only in some cases. Perhaps other models of short-term exchange-rate
determination should be developed to explain this stylized fact. Moreover, currency returns
may exhibit statistically significant higher-order moments. This may explain why GARCH
filtering does not capture all the nonlinearity in currency returns. A similar result was
reported in Scheinkman, J., and LeBaron, (1989) for stock returns. Finally, alternative
parameterized asymmetric multivariate GARCH models could be employed. These models
would accommodate the asymmetric impact of unconditional shocks on the conditional
variances.
16
B. Oil derivatives Markets
1B. Introduction
The role of futures markets in providing an efficient price discovery mechanism has
been an area of extensive empirical research. Several studies have dealt with the lead–lag
relationships between spot and futures prices of commodities with the objective of
investigating the issue of market efficiency. Garbade and Silber (1983) first presented a
model to examine the price discovery role of futures prices and the effect of arbitrage on price
changes in spot and futures markets of commodities. The Garbade-Silber model was applied
to the feeder cattle market by Oellermann et al. (1989) and to the live hog commodity market
by Schroeder and Goodwin (1991), while a similar study by Silvapulle and Moosa (1999)
examined the oil market. Bopp and Sitzer (1987) tested the hypothesis that futures prices
are good predictors of spot prices in the heating oil market, while Serletis and Banack (1990),
Cologni and Manera (2008) and Chen and Lin (2004) tested for market efficiency using
cointegration analysis. Crowder and Hamed (1993) and Sadorsky (2000) also used
cointegration to test the simple efficiency hypothesis and the arbitrage condition for crude oil
futures. Finally, Schwarz and Szakmary (1994) examined the price discovery process in the
markets of crude and heating oil.
The recent empirical evidence on causality is invariably based on the Granger test
(Granger, 1969). The conventional approach of testing for Granger causality is to assume a
parametric linear, time series model for the conditional mean. Although it requires the
linearity assumption this approach is appealing, since the test reduces to determining
whether the lags of one variable enter into the equation for another variable. Moreover, tests
based on residuals will be sensitive only to causality in the conditional mean while
covariables may influence the conditional distribution of the response in nonlinear ways.
Baek and Brock (1992) noted that parametric linear Granger causality tests have low power
against certain nonlinear alternatives.
17
Recent work has revealed that nonlinear structure indeed exists in spot and futures
returns. These nonlinearities are normally attributed to nonlinear transaction cost functions,
the role of noise traders, and to market microstructure effects (Abhyankar, 1996; Chen and
Lin, 2004; Silvapulle and Moosa, 1999). In view of this, nonparametric techniques are
appealing because they place direct emphasis on prediction without imposing a linear
functional form. Various nonparametric causality tests have been proposed in the literature.
The test by Hiemstra and Jones (1994), which is a modified version of the Baek and Brock
(1992) test, is regarded as a test for a nonlinear dynamic causal relationship between a pair
of variables. The Hiemstra and Jones test relaxes Baek and Brock’s assumption that the
time series to which the test is applied are mutually and individually independent and
identically distributed. Instead, it allows each series to display weak (or short-term)
temporal dependence. When applied to the residuals of vector autoregressions, the Hiemstra
and Jones test is intended to determine whether nonlinear dynamic relations exist between
variables by testing whether the past values influence present and future values. However,
Diks (2005, 2006) demonstrate that the relationship tested by Hiemstra and Jones test is not
generally compatible with Granger causality, leading to the possibility of spurious rejections
of the null hypothesis. As an alternative we developed a new test statistic that overcomes
these limitations.
Empirically it is important to take into account the possible effects of cointegration
on both linear and nonlinear Granger causality tests. Controlling for cointegration is
necessary because it affects the specification of the model used for causality testing. If the
series are cointegrated, then causality testing should be based on a Vector Error Correction
model (VECM) rather than an unrestricted VAR model (Engle and Granger, 1987). When
cointegration is not modelled, evidence may vary significantly towards detecting linear and
nonlinear causality between the predictor variables. Specifically, the absence of cointegration
could mean the violation of the necessary condition for the simple efficiency hypothesis
(Dwyer and Wallace, 1992), which implies that the futures price is not an unbiased predictor
18
of the spot price at maturity. This implies an absence of a long-run relationship between spot
and futures prices, as it was reported in works of Chowdhury (1991), Krehbiel and Adkins
(1993), Crodwer and Hamed (1993). Alternatively, based on the cost-of-carry relationship, a
failure to find cointegration may be attributed to the nonstationarity of the other components
of this relationship such as the interest rate or the convenience yield (Moosa and Al-
Loughani, 1995, and Moosa, 1996).
The aim of the present study is to test for the existence of linear and nonlinear causal
lead–lag relationships between spot and futures prices of West Texas Intermediate (WTI)
crude oil, which is used as an indicator of world oil prices and is the underlying commodity of
New York Mercantile Exchange's (NYMEX) oil futures contracts. We apply a three-step
empirical framework for examining dynamic relationships between spot and futures prices.
First, we explore nonlinear and linear dynamic linkages applying the new nonparametric
causality test, and after controlling for cointegration, a parametric linear Granger causality
test. In the second step, after filtering the return series using the properly specified VAR or
VECM model, the series of residuals are examined by the nonparametric causality test. In
addition to applying the usual bivariate VAR or VECM model to each pair of time series, we
also consider residuals of a full five-variate model to account for the possible effect of the
other variables. This step ensures that any remaining causality is strictly nonlinear in
nature, as the VAR or VECM model has already purged the residuals of linear dependence.
Finally, in the last step, we investigate the null hypothesis of nonlinear non-causality after
controlling for conditional heteroskedasticity in the data using a GARCH-BEKK model,
again both in a bivariate and in a five-variate representation. Our approach incorporates the
entire variance-covariance structure of the spot and future prices interrelationship. The
empirical methodology employed with the multivariate GARCH-BEKK model can not only
help to understand the short-run movements, but also explicitly capture the volatility
persistence mechanism. Improved knowledge of the direction and nature of causality and
interdependence between the spot and futures markets, and consequently the degree of their
19
integration, will expand the information set available to policymakers, international portfolio
managers and multinational corporations for decision-making.
The remainder of the 2nd part of the paper is organized as follows. Section 2B
provides an introduction to the theoretical considerations and existing empirical evidence on
the relationship between spot and futures prices. Section 3B briefly reviews the linear
Granger causality framework and provides a description of the new nonparametric test for
nonlinear Granger causality. Section 4B describes the data used and Section 5B presents the
results. Section 6B concludes with a summary and suggestions for future research.
2B. Theory and Evidence
In theory, since both futures and spot prices “reflect” the same aggregate value of the
underlying asset and considering that instantaneous arbitrage is possible, futures should
neither lead nor lag the spot price. However, the empirical evidence is diverse, although the
majority of studies indicate that futures influence spot prices but not vice versa. The usual
rationalization of this result is that the futures prices respond to new information more
quickly than spot prices, due to lower transaction costs and flexibility of short selling. With
reference to the oil market, if new information indicates that oil prices are likely to rise,
perhaps because of an OPEC decision to restrict production, or an imminent harsh winter, a
speculator has the choice of either buying crude oil futures or spot. Whilst spot purchases
require more initial outlay and may take longer to implement, futures transactions can be
implemented immediately by speculators without an interest in the physical commodity per
se and with little up-front cash. Moreover, hedgers who are interested for the physical
commodity and have storage constraints will buy futures contracts. Therefore, both hedgers
and speculators will react to the new information by preferring futures rather than spot
transactions. Spot prices will react with a lag because spot transactions cannot be executed
so quickly (Silvapulle and Moosa, 1999).
20
Furthermore, the price discovery mechanism, as illustrated by Garbade and Silber
(1983), supports the hypothesis that futures prices lead spot prices. Their study of seven
commodity markets indicated that, although futures markets lead spot markets, the latter do
not just echo the former. Futures trading can also facilitate the allocation of production and
consumption over time, particularly by providing a market scheme in inventory holdings
(Houthakker, 1992). In this case, if futures prices for late deliveries are above those for early
ones, delay of consumption becomes attractive and changes in futures prices result in
subsequent changes in spot prices. According to Newberry (1992) futures markets provide
opportunities for market manipulation by the better informed or larger at the expense of
other market participants. For example, it is profitable for the OPEC to intervene in the
futures market to influence the production decisions of its competitors in the spot market.
Finally, support for the hypothesis that causality runs from futures to spot prices can
also be found in the model of determination of futures prices proposed by Moosa and Al-
Loughani (1995). In their model the futures price is determined by arbitrageurs whose
demand depends on the difference between the arbitrage and actual futures price and by
speculators whose demand for futures contracts depends on the difference between the
expected spot and the actual futures price. The reference point in both cases is the futures
price and not the spot price (Silvapulle and Moosa, 1999).
There is also empirical evidence that spot prices lead futures prices. Specifically, in
the study of Moosa (1996) a spot price change triggers action from all kinds of market
participants and this subsequently changes the futures price. Initially, arbitrageurs will
react to the violation of the cost-of-carry condition3 and then speculators will revise their
expectation of the spot price and respond to the disparity between expected spot and futures
price. Similarly, speculators who act upon the expected futures price will revise their
expectation responding to the disparity between current and expected futures prices. Finally,
3 The relationship between futures and spot prices can be summarized as
−= ( )c y TF Se in terms of what is known as
the cost-of-carry. In that, y is the convenience yield (market’s expectations of the future availability of the commodity), T is the period to maturity, and c the cost-of-carry which equals the storage cost plus the cost of financing a commodity minus the income earned on the commodity (Hull, 2000).
21
in few studies causality is reported to be bi-directional. Kawaller et al. (1988) introduced the
principle that both spot and futures prices are affected by their past history, as well as by
current market information. They argue that potential lead - lag patterns dynamically
change as new information arrives. At any time point each may lead the other, as market
participants filter information relevant to their positions, which may be spot or futures. So
far, the hypothesis that futures prices lead spot prices is stronger in terms of empirical
evidence and more compelling. Thus, further empirical testing is required to infer on this
issue with respect to the crude oil market.
3B. Data and preliminary analysis
The data consist of time series of daily spot and futures prices for maturities of one,
two, three and four months of West Texas Intermediate (WTI), also known as Texas Light
Sweet, which is a type of crude oil used as a benchmark in oil pricing and the underlying
commodity of New York Mercantile Exchange's (NYMEX) oil futures contracts. The NYMEX
futures price for crude oil represents, on a per-barrel basis, the market-determined value of a
futures contract to either buy or sell 1,000 barrels of WTI at a specified time. The NYMEX
market provides important price information to buyers and sellers of crude oil around the
world, although relatively few NYMEX crude oil contracts are actually executed for physical
delivery.
The data cover two equally sampled periods, namely PI which spans October 21,
1991 to October 29, 1999 (2061 observations) and PII November 1, 1999 to October 30, 2007
(2061 observations). The segmentation of the sample corresponds roughly to the reduction in
OPEC spare capacity (defined as the difference between sustainable capacity and current
OPEC crude oil production) and to the increase in the United States’ gasoline consumption
and imports, both of which occurred after 1999. The effect of these events on price dynamics
is evident and it can be summarized in the accelerated rise of the average level of oil prices
and in the increased volatility (Regnier, 2006). Additionally, in PII markets witnessed more
22
occasional spikes in crude prices. The following notation is used: “WTI Spot” is the spot price
and “WTI F1”, “WTI F2”, “WTI F3” and “WTI F4” are the futures prices for maturities of one,
two, three and four months respectively. Descriptive statistics for WTI spot and futures log-
daily returns are reported in Table 1B. Specifically, the returns are defined as
−= − 1ln( ) ln( )t t tr P P , where Pt is the closing price on day t. The differences between the two
periods are quite evident in Table 1B where a significant increase in variance can be
observed as well as a higher dispersion of the returns distribution in Period II reflected in
the lower kurtosis. Additionally, Period II witnessed many occasional negative spikes as it
can be also inferred from the skewness. The results from testing nonstationarity are
presented in Table 2B.
[ Insert Table 1B here ]
[ Insert Table 2B here ]
Specifically, Table 2B reports the Augmented Dickey-Fuller (ADF) test for the logarithmic
levels and log-daily returns. The lag lengths which are consistently zero in all cases were
selected using the Schwartz Information Criterion (SIC). All the variables appear to be
nonstationary in log-levels and stationary in log-returns based on the reported p-values.
Table 1B also reports the correlation matrix at lag 0 (contemporaneous correlation) for both
periods. Significant sample cross-correlations are noted for spot and futures returns
indicating a high interrelationship between the two markets. However, since linear
correlations cannot be expected to fully capture the long-term dynamic linkages in a reliable
way, these results should be interpreted with caution. Consequently, what is needed is a
long-term causality analysis.
4B. Empirical results
The empirical methodology comprises three steps. In the first pre-filtering step, we
explore the linear and nonlinear dynamic linkages applying a Granger causality test based
on a VECM specification on the log-price levels and the new nonparametric test on the log-
23
differenced time series of the spot and futures prices. Then, we implement both pairwise and
five-variate VECM filtering on the log-price series, and the residuals are examined by the
new test. Finally, we investigate the hypothesis of nonlinear Granger non-causality after
controlling for conditional heteroskedasticity using a GARCH-BEKK filter, again in a
bivariate and a five-variate representation. Additionally, in the last two steps we
consistently apply a linear Granger causality test on the “whitened” residuals via a VAR
specification (i.e., no cointegration detected on the residuals) in order to investigate whether
any remaining causality is strictly nonlinear in nature or not.
The results are reported in the corresponding panels of Tables 3B and 4B. In order to
overcome the difficulty of presenting large tables with numbers we use the following
simplifying notation: “ ** ” indicating that the corresponding p-value of a particular causality
test is smaller than 1% and “ * ” that the corresponding p-value of a test is in the range 1-5%;
Directional causalities will be denoted by the functional representation →.
4B.1 Causality testing on raw data
The linear Granger causality test is usually constructed in the context of a reduced-
form vector autoregression (VAR). Let tY the vector of endogenous variables and ℓ number of
lags. Then the VAR( ℓ ) model is given as follows:
ε−
=
= +∑ℓ
1t s t s t
s
Y A Y (10)
where [ ]=ℓ1 ,...,t t tY YY the ×ℓ 1 vector of endogenous variables, sA the ×ℓ ℓ parameter matrices
and ε t the residual vector, for which εε ε ε =
= = ≠
∑' ( ) 0 , ( )
0 t t s
t sE E
t s. Specifically, in case of
two time series { }tX and{ }~ (1)tY I (for example log-prices) the bivariate VAR model is given
by:
24
ε
ε∆
∆
∆ = ∆ + ∆ +=
∆ = ∆ + ∆ +
ℓ ℓ
ℓ ℓ
,
,
( ) ( ) 1,2,...,
( ) ( )t t X t
t t Y t
X A X B Yt N
Y C X D Y (11)
where ∆ tX , ∆ tY the first differences of{ }tX ,{ }tY and ( ), ( ), ( ), ( )A B C Dℓ ℓ ℓ ℓ are all polynomials
in the lag operator with all roots outside the unit circle. The error terms are separate i.i.d.
processes with zero mean and constant variance. The test whether Y strictly Granger causes
X is simply a test of the joint restriction that all the coefficients of the lag polynomial ℓ( )B
are zero, whilst similarly, a test of whether X strictly Granger causes Y is a test
regarding ℓ( )C . In each case, the null hypothesis of no Granger causality is rejected if the
exclusion restriction is rejected. If both joint tests for significance show that ℓ( )B and ℓ( )C are
different from zero, the series are bi-causally related. However, in order to explore effects of
possible cointegration, a VAR in error correction form (Vector Error Correction Model-VECM)
is estimated using the methodology developed by Engle and Granger (1987) and expanded by
Johansen (1988) and Johansen and Juselius (1990). The bi-variate VECM model has the
following form:
[ ]
[ ]
λ ε
λ ε
−
∆
−
−
∆
−
∆ = − − ⋅ + ∆ + ∆ +
=
∆ = − − ⋅ + ∆ + ∆ +
ℓ ℓ
ℓ ℓ
1
1 ,
1
1
2 ,
1
1 ( ) ( )
1,2,...,
1 ( ) ( )
t
t t t X t
t
t
t t t Y t
t
YX p A X B Y
Xt N
YY p C X D Y
X
(12)
where [ ]λ−1 the cointegration vector and λ the cointegration coefficient. Thus, in case of
cointegrated time series, linear Granger causality should be investigated via the VECM
specification.
For the pairwise implementation the linear causality testing was carried out using
the Granger’s test based on a VECM model of the log-prices because all series were found to
be cointegrated. The lag lengths of the VECM specification were set using the Wald exclusion
criterion and for each pair in PI are (in parenthesis): WTI Spot - WTI F1 (3), WTI Spot - WTI
F2 (7), WTI Spot - WTI F3 (3) and WTI Spot - WTI F4 (3). Similarly, in period PII: WTI Spot
25
- WTI F1 (3), WTI Spot - WTI F2 (6), WTI Spot - WTI F3 (6) and WTI Spot - WTI F4 (4). In
addition, in PI for all pairs the Johansen test identified two (2) cointegrating vectors using
the trace statistic and in PII one (1) cointegrating vector. In case of the five-variate
implementation cointegration was also detected and in particular in PI the Johansen test
identified five (5) cointegrating vectors while in PII three (3). The number of lags for the 5x5
system in PI was eleven (11) and in PII nine (9).
For the non-parametric test, in what follows we discuss results for lags = =ℓ ℓ 1X Y .
Moreover, the test was applied directly on log-returns. To implement the test, the constant C
for the bandwidth ε n was set at 7.5, which is close to the value 8.0 for ARCH processes
suggested by Diks and Panchenko (2006). With the theoretical optimal rate β = 2 7 this
implies a bandwidth value of approximately one times the standard deviation of the time
series for both PI and PII. Selecting bandwidth values smaller (larger) than one times the
standard deviation resulted, in general, in larger (smaller) p-values.
The results presented in Tables 3B and 4B allow for the following remarks: In the
pairwise implementation of the linear Granger tests (VECM), strong bi-directional Granger
causality between spot and futures prices was detected in both periods with small differences
regarding the degree of statistical significance. An exception could be that WTI Spot and
WTI F4 present only unidirectional linear relationship WTI Spot→WTI F4. On the contrary,
the linear causality for the five-variate implementation appears to be uni-directional, mainly
in the more volatile and trending period PII and from spot to futures prices regardless of
maturity, providing evidence that spot tend to lead futures prices. This indicates that spot
prices can be useful in the prediction of futures prices under a 5x5 VECM formulation, i.e.,
accounting for the contributions of all maturities in the causality detection. Further, there is
a causal relationship in PI of WTI Spot→WTI F1, WTI F3→WTI Spot and WTI F4→WTI
Spot. The nonlinear causality test revealed a bi-directional nonlinear relationship in PI,
26
whereas in PII only uni-directional causality was detected from Spot to WTI F1, WTI F2 and
WTI F3 returns, excluding WTI F4.
[ Insert Table 3B here ]
[ Insert Table 4B here ]
4B.2 Causality testing on VECM-filtered residuals
The results from the previous step suggest that there are significant and persistent
linear and nonlinear causal linkages between the spot and futures prices. However, even
though we found nonlinear causality, the new test should be reapplied to the filtered VECM-
residuals to ensure that any causality found is strictly nonlinear in nature. The number of
lags and the number of cointegrating vectors identified for the VECM specification were
reported in the previous section. Moreover, a linear Granger test is applied to the filtered
residuals to conclude on a remaining linear structure even after filtering. The causality on
the filtered residuals was investigated with a VAR specification (the null of no cointegration
was not rejected) and the lags were determined using the Schwartz Information Criterion
(SIC).
The pairwise implementation of the Granger tests after VECM filtering shows that
the linear causal relationships detected on the raw returns have now disappeared. In fact
none of the previously mentioned causalities appear or any other new ones have emerged
after linear filtering. Similarly, no causal relationship could be detected after five-variate
filtering. The application of the nonlinear test on the VECM residuals, both in the bivariate
and five-variate implementation, points towards the preservation of the bi-directional
causality reported in PI on the raw log-returns. In PII the nonlinear causal relationships
WTI Spot→WTI F2, WTI Spot→WTI F3 have vanished, while WTI Spot→WTI F1 remains,
albeit statistically less significant. Interestingly, in the same period, a uni-directional
causality from futures to spot returns has now emerged for all maturities.
27
The nature and source of the detected nonlinearities are different from that of the
linear Granger causality and may also imply a temporary, or long-term, causal relationship
between the spot and futures markets. For instance, excess volatility in PII might have
induced nonlinear causality. The nature of the volatility transmission mechanism can be
investigated after controlling for conditional heteroskedasticity using a GARCH-BEKK
model, in a bi-variate and five-variate representation.
4B.3 Causality testing on GARCH-BEKK filtered VECM-residuals
The use of the non-parametric test on filtered data with a multivariate GARCH
model enables one to determine whether the posited model is sufficient to describe the
relationship among the series. If the statistical evidence of nonlinear Granger causality lies
in the conditional variances and covariances then it would be strongly reduced when the
appropriate multivariate GARCH model is fitted to the raw or linearly filtered data.
However, failure to accept the no-causality null hypothesis may also constitute evidence that
the selected multivariate GARCH model was incorrectly specified. This line of analysis is
similar to the use of the univariate BDS test on raw data and on GARCH models (Brock et
al., 1996; Brooks, 1996; Hsieh, 1989). Many GARCH models can be used for this purpose. In
the present study the GARCH-BEKK model of Engle and Kroner (1995) is used. The BEKK
(p,q) model is defined as:
ε ε− − −= =
′ ′ ′ ′= + +∑ ∑1 1
q p
tj j
jk t j t j jk jk t j jkH C C A A G H G , ε = 1/2t t tH v (13)
where jkC,A and jkG are (NxN) matrices and C is upper triangular. tH is the conditional
covariance matrix of { }ε t with ε −Φ 1| ~ (0 )t t t, H and −Φ 1t the information set at time t − 1. The
residuals are obtained by the whitening matrix transformation ε1/2 tH . Gourieroux (1997)
gives sufficient conditions for tA and tG in order to guarantee that tH is positive definite.
Tables 3B and 4B show results before and after GARCH-BEKK (1,1) filtering. The
order parameters were determined for the time series in terms of the minimal SIC. The
28
linear Granger causality interdependencies remain absent, exactly as after VECM filtering
in both periods and for both representations i.e., bivariate and five-variate. After the
nonlinear causality testing in some cases the statistical significance is weaker after filtering,
particularly in the five-variate GARCH-BEKK implementation. These differences in
statistical significance indicate that the nonlinear causality is partially due to simple
volatility effects. However, this is not indicative of a general conclusion. Instead, significant
nonlinear interdependencies remain after the bi-variate and five-variate GARCH-BEKK
filtering, revealing that volatility effects and spillovers are probably not the only ones
inducing nonlinear causality. This of course does not apply to all the pairs of spot and futures
returns but some main results can be drawn for specific relationships. These are also
depicted graphically in Figure 1B where strong causality (“**”) is denoted by a “double
arrow”.
In particular, the pairwise nonlinear causality reveals the bi-directional linkages
WTI Spot↔WTI F1, WTI Spot↔WTI F3 and WTI Spot↔WTI F4 in PI, and WTI Spot→WTI
F1 in PII. In fact, these relationships remain roughly unchanged from the previous VECM
filtering stage. Yet, there are two significant changes; the bi-directional causality WTI
Spot↔WTI F2 in PI is reduced to a weakened WTI Spot→WTI F2 linkage, and most
importantly in PII the uni-directional causality from futures to spot for all maturities, has
now vanished. Thus, there is strong evidence that the latter nonlinear causal relationship
can be attributed to second moment effects.
Now, incorporating the “contribution” of all variables in a five-variate GARCH-BEKK
framework, the whitened residuals present different causal relationships than before.
Specifically, in PI the bi-directional linkages WTI Spot↔WTI F1, WTI Spot↔WTI F3 and
WTI Spot↔WTI F4 are reduced to uni-directional and the WTI Spot→WTI F2 has
disappeared. It seems that the nonlinear causality from futures to spot returns which
persisted even after the five-variate VECM filtering was induced by conditional
heteroskedasticity and thus a five-variate and not a bi-variate GARCH-BEKK filtering of the
29
VECM-residuals is better at “capturing” the volatility transmission mechanism. Instead, in
PII the uni-directional linkages WTI F1→WTI Spot and WTI F4→WTI Spot were not
entirely removed as in the bi-variate GARCH-BEKK filtering of the VECM-residuals.
Eventually, in all results, third or higher-order causality may be a significant factor of the
remaining interdependence.
6B. Conclusions for oil derivatives markets
In the second part of the paper we investigated the existence of linear and nonlinear
causal relationships between the daily spot and futures prices for maturities of one, two,
three and four months of West Texas Intermediate (WTI), which is the underlying
commodity of New York Mercantile Exchange's (NYMEX) oil futures contracts. The data
covered two separate periods, namely PI: 10/21/1991-10/29/1999 and PII: 11/1/1999-
10/30/2007, with the latter being significantly more turbulent. The study contributed to the
literature on the lead–lag relationships between the spot and futures markets in several
ways. In particular, it was shown that the pairwise VECM modeling suggested a strong bi-
directional Granger causality between spot and futures prices in both periods, whereas the
five-variate implementation resulted in a uni-directional causal linkage from spot to futures
prices only in PII. This empirical evidence appears to be in contrast to the results of
Silvapulle and Moosa (1999) on the futures to spot prices uni-directional relationship.
Additionally, whilst the linear causal relationships have disappeared after the cointegration
filtering, nonlinear causal linkages in some cases were revealed and more importantly
persisted even after multivariate GARCH filtering during both periods. Interestingly, it was
shown that the five-variate implementation of the GARCH-BEKK filtering, as opposed to the
bi-variate, captured the volatility transmission mechanism more effectively and removed the
nonlinear causality due to second moment spillover effects.
Moreover, the results imply that if nonlinear effects are accounted for, neither
market leads or lags the other consistently, or in other words the pattern of leads and lags
30
changes over time. Given that causality can vary from one direction to the other at any point
in time, a finding of bi-directional causality over the sample period may be taken to imply a
changing pattern of leads and lags over time, providing support to the Kawaller et al. (1988)
hypothesis. According to that hypothesis, market participants filter information relevant to
their positions as new information arrives and, at any time point, spot may lead futures and
vice versa. Hence it can be safely concluded that, although in theory the futures market play
a bigger role in the price discovery process, the spot market also plays an important role in
this respect. The empirical evidence of a causal linkage from spot to futures prices can be
attributed to the sequence of actions taken from the three market participants following a
spot price change, as described in the Moosa model (1996). In that, arbitrageurs respond to
the violation of the cost-of-carry condition and then speculators acting upon the expected spot
price will revise their expectation. In the same way, speculators who act upon the expected
futures price will revise their expectation and react to the disparity between current and
expected futures prices. These conclusions, apart from offering a much better understanding
of the dynamic linear and nonlinear relationships underlying the crude oil spot and futures
markets, may have important implications for market efficiency. For instance, they may be
useful in future research to quantify the process of market integration or may influence the
greater predictability of these markets.
An interesting subject for future research is the nature and source of the nonlinear
causal linkages. As presented, volatility effects may partly account for nonlinear causality.
The GARCH-BEKK model partially captured the nonlinearity in daily spot and future
returns, but only in some cases. An explanation could be that spot and futures returns may
exhibit statistically significant higher-order moments. A similar result was reported by
Scheinkman and LeBaron, (1989) for stock returns. Alternatively, parameterized asymmetric
multivariate GARCH models could be employed in order to accommodate the asymmetric
impact of unconditional shocks on the conditional variances.
31
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Blanchard, O., Watson, M.W., 1982. Bubbles, rational expectations, and financial
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37
Table 1A: Descriptive Statistics
Period I (3/20/1991-3/20/1997)
AUD/USD USD/CAD EUR/USD GBP/USD USD/CHF USD/JPY
Mean 1.47E-05 0.000113 -4.8E-05 -7.1E-05 1.6E-05 -6.9E-05
Standard Deviation 0.004764 0.002721 0.006599 0.006731 0.007806 0.006674
Sample Variance 2.27E-05 7.4E-06 4.35E-05 4.53E-05 6.09E-05 4.45E-05
Kurtosis 1.702 2.265 4.665 4.032 2.422 8.091
Skewness -0.303 0.070 -0.307 -0.274 -0.049 -0.693
Correlation Matrix
AUD/USD USD/CAD EUR/USD GBP/USD USD/CHF USD/JPY
AUD/USD 1
USD/CAD -0.166 1
EUR/USD -0.004 0.010 1
GBP/USD 0.088 -0.008 0.600 1
USD/CHF 0.023 -0.082 -0.659 -0.717 1
USD/JPY 0.108 -0.154 -0.345 -0.408 0.565 1
Period II (3/20/2003-3/20/2007)
AUD/USD USD/CAD EUR/USD GBP/USD USD/CHF USD/JPY
Mean 0.000294 -0.00023 0.000218 0.000216 -0.00013 -2.4E-05
Standard Deviation 0.006549 0.005063 0.005597 0.005368 0.006582 0.005423
Sample Variance 4.29E-05 2.56E-05 3.13E-05 2.88E-05 4.33E-05 2.94E-05
Kurtosis 1.168 0.520 0.399 0.570 0.765 1.016
Skewness -0.430 0.049 0.002 -0.077 -0.183 -0.174
Correlation Matrix
AUD/USD USD/CAD EUR/USD GBP/USD USD/CHF USD/JPY
AUD/USD 1
USD/CAD -0.538 1
EUR/USD 0.567 -0.407 1
GBP/USD 0.647 -0.443 0.647 1
USD/CHF -0.608 0.460 -0.752 -0.757 1
USD/JPY -0.514 0.319 -0.444 -0.487 0.544 1
38
Table 2A: Unit root tests
Variables ADF-statistic (PI) ADF-statistic (PII)
EUR/USD (0) 0.179 0.081
r EUR/USD (0) 0.000** 0.000**
GBP/USD (0) 0.207 0.242
r GBP/USD (0) 0.000** 0.000**
USD/JPY (0) 0.505 0.268
r USD/JPY (0) 0.000** 0.000**
USD/CHF (0) 0.491 0.070
r USD/CHF (0) 0.000** 0.000**
AUD/USD (0) 0.451 0.023
r AUD/USD (0) 0.000** 0.000**
USD/CAD (0) 0.595 0.204
r USD/CAD (0) 0.000** 0.000**
All variables are in logarithms and reported numbers are p-values. The number of lags in parenthesis is selected using the SIC. (**) denotes p-value corresponding to 99% confidence level.
39
Ta
ble
3A
: C
au
sali
ty r
esu
lts
(Pair
wis
e)
Pa
ir
Lin
ea
r G
ra
ng
er
Ca
usa
lity
N
on
Lin
ea
r C
au
sa
lity
Ra
w d
ata
R
aw
da
ta
VA
R
GA
RC
H-B
EK
K
X→
Y
Y→
X
X→
Y
Y→
X
X→
Y
Y→
X
X→
Y
Y→
X
X
Y
PI
PII
P
I P
II
PI
PII
P
I P
II
PI
PII
P
I P
II
PI
PII
P
I P
II
EU
R
GB
P (
1)
**
**
*
**
* **
**
**
EU
R
JP
Y (
1)
**
*
**
*
* *
EU
R
CH
F (
2)
**
**
**
**
**
*
**
**
EU
R
AU
D (
1)
**
*
EU
R
CA
D (
1)
*
*
*
GB
P
JP
Y (
1)
GB
P
CH
F (
1)
*
GB
P
AU
D (
1)
*
GB
P
CA
D (
1)
*
JP
Y
CH
F (
1)
*
*
JP
Y
AU
D (
1)
*
JP
Y
CA
D (
1)
**
**
* *
* *
CH
F
AU
D (
1)
CH
F
CA
D (
1)
**
AU
D
CA
D (
1)
**
*
**
*
*
X→
Y: r X
doe
s n
ot
Gra
nger
Cau
se r
Y. S
tati
stic
al
sign
ific
an
ce 5
% (
*),
1%
(**)
1=
=Y
Xℓ
ℓ
Th
e la
g l
en
gth
s of
VA
R s
pec
ific
ati
on
usi
ng t
he S
chw
art
z In
form
ati
on
Cri
teri
on (
SIC
) are
pre
sen
ted
in
pare
nth
esi
s.
Th
e f
ore
ign
exch
an
ge r
ate
s E
uro
(E
UR
), G
reat
Bri
tain
Pou
nd
(G
BP
), J
ap
an
ese
Yen
(JP
Y),
Sw
iss
Fra
nk
(C
HF
), A
ust
rali
an
Doll
ar
(AU
D)
an
d C
an
ad
ian
Doll
ar
(CA
D)
are
d
en
ote
d r
ela
tive
to U
nit
ed S
tate
s d
oll
ar
(US
D).
Th
e exact
rati
os
rep
rese
nt
EU
R/U
SD
, G
BP
/US
D, U
SD
/JP
Y, U
SD
/CH
F, A
UD
/US
D a
nd
US
D/C
AD
resp
ect
ively
.
40
Ta
ble
4A
: C
au
sali
ty r
esu
lts
(six
-vari
ate
)
Pa
ir
Lin
ea
r G
ra
ng
er
Ca
usa
lity
N
on
Lin
ea
r C
au
sa
lity
Ra
w d
ata
R
aw
da
ta
VA
R
GA
RC
H-B
EK
K
X→
Y
Y→
X
X→
Y
Y→
X
X→
Y
Y→
X
X→
Y
Y→
X
X
Y
PI
PII
P
I P
II
PI
PII
P
I P
II
PI
PII
P
I P
II
PI
PII
P
I P
II
EU
R
GB
P
**
**
*
**
* **
**
**
**
*
EU
R
JP
Y
**
*
**
EU
R
CH
F
**
**
**
**
**
**
*
EU
R
AU
D
**
EU
R
CA
D
*
*
**
**
*
GB
P
JP
Y
GB
P
CH
F
*
GB
P
AU
D
*
GB
P
CA
D
*
*
JP
Y
CH
F
*
*
JP
Y
AU
D
*
JP
Y
CA
D
**
**
* *
*
**
CH
F
AU
D
CH
F
CA
D
**
*
AU
D
CA
D
**
*
**
*
*
X→
Y: r X
doe
s n
ot
Gra
nger
Cau
se r
Y .
Sta
tist
ical
sign
ific
an
ce 5
% (
*),
1%
(**).
=
=ℓℓ1
XY
.
Th
e f
ore
ign
exch
an
ge r
ate
s E
uro
(E
UR
), G
reat
Bri
tain
Pou
nd
(G
BP
), J
ap
an
ese
Yen
(JP
Y),
Sw
iss
Fra
nk
(C
HF
), A
ust
rali
an
Doll
ar
(AU
D)
an
d C
an
ad
ian
Doll
ar
(CA
D)
are
d
en
ote
d r
ela
tive
to U
nit
ed S
tate
s d
oll
ar
(US
D).
Th
e exact
rati
os
rep
rese
nt
EU
R/U
SD
, G
BP
/US
D, U
SD
/JP
Y, U
SD
/CH
F, A
UD
/US
D a
nd
US
D/C
AD
resp
ect
ively
.
41
Figure 1A: Diagrammatical representation of directional causalities on GARCH-BEKK filtered VAR residuals (nonlinear causality) Notation: denote unidirectional and bi-directional causality corresponding to 5% ≤ p-value < 1%
denote unidirectional and bi-directional causality corresponding to p-value ≤ 1%
42
Table 1B: Descriptive Statistics
Period I (10/21/1991–10/29/1999)
WTI Spot WTI F1 WTI F2 WTI F3 WTI F4
Mean -0.00004 -0.00005 -0.00005 -0.00004 -0.00004
Standard Deviation 0.02060 0.01975 0.01702 0.01542 0.01432
Sample Variance 0.00042 0.00039 0.00029 0.00024 0.00021
Kurtosis 6.14 4.47 4.72 4.29 4.51
Skewness -0.01979 0.17027 0.16022 0.09616 0.10314
Correlation Matrix
WTI Spot WTI F1 WTI F2 WTI F3 WTI F4
WTI Spot 1
WTI F1 0.848 1
WTI F2 0.835 0.955 1
WTI F3 0.824 0.936 0.993 1
WTI F4 0.813 0.917 0.983 0.996 1
Period II (11/1/1999 – 10/30/2007)
WTI Spot WTI F1 WTI F2 WTI F3 WTI F4
Mean 0.00069 0.00069 0.00069 0.00068 0.00069
Standard Deviation 0.02388 0.02276 0.02083 0.01945 0.01879
Sample Variance 0.00057 0.00052 0.00043 0.00038 0.00035
Kurtosis 4.04 3.08 2.65 1.86 3.05
Skewness -0.58017 -0.56054 -0.44623 -0.35836 -0.44760 Correlation Matrix
WTI Spot WTI F1 WTI F2 WTI F3 WTI F4
WTI Spot 1
WTI F1 0.871 1
WTI F2 0.859 0.970 1
WTI F3 0.849 0.957 0.994 1
WTI F4 0.828 0.932 0.973 0.983 1
43
Table 2B: Unit root tests
Variables ADF-statistic (PI) ADF-statistic (PII)
WTI Spot (0) 0.039 0.943
r WTI Spot (0) 0.000 ** 0.000**
WTI F1(0) 0.044 0.967
r WTI F1 (0) 0.000** 0.000**
WTI F2 0) 0.070 0.974
r WTI F2 (0) 0.000** 0.000**
WTI F3 0) 0.085 0.978
r WTI F3 (0) 0.000** 0.000**
WTI F4 0) 0.089 0.979
r WTI F4 (0) 0.000** 0.000**
All variables are in logarithms and reported numbers for the augmented Dickey–Fuller test are p-values. The number of lags in parenthesis is selected using the SIC. (**) denotes p-value corresponding to 99% confidence level. PI: 10/21/1991–10/29/1999; PII: 11/1/1999 – 10/30/2007
44
Ta
ble
3B
: C
au
sali
ty r
esu
lts
(Pair
wis
e)
Va
ria
ble
s
Pa
ne
l A
: L
ine
ar G
ra
ng
er C
au
sa
lity
P
an
el
B:
No
n-L
inea
r C
au
sa
lity
Ra
w d
ata
V
EC
M
resid
ua
ls
GA
RC
H-B
EK
K
resid
ua
ls
Ra
w d
ata
V
EC
M
resid
ua
ls
GA
RC
H-B
EK
K
resid
ua
ls
X→
Y
Y→
X
X→
Y
Y→
X
X→
Y
Y→
X
X→
Y
Y→
X
X→
Y
Y→
X
X→
Y
Y→
X
X
Y
PI
PII
P
I P
II
PI
PII
P
I P
II
PI
PII
P
I P
II
PI
PII
P
I P
II
PI
PII
P
I P
II
PI
PII
P
I P
II
WT
I S
pot
WT
I F
1
**
**
**
*
**
**
**
**
* **
*
**
* **
WT
I S
pot
WT
I F
2
**
**
**
**
**
**
**
**
**
**
*
WT
I S
pot
WT
I F
3
**
**
* **
**
*
**
**
**
**
**
**
WT
I S
pot
WT
I F
4
**
**
**
**
*
**
**
**
**
(*),
(**)
Den
ote
s p
-valu
e s
tati
stic
al
sign
ific
an
ce a
t 5%
an
d 1
%
level.
X→
Y:
r X d
oes
not
Gra
nger
Cau
se r
Y. P
I: 1
0/2
1/1
991–10/2
9/1
999;
PII
: 11/1
/1999 –
10/3
0/2
007.
Pan
el
A: L
inear
Gra
nger
Cau
sali
ty
All
data
(lo
g-l
evels
) w
ere f
ou
nd
to b
e c
oin
tegra
ted
an
d t
he l
ag l
ength
s of
VE
CM
sp
eci
fica
tion
are
set
usi
ng t
he W
ald
excl
usi
on c
rite
rion
. T
he n
um
ber
of
lags
(in
pare
nth
esi
s)
iden
tifi
ed
in
per
iod P
I are
: W
TI
Sp
ot
- W
TI
F1 (
3),
WT
I S
pot
- W
TI
F2 (
7),
WT
I S
pot
- W
TI
F3 (
3),
WT
I S
pot
- W
TI
F4 (
3)
an
d i
n p
eri
od
PII
: W
TI
Spot
- W
TI
F1 (
3),
WT
I S
pot
- W
TI
F2 (
6),
WT
I S
pot
- W
TI
F3 (
6),
WT
I S
pot
- W
TI
F4 (
4).
In
peri
od
PI
for
all
pair
s th
e J
oh
an
sen
test
id
en
tifi
ed t
wo
(2)
coin
tegra
tin
g v
ect
ors
usi
ng t
he
trace
sta
tist
ic a
nd
in
P
II o
ne (
1)
coin
tegra
tin
g v
ecto
r again
for
all
pair
s. T
he
cau
sali
ty o
n t
he V
EC
M r
esi
du
als
was
invest
igate
d w
ith
a V
AR
sp
ecif
icati
on
(th
e n
ull
of
no c
oin
tegra
tion
was
not
reje
cted
) an
d t
he l
ags
wer
e d
ete
rmin
ed
usi
ng t
he S
chw
art
z In
form
ati
on
Cri
teri
on (
SIC
). T
he n
um
ber
of
lags
iden
tifi
ed
for
all
vari
able
s in
all
per
iod
s is
on
e (
1).
Th
e se
con
d
mom
en
t fi
lteri
ng w
as
per
form
ed
wit
h a
GA
RC
H-B
EK
K (
1,1
) m
od
el.
P
an
el
B: N
on
-Lin
ear
Cau
sali
ty
Th
e n
um
ber
of
lags
use
d f
or t
he
non
lin
ear
cau
sali
ty t
est
are
==
ℓℓ1
XY
. T
he
data
use
d a
re l
og-r
etu
rns.
Sin
ce t
he
log-l
evels
wer
e fo
un
d t
o be
coin
tegra
ted (
Pan
el
A)
the
non
lin
ear
cau
sali
ty w
as
inves
tigate
d o
n t
he
VE
CM
resi
du
als
. T
he
nu
mber
of
lags
(Wald
excl
usi
on c
rite
rion
) an
d t
he n
um
ber
of
coin
tegra
tin
g v
ecto
rs (
Joh
an
sen
tra
ce s
tati
stic
te
st)
iden
tifi
ed f
or
the V
EC
M s
pec
ific
ati
on a
re r
ep
ort
ed
in
Pan
el
A.
Th
e se
con
d m
om
en
t fi
lter
ing w
as
perf
orm
ed
wit
h a
GA
RC
H-B
EK
K (
1,1
) m
od
el.
45
Ta
ble
4B
: C
au
sali
ty r
esu
lts
(fiv
e-v
ari
ate
)
Va
ria
ble
s
Pa
ne
l A
: L
ine
ar G
ra
ng
er C
au
sa
lity
P
an
el
B:
No
n-L
inea
r C
au
sa
lity
Ra
w d
ata
V
EC
M
resid
ua
ls
GA
RC
H-B
EK
K
resid
ua
ls
Ra
w d
ata
V
EC
M
resid
ua
ls
GA
RC
H-B
EK
K
resid
ua
ls
X→
Y
Y→
X
X→
Y
Y→
X
X→
Y
Y→
X
X→
Y
Y→
X
X→
Y
Y→
X
X→
Y
Y→
X
X
Y
PI
PII
P
I P
II
PI
PII
P
I P
II
PI
PII
P
I P
II
PI
PII
P
I P
II
PI
PII
P
I P
II
PI
PII
P
I P
II
WT
I S
pot
WT
I F
1
* **
**
**
**
**
* **
*
* *
*
WT
I S
pot
WT
I F
2
**
**
**
**
**
**
*
WT
I S
pot
WT
I F
3
**
**
**
* **
**
**
**
**
WT
I S
pot
WT
I F
4
**
**
**
**
**
**
**
**
*
(*),
(**)
Den
ote
s p
-valu
e s
tati
stic
al
sign
ific
an
ce a
t 5%
an
d 1
%
level.
X→
Y:
r X d
oes
not
Gra
nger
Cau
se r
Y. P
I: 1
0/2
1/1
991–10/2
9/1
999;
PII
: 11/1
/1999 –
10/3
0/2
007.
Pan
el
A: L
inear
Gra
nger
Cau
sali
ty
Th
e 5x5 s
yst
em o
f th
e d
ata
(lo
g-l
evels
) w
as
fou
nd t
o be
coin
tegra
ted
an
d t
he
lag l
en
gth
of
the V
EC
M s
pec
ific
ati
on
was
set
usi
ng t
he
Wald
excl
usi
on c
rite
rion
. T
he
nu
mber
of
lags
iden
tifi
ed
in
PI
is e
leven
(11)
an
d i
n P
II i
s n
ine (
9).
In
peri
od
PI
the
Joh
an
sen
test
id
enti
fied
fiv
e (
5)
coin
tegra
tin
g v
ect
ors
usi
ng t
he t
race
sta
tist
ic a
nd
in
PII
th
ree (
3).
Th
e ca
usa
lity
on
th
e V
EC
M r
esid
uals
was
inves
tigate
d w
ith
a V
AR
sp
eci
fica
tion
(th
e n
ull
of
no c
oin
tegra
tion
was
not
reje
cted
) an
d t
he l
ags
were
dete
rmin
ed
usi
ng t
he S
chw
art
z In
form
ati
on C
rite
rion
(S
IC).
Th
e n
um
ber
of
lags
iden
tifi
ed
for
all
vari
able
s in
all
per
iod
s is
on
e (1
). T
he
seco
nd
mom
en
t fi
lter
ing w
as
per
form
ed
wit
h a
GA
RC
H-B
EK
K (
1,1
) m
odel.
P
an
el
B: N
on
-Lin
ear
Cau
sali
ty
Th
e n
um
ber
of
lags
use
d f
or
the
non
lin
ear
cau
sali
ty t
est
are
==
ℓℓ1
XY
. T
he d
ata
use
d a
re l
og-r
etu
rns.
Sin
ce t
he
5-v
ari
ate
syst
em
of
log-l
evels
was
fou
nd
to
be
coin
tegra
ted
(Pan
el
A)
the n
on
lin
ear
cau
sali
ty w
as
invest
igate
d o
n t
he V
EC
M r
esid
uals
. T
he n
um
ber
of
lags
(Wald
excl
usi
on
cri
teri
on)
an
d t
he n
um
ber
of
coin
tegra
tin
g v
ect
ors
(Joh
an
sen
tr
ace
sta
tist
ic t
est
) id
en
tifi
ed
for
th
e V
EC
M s
peci
fica
tion
are
rep
orte
d i
n P
an
el
A. T
he
seco
nd
mom
en
t fi
lteri
ng w
as
per
form
ed
wit
h a
GA
RC
H-B
EK
K (
1,1
) m
od
el.
46
Figure 1B: Diagrammatical representation of directional causalities on GARCH-BEKK filtered VECM residuals Notation: denote unidirectional and bi-directional causality corresponding to 5% ≤ p-value < 1%
denote unidirectional and bi-directional causality corresponding to p-value ≤ 1%