The Estimation of Leverage Effect With High-Frequency Data · based on the “leverage effect” hypothesis: A drop in the value of the stock (negative return) increases the ﬁnancial

The Estimation of Leverage Effect WithHigh-Frequency Data

Christina D. WANG and Per A. MYKLAND

The leverage effect has become an extensively studied phenomenon that describes the (usually) negative relation between stock returnsand their volatility. Although this characteristic of stock returns is well acknowledged, most studies of the phenomenon are based oncross-sectional calibration with parametric models. On the statistical side, most previous works are conducted over daily or longer returnhorizons, and few of them have carefully studied its estimation, especially with high-frequency data. However, estimation of the leverageeffect is important because sensible inference is possible only when the leverage effect is estimated reliably. In this article, we providenonparametric estimation for a class of stochastic measures of leverage effect. To construct estimators with good statistical properties, weintroduce a new stochastic leverage effect parameter. The estimators and their statistical properties are provided in cases both with andwithout microstructure noise, under the stochastic volatility model. In asymptotics, the consistency and limiting distribution of the estimatorsare derived and corroborated by simulation results. For consistency, a previously unknown bias correction factor is added to the estimators.Applications of the estimators are also explored. This estimator provides the opportunity to study high-frequency regression, which leads tothe prediction of volatility using not only previous volatility but also the leverage effect. The estimator also reveals a theoretical connectionbetween skewness and the leverage effect, which further leads to the prediction of skewness. Furthermore, adopting the ideas similar to theestimation of the leverage effect, it is easy to extend the methods to study other important aspects of stock returns, such as volatility ofvolatility.

KEY WORDS: Consistency; Discrete observation; Efficiency; Ito process; Microstructure noise; Realized volatility; Skewness; Stableconvergence.

1. INTRODUCTION

The leverage effect has become an extensively studied empir-ical phenomenon in the form of the (usually negative) correla-tion between (current) returns and (current and future) volatil-ity (Engle and Ng 1993; Zakoian 1994; Wu and Xiao 2002,etc.). It is one of the many stylized facts of the security re-turn distribution, along with the well-known fat tails, skew-ness, excess kurtosis, and heteroscedasticity. The discovery ofleverage effect closely relates to the study of stochastic volatil-ity. Although for very low frequency data, such as monthly oryearly asset returns, the assumption of homogeneity seems notto be entirely unreasonable (Mandelbrot 1963; Fama 1965; Of-ficer 1973), the increasing frequency of observed data in stud-ies suggests heterogeneity in volatility, in other words, time-varying volatility (Engle 1982; Bollerslev 1986; Andersen andBollerslev 1998; Engle 2000; Andersen et al. 2001). This findinghas had profound implications in both the theory and practiceof financial economics and econometrics. It has inspired newmodel building, such as the emergence of ARCH models and thelater stochastic volatility models. Modeling volatility as a sepa-rate process allows the study of its relation with the associated re-turn process, which leads to the discovery of asymmetric volatil-ity. Time varying volatility is also of substantial importance inmodeling for options pricing, as in Hull and White (1987), Steinand Stein (1991), Heston (1993), and Ball and Roma (1994).

Black (1976) and Christie (1982) were among the first todocument the volatility asymmetry, and gave an explanation

Christina D. Wang is Post Doctoral Research Fellow, Department of Oper-ations Research and Financial Engineering and Bendheim Center for Finance,Princeton University, Princeton, NJ 08544 (E-mail: [email protected]). PerMykland is Professor, Department of Statistics and the College, University ofChicago, Chicago, IL 60637 (E-mail: [email protected]). Finan-cial support From the National Science Foundation Under grants DMS 06-04758, SES 06-31605, SES 11-24 526 and from the Oxford-Man Institute isgratefully acknowledged. The authors thank Neil Shephard, Kevin Sheppard,and Lan Zhang for their helpful comments and suggestions.

based on the “leverage effect” hypothesis: A drop in the valueof the stock (negative return) increases the financial leverage(debt-to-equity ratio), which makes the stock riskier and in-creases its volatility. Since then, leverage effect has been takento be synonymous with asymmetric volatility. Financial lever-age itself, however, seems not enough to explain either the largemagnitude of the effect of declines in current price on futurevolatility (Figlewski and Wang 2001), or the phenomenon thatthe asymmetry of market index returns is generally larger thanthat of individual stocks (Kim and Kon 1994; Tauchen andZhang 1996; Andersen et al. 2001). In another point of view,the asymmetric nature of volatility to return shocks simply re-flects time-varying risk premium (Pindyck 1984; French et al.1987; Campbell and Hentschel 1991). This explanation is of-ten referred to as the “volatility feedback effect”: If volatility ispriced, an anticipated increase in volatility raises the requiredreturn on equity, leading to an immediate stock price decline.

Many later works either compare the two effects or seekto argue that they can both be at work (Nelson 1991; Engleand Ng 1993; Glosten et al. 1993; Bekaert and Wu 1997; Wu2001; Hasanhodzic and Lo 2011). While there is little agreementconcerning the fundamental causes behind the leverage effect,that is not the focus of this article.

As most early studies are conducted over daily or longertime horizons, it is worthwhile to examine this phenomenonwith high frequency data, which provides the opportunity toexplore more closely the relation between stock price and its ownvolatility. Some recent work has demonstrated that the volatilityasymmetry still appears over fairly small time intervals. Butsome new aspects are added as both very good and bad newsincrease volatility, with the latter having a more severe effect

© 2014 American Statistical AssociationJournal of the American Statistical Association

March 2014, Vol. 109, No. 505, Theory and MethodsDOI: 10.1080/01621459.2013.864189

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198 Journal of the American Statistical Association, March 2014

(Barndorff-Nielsen et al. 2008b; Chen and Ghysels 2011). Also,the leverage effect decays exponentially as the time lag betweenreturn and volatility increases. In the literature, the peak effectis obtained at the instantaneous correlation between return andvolatility (Bouchaud et al. 2001; Bollerslev et al. 2006).1 Thiscorresponds to our definition of the leverage effect as beinginstantaneous. For further discussion of this effect; see the endof Section 2.2.

Although many papers deal with the source or new propertiesof the leverage effect, few have tried to rigorously estimate it,which is critical for supporting any conclusive claims. Simplecorrelation estimators may be applied to the estimation of theleverage effect with caution. Those estimators lose consistencywith high-frequency data (Aıt-Sahalia et al. 2013). It is thepurpose of this article to construct nonparametric estimatorsof leverage effects in the stochastic volatility model. To studythe estimation, we define a new leverage effect parameter asthe covariance (a covariation, to be precise) between the stockreturn and a function of its volatility. To construct the estimatorfor the new parameter, we first study the classical equi-distantsampling case without microstructure noise as the foundationfor the study (later in the article) of more complicated cases,such as the case with microstructure noise. As is emphasizedin several studies (Mykland and Zhang 2006; Barndorff-Nielsenet al. 2008a; Renault and Werker 2011), it is more natural to workwith irregularly spaced data in practice. Based on the resultsin equidistant cases, the extension of estimators to irregularlyspaced data can be constructed in a similar way, with someadaptations.

Even with equidistant observations without microstructurenoise, we discover a previously unknown bias correction factor.This bias correction factor is critical to obtaining consistency.The factor may have a substantial impact on the estimated valuesince it functions as a magnifier, especially when estimates areclose to zero. The bias correction factor may play an even moreimportant role in the estimation, when the situation becomesmore complicated as market microstructure noise is present. In-deed, in the case with microstructure noise, the bias correctionfactor is found to be bigger than that in the case with uncontam-inated continuous price paths.

Statistical properties such as consistency and asymptotic dis-tribution are carefully studied in different settings. The theo-retical findings of these statistical properties are corroboratedby the simulation results. These asymptotic properties have ap-plications to hypothesis testing (e.g., for model checking) andconstructing confidence intervals.

There are many ways to apply the estimators of the leverageeffect depending on the practical purpose. One way to explorethe potential application of the estimators is embedded in thedefinition of the stochastic parameter of the leverage effect. Ac-cording to the definition, one specific choice of the functionimposed on volatility gives rise to a unique relation between theleverage effect and skewness, which will help to estimate skew-ness consistently. This relation may introduce further applica-tions in hedging strategy or new product design; see Neuberger

1The relative change between time lag and lagged leverage effect should bemaintained, even if the consistency may be an concern, since the consistencycan be achieved by a bias correction multiplier from our study shown later.

(2011). Another carefully chosen function of volatility can sim-plify the estimation of high-frequency regression coefficients.This leads to an interesting discovery in volatility prediction.The empirical study with Microsoft stock data (2008–2011)shows strong predictive power of a term containing the leverageeffect on the next period volatility. The power is comparable tothat of current period volatility which is believed to be the mostsignificant term in volatility prediction.

The main results of this article will be given in Sections 2 and4. The data-generating mechanism and model setting can befound in Section 2.1. The (stochastic) parameter of the leverageeffect is defined in Section 2.2. Based on this, for the casewithout microstructure noise, the estimator and limit theoremscan be found in Sections 2.3 and 2.4. Simulation results areprovided in Section 3. Results that corroborate the theorems canbe found in Section 3.1. Section 4 studies the case where marketmicrostructure noise is present in the data. The estimator andlimit theorems for this case are provided in Sections 4.1 and4.2. Simulation results for this case are provided in Section 5.The extension to irregularly spaced data can be found in Section6. The relation between leverage effect and skewness is shownin Section 7. An application of the leverage effect in high-frequency regression is implemented in Section 8. The detailsof empirical studies are in Section 9. The conclusion is providedin Section 10. Proofs are in the Appendix.

2. MAIN RESULTS

2.1 Data-Generating Mechanism

In general, we shall work with a broad class of continuoussemimartingales, namely Ito processes. In econometrics andfinancial mathematics studies, this is the most popular model forlog price processes due to nonarbitrage considerations (Delbaenand Schachermayer 1994, 1995, 1998).

Definition 1. A process Xt is called an Ito process providedit satisfies

dXt = µt dt + !t dWt ,X0 = x0, (1)

where µt and !t are adapted cadlag locally bounded randomprocesses, and Wt is a Wiener process. The underlying filtrationwill be called (Ft ). The probability measure will be called P.

The integrated variance process is given as

!X,X"t =! t

0! 2

u du. (2)

The process (2) is also known as the quadratic variation of X.We shall sometimes also use the term “integrated volatility.”

We further assume that !t is also an Ito process (see the nextsection for discussion of this)

d!t = at dt + ft dWt + gt dBt , (3)

where Bt is another Wiener process independent of Wt , and at ,ft , and gt are all assumed to be Ito processes.

Clearly, in this stochastic volatility (SV) model, Xt corre-sponds to the log price process and !t is its own volatility pro-cess. Both processes have a common driving Wiener processWt , which accommodates the leverage effect.

Wang and Mykland: The Estimation of Leverage Effect With High-Frequency Data 199

To summarize the technical requirements, we specify exactconditions as follows:

Assumption 1. The system satisfies (1) and (3), where X and! are continuous processes (the continuous modification). Weassume that all the coefficients ft , gt , at , µt are locally boundedin absolute value. We also assume that !t is locally boundedaway from zero.2

2.2 The Parameter: A Definition of Leverage Effect

As we have seen in the Introduction, the literature offers var-ious perspectives on how to specify a parameter for this effect;see also the discussion below in this section. We concentrate onthe estimation of the following stochastic parameter:

Definition 2. The stochastic parameter of the contempora-neous leverage effect is defined as the quadratic covariationbetween Xt and F (! 2

t )

!X,F (! 2)"T =! T

02F #"! 2

t

#! 2

t ft dt, (4)

where we suppose that

Assumption 2. x $% F (x) is twice continuously differen-tiable, monotone on (0,&).

The incorporation of the function F allows more flexibil-ity and a wider range of applications when different forms ofvolatility are of interest, such as log-volatility processes whichtend to be more stationary over time as implied by many empir-ical studies. The actual choice of F will depend on the practicalpurpose and empirical evidence. The inclusion of this func-tion can also help reveal some interesting connections betweenleverage effects and other statistics. Further interpretation of thisspecification will be seen in Sections 7 and 8.

We stop here for a moment to reflect on the definitions. Firstof all, we work with continuous processes. The interface withjump processes remains to be explored. The latter permits ad-ditional concepts of asymmetry, in particular the semivarianceof Barndorff-Nielsen, Kinnebrock, and Shephard (2008b). Theconnection between semivariance and the leverage effect (andskewness, see Section 7) in this article is an important questionwhich we leave for future investigation. This is necessarily acomplex matter, as it involves a different model of the priceprocess (continuous paths vs. jumps).

Once one works with continuous paths, the assumption thatthe leverage effect is instantaneous is both natural in a semi-martingale model for !t , and is empirically supported by thefinding in Bollerslev, Litvinova, and Tauchen (2006), where itwas shown that the connection between return and volatility ismost significant when the time lag is 0. This does not contra-dict the fact that the effect can appear at a greater time lag, asdocumented by Chen and Ghysels (2011).

As far as the Ito process (continuous semimartingale) assump-tion is concerned, this assumption appears frequently both on

2To get from local boundedness to results that cover the whole time interval, usearguments as in chap. 2.4.5 (p. 160-161) of Mykland and Zhang (2012). |!t | islocally bounded from above by continuity. The assumptions guarantee that theequivalent martingale measure for X exists locally. This is used in the proofs;see the beginning of Section A.1.

the options pricing side (Hull and White 1987; Stein and Stein1991; Heston 1993; Ball and Roma 1994), and on the econ-metric side (Barndorff-Nielsen and Shephard 2002; Barndorff-Nielsen et al. 2006; Jacod 2008; Barndorff-Nielsen and Veraart2009, Aıt-Shalia and Jacod 2009; Mykland and Zhang 2011b).A parallel development can be carried out under assumptions offractional Brownian motion (Comte and Renault 1998; Gloterand Hoffmann 2004; Brockwell and Marquardt 2005; Nualart2006; Comte et al. 2010).

The above is, of course, a set of theoretical considerations.We finally appeal to the results in Section 9 to show that ourcurrent definition of volatility asymmetry does find somethingempirically relevant; we substantially improve the prediction ofnext-period volatility using the current-period leverage effect.

2.3 Estimation in the Absence of Microstructure Noise

As the first step, we shall work with the equally spacedcase for the process (Xt ); specifically it is observed every"tn,i+1 = "t = T

nunits of time, at times 0 = tn,0 < tn,1 <

tn,2 < · · · < tn,n = T . Furthermore, we divide observed valuesinto Kn blocks, with block size Mn = [c

'n] (except possi-

bly for the first and last block, which does not matter for theasymptotics), for some constant c. The boundary points areon the grid H = {0 < #n,1 < #n,2 < · · · < #n,Kn(1 ) T }, whereKn = [ n

Mn].

Define3

!!X,F (! 2)"T = 2Kn(2$

i=0

"X#n,i+1 ( X#n,i

#"F

"! 2

#n,i+1

#( F

"! 2

#n,i

##,

(5)

and

! 2#n,i

= 1Mn * "t

$

tn,j +(#n,i ,#n,i+1]

"Xtn,j+1 ( Xtn,j

#2.

The factor 2 in the first part of Equation (5) might look unnat-ural to be included. However, it is crucial for the consistency ofthe estimator (see Remark 3 for a discussion of this previouslyunknown factor).

Theorem 1. Under Assumptions 1–2, as n % & and T fixed,

n1/4( !!X,F (! 2)"T ( !X,F (! 2)"T )L%Z

%16c

! T

0

"F #"! 2

t

##2! 6

t dt

+ cT

! T

0

"F #"! 2

t

##2! 4

t

%443

f 2t + 32

3g2

t

&dt

&1/2

, (6)

stably in law,4 where Z is a standard normal random variableand independent of FT .

3One can also consider a kernel estimator of the spot volatility in (5), by applyingthe methods in Kristensen (2010), with some adaptation. A detailed study isbeyond the scope of this article.4Suppose that all relevant processes (Xt , !t , etc.) are adapted to the filtration(Ft ). Let Zn be a sequence of Ft -measurable random variables. We say thatZn converges stably in law to Z as n % & if Z is measurable with respect toan extension of FT so that for all A + FT and for all bounded continuous g,EIAg(Zn) % EIAg(Z) as n % &. The same definition applies to triangulararrays.


Another natural estimator analogous to !!X,F (! 2)"T is

"!X,F (! 2)"T = 2Kn(2$

i=0

"X#n,i+1 ( X#n,i

#"F

"'! 2

#n,i+1

#( F

"'! 2

#n,i

##,

'! 2#n,i

= 1(Mn ( 1) * "t

$

tn,j +(#n,i ,#n,i+1]

*""Xtn,j+1 ( "X#n,i+1

#2, (7)

and

"X#n,i+1 = 1Mn

$

tn,j +(#n,i ,#n,i+1]

"Xtn,j+1 = 1Mn

"X#n,i+1 ( X#n,i

#.

Noticing the relation between the two estimators when

F (x) = x: "!X, ! 2"T = Mn

Mn(1!!X, ! 2" (

(i

2Mn(Mn(1)"t

"X#n,i+1

("X#n,i+2 )2 +(

i2

Mn(Mn(1)"t("X#n,i+1 )3, the following theorem

can be easily derived:

Theorem 2. Under Assumptions 1–2 as in Theorem 1, asn % & and T fixed,

n1/4( "!X,F (! 2)"T ( !X,F (! 2)"T )L%Z

%16c

! T

0

"F #"! 2

t

##2! 6

t dt

+ cT

! T

0

"F #"! 2

t

##2! 4

t

%443

f 2t + 32

3g2

t

&dt

&1/2

, (8)


Remark 1. From the limit theorems, it is not hard to see thatby properly choosing c, one can minimize the limit variance.The optimal value is

c2 =16

) T

0

"F #"! 2

t

##2! 6

t dt

T) T

0

"F #

"! 2

t

##2! 4

t

" 443 f 2

t + 323 g2

t

#dt

. (9)

See Section 2.4 for an estimator6 of c.

Remark 2. The two estimators have the same asymptoticproperties. Even though the centered version gives slightly moresymmetric results, it does not behave very differently from thenoncentered version. In practice, the noncentered version can beapplied with less programming effort. Therefore, our later sim-ulation mainly adopts the noncentered version of estimators.

Remark 3. The origin of the factor 2 in the estimator can befound in the proof of Theorem 1. For intuition, however, wegive here a verbal explanation of the source of this adjustmentconstant. Let us consider the case where F (x) = x. Since ! 2

t is aconsistent estimator of ! 2

t , then the first multiplication (X#i+1 (X#i

)(! 2#i+1

( ! 2#i

) already gives a consistent (though infeasible)estimator of the leverage effect in the interval (#i , #i+1]. Thenone may expect the remainder term (X#i+1 ( X#i

)(! 2#i

( ! 2#i

) tohave mean zero. However, since ! 2

#iemploys data in the time

interval (#i , #i+1], as does (X#i+1 ( X#i), the product does not

5See Footnote 4.6Here and in the continuation of Remark 1, we assume that the denominator in(9) is nonzero.

converge to zero but to one half of the leverage effect. To see

why it is one half, note that each increment"X2

tj

"tterm is roughly

an (inconsistent) estimator of ! 2tj

. Thus the cross product gives anaverage of leverage effects over (#i , t1], (#i , t2], . . . , (#i , #i+1]. If!X, ! 2"#t is considered to be constant over (#i , #i+1], that averageof those leverage effects will give a value of about half of theleverage effect over the entire interval. Hence, we have reducedthe estimation of leverage effect by half. An adjustment factorof 2 therefore needs to be added to achieve consistency.

2.4 Estimation of Asymptotic Variance

Let

G1n = 2n

12

Kn(2$

i=0

(X#n,i+1 ( X#n,i)2"F

"*! 2

#n,i+1

#( F

"*! 2

#n,i

##2,

and (10)

G2n = 2n

12 Mn"t

Kn(2$

i=0

*! 2#n,i

"F

"*! 2

#n,i+1

#( F

"*! 2

#n,i

##2.

By the same methods as in the proof of Theorem 1, we havethe following convergences in probability:

G1n

p% 8c

! T

0

"F #"! 2

t

##2! 6

t dt

+ cT

! T

0

"F #"! 2

t

##2! 4

t

%283

f 2t + 16

3g2

t

&dt, (11)

G2n

p% 8c

! T

0

"F #"! 2

t

##2! 6

t dt

+ cT

! T

0

"F #"! 2

t

##2! 4

t

163

"f 2

t + g2t

#dt, (12)

and

G1n + G2

n

p% 16c

! T

0

"F #"! 2

t

##2! 6

t dt

+ cT

! T

0

"F #"! 2

t

##2! 4

t

%443

f 2t + 32

3g2

t

&dt.

(13)

Equation (13) gives the estimation of the asymptotic vari-ance. With this estimation, a feasible version of the central limitdistribution can be derived.


n1/4( !!X,F (! 2)"T ( !X,F (! 2)"T )+

G1n + G2

n

L%Z1,

and (14)

n1/4( "!X,F (! 2)"T ( !X,F (! 2)"T )+

G1n + G2

n

L%Z1

stably in law,7 where Z1 is a standard normal random variableand independent of FT .

Notice that the limiting distribution Z1 is the same in bothlimits. In other words, the difference between the two statistics

7See Footnote 4.


Table 1. The summary statistics do exhibit the target normality

MSE Mean Median Q1 Q3

n = 390, T = 1/250 infeasible 1.097112 (0.02108 (0.005707 (0.676 0.6496n = 390, T = 1/250 feasible 1.051725 0.006831 (0.004835 (0.754 0.7399n = 23400, T = 1/250 infeasible 1.009285 (0.006430 0.003727 (0.6982 0.6796n = 23400, T = 1/250 feasible 1.002125 0.002964 0.004267 (0.6964 0.6917n % &, fixed T (asymptotic value) 1 0 0 (0.674 0.674

NOTE: This corroborates the theorems and shows that the asymptotics can predict small sample behavior. For sample size 390, both the mean and median are very close to 0. The MSEis close to 1 and the quartiles are close to the theoretical values from N(0, 1). As sample size increases, the MSE decreases further closer to 1.

converges to zero in probability. With this feasible CLT, one canconduct hypothesis testing and construct confidence interval forthe leverage effect parameter.

Remark 1 (continued). The result (13) opens paths to es-timating the tuning parameter c in (9). We here outline twoapproaches.

Method 1: The conceptually simplest possibility is to pickc = arg min{G1

n + G2n} over a suitable grid of c’s. If the grid

is nested and becomes dense as n % &, this automaticallyprovides a consistent estimator of c.

Method 2: Since Method 1 is computationally heavy, wehere also propose an alternative two-step method. Fix an ini-tial value c0, and compute (G1

n + G2n)(c0). On the other hand,

we can reduce estimation of $ 2 =) T

0 (F #(! 2t ))2! 6

t dt to the lo-cal estimation of volatility by the methods in Section 4.1 inMykland and Zhang (2009). Call this latter estimate $ 2. We thusobtain that c(1

0 (G1n + G2

n)(c0) ( 16$ 2/c20 consistently estimates

T) T

0 (F #(! 2t ))2! 4

t ( 443 f 2

t + 323 g2

t ) dt . A consistent estimate of c2

is thus given from (9) as

c2 = 16$ 2 "c(1

0

"G1

n + G2n

#(c0) ( 16$ 2/c2

0

#(1.

The two methods can be used together, with the second provid-ing a starting point for seaching for a minimum in Method 1.

3. SIMULATION RESULTS

All simulation results are based on 10,000 sample paths whilevarying the sample size n, function F, and optimal choice ofc (path dependent). In the simulation, the properties of the es-timator are studied with the Heston model (Heston 1993). Toexamine the theoretical limit distribution, the distribution of thestatistics in Theorem 1

n1/4( !!X,F (! 2)"T ( !X,F (! 2)"T )" 16

cT

) T

0 F #"! 2

t

#2! 6

t dt + c) T

0

"F #

"! 2

t

##2! 4

t

" 443 f 2

t + 323 g2

t

#dt

#1/2 ,

and the statistics in Theorem 3,

n1/4( !!X,F (! 2)"T ( !X,F (! 2)"T )+

G1n + G2

n

and

n1/4( "!X,F (! 2)"T ( !X,F (! 2)"T )+

G1n + G2

n

,

are studied. So if the asymptotics correctly predict small samplebehavior, the distributions should be close to the standard normaldistribution.

The Heston model used in the simulation is of the form:

dXt = !t dWt ,

d! 2t = %

"& ( ! 2

t

#dt + $ !t (' dWt +

+1 ( '2 dBt ),

where Wt ,, Bt . (15)

3.1 Normality Demonstration

In the simulation, the true log price is simulated from the He-ston model with broadly realistic parameter values: % = 5, & =0.04, $ = 0.5, ' = (

'0.5 over 1 trading day. Two different

sampling frequencies are studied to examine the small samplebehavior. The first is when the data are observed at 1-min fre-quency, which corresponds to sample size 390. The second iswhen the data are observed at every second, which correspondsto sample size 23,400. The results are given in Table 1.

4. ESTIMATION WITH MICROSTRUCTURE NOISE

It is well known that markets are not so ideal that log price pro-cesses can be simply represented by pure semimartingales. Thishas long been thought about as “microstructure” see, e.g., Roll(1984), O’Hara (1995), Harris (1990), and Hasbrouck (1996).In the context of high-frequency data, such microstructure wasoriginally observed through the so-called signature plot (intro-duced by Andersen et al. (2000); see also the discussion inMykland and Zhang (2005)). This led researchers to investigatea model where the efficient price is latent, and one actuallyobserves

Yt = Xt + (t . (16)

Several approaches8 seek to deal with microstructure noisewhile estimating integrated volatility, and they shed light on howto proceed in the estimation of leverage effects in the similarsituation. Among these approaches, we have focused on preav-eraging. The preaveraging method (Jacod et al. 2009; Podolskijand Vetter 2009a; Mykland and Zhang 2011a) provides a plausi-ble way to solve the problem with microstructure. Therefore, allof the following discussion will be in the framework of preav-eraging and the blocking method will be adjusted as follows:

8Such as Zhang, Mykland, and Aıt-Sahalia (2005), Zhang (2006), Barndorff-Nielsen et al. (2008a), Reiss (2010), and Xiu (2010), as well as the preaveragingpapers cited in the text.


The contaminated log price process (Yt ) is observed every"tn,i = T

nunits of time, at times 0 = tn,0 < tn,1 < tn,2 < · · · <

tn,n = T .

Assumption 3.

Yt = Xt + (t , where (t ’s are iid. N (0, a2) and(t ,, the W and B processes, for all t . (17)

We also assume that (t ’s have finite fourth moment, and areindependent of both return and volatility processes.

We can also relax these assumptions; see the development inMykland and Zhang (2011a).

Here two nested levels of blocks will be required. The firstlevel of blocks defines the range of preaveraging and the sec-ond one implements a blocking idea similar to that in the casewithout noise in Section 2.3.

Blocks are defined on a much less dense grid of #n,i , alsospanning [0, T ], so that

block # i = {tn,j : #n,i ) tn,j < #n,i+1} (18)

(the last block, however, includes T). We define the block sizeby

Mn,i = #{j : #n,i ) tn,j < #n,i+1}. (19)

In principle, the block size Mn,i can vary across the tradingperiod [0, T ], but for this development we take Mn,i = Mn; itdepends on the sample size n, but not on the block index i.

We then use as an estimated value of the efficient price in thetime period [#n,i , #n,i+1):

X#n,i= 1

Mn

$

tn,j +[#n,i ,#n,i+1)

Ytn,j.

Treating the estimated efficient price as a new data frame, weproceed as in Section 2.3 but with Xt replaced by Xt , n by n# =n/Mn (up to rounding), and tn,i by #n,i . Furthermore, we divideXt values into Kn blocks, with block size L = Ln = [c

'n#]

(except possibly for the first and last block, which does notmatter for the asymptotics), for some constant c. The boundarypoints are on the grid G = {0 < )n,1 < )n,2 < · · · < )n,Kn(1 )T } - H.

4.1 The Case With Microstructure Noise

In the case with microstructure noise, the data blockingmechanism will be similar to that just stated, but less compli-cated where M = Mn = [c1

'n], #n,i = iMn

Tn

, and L = Ln =[ cn1/4

'c1

]. The interval between successive observations is now"t = "tn = tn,j+1 ( tn,j = T/n.

Define

!!X,F (! 2)"T = 3Kn(2$

i=0

"X)n,i+1 ( X)n,i

#"F

"! 2

)n,i+1

#( F

"! 2

)n,i

##,

X#n,i= 1

M

$

tn,j +[#n,i ,#n,i+1)

Ytn,j, (20)

and

! 2)n,i

= 1L * M * "t

$

#n,j+1+()n,i ,)n,i+1]

"X#n,j+1 ( X#n,j

#2.

Note that the factor 2 in the previous proposed estimator inEquation (5) is now changed to 3 instead. This change is dueto the preaveraging method we adopted first to asymptoticallyeliminate the impact of noise on the estimation. The change isconsistent with the adjustment to the realized volatility estimatedby preaveraging; see Jacod et al. (2009).


n1/8( !!X,F (! 2)"T ( !X,F (! 2)"T )

L%Z

,

-c'

c1T

! T

0

"F #"! 2

t

##2! 4

t

%443

f 2t + 32

3g2

t

&dt

+ 16'

c1

c

! T

0

"F #"! 2

t

##2! 6

t dt

+ 96a2

cc3/21 T

! T

0

"F #"! 2

t

##2! 4

t dt

+ 216a4

cc7/21 T 2

! T

0

"F #"! 2

t

##2! 2

t dt

.

/1/2

, (21)


The optimal c and c1 that minimize the asymptotic varianceare derived as follows:

c =

0(C2 + 12AD + C

'C2 + 12AD

9BD, (22)

and

c1 =

0C +

'C2 + 12AD

2A, (23)

where A = 16) T

0 (F #(! 2t ))2! 6

t dt , B = T) T

0 (F #(! 2t ))2! 4

t

( 443 f 2

t + 323 g2

t ) dt , C = 96a2

T

) T

0 (F #(! 2t ))2! 4

t dt , and D = 216a4

T 2) T

0 (F #(! 2t ))2! 2

t dt .In practice, c and c1 can be estimated by minimizing G1

n + G2n

defined in the next section, over a suitable grid of c’s and c1’s.If the grid is nested and becomes dense as n % &, this auto-matically provides a consistent estimator of c and c1.

4.2 Estimation of Asymptotic Variance

Let

G1n = 9

2n

14

Kn(2$

i=0

"X)n,i+1 ( X)n,i

#2"F

"! 2

)n,i+1

#( F

"! 2

)n,i

##2,

and (24)

G2n = 9

2n

14 LnMn"t

Kn(2$

i=0

*! 2)n,i+1

"F

"*! 2

)n,i+1

#( F

"*! 2

)n,i+1

##2.

9See Footnote 4.


Table 2. Because of the much slower convergence rate, the simulation results are not as good as in the case without microstructure noise

MSE Mean Median Q1 Q3

n = 5 days, T = 1/50 infeasible 1.315581 (0.05502 (0.01236 (0.71910 0.66330n = 5 days, T = 1/50 feasible 1.142911 0.02566 (0.02703 (0.79680 0.80940n = 20 days, T = 2/25 infeasible 1.193074 (0.03032 (0.003359 (0.6793 0.6578n = 20 days, T = 2/25 feasible 1.125859 0.02167 (0.05247 (0.77740 0.76390n % &, fixed T (asymptotic value) 1 0 0 (0.674 0.674

NOTE: Even so, with reasonably large sample size, the mean and median are still close to 0. The MSE is not very far from 1, and the quartiles are reasonably close to the theoreticalvalues from the standard normal distribution.

By the same methods as in the proof of Theorem 1, we have thefollowing convergences in probability

G1n

p% 8'

c1

c

! T

0

"F #"! 2

t

##2! 6

t dt

+ c'

c1

! T

0

"F #"! 2

t

##2! 4

t

%283

f 2t + 16

3g2

t

&dt

+ 48a2

cc3/21 T

! T

0

"F #"! 2

t

##2! 4

t dt

+ 108a4

cc7/21 T 2

! T

0

"F #"! 2

t

##2! 2

t dt, (24)

G2n

p% 8'

c1

cT

! T

0

"F #"! 2

t

##2! 6

t dt

+ c'

c1

! T

0

"F #"! 2

t

##2! 4

t

163

"f 2

t + g2t

#dt

+ 48a2

cc3/21 T 2

! T

0

"F #"! 2

t

##2! 4

t dt

+ 108a4

cc7/21 T 3

! T

0

"F #"! 2

t

##2! 2

t dt, (25)

and

G1n + G2

n

p% 16'

c1

cT

! T

0

"F #"! 2

t

##2! 6

t dt

+ c'

c1

! T

0

"F #"! 2

t

##2! 4

t

%443

f 2t + 32

3g2

t

&dt

+ 96a2

cc3/21 T 2

! T

0

"F #"! 2

t

##2! 4

t dt

+ 216a4

cc7/21 T 3

! T

0

"F #"! 2

t

##2! 2

t dt. (26)

Equation (27) gives the estimation of the asymptotic vari-ance. With this estimation, a feasible version of the central limitdistribution can be derived.


n1/8( !!X,F (! 2)"T ( !X,F (! 2)"T )+

G1n + G2

n

L%Z1, (28)

stably in law,10 where Z1 is a standard normal random variableand independent of FT .

10See Footnote 4.

5. SIMULATION FOR THE CASE WITHMICROSTRUCTURE NOISE

Similarly to the case without microstructure noise, the smallsample behavior of the asymptotic normality can be demon-strated by simulating the statistics

n1/8( !!X,F (! 2)"T ( !X,F (! 2)"T )1222223

16'

c1

c

) T

0 (F #(! 2t ))2! 6

t dt + c'

c1T) T

0 (F #(! 2t ))2

! 4t ( 44

3 f 2t + 32

3 g2t ) dt + 96a2

cc3/21 T

) T

0 (F #(! 2t ))2! 4

t dt

+ 216a4

cc7/21 T 2

) T

0 (F #(! 2t ))2! 2

t dt

and

n1/8( !!X,F (! 2)"T ( !X,F (! 2)"T )+

G1n + G2

n

.

The Heston model is once again adopted in the simulation.The parameterization is the same with % = 5, & = 0.04, $ =0.5, ' = (

'0.5. The true log-price process is latent. It is con-

taminated by market microstructure as in Equation (17). Thestandard deviation of noise is set to be a = 0.0005. This is alsoa realistic value in practice. Since the first step of preaveragingconsumes part of the data and reduces the sample size for thesecond step of estimation, the choices of n are bigger than thosein the case without noise. The frequency is chosen as 1 sec,which produces 23, 400 observations in each trading day. Theresults corroborate the theorem and are demonstrated in Table 2.

Even though only the simulations with F (x) = x are pre-sented, the results with other functions satisfying the conditionin definition (2) such as F (x) = log x have been investigated.The results look very similar and the tables are omitted for thereasons of space.

6. IRREGULARLY SPACED DATA

So far our analysis in cases both with and without microstruc-ture noise has been based on measuring prices in regularlyspaced intervals. In some ways it is more natural to work withprices measured in tick time and so it would be desirable to ex-tend the above theory to cover irregularly spaced data. Thisis emphasized by Zhang, Mykland, and Aıt-Sahalia (2005),Barndorff-Nielsen et al. (2008a), and Renault and Werker (2011)in their studies. We here use the framework from Barndorff-Nielsen et al. (2008a).


Assumption 4. The observation times (tn,i) satisfy thecondition

tn,i = G

%iT

n

&=

! i Tn

0G#(s) ds, i = 0, 1, . . . , n,

where G : [0, T ] % [0, T ] is a strictly increasing, twice differ-entiable function with G(0) = 0, G(T ) = T . G#(s) is locallybounded away from 0, and G## is bounded.

With the change of time under the Assumption 4, the stochas-tic volatility model can be written as:

dZt = dX . G(t) = µG(t)G#(t) dt + !G(t)

+G#(t) dWe

t ,

and (29)

dst = d! . G(t) = aG(t)G#(t) dt + fG(t)

+G#(t) dWe

t

+ gG(t)

+G#(t) dBe

t ,

where We and Be are independent Wiener processes.

Proposition 1. The leverage effect in Definition 2 satisfies!X,F (! 2)"T = !Z,F (s2)"T .

Proof:

!Z,F (s2)"T =! T

0F #(s2)d!s2, Z"t

=! T

0F #"! 2

G(t)

#2! 2

G(t)fG(t)G#(t) dt

=! T

0F #"! 2

G(t)

#2! 2

G(t)fG(t) dG(t)

=! T

0F #"! 2

v

#2! 2

v fv dv

= !X,F (! 2)"T .

With all index notation kept the same as in Section 2.3, theestimator of !Z,F (s2)"T can be constructed similar to Equa-tion (5), with one adaptation:

!!Z,F (s2)"T = 2Kn(2$

i=0

(X . G((i + 1)Mn"t) ( X . G(iMn"t))

*"F

"s2

(i+1)Mn"t

#( F

"s2iMn"t

##

and (30)

s2iMn"t = 1

"#n,i+1

$

tn,j+1 +(iMn"t, (i + 1)Mn"t]

* (X . G(tn,j+1) ( X . G(tn,j ))2.

The CLT, estimation of asymptotic variance, and feasibleCLT follow for the estimator !!Z,F (s2"T as in the equidistantcase without microstructure noise. The results for the case withmicrostructure noise can be derived analogously. Consideringthe contaminated process Y . G(t) = X . G(t) + ( . G(t) =Zt + ( . G(t), with all index notation kept the same as in sec-tion 4, the estimator of the leverage effect can be constructed as

follows:

!!Z,F (s2)"T = 3Kn(2$

i=0

"Z)n,i+1 ( Z)n,i

#"F

"s2)n,i+1

#( F

"s)n,i

##,

Z#n,j= 1

M

$

tn,p+1 +(jM"t, (j + 1)M"t]

(Y .G(tn,p+1) ( Y .G(tn,p)),

and

s2)n,i

= 1")n,i+1

$

#n,j+1 +(iLM"t, (i + 1)LM"t]

(Z#n,j+1 ( Z#n,j)2. (30)

We emphasize that this estimator is feasible (observable), sinceboth contaminated price Yt and observation times tn,i and )n,i

are directly observable from the market.The CLT, estimator of asymptotic variance and feasible CLT

can be derived in a similar manner as when observations areregularly spaced.

7. LEVERAGE EFFECT AND SKEWNESS

From sec. 2 of Mykland and Zhang (2009), leverageeffect(F (x) = x) and skewness have a close relationship. Forequidistant data, the skewness of returns in high-frequency datasatisfies (as n % &)

n

Tlim

$

tn,i+1)T

"X3tn,i+1

L%32!! 2, X"T + 3

! T

0! 3

t

"dWt + !(1

t µt dt#

+%

6! T

0! 6

t dt

&1/2

Z,

where Z is a standard normal random variable. This is a biasedand inconsistent estimator, but it is interesting to find that lever-age effect appears on the right-hand side. When the mean isremoved from blocks of size M, this empirical skewness con-verges to the leverage effect plus a mixed normal error:

n

Tlim

$

tn,i+1)T

("Xtn,i+1 ( local mean of X)3

L%32!! 2, X"T +

%M ( 1

M

%6 + 18

M( 15

M2

&! T

0! 6

t dt

&1/2

Z.

M is chosen differently in Mykland and Zhang (2009) from thatin this article. It is a constant instead (i.e., M does not grow withn). This relationship tells us that in the case where skewness ishard to estimate directly, the consistent estimation of leverageeffect proposed by this article provides an alternative way toestimate skewness.

To further emphasize that we are indeed estimating a form ofskewness by the leverage effect, we now consider the predictableinstantaneous skewness:

p-skew := n

T

$

tn,i+1)T

E""X3

tn,i+1|Fti

#.


We obtain

Proposition 2. Subject to regularity conditions, as n % &,

p-skewp%3

2!! 2, X"T .

It should be noted that since E("X3tn,i+1

|Fti ) is an unobserv-able quantity, this proposition does not yield a method of estima-tion. It does, however, clarify the relationship between skewnessand leverage effect.

The existence of a connection between skewness and theleverage effect has previously been noted in Meddahi andRenault (2004); see the discussion following Proposition 3.4(p. 370).

8. LEVERAGE EFFECTS AND REGRESSIONS

The estimation of leverage effects also has an application toestimating the regression coefficient of the volatility on its ownlog return. On one hand, the existence of leverage effect impliesthe relation of volatility and the log return as stated below:

d! 2t = 2ft dXt + 2!t gt dBt +

"2!t at ( 2ftµt + f 2

t + g2t

#dt,

(31)

and

d4X, ! 2

t

5t

d4X,X

5t

= 2ft . (32)

On the other hand, the leverage effect specified as !X, log ! "takes the following form:

2d!X, log ! "dt

= 2ft . (33)

Equations (33) and (34) suggest two ways of applying the esti-mation of leverage effects (F (x) = x and F (x) = 1

2 log(x)) toestimating the regression coefficient of the volatility process onits own log-return process. The second method only involveslower orders of the volatility process, and is thus comparativelyrobust. We will use this method in the next section.

9. EMPIRICAL STUDY

In the empirical study, we employ Microsoft stock tradesdata from the New York Stock Exchange (NYSE TAQ). Theyears under study are 2008 through 2011. Even though thestock is traded between 9:30 am and 4:00 pm, the window9:45 am–3:45 pm is chosen in the empirical analysis. The reasonfor choosing this window is that a vast body of empirical studiesdocuments increased return volatility and trading volume at theopen and close of the stock market (Wood et al. 1985; Chanet al. 2000). A 15-min cushion at the open and close may strikea good balance between avoiding abnormal trading activities inthe market and preserving enough data points to perform theestimation procedures in a consistent way. On average, there arecurrently several hundred thousand trades of Microsoft duringeach trading day. There are frequently multiple trades in eachsecond.

In Section 8, we explored intraday high-frequency regression.It is clear that two forms of the leverage effect reveal the relation

between volatility and return in the regression model. One wayto extrapolate this intraday behavior to between-day volatilityprediction is to include the previous day’s return but scaled by atime-varying leverage effect. Technically, all regressors are nowin the drift term.

Since we are not trying to discover the best model calibrationfor volatility prediction, but rather to investigate the predictivepower of return scaled by leverage effect, the prediction modelis simply a linear regression (or AR(2)). Though this may notbe a very sophisticated model, the results can still improveunderstanding the role of leverage effects in volatility prediction:! ti+1

ti

! 2t dt = *0 + *1

! ti

ti(1

! 2t dt + *2

! ti(1

ti(2

! 2t dt + *3"X2

ti(

+ *4

! ti

ti(1

2ft dt * "Xti + (i .

• The integrated volatility) ti+1

ti! 2

t dt can be estimated byvarious methods. In this empirical study, the preaveragingmethod (Jacod et al. 2009) is adopted.

• "Xti( denotes the overnight log return.•

) titi(1

2ft dt can be estimated by the proposed leverage effectestimator in this article by setting F (x) = 1

2 log(x).• The inclusion of lagged volatilities and overnight returns

is due to the empirical finding of volatility clustering.

In this study, since we do not consider the case with jumpsinvolved, we first remove the days with jump activities by thejump test from Lee and Mykland (2012).11 Alternatively, onecan apply the jump tests as in Aıt-Shalia and Jacod (2009),Barndorff-Nielsen and Shephard (2004), Barndorff-Nielsen andShephard (2006), Mancini (2001), Podolskij and Ziggel (2010),and other works by the same authors.

We also follow the convention by preaveraging the data overevery 5 min in the first step. After preaveraging, we take c = 1.We explore the volatility prediction over a two-day period, alonger period than one day, because of the comparatively slowconvergence rate of the estimators as discussed in Section 5. Tocheck the robustness, we also repeat the regression replacingreturn scaled the leverage effect by return itself. The predictionresults are shown in Table 3, and the time series plot of theestimated leverage effect is given in Figure 1.

“SS explained” is the main indicator to show whether returnscaled by leverage effect has a big contribution to the predictionof leverage effect. For each year, the first column of p-valuedbased on t-test is given only as reference. Of course, since thecovariates are not independent from the response variable, thesestatistics cannot really tell whether the corresponding covariateshould be included in the model. The third column gives thecollinearity diagnostic by variance inflation factors (vif) (see,e.g., Weisberg 2004).

Since almost all vif values are close to 1, one can consider thecovariates not to be collinear with each other. The sum of squaresexplained by the return scaled by the leverage effect (RLE)are substantial even when this term is included into the model

11The total numbers of days removed are 21 for 2008, 27 for 2009, 60 for 2010,and 43 for 2011.


Table 3. Two-day ahead volatility prediction results with Microsoft 2007–2010 data

2008 2009

P(> |t |) SS explained vif P(> |t |) SS explained vif

*0 0.003909 0.035226RVt(1 1.75 · 10(7 7.05 · 10(5* 2.050335 0.000752 1.7633 · 10(5 / // 5.603527RVt(2 0.369974 1.05 · 10(5* 1.461129 0.658205 1.9960 · 10(6 1.349047R2

t( 0.0493 9.198 · 10(6 1.142936 0.570849 7.09 * 10(7 1.062036RLEt(1 0.000454 2.0697 · 10(5/ 1.551306 0.027969 4.827 · 10(6/ 4.855324or Rt(1 0.821 9.0 · 10(8/ 1.003209 0.280060 1.1866 · 10(6 1.198528

2010 2011

P(> |t |) SS explained vif P(> |t |) SS explained vif

*0 0.000706 0.000747RVt(1 0.833737 1.18 · 10(7 1.453209 5.61 · 10(5 2.2341 · 10(6 / // 1.622136RVt(2 0.180635 2.066 · 10(7 1.008225 0.215383 8.6345 * 10(7/ 1.178455R2

t( 4.32 · 10(10 1.4117 · 10(5 / // 1.011785 0.270385 1.7367 · 10(7 1.034753RLEt(1 0.5254 1.172 · 10(7 1.455641 0.000486 2.1891 · 10(6 / // 1.463627or Rt(1 0.870061 7.8 · 10(9 1.068818 0.102010 5.0273 · 10(7 1.049329

RVt denotes the estimated integrated volatility at day t; Rt( denotes overnight return for day t; RLE denotes the log return scaled by leverage effect at day t (estimated leverage effect *log return). Rt(1 denotes the previous period return itself without scaling. “SS explained” denotes the sum of squares gained by adding each covariate in the order presented in the table.

0 20 40 60 80 100 120

-2.0

-1.0

0.0

1.0

tsplot for the leverage effect in 2008

time

the

leve

rage

effe

ct

0 20 40 60 80 100

-3-2

-10

1tsplot for the leverage effect in 2009

time

the

leve

rage

effe

ct

0 20 40 60 80 100

0.0

0.5

1.0


time

the

leve

rage

effe

ct

0 20 40 60 80 100

-0.2

0.0

0.2

0.4


time

the

leve

rage

effe

ct

Figure 1. TS-plot of the estimated leverage effects: The black curves present the time series plots of the estimated leverage effects. The redcurves give the 95% confidence intervals of the estimated values. The values on the vertical axes are different from one year to another. That isdue to the different magnitudes of the estimated leverage effects. Apparently year 2008 and 2009 display the biggest negative leverage effects.This observation coincides with the empirical fact during the financial crisis.


last. Most of these sums of squares are comparable to the sumof squares explained by the previous day’s volatility, which isbelieved to be the most significant factor for volatility prediction(Engle 1982; Bollerslev 1986). In all cases, RLE has strongerpredictive power than the two-period ahead integrated volatilitydoes. This strongly suggests the inclusion of return scaled byleverage effect into any model trying to predict next-period’svolatility. The predictive power of a time-varying leverage effectestimator12 is consistent with the earlier work by Engle and Ng(1993) and Chen and Ghysels (2011), but here appears in a newform. In addition, the previous period return does not contributeto the sum of squares as much as the one scaled by the leverageeffect. In some case, the previous period return is not significantwhile the return scaled by the leverage effect explains significantamount of sum of squares.

10. CONCLUSION

This article provides nonparametric estimators of the leverageeffect, and analyzes them both theoretically and in simulation.The definition of the stochastic parameter of the leverage effectinvolves a twice differentiable monotone function. Even thoughthe reliance of the estimation on higher moments of volatilityis of concern in practice, the carefully chosen function F canhelp to reduce the order of moments required and provide ro-bust results. Other benefits of this function can be easily seenin the discussion of the connection between the leverage ef-fect and skewness. While the sum of intraday cubic returns isnot a consistent estimator of skewness, the p-skewness (Sec-tion 7) can instead be estimated consistently by the leverageeffect estimators proposed by this article. The related propertiesof p-skewness can also be studied by the properties of the lever-age effect. Clever choices of the function F can also reduce thework of estimation, such as the estimation of high-frequency re-gression coefficients. Instead of estimating both leverage effectsand realized volatility, a different form of the leverage effect canserve as the estimated coefficient (Section 8). If the propertiesof the estimated coefficient are of interest, it is more attractiveto apply the method in Equation (34), whose statistical proper-ties have already been studied in this article, than to apply thefirst ratio statistic in Equation (33) whose statistical propertiesrequire further efforts to investigate.

The bias correction factors in the estimators contribute toan important finding in this article and are previously unknown.

12The main motivation for us to include “leverage effect scaled returns” is theearlier empirical findings on asymmetric impact of positive or negative returnson the volatility process. To capture this asymmetric impact in the predictionmodel, we include the extra term. We realize that Jacod (1994, 1996), Barndorff-Nielsen, and Shephard (2005), and others, showed that the correlation (leverageeffect) has no impact on the asymptotic distribution. These observations seemto suggest that asymmetries do not matter for forecasting, but that is not so.The concept of the news impact curve (Engle and Ng 1993) was originallyformulated within the context of daily ARCH-type models. In the models ofEngle and Ng (1993), the returns are included in the volatility prediction modelsby differently scaling the positive or negative returns. Here we applied the intra-day data over small time interval to estimate integrated volatility, but predictthe next volatility over a much longer time interval (two-day ahead). Therefore,the prediction falls into a comparative low frequency setting. The leverageeffect turns out to have a significant impact on the volatility prediction whichis supported by our empirical finding. This inclusion of leverage effect scaledreturns in the volatility prediction model is further supported by the findings inChen and Ghysels (2011), where they are dealing with two different frequenciesand reaching a similar conclusion.

They not only provide the consistency of the estimation, but alsoimply that simple covariance estimators tend to underestimatethe leverage effect, especially when the values of the leverageeffect are close to zero. The amplifying factors play a vital roleof bias correction in the estimation.

The empirical studies demonstrate the importance of theleverage effect in volatility prediction. Even though the sim-ple regression (or AR(2)) model is adopted in the study, theexplanatory power of RLE is surprisingly high. The power isalmost of the same magnitude as the predictive power of the pre-vious period volatility which is widely considered to be the mainsource of variation in volatility prediction. This high explana-tory power suggests that time-varying leverage effects shouldbe included additionally to explain the variation and clusteringin volatility prediction models.

Even though we have provided a way to deal with irregularlyspaced data, it is important to study the estimation of leverageeffects and the asymptotic properties of estimators when time isendogenous (as in Li et al. 2013). Different methods of dealingwith microstructure noise should also be studied and comparedwith the ones in this article. Our findings create the importantnecessary foundation for further analysis both theoretically andempirically, as well as an investigation of how to carry out riskmanagement in the presence of leverage effects.

Finally, as discussed in Section 2.2, many open questionsremain in terms of model specification (continuous vs. jumps,semimartingale vs. long range dependence), with reference tothe papers cited in that section. In particular, the connection tosemivariance remains to be explored.

APPENDIX A: PROOF OF THEOREM 1

A.1 Preliminaries

In the following, by Fj we mean Ftn,j, and p is a positive integer.

Without loss of generality, we will set µt = 0, see Mykland and Zhang(2009), sec. 2.2, as well as our current Assumption 1 and associatedFootnote 1. Recall that

!!X, F (! 2)"T = 2$

i

"X#n,i+1 ( X#n,i

#"F

"! 2

#n,i+1

#( F

"! 2

#n,i

##(A.1)

and

! 2#n,i

= 1Mn * "t

$

tn,j+1+(#n,i ,#n,i+1]

"X2tn,j+1

. (A.2)

Lemma 1. Under Assumption 1,

E""

Xtn,k( Xtn,j

#2!

ptn,m

66Fj

#

= !p+2tn,j

(tn,k ( tn,j ) + p2f 2tn,j

!ptn,j

(tn,k ( tn,j )2

+ p!ptn,j

!X, f "tn,j(tn,k ( tn,j )2

+ 12p(p ( 1)

"f 2

tn,j+ g2

tn,j

#!

ptn,j

(tn,k ( tn,j )(tn,m ( tn,j )

+%

p + 12

& "f 2

tn,j+ g2

tn,j

#!

ptn,j

(tn,k ( tn,j )2

+ atn,j!

p+1tn,j

(tn,k ( tn,j )2

+ patn,j!

p+1tn,j

(tn,k ( tn,j )(tn,m ( tn,j ) + Op("t5/2), (A.3)

E""

Xtn,k( Xtn,j

#!

ptn,m

66Fj

#= p!

ptn,j

ftn,j(tn,k ( tn,j ) + Op("t3/2),

(A.4)


and

E"2"Xtn,j

( Xtn,i

#! 2

tn,j+1

"Xtn,k

( Xtn,j

#! 2

tn,k+1

66Fi

#

= 16! 4tn,i

f 2tn,i

(tn,k ( tn,j+1)(tn,j ( tn,i)

+ 4! 4tn,i

!X, f "tn,i(tn,k ( tn,j+1)(tn,j ( tn,i) + Op("t5/2), (A.5)

where tn,i < tn,j < tn,k ) tn,m, tn,m ( tn,j = Op("t12 ) and "t = T

n.

Proof of Lemma 1:

1. For (A.3), note first that

!ptn,m

( !ptn,j

= p

! tn,m

tn,j

!p(1t d!t + p(p ( 1)

2

! tn,m

tn,j

!p(2t

"f 2

t + g2t

#dt,

and"Xtn,k

( Xtn,j

#2

=! tn,k

tn,j

2"Xt ( Xtn,j

#dXt +

! tn,k

tn,j

! 2t dt

=! tn,k

tn,j

2"Xt ( Xtn,j

#dXt +

! tn,k

tn,j

(tn,k ( t)d"! 2

t ( ! 2tn,j

#

+ ! 2tn,j

(tn,k ( tn,j )

=! tn,k

tn,j

2"Xt ( Xtn,j

#dXt +

! tn,k

tn,j

(tn,k ( t)2!t d!t

+! tn,k

tn,j

(tn,k ( t)"f 2

t + g2t

#dt

+ ! 2tn,j

(tn,k ( tn,j ).

Hence,

E""

Xtn,k( Xtn,j

#2!

ptn,m

66Fj

#

= E

78

!ptn,j

+ p(p ( 1)2

! tn,m

tn,j

!p(2t

"f 2

t + g2t

#dt

9

*8! tn,k

tn,j

2"Xt ( Xtn,j

#dXt + ! 2

tn,j(tn,k ( tn,j )

+! tn,k

tn,j

(tn,k ( t)2!t at dt +! tn,k

tn,j

(tn,k ( t)"f 2

t + g2t

#dt

9

+ p

! tn,k

tn,j

2"Xt ( Xtn,j

#!

p(1t d!X, ! "t

+ p

! tn,k

tn,j

2(tn,k ( t)! pt d!!, ! "t + ! 2

tn,j(tn,k ( tn,j )

* p

! tn,m

tn,j

!p(1t at dt |Fj

:

+ Op("t5/2)

= E

%!

p+2tn,j

(tn,k ( tn,j ) + !p+1tn,j

atn,j(tn,k ( tn,j )2

+ 12

"f 2

tn,j+ g2

tn,j

#!

ptn,j

(tn,k ( tn,j )2

+ p(p ( 1)2

"f 2

tn,j+ g2

tn,j

#!

ptn,j


+ pftn,j

! tn,k

tn,j

2"Xt ( Xtn,j

#!

pt dt

+ p!ptn,j

! tn,m

tn,j

2"Xt ( Xtn,j

#"ft ( ftn,j

#dt

+ p

! tn,k

tn,j

2(tn,k ( t)! pt

"f 2

t + g2t

#dt

+ p!p+1tn,j

atn,j(tn,k ( tn,j )(tn,m ( tn,j )|Fj

&+ Op("t5/2)

= E

,

;-!p+2tn,j

(tn,k ( tn,j ) + !p+1tn,j

atn,j(tn,k ( tn,j )2

+%

12

+ p

& "f 2

tn,j+ g2

tn,j

#!

ptn,j

(tn,k ( tn,j )2

+ p(p ( 1)2

"f 2

tn,j+ g2

tn,j

#!

ptn,j


+ p2ftn,j

! tn,k

tn,j

(tn,k ( t)2!pt ft dt

+ p!ptn,j

!X, f "tn,j(tn,k ( tn,j )2 + p!

p+1tn,j

atn,j(tn,k ( tn,j )

*(tn,m ( tn,j )|Fj

.

</ + Op("t5/2)

= !p+2tn,j

(tn,k ( tn,j ) + atn,j!

p+1tn,j

(tn,k ( tn,j )2

+%

p + 12

& "f 2

tn,j+ g2

tn,j

#!

ptn,j

(tn,k ( tn,j )2

+ p(p ( 1)2

"f 2

tn,j+ g2

tn,j

#!

ptn,j


+ p2f 2tn,j

!ptn,j

(tn,k ( tn,j )2

+ p!ptn,j

!X, f "tn,j(tn,k ( tn,j )2

+ patn,j!

p+1tn,j

(tn,k ( tn,j )(tn,m ( tn,j ) + Op("t5/2).

2. For (A.4):

E"(Xtn,k

( Xtn,j)! p

tn,m

66Fj

#

= E

7

p

! tn,k

tn,j

!p(1t d!X, ! "t |Fj

:

+ Op("t3/2)

= E

7

p

! tn,k

tn,j

!pt ft dt |Fj

:

+ Op("t3/2)

= p!ptn,j

ftn,j(tn,k ( tn,j ) + Op("t3/2).

3. (A.5) is the direct consequence of (A.3) and (A.4). This com-pletes the proof.

Later in the derivation of limit theorem, it is easy to see that the proofand calculations strongly depend on Lemma 1, which will be appliedrecursively.

A.2 Main Martingale Representation and Argumentto Prove Theorem

The proof of Theorem 1 is provided in this section, with supportingdetails in the following sections. Construct an approximate martingale(MG) on the grid of the #n,i’s as follows:

1'Mn"t

!!X,F (! 2)")t

= 2'

Mn"t

=>

?

Kn(2$

i=0,#n,i+1)t

"X#n,i+1 ( X#n,i

#"F

"! 2

#n,i+1

#( F

"! 2

#n,i

##@A

B

= 2'Mn"t

=>

?

Kn(2$

i=0,#n,i+1)t

"X#n,i+1 ( X#n,i

#"F

"! 2

#n,i+1

#( F

"! 2

#n,i+1

#

+F"! 2

#n,i+1

#( F

"! 2

#n,i

#+ F

"! 2

#n,i

#( F

"! 2

#n,i

##@CA

CB


= 2'

Mn"t

=>

?

Kn(2$

i=0,#n,i+1)t

"X#n,i+1 ( X#n,i

#"F

"! 2

#n,i+1

#( F

"! 2

#n,i

##

+Kn(2$

i=0,#n,i+1)t

"X#n,i+1 ( X#n,i

#"F

"! 2

#n,i+1

#( F

"! 2

#n,i+1

##

(Kn(2$

i=0,#n,i+1)t

"X#n,i+1 ( X#n,i

#"F

"! 2

#n,i

#( F

"! 2

#n,i

#@A

B

= 2'Mn"t

=>

?

Kn(2$

i=0,#n,i+1)t

"X#n,i+1 ( X#n,i

#"F

"! 2

#n,i+1

#( F

"! 2

#n,i

##

+Kn(1$

i=1,#n,i+1)t

"X#n,i

( X#n,i(1

#"F

"! 2

#n,i

#( F

"! 2

#n,i

##

(Kn(2$

i=0,#n,i+1)t

"X#n,i+1 ( X#n,i

#"F

"! 2

#n,i

#( F

"! 2

#n,i

##@A

B

= 2'

Mn"t

=CCCCC>

CCCCC?

Kn(2$

i=0,#n,i+1)t

"X#n,i+1 ( X#n,i

#"! 2

#n,i+1( ! 2

#n,i

#F #"! 2

#n,i

#

D EF G(1)

+Kn(1$

i=1,#n,i+1)t

"X#n,i

( X#n,i(1

#"! 2

#n,i( ! 2

#n,i

#F #"! 2

#n,i

#

D EF G(2)

(Kn(2$

i=0,#n,i+1)t

"X#n,i+1 ( X#n,i

#"! 2

#n,i( ! 2

#n,i

#F #"! 2

#n,i

#

D EF G(3)

@CCCCCA

CCCCCB

+ op(1).

It follows that

1'Mn"t

( !!X,F (! 2)")t ( !X,F (! 2)")t ) =$

i=0,#n,i+1)t

"V n#n,i+1

+ op(1),

where, except for a few terms at the edge,

"V n#n,i+1

= 2'Mn"t

=CC>

CC?

"X#n,i+1 ( X#n,i

#"! 2

#n,i+1( ! 2

#n,i

#F #"! 2

#n,i

#

D EF G(1)

+"X#n,i

( X#n,i(1

#"! 2

#n,i( ! 2

#n,i

#F #"! 2

#n,i

#

D EF G(2)

("X#n,i+1 ( X#n,i

#"! 2

#n,i( ! 2

#n,i

#F #"! 2

#n,i

#

D EF G(3)

(! #i+1

#i

F #"! 2t

#! 2

t ft dt

D EF G(4)

.

And so the martingale increment becomes

"Mn#i+1

= "V n#n,i+1

( E""V n

#n,i+1|Fi

#. (A.6)

The martingale up to time t is

Mnt =

$

#n,i+1)t

H"V n

#n,i+1( E

""V n

#n,i+1

66Fi

#I. (A.7)

Although Mn above is observed in discrete time, we can interpolatethe martingale into a continuous martingale up to any time t; see, in

particular, Heath (1977) as well as Mykland (1995) and the referencestherein. This interpolation is closely related to Skorokhod embeddingin Brownian motion; see, for example, Appendix I of Hall and Heyde(1980). Then we only need to prove the CLT for the interpolated con-tinuous martingale. Therefore, we can apply Theorem 2.28 (p. 152) inMykland and Zhang (2012) to prove the CLT. The conditions of citedtheorem will follow from the development in the rest of Appendix 10,in particular (A.16) and (A.19), and the approximation in Lemma 2.Alternatively, one can develop a functional argument for the CLT asin Jacad (2009), Jacod and Shiryaev (2003), Jacod and Protter (2011),and Podolskij and Vetter (2009b).

A.3 The Aggregate Conditional Variance

To calculate the quadratic variation, we will calculate the aggregateconditional variance of Mn

t

$

#i+1)t

var""Mn

#i+1

66Fi

#

=$

#i+1)t

E""

"Mn#i+1

#266Fi

#

=$

#i+1)t

HE

"""V n

#n,i+1

#266Fi) ("E

""V n

#n,i+1

66Fi

##2I. (A.8)

We will first prove that(

#i+1)t (E("V n#n,i+1

|Fi))2 is negligible, of orderOp(M2

n"t2). Except a few terms at the edge, the following could beestablished:

E""V n

#n,i+1

66Fi

#

= 2'Mn"t

E""

X#n,i+1 ( X#n,i

#"! 2

#n,i+1( ! 2

#n,i

#F #"! 2

#i

#

+"X#n,i

( X#n,i(1

#"! 2

#n,i( ! 2

#n,i

#F #"! 2

#i

#

("X#n,i+1 ( X#n,i

#"! 2

#n,i( ! 2

#n,i

#F #"! 2

#i

#66Fi

#

=2F #"! 2

#n,i

#'

Mn"tE

% ! #i+1

#i

! 2t ft dt (

"X#n,i+1 ( X#n,i

#"! 2

#n,i( ! 2

#n,i

#

+ (X#n,i( X#n,i(1 )

"! 2

#n,i( ! 2

#n,i

#|Fi

&+ Op(Mn"t)

=2F #"! 2

#n,i

#'

Mn"tE

%! #i+1

#i

"! 2

t ft ( ! 2#n,i

f#n,i

#dt

(M$

j=1

! tn,j

#n,i

2"! 2

t ft ( ! 2#n,i

f#n,i

#dt |Fi

.

/

+2F #"! 2

#n,i

#'

Mn"t

"X#n,i

( X#n,i(1

#E

7! #n,i+1

#n,i

! 2t ( ! 2

#n,idt |Fi

:

+ Op(Mn"t). (A.9)

We derive that

E

,

-$

#n,i+1)t

"E

""V n

#n,i+1

66Fi

##2

.

/

)12F #"! 2

#i

#2

Mn"t

$

#i+1)t

E

J%E

! #i+1

#i

"! 2

t ft ( ! 2#n,i

f#n,i

#dt

66Fi

#&2

+

,

-M$

j=1

E

7! tn,j

#n,i

2"! 2

t ft ( ! 2#n,i

f#n,i

#dt

66Fi

:.

/2

+"X#n,i

( X#n,i(1

#2

7

E

7! #n,i+1

#n,i

! 2t ( ! 2

#n,idt

66Fi

::2@A

B

+ Op((Mn"t)2)) K(Mn"t)2. (A.10)


With this result, we can calculate the conditional variance by onlyconsidering conditional second moments instead.

A.4 Aggregate Conditional Second Moment

$

#n,i+1)t

E""

"V n#i+1

#266Fi

#

= 4Mn"t

$

#i+1)t

E

,

;-

K

LM"X#n,i+1 ( X#n,i

#"! 2

#n,i+1( ! 2

#n,i

#F #"! 2

#i

#

+"X#n,i

( X#n,i(1

#"! 2

#n,i( ! 2

#n,i

#F #"! 2

#i

#

("X#n,i+1 ( X#n,i

#"! 2

#n,i( ! 2

#n,i

#F #"! 2

#i

#

(! #i+1

#i

F #"! 2t

#! 2

t ft dt

N

OP

26666Fi

.

</

=4F #"! 2

#i

#2

Mn"t

$

#i+1)t

E((X#n,i+1 ( X#n,i)2"! 2

#n,i+1( ! 2

#n,i

#2

+"X#n,i

( X#n,i(1

#2"! 2

#n,i( ! 2

#n,i

#2

+"X#n,i+1 ( X#n,i

#2"! 2

#n,i( ! 2

#n,i

#2

( 2"X#n,i+1 ( X#n,i

#2"! 2

#n,i+1( ! 2

#n,i

#"! 2

#n,i( ! 2

#n,i

#

+%! #i+1

#i

! 2t ft dt

&2

( 2(X#n,i+1 ( X#n,i)"! 2

#n,i+1( ! 2

#n,i

#

*! #i+1

#i

! 2t ft dt |Fi

&+ Op(Mn"t). (A.11)

Applying Lemma (1), we can prove the following results

1Mn"t

$

#n,i+1)t

E""

X#n,i( X#n,i(1

#2"! 2

#n,i( ! 2

#n,i

#266Fi

#

= 2M2

n"t

$

#i+1)t

! 6#n,i

Mn"t + 43

$

#i+1)t

! 4#n,i

"f 2

#n,i+ g2

#n,i

#Mn"t

+ Op(Mn"t), (A.12)

1Mn"t

$

#n,i+1)t

E

7%! #i+1

#i

! 2t ft dt

&2

( 2(X#n,i+1 ( X#n,i)

*"! 2

#n,i+1( ! 2

#n,i

# ! #i+1

#i

! 2t ft dt |Fi

&

= ($

#i+1)t

! 4#n,i

f 2#n,i

Mn"t + Op(Mn"t), (A.13)

1Mn"t

$

#n,i+1)t

E""

X#n,i+1 ( X#n,i

#2"! 2

#n,i+1( ! 2

#n,i

#2

( 2"X#n,i+1 ( X#n,i

#2"! 2

#n,i+1( ! 2

#n,i

#"! 2

#n,i( ! 2

#n,i

#66Fi

#

= Op(Mn"t), (A.14)

and

1Mn"t

$

#n,i+1)t

E""

X#n,i+1 ( X#n,i

#2"! 2

#n,i( ! 2

#n,i

#266Fi

#

=$

#i+1)t

2M2

n"t! 6

#n,iMn"t + 4

3! 4

#n,i

"f 2

#n,i+ g2

#n,i

#Mn"t

+ 2! 4#n,i

f 2#n,i

Mn"t + Op(Mn"t), (A.15)

The above three equations lead to the following convergence:

$

#i+1)t

var""Mn

#i+1

66Fi

# p% 16c2t

! t

0

"F #"! 2

s

##2! 6

s ds

+! t

0

"F #"! 2

s

##2! 4

s

%443

f 2s + 32

3g2

s

&ds.

(A.16)

To finalize the proof of Theorem 1, we need the quadratic varia-tion of the interpolated martingale. The following lemma shows thatthe asymptotics of the quadratic variation is the same as that of theaggregate conditional variance as in Equation (A.16)

Lemma 2.

[Mn, Mn]t ($

#n,i)t

var""Mn

#n,i

66Fi

#= op(1) (A.17)

Proof of Lemma 2:

E

,

-[Mn,Mn]t ($

#n,i)t

var""Mn

#n,i

66Fi

#.

/2

= E

,

-$

#n,i)t

""Mn

#n,i

#2 ($

#n,i)t

var""Mn

#n,i

66Fi

#.

/2

=$

#n,i)t

E""

"Mn#n,i

#2 ( EQ"

"Mn#n,i

#266Fi

R#2

(by MG property, cross product term has expectation 0)

)$

#n,i)t

E""Mn

#n,i

#4 + E"E

Q("Mn

#n,i)2

66Fi

R#2

)$

#n,i)t

E""Mn

#n,i

#4(by Jensen’s inequality for

conditional expectation)

) E"

sup""Mn

#n,i

#2 *QMn, Mn

Rt

#

% 0, (A.18)

since !#i, f#i

, g#i, [Mn, Mn]t are assumed to be bounded, Mn"t =

Op(n(1/2).

A.5 Elimination of the Bias Term

The bias in the limit of !!X,F (! 2)"T also depends on the limit of[Mn, W (i)]t , where either W

(i)t = Wt or W (i) is orthogonal to Wt , for

any t + (0, T ]. For the second case, it is obvious that [Mn, W (i)]t = 0.We only need to study the first case W

(i)t = Wt . In this case, since the

covariance between Op(Mn"t) terms and Wt will be of even higherorder (at least Op((Mn"t)3/2)), those are negligible in the limit. Thus weonly need to consider the following aggregate conditional expectation:

1'Mn"t

$

#i+1)t

cov""Mn

#n,i+1, "W#n,i+1

66Fi

#

= 1'Mn"t

F #"! 2#n,i

#

*$

#i+1)t

E

7$

i

"W#n,i+1

"X#n,i+1 ( X#n,i

#"! 2

#n,i+1( ! 2

#n,i

#

+ "W#n,i+1

"X#n,i

( X#n,i(1

#"! 2

#n,i( ! 2

#n,i

#

( "W#n,i+1

"X#n,i+1 ( X#n,i

#"! 2

#n,i( ! 2

#n,i

#

+ Op((Mn"t)3/2)= Op((Mn"t)3/2) (By Ito’s formula and Lemma 1). (A.19)

Thus, [Mn,W ]t = Op((Mn"t)3/2) for any t + (0, T ].


All the proofs above can be easily extended to the case for anyt + (0, T ]. Thus, as outlined at the end of Section A.2, with (A.3),(A.16), (A.18), and (A.19), the proof of Theorem 1 is completed byapplying the Central Limit Theorem for semimartingales (refer to theversion in Mykland and Zhang 2012, thm 2.28).

APPENDIX B: PROOF OF THEOREM 2

With Taylor expansion, the first-order difference of thetwo estimators is:

(#i+1)t

2Mn(Mn(1)"t

F #(!#n,i)("X#n,i+1 )3 and

(#i+1)t

2Mn(Mn(1)"t

F #(!#n,i)"X#n,i+1 ("X#n,i+2 )2. In order for any term

to make a difference in the limit, the term must be of order lower thanOp(Mn"t). However, the difference terms are of order higher thanOp(Mn"t) by BDG inequality as shown in the proof of Proposition 2at the end of the article:

$

#i+1)t

2Mn(Mn ( 1)"t

F #"!#n,i

#E

"""X#n,i+1

#366Fi

#= Op

"n( 1

2#,

$

#i+1)t

2Mn(Mn ( 1)"t

F #"!#n,i

#"X#n,i+1

""X#n,i+2

#2 = Op

"n( 1

2#.

Theorem 2 is easily proved with this result and a slight variation of theproof of Theorem 1.

APPENDIX C: PROOF OF THEOREM 3

As we saw in the proof of Theorem 1 that

$

i

1Mn"t

E""

X#n,i+1 ( X#n,i

#2"! 2

#n,i+1( ! 2

#n,i

#266Fi

#

p% 4c2T

! T

0

"F #"! 2

t

##2! 6

t dt +! T

0

"F #"! 2

t

##2! 4

t

%143

f 2t + 8

3g2

t

&dt.

To prove the consistency of Equation (11), we can again apply themartingale convergence argument. According to Lemma 2, it sufficesto check wether the conditional variance of the martingale convergesto 0. By applying Lemma 1, we can obtain:

$

i

E

%%1

Mn"t

"X#n,i+1 ( X#n,i

#2"! 2

#n,i+1( ! 2

#n,i

#2

( 1Mn"t

E""

X#n,i+1 ( X#n,i

#2"! 2

#n,i+1( ! 2

#n,i

#266Fi

##26666Fi

&

=$

i

E

%1

(Mn"t)2

"X#n,i+1 ( X#n,i

#4"! 2

#n,i+1( ! 2

#n,i

#46666Fi

&

(E

%1

Mn"t

"X#n,i+1 ( X#n,i

#2"! 2

#n,i+1( ! 2

#n,i

#26666Fi

&2

= Op

"M2

n"t2#. (C.20)

This proves G1n

p% 8c

) T

0 (F #(! 2t ))2! 6

t dt + cT) T

0 (F #(! 2t ))2! 4

t

( 283 f 2

t + 163 g2

t ) dt.

By similar argument, one can prove G2n

p% 8c

) T

0 (F #(! 2t ))2! 6

t dt +cT

) T

0 (F #(! 2t ))2! 4

t163 (f 2

t + g2t ) dt .

These two convergences, together with Theorem 1, prove the stableconvergence in Theorem 3 .

APPENDIX D: PROOF OF THEOREM 4

The proof of Theorem 4 will be done similarly to that of Theorem1 in a manner to compare the difference between (Xtn,j+1 ( Xtn,j

) and(Xtn,j+1 ( Xtn,j

) in each step. As seen in the case without microstructurenoise, it is enough to prove the case F (x) = x. For the convenience

of later calculations, we always assume #n,j+1 + ()n,i , )n,i+1], # #n,j+1 +

()n,i+1, )n,i+2] and #n,j+1 and # #n,j+1 are corresponding (j + 1)th obser-

vation time in the consecutive two big )-blocks.

D.1 Aggregate Conditional Expectation of the Estimator

According to the contiguity to Gaussian noise (Mykland and Zhang2011a), the process can be simplified as follows:

X#n,i= 1

M

$

tn,j +[#i ,#i+1)

Ytn,j= 1

M

$

tn,j +[#i ,#i+1)

Xtn,j+ M(1/2Z#n,i

= X#n,i+ M(1/2Z#n,i

,

! 2)n,i+1

= 1LM"t

$

#n,j+1+()n,i ,)n,i+1]

"X#n,j+1 ( X#n,j

#2

= 1LM"t

$

#n,j+1+()n,i ,)n,i+1]

""X#n,j+1 + "Z#n,j+1M

(1/2#2,

(D.21)

and

!!X, ! 2"T = 3Kn(2$

i=0

"X)n,i+1 ( X)n,i

#"! 2

)n,i+1( ! 2

)n,i

#

= 3LM"t

Kn(2$

i=0

""X)n,i+1 + "Z)n,i+1M

(1/2#

*

,

-$

# #n,j+1

("X# #n,j+1

+ "Z# #n,j+1

M(1/2)2

($

#n,j+1

""X#n,j+1 + "Z#n,j+1M

(1/2#2

.

/ ,

where the Z#n,i’s are iid N (0, a2). This iid normality leads to the fol-

lowing aggregate conditional expectation of the estimator:

E

,

- 3LM"t

Kn(2$

i=0

"X)n,i+1

,

-$

# #n,j+1+()n,i+1,)n,i+2]

"X2# #n,j+1

($

#n,j+1+()n,i ,)n,i+1]

"X2#n,j+1

.

/

6666666Fi

.

</ + Op(LM"t)

= 3LM"t

Kn(2$

i=0

E

,

;;;;;;;;;;;;;-

1M3

$

# #n,j+1 + ()n,i+1, )n,i+2]

tn,l + [)n,i , #n,1)t #n,l + [)n,i+1, #

#n,1)

tn,k + [# #n,j , #

#n,j+1)

t #n,k + [# #n,j+1, #

#n,j+2)

*"Xt #n,l

( Xtn,l

#"Xt #n,k

( Xtn,k

#2

( 1M3

$

tn,l + [)n,i , #n,1)t #n,l + [)n,i+1, #

#n,1)

"Xt #n,l

( Xtn,l

#


*

,

;;;;;;-

$

#n,j+1 + ()n,i , )n,i+1]tn,k + [#n,j , #n,j+1)

t #n,k + [#n,j+1, #n,j+2)

"Xt #n,k

( Xtn,k

#

.

<<<<<</

2

666666666666666666

Fi

.

<<<<<<<<<<<</

+ Op(LM"t)

=Kn(2$

i=0

2! 2)n,i

f)n,iLM"t + Op(LM"t). (D.22)

The last step applies Lemma 1 (A.3) and (A.4), and we conclude that:

!!X, ! 2"T

p%! T

02! 2

t ftdt = !X, ! 2"T . (D.23)

There is another way to look at "X#n,j+1 , which leads to a continuousapproximation and provides a simpler way to prove the results.

X#n,j+1 ( X#n,j

= X#n,j+1 ( X#n,j+

M$

k = 1t #n,k + [#n,j+1, #n,j+2)

%M ( k

M

& "Xt #n,k+1

( Xt #n,k

#

(M$

k = 1tn,k + [#n,j , #n,j+1)

%M ( k

M

& "Xtn,k+1 ( Xtn,k

#

0 "X#n,j+1 + 1M"t

! #n,j+2

#n,j+1

(#n,j+2 ( t) dXt

( 1M"t

! #n,j+1

#n,j

(#n,j+1 ( t) dXt , (D.24)

and

X)n,i+1 ( X)n,i

= X)n,i+1 ( X)n,i+

M$

k = 1t #n,k + [)n,i+1, #

#n,1)

%M ( k

M

& "Xt #n,k+1

( Xt #n,k

#

(M$

k = 1tn,k + [)n,i , #n,1)

%M ( k

M

& "Xtn,k+1 ( Xtn,k

#

0 X)n,i+1 ( X)n,i+ 1

M"t

! # #n,1

)n,i+1

(# #n,1 ( t) dXt

( 1M"t

! #n,1

)n,i

(#n,1 ( t) dXt . (D.25)

One can verify that, to the relevant order, the aggregate conditionalexpectation by this continuous approximation gives the same result butless complicated calculations.

D.2 Conditional Variance of the Approximate Martingale

Similarly as in the proof of Theorem 1,the martingale is constructedas follows:

Up to time t,

Mnt =

$

)n,i+1)t

H"*V n

)n,i+1( E

""*V n

)n,i+1

66Fi

#I, (D.26)

where, except a few terms at the edge,

"*V n)n,i+1

= 3'

LM"t

=C>

C?

"*X)n,i+1 ( *X)n,i

#"! 2

)n,i+1( ! 2

)n,i

#

+"*X)n,i

( *X)n,i(1

#,

- 1LM"t

$

#i+()n,i ,)n,i+1]

"X2#i

( ! 2)n,i

.

/

("*X)n,i+1 ( *X)n,i

#,

- 1LM"t

$

#i+()n,i ,)n,i+1]

"X2#i

( ! 2)n,i

.

/

(! )i+1

)i

23! 2

t ft dt

+"*X)n,i+1 ( *X)n,i

# 1LM"t

,

-$

#i+()n,i+1,)n,i+2]

"Z2#iM(1

($

#i+()n,i ,)n,i+1]

"Z2#iM(1

.

/

+ (*X)n,i(*X)n,i(1 )

,

- 1LM"t

$

#i+()n,i ,)n,i+1]

"X#i"Z#i

M(1/2

.

/

( (*X)n,i+1 ( *X)n,i)

*

,

- 1LM"t

$

#i+()n,i ,)n,i+1]

"X#i"Z#i

M(1/2

.

/

@CA

CB.

Again, we will interpolate this martingale into a continuous martin-gale up to any time t + (0, T ] as in the proof of Theorem 1.

From this point on, the continuous approximation of X with inte-gral will be adopted for simplicity and transparency. However, all thecalculation can be checked by applying Lemma 1 to the direct proofwithout continuous approximation.

Lemma 3.

E

,

;-

71

M"t

! #n,j+1

#n,j

(#n,j+1 ( t) dXt

:2

! p#n,m

6666666Fj

.

</

= 13! p+2

#n,j(#n,j+1 ( #n,j ) +

Sp(p ( 1)

6(#n,m ( #n,j )(#n,j+1 ( #n,j )

+ p2

4! p

#n,jf 2

#n,j(#n,j+1 ( #n,j )2

+ 2p + 112

(#n,j+1 ( #n,j )2

T! p

#n,j

"f#n,j

+ g2#n,j

#

+ p

4! p

#n,j!X, f "#n,j

(#n,j+1 ( #n,j )2 + Op

""t

32#, (D.27)

E

,

;-"X#n,j

( X#n,k

# 1M"t

! #n,k+1

#n,k

(#n,k+1 ( t) dXt !p#n,m

6666666Fk

.

</

= 12! p+2

#n,k(#n,k+1 ( #n,k) + ! p

#n,k

%p2

2f 2

#n,k+ p

2!X, f "#n,k

&

*Q(#n,k+1 ( #n,k)2 + (#n,k+1 ( #n,k)(#n,j ( #n,k+1)

R

+ p(p ( 1)4

! p#n,k

"f 2

#n,k+ g2

#n,k

#(#n,k+1 ( #n,k)(#n,m ( #n,k)

+ 2p + 16

! p#n,k

"f 2

#n,k+ g2

#n,k

#(#n,k+1 ( #n,k)2 + Op

""t

32#,

(D.28)


E

,

;-"X#n,j

( X#n,k

# 1M"t

! #n,j+1

#n,j

(#n,j+1 ( t) dXt !p#n,m

6666666Fk

.

</

= ! p#n,k

%p2

2f 2

#n,k+ p

2!X, f "#n,k

&(#n,j+1 ( #n,j )(#n,j ( #n,k)

+ Op

""t

32#, (D.29)

and

E

71

M"t

! #n,k+1

#n,k

(#n,k+1 ( t) dXt

* 1M"t

! #n,j+1

#n,j

(#n,j+1 ( t) dXt !p#n,m

6666666Fk

.

</

= ! p#n,k

%p2

4f 2

#n,k+ p

4!X, f "#n,k

&(#n,j+1 ( #n,j )(#n,k+1 ( #n,k)

+ Op

""t

32#, (D.30)

where #n,k < #n,j ) #n,m, Fk = Fn,k .

The proof of this Lemma is similar to that of Lemma 1.It is easy to see that the aggregate conditional expectation term in

Mnt is of order Op(LM"t) from (D.22). So the aggregate conditional

variance will be the same as the second moment of the term before theaggregate conditional expectation.

$

)n,i+1)t

var""Mn

)n,i+1

66Fi

#

= 9LM"t

$

)n,i+1)t

E

,

;-"*X)n,i+1 ( *X)n,i

#2"! 2

)n,i+1( ! 2

)n,i

#2

+"X)n,i

( X)n,i(1

#2"! 2

)n,i( ! 2

)n,i

#2

+"X)n,i+1 ( X)n,i

#2"! 2

)n,i( ! 2

)n,i

#2

( 2"X)n,i+1 ( X)n,i

#2"! 2

)n,i( ! 2

)n,i

#"! 2

)n,i( ! 2

)n,i

#

+%! )i+1

)i

23! 2

t ft dt

&2

( 43

%! )i+1

)i

! 2t ft dt

&

*"*X)n,i+1 ( *X)n,i

#"! 2

)n,i+1( ! 2

)n,i

#

+"*X)n,i+1 ( *X)n,i

#2 1L2M2"t2

,

-$

#i+()n,i+1,)n,i+2]

"Z2#iM(1

($

#i+()n,i ,)n,i+1]

"Z2#iM(1

.

/2

+"*X)n,i

( *X)n,i(1

#2

,

- 1LM"t

$

#i+()n,i ,)n,i+1]

"X#i"Z#i

M(1/2

.

/2

+ (*X)n,i+1 (*X)n,i)2

,

- 1LM"t

$

#i+()n,i ,)n,i+1]

"X#i"Z#i

M(1/2

.

/26666666Fi

.

</

+ Op(LM"t).

We will separately calculate the conditional variance involving themicrostructure noise "ZM(1/2 and the variance involving only thesemimartingale process, denoted accordingly as var("Mn,noise

)n,i+1|Fi) and

var("Mn,process)n,i+1

|Fi). Applying Lemma 1 and Lemma 3, we can derive:

$var

""Mn, noise

)n,i+1

66Fi

#

= 9L3M3"t3

$

)n,i+1)t

EU"*X)n,i+1 ( *X)n,i

#2

*

,

-$

#i+()n,i+1,)n,i+2]

"Z2#iM(1 (

$

#i+()n,i ,)n,i+1]

"Z2#iM(1

.

/2

+"*X)n,i

( *X)n,i(1

#2

,

-$

#i+()n,i ,)n,i+1]

"X#i"Z#i

M(1/2

.

/2

+"*X)n,i+1 ( *X)n,i

#2

,

-$

#i+()n,i ,)n,i+1]

"X#i"Z#i

M(1/2

.

/26666666Fi

.

</

+ Op(LM"t)

=$

)n,i)t

96a2

L3M4"t3! 4

)n,i(LM"t)2

+ 324a4

L2M5"t3

$

)n,i+1)t

E""X2

)n,i+1

66Fi

#+ Op(LM"t)

p% 96a2

c2c21T

2

! T

0! 4

t dt + 216a4

c2c41T

3

! T

0! 2

t dt, (D.31)

and

$var

""M

n, process)n,i+1

|Fi

#

= 9LM"t

$

)n,i+1)t

E

,

;-"*X)n,i+1 ( *X)n,i

#2"! 2

)n,i+1( ! 2

)n,i

#2

+"X)n,i

( X)n,i(1

#2"! 2

)n,i( ! 2

)n,i

#2

+"X)n,i+1 ( X)n,i

#2"! 2

)n,i( ! 2

)n,i

#2

( 2"X)n,i+1 ( X)n,i

#2"! 2

)n,i( ! 2

)n,i

#"! 2

)n,i( ! 2

)n,i

#

+%! )i+1

)i

23! 2

t ft dt

&2

( 43

%! )i+1

)i

! 2t ft dt

& "*X)n,i+1 ( *X)n,i

#

*"! 2

)n,i+1( ! 2

)n,i

#|Fi

.

</ + Op(LM"t)

=$

)n,i)t

%16

L2M"t! 6

)n,iLM"t + 44

3! 4

)n,if 2

)n,iLM"t

+ 323

! 4)n,i

g2)n,i

LM"t

&+ Op(LM"t). (D.32)

These results prove that, for any t + (0, T ]:

[M,M]tp% 16

c2t

! t

0! 6

s ds +! t

0! 4

s (443

f 2s + 32

3g2

s ) ds

+ 96a2

c2c21t

2

! t

0! 4

s ds + 216a4

c2c41t

3

! t

0! 2

s ds. (D.33)

By similar methods, we can prove that the bias term also convergesto zero (here we omit the lengthy proof):

1'LM"t

$

i

Cov""Mn

)n,i+1, "W)n,i+1

66Fi

#= Op((LM"t)3/2).

(D.34)


Thus, [Mn, W ]t = Op((LM"t)3/2) for any t + (0, T ]. This completesthe proof of Theorem 4.

APPENDIX E: PROOF OF THEOREM 5

Theorem 5 can be easily proved by the consistency of asymptoticvariance (27) and stable convergence in Theorem 4. We only needto establish the convergence in probability of Equation (27). Thisconvergence can be proved by adopting the similar martingaletechnique as in the proof of Theorme 3, and by applying Lemma 1,Lemma 2, and Lemma 3. We can prove: E( 81

4L2M2"t2

(i(X)n,i+1 (

X)n,i)4(F (! 2

)n,i+1) ( F (! 2

)n,i))4|Fi)((E( 9

2LM"t

(i(X)n,i+1 (X)n,i

)2(F(! 2

)n,i+1) ( F (! 2

)n,i))2|Fi))2 = Op(LM"t). This equation gives the

consistency of Gn,1 in Equation (27). By similar argument, theconsistency of Gn,2 can be proved and consequently completing theproof of the convergence of Equation (27) and Theorem 5.

APPENDIX F: PROOF OF PROPOSITION 2

We will assume the equivalent measure where both Xt and !t aremartingales. This can be done in analogy with the development inMykland and Zhang (2009), sec. 2.2, by a stopping argument and solong as the instantaneous correlation between Xt and !t is not ±1. (Inthe latter case, the proof is different but straightforward.)

By the third Bartlett identity (Mykland 1994)

E""X3

tn,i+1

66Fti

#= 3E

""Xtn,i+1"!X,X"|Fti

#

= 3E

7

"Xtn,i+1

! tn,i+1

tn,i

! 2u du|Fti

:

= 3E

7

"Xtn,i+1

! tn,i+1

tn,i

"! 2

u ( ! 2tn,i

#du|Fti

:

(by MG property of Xt ). (F.35)

By the Ito formula, d(tn,i+1 ( t)(! 2u ( ! 2

tn,i) = ((! 2

u ( ! 2tn,i

) dt +(tn,i+1 ( t) d! 2

u , we obtain:

(F.35) = 3E

7

"Xtn,i+1

! tn,i+1

tn,i

(tn,i+1 ( t) d! 2u |Fti

:

= 3E

7! tn,i+1

tn,i

(tn,i+1 ( t) d4X, ! 2

u

5|Fti

:

.

Hence,

n

T

$

tn,i+1)T

E""X3

tn,i+1

66Fti

#= n

T

32

$

tn,i+1)T

"t2n,i+1!X, ! 2"#

tn,i

+ higher order termsp% 3

2

! T

0!X, ! 2"#

u du

= 32!X, ! 2"T .

We have here used that "ti = Tn

(equidistant spacing); for a moregeneral case, the expression will involve the quadratic variation oftime.

[Received September 2011. Revised August 2013.]

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The Estimation of Leverage Effect With High-Frequency Data · based on the “leverage effect” hypothesis: A drop in the value of the stock (negative return) increases the ﬁnancial