Phs Realized Garch 10

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JOURNAL OF APPLIED ECONOMETRICSJ. Appl. Econ.(2011)Published online in Wiley Online Library(wileyonlinelibrary.com) DOI: 10.1002/jae.1234

REALIZED GARCH: A JOINT MODEL FOR RETURNSAND REALIZED MEASURES OF VOLATILITY

PETER REINHARD HANSEN,a,b* ZHUO HUANGc AND HOWARD HOWAN SHEKa,d

a Department of Economics, Stanford University, Stanford, CA, USAb CREATES, Aarhus, Denmark

c China Center for Economic Research, National School of Development, Peking University, Beijing,China

d iCME, Stanford University, Stanford, CA, USA

SUMMARY

We introduce a new framework, Realized GARCH, for the joint modeling of returns and realized measures ofvolatility. A key feature is ameasurement equationthat relates the realized measure to the conditional varianceof returns. The measurement equation facilitates a simple modeling of the dependence between returns andfuture volatility. Realized GARCH models with a linear or log-linear specification have many attractivefeatures. They are parsimonious, simple to estimate, and imply an ARMA structure for the conditionalvariance and the realized measure. An empirical application with Dow Jones Industrial Average stocksand an exchange traded index fund shows that a simple Realized GARCH structure leads to substantialimprovements in the empirical fit over standard GARCH models that only use daily returns. Copyright 2011 John Wiley & Sons, Ltd.

Supporting information may be found in the online version of this article.

1. INTRODUCTION

The latent volatility process of asset returns is relevant for a wide variety of applications, such as

option pricing and risk management, and generalized autoregressive conditional heteroskedasticity(GARCH) models are widely used to model the dynamic features of volatility. This has sparked

the development of a large number of ARCH and GARCH models since the seminal paper by

Engle (1982). Within the GARCH framework, the key element is the specification for conditional

variance. Standard GARCH models utilize daily returns (typically squared returns) to extract

information about the current level of volatility, and this information is used to form expectations

about the next periods volatility. A single return only offers a weak signal about the current

level of volatility. The implication is that GARCH models are poorly suited for situations where

volatility changes rapidly to a new level. The reason is that a GARCH model is slow at catching

up and it will take many periods for the conditional variance (implied by the GARCH model) to

reach its new level, as discussed in Andersenet al. (2003).

High-frequency financial data are now widely available and the literature has recently introduced

a number of realized measures of volatility, including realized variance, bipower variation, the

realized kernel, and many related quantities (see Andersen and Bollerslev, 1998; Andersen et al.,

2001; Barndorff-Nielsen and Shephard, 2002, 2004; Andersen et al., 2008; Barndorff-Nielsen

et al., 2008; Hansen and Horel, 2009; and references therein). Any of these measures is far more

informative about the current level of volatility than is the squared return. This makes realized

measures very useful for modeling and forecasting future volatility. Estimating a GARCH model

Correspondence to: Peter Reinhard Hansen, Department of Economics, Stanford University, 579 Serra all, Stanford, CA94305-6072, USA. E-mail: [email protected]

Copyright 2011 John Wiley & Sons, Ltd.


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P. R. HANSEN, Z. HUANG AND H. H. SHEK

that includes a realized measure in the GARCH equation (known as a GARCH-X model) provides

a good illustration of this point. Such models were estimated by Engle (2002), who used the

realized variance (see also Forsberg and Bollerslev, 2002). Within the GARCH-X framework no

effort is paid to explain the variation in the realized measures, so these GARCH-X models are

partial (incomplete) models that have nothing to say about returns and volatility beyond a singleperiod into the future.

Engle and Gallo (2006) introduced the first complete model in this context. Their model

specifies a GARCH structure for each of the realized measures, so that an additional latent

volatility process is introduced for each realized measure in the model. The model by Engle

and Gallo (2006) is known as the multiplicative error model (MEM), because it builds on the

MEM structure proposed by Engle (2002). Another complete model is the HEAVY model by

Shephard and Sheppard (2010), which, in terms of its mathematical structure, is nested in the

MEM framework. Unlike the traditional GARCH models, these models operate with multiple latent

volatility processes. For instance, the MEM by Engle and Gallo (2006) has a total of three latent

volatility processes and the HEAVY model by Shephard and Sheppard (2010) has two (or more)

latent volatility processes. Within the context of stochastic volatility models, Takahashiet al. (2009)

proposed a joint model for returns and a realized measure of volatility. Importantly, the economic

and statistical gains from incorporating realized measures in volatility models are typically found

to be large (see, for example, Christoffersen et al., 2010; Dobrev and Szerszen, 2010).

In this paper we introduce a new framework that combines a GARCH structure for returns with

an integrated model for realized measures of volatility. Models within our framework are called

Realized GARCH models, a name that transpires both the objective of these models (similar to

GARCH) and the means by which these models operate (using realized measures).

To illustrate our framework and fix ideas, consider a canonical version of the Realized GARCH

model that will be referred to as the RealGARCH(1,1) model with a linear specification. This

model is given by the following three equations:

rtD htzt,htD C ht1 C xt1,xtD C ht C zt C ut

where rt is the return, xt a realized measure of volatility, zt i.i.d.0, 1, ut i.i.d.0, 2u , and

htD varrtjFt1with FtD rt, xt, rt1, xt1, . . . . The last equation relates the observed realizedmeasure to the latent volatility and is therefore called the measurement equation. This equation

is natural when xt is a consistent estimator of the integrated variance, because the integrated

variance may be viewed as the conditional variance plus a random innovation. The latter is, in

our model, absorbed by zt

Cut. It is easy to verify that ht is an autoregressive process of

order one, htD C ht1 C wt1, where D C , D C , and wtD zt C ut. Thusit is natural to adopt the nomenclature of GARCH models. The inclusion of the realized measure

in the model and the fact that xt has an autoregressive moving average (ARMA) representation

motivate the name Realized GARCH. A simple yet potent specification of the leverage function is

z D 1z C 2z2 1, which can generate an asymmetric response in volatility to return shocks.The simple structure of the model makes the model easy to estimate and interpret, and leads to a

tractable analysis of the quasi maximum likelihood estimator. The framework allows us to use a

realized measure that is computed from a shorter period (e.g. 6.5 hours) than the interval that the

conditional variance refers to (e.g. 24 hours). In such instances we should expect


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REALIZED GARCH

We apply the Realized GARCH framework to Dow Jones Industrial Average (DJIA) stocks and

an exchange traded index fund, SPY. We find, in all cases, substantial improvements in the log-

likelihood function (both in-sample and out-of-sample) when benchmarked to a standard GARCH

model. This is not too surprising, because the standard GARCH model is based on a more limited

information set that only includes daily returns. The empirical evidence strongly favors inclusionof the leverage function, and the parameter estimates are remarkably similar across stocks.

The paper is organized as follows. Section 2 introduces the Realized GARCH framework as a

natural extension to GARCH. We focus on linear and log-linear specification and show that squared

returns, the conditional variance, and realized measures have ARMA representations in this class

of Realized GARCH models. Our Realized GARCH framework is compared to MEM and related

models in Section 3. Likelihood-based inference is analyzed in Section 4, where we derive the

asymptotic properties of the quasi-maximum likelihood estimator (QMLE). Our empirical analysis

is given in Section 5. We estimate a range of Realized GARCH models using time series for 28

stocks and an exchange-traded index fund. In Section 6 we derive results related to forecasting

and the skewness and kurtosis of returns over one or more periods. The latter shows that the

Realized GARCH is capable of generating substantial skewness and kurtosis. Concluding remarks

are given in Section 7, and the Appendix presents all proofs.

2. REALIZED GARCH

In this section we introduce the Realized GARCH model. The key variable of interest is the

conditional variance, htD varrtjFt1, wherefrtgis a time series of returns. In the GARCH(1,1)model the conditional variance, ht, is a function of ht1 and r2t1.In the present framework, htwill also depend on xt1, which represents a realized measure of volatility, such as the realizedvariance. More generally,xtwill denote a vector of realized measures, such as the realized variance,

bipower variation, intraday range, and squared return. A measurement equation, which ties the

realized measure to the latent volatility, completes the model. Thus the Realized GARCH model

fully specifies the dynamic properties of both returns and the realized measure.To simplify the exposition we will assume ErtjFt1 D 0. A more general specifications for

the conditional mean, such as a constant or the GARCH-in-mean by Engle et al. (1987), is

accommodated by reinterpreting rtas the return less its conditional mean. The general framework

for the Realized GARCH model is presented next.

2.1. The General Formulation

The general structure of the RealGARCH(p, q) model is given by

rtD

htzt 1

htD vht1, . . . , htp, xt1, . . . , x tq 2xtD mht, zt, ut 3

where zt i.i.d.0, 1and ut i.i.d.0, 2u , with ztand utbeing mutually independent.

We refer to the first two equations as the return equation and the GARCH equation, and

these define a class of GARCH-X models, including those that were estimated by Engle (2002),

Barndorff-Nielsen and Shephard (2007), and Visser (2011). The GARCH-X acronym refers to the

the fact that xt is treated as an exogenous variable. The HYBRID GARCH framework by Chen

et al. (2009) includes variants of the GARCH-X models and some related models.

Copyright 2011 John Wiley & Sons, Ltd. J. Appl. Econ.(2011)DOI: 10.1002/jae


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We shall refer to (3) as the measurement equation, because the realized measure,xt, can often

be interpreted as a measurement of ht. The simplest example of a measurement equation is

xtD htC ut. The measurement equation is an important component because it completes themodel. Moreover, the measurement equation provides a simple way to model the joint dependence

betweenrtand xt, which is known to be empirically important. This dependence is modeled thoughthe presence of zt in the measurement equation, which we find to be highly significant in our

empirical analysis.

It is worth noting that most (if not all) variants of ARCH and GARCH models are nested in the

Realized GARCH framework. See Bollerslev (2009) for a comprehensive list of such models. The

nesting can be achieved by setting xtD rtor xtD r2t, and the measurement equation is redundantfor such models, because it is reduced to a simple identity. Naturally, the interesting case is when

xt is a high-frequency-based realized measure, or a vector containing several realized measures.

Next we consider some particular variants of the Realized GARCH model.

2.2. Realized GARCH with a Log-Linear Specification

The Realized GARCH model with a simple log-linear specification is characterized by thefollowing GARCH and measurement equations:

log htD Cp

iD1 ilog hti Cq

jD1 jlogxtj 4

logxtD C log htC zt C ut 5

whereztD rt/p

ht i.i.d.0, 1,ut i.i.d.0, 2u , andzis called theleverage function. Without

loss of generality we assume Ezt D 0.A logarithmic specification for the measurement equation seems natural in this context. The

reason is that (1) implies that

log r2t

Dlog ht

Clogz2t 6

and a realized measure is in many ways similar to the squared return, r2t, albeit a more accurate

measure of ht. It is therefore natural to explore specifications where logxt is expressed as a

function of log ht and zt, such as (5). A logarithmic form for the measurement equation makes

it convenient to specify the GARCH equation with a logarithmic form, because this induces a

convenient ARMA structure.

In our empirical application we adopt a quadratic specification for the leverage function, zt.

The conditional variance, ht, is adapted to Ft1. Therefore Ft must be such that xt2 Ft (unlessD 0). This requirement is satisfied by FtD rt, xt, rt1, xt1, . . . , but Ft could in principlebe an even richer -field. Also, note that the measurement equation does not require xt to be

an unbiased measure of ht. For instance, xt could be a realized measure that is computed with

6.5 hours of high-frequency data, while the return is a close-to-close return that spans 24 hours.

An attractive feature of the log-linear Realized GARCH model is that it preserves the ARMAstructure that characterizes some of the standard GARCH models. This shows that the ARCH

nomenclature is appropriate for the Realized GARCH model. For the sake of generality we derive

the result for the case where the GARCH equation includes lagged squared returns. Thus consider

the following GARCH equation:

log htD Cp

iD1 ilog hti Cq

jD1 jlogxtj Cq

jD1 jlog r2tj 7

where qD maxifi, i 6D 0, 0g.Copyright 2011 John Wiley & Sons, Ltd. J. Appl. Econ.(2011)

DOI: 10.1002/jae


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REALIZED GARCH

Proposition 1. Define wtD zt C ut and vtD logz2t , where D Elogz2t. The RealizedGARCH model defined by (5) and (7) implies

log htD h Cp_qiD1

i C i C i log hti Cq

jD1jwtj C jvtj,

log xtD xCp_qiD1

i C i C i logxti C wt Cp_qjD1

fj C jwtj C jvtjg,

log r2tD rCp_qiD1

i C i C i log r2ti C vtCp_qjD1

fiwtj vtj jvtjg

where h

D

C

C

, x

D

C

C1

, and r

D

C

C1

, withD qjD1 j,D piD1i, andD qjD1 j, and the conventionsiD jD jD0 for i > pand j > q.

Thus the log-linear Realized GARCH model implies that log htis ARMA(p _ q, q 1), whereaslog r2t and logxtare ARMA(p _ q,p _ q). If1D . . . D qD 0, then logxtis ARMA(p _ q,p).

From Proposition 1 we see that the persistence of volatility is summarized by a persistence

parameter:

Dp_qiD1

i C i C i D C C

Example 1. For the case p D qD 1 we have log htD C log ht1 C logxt1 and logxtDC log ht C zt C ut, so that log ht AR(1) and logxt ARMA(1,1). Specifically log htDh C log ht1 C wt1and logxtD xC logxt1 C wt wt1, where D C .

The measurement equation induces a GARCH structure that is similar to an EGARCH with

a stochastic volatility component. Take the case in Example 1 where log htD h C log ht1 Czt1 C ut1. Note thatzt1captures the leverage effects, whereasut1adds an additionalstochastic component that resembles that of stochastic volatility models. Thus the Realized GARCH

model can induce a flexible stochastic volatility structure, similar to that in Yu (2008), but

does in fact have a GARCH structure because ut1 is Ft1-measurable. Interestingly, for thepurpose of forecasting (beyond one-step-ahead predictions), the Realized GARCH is much like

a stochastic volatility model since future values ofutare unknown. This analogy does not apply

to one-step-ahead predictions because the lagged values, zt1 and ut1, are known at timet

1.

An obvious advantage of using a logarithmic specification is that it automatically ensures apositive variance. Here it should be noted that the GARCH model with a logarithmic specification,

known as LGARCH (see Geweke, 1986; Pantula, 1986; Milh j, 1987), has some practical

drawbacks. These drawbacks may explain that the LGARCH is less popular in applied work

than the conventional GARCH model that uses a specification for the level of volatility (see

Terasvirta, 2009). One drawback is that zero returns are occasionally observed, and will cause

havoc for the log-specification unless we impose some ad hoc censoring. Within the Realized

GARCH framework, zero returns are not problematic, because log r2t1 does not appear in itsGARCH equation.



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2.2.1 The Leverage Function

The function z is called the leverage function because it captures the dependence between

returns and future volatility, a phenomenon that is referred to as the leverage effect. We normalize

such functions by Ezt D 0, and we focus on those that have the form

zt D 1a1zt C . . . C kakzt, where Eakzt D 0, for allk

so that the function is linear in the unknown parameters. We shall see that the leverage function

induces an EGARCH type structure in the GARCH equation, and we note that the functional form

used in Nelson (1991), zt D 1z C Cjztj Ejztj, is within this class of leverage functions. Inthis paper we focus on leverage functions that are constructed from Hermite polynomials, i.e.

zD 1z C 2z2 1 C 3z3 3z C 4z4 6z2 C 3 C . . .

and our baseline choice for the leverage function is a simple quadratic form: zt D 1zt C 2z2t1. This choice is convenient because it ensures that Ezt D 0, for any distribution with EztD 0and varzt D 1. The polynomial form is also convenient in our quasi-likelihood analysis, and inour derivations of the kurtosis of returns generated by this model.

The leverage function z is closely related to the news impact curve (see Engle and Ng,

1993), which maps out how positive and negative shocks to the price affect future volatility.

We can define the news impact curve by zD Elog htC1jztD z Elog htC1, so that 100zmeasures the percentage impact on volatility as a function of the Studentized return. From the

ARMA representation in Proposition 1 it follows that zD 1z.

2.3. Realized GARCH with a Linear Specification

In this section we adopt a linear structure that is more similar to the original GARCH model by

Bollerslev (1986). One advantage of this formulation is that the measurement equation is simple to

interpret in this model. For instance, ifxtis computed from intermittent high-frequency data (i.e.over 6.5 hours) whereas rt is a close-to-close return that spans 24 hours, then we would expect

to reflect how much of the daily volatility occurs during trading hours. The linear Realized

GARCH model is defined by

xthtD Cp

iD1 ihti Cq

jD1 jxtj, and D C ht C zt C ut

As is the case for the GARCH(1,1) model the RealGARCH(1,1) model with the linear

specification implies that ht has an AR(1) representation htD C C C ht1 C wt1,wherewtD ut C ztis an i.i.d. process, and that the realized measure, xt, is ARMA(1,1), which isconsistent with the time series properties of realized measures in this context (see Meddahi, 2003).

3. COMPARISON TO RELATED MODELS

In this section we relate the Realized GARCH model to the multiplicative error model (MEM)

by Engle and Gallo (2006) and the HEAVY model by Shephard and Sheppard (2010),1 and some

related approaches.

1 The Realized GARCH model was conceptualized and developed concurrently and independently of Shephard andSheppard (2010). However, in our current presentation of the model we have adopted some terminology from Shephardand Sheppard (2010).



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REALIZED GARCH

The MEM by Engle and Gallo (2006) utilizes two realized measures in addition to the squared

returns. These are the intraday range (high minus low) and the realized variance, whereas the

HEAVY model by Shephard and Sheppard (2010) uses the realized kernel (RK) by Barndorff-

Nielsenet al. (2008). These models introduce an additional latent volatility process for each of the

realized measures. Thus the MEM and the HEAVY digress from the traditional GARCH modelsthat only have a single latent volatility factor. Key model features are given in Table I. We have

included the level specification of the Realized GARCH model because it is most similar to the

GARCH, MEM, and HEAVY models. Based on our empirical analysis in Section 5 we recommend

the log-linear specification in practice.

Brownless and Gallo (2010) estimates a restricted MEM model that is closely related to the

Realized GARCH with the linear specification. They utilize a single realized measure, which

leads to two latent volatility processes in their model: htD Er2t jFt1 and tD ExtjFt1.However, their model is effectively reduced to a single-factor model as they introduce the constraint

htD c C dt.The usual MEM formulation is based on a vector of non-negative random innovations, t,

that are required to have mean Et

D1, . . . , 10. The literature has explored distributions with

this property such as certain multivariate Gamma distributions, and Cipollini et al. (2009) usecopula methods that entail a very flexible class of distributions with the required structure. (A

perhaps simpler way to achieve this structure is by setting tD ZtZt, and working with thevector of mean-zero unit-variance random variables, Zt, instead.) The estimates in Engle and

Gallo (2006) and Shephard and Sheppard (2010) are based on a likelihood where the elements

of t are independent 2-distributed random variables with one degree of freedom, which maps

into Zt N0, I. We have used the alternative formulation in Table I so that z2t , z

2R,t, z

2RV,t

0

corresponds to tin the MEM by Engle and Gallo (2006).

3.1. Multi-factor Realized GARCH Models

The Realized GARCH framework can be extended to a multi-factor structure. For instance, with

mrealized measures (including the squared return) we could specify a model with km latentvolatility factors. The Realized GARCH model introduced in this paper has kD 1, whereas theMEM has mD k. This hybrid framework with 1 km, provides a way to bridge the RealizedGARCH models with the MEM framework. All these models can be viewed as extensions of

standard GARCH models, where the extensions are achieved by incorporating realized measures

into the model in various ways.2

4. QUASI-MAXIMUM LIKELIHOOD ANALYSIS

In this section we discuss the asymptotic properties of the quasi-maximum likelihood estimator

within the Realized GARCH (p, q) model. The structure of the QMLE analysis is very similar

to that of the standard GARCH model (see Bollerslev and Wooldridge, 1992; Lee and Hansen,1994; Lumsdaine, 1996; Jensen and Rahbek, 2004a,b; Straumann and Mikosch, 2006). Both Engle

and Gallo (2006) and Shephard and Sheppard (2010) justify the standard errors they report, by

referencing existing QMLE results for GARCH models. This argument hinges on the fact that

the joint log-likelihood in the MEM is decomposed into a sum of univariate GARCH-X mod-

els, whose likelihood can be maximized separately. The factorization of the likelihood is achieved

2 A realized measure simply refers to a statistic that is constructed from high-frequency data. Well-known examples includerealized variance, realized kernel, intraday range, number of transactions, and trading volume.



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TableI.Keymode

lfeaturesataglance:therealizedme

asures,

Rt,

RV

t,and

xt,

denotetheintradayrange,realizedvariance,andre

alizedkernel,

respectively.

IntheRealizedGARCHmodel,

thedependencebetweenreturnsandinnovation

stothevolatility(leverageeffect)ismodeledwith

z

t,suchaszD

1zC2

z21,sothatEz

tD0,w

henzt

0,

1

Latentvariablesa

Observables

Distributionb

GARCH(1,1)(Bollerslev,1986)

htD

Cr

2 t1Ch

t1

rtDph

tzt

zt

i.i.d.N

0

,1

MEM

(EngleandGallo,2006)

htDCr

2 t1Ch

t1C

rt1C

R2 t1

hR

,tDRC

R

R2 t1C

Rh

R,t1C

Rrt1

r2 tD

htz

2 t

hRV

,tDRV

CRV

RV

t1C

RV

hRV

,t1

R2 tD

hR

,tz

2 R,t

zt zR

,t

zRV

,t

i.i.d.N

0

,I

C

RVrt1C

RV

RV

t11

rt1

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