Stochastic Volatility, Trading Volume, and the Daily …jfleming/pub/jb0505.pdf1554 Journal of...

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[Journal of Business, 2006, vol. 79, no. 3]� 2006 by The University of Chicago. All rights reserved.0021-9398/2006/7903-0018$10.00

Jeff FlemingRice University

Chris KirbyClemson University

Barbara OstdiekRice University

Stochastic Volatility, TradingVolume, and the Daily Flow ofInformation*

I. Introduction

Empirical studies of the relation between pricechanges and trading volume have a long history infinance. Karpoff (1987) identifies 19 articles in hissurvey of the early empirical work in this area (seehis table 1). Although much of the research documentsa positive correlation between absolute returns andtrading volume, there is some debate about whetherwe can reconcile the observed relation with the pre-dictions of theory (see, e.g., Tauchen and Pitts 1983;Richardson and Smith 1994; Foster and Viswanathan1995; Andersen 1996). In this paper, we focus on onetheory in particular: the mixture of distributions hy-pothesis (MDH). Once we specify the nature of theinformation arrival process, the MDH provides a fulldynamic representation for returns and trading vol-ume. We use the MDH to examine the relation be-

* The comments of an anonymous referee substantially improvedthe paper. Kirby received support for this research under the Aus-tralian Research Council SPIRT grant C00001858. Contact the cor-responding author, Jeff Fleming, at [email protected].

We use state-space meth-ods to investigate the re-lation between volume,volatility, and ARCH ef-fects within a mixture ofdistributions hypothesis(MDH) framework. Mostrecent studies of theMDH fit AR(1) specifica-tions that require the in-formation flow to behighly persistent. Using amore general specifica-tion, we find evidence ofa large nonpersistentcomponent of volatilitythat is closely related tothe contemporaneousnonpersistent componentof volume. However, incontrast to studies that fitvolume-augmentedGARCH models, we findno evidence that volumesubsumes ARCH effects.Since volume-augmentedGARCH models are sub-ject to simultaneity bias,our findings should bemore robust than theseprior results.

1552 Journal of Business

tween stochastic volatility, volume dynamics, and autoregressive conditionalheteroscedasticity (ARCH) effects in returns.

Several variants of the MDH appear in the literature. We frame our analysisin terms of the “modified MDH” developed by Andersen (1996). The keyprediction of the MDH is that daily returns and trading volume are jointlysubordinate to an unobserved directing variable that measures the daily rateof information flow to the market. If the daily number of information arrivalsis positively correlated across days, then the model predicts positive serialcorrelation in the squared daily returns. Thus, by modeling the informationflow as an autoregressive process, we obtain a specification that can providea trading-based explanation for ARCH effects. Andersen (1996) and mostsubsequent studies of the modified MDH specify AR(1) dynamics for eitherthe information flow or its logarithm. This yields a simple stochastic auto-regressive volatility (SARV) representation for daily returns in which eitherthe variance or its logarithm follows an AR(1) process.

We argue that AR(1) specifications are unlikely to be consistent with thedata. To illustrate why, we consider the forecasting properties of a SARVmodel in which the return variance follows an AR(1) process. First, we developa linear state-space representation of the model that allows us to constructminimum mean square (MMS) linear forecasts of the daily return variancesbased on the lagged squared daily returns. Next, we show that these forecastshave the same recursive structure as the conditional variances implied by ageneralized ARCH (GARCH) process. In other words, the MDH implies thatthe persistence in the variance forecasts mimics the persistence in the infor-mation flow. This finding raises questions about the appropriateness of as-suming an AR(1) process for information flow. Although we might expectsome serial correlation in information arrivals, the strong persistence typicallyassociated with GARCH conditional variances seems unlikely.

To develop a more plausible specification of the MDH, we model theinformation flow as the sum of an AR(1) process and white noise. This breaksthe link between the persistence in the MMS variance forecasts and the per-sistence in the information flow. Moreover, it allows the return volatility tocontain a nonpersistent component that is uncorrelated with lagged squaredreturns. If this nonpersistent component of volatility exists, then the MDHpredicts that it should be closely related to the nonpersistent component oftrading volume. Thus, by adopting a more general model of the informationflow process, we obtain a bivariate SARV specification for returns and volumein which contemporaneous volume captures a component of volatility that isunrelated to ARCH effects.

The findings of Lamoureux and Lastrapes (1990) indirectly support thistype of specification. They report that ARCH effects remain significant whenlagged volume is included as an explanatory variable in a GARCH(1, 1)model for daily stock returns, but ARCH effects largely disappear if contem-poraneous volume is used instead. This suggests that contemporaneous volumecaptures a large nonpersistent component of return volatility that overshadows

Volatility, Volume, and Information 1553

the persistence captured by lagged squared returns. Volume-augmentedGARCH models, however, suffer from econometric problems that make itdifficult to evaluate the robustness of this evidence. Adding contemporaneousvolume to a GARCH model requires treating volume as exogenous. This runscontrary to most trading models, including the MDH. If volume is endogenous,then volume-augmented GARCH models are subject to a specification biasthat renders the maximum likelihood estimator inconsistent. Consequently, itis impossible to draw firm inferences about the significance of the results.

With our state-space methodology, on the other hand, it is straightforwardto obtain consistent parameter estimates and assess their statistical significance.Instead of adopting a GARCH perspective, we work with the univariate andbivariate SARV models implied by the MDH. After expressing the modelsin state-space form, we use linear filters to extract MMS forecasts and filteredestimates of the return variances. In the case of the bivariate model, the filteredvariance estimates are similar in form to the conditional variances implied bya volume-augmented GARCH(1, 1) model. This makes our approach wellsuited to investigating the extent to which contemporaneous volume explainsARCH effects. Moreover, by working with linear rather than nonlinear filters,we facilitate estimation and inference using standard econometric techniques.

The empirical analysis covers the 20 stocks in the major market index(MMI) for the period from January 1, 1988, to November 30, 2000. We usedaily data that have been adjusted to remove volatility and volume trends.First, we fit a GARCH(1, 1) to the returns to confirm that ARCH effects areimportant for the MMI firms. The results are consistent with previous research.They point to strong volatility persistence and suggest that ARCH effects area prominent feature of the data. Next, we fit a univariate SARV model to thereturns in which the variance (information flow) displays ARMA(1, 1) dy-namics. The results indicate that the dynamics of the persistent component ofvolatility captured by the SARV model are similar to those of the GARCH(1,1) conditional variances. This is consistent with our theoretical analysis ofthe properties of the MMS variance forecasts. Overall the SARV model ap-pears to perform slightly better than the GARCH model, although we findsome evidence of serial correlation in the forecast errors for both models.

The most important difference between the SARV and GARCH results isthat the SARV model provides clear evidence of a substantial nonpersistentcomponent of return volatility. The SARV parameter estimates suggest thatthis component accounts for at least half of the variability in the return var-iances for a majority of the MMI firms. As a consequence, the SARV modeltypically implies much less volatility persistence than the GARCH model. Incontrast, the GARCH model fails to capture the nonpersistent component ofvolatility because it assumes that the return variance is completely determinedby the past history of returns.

According to the MDH, the nonpersistent component of volatility identifiedby fitting the univariate SARV model should be closely related to the non-persistent component of trading volume since both variables are driven by


the nonpersistent component of the information flow. To determine whetherthis implication of the MDH is supported by the data, we fit the bivariateSARV model to returns and trading volume. We find mixed results. Thenonpersistent component of volume does explain a large fraction of the non-persistent component of volatility, but the diagnostics indicate that the SARVmodel is too restrictive to adequately characterize the joint dynamics of volumeand volatility. Specifically, we find evidence of significant serial correlationin the forecast errors for both squared returns and volume for a large majorityof the firms.

To investigate this further, we consider a less restrictive bivariate SARVspecification that nests the MDH-based model as a special case. In particular,we modify the bivariate specification to allow for persistence in return vol-atility that is unrelated to persistence in trading volume. This specificationperforms significantly better in terms of goodness-of-fit measures for most ofthe firms. The evidence of serially correlated forecast errors for squared re-turns, for example, drops to a level comparable to that observed with theGARCH(1, 1) model. The improved performance indicates that a significantportion of the variation in the persistent component of return volatility isunrelated to variation in either the persistent or nonpersistent components oftrading volume. In other words, we find no evidence that ARCH effectsdisappear once we account for the dynamics of trading volume.

Both of the bivariate SARV specifications indicate a strong positive cor-relation between the nonpersistent components of squared returns and tradingvolume. This has important implications for the econometric evaluation ofmarket microstructure models. In the case of the MDH, for example, a nonzerocorrelation between the nonpersistent components of volume and squaredreturns would be interpreted as evidence against the model under an AR(1)specification for the information flow. However, any test of the MDH is ajoint test of the predictions of the model and the specification of the infor-mation flow process. Our analysis indicates that the nonpersistent componentof return volatility accounts for a large fraction of the total variation in vol-atility. Thus empirical tests that fail to account for this feature of the data areunlikely to yield reliable results.

The rest of the paper is organized as follows. In Section II we present theMDH and use linear state-space methods to analyze its implications concerningSARV dynamics, trading volume, and ARCH effects. In Section III we de-scribe the data and discuss the econometric methodology. In Section IV wepresent the results of the empirical analysis, and in Section V we summarizeour key findings and provide suggestions for future research.

II. Modeling the Volume-Volatility Relation

Studies in the market microstructure literature suggest that a variety of char-acteristics influence the dynamics of return volatility and trading volume.Some of these include the informativeness of market prices, the presence or


absence of liquidity traders in the market, and the manner in which news isdisseminated to market participants. Under most circumstances, however, the-ory points to the rate of information arrival to the market as the primaryvariable of interest. Andersen (1996), for example, combines the theoreticalframework of Glosten and Milgrom (1985) with the subordinated stochasticprocess representation of Clark (1973) to obtain a model in which daily returnsand trading volume are jointly subordinate to an unobserved directing variable:the daily information flow. This model, which is known as the modified MDH,provides the theoretical foundation for our work. In this section, we analyzethe predictions of the modified MDH, focusing in particular on the relationbetween daily trading volume and ARCH effects in daily returns.

A. The Data-Generating Process

Let and denote the demeaned return and detrended trading volume forr vt t

day t. Under Andersen’s (1996) modified MDH, these variables are generatedby a process of the form

r p j z (1)t t rt

and

2 2 1/2v p t(m � lj ) � t(m � lj ) z , (2)t t t vt

where is an independently and identically distributed (i.i.d.) N(0, 1) stan-zrt

dardized innovation, is an i.i.d. standardized innovation such that 2z vFj ∼vt t t

, and is independent of for all t and s. That is, the2t 7 Po(m � lj ) z zt rt vs

dynamics of and are described by a bivariate stochastic volatility model.r vt t

It may appear that this model fails to place any restrictions on the dynamicsof the information flow. However, like the speculative trading model ofTauchen and Pitts (1983), the modified MDH implies that the return varianceis indistinguishable from the information flow.

Equations (1) and (2) reveal three key predictions of the modified MDH.First, demeaned returns exhibit stochastic volatility. Second, the conditionalmean of detrended volume is a linear function of the conditional variance ofdemeaned returns. Third, squared demeaned returns are uncorrelated withdetrended volume after conditioning on the return variance. To more fullydevelop the empirical content of these predictions, we now address a numberof issues relating to the specification, parameterization, and estimation of themodel. We first show how persistence in the information flow can generateGARCH(1, 1) dynamics in the MMS forecasts of return variances. Then,analyzing the relation between trading volume and ARCH effects in dailyreturns follows naturally.

B. Stochastic Volatility and ARCH Effects

Initially we ignore equation (2) and focus strictly on the data-generating pro-cess for demeaned daily returns. If we assume that the information flow is


serially correlated, then equation (1) implies that is described by a SARVrt

model. The econometric analysis of SARV models is typically accomplishedusing computationally intensive Monte Carlo methods (see, e.g., Chib, Nar-dari, and Shephard 2002). Fleming and Kirby (2003) show, however, that itis often possible to obtain comparable results by applying the more tractableframework of the Kalman filter. In addition, they provide a straightforwardcharacterization of the relation between GARCH and SARV models that isuseful for illustrating the empirical implications of the modified MDH.

To see how we can adapt their approach to our setting, consider a modelin which is generated by a one-factor SARV model of the formrt

r p j z (3)t t rt

and

2 2j p q � fj � gh , (4)t t�1 t

where , ht is white noise with , and ht is2z ∼ NID(0, 1) Var (h ) p 1 � frt t

independent of for all t and s. Without loss of generality, we can expresszrs

the date t return variance as , where and2j p � � gb � p q/(1 � f) b pt t t

. It follows, therefore, that the dynamics of are governed by the2fb � h rt�1 t t

system

b p fb � h (5)t t�1 t

and

2r p � � gb � � , (6)t t t

where is white noise with . It is2 2 2� p (� � gb )(z � 1) Var (� ) p 2(� � g )t t rt t

easy to verify that, under our maintained assumptions,

Cov (� , h ) p Cov (� , b ) p 0 G t, s (7)t s t s

and

2 2Cov (h , b ) p Cov (h , r ) p Cov (� , r ) p 0 G t 1 s. (8)t s t s t s

Therefore, the system in equations (5) and (6) constitutes a linear state-spacerepresentation (see, e.g., Hamilton 1994).1 By recasting the SARV process asa dynamic latent factor model for the squared demeaned returns, we obtaina specification that allows estimation and inference using the Kalman filter.

The most common use of the Kalman filter is to construct MMS linear

1. There is one aspect of this system that is unusual for a linear state-space representation: �t,the innovation to the observation equation, is conditionally heteroscedastic. Note, however, thatwe can accommodate this feature without altering the way in which we apply the Kalman filter.To see this, recall that the filter is simply a convenient algorithm for recursively updating a linearprojection. As in linear regression, the projection coefficients depend on unconditional momentsthat remain well defined in the presence of unmodeled conditional heteroscedasticity. Of course,the Kalman filter will produce less precise variance forecasts than an optimal nonlinear filterthat accounts for this conditional heteroscedasticity, but the precision loss appears to be smallfor the type of SARV model considered here. See Fleming and Kirby (2003) for details.


forecasts of the state of the process based on past observables. For the systemin equations (5) and (6), the filter delivers MMS linear forecasts of givenbt�1

for . These in turn imply MMS linear forecasts2 2 2{r , r , … , r } t p 1, 2, … , Tt t�1 1

of for . The date t forecast of , which we denote by2 2j t p 1, 2, … , T jt�1 t�1

, is given by2jt�1Ft

2 2 2 2j p � � f(j � �) � gK (r � j ), (9)t�1Ft tFt�1 t t tFt�1

where is the filter gain. We can also express this forecast asKt

2 2 2j p q � bj � a r , (10)t�1Ft t tFt�1 t t

where and .b p f � gK a p gKt t t t

The time variation in the coefficients in equation (10) reflects the absenceof an observed history when we initialize the filter at time . For at p 0covariance stationary process, these coefficients quickly converge to constantsas t increases and we approach steady state. In steady state, the varianceforecasts produced by the Kalman filter have the same recursive structure asthe conditional variances in a GARCH(1, 1) model. Thus, by applying linearfiltering techniques, we uncover a simple relation between SARV dynamics,information flow persistence, and ARCH effects in returns. Specifically, if weassume that follows an AR(1) process, then our MMS linear forecasts of2jt

display the same persistence as the information flow, and these forecasts2jt

take the same form as the conditional variances implied by the most widelyused GARCH model.

This close link between the persistence in the GARCH conditional variancesand the persistence in the information flow is problematic from an empiricalperspective. Typically, fitting a GARCH model to asset returns yields param-eter estimates that imply that the half-life of a variance shock is between twoand six months (see Bollerslev, Chou, and Kroner 1992). Since we wouldexpect our MMS linear forecasts of to imply a similar degree of persistence,2jt

we can infer that, under an AR(1) specification, the half-life of an informationflow shock is around two to six months. Although it is natural to expect someserial correlation in information arrivals, it is difficult to conjecture plausiblescenarios that would generate this level of persistence. To break the linkbetween the persistence in variance forecasts and the persistence in the in-formation flow, we must consider models that generate more complex dy-namics for .2jt

A natural way to do this is to consider multifactor SARV models. To thisend, let denote the date t information set. Without loss of generality, weFt

can decompose as2jt

2j p h � rc , (11)t t t

where denotes the conditional variance and is a white-2h p E(j FF ) ct t t�1 t

noise innovation such that . This decomposition suggests that weVar (c ) p 1t

could treat the expected and unexpected components of as uncorrelated2jt

latent factors. Suppose, for example, that we write the conditional variance


as and again assume that follows an AR(1) process. In thish p � � gb bt t t

case, we obtain a linear state-space representation for of the form2rt

b p fb � h , (12)t t�1 bt

c p h , (13)t ct

and

2r p � � gb � rc � � , (14)t t t t

where , , and .2 2 2 2Var (h ) p 1 � f Var (h ) p 1 Var (� ) p 2(� � g � r )bt ct t

This two-factor specification implies that displays ARMA(1, 1) dynamics.2jt

Consequently, the model can accommodate strong persistence in the condi-tional variances without requiring strong persistence in the daily informationflow.

To see the impact of this flexibility, we have to look beyond the forecastingproperties of the model. As in the one-factor specification, the model impliesthat volatility persistence is generated by a single latent factor that followsan AR(1) process. This means that our date t MMS linear forecast of 2jt�1

takes the same general form as equation (9). However, suppose that, insteadof forecasting , we want to construct an estimate of given2 2 2 2j j {r , r ,t�1 t t t�1

, that is, a filtered variance estimate. Our one-factor model implies2… , r }1

that the date t MMS linear estimate of , which we denote by , is given2 2j jt tFt

by

2 2j p j � gk e , (15)tFt tFt�1 t t

where is the error in forecasting and is a projection2 2 2e p r � j r kt t tFt�1 t t

coefficient that is proportional to the filter gain: . In contrast, theK p fkt t

two-factor model implies

2 2j p j � gk e � rk e , (16)tFt tFt�1 bt t ct t

where kbt and kct are projection coefficients such that , the first element ofKbt

the filter gain vector, is given by . The contribution of the secondK p fkbt bt

factor is reflected in the last term of equation (16). This term, which is absentfrom the expression for , captures the nonpersistent component of .2 2j jt�1Ft t

Note that identification is not a problem for the two-factor model becausethe modified MDH implies a specific distribution for . This allows us tozt

express the variance of �t in terms of the other parameters, which in turnidentifies r. However, there is a related issue that affects our interpretation ofthe empirical results. Because the variance of �t depends on the kurtosis of

, a statistically significant estimate of r could arise in at least two ways.zrt

We could be fitting a correctly specified model in which andz ∼ NID(0, 1)rt

displays ARMA(1, 1) dynamics. Alternatively, we could be fitting a mis-2jt

specified model in which displays AR(1) dynamics and is drawn from2j zt rt

a distribution that has fatter tails than a standard normal. This could occur,


for example, if part of the nonpersistent information flow consists of publicnews that causes prices to change without any trading because the informationcontent is readily interpretable to all market participants.2 To resolve thisambiguity, we bring trading volume into the analysis.

C. Stochastic Volatility, Trading Volume, and ARCH Effects

We now incorporate the implications of equation (2), which we previouslyignored. Let denote the innovation to detrended trading2 1/2� p t(m � lj ) zvt t vt

volume. Under our two-factor model, the dynamics of and are governed2r vt t

by a system of the form

b p fb � h , (17)t t�1 bt

c p h , (18)t ct

2r p � � g b � r c � � , (19)t r r t r t rt

and

v p � � g b � r c � � , (20)t v v t v t vt

where , , , ,2� p t(m � l� ) g p tlg r p tlr Var (h ) p 1 � fv r v r v r bt

, , and .2 2 2 2Var (h ) p 1 Var (� ) p 2(� � g � r ) Var (� ) p t (m � l� )ct rt r r r vt r

Since �vt is white noise with

Cov (� , h ) p Cov (� , b ) p Cov (� , h ) p Cov (� , c ) p 0 G t, svt bs vt s vt cs vt s

(21)

and

2Cov (� , r ) p Cov (� , v ) p 0 G t 1 s, (22)vt s vt s

we can easily verify that this system constitutes a linear state-spacerepresentation.

The cross-sectional implications of the modified MDH show up clearly inthis bivariate specification. The first factor, which follows an AR(1) process,captures the common persistent component of and , whereas the second2r vt t

serially uncorrelated factor captures the common nonpersistent component ofthese variables. Under the restriction , the model reduces to ar p r p 0r v

one-factor specification of the MDH similar to those examined in prior studies(see, e.g., Lamoureux and Lastrapes 1994; Liesenfeld 1998; Bollerslev andJubinski 1999; Watanabe 2000). Under the restriction , the model isr p 0v

consistent with a scenario in which displays AR(1) dynamics and is2j zt rt

drawn from a distribution that has fatter tails than a standard normal. By fittingthe model with rr and rv unrestricted, we can isolate the component of and2rt

2. See Andersen (1996) for further discussion.


that, according to the modified MDH, reflects the nonpersistent componentvt

of the daily information flow.The bivariate specification also provides insights into the relation between

daily trading volume and ARCH effects in daily returns. Suppose that u pt

denotes the error in forecasting . Using the Kalman filter,2v � tm � tlj vt tFt�1 t

we can show that the date t MMS linear forecast of is given by2jt�1

2 2j p � � f(j � � ) � g K e � g K u , (23)t�1Ft r tFt�1 r r br ,t t r bv,t t

where and are the elements of the first row of the Kalman filter gainK Kbr,t bv,t

matrix.3 We can also express this forecast as

2 2 2j p q � bj � a r � a v , (24)t�1Ft t t tFt�1 rt t vt t

where , ,q p � (1 � f) � K (g� � g � ) b p f � g K � g K a pt r bv,t r v v r t r br,t v bv,t rt

, and . Thus, in steady state, the variance forecasts haveg K a p g Kr br,t vt r bv,t

the same recursive structure as the conditional variances implied by aGARCH(1, 1) model that includes lagged volume as an additional explanatoryvariable.

Of course, this does not necessarily imply that fitting the bivariate speci-fication would produce results similar to those obtained by fitting a volume-augmented GARCH model. To understand why, note that the coefficients inequation (24) depend on elements of the filter gain matrix, which in turndepend in a complex nonlinear fashion on the parameters of the SARV model.The reason is that we use the underlying structure of the state-space repre-sentation to infer the values of the projection coefficients rather than estimatingtheir values directly. The volume-augmented GARCH model imposes no suchstructure, so it could easily yield different results.

We gain additional insights into the relation between volume and volatilityunder the model by constructing a filtered estimate of . This estimate takes2jt

the form

2 2j p j � (g k � r k )e � (g k � r k )u . (25)tFt tFt�1 r br,t r cr,t t r bv,t r cv,t t

To see the impact of volume more clearly, suppose that . In this case,e p 0t

the difference between and is determined by the last term in equation2 2j jtFt tFt�1

(25). If is positive (negative), which occurs when the date t volume is higherut

(lower) than forecast, then we obtain our estimate of the date t variance byadjusting upward (downward). The extent of the required adjustment is2jtFt�1

determined by the expression multiplying . The first term in this expression,ut

, measures the sensitivity of the persistent component of to volume2g k jr bv,t t

changes, and the second term, , measures the sensitivity of the nonper-r kr cv,t

sistent component of to these changes.2jt

If we rearrange the steady-state version of equation (25) to isolate the termsinvolving contemporaneous volume, we see parallels between the expression

3. Note that this forecast actually depends on all elements of the gain matrix. In deriving eq.(23), however, we recognize that the second-row elements are zero because is white noise.ct


for and the conditional variance process implied by the volume-augmented2jtFt

GARCH(1, 1) model of Lamoureux and Lastrapes (1990). Equation (25),however, is well specified under the modified MDH, unlike the volume-aug-mented GARCH model. Inserting contemporaneous volume into a GARCHmodel is problematic because this variable is neither predetermined nor ex-ogenous. Therefore, the resulting model will produce biased coefficient es-timates. In contrast, our approach for constructing filtered variance estimatesexplicitly accounts for the endogenous nature of trading volume. Althoughthese estimates are described by a GARCH-like recursion, the associatedparameters can be estimated consistently using standard techniques.

D. A Generalization of the Bivariate Model

Although we have gained some flexibility over the models used in previousresearch, our two-factor model may still be too restrictive to adequately capturethe dynamics of volume and volatility. As an alternative, we consider a three-factor specification of the form

a p f a � h , (26)t a t�1 at

b p f b � h , (27)t b t�1 bt

c p h , (28)t ct

2r p � � da � g b � r c � � , (29)t r t r t r t rt

and

v p � � g b � r c � � , (30)t v v t v t vt

where hat, hbt, and hct are mutually uncorrelated white-noise innovations. Byincluding the third factor, , we allow for a persistent component of volatilityat

that is linearly unrelated to the persistent component of volume.We can motivate this specification with either of two arguments. First, we

could view the third factor as an ad hoc addition to the two-factor model.From this perspective, a statistically significant estimate of d would provideevidence against the modified MDH. Alternatively, we could view the thirdfactor as an additional component of the information flow: a persistent streamof public news that causes prices to change without any trading. From thisperspective, the model is consistent with a generalized version of the modifiedMDH that incorporates both public and private information arrivals. As amatter of convenience, we follow previous research and adopt the first per-spective when discussing the empirical results. However, whether we viewthe model as consistent or inconsistent with the MDH, it does shed light onthe relation between ARCH effects and trading volume: if adding the thirdfactor improves the performance of the model, then we would conclude that


daily squared returns capture a persistent component of volatility that is notcaptured by volume. In other words, ARCH effects remain after accountingfor trading volume.

III. Data and Econometric Methodology

In this section we discuss the data used for the empirical analysis and wedescribe our empirical methods. We begin with a brief description of the dataset and follow with a description of our detrending procedure and a discussionof the statistical properties of the detrended series. We then provide an over-view of our model-fitting procedure.

A. The Data

The data set consists of daily total returns and daily split-adjusted tradingvolume for the 20 stocks in the MMI.4 Our sample period is January 1, 1988,to November 30, 2000. We examine the MMI stocks because they are widelyheld by both individual and institutional investors and they generally exhibita high level of trading activity. For instance, over the nine-year interval from1993 to 2000, the average number of daily trades for the MMI stocks rangesfrom 626 for Dow Chemical to 3,538 for General Electric. We obtain thereturns from the daily stock file of the Center for Research in Security Prices(CRSP).

The trading volume is drawn from two sources: CRSP and the Trade andQuote (TAQ) files of the New York Stock Exchange (NYSE). We use twosources because there is some contamination in the CRSP volume files. Thereare several instances in which CRSP reports zero trading volume for the dayand a number of others with clear transcription errors such as omitting thefinal two zeros. Because the TAQ files go back to only 1993, we rely onCRSP volume for the years 1988–92. All the obvious errors in the CRSP dataare confined to the period covered by TAQ, so the benefits of having fiveadditional years of data outweigh the potential downside of any remainingcontamination.

B. Detrending Procedures

Before fitting the modified MDH to the data, we first detrend the volumeseries using an approach similar to that used by Gallant, Rossi, and Tauchen(1992). Specifically, we fit a quadratic time trend to log volume using ordinaryleast squares (OLS), exponentiate the OLS residuals, and then scale the re-

4. The firms in the MMI are American Express (AXP), AT&T (T), Chevron (CHV), Coca-Cola (KO), Disney (DIS), Dow Chemical (DOW), DuPont (DD), Eastman Kodak (EK), Exxon-Mobil (XOM), General Electric (GE), General Motors (GM), International Business Machines(IBM), International Paper (IP), Johnson & Johnson (JNJ), McDonald’s (MCD), Merck (MRK),3M (MMM), Philip Morris (MO), Procter and Gamble (PG), and Sears (S).


sulting series to have the same mean as the original series. While other methodsoffer more flexibility in fitting the trend, they would be more susceptible tooverfitting. This could lead to removing components of volume that are im-portant to the volume-volatility relation.5

We also remove volatility trends. Although the modified MDH is silent onthis issue, recent studies such as Campbell et al. (2001) report that idiosyncraticvolatility has increased over time. Since a large fraction of the volatility forindividual stocks is idiosyncratic, volatility trends are likely to be relevant toour analysis. To remove any trends in volatility, we apply the regressionapproach described above to the log squared demeaned returns. That is, wefit a quadratic time trend to the log squared demeaned returns using OLS,exponentiate the residuals, and then scale the resulting series to have the samemean as the original series. For the remainder of the paper, all references toreturns and trading volume are to the trend-adjusted variables.

Table 1 presents summary statistics for the data. The first set of columnsreport the sample mean, standard deviation, coefficient of skewness, and co-efficient of kurtosis of the returns (panel A) and trading volume (panel B).The second set of columns report the first-, second-, third-, fifth-, and tenth-order sample autocorrelations, and the final column reports the Ljung-Box(1978) statistic for serial correlation to 10 lags. A single (double) asteriskindicates that the null of no serial correlation is rejected at the 5% (1%) level.

Panel A contains few surprises. The mean and standard deviation valuesare typical of those for daily individual stock returns, and excess kurtosis,which is characteristic of financial market data, is evident. In the case ofEastman Kodak, Philip Morris, and Procter and Gamble, we see particularlylarge kurtosis values—13.68, 23.56, and 16.01—that are accompanied bynegative skewness. This suggests that these firms experienced unusually largenegative returns during our sample period. As expected, the autocorrelationsof the squared returns point to the presence of ARCH effects. The Ljung-Boxstatistics indicate that we can reject the null of no serial correlation at the 1%level for all but two of the firms: Eastman Kodak and Philip Morris. Thesmall test statistics for these firms likely reflect the impact of the large negativereturns.

The statistics in panel B parallel those in panel A on several dimensions.First, trading volume displays substantial skewness and excess kurtosis. Thereported values suggest that four of the firms—Chevron, General Motors,Procter and Gamble, and Sears—experienced unusually high volume at sometime during our sample period. Second, there is clear evidence of positiveserial correlation in trading volume, with the first-order autocorrelation ex-ceeding 0.5 for many of the firms. Moreover, the serial correlation dies outslowly: the tenth-order autocorrelation is often greater than 0.1. We reject thenull of no serial correlation in volume at 1% for all the firms.

5. See Lo and Wang (2000) for more discussion of this issue.

1564Journal

ofB

usiness

TABLE 1 Descriptive Statistics for the Trend-Adjusted Series

A. Adjusted Demeaned Returns

Distributional Properties of rt Autocorrelation in 2rt

Firm MeanStandardDeviation Skewness Kurtosis Lag 1 Lag 2 Lag 3 Lag 5 Lag 10 Q

AXP .00 2.09 .20 5.16 .19 .09 .10 .10 .09 389.9**T .01 1.82 .21 6.33 .14 .04 .03 .05 �.01 90.1**CHV .00 1.46 .07 5.00 .10 .03 .02 .04 .07 99.6**KO .00 1.67 .16 5.01 .13 .09 .03 .03 .06 126.4**DIS .00 1.84 .27 6.81 .09 .05 .02 .03 .06 73.1**DOW .00 1.67 .14 6.24 .07 .07 .03 .06 .06 187.9**DD .01 1.73 .14 4.84 .07 .03 .02 .02 .03 39.7**EK .00 1.79 �.54 13.68 .04 .02 .01 .00 .01 12.6XOM .00 1.38 .16 4.55 .11 .05 .08 .04 .03 95.5**GE .00 1.51 .06 4.53 .14 .09 .04 .06 .08 196.6**GM .00 1.90 .14 4.35 .10 .07 .07 .04 .06 151.2**IBM �.01 1.92 .10 8.40 .08 .02 .01 .05 .00 55.5**IP .01 1.85 .10 5.16 .07 .03 �.01 .05 .03 53.3**JNJ .00 1.61 .09 4.88 .14 .06 .02 .04 .07 165.0**MCD .00 1.67 .10 5.20 .15 .02 .00 .12 .02 135.2**MRK .00 1.69 .07 4.44 .10 .04 .06 .03 .04 104.4**MMM .00 1.46 �.20 6.40 .08 .03 .02 .00 .00 39.2**MO .00 1.89 �1.12 23.56 .02 .03 .02 .01 .00 7.4PG .00 1.68 �.80 16.01 .03 .01 .05 .02 .01 28.8**S .00 2.03 .29 6.42 .10 .06 .04 .13 .05 216.8**

Volatility,

Volum

e,and

Information

1565

B. Adjusted Trading Volume

Distributional Properties of vt Autocorrelation in vt

Firm MeanStandardDeviation Skewness Kurtosis Lag 1 Lag 2 Lag 3 Lag 5 Lag 10 Q

AXP 4.16 2.30 2.71 17.84 .51 .35 .30 .24 .21 2,669.8**T 5.40 3.11 4.05 36.34 .40 .25 .22 .17 .04 1,186.1**CHV 1.14 .72 12.66 356.34 .42 .33 .29 .23 .14 2,154.4**KO 3.68 1.66 2.18 11.51 .43 .27 .17 .11 .07 1,143.7**DIS 5.07 2.97 2.73 15.54 .56 .38 .32 .26 .21 3,045.9**DOW 2.29 1.27 4.35 59.24 .40 .26 .21 .17 .14 1,490.0**DD 2.35 1.29 3.31 23.91 .51 .35 .28 .22 .17 2,327.0**EK 1.16 .96 6.97 83.12 .49 .27 .21 .14 .09 1,529.7**XOM 2.83 1.12 2.29 12.60 .39 .26 .17 .07 .09 935.9**GE 14.64 5.97 1.82 8.94 .49 .34 .27 .21 .19 2,150.8**GM 2.16 2.88 28.75 1,046.65 .23 .07 .06 .05 .05 283.4**IBM 8.62 4.71 3.34 25.04 .56 .35 .29 .22 .20 2,555.8**IP 1.24 .69 3.68 44.03 .49 .30 .22 .15 .11 1,673.1**JNJ 2.60 1.22 2.76 21.17 .49 .31 .23 .19 .18 2,115.6**MCD 3.61 1.79 2.28 13.27 .53 .36 .30 .21 .17 2,507.9**MRK 5.10 2.35 2.10 11.73 .52 .32 .27 .25 .19 2,519.2**MMM .87 .46 3.54 29.10 .39 .21 .16 .09 .09 943.1**MO 7.19 4.70 6.21 84.43 .56 .36 .28 .19 .08 2,243.7**PG 2.32 1.50 11.31 288.30 .50 .33 .30 .21 .16 2,258.0**S 1.13 1.22 31.11 1,395.81 .18 .14 .13 .11 .06 422.5**

Note.—The table reports descriptive statistics for the trend-adjusted series used to fit the GARCH and SARV models. We construct these series as follows. First, we identify long-runtrends in volume and volatility by regressing the log trading volume and log squared demeaned returns on a constant, a time index, and the square of the time index. Next, we construct theadjusted trading volume, , and adjusted squared demeaned returns, , by exponentiating the regression residuals and scaling the resulting series to have the same sample mean as the2v rt t

original series. Finally, we obtain the adjusted returns, , by attaching the sign of each demeaned return to the corresponding adjusted absolute demeaned return. The first set of columnsrt

report the mean, standard deviation, coefficient of skewness, and coefficient of kurtosis for (panel A) and (panel B). The next set report autocorrelations for (panel A) and (panel2r v r vt t t t

B). The final column reports the Ljung-Box statistic (Q) for serial correlation to 10 lags of (panel A) and (panel B). The returns are expressed in percent, and the trading volume is2r vt t

measured in millions of shares. The sample period is January 1, 1988, to November 30, 2000.* Statistically significant at the .05 level.** Statistically significant at the .01 level.


C. Parameter Estimates and Standard Errors

We fit the various specifications of the modified MDH by Gaussian quasimaximum likelihood (QML). This is accomplished using the prediction errordecomposition of the log likelihood implied by the Kalman filter. Suppose,for example, that we want to fit the univariate two-factor specification inequations (12), (13), and (14). Let and′ 2 2v p (f, �, g, r) y p g P �tFt�1 b,tFt�1

, where denotes the mean squared error (MSE) of2 2 2 2r � 2(� � g � r ) Pb,tFt�1

. If hbt, hct, and �t were distributed multivariate normal, then the conditionalbtFt�1

distribution of would be normal with mean and variance . Hence,2 2 2r j yt tFt�1 tFt�1

we construct the quasi log likelihood for the model as , whereTL(v) p � l (v)ttp1

2 2 21 1 1 r � jt tFt�12l (v) p � log (2p) � log y � , (31)t tFt�1 ( )2 2 2 ytFt�1

and T is the number of observations in the data set.6 The value of rapidlyPb,tFt�1

converges to a constant as the impact of the initial conditions on our forecastsdies out. It follows, therefore, that the QML estimator is essentially obtainedby minimizing the MSE in forecasting .2rt

To obtain robust standard errors for the parameter estimates, we nest theQML estimator in a generalized method of moments framework. This isaccomplished by treating the restriction as a vector of sample′�L(v)/�v p 0moment conditions. To illustrate, let denote the QML estimator of v. Wev

have

d ′ �1 �1ˆ�T(v � v) r N(0, (D S D) ), (32)

where D and S are the second-derivative and outer-product forms of the quasiinformation matrix. If we let

T1 �l (v)tD p (33)� ′ F

ˆT �v�vtp1 vpv

and

′T1 �l (v) �l (v)t t

S p , (34)� F[ ][ ] ˆT �v �vtp1 vpv

then and under standard regularity conditions. See Hamiltonp pˆD r D S r S(1994) for more details.

6. Note that the initial values of and are given by and .b P b p 0 P p 1tFt�1 b,tFt�1 1F0 b,1F0


IV. Model Fitting Results

To lay the foundation for evaluating the empirical performance of the SARVspecifications, it is useful to first estimate a GARCH(1, 1) model for thereturns. We fit the model

�r p h z ,t t t

2h p q � bh � ar , (35)t t�1 t�1

by Gaussian QML; that is, we treat the standardized innovation as i.i.d.zt

N(0, 1) for the purpose of constructing the log likelihood. Bollerslev andWooldridge (1992) show that the QML estimator is consistent and asymp-totically normal provided that certain regularity conditions are satisfied. Wefollow their methodology for constructing robust standard errors for the pa-rameter estimates.

Table 2 summarizes the model-fitting results. The first set of columns con-tain the parameter estimates and standard errors. The second set of columnscontain the maximized quasi log likelihood and selected model diagnostics:the estimated first-order autocorrelation coefficient for , the estimated var-ht

iance of as a fraction of the estimated variance of , the Ljung-Box statistic2h rt t

for the first 10 sample autocorrelations in , and the Ljung-Box statistic2r /ht t

for the first 10 sample autocorrelations in the forecast error .7 The2r � ht t

parameter estimates are similar to those reported in other studies for individualstock returns (e.g., Kim and Kon 1994). Typically, the estimate for a is lessthan 0.1, the estimate for b is greater than 0.8, and the standard errors implyt-ratios greater than two. This is consistent with the presence of ARCH effects,which is not surprising given the long literature on time-varying volatility infinancial markets.

We also find evidence of strong persistence in the conditional variances:the estimated autocorrelation in exceeds 0.9 for 15 of the 20 firms and isht

greater than 0.78 for all of them. But the results indicate that the GARCH(1,1) model has low explanatory power. The estimated variance of is typicallyht

between 1% and 6% of the estimated variance of . Dow Chemical, with an2rt

estimated variance ratio of 0.17, is the sole exception. We can understandthese results using the expression for derived in Andersen2Var (h )/ Var (r )t t

and Bollerslev (1998). They show that this ratio is given by 2 2a /[1 � a �. Thus, for a fixed value of a, the value of is2 2(a � b) ] Var (h )/ Var (r )t t

increasing in the first-order autocorrelation of and becomes more sensitiveht

to a given change in this parameter as the autocorrelation approaches one.However, if the autocorrelation is such that , then neither2 2(a � b) ≥ 1 � 2a

nor exists. This implies that the upper bound on the variance2Var (h ) Var (r )t t

7. Under a GARCH(1, 1) data-generating process, it is straightforward to compute the momentsthat determine the variance ratio: , , , and2 4 2E(r ) p E(h ) E(r ) p 3E(h ) E(h ) p q/(1 � a � b)t t t t t

. See Bollerslev (1986) for more2 2 2 2E(h ) p q (1 � a � b)/[(1 � a � b)(1 � b � 2ab � 3a )]t

details.


TABLE 2 Results of Fitting a GARCH(1, 1) Model to Daily Returns

Firm q b a

Log Likelihood and Diagnostics

LT AutocorrelationVariance

Ratio Q1 Q2

AXP .12(.08)

.92(.04)

.05(.02)

�6,897.2 .97 .05 20.68* 32.43**

T .68(.19)

.68(.07)

.12(.02)

�6,512.9 .80 .04 4.34 26.77**

CHV .07(.03)

.93(.02)

.04(.01)

�5,824.4 .97 .03 16.57 27.40**

KO .14(.08)

.90(.04)

.05(.02)

�6,241.5 .95 .02 13.92 28.83**

DIS .04(.14)

.96(.07)

.03(.02)

�6,548.0 .99 .04 14.23 17.11

DOW .02(.03)

.95(.02)

.04(.01)

�6,132.8 1.00 .17 19.57* 27.25**

DD .02(.03)

.98(.02)

.02(.01)

�6,381.1 .99 .02 12.24 13.99

EK .74(.21)

.70(.07)

.08(.03)

�6,510.9 .78 .01 2.43 14.26

XOM .09(.17)

.91(.13)

.05(.04)

�5,636.5 .96 .03 8.65 14.18

GE .04(.03)

.95(.02)

.04(.01)

�5,896.1 .98 .04 22.84* 40.28**

GM .05(.24)

.95(.13)

.04(.05)

�6,649.8 .99 .04 11.39 14.05

IBM .31(.13)

.87(.04)

.05(.02)

�6,724.0 .92 .02 9.85 22.60*

IP .42(.16)

.81(.06)

.07(.02)

�6,606.5 .88 .02 12.07 19.06*

JNJ .09(.04)

.92(.03)

.05(.01)

�6,095.5 .97 .04 19.81* 36.08**

MCD .07(.06)

.94(.04)

.04(.02)

�6,252.1 .98 .03 34.77** 65.75**

MRK .12(.05)

.91(.03)

.05(.01)

�6,279.2 .96 .03 8.34 11.81

MMM .28(.13)

.82(.07)

.05(.02)

�5,833.1 .87 .01 4.88 14.73

MO .51(.28)

.78(.09)

.08(.03)

�6,648.4 .86 .03 .55 22.70*

PG .04(.06)

.95(.02)

.04(.01)

�6,239.5 .99 .05 1.09 10.11

S .28(.10)

.84(.04)

.09(.02)

�6,808.9 .93 .06 9.40 46.57**

Note.—The table reports the results of using Gaussian QML to fit a GARCH(1, 1) model to the adjusteddemeaned returns ( ) for the MMI firms. The model is given by eq. (35) in the text. The first set of columnsrt

report the parameter estimates and asymptotic standard errors (in parentheses). The second set of columnsreport the quasi log likelihood ( ) along with selected model diagnostics: the estimated first-order autocor-LT

relation in , the variance ratio obtained by dividing the estimated variance of by the sample variance ofh ht t

, and Ljung-Box statistics (Q1 and Q2) based on the first 10 sample autocorrelations of and .2 2 2r r /h r � ht t t t t

* Statistically significant at the .05 level.** Statistically significant at the .01 level.


ratio for a GARCH(1, 1) process whose first four moments are well definedis . As a result, the estimated variance ratio tends to be small unless the1/3estimated autocorrelation is close to one, as with Dow Chemical.

Despite its popularity, the GARCH(1, 1) model fails to completely capturethe higher-order serial dependence in the adjusted demeaned returns. Specif-ically, 12 of the Ljung-Box statistics for are significant at the 5% level2r � ht t

and nine are significant at the 1% level. The Ljung-Box statistics for serialcorrelation in indicate fewer rejections. Five of these statistics are sig-2r /ht t

nificant at the 5% level and only one is significant at the 1% level. We focuson the first statistic in evaluating the model because diagnostics based onforecast errors provide better benchmarks for assessing the performance ofthe SARV specifications. If our MDH-inspired SARV model is well specified,then the errors generated by our MMS forecasts of should be white noise.2rt

Consequently, we would expect the sample autocorrelation in the SARV fore-cast errors to be lower than the sample autocorrelation in if the MDH2r � ht t

accurately describes the data-generating process.

A. The Two-Factor SARV Specification for Returns

We now turn to the first SARV specification: the two-factor model withreturn variance (information flow) displaying ARMA(1, 1) dynamics. Asa first step, we fit the model to using the linear state-space representation2rt

in equations (12)–(14). Table 3 summarizes the results. Panel A reportsthe parameter estimates, standard errors, and maximized quasi log like-lihood. Panel B reports selected model diagnostics: the estimated first-order autocorrelation in ; the estimated values of ,2 2 2j Var (j )/ Var (r )t t t

, and ; the estimated coefficient of2 2 2 2Var (j )/ Var (j ) Var (j )/ Var (j )tFt�1 t tFt t

determination for a linear regression of on ; and the2 2 2 2j � j r � jt tFt�1 t tFt�1

Ljung-Box statistics for the first 10 sample autocorrelations in and2 2r /jt tFt�1

the first 10 sample autocorrelations in .2 2r � jt tFt�1

First consider the parameter estimates and standard errors in panel A. Theestimated value of f points to strong persistence in : the estimated valuebt

exceeds 0.9 for 14 of the firms and 0.8 for another five. The standard errorsare generally an order of magnitude smaller. Consistent with the theoreticalresults presented in Section II, we typically find that the estimate of f is closeto the sum of the a and b estimates reported in table 2. Since the two-factormodel implies that the MMS linear forecasts of take the same general form2jt

as in the GARCH(1, 1) model, it is not surprising to find that these forecastsht

display dynamics similar to the GARCH(1, 1) conditional variances. We ex-pect the persistent factor, , to capture ARCH effects in daily returns, andbt

this is the case.At the same time, the nonpersistent factor, , appears to capture an unre-ct

lated, but important, component of the return variances. The estimated valueof r, which reflects the contribution of the nonpersistent factor to , is2jt

statistically significant at the 5% level for 18 of the MMI firms. To understand


TABLE 3 Results of Fitting a Two-Factor SARV Model to Daily Returns

Firm

A. Estimation Results

f � g r LT

AXP .97(.03)

4.39(.45)

3.21(.86)

1.80(1.24)

�11,663.5

T .58(.10)

3.28(.18)

3.49(.53)

.01(.04)

�11,231.8

CHV .96(.03)

2.16(.16)

1.06(.38)

1.38(.29)

�9,351.6

KO .93(.05)

2.80(.19)

1.64(.43)

1.56(.39)

�10,195.5

DIS .92(.16)

3.39(.23)

2.10(1.07)

3.16(.80)

�11,453.4

DOW .99(.01)

2.82(.43)

1.63(.41)

2.35(.33)

�10,611.9

DD .99(.00)

3.05(.44)

1.02(.33)

2.06(.31)

�10,388.9

EK .72(.23)

3.22(.23)

2.62(.97)

5.48(1.29)

�12,593.5

XOM .94(.07)

1.93(.13)

1.03(.29)

.90(.32)

�8,780.8

GE .98(.01)

2.31(.26)

1.30(.34)

.94(.47)

�9,312.2

GM .97(.03)

3.66(.32)

1.75(.42)

1.60(.49)

�10,742.7

IBM .86(.09)

3.70(.24)

2.62(.85)

4.20(.71)

�12,150.3

IP .89(.05)

3.43(.18)

1.71(.32)

2.33(.42)

�10,954.9

JNJ .97(.02)

2.61(.25)

1.44(.33)

1.42(.31)

�9,884.1

MCD .97(.03)

2.82(.23)

1.33(.46)

1.99(.26)

�10,308.6

MRK .97(.02)

2.87(.22)

1.36(.25)

1.40(.28)

�10,020.3

MMM .86(.17)

2.12(.11)

1.16(.45)

1.93(.45)

�9,827.1

MO .81(.17)

3.57(.34)

3.12(1.81)

8.80(3.78)

�13,868.2

PG .97(.01)

2.84(.34)

1.85(.76)

5.60(2.41)

�12,443.8

S .95(.02)

4.16(.41)

3.06(.73)

3.10(.78)

�11,941.2

Firm

B. Model Diagnostics

Autocorrelation VR0 VR1 VR22Rr Q1 Q2

AXP .73 .17 .37 .44 .11 16.92 28.74**T .58 .21 .09 .27 .19 6.23 13.10CHV .36 .16 .12 .25 .15 16.12 27.52**KO .49 .17 .16 .28 .14 13.59 27.14**DIS .28 .22 .07 .26 .21 7.39 13.26DOW .32 .20 .19 .33 .17 27.51** 29.03**DD .19 .15 .11 .23 .14 15.91 13.76EK .13 .28 .01 .29 .28 2.47 3.54XOM .53 .14 .17 .27 .12 8.94 12.85GE .64 .14 .33 .39 .10 15.18 40.17**GM .53 .13 .24 .31 .10 8.24 11.52


TABLE 3 (Continued)

Firm


Autocorrelation VR0 VR1 VR22Rr Q1 Q2

IBM .24 .24 .04 .27 .24 7.22 20.65*IP .31 .17 .06 .21 .16 12.43 15.62JNJ .49 .16 .23 .32 .13 24.69** 35.30**MCD .30 .17 .12 .25 .16 40.95** 65.80**MRK .47 .14 .20 .29 .11 7.51 12.31MMM .23 .21 .03 .23 .20 6.01 14.41MO .09 .30 .01 .31 .30 .81 .62PG .10 .29 .03 .30 .28 1.46 7.65S .47 .21 .19 .33 .17 6.24 36.24**

Note.—The table reports the results of using the Kalman filter to fit a two-factor SARV model to theadjusted demeaned returns ( ) for the MMI firms. The model has a linear state-space representation given byrt

eqq. (12)–(14). Panel A reports the parameter estimates, asymptotic standard errors (in parentheses), and quasilog likelihood ( ). Panel B reports selected model diagnostics based on the parameter estimates in panel ALT

along with the associated forecast and filtered estimate of for each date in the sample: the2j p � � gb � rct t t

estimated first-order autocorrelation in ; the variance ratio (VR0) obtained by dividing the estimated variance2jt

of by the sample variance of ; the variance ratios (VR1 and VR2) obtained by dividing the estimated2 2j rt t

variance of the forecasts and filtered estimates of , respectively, by the estimated variance of ; the estimated2 2j jt t

coefficient of determination for a regression of the error in forecasting on the error in forecasting ; and2 2j rt t

Ljung-Box statistics (Q1 and Q2) based on the first 10 sample autocorrelations of and .2 2 2 2r /j r � jt tFt�1 t tFt�1


the magnitude of this effect, note that the unconditional variance of is2jt

given by . If our estimate of r is larger than our estimate of g, which2 2g � r

it is for 13 of the firms, then this suggests that accounts for more than halfct

the variance of . Hence, by fitting the two-factor specification, we capture2jt

a large nonpersistent component of the return variances that would be missedby either a one-factor SARV specification or a standard GARCH model.

The diagnostics in panel B provide more insights into this aspect of thetwo-factor specification. As anticipated, the model implies lower levels ofvolatility persistence than the GARCH(1, 1) model. The estimated first-orderautocorrelation in ranges from 0.09 for Philip Morris to 0.73 for American2jt

Express, with most of the estimates between 0.3 and 0.6. This also implieshigher estimates of : 0.13 to 0.30.8 These findings highlight2 2Var (j )/ Var (r )t t

the impact of including the second factor. Like the GARCH(1, 1) model,the two-factor specification imposes an upper bound of on1/3

.9 However, it does so within a framework that allows to2 2 2Var (j )/ Var (r ) jt t t

contain both persistent and nonpersistent components without restricting themagnitude of either. This makes it possible to capture large unpredictablechanges in that show up in the GARCH model as excess kurtosis in .2j zt t

The net effect is a drop in volatility persistence and an increase in explanatorypower.

We can better understand the contribution of the second factor to the ex-

8. Under the model, the first-order autocorrelation in is given by .2 2 2 2j fg /(g � r )t

9. This follows from the assumption that is Gaussian. To see why, note that 4z E(r ) pt t

. Thus , implying that cannot exceed4 4 2 2 2 2 2 2E(z )E(j ) Var (r ) p 3 Var (j ) � 2E(j ) Var (j )/ Var (r )t t t t t t t

.1/3


planatory power of the model by examining the ratios 2 2Var (j )/ Var (j )tFt�1 t

and .10 For the first ratio, which measures the extent to which2 2Var (j )/ Var (j )tFt t

is predictable using past observables, our estimates vary widely from firm2jt

to firm, ranging from 0.01 for Eastman Kodak and Philip Morris to 0.37 forAmerican Express. Not surprisingly, there is a relation between the magnitudeof the estimated ratio and the relative size of the g and r estimates. Firmswith large g estimates and small r estimates tend to have higher estimatedvariance ratios because g determines the variance of the persistent factor of

. To compare the forecasting performance of the two-factor specification to2jt

that of the GARCH(1, 1) model, we need to multiply our estimate ofby our estimate of to obtain an estimate2 2 2 2Var (j )/ Var (j ) Var (j )/ Var (r )tFt�1 t t t

of . These results are not reported in the table, but they2 2Var (j )/ Var (r )tFt�1 t

are generally comparable to those reported in table 2. The main exception isDow Chemical, which implies an estimate (0.04) that is no longer out of linewith those for the other firms.

Turning to the second variance ratio, which reveals how well we track thevariation in using both current and past observables, we find that our2jt

estimates are relatively stable across firms. All but two of them are between0.21 and 0.33. The difference in the estimates of the first and second varianceratios is an indication of how much information has about the nonpersistent2rt

component of . To assess this issue more directly, we examine estimates of2jt

the coefficient of determination for a regression of the error in forecastingon the error in forecasting .11 About three-quarters of these estimates are2 2j rt t

between 0.1 and 0.2 and the rest fall between 0.2 and 0.3. The implicationis that the unpredictable component of is a relatively noisy proxy for the2rt

nonpersistent component of the date t return variance. Of course, if the mod-ified MDH is well specified, then the unpredictable part of the date t tradingvolume should provide valuable additional information about this componentof . We will consider the evidence on this shortly.2jt

The final two diagnostics in panel B address goodness of fit. The Ljung-Box statistics for serial correlation in indicate that we reject the null2 2r /jt tFt�1

of no serial correlation for three of the firms: Dow Chemical, Johnson &Johnson, and McDonald’s. Recall that these same three firms, and two others,produced rejections under the GARCH(1, 1) model. The Ljung-Box statistics

10. To see how we construct these estimates, note that and2 2 2Var (j ) p g � r � MSEtFt�1 For

, where and denote the MSEs of and . In2 2 2 2 2Var (j ) p g � r � MSE MSE MSE j jtFt Fil For Fil tFt�1 tFt

steady state, we have and , where′ ′MSE p (g, r)P (g, r) MSE p (g, r)P (g, r) P pFor For Fil Fil For

and denote the steady-state values of and , the MSElim P P p lim P P PTr� TFT�1 Fil Tr� TFT tFt�1 tFt

matrices of our MMS forecasts and filtered estimates of . We estimate and by′(b , c ) P Pt t For Fil

using the QML parameter estimates and iterating on the Kalman filter MSE recursions untilconvergence.

11. We obtain an estimate of this coefficient as a by-product of fitting the model using theKalman filter. The variance of the error in forecasting is given by (see n. 10); the2j MSEt For

variance of the error in forecasting is given by , where2 2r MSE � Var (� ) Var (� ) p 2(� �t For t t

; and the covariance between the errors is obtained as part of computing the gain matrix2 2g � r )for the filter. To get the coefficient of determination, we just square the covariance and divideby the product of the variances.


for serial correlation in indicate that we reject the null of no serial2 2r � jt tFt�1

correlation for nine of the firms at the 5% level. For the GARCH(1, 1) model,we rejected the null at this level for 12 of the firms. Since these findingssuggest that the two-factor SARV specification performs at least as well asthe GARCH(1, 1) model, we conclude that the model does an adequate jobof capturing ARCH effects.

B. The Two-Factor SARV Specification for Returns and Volume

Our findings thus far support the view that the return variances contain bothpersistent and nonpersistent components. It remains to be seen, however,whether these findings are consistent with the predictions of the modifiedMDH. To address this issue, we refit the two-factor specification using bothreturns and trading volume. Table 4 summarizes the results. Panel A reportsthe parameter estimates, standard errors, and maximized quasi log likelihood.Panel B reports selected model diagnostics.

In table 4, notice first that for all but one of the firms the estimated valueof f is smaller than that reported for the univariate SARV specification intable 3. None of the f estimates exceeds 0.9, and only three exceed 0.8.Moreover, the 95% confidence intervals for the estimates in table 4 are gen-erally too narrow to encompass the estimated value of f reported in table 3.This provides our first indication that the predictions of the modified MDHmay not be supported by the data. According to the modified MDH, thepersistence in both the variance forecasts and the volume forecasts is due topersistence in the information flow, which is captured by under our two-bt

factor SARV specification. It follows, therefore, that if the estimated persis-tence of falls when we bring volume into the analysis, then the modifiedbt

MDH may be too restrictive to explain the observed dynamics of returns andtrading volume. We will explore this issue in more detail in Section IV.C.

Unlike the estimated value of f, the estimated value of g does not changedefinitively. About three-quarters of the estimates in table 4 are greater thanin table 3. However, many of the 95% confidence intervals for these estimatescontain the values reported in table 3. Perhaps more significantly, bringingvolume into the analysis has little impact on the estimated value of r. Whenwe fit the two-factor specification using only the returns, our r estimatesdepend on the estimated excess kurtosis in the standardized returns. Althoughthe estimates in table 3 indicate the presence of significant excess kurtosis,they provide no information about its source. In contrast, the r estimates forthe bivariate system in table 4 reflect the degree to which the nonpersistentcomponents of the return variances and trading volumes tend to move together.Since many of the estimates in tables 3 and 4 are similar, our results suggestthat much of the excess kurtosis in the standardized returns is linked to non-persistent volatility that is in turn related to nonpersistent trading volume.This is broadly consistent with what we would expect to find if there is alarge nonpersistent component to the information flow.


TABLE 4 Results of Fitting a Two-Factor SARV Model to Daily Returns andTrading Volume

Firm


f �r gr rr �v gv rv w LT

AXP .79(.05)

4.18(.30)

2.82(.42)

2.60(.52)

4.10(.10)

1.80(.14)

1.41(.13)

.00(.00)

�18,235.3

T .76(.05)

3.04(.27)

1.88(.34)

3.09(.68)

5.28(.15)

2.19(.18)

2.17(.22)

.00(.00)

�18,828.0

CHV .87(.05)

2.14(.11)

1.13(.43)

1.34(.34)

1.14(.03)

.49(.14)

.37(.12)

.38(.09)

�12,328.0

KO .62(.04)

2.77(.14)

2.18(.33)

.67(.78)

3.68(.05)

1.39(.10)

.89(.45)

.21(1.98)

�15,894.2

DIS .78(.04)

3.07(.21)

2.45(.37)

3.10(.65)

4.92(.14)

2.44(.18)

1.63(.17)

.00(.00)

�18,567.0

DOW .83(.09)

2.78(.12)

.83(.44)

2.77(.37)

2.29(.05)

.84(.15)

.54(.16)

.78(.13)

�15,622.5

DD .78(.04)

3.00(.11)

.72(.21)

2.23(.26)

2.35(.05)

1.03(.10)

.75(.11)

.26(.25)

�15,159.9

EK .64(.08)

2.77(.34)

4.22(1.33)

4.44(1.82)

1.12(.04)

.82(.14)

.48(.14)

.00(.00)

�16,020.1

XOM .65(.04)

1.91(.08)

1.24(.18)

.60(.43)

2.83(.04)

.86(.06)

.53(.27)

.47(.30)

�13,306.7

GE .75(.04)

2.15(.12)

1.57(.19)

.76(.24)

14.42(.22)

4.74(.27)

3.56(.23)

.00(.00)

�19,051.8

GM .44(.26)

3.60(.13)

1.96(.49)

�1.42(.68)

2.16(.06)

2.03(1.22)

.99(1.38)

1.78(1.68)

�18,741.7

IBM .73(.04)

3.03(.21)

3.12(.57)

4.11(.71)

8.24(.17)

3.96(.30)

2.34(.33)

.00(.00)

�20,663.0

IP .65(.05)

3.42(.14)

1.90(.33)

2.19(.57)

1.24(.02)

.60(.09)

.14(.10)

.32(.05)

�13,785.7

JNJ .68(.05)

2.43(.18)

1.84(.26)

1.14(.39)

2.56(.05)

1.02(.08)

.65(.07)

.00(.00)

�14,389.6

MCD .78(.03)

2.60(.15)

1.13(.18)

2.28(.30)

3.54(.07)

1.43(.10)

1.06(.06)

.00(.00)

�15,972.7

MRK .73(.06)

2.53(.12)

1.69(.24)

1.48(.31)

4.94(.08)

1.93(.15)

1.30(.17)

.00(.00)

�16,597.5

MMM .61(.06)

2.12(.10)

1.58(.28)

1.61(.53)

.87(.01)

.37(.03)

.26(.09)

.11(.13)

�11,328.1

MO .71(.08)

3.31(.61)

5.23(1.64)

7.71(3.86)

7.10(.27)

4.10(.48)

2.21(.84)

.00(.00)

�22,397.7

PG .79(.09)

1.88(.60)

3.02(1.15)

5.04(2.27)

2.14(.09)

1.08(.22)

.92(.28)

.00(.00)

�17,035.8

S .88(.05)

4.11(.22)

2.09(.95)

3.85(1.06)

1.13(.04)

.54(.19)

.43(.15)

1.01(.48)

�17,077.1

Firm


Autocorrelation VR0 VR1 VR22Rr

2Rv2Rrv Qr Qv

AXP .43 .19 .24 1.00 .15 .99 .99 165.62** 42.23**T .20 .23 .10 .95 .21 .95 .95 48.47** 31.47**CHV .36 .17 .21 .71 .14 .61 .64 35.59** 34.26**KO .56 .17 .25 .92 .13 .90 .90 32.29** 15.48DIS .30 .24 .18 .93 .20 .91 .91 20.89* 59.61**DOW .07 .21 .03 .46 .20 .34 .44 184.04** 41.00**DD .07 .16 .04 .71 .15 .68 .69 38.90** 40.73**EK .30 .29 .15 .93 .26 .92 .92 20.63* 45.63**XOM .53 .15 .23 .83 .12 .78 .78 29.74** 26.04**GE .61 .17 .32 .97 .12 .95 .96 40.04** 41.41**


TABLE 4 (Continued)

Firm



2Rv2Rrv Qr Qv

GM .29 .13 .07 .27 .13 .11 .21 95.49** 27.17**IBM .27 .26 .15 .89 .24 .86 .87 33.19** 90.66**IP .28 .17 .14 .56 .15 .44 .49 35.30** 33.34**JNJ .49 .18 .25 1.00 .14 1.00 1.00 45.09** 58.25**MCD .15 .20 .09 .84 .18 .82 .83 107.49** 47.99**MRK .41 .18 .22 .99 .15 .98 .98 29.07** 77.76**MMM .30 .21 .12 .92 .19 .90 .90 17.99 27.54**MO .22 .30 .13 .83 .28 .79 .80 7.00 48.06**PG .21 .29 .12 .92 .29 .91 .91 17.85 67.90**S .20 .21 .08 .41 .20 .24 .36 118.45** 4.60

Note.—The table reports the results of using the Kalman filter to fit a two-factor SARV model to theadjusted demeaned returns ( ) and trading volume ( ) for the MMI firms. The model has a linear state-spacer vt t

representation given by eqq. (17)–(20). Panel A reports the parameter estimates, asymptotic standard errors(in parentheses), and quasi log likelihood ( ). Panel B reports selected model diagnostics based on the parameterLT

estimates in panel A along with the associated forecasts and filtered estimates of : the2j p � � g b � r ct r r t r t

estimated first-order autocorrelation in ; the variance ratio (VR0) obtained by dividing the estimated variance2jt

of by the sample variance of ; the variance ratios (VR1 and VR2) obtained by dividing the estimated2 2j rt t

variance of the forecasts and filtered estimates of , respectively, by the estimated variance of ; the estimated2 2j jt t

coefficient of determination ( ) for a regression of the error in forecasting on the error in forecasting ;2 2 2R j rr t t

the estimated coefficient of determination ( ) for a regression of the error in forecasting on the error in2 2R jv t

forecasting ; the estimated coefficient of determination ( ) for a regression of the error in forecasting2 2v R jt rv t

on the errors in forecasting and ; and Ljung-Box statistics (Qr and Qv) based on the first 10 sample2r vt t

autocorrelations of the errors in forecasting and .2r vt t


The diagnostics in table 4 provide additional insights regarding the non-persistent component of volatility and its relation to trading volume. In general,the volatility persistence implied by the bivariate system is similar to thatobtained by fitting the two-factor specification to returns alone. The estimatedfirst-order autocorrelation in ranges from 0.07 for Dow Chemical and2jt

DuPont to 0.61 for General Electric, and the average value of 0.31 in table4 is close to the average value of 0.37 in table 3. The estimates of

are also similar to those reported in table 3. Seven of the2 2Var (j )/ Var (r )t t

estimates do not change at all, and none changes by more than 0.04. Thus,for most of the firms, bringing volume into the analysis does not have a majorimpact on our inference regarding either the persistence in volatility or itsvariability through time.

But, adding volume does improve the forecasting and filtering performanceof the model. Many of the estimates of in table 4 differ2 2Var (j )/ Var (j )tFt�1 t

from the values reported in table 3. The largest change is a drop from 0.24to 0.07 for General Motors. On average, however, the value of the ratioincreases by 0.02, which suggests that bringing in volume leads to a slightimprovement in forecasting performance. The effect of volume on the model’sfiltering performance is more definitive. Most of the estimates of

are greater than 0.7 in table 4 and most of the estimates2 2Var (j )/ Var (j )tFt t

in table 3 are less than 0.4. The average value of the ratio increases by 0.51,and half of the estimates are now greater than 0.9. Thus contemporaneous


trading volume appears to be very informative regarding the nonpersistentcomponent of daily volatility.

To assess this more directly, we consider the coefficients of determinationfor (i) a regression of the error in forecasting on the error in forecasting2jt

, (ii) a regression of the error in forecasting on the error in forecasting2 2r jt t

, and (iii) a regression of the error in forecasting on the errors in forecasting2v jt t

and . While the coefficient of determination is always less than 0.3 for2r vt t

the first regression, it often exceeds 0.9 for the second regression. This in-dicates that the unpredictable component of is a much better proxy for thevt

nonpersistent component of the date t return variance than the unpredictablecomponent of . The third regression indicates that, after we account for the2rt

unpredictable component of , the unpredictable component of has little2v rt t

additional explanatory power. The coefficient of determination is the same asfor the second regression for half the firms and shows only a small increasefor the other half.

Of course, all our conclusions are tempered by the fact that the restrictionsof the two-factor model are not supported by the data. We can see this byexamining the Ljung-Box statistics for serial correlation in the forecast errors.The statistic for the squared returns rejects at the 5% significance level for17 of the firms, which is almost twice the rejection rate observed in table 3.Thus the quality of the squared return forecasts deteriorates when we bringvolume into the analysis. If volume is a more precise volatility proxy thansquared returns, we would expect some deterioration in the quality of thesquared return forecasts even if the model were correctly specified. However,we would not expect the statistic for serial correlation in the volume forecasterrors to reject at a similar rate, which is what we find with 18 of the firmsrejecting at the 5% level.

C. The Three-Factor SARV Specification

Since fitting the two-factor model to both trading volume and squared returnscauses the quality of the squared return forecasts to deteriorate, it appears thatthe modified MDH fails to adequately characterize the joint dynamics ofvolume and volatility. To investigate further, we fit a less restrictive bivariateSARV specification that nests the two-factor model as a special case. Inparticular, we add a third factor to the model that affects only the squaredreturns process, thereby allowing for persistence in return volatility that isunrelated to persistence in trading volume. If this leads to a statistically sig-nificant improvement in goodness of fit, then it follows that the MMS varianceforecasts contain a component that is specific to information contained inlagged squared returns. In other words, the model provides direct evidenceon whether ARCH effects remain statistically significant after we account forthe information content of trading volume.

Table 5 summarizes the model-fitting results. We begin with the parameterestimates, standard errors, and maximized quasi log likelihood in panel A. In

Volatility,

Volum

e,and

Information

1577

TABLE 5 Results of Fitting a Three-Factor SARV Model to Daily Returns and Trading Volume

Firm


fa fb �r d gr rr �v gv rv w LT

AXP .98(.01)

.79(.05)

3.21(.54)

2.27(.60)

2.82(.43)

2.52(.51)

4.08(.10)

1.79(.14)

1.41(.13)

.00(.07)

�18,168.8

T .99(.01)

.75(.05)

2.43(.80)

1.39(.34)

2.20(.36)

2.97(.83)

5.31(.11)

2.22(.18)

2.14(.22)

.00(.11)

�18,801.4

CHV .99(.01)

.87(.05)

2.16(.17)

.44(.17)

1.15(.43)

1.22(.38)

1.14(.03)

.49(.14)

.40(.14)

.34(.11)

�12,323.1

KO .98(.01)

.62(.04)

2.50(.24)

1.09(.27)

2.12(.29)

.73(.44)

3.66(.05)

1.37(.09)

.92(.09)

.00(.35)

�15,864.3

DIS .99(.01)

.77(.04)

2.50(1.01)

1.23(.76)

2.76(.50)

3.00(.75)

4.98(.17)

2.46(.20)

1.60(.17)

.00(.05)

�18,553.5

DOW 1.00(.00)

.79(.09)

2.89(1.33)

1.72(.79)

1.62(.43)

1.67(1.85)

2.29(.05)

.87(.14)

.69(.80)

.60(.93)

�15,532.7

DD 1.00(.00)

.76(.04)

2.83(.23)

1.24(.34)

1.29(.23)

1.80(.32)

2.35(.05)

1.05(.09)

.76(.06)

.00(.07)

�15,110.1

EK 1.00(.01)

.63(.08)

1.36(5.87)

1.59(1.95)

4.59(1.37)

4.34(2.05)

1.15(.05)

.83(.13)

.46(.15)

.00(.02)

�16,010.1

XOM .99(.01)

.65(.04)

1.81(.15)

.68(.28)

1.22(.17)

.47(.20)

2.82(.03)

.86(.06)

.71(.04)

.00(.23)

�13,281.5

GE .98(.01)

.75(.04)

1.96(.15)

.76(.22)

1.53(.21)

.79(.25)

14.45(.22)

4.75(.28)

3.56(.24)

.01(.22)

�19,031.9

GM .99(.01)

.41(.21)

3.68(.50)

1.39(.44)

1.69(.47)

�.81(1.13)

2.16(.07)

2.16(1.28)

1.33(.88)

1.36(1.50)

�18,700.8

IBM 1.00(.00)

.70(.04)

.83(.24)

1.84(.34)

3.79(.62)

3.96(.76)

8.54(.17)

4.04(.29)

2.19(.36)

.00(.04)

�20,618.5

IP .99(.01)

.64(.05)

3.51(.60)

1.18(.64)

2.15(.37)

1.48(.95)

1.24(.02)

.60(.09)

.16(.17)

.30(.07)

�13,765.1

1578Journal

ofB

usiness

TABLE 5 (Continued )

Firm


fa fb �r d gr rr �v gv rv w LT

JNJ .99(.01)

.68(.05)

2.15(.28)

.89(.24)

1.87(.30)

1.13(.39)

2.57(.04)

1.02(.08)

.65(.07)

.00(.02)

�14,367.0

MCD .99(.01)

.77(.03)

2.06(.45)

1.19(.34)

1.38(.22)

2.24(.32)

3.58(.07)

1.45(.10)

1.03(.06)

.00(.02)

�15,935.2

MRK 1.00(.00)

.72(.06)

1.80(.45)

1.19(.42)

1.97(.28)

1.46(.36)

5.07(.09)

1.97(.15)

1.27(.17)

.00(.17)

�16,556.7

MMM .99(.01)

.60(.06)

2.07(.21)

.71(.21)

1.72(.29)

1.36(.60)

.87(.01)

.37(.03)

.27(.04)

.00(.00)

�11,316.6

MO .99(.01)

.71(.07)

3.16(.91)

.88(.22)

5.34(1.65)

7.65(3.94)

7.11(.25)

4.10(.47)

2.20(.82)

.00(.15)

�22,396.7

PG 1.00(.00)

.78(.09)

.49(.34)

1.19(.16)

3.10(.16)

5.09(2.16)

2.27(.05)

1.08(.23)

.91(.28)

.00(.00)

�17,019.7

S .98(.01)

.88(.05)

4.20(.62)

2.65(.86)

1.93(.48)

2.88(.92)

1.13(.04)

.54(.21)

.65(.24)

.88(.55)

�17,022.1

Firm



2Rv2Rrv Qr Qv

AXP .58 .25 .31 .87 .18 .81 .82 28.38** 42.22**T .35 .27 .16 .92 .23 .90 .90 33.45** 30.06**CHV .45 .16 .24 .75 .13 .66 .67 33.62** 34.23**KO .64 .20 .29 .85 .15 .78 .79 11.24 14.85DIS .41 .27 .23 .92 .22 .89 .89 16.04 58.46**DOW .60 .21 .38 .77 .13 .60 .62 39.01** 35.53**DD .43 .19 .27 .87 .14 .81 .82 14.77 36.71**EK .37 .32 .19 .93 .28 .91 .91 27.27** 43.82**XOM .66 .17 .30 .83 .12 .75 .76 8.95 25.64**GE .65 .19 .33 .89 .14 .83 .83 31.21** 41.33**GM .57 .13 .26 .43 .10 .15 .22 13.32 27.65**IBM .40 .33 .25 .91 .27 .87 .88 34.01** 86.19**

Volatility,

Volum

e,and

Information

1579

IP .53 .17 .27 .70 .13 .56 .59 18.71* 33.93**JNJ .57 .22 .28 .92 .16 .89 .89 29.51** 58.26**MCD .34 .25 .20 .84 .21 .79 .80 65.15** 45.72**MRK .56 .27 .34 .95 .19 .92 .92 18.65* 76.88**MMM .43 .22 .17 .94 .19 .93 .93 15.61 27.97**MO .24 .31 .13 .83 .28 .79 .81 9.05 47.73**PG .24 .31 .14 .92 .30 .90 .91 21.84* 66.70**S .53 .21 .25 .55 .16 .32 .40 36.86** 3.61

Note.—The table reports the results of using the Kalman filter to fit a three-factor SARV model to the adjusted demeaned returns ( ) and trading volume ( ) for the MMI firms. Ther vt t

model has a linear state-space representation given by eqq. (26)–(30). Panel A reports the parameter estimates, asymptotic standard errors (in parentheses), and quasi log likelihood ( ).LT

Panel B reports selected model diagnostics based on the parameter estimates in panel A along with the associated forecasts and filtered estimates of . These diagnostics2j p � � da � g b � r ct r t r t r t

are defined in the note to table 4.* Statistically significant at the .05 level.** Statistically significant at the .01 level.


most cases, the estimates of the parameters that are present in the two-factormodel show little change from the values reported in table 4. The estimateof fb, which measures the autocorrelation in the factor that captures commonpersistence in squared returns and trading volume, is quite stable. The largestchange is a drop from 0.83 to 0.79 for Dow Chemical. Moreover, the coef-ficients on this factor, gr and gv, and the coefficients on the nonpersistentfactor, rr and rv, are reasonably stable. This explains why we see the dete-rioration in the quality of the squared return forecasts noted in Section IV.B.When we fit the two-factor specification by QML, the forecast errors for thesquared returns receive relatively little weight because the forecast errors forvolume provide a much more precise signal about volatility. As a result,introducing a new factor that is specific to the squared returns has little impacton the estimates of the parameters present in the two-factor model.

On the other hand, the estimates of the remaining parameters suggest thatthe three-factor model captures an important component of return volatilitythat is missed by the two-factor specification. The third factor always entersthe model with a statistically significant coefficient, and in most cases the t-ratio for d is greater than three. In addition, it appears to be much morepersistent than the factor that is common to squared returns and trading vol-ume. The estimated autocorrelation in is at least 0.98 for every firm. Thusat

the model indicates that return volatility contains a highly persistent com-ponent that cannot be explained by common persistence in squared returnsand trading volume. This is inconsistent with the modified MDH and is in-consistent with the hypothesis that volume explains ARCH effects.

In light of these results, many of the changes to the diagnostics in panelB are predictable. For example, the estimated first-order autocorrelation in

is always higher than the value reported in table 4. It ranges from 0.242jt

for Philip Morris and Procter and Gamble to 0.66 for Exxon-Mobil. In addition,the estimate of is at least as large as the value reported2 2Var (j )/ Var (j )tFt�1 t

in table 4 for every firm. The lowest estimate is 0.13 (Philip Morris), and thehighest is 0.38 (Dow Chemical). These findings are consistent with the ob-served increase in the quasi log likelihood relative to the two-factor specifi-cation. Adding a third factor that is specific to squared returns allows themodel to capture more volatility persistence and improves the explanatorypower of the MMS variance forecasts.

Adding the third factor also reduces the cross-sectional variation in ourestimates of . With the exception of General Motors and2 2Var (j )/ Var (j )tFt t

Sears, all the estimates are between 0.70 and 0.95. This highlights the extentto which knowledge of the contemporaneous realizations of and resolves2r vt t

uncertainty regarding the value of . Most of the additional information about2jt

is contained in volume. Our parameter estimates imply that the coefficient2jt

of determination for a regression of the error in forecasting on the error2jt

in forecasting exceeds 0.6 in most cases. Moreover, if we add the error invt

forecasting as a second explanatory variable in this regression, the increase2rt


in the coefficient of determination is 0.01 or less for three-quarters of thefirms.

The improved forecasting performance of the model is reflected in theLjung-Box statistic for the squared returns. The number of rejections at the5% significance level drops from 17 in table 4 to 13 in table 5. Although thismay not be impressive in an absolute sense, it is on par with the performanceof the GARCH(1, 1) model reported in table 2. Thus we conclude that froma volatility forecasting perspective, the three-factor specification performsabout as well as standard volatility models. The evidence with respect tovolume forecasting is less favorable. As in table 4, the Ljung-Box statisticfor serial correlation in the forecast errors is significant at the 5% significancelevel for 18 of the firms, which indicates that the autocorrelation structure ofvolume is more complex than that implied by an ARMA(1, 1) model. Althoughfitting a more complex specification for the volume process might improvethe forecasting performance, we would not expect this to affect our conclusionsregarding the relation between trading volume and ARCH effects.

D. Three-Factor Forecasts and Filtered Estimates

We get a broader perspective on the performance of the three-factor modelby examining the output from the Kalman filter. Figure 1, for example, showsthe three-factor forecasts (top panel) and filtered estimates (bottom panel) ofthe return variances for the first stock alphabetically in the MMI index: Amer-ican Express. The plots look much like we expect given the discussion thusfar. The forecasts in the top panel are relatively persistent, but they displaymore high-frequency variation than is typical for the fitted conditional vari-ances from a GARCH model. We also see a prolonged period of higher thanaverage volatility during 1990–91 and a shorter volatility spike in 1998. Asimilar pattern is evident for the filtered estimates in the bottom panel, butthese estimates are much less persistent.

The only surprise in figure 1 is the prevalence of negative variance estimatesin the bottom panel. We see this for all the firms to one degree or another.While it might be taken as clear evidence of model misspecification, thepropensity of the filter to produce negative estimates is likely driven by vol-atility dynamics. Because changes in return volatility explain only a smallfraction of the variation in the squared returns and trading volume, the ef-fectiveness of the Kalman filter is limited by the low signal-to-noise ratio. Inaddition, because volatility is relatively persistent, it can spend protractedperiods below its long-run mean. The combination of these two propertiesincreases the likelihood that the filter will generate negative estimates overcertain periods.

A similar situation arises when one estimates conditional expected excessmarket returns. Asset pricing researchers typically model conditional expec-tations as linear in a set of instrumental variables because linear models aresimple to implement and the results are easy to interpret. However, linear


Fig. 1.—Forecasts (top panel) and filtered estimates (bottom panel) of the dailyreturn variance for American Express based on the parameter estimates for the three-factor specification.

models frequently produce negative fitted values.12 Investors generally wouldnot expect that the monthly market return is less than the one-month T-billrate. But linear models remain popular because we know that the variationin conditional expected returns is a small fraction of the overall variation inreturns. Therefore, we could easily obtain negative estimates of the conditionalexpected excess return from a correctly specified linear model in periods inwhich the equity risk premium is small. The circumstances in our analysis

12. Harvey (2001), e.g., fits a predictive regression to monthly excess returns on the value-weighted NYSE index and finds that 29% of the fitted values are negative for the years 1947–88.


Fig. 2.—Components of the daily variance forecasts for American Express basedon the parameter estimates for the three-factor SARV specification. The top panelplots the component of the variance forecast associated with the persistent factor thatis specific to squared returns. The middle panel plots the component of the varianceforecast associated with the persistent factor that is common to squared returns andtrading volume. These forecast components include the contribution of the long-runmean of the variance series. The bottom panel plots the forecast based on both factors.GARCH(1, 1) forecasts are included for comparison.

are similar: in periods with low volatility and volume, a well-specified linearfilter could easily produce negative filtered variance estimates. Since the neg-ative estimates are not a problem for our investigation of trading volume andARCH effects, we do not pursue this issue further.

Figure 2 examines the relation between trading volume and ARCH effectsunder the three-factor specification for American Express. The top panel plotsthe component of the variance forecasts related to the dynamics of , theat


persistent factor specific to squared returns. The middle panel plots the com-ponent of the variance forecasts related to , the persistent factor common tobt

squared returns and trading volume. The bottom panel shows the overallforecasts. In each part, we also include the fitted conditional variances fromthe GARCH(1, 1) model for comparison.

The forecast component in the top panel behaves much like the GARCHconditional variances. It generally tracks the GARCH series and displays asimilar level of persistence. This suggests that ARCH effects are closelyassociated with the persistent factor specific to squared returns. In contrast,the forecast component in the middle panel does not track the GARCH seriesvery well. It is much less persistent and spikes upward in many places inwhich the GARCH variance does not. Thus, even though the persistent factorcommon to squared returns and trading volume is important for explainingvolatility dynamics, it does not explain ARCH effects. When we combine thetwo components to generate the overall forecast in the bottom panel, the resultlooks like a less persistent version of the GARCH series.

Table 6 provides a closer examination of the relation between the forecasts,filtered estimates, and GARCH conditional variances. Under the model, wecan decompose the forecast, , and filtered estimate, , into three com-2 2j jtFt�1 tFt

ponents: a persistent component associated with common movements insquared returns and trading volume, a persistent component associated withmovements specific to squared returns, and a nonpersistent component as-sociated with common movements in squared returns and trading volume.Panel A of the table reports the sample variance of the forecasts, the samplevariance of the forecast components, and the sample correlation of each serieswith the GARCH conditional variances. Panel B reports the same quantitiesfor the filtered estimates.

Panel A reveals that the sample variance of varies widely from firm2jtFt�1

to firm. It ranges from a low of 0.63 for Exxon-Mobil to a high of 11.75 forPhilip Morris. Most of the variation is due to the component associated withcommon persistence in squared returns and trading volume, , ratherg br tFt�1

than the component associated with persistence specific to squared returns,. The sample variance of ranges from 0.28 to 11.90, comparedda g btFt�1 r tFt�1

to a range of 0.06 to 3.21 for the sample variance . The variance of thedatFt�1

third component, , is always zero because the nonpersistent factor, ,r c cr tFt�1 t

is unpredictable using lagged observables. Since fitting a dynamic factor modelby QML does not guarantee orthogonality of the extracted factors, there isusually a discrepancy between the variance of and the sum of the com-2jtFt�1

ponent variances. However, the differences are not large enough to have ameaningful effect on our inferences.

Consistent with the plots in figure 2, we find that the correlation betweenand the GARCH conditional variance, , tends to be relatively low.g b hr tFt�1 t

It is less than 0.5 for 15 of the firms and less than 0.3 for five of them. Incontrast, the correlation of with is greater than 0.5 for 13 firms andda htFt�1 t

greater than 0.8 for five. This correlation also equals or exceeds the correlation


between and for seven of the firms. These results support our earlier2j htFt�1 t

findings regarding the relation between trading volume and ARCH effects.Although trading volume contains information about future return volatility,accounting for this information does little to diminish the importance of ARCHeffects.

Panel B of the table conveys a similar message. The variability of differs2jtFt

widely across firms, ranging from 1.78 for Exxon-Mobil to 75.65 for PhilipMorris. Most of the variation is due to and , the components as-g b r cr tFt r tFt

sociated with common movements in squared returns and trading volume.And, as before, tends to have a lower correlation with the GARCHg br tFt

conditional variance than does. In fact, if we compare the results fordatFt

these two components, we find that most of the correlations in panel B arewithin 0.1 of the corresponding correlations in panel A. This is a consequenceof the persistence in and . Since a large fraction of the variation in thesea bt t

factors is predictable, we obtain a limited amount of new information aboutthe values of and by observing and .2da g b r vt r t t t

Note that the variability of generally accounts for only 5%–35% ofr cr tFt

the total variability of . This is lower than we would expect given the2jtFt

parameter estimates. To see why it is lower, recall that is obtained by2jtFt

adjusting up or down to reflect the error in forecasting and . With2 2j r vtFt�1 t t

only two forecast errors, we cannot decompose into three mutually un-2jtFt

correlated components. Since the correlation between and is rea-g b r cr tFt r tFt

sonably large, the sample variance of is often 20%–30% larger than the2jtFt

sum of the component variances. Although this does not affect our interpre-tation of , it does suggest that the parameter estimates and diagnostics2jtFt

provide a more precise picture of the relative importance of the differentcomponents of volatility.

Nonetheless, we can see from the remaining entries in panel B that thefiltered estimates capture an important aspect of volatility dynamics that ismissed by standard GARCH models. For every firm, the correlation between

and is less than 0.1 in absolute value. Moreover, the correlation betweenr c hr tFt t

and is always less than the correlation between and . So, regardless2 2j h j htFt t tFt�1 t

of how we look at the evidence, it points to the existence of a significantnonpersistent component of that can be captured only by using the con-2jt

temporaneous realizations of and .2r vt t

E. Additional Insights

Our empirical analysis reveals that a central prediction of the modified MDHis clearly at odds with the data. According to the MDH, a latent variabledirects daily changes in the mean of trading volume and the variance of returns.The evidence, however, points to a more complex dynamic relation betweenthe two series. Specifically, it indicates that there is a predictable componentof the return variance that is unrelated to the common variation in laggedsquared returns and lagged trading volume. To capture this component, we


TABLE 6 Decomposition of the Three-Factor Forecasts and Filtered Estimates

A. SARV Forecasts

Variance of Forecast/Component Correlation with GARCH

Firm 2jtFt�1 datFt�1 g br tFt�1 r cr tFt�12jtFt�1 datFt�1 g br tFt�1 r cr tFt�1

AXP 6.42 2.64 3.57 .00 .85 .81 .44 .00T 2.34 .86 1.70 .00 .70 .44 .50 .00CHV .69 .06 .66 .00 .69 .49 .56 .00KO 1.76 .51 1.27 .00 .70 .75 .36 .00DIS 3.57 .62 3.52 .00 .57 .42 .40 .00DOW 2.56 2.24 1.00 .00 .86 .84 .11 .00DD 1.36 1.09 .70 .00 .77 .91 �.06 .00EK 7.09 .88 6.81 .00 .69 .16 .65 .00XOM .63 .22 .42 .00 .72 .72 .36 .00GE 1.22 .22 .95 .00 .71 .74 .45 .00GM 1.45 1.10 .28 .00 .89 .95 .15 .00IBM 6.28 1.98 5.95 .00 .62 .26 .48 .00IP 2.00 .68 1.49 .00 .63 .52 .37 .00JNJ 1.56 .32 1.24 .00 .71 .69 .45 .00MCD 1.38 .74 .85 .00 .76 .76 .26 .00MRK 1.97 .66 1.51 .00 .68 .56 .41 .00MMM .85 .17 .73 .00 .56 .68 .28 .00MO 11.75 .15 11.90 .00 .59 .18 .57 .00PG 5.08 .48 4.63 .00 .64 .36 .55 .00S 4.49 3.21 1.25 .00 .94 .84 .43 .00

B. SARV Filtered Estimates

Variance of Estimate/Component Correlation with GARCH

Firm 2jtFt datFt g br tFt r cr tFt2jtFt datFt g br tFt r cr tFt

AXP 17.73 2.74 5.73 3.37 .56 .79 .39 .07T 14.24 .88 3.05 5.37 .36 .43 .44 .09CHV 2.23 .06 .86 .63 .43 .48 .51 .05KO 5.28 .52 3.30 .20 .44 .74 .25 .04DIS 16.00 .63 5.97 4.20 .32 .41 .35 .05DOW 5.39 2.26 1.59 1.03 .57 .84 .07 �.02DD 4.90 1.10 1.21 1.60 .35 .90 �.09 �.06EK 39.26 .88 17.39 6.81 .37 .14 .49 .05XOM 1.78 .22 .98 .11 .45 .71 .26 .04GE 3.25 .23 1.70 .30 .50 .72 .39 .09GM 2.37 1.12 1.72 .17 .72 .94 .10 �.04IBM 28.47 1.98 12.20 6.10 .31 .26 .36 .00IP 5.40 .69 3.61 .21 .40 .51 .26 .01JNJ 5.17 .33 2.65 .49 .46 .67 .37 .08MCD 6.71 .75 1.42 2.57 .36 .75 .21 .01MRK 6.38 .66 2.94 .88 .42 .55 .33 .04MMM 4.90 .18 2.05 .77 .28 .66 .20 .05MO 75.65 .15 23.83 24.80 .29 .17 .47 .03PG 38.27 .48 7.52 17.71 .29 .35 .49 .05S 10.08 3.37 1.61 2.83 .62 .81 .38 .00


Note.—The table reports selected sample statistics for the forecasts and filtered estimates of the returnvariances obtained by fitting the three-factor SARV model to the adjusted demeaned returns and trading volumefor the MMI firms. Under the three-factor model, the return variance for date t is given by 2j p � � da �t r t

. To construct the table, we use the Kalman filter to obtain a MMS forecast and filtered estimate ofg b � r cr t r t

for each date in the sample. Panel A reports the sample variance of the MMS forecasts, the sample variance2jt

of each component of these forecasts, the sample correlation of these forecasts with the conditional returnvariances from the GARCH(1, 1) model, and the sample correlation of each component of these forecasts withthe GARCH(1, 1) conditional variances. Panel B reports the sample variance of the MMS filtered estimates,the sample variance of each component of these estimates, the sample correlation of these estimates with theconditional return variances from the GARCH(1, 1) model, and the sample correlation of each component ofthese estimates with the GARCH(1, 1) conditional variances. The parameter estimates used to construct theGARCH conditional variances are shown in table 2, and the estimates used to construct the MMS forecastsand filtered estimates are shown in table 5.

must use information that is specific to the lagged squared returns. In otherwords, we find that ARCH effects explain part of the persistence in returnvolatility even after we account for the information about volatility containedin trading volume.

This finding may seem surprising given the results of Lamoureux andLastrapes (1990). They report that lagged squared returns contain little, if any,information about return volatility after accounting for the information con-tained in trading volume. However, Fleming, Kirby, and Ostdiek (2006) showthat this is due to their econometric specification. Specifically, with theirvolume-augmented GARCH(1, 1) model, it is impossible to simultaneouslyallow for strong ARCH effects and capture the contemporaneous relationbetween volume and volatility. Using a more flexible volume-augmentedGARCH model, Fleming et al. find that ARCH effects remain strong even inthe presence of contemporaneous trading volume. Thus, either way we lookat the evidence, it does not support the view that trading volume subsumesARCH effects in daily returns.

Nonetheless, the model-fitting results provide a useful perspective on thelinkages between stochastic volatility, trading volume, and ARCH effects. Thefirst important insight is that the nonpersistent component of volatility isresponsible for a large fraction of its variability through time. For example,the parameter estimates for the three-factor model suggest that this nonper-sistent component typically explains over two-thirds of the variability in thereturn variance. Because our analysis reveals that the MDH is misspecified,we cannot argue that this component of volatility reflects the impact of un-expected information arrivals. We can, however, make a strong case that anyspecification of the MDH in which the information flow is modeled as highlypersistent has little chance of success.

Our results also indicate that conventional GARCH models fail to capturea large component of return volatility. This conclusion is supported by recentevidence from the realized volatility literature. Andersen and Bollerslev(1998), for example, construct the realized variances of daily currency returnsby cumulating the squared five-minute returns over the course of the tradingday. They find that the conditional variances obtained by fitting a GARCH(1,1) model to the daily returns explain only around 40%-50% of the dailyvariability in the realized variances. We would expect to find some unexplained


variability due to measurement error, but the results seem too dramatic toreflect this alone. For instance, if the returns were generated by a SARVdiffusion, then the variance of the measurement error would be less than 2.5%of the variance of the latent volatility factor obtained by integrating over thesample path of the instantaneous return variance (Andersen and Bollerslev1998).

Another important insight from our results is that the nonpersistent com-ponent of return volatility is closely tied to the nonpersistent component oftrading volume. This is broadly consistent with the predictions of the MDH.We do not, however, need to impose the restrictions of the MDH to establishthis link. Using the three-factor specification, which allows for the possibilitythat volume and volatility are completely unrelated, we find that the nonper-sistent component of trading volume explains 60%–80% of the daily vari-ability in the nonpersistent component of return variance. Therefore, the resultssuggest that we can use trading volume to construct better estimates of returnvolatility than could be obtained using conventional GARCH models.

There is one caveat regarding these findings. Our analysis is based on linearmethods of estimation and inference to highlight the relation between tradingvolume and ARCH effects. As a consequence of using these methods, weessentially obtain linear least-squares estimates of both the fixed parametersand the unobserved return variances. One of the undesirable properties ofleast-squares estimates is that they do not display robustness to outliers. Sincewe occasionally see extreme realizations for trading volume, our results mightoverstate the explanatory power of this variable.

On the other hand, we cannot rule out the possibility that trading volumecarries more information about return volatility than our results suggest. Per-haps our approach of specifying linear models for the mean of trading volumeand the variance of returns is inappropriate. It might be better, for example,to model the logarithm of these variables as linear in a set of unobservedfactors, thereby constraining the original variables to be strictly nonnegative.This would impose large computational demands since it would almost cer-tainly require the use of simulation techniques to fit the models. But it wouldalso offer the opportunity to examine the robustness of the methodologydeveloped here.

V. Conclusions

The modified MDH provides a complete description of the joint dynamics ofdaily return volatility and trading volume once the dynamics of the informationflow are specified. Most recent work assumes that the information flow or itslogarithm follows an AR(1) process. We show that this is not likely to besatisfactory. If the information flow follows an AR(1) process, then it mustdisplay the same level of persistence as the MMS linear forecasts of the returnvariances, and these forecasts are identical in form to the conditional variances


implied by a GARCH(1, 1) model. Although we would expect to observesome clustering in information arrivals, it is unlikely that the information flowwould display the strong persistence typically implied by GARCH models.

To overcome this problem, we propose a less restrictive specification inwhich the information flow may have both persistent and nonpersistent com-ponents. Fitting this specification to the data, we find that a substantial com-ponent of daily return volatility is unpredictable. Moreover, the unpredictablecomponent of volatility is closely tied to the unpredictable component ofcontemporaneous trading volume. These findings are broadly consistent witha version of the MDH in which the innovation to squared returns is correlatedwith the innovation to trading volume because a substantial portion of theinformation flow is unanticipated news. Since modeling the information flowas an AR(1) process implies that the correlation between these innovationsshould be zero, our results may explain some of the evidence against themodified MDH reported in the recent literature.

Ultimately, our analysis indicates that a two-component specification failsto adequately characterize the joint dynamics of volume and volatility. Ap-parently a significant portion of the variation in the persistent component ofvolatility does not coincide with the variation in the persistent component ofvolume. In other words, we capture some of the predictable variation involatility using information that is common to lagged squared returns andlagged trading volume, but the remainder is related to information that isspecific to lagged squared returns. Thus, in contrast to previous studies, wefind no evidence that ARCH effects disappear once we account for the in-formation about volatility contained in trading volume.

With respect to future research, our analysis suggests that it would beinteresting to consider a more general specification of the modified MDH thatdoes not require all price changes to be accompanied by trading volume. Thiscould be accomplished using an approach like that of Tauchen and Pitts (1983).They assume that after the release of new information, part of the change ineach trader’s reservation price is common to all market participants. Sincethe common part of the change in reservation prices plays no role in generatingtrade, this leads to an MDH with some fraction of the daily return volatilityunrelated to daily trading volume. Adding this feature to the modified MDHshould bring the predictions of the model more in line with the data.

Future research could also exploit recent advances in econometric methodswhen exploring this issue. Although our methodology makes it easy to un-derstand the predictions of the MDH regarding trading volume and ARCHeffects, a fully parametric approach would be more informative when inves-tigating other aspects of the model. Because it is now commonplace to estimatemultivariate SARV models by Markov chain Monte Carlo methods, likeli-hood-based estimation and inference are a tractable proposition for the MDH.This raises the possibility of in-depth model comparisons that would provideinsights beyond those developed here.


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