can idiosyncratic volatility help forecast stock market volatility?

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International Journal of Foreca

Can idiosyncratic volatility help forecast stock market volatility?

Nicholas Taylor

Cardiff Business School, Cardiff University, Cardiff CF10 3EU, United Kingdom

Abstract

This paper examines the predictive power of idiosyncratic volatility in the context of daily stock market volatility dynamics.Specifically, the relative performance of various models of market volatility is considered with respect to whether idiosyncraticvolatility is excluded or included as an explanatory variable in such models. Using high frequency data covering the thirtystocks within the Dow Jones Industrial Average (DJIA) index, the results indicate that the inclusion of idiosyncratic volatilityleads to significant in-sample and out-of-sample improvements in the fit of all the volatility models considered. These results areshown to be relatively robust to the loss function adopted by the forecaster, with reasonable forecast accuracy improvementsavailable to such forecasters.© 2008 International Institute of Forecasters. Published by Elsevier B.V. All rights reserved.

JEL Classification: C53; G14

Keywords: Financial markets; Volatility forecasting; Loss function

1. Introduction

The importance of measuring and forecasting riskin financial markets has motivated a vast body ofliterature on the dynamics of asset return volatility (seePoon and Granger, 2003, for a comprehensive review).All of the models proposed in this body of literatureare, without exception, based on the highly time-dependent nature of volatility in each of the marketsconsidered. However, they differentiate themselvesfrom each other by innovating in terms of modelspecification, by using alternative definitions ofvolatility, or by enriching the informational contentof the model (see Franses and McAleer, 2002, for an

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overview of the models used in the context of financialmarkets). It is to the latter tranche of the literature thatthis paper contributes. In particular, we introduce andexamine volatility models of DJIA index returns thatexplicitly allow for intraday variation in the overallamount of private information flow in the relevantmarket. We demonstrate that improved forecasts ofmarket volatility can be obtained by doing this.

The information content of the volatility modelsproposed in the literature is ultimately based on one ofthree models, viz., the mixture of distributions model(Andersen, 1996; Clark, 1973; Epps & Epps, 1976;Foster & Viswanathan, 1993, 1995; Harris, 1987;Liesenfeld, 2001; and Tauchen & Pitts, 1983), thesequential information arrival model (Copeland,1976; Jennings, Starks, & Fellingham, 1981), or the

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2 Kelly (2007) provides counter-evidence that suggests thatidiosyncratic volatility is unrelated to private information.3 Campbell, Lettau, Malkiel, and Xu (2001) and Duffee (2001)

both find, inter alia, that monthly frequency market volatility ispositively associated with (aggregate) idiosyncratic volatility — aresult consistent with the arguments that idiosyncratic volatility and

463N. Taylor / International Journal of Forecasting 24 (2008) 462–479

no-arbitrage martingale model (Ross, 1989). Despitedifferences in their underlying motivations, all of thesemodels predict that the return volatility will be pro-portional to the (unobservable) rate of informationarrival (i.e., information flow). Given this motivation,a number of studies have demonstrated that theperformance of volatility models can be greatlyimproved by incorporating proxies for informationflow in their specification. Perhaps most notably, theinclusion of contemporaneous trading volume withinthe specification of such models has been shown tolead to significant improvements in their fit to the data(see Karpoff, 1987, for an early survey, and Bessem-binder & Seguin, 1993; Bollerslev & Jubinski, 1999;Lamoureux & Lastrapes, 1990; and Luu & Martens,2003, for more recent examples). Despite thesesuccesses, trading volume may not be the mostaccurate measure of information flow. This is becausetrading volume may be driven by factors other thaninformation flow; for example, trading may be liquid-ity motivated, and/or may be the result of divergenttrader opinion. Indeed, many studies have found thatlagged trading volume is not helpful in forecastingvolatility (see, e.g., Brooks, 1998; Donaldson &Kamstra, 2005; Heimstra & Jones, 1994; Lamoureux& Lastrapes, 1994; and Richardson & Smith, 1994).For these reasons, an alternative measure of informa-tion flow is considered in the current paper.

The noisiness of the trading volume measure ofinformation flow has motivated a number of authors topropose alternative measures of information flow.Mostnotably, in the context of volatility models, a number ofstudies have used either firm-specific news headlines(see, e.g., Berry & Howe, 1993; Kalev, Liu, Pham, &Jarnecic, 2004; Melvin & Yin, 2000; and Mitchell &Mulherin, 1994) or macroeconomic announcementdata (see, e.g., Andersen & Bollerslev, 1998; Flannery& Protopapadakis, 2002; and Jones, Lamont, &Lumsdaine, 1998) as inputs into information flowmeasures. The disappointing performance of modelsbased on these measures is often explained with refer-ence to the nature of the information flow considered;specifically, these measures attempt to proxy public, asopposed to private, information flow.1 Consequently,

1 The key distinction between public and private information is thatthe former affects prices as soon as it becomes known, while the latter isrevealed over time through the action of trading (French&Roll, 1986).

the large proportion of unexplained volatility is arguedto be due to the effects of private information flow; seeFrench and Roll (1986), Barclay, Litzenberger, andWarner (1990), and Jones, Kaul, and Lipson (1994),who find that return volatility is primarily driven byprivate information; and Darrat, Zhong, and Cheng(2005), who provide evidence in favour of thisconjecture in the context of the relationship betweenreturn volatility and trading volume. Given thisevidence, we consider a measure of private informationflow, and examine its importance with regard to thedynamics of market return volatility.

As private information is more common with respectto firms and industries than to the broadmarket, we use ameasure of idiosyncratic volatility as our proxy forprivate information flow. This particular reasoning iscommonly associatedwithRoll's (1988) conjectures thatstocks with high (low) levels of idiosyncratic volatilityare associated with either high (low) rates of ‘privateinformation [flow]’ or an ‘occasional frenzy unrelated toconcrete information’. Despite a large number of papersdemonstrating the empirical validity of the formerconjecture with ever richer cross-sectional (firm-speci-fic) datasets (Durnev, Morck, Yeung, & Zarowin, 2003;Durnev, Morck, & Yeung, 2004; Ferreira & Laux, 2007;Jin & Myers, 2006; and Morck, Yeung, & Yu, 2000),2

few (if any) studies have examined the issue usingaggregate intraday time series data.3 This is somewhatsurprising, given that if there is indeed a positive cross-sectional association between private information flowand idiosyncratic volatility, then this relationship shouldhold in aggregate (and over time), and hence we shouldexpect to observe a positive time series relationshipbetween aggregate idiosyncratic volatility and aggregateprivate information flow. Furthermore, given that returnvolatility is a positive function of private information

private information flow are positively related, and that markevolatility is positively associated with private information flowHowever, these studies choose to rationalise their results via theargument that both market and idiosyncratic volatilities co-vary withfuture economic conditions.

t.

464 N. Taylor / International Journal of Forecasting 24 (2008) 462–479

flow (see, e.g., Ross, 1989), it follows that we shouldobserve a positive relationship between aggregate returnvolatility (i.e., market volatility) and aggregate idiosyn-cratic volatility, with the former volatility dependent onthe latter volatility. It is this hypothesised relationshipthat is examined in this paper.

While many studies have used idiosyncratic vola-tility to explain cross-sectional variation in expectedreturns (see Chua, Goh, & Zhang, 2007; and Guo &Savickas, 2006, for recent examples), few haveconsidered the explanatory power of idiosyncraticvolatility with respect to future market volatility. Ofthe studies that have been conducted, all haveexamined the relationship between market and idio-syncratic volatility using low frequency data (see, e.g.,Campbell et al., 2001; Duffee, 2001). As an alternativeto these papers, we choose to examine this relationshipusing high frequency intraday data. This is because itis at this frequency that the economic theory under-lying this relationship is likely to hold. For instance,the relationship between return volatility and informa-tion flow can be based on Ross's (1989) no-arbitragemartingale model. In this model, disequilibrium isassumed to be short-lived, with its removal achievedvia the process of arbitrage. Given the speed at whichthese arbitrageurs operate, it follows that the centralprediction of this model (i.e., that return volatility isa positive function of information flow) should beobserved when using high frequency data. Thisreasoning predicts a positive relationship betweenhigh frequency measures of idiosyncratic volatilityand private information flow, and hence between highfrequency measures of market and idiosyncraticvolatility.

Given the importance of appropriatemarket volatilityforecasting in areas such as risk management and optionpricing, the relationship between market and idiosyn-cratic volatility is examined in a forecasting context.Specifically, we investigate whether the inclusion ofidiosyncratic volatility in market volatility models leadsto improved forecasts of daily market volatility. Toanticipate some of the results, we find that marketvolatility is positively associated with idiosyncraticvolatility. Moreover, market volatility forecasts pro-duced bymodels that incorporate idiosyncratic volatilityare more accurate than the forecasts produced byexisting (and competing) volatility models. In addition,these results are found to be relatively robust to variation

in the loss function employed, and are shown to providenon-trivial utility improvements to forecasters.

The rest of this paper is organised as follows. Thenext section provides definitions of idiosyncratic andmarket volatility, and is followed by a section thatcontains the modelling framework used to incorporateidiosyncratic volatility into the dynamics of marketvolatility. Section 4 contains results pertaining tovarious aspects of the estimated volatility modelsconsidered, and the final section concludes.

2. Definitions

This section is divided into three parts. The first twoparts provide concise definitions of idiosyncratic andmarket volatility, respectively; while the third partcontains a statement of this study's central hypothesisrelating these measures of volatility.

2.1. Idiosyncratic volatility

The measure of idiosyncratic volatility used in thispaper is based on the average volatility of the errorsfrom the CAPM single-factor market model. Thismeasure of idiosyncratic volatility is closely related tothe ‘R2-based firm-specific variation’ measure (see,e.g., Durnev et al., 2003; Ferreira & Laux, 2007;Morck et al., 2000; Roll, 1988), the ‘non-marketvolatility’ measure (Duffee, 2001), the ‘average firm-specific volatility’measure (Campbell et al., 2001), the‘firm-level return dispersion’ measure (see, e.g.,Stivers, 2003), and the ‘average stock variance’ mea-sure (Goyal & Santa-Clara, 2003). Of these measures,it is the first measure that has been most commonlyemployed in the literature. However, the nature of thenull hypothesis examined in the current paper requiresthe use of a high frequency time-varying measure ofaggregate idiosyncratic volatility. As the R2-basedmeasure uses the full sample variance of the marketmodel errors in its construction (scaled by the fullsample return variance), this requirement cannotbe fulfilled by this particular measure. Rather, thevariance of the market model errors must be con-structed over short periods of time. Therefore, wefollow Campbell et al. (2001) and Duffee (2001), andconstruct high frequency versions of their low fre-quency aggregate measures of the instantaneousvolatility of the market model errors.


The CAPM single-factor market model states thatreturns to the jth constituent stock are a function ofcontemporaneous returns to the market portfolio and tothe industry portfolio to which each stock belongs. Inaddition, given the likely presence of bid-ask bounceeffects within the high frequency data used (Roll,1984), the error term in the model is assumed to followan MA(1) process.4 Such a model has the followingrepresentation:

Rj;s ¼ wj;0 þ wj;1RM ;s þ wj;2RP;s þ 1þ hjL� �

�j;s; ð1Þwhere Rj,s is the return to the jth constituent stock attime s; RM,s is the return to the market portfolio(calculated as the weighted average of all stocks withinthe index, but with the jth stock excluded); RP,s is thereturn to the industry portfolio to which the jth stockbelongs (calculated as the weighted average of allstocks in the same industry as the jth stock, but withthe jth stock excluded); L is a lag operator; and εj,s is asuitably defined (orthogonal) innovation.5

It is a measure of the variance of the error term inEq. (1) that forms the basis of the measure of realisedidiosyncratic volatility used in this paper. Specifically,we define realised idiosyncratic volatility as (the

4 We are following Zhou (1996), Corsi, Zumbach, Müller, andDacorogna (2001), and Andersen, Bollerslev, Diebold, and Labys(2003), in using a simple moving-average process to allow for themicrostructure noise present in the return process. In doing this, weare explicitly assuming that the relative amounts of return volatilityand noise volatility remain constant over time. This is clearlysomewhat restrictive, as it is known that, while the noise volatilityremains fairly constant, the return volatility varies considerably overtime. Indeed, this observation provides the main motivation foralternative methods of constructing realised volatility measures inthe presence of microstructure noise; see Hansen and Lunde (2006)for an overview. For example, Owens and Steigerwald (2006) use aKalman filter to weight each observation according to itscontribution to the return-noise volatility ratio. While such methodsare undoubtedly more efficient than most competing approaches,the computational ease associated with using simple moving-average filters means that they remain a popular alternative.5 Idiosyncratic volatility can also be constructed using the three-

factor model of Fama and French (1993); see, e.g., Guo andSavickas (2006), and Chua et al. (2007). However, we construct arealised measure of daily idiosyncratic volatility using intraday data;specifically, the residuals from Eq. (1). As the variables used in theFama-French three-factor regression model are only available overdaily (or lower) frequencies (in particular, the returns to the twozero-investment portfolio variables in the model), a realisedmeasure of daily idiosyncratic volatility based on such a model isnot available in the current context.

square root of) the weighted average of the sum ofintraday squared market model residuals to each stock,that is,

r̂I ;tuXjac

wj;t

Xsat

�̂2j;s

!1=2

; ð2Þ

where c ∊ {1,2,…,C} is the set of all C stocks withinthe market portfolio; wj,t represents the weightingscheme and is dictated by the type of market portfolioused (price-weighting in the current application, asstocks within in the DJIA index are considered); ε ̂j,s isthe residual in Eq. (1) measured using time-sfrequency data; s ∊ (t−1,t]; and t corresponds to datafrequencies that are lower than or equal to the time-sdata frequency.

2.2. Market volatility

Using the same construction principles as above,market volatility is defined as (the square root of) thesum of squared time-s market returns, that is,

r̂M ;tuXsat

R2M ;s

!1=2

; ð3Þ

where RM,s is the market portfolio return at time s.6 Inthe current application, we use five-minute (time-s)frequency data in Eqs. (1), (2), and (3), and integrateover the trading day to give daily (time-t) frequencymeasures of realised idiosyncratic and marketvolatility.

To illustrate how the above measures are con-structed using these data frequencies, consider thefollowing calculation of daily frequency measures ofrealised market volatility. First, market returns aresampled every 5 minutes within the sample used.These returns are then squared, and the daily sums ofthese squared returns are calculated. Finally, the squareroot of these values is taken to give the above dailymeasure of realised market volatility.

6 See Andersen, Bollerslev, Diebold, and Labys (2001, 2003), andBarndorff-Nielson and Shephard (2001, 2002), for the theoreticaljustification of the use of daily integrated volatility measures basedon intraday squared returns, and Hansen and Lunde (2006) andBandi and Russell (2008) for a discussion of the empirical issuespertaining to the construction of these measures.

7 See Meddahi (2003) for a theoretical justification of an ARMA-based representation of realised volatility. Moreover, realisedvolatility models based on ARMA-based representations have beenshown to deliver more accurate volatility forecasts than those basedon GARCH-based representations; see, e.g., Bollerslev and Wright(2001), and Galbraith and Kisinbay (2005).8 Chan and Fong (2006) show that the number of trades has more

explanatory power than alternative measures of trading volume inthe context of return volatilities of individual stocks within the DJIAindex. However, to examine the validity of the traditionally-useddollar trading volume within the context of market volatilitymodels, we refrain from using such a measure of trading volume.


2.3. The central hypothesis

Given the above definitions of idiosyncratic andmarket volatility, we test the null hypothesis that thesemeasures are unrelated against the alternative hypoth-esis that they are positively related. A rejection of thisnull in favour of the alternative hypothesis is consistentwith the economic arguments outlined in the previoussection; most notably, that idiosyncratic volatility andprivate information flow are positively related. Tests ofthis null hypothesis are performed via various marketvolatility models. Specifically, the performances ofmarket volatility models that include idiosyncraticvolatility within their specification are compared withthose of market volatility models that do not depend onidiosyncratic volatility. The primary means of compar-ison used in this paper is the quality of the forecastsproduced by these competing models.

3. The volatility models

Several models of volatility have been utilisedwithin the areas of asset pricing and risk management.The vast majority of these models assume thatabnormal market returns are adequately representedby its first two conditional moments; specifically, thiscan be formally stated as

ntjF t�1fD 0; r2M ;t

� �;

where the abnormal market return nt ¼ RM ;t �E RM ;tjF t�1

� �;F t�1 represents the information set

available at time t−1, and σM,t is the conditionalvolatility of (abnormal) market returns.

Of the models based on the above conditionalrepresentation, the class of econometric model that hastraditionally received the most attention has been theautoregressive conditional heteroscedastic (ARCH)model (Engle, 1982), and its generalised (GARCH)version (Bollerslev, 1986). However, recent innova-tions in the measurement of conditional volatility haveresulted in a new class of volatility models. Thesemodels, referred to as realised volatility models,circumvent the problem of not being able to directlyobserve conditional volatility in the GARCH class ofmodels. This is achieved by firstly estimating marketvolatility via the realised market volatility measuredefined in Eq. (3), and then using this estimate as the

dependent variable in the volatility model. In doingthis, an estimate of market volatility is used that can bemodelled using conventional statistical techniques.Specifically, given the strong time dependencies ob-served in realised volatility (see, e.g., Andersen et al.,2003), autoregressive moving average (ARMA) mod-els are often employed.7 We also adopt this assumptionby considering the following dynamic model of dailymarket volatility:

ln r̂M ;t ¼ l0þl1Dt þ g1 Lð Þln r̂I ;t�1þ g2 Lð ÞVt�1þ gt;

ð4aÞ

1� / Lð Þð Þgt ¼ 1þ h Lð Þð Þυt; ð4bÞ

where Dt is a dummy variable that equals unity for alldays following a non-trading day, and zero otherwise;Vt is the Box-Cox adjusted trading volume; ηt is anARMA(p,q) error term; ϕ(L) and θ(L) are p- and q-order lag polynomials, respectively; γ1(L) and γ2(L)are r-order lag polynomials; υt is a suitably definedinnovation; and natural logarithms of realised volati-lities are taken, as this transformation is commonlyfound to improve the forecasting performance of suchmodels (see, e.g., Martens and Zein, 2004), and alsoensures that forecasts of market volatility are alwayspositive.8 Henceforth, the family of models describedby Eqs. (4a) and (4b) are referred to as the ARMAX(p,q,r) model class.

4. Empirical results

Having defined the market and volatility modelsused in this paper, we proceed with an empiricalanalysis of the explanatory power of idiosyncratic


volatility within this framework. This begins with adescription of the data used, is followed by an in-sample evaluation of the volatility models, and cul-minates in an examination of the accuracy of theforecasts generated by these models. In the followinganalysis, in-sample and out-of-sample performance areassessed with respect to whether or not past idiosyn-cratic volatility is included within the specification ofthe model.

4.1. Data

We make use of various pieces of informationconcerning all trades in the thirty stocks within theDJIA index over the period January 2, 2000 toDecember31, 2005. In particular, transaction prices (adjusted forstock splits); returns, defined as the log change intransaction prices; and trading volume, defined as thedollar volume of trading, were obtained for the thirtystockswithin the index as of January 2, 2000.9 In order toachieve the longest span of data, data for the abovestocks were obtained over the entire sample period,regardless of whether a stock was removed from theindex. In the current application, this requirement isaffected by the replacement of AT&T, Eastman Kodak,and International Paper, with Pfizer, Verizon, and AIG,on April 8, 2004. Consequently, we assume that themarket portfolio is given by a pseudo (price-weighted)DJIA index, with all of the caveats associated with theuse of such an index applying. All of these data wereobtained from Price-Data.com, Inc., with stock splitdata supplied by Briefing.com, Inc.

The above transaction data are then converted tofive-minute frequency data in preparation for conver-sion into the daily measures of idiosyncratic andmarket volatility described previously.10 This fre-quency is deemed to be low enough to avoid stale data,and high enough to avoid loss of information. In

9 A more satisfactory measure of prices would have been the mid-points of the bid-ask quotes. This is because this particular pricemeasure is less affected by microstructure effects such as bid-askbounce (though it may be adversely affected by the inventory needsof specialists; see O'Hara, 1995, for empirical evidence). However,data limitations mean that transaction prices are used instead. Giventhe use of these data, we include an MA(1) term in Eq. (1) tomitigate the microstructure contamination.10 The DJIA index data used in this paper are available over theintraday period, 9.30 a.m. to 4.00 p.m. (EST).

addition, each stock is assigned to one of five industryportfolios according to its SIC code as described byKenneth French's website, viz. Consumer (includingconsumer durables, nondurables, wholesale, retail, andsome services);Manufacturing (including manufactur-ing, energy, and utilities); HiTec (including businessequipment, telephone, and television transmission);Health (including healthcare, medical equipment, anddrugs); and Other (including mines, construction,building materials, transport, hotels, business services,entertainment and finance).11 The use of this narrowindustrial classification is necessary, due to the limitednumber of stocks within the DJIA index.

The exclusive use of stocks within the DJIA indexis motivated by a number of factors. First, thecomponents of this index are substantial blue-chipcompanies with a wide following amongst investors.Consequently, the constituent stocks are very activelytraded, and hence the intraday returns to these stocksare suitable for use as inputs into the realised volatilitymeasures considered in the current paper. Second, asonly thirty stocks are included in the DJIA index, theconstruction of idiosyncratic volatility is computation-ally less demanding than it would be if a broader indexwere used. The downside to this saving is that there isthe potential for the DJIA index to represent a portfolioof stocks that contains a significant amount of non-diversifiable risk. However, it has been well docu-mented that only ten to forty stocks are required toachieve a well-diversified portfolio (see, e.g., Evans &Archer, 1968; Statman, 1987; and Tole, 1982). Theseresults suggest (but do not necessarily imply) that theDJIA index may represent a reasonable approximationto the market portfolio, and that the idiosyncraticvolatility measure used in this paper can be consideredfairly representative. Finally, it is also worth notingthat, while the price-weighting scheme used in theconstruction of the DJIA index is generally accepted tobe a poor way of representing the different sectorswithin the market, it is still highly correlated withcompeting equal-weighted and value-weighted equityindices such as the S&P 500 index.12

11 See http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/DataLibrary/det5indport.html.12 The Dow Jones website provides further details of theadvantages of using the DJIA index over competing indices(http://www.djindexes.com).

http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/DataLibrary/det5indport.html

http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/DataLibrary/det5indport.html

http://www.djindexes.com

Table 1Summary statistics

Stock Series characteristic

Ticker Industry Mean S.D. SK KT ARCH ρ1̂ ρ1̂2

Panel A: Five-minute frequency seriesAlcoa AA Manuf −0.0755 0.3753⁎⁎⁎ −0.6372⁎⁎⁎ 80.5241⁎⁎⁎ 54.7166⁎⁎⁎ −0.0320⁎⁎⁎ 0.0061⁎⁎

American Express AXP Other −0.0158 0.3627⁎⁎⁎ −3.8644⁎⁎⁎ 338.2734⁎⁎⁎ 9.4707⁎⁎⁎ −0.0253⁎⁎⁎ −0.0086⁎⁎⁎Boeing BA Manuf 0.0812 0.3538⁎⁎⁎ −4.4305⁎⁎⁎ 180.4030⁎⁎⁎ 3.0282⁎ −0.0395⁎⁎⁎ −0.0039Citigroup C Other 0.0265 0.3568⁎⁎⁎ −0.3542⁎⁎⁎ 283.3623⁎⁎⁎ 139.0075⁎⁎⁎ −0.0495⁎⁎⁎ 0.0040Caterpillar CAT Manuf 0.1450 0.3338⁎⁎⁎ −2.5024⁎⁎⁎ 152.4148⁎⁎⁎ 58.6323⁎⁎⁎ −0.0173⁎⁎⁎ 0.0057⁎

Du Pont DD Manuf −0.0758 0.3164⁎⁎⁎ −0.7752⁎⁎⁎ 54.6348⁎⁎⁎ 131.8267⁎⁎⁎ −0.0338⁎⁎⁎ 0.0063⁎⁎

Walt Disney DIS Other −0.0258 0.4059⁎⁎⁎ −2.2445⁎⁎⁎ 426.0993⁎⁎⁎ 7.4641⁎⁎⁎ −0.0608⁎⁎⁎ −0.0065⁎⁎Eastman Kodak EK Manuf −0.1908 0.3646⁎⁎⁎ −2.6861⁎⁎⁎ 268.0285⁎⁎⁎ 62.6855⁎⁎⁎ 0.0145⁎⁎⁎ 0.0048General Electric GE Manuf −0.0606 0.3245⁎⁎⁎ −0.0311⁎⁎⁎ 86.1313⁎⁎⁎ 194.8303⁎⁎⁎ −0.0411⁎⁎⁎ 0.0032General Motors GM HiTec −0.1893 0.3359⁎⁎⁎ −1.1316⁎⁎⁎ 112.9189⁎⁎⁎ 15.0818⁎⁎⁎ 0.0313⁎⁎⁎ 0.0073⁎⁎⁎

Home Depot HD Cnsmr −0.0823 0.3932⁎⁎⁎ −9.1492⁎⁎⁎ 898.8306⁎⁎⁎ 1.2907 −0.0225⁎⁎⁎ 0.0023Honeywell Intl. HON Manuf −0.0780 0.4308⁎⁎⁎ −0.6644⁎⁎⁎ 438.1897⁎⁎⁎ 9.7452⁎⁎⁎ −0.0363⁎⁎⁎ 0.0123⁎⁎⁎

Hewlett-Packard HPQ HiTec −0.1128 0.5018⁎⁎⁎ −2.8072⁎⁎⁎ 263.2302⁎⁎⁎ 63.7755⁎⁎⁎ −0.0380⁎⁎⁎ 0.0004Intl. BusinessMachines

IBM HiTec −0.0422 0.3323⁎⁎⁎ −1.0204⁎⁎⁎ 384.9193⁎⁎⁎ 31.3885⁎⁎⁎ −0.0184⁎⁎⁎ 0.0094⁎⁎⁎

Intel INTC HiTec −0.0857 0.5160⁎⁎⁎ −4.9149⁎⁎⁎ 425.3044⁎⁎⁎ 2.3234 −0.0292⁎⁎⁎ −0.0039Intl. Paper IP Manuf −0.1044 0.3406⁎⁎⁎ 1.0092⁎⁎⁎ 91.6985⁎⁎⁎ 112.1801⁎⁎⁎ −0.0402⁎⁎⁎ 0.0021⁎⁎⁎

Johnson & Johnson JNJ Hlth 0.0498 0.2650⁎⁎⁎ −7.7620⁎⁎⁎ 726.4611⁎⁎⁎ 16.8970⁎⁎⁎ −0.0501⁎⁎⁎ −0.0085⁎⁎⁎J.P. Morgan Chase JPM Other 0.0191 0.3958⁎⁎⁎ −1.8644⁎⁎⁎ 171.0479⁎⁎⁎ 48.8087⁎⁎⁎ −0.0262⁎⁎⁎ 0.0073⁎⁎⁎

Coca-Cola KO Cnsmr −0.0529 0.2734⁎⁎⁎ −1.0286⁎⁎⁎ 60.0855⁎⁎⁎ 137.9433⁎⁎⁎ −0.0346⁎⁎⁎ −0.0035McDonald's MCD Cnsmr −0.0319 0.3352⁎⁎⁎ −2.2569⁎⁎⁎ 135.0846⁎⁎⁎ 53.5498⁎⁎⁎ −0.0469⁎⁎⁎ −0.0061⁎⁎3M MMM Manuf 0.0822 0.2687⁎⁎⁎ 0.3445⁎⁎⁎ 53.1368⁎⁎⁎ 107.8465⁎⁎⁎ −0.0306⁎⁎⁎ −0.0021Altria Group MO Cnsmr 0.1874 0.3281⁎⁎⁎ 0.7329⁎⁎⁎ 236.9389⁎⁎⁎ 161.7567⁎⁎⁎ −0.0537⁎⁎⁎ 0.0040Merck MRK Hlth −0.1385 0.3240⁎⁎⁎ −22.1719⁎⁎⁎ 2925.7346⁎⁎⁎ 0.2014 −0.0244⁎⁎⁎ 0.0038Microsoft MSFT HiTec −0.1255 0.3746⁎⁎⁎ −1.9811⁎⁎⁎ 258.5535⁎⁎⁎ 12.7749⁎⁎⁎ −0.0231⁎⁎⁎ 0.0001Procter and Gamble PG Manuf 0.0114 0.3045⁎⁎⁎ −57.6927⁎⁎⁎ 10877.4408⁎⁎⁎ 0.0083 −0.0411⁎⁎⁎ 0.0070⁎⁎⁎

SBCCommunications SBC HiTec −0.1174 0.3598⁎⁎⁎ −2.2349⁎⁎⁎ 109.0419⁎⁎⁎ 71.7821⁎⁎⁎ −0.0492⁎⁎⁎ 0.0058⁎⁎

AT&T T HiTec −0.2270 0.4093⁎⁎⁎ −1.3163⁎⁎⁎ 262.0112⁎⁎⁎ 49.4529⁎⁎⁎ −0.0367⁎⁎⁎ 0.0032United Technologies UTX Manuf 0.0859 0.3226⁎⁎⁎ −3.5337⁎⁎⁎ 307.8370⁎⁎⁎ 7.4862⁎⁎⁎ 0.0080⁎⁎⁎ −0.0006Wal-Mart Stores WMT Cnsmr −0.0551 0.3218⁎⁎⁎ −0.2163⁎⁎⁎ 62.3724⁎⁎⁎ 258.1399⁎⁎⁎ −0.0239⁎⁎⁎ −0.0034Exxon Mobil XOM Manuf 0.0641 0.2598⁎⁎⁎ −0.3174⁎⁎⁎ 29.9171⁎⁎⁎ 693.4961⁎⁎⁎ −0.0585⁎⁎⁎ −0.0014Market IndexReturns DJIA −0.0205 0.1757⁎⁎⁎ −0.5382⁎⁎⁎ 89.2267⁎⁎⁎ 56.0450⁎⁎⁎ 0.0141⁎⁎⁎ 0.0016

Panel B: Daily frequency seriesMarket index returns −0.0205 0.1835⁎⁎⁎ −0.0258⁎⁎⁎ 6.4609⁎⁎⁎ 12.3264⁎⁎⁎ −0.0124 0.0491⁎

Market volatility 0.1549⁎⁎⁎ 0.6668⁎⁎⁎ 0.4990⁎⁎⁎

Idiosyncratic volatility 0.2457⁎⁎⁎ 0.8937⁎⁎⁎ 0.7570⁎⁎⁎

This table contains summary statistics pertaining to the five-minute frequency returns to each of the thirty DJIA stocks (Panel A), and dailyfrequency (price-weighted) measures of DJIA index returns and volatility (Panel B). Returns are defined as the log of the first difference in prices,with means and standard deviations given in annualised terms. The significance of the mean and standard deviation (S.D.) are assessed using astandard asymptotic test obtained under the null hypothesis that each of the measures equals zero. Skewness (SK) and (excess) kurtosis (KT) aretested under the (asymptotic) assumption that these moments have, respectively, N (0,6/T) and N (0,24/T) distributions under the appropriate nullhypotheses. The ARCH test statistic is given by T×R2, where R2 is obtained from an OLS regression of squared returns on lagged squaredreturns. Autocorrelation is measured as the first-order autoregressive coefficient (ρ̂1) and the twelfth-order autoregressive coefficient (ρ1̂2), withthe associated t-tests being used to assess its significance. All information contained is this table is based on the use of data measured over thesample period January 2, 2000, to December 31, 2005. The significance of each moment or test is denoted by *** (1% significance), ** (5%significance), and * (10% significance).



4.2. Summary statistics

A selection of descriptive statistics pertaining to five-minute frequency returns for each constituent stock, anddaily frequency aggregate return and volatility mea-sures, are given in Table 1. The results highlight fourmain characteristics of the data. First, mean returns toeach stock are uniformly equal to zero, reflecting the falland subsequent recovering of prices over the sampleperiod (see panel (a) of Fig. 1). Second, the resultsindicate that the data appear to be highly non-normal,with negative skewness and excess kurtosis beingevident. It is possible that this is due to time-variationin the conditional volatility of returns (as indicated bythe tests for no ARCH effects), or deficiencies in thetesting procedure. Nevertheless, some attention must bepaid to the validity of the normality assumption whenestimating the models described previously. Third,returns to the vast majority of stocks are negativelyfirst-order autocorrelated, while DJIA index returns arepositively first-order autocorrelated — featuresundoubtedly due to bid-ask bounce within individual

Fig. 1. Summary information. This figure contains full sample five-minuteindex (panel (a)), and annualised DJIA market and idiosyncratic volatilitvolatility is given by the square root of the price-weighted average of integrais given by the square root of the integrated squared intraday DJIA index

transaction prices, and non-synchronous tradingamongst the stocks within the index, respectively.Fourth, market volatility is highly time-dependent, witha slowly decaying autocorrelation function (ρ̂1=0.6668and ρ̂12=0.4990). This particular feature of the datasuggests that an AR(1) model of market volatility maynot sufficiently account for such persistence. Furtherinsight into the nature of the data used is provided by theplots in panel (b) of Fig. 1. It is apparent that index returnsaremore volatile during the first half of the sample period(corresponding to the bear market of 2001 through2003). Moreover, this volatility appears to be primarilycomposed of idiosyncratic volatility.

4.3. Market model estimation

To generate the measure of idiosyncratic volatilityused in this paper, the market model given by Eq. (1) isestimated via ordinary least squares (OLS) using five-minute frequency constituent stock data. Spacelimitations prevent the presentation of all of the marketmodel results; however, a description of the most

frequency time series plots of the level of the price-weighted DJIAy (panel (b)). With regard to the volatility measures, idiosyncraticted squared intraday DJIA market model errors, and market volatilityreturns.

14 The adequacy of the models used is also examined by applyingthe KPSS (Kwiatkowski, Phillips, Schmidt, & Shin, 1992) test tothe residuals from each model. This test has been shown to be avalid test of the null hypothesis that a series follows a short-memoryprocess against the alternative that it follows a long-memory process(Lee & Schmidt, 1996). The results indicate a clear rejection (at the1% level) of this null when the ARMAX(1,0,1) model is used. Bycontrast, only marginal rejections of the null are possible whenusing the ARMAX(1,1,1) and ARMAX(2,1,1) models, with norejection possible (at any reasonable level) when the ARMAX(2,1,2) model is assumed. This finding suggests that the strongpersistence in market volatility is adequately captured by an ARMA(2,1) error in Eq. (4a), thus rendering long-memory models


notable features is provided. In particular, the results(available upon request) indicate that the market modelappears to provide a good representation of the returnsof each constituent stock. This is evinced by theadjusted R2 values, and the magnitude (and signifi-cance) of each coefficient. Regarding the formerobservation, it is noticeable that there exists highcross-sectional variation in the degree of fit across theequations: 9.13% (Altria Group) to 44.26% (GeneralElectric) — a result that suggests considerablevariation in the level of systematic risk captured bythe market model (see Roll, 1988, for similar resultsusing low frequency data). In terms of the coefficientsthemselves, their magnitudes are in accordance withprior expectations. Specifically, the average of the sumof the coefficients of market and industry portfolioreturns is close to unity, while the significant MA(1)coefficients indicate the presence of strong marketmicrostructure (bid-ask bounce) effects within thedata.13

4.4. Volatility model estimation

Having estimated the market model for eachconstituent stock, daily frequency measures of idio-syncratic volatility can be constructed via Eq. (2). It isthe incremental explanatory power of this particularmeasure of idiosyncratic volatility that is considered inthe context of the volatility models given by Eqs. (4a)and (4b). The results of this investigation are describedin the following subsections.

4.4.1. In-sample performanceWe examine the impact of idiosyncratic volatility

upon the future dynamics of market volatility via fourdifferent ARMAX(p,q,r) models; specifically, theARMAX(1,0,1), ARMAX(1,1,1), ARMAX(2,1,1), andARMAX(2,1,2) models, based on Eqs. (4a) and (4b),each estimated with all exogenous variables (includingidiosyncratic volatility) contained in its specification.These four models represent a broad range of modelswith a lag structure that coincides with that of the modelsfrequently employed in the literature. Most notably, the

13 Inference is conducted using Huber–White robust standarderrors. These robust standard errors are based on the ‘sandwich(robust covariance matrix) estimator’; see Kauermann and Carroll(2001) for further details of this estimator.

assumption of an ARMA(2,1) error term in Eq. (4a) iscommonly motivated by Gallant, Hsu, and Tauchen's(1999) finding that the strong persistence present in assetreturn volatility can be adequately represented by thesum of two separate AR(1) processes — a sum that isrepresented identically by an ARMA(2,1) process(Granger & Newbold, 1976). Moreover, models basedon this assumption have been shown to produce volatilityforecasts that are equally as good as the forecasts basedon more sophisticated long-memory models (see, e.g.,Pong, Shackleton, Taylor, & Xu, 2004).

The estimated coefficients, associated Newey–Weststandard errors, and measures of model fit given inTable 2 indicate that the majority of the coefficients taketheir expected values, with the ARMAX(1,0,1) modelproviding the most inadequate representation of thedata.14 Exceptions to this finding occurwhen consideringthe coefficient on trading volume. In particular, we findthat this coefficient is insignificantly different from zero(except when the poorly fitting ARMAX(1,0,1) model isassumed).15 By contrast, the coefficient of first-orderlagged idiosyncratic volatility is positive and significantat the 1% level, a result that holds for all modelsconsidered. Thus, these results provide strong support forthe notion that lagged idiosyncratic volatility hasexplanatory power with respect to daily market volatility.

To examine the robustness of the above results tovariation in the sample period used, we also estimatethe models over the period January 2, 2000 to April 7,2004. This sample period corresponds to a period inwhich there were no changes in the stocks making up

superfluous in this instance.15 It is also noticeable that the coefficients of the trading-breakdummy variable are significantly less than zero, implying that thevolatility is lower after a trading break. This result contradictstheory, which suggests that volatility should be higher on such days(French & Roll, 1986).

Table 2Realised volatility model results

Parameter Volatility model

ARMAX(1,0,1)

ARMAX(1,1,1)

ARMAX(2,1,1)

ARMAX(2,1,2)

μ ̂0 −2.5075⁎⁎⁎ −3.3682⁎⁎⁎ −3.2616⁎⁎⁎ −2.7107⁎⁎⁎(0.3618) (0.4316) (0.4366) (0.4911)

μ1̂ −0.0409⁎⁎⁎ −0.0637⁎⁎⁎ −0.0664⁎⁎⁎ −0.0675⁎⁎⁎(0.0156) (0.0165) (0.0167) (0.0168)

ϕ̂1 0.6740⁎⁎⁎ 0.9852⁎⁎⁎ 0.9268⁎⁎⁎ 0.9610⁎⁎⁎

(0.0344) (0.0056) (0.0373) (0.0406)ϕ̂2 0.0564 0.0232

(0.0365) (0.0393)θ̂1 −0.8054⁎⁎⁎ −0.7909⁎⁎⁎ −0.8142⁎⁎⁎

(0.0219) (0.0258) (0.0261)γ̂1,1 0.3481⁎⁎⁎ 0.3260⁎⁎⁎ 0.3556⁎⁎⁎ 0.3287⁎⁎⁎

(0.0689) (0.0805) (0.0792) (0.0827)γ̂1,2 0.1377⁎

(0.0730)γ̂2,1 −0.0142⁎⁎⁎ 0.0001 0.0006 0.0002

(0.0024) (0.0022) (0.0022) (0.0023)γ̂2,2 −0.0011

(0.0021)R̅ 2 54.71% 65.34% 65.26% 65.31%AIC 0.4458 0.1824 0.1732 0.1823SIC 0.4638 0.2039 0.2019 0.2146HQC 0.4525 0.1904 0.1911 0.1943KPSS 2.4912⁎⁎⁎ 0.3564⁎ 0.3748⁎ 0.3111

This table contains all parameter estimates, associated Newey–Wesrobust standard errors (with the truncation lag set to floor[4(T/100)2/9])associated with the following ARMAX(p,q,r) model:

ln r̂M ;t ¼ l0 þ l1Dt þ g1ln r̂I ;t�1 þ g2Vt�1 þ gt ;1� / Lð Þð Þgt ¼ 1þ h Lð Þð Þυt ;

where σ̂M,t is a measure of realisedmarket volatility; σ̂I,t is a measure orealised idiosyncratic volatility; Dt is a dummy variable that equalsunity for all days following a non-trading day, and zero otherwise;Vt isthe Box-Cox adjusted trading volume; ηt is an ARMA (p,q) error termϕ(L) =ϕ1L+ϕ2L

2+…+ϕpLp; θ(L) =θ1L+θ2L2+…+θqL

q; γ1(L) =γ1,1L+γ1,2L

2+…+γ1,rLr; γ2(L)=γ2,1L +γ2,2L

2+…+γ2,rLr; and υt is a

suitably defined innovation. All models are estimated using theMarquardt non-linear least squares algorithm, under the normal errodistribution assumption. Model fit is measured via the adjusted R2

statistic, the Akaike information criterion (AIC), the Schwartzinformation criterion (SIC), and the Hannan–Quinn criterion (HQC)In addition, the KPSS (Kwiatkowski et al., 1992) test statistic, whichgives an indication of the validity of the null hypothesis that theresiduals follow a short-memory process against the alternative thathey follow a long-memory process, is also provided. Following therecommendation of Hobijn, Franses, and Ooms (2004), this tesstatistic is calculated using the quadratic spectral kernel, with thebandwidth set equal to floor[8(T/100)2/25]. All information contained inthis table is based on the use of data measured over the sample periodJanuary 2, 2000 to December 31, 2005. Realisations of (squaredmarket volatility are given by integrated squared intraday markereturns. Significance is denoted in the usual way.

16 For popular appeal, we analyse the forecasts of σ̂M,t2 , viz., hM,t.

However, for the econometric reasons described previously, weactually model lnσ̂M,t. This mismatch may introduce a transformationbias into the forecasts. To avoid such bias, we assume that the errors inEq. (4a) are normally distributed, specifically υt223CN(0,συ

2), andconstruct forecasts of the realised market variance as follows:

hM ;t ¼ exp 2 ln r̂M ;t � �̂t þ �̂2�

� �h i;

where υt̂ is the residual from Eq. (4a), and σ̂υ is the standard deviationof these residuals. SeeGiot andLaurent (2004) for a similar approach inthe context of Value-at-Risk (VaR) analysis.


t,

f

;

r

.

t

t

,)t

the DJIA index. The results (available upon request)are largely in line with those obtained over the fullsample period. In particular, when using models thatadequately represent the data (i.e., all models exceptthe ARMAX(1,0,1) model), the coefficient of the first-order lagged idiosyncratic volatility is positive andsignificant at the 5% level.

4.4.2. Out-of-sample performance (Mincer–Zarnowitz tests)A superior test of the explanatory power of idiosyn-

cratic volatility is achieved via an examination of thequality of the out-of-sample market volatility forecastsgenerated by volatility models that make various as-sumptions regarding their use of idiosyncratic volatility.Ergo, the out-of-sample performance of various volatilitymodels that exclude and include idiosyncratic volatilitywithin their specifications is assessed via a comparison ofthe statistical accuracy of their forecasts.

The first method by which the relative accuracy ofthe forecasts generated by such models is assessed isvia the following Mincer–Zarnowitz regression (Min-cer & Zarnowitz, 1969):

r̂2M ;t ¼ Y 0 þ Y 1hM ;t þ ft; ð5Þ

where σM̂,t2 is the realised (squared) market volatility

(i.e., the market variance), hM,t is the 1-step-aheadmodel-based forecast of realised market variance, andζt is an error term. Under the null hypothesis of fore-cast rationality, Y 0 should equal zero, and Y 1 shouldequal unity. Equivalently, under this null hypothesis,the forecasts are unbiased under quadratic loss.

The forecasts in Eq. (5) are generated via theARMAX(1,0,1), ARMAX(1,1,1), ARMAX(2,1,1),and ARMAX(2,1,2) models.16 In addition, to providea benchmark against which to compare this set of


forecasts, we also consider the forecasts generated by adaily GARCH(1,1) model, augmented to include all ofthe (first-order) explanatory variables used in theARMAX models (henceforth referred to as theGARCHX(1,1,1) model).17,18 To isolate the contribu-tion of idiosyncratic volatility, we estimate all of theabove models with this volatility both excluded fromand included within their specifications. Each of thesemodels is estimated over the period January 2, 2000 toDecember 31, 2004, with out-of-sample 1-step-aheadforecasts of daily market volatility generated over theperiod January 2, 2005 to December 31, 2005.

The OLS estimated coefficients, associatedNewey–West standard errors, and adjusted R2 valuesobtained from Eq. (5), are presented in Table 3. Theseresults indicate that, with the exception of both theGARCHX(1,1,1) and the ARMAX(1,0,1) models, themodels pass the Mincer–Zarnowitz test. Thus, the testscannot generally differentiate between the quality ofthe models' forecasts with respect to whether idiosyn-cratic volatility is excluded from or included withintheir specification. However, it is noticeable thatforecasts based on models that include idiosyncraticvolatility exhibit superior Mincer–Zarnowitz features,i.e., values of Y 0̂ closer to zero, values of Y ̂1 closer tounity, and higher R̅ 2 values.

4.4.3. Out-of-sample performance (forecast encom-passing tests)

An alternative method of comparing the perfor-mance of the volatility models is to examine whetherone set of forecasts encompasses a competing set.Specifically, we examine whether forecasts generatedby volatility models that exclude idiosyncratic volati-lity from their specification encompass forecastsgenerated by volatility models that include idiosyn-cratic volatility within their specification (and viceversa); see Chong and Hendry (1986) for a seminal

17 Hansen and Lunde (2005) document thorough empiricalevidence regarding the excellent forecasting performance of theGARCH(1,1) model. For this reason, we only consider augmentedversions of this particular GARCH model.18 The GARCHX(1,1,1) model assumes that market returns followan AR(1) process, and is estimated by maximum likelihood underthe assumption that the errors are drawn from a generalised errordistribution. Furthermore, this model is estimated with laggedrealised idiosyncratic variance, σ̂I,t− 1

2 , both excluded from andincluded within its specification.

paper in this area. This is achieved using the Ericsson(1992) regression-based testing procedure. The regres-sion equation used in this test is given by

eA;t ¼ H0 þH1 eA;t � eB;t� �þ mt; ð6Þ

where eA,t is the 1-step-ahead forecast error (squaredrealised market volatility minus the respective fore-cast) associated with volatility models that excludeidiosyncratic volatility from their specification; eB,t isthe 1-step-ahead forecast error associated with volati-lity models that include idiosyncratic volatility withintheir specification; and νt is an error term. Under thenull hypothesis that the forecast associated with amodel that excludes idiosyncratic volatility from itsspecification (referred to as forecast A), encompassesthe forecast associated with a model that includesidiosyncratic volatility within its specification(referred to as forecast B), Θ1 will be less than orequal to zero; otherwise, forecast A contains no addedinformation. This null is tested using the OLS-basedstandard t-statistic associated with Θ1.

19 Moreover,we also consider the modified version of this t-testproposed by Harvey, Leybourne, and Newbold (1998).

The results in Table 4 indicate that forecasts based onvolatility models that include idiosyncratic volatilitywithin their specification, tend to encompass theforecasts based on volatility models that excludeidiosyncratic volatility from their specification. Specifi-cally, one can reject the null hypothesis that forecast Aencompasses forecast B at the 1% level, but not the nullhypothesis that forecastB encompasses forecastA at anyreasonable level. Therefore, one can conclude that usingidiosyncratic volatility leads to market volatility fore-casts that are superior to those generated by volatilitymodels that do not make use of idiosyncratic volatility.

4.4.4. Out-of-sample performance (mean losscomparison tests)

To further examine the explanatory power ofidiosyncratic volatility, we examine the mean lossesassociated with all the market volatility forecastsconsidered previously, and test whether the mean lossesassociated with forecasts based on models that exclude

19 To test the null that forecast B encompasses forecast A, thereverse of the regression in Eq. (6) is performed, and the analysis isrepeated.

Table 3Volatility forecast accuracy (Mincer–Zarnowitz regression results)

Volatility model Ex. idiosyncratic volatility Inc. idiosyncratic volatility

Y 0̂ Y 1̂ χ2-statistic R̅ 2Y ̂0 Y ̂1 χ2-statistic R̅ 2

GARCHX(1,1,1) 0.2114⁎⁎⁎ 0.3223⁎⁎⁎ 99.4719⁎⁎⁎ 7.02% 0.2380⁎⁎⁎ 0.3401⁎⁎⁎ 33.5209⁎⁎⁎ 7.16%(0.0706) (0.1267) (0.0583) (0.1281)

ARMAX(1,0,1) 0.1942⁎⁎⁎ 0.4543⁎⁎⁎ 33.5000⁎⁎⁎ 13.05% 0.1923⁎⁎⁎ 0.4587⁎⁎⁎ 32.7137⁎⁎⁎ 13:20%(0.0465) (0.1046) (0.0468) (0.1053)

ARMAX(1,1,1) 0.0780 0.8394 3.5009 22.09% 0.0558 0.8999 2.7406 25.01%(0.0516) (0.1380) (0.0527) (0.1409)

ARMAX(2,1,1) 0.0782 0.8371 3.3620 22.24% 0.0577 0.8919 2.5334 25.47%(0.0516) (0.1374) (0.0526) (0.1397)

ARMAX(2,1,2) 0.0789 0.8358 3.4351 21.94% 0.0604 0.8893 3.0592 25.45%(0.0521) (0.1390) (0.0512) (0.1360)

This table contains the OLS parameter estimates, the associated Newey–West robust standard errors (with the truncation lag set to floor[4(T/100)2/9], and the measures of model fit, associated with the following Mincer–Zarnowitz regression:

r̂2M ;t ¼ Y 0 þ Y 1hM ;t þ ft ;

where σM̂,t is the realised market volatility, √hM,t is the 1-step-ahead model-based forecast of market volatility, and ζt is an error term. In addition,the χ2-statistic associated with the null hypothesis that Y 0=0 and Y 1=1 is provided. The forecasts used in the above regression are based onvolatility models that both exclude and include idiosyncratic volatility within their specification. Each of these models is estimated over the periodJanuary 2, 2000 to December 31, 2004, and out-of-sample 1-step-ahead forecasts of market volatility are generated over the period January 2,2005 to December 31, 2005. Realisations of (squared) market volatility are given by integrated squared intraday market returns. The significanceof the null hypothesis that Y 0=0 and Y 1=1 is denoted in the usual way.

Table 4Volatility forecast accuracy (forecast encompassing tests)

Null hypothesis Volatility model Test statistic

ENC1 ENC2

H0: Forecast A encompasses Forecast B GARCHX(1,1,1) 12.4732⁎⁎⁎ 11.4667⁎⁎⁎

H0: Forecast B encompasses Forecast A −5.4721 −5.4041H0: Forecast A encompasses Forecast B ARMAX(1,0,1) 5.5101⁎⁎⁎ 3.4609⁎⁎⁎




H0: Forecast B encompasses Forecast A −0.8372 −0.7139

This table contains information pertaining to the following regression equation:

eA;t ¼ H0 þH1 eA;t � eB;t� �þ mt;

where eA,t is the forecast error (squared realised market volatility minus the respective forecast) associated with volatility models that excludeidiosyncratic volatility from their specification; eB,t is the forecast error associated with volatility models that include idiosyncratic volatility withintheir specification; and νt is an error term.Under the null hypothesis that the forecast associatedwith amodel that excludes idiosyncratic volatility fromits specification (referred to as forecast A) encompasses the forecast associated with a model that includes idiosyncratic volatility within itsspecification (referred to as forecast B), Θ1 will be less than or equal to zero; otherwise forecast A contains no added information. This null is testedusing anOLS-based standard t-statistic associated withΘ1 (denoted ENC1), and theHarvey–Leybourne–Newboldmodified t-statistic associated withΘ1 (denoted ENC2). To test the null that forecast B encompasses forecast A, forecast A is replaced by forecast B (and vice versa), and the analysis isrepeated. Each volatility model is estimated over the period January 2, 2000 toDecember 31, 2004, and out-of-sample 1-step ahead forecasts ofmarketvolatility are generated over the period January 2, 2005 to December 31, 2005. Realisations of (squared) market volatility are given by integratedsquared intraday market returns. Significance is denoted in the usual way.


21 Though similar results are found for the other mean loss


idiosyncratic volatility from their specification equalthose produced by models that include idiosyncraticvolatility within their specification. This is achieved bythe use of the Diebold–Mariano test statistic (Diebold &Mariano, 1995). Various mean loss functions are used inconjunction with this test. Specifically, mean loss ismeasured using the following loss functions: mean lossimplied by a Gaussian likelihood function (QLIKE);mean square forecast error based on squared volatilityforecast/realisation measures (MSFE); mean squareforecast error based on log squared volatility forecast/realisation measures (MSFE-ln); mean square forecasterror based on volatility forecast/realisation measures(MSFE-sd); mean square forecast error based onproportional squared volatility forecast/realisation mea-sures (MSFE-prop); and the mean absolute counterpartsto these mean loss functions. See Patton (2007) fordetails of these mean loss functions.

The results in Table 5 indicate that, for all lossfunctions adopted by the forecaster, the most accurateforecasts are generated by the models that includeidiosyncratic volatility within their specification. More-over, the improvements in mean loss achieved whenidiosyncratic volatility is included in these models arestatistically significant for almost all of the loss func-tions, a result that confirms the forecast encompassingresults. It is also noticeable from this table that theARMAX models deliver mean losses that are univer-sally lower than the mean losses achieved when usingthe GARCHX(1,1,1) model. Moreover, the mean lossesassociated with the former set of models (except theARMAX(1,0,1) model) are generally significantlylower than those obtained from forecasts based on thecommonly employed RiskMetrics exponentially-weighted moving-average (EWMA) volatility model.20

Themagnitudes of themean loss improvements givenin Table 5 highlight the practical significance of includingidiosyncratic volatility within a forecasting framework.Specifically, for all adequately-fitting volatility models,

20 The EWMA volatility model considered in this paper assumesthat market volatility forecasts are given by

hM ;t ¼ 1� kð ÞXt�1

j¼1

kj�1R2M ;t�j;

where RM,t is the daily DJIA index return, and the decay parameter,λ, is set equal to 0.94 (as originally proposed by RiskMetrics). Forfurther details of the various return volatility models employed byRiskMetrics, see Mina and Xiao (2001).

consistent mean loss improvements of around 4% areachieved. For instance, for forecasters who adopt anMSFE loss function and theARMAX(2,1,1) model, theirmean loss will decrease by 4.62% simply by includingidiosyncratic volatility in their volatility model.21 Giventhe inverse relationship between loss and utility, theseresults suggest that non-trivial gains in utility can beachieved under this directive at little extra cost, comparedto competing modelling strategies.22

In using a symmetric loss function (as in themajority of loss functions considered above), positiveand negative forecast errors (of the same absolutemagnitude) receive equal weight in the loss function. Inthe current context this may not be appropriate, as bothBrailsford and Faff (1996) and Bystrom (2004) arguethat under-prediction of return volatility should bemore heavily penalised than over-prediction. This needto protect against under-prediction can be demon-strated in the context of VaR analysis. Specifically, it isobvious in this context that under-prediction of returnvolatility is to be avoided if the solvency of the port-folio owner is to be maintained. Therefore, some formof asymmetric loss function is required. To this end, weassume the following family of robust loss functionsintroduced by Patton (2007):

L r̂M ;t ; hM ;t ; b� �

¼

r̂2M ;t

hM ;t� ln

r̂2M ;t

hM ;t� 1; if b ¼ �2:

hM ;t � r̂2M ;t þ r̂2M ;t lnr̂2M ;t

hM ;t; if b ¼ �1;

1

bþ 1ð Þ bþ 2ð Þ r̂2bþ4M ;t � hbþ2

M ;t

� �� 1

bþ 1hbþ1M ;t r̂2M ;t � hM ;t

� �otherwise;

8>>>>>>>>>>><>>>>>>>>>>>:

ð7Þwhere b governs the shape of the loss function, withbb0 indicating heavier penalisation of under-prediction,

functions and volatility models, improvements are fairly insubstan-tial when the (inadequate) ARMAX(1,0,1) model is assumed — aresult that is in line with the Mincer-Zarnowitz regression results.22 These results do not necessarily imply that idiosyncratic volatilitycan be used to generate abnormal profits from a trading strategy (suchas one based on a volatility timing strategy). Such evidence wouldrequire explicit knowledge of the exact strategy adopted, and the riskpreferences of the investor (see, e.g., Fleming, Kirby,&Ostdiek, 2001).Instead, we present evidence on a variety of generally accepted(statistically-motivated) loss functions, rather than on loss functionsthat depend on the specific characteristics of investors.

Table 5Volatility forecast accuracy (mean loss comparisons)

Mean loss×102

Loss function Volatility model Ex. idiosyncratic volatility Inc. idiosyncratic volatility DM statistic Improvement

QLIKE GARCHX(1,1,1) 19.5256 16.9968 1.3350 12.95%MSFE 9.6337 6.4366 4.8480⁎⁎⁎ 33.19%MSFE-ln 52.7978 39.8336 3.6930⁎⁎⁎ 24.55%MSFE-sd 5.2315 3.6697 4.4999⁎⁎⁎ 29.85%MSFE-prop 24.2887 33.5070 −1.2149 −37.95%MAFE 25.6312 20.1556 6.0703⁎⁎⁎ 21.36%MAFE-ln 58.3622 49.7769 4.0571⁎⁎⁎ 14.71%MAFE-sd 18.9240 15.4785 5.2970⁎⁎⁎ 18.21%MAFE-prop 42.2473 43.3265 −0.3670 −2.55%QLIKE ARMAX(1,0,1) 13.2478 13.2030 3.7971⁎⁎⁎ 0.34%MSFE 5.3313 5.2986 3.2907⁎⁎⁎ 0.61%MSFE-ln 29.1852 29.0759 4.1949⁎⁎⁎ 0.37%MSFE-sd 2.8248 2.8110 3.7810⁎⁎⁎ 0.49%MSFE-prop 30.3168 30.2149 2.0249⁎⁎ 0.34%MAFE 17.3240 17.2756 3.3140⁎⁎⁎ 0.28%MAFE-ln 42.0560 41.9731 3.5255⁎⁎⁎ 0.20%MAFE-sd 13.1739 13.1430 3.4874⁎⁎⁎ 0.23%MAFE-prop 38.6135 38.5411 3.0974⁎⁎⁎ 0.19%QLIKE ARMAX(1,1,1) 11.1046 10.6680 2.3858⁎⁎ 3.93%MSFE 3.7242⁎⁎ 3.5677⁎⁎⁎ 2.0398⁎⁎ 4.20%MSFE-ln 21.7413⁎⁎⁎ 20.7480⁎⁎⁎ 3.2639⁎⁎⁎ 4.57%MSFE-sd 2.0011⁎⁎⁎ 1.9104⁎⁎⁎ 2.6259⁎⁎⁎ 4.53%MSFE-prop 31.2777 30.2023 1.0590 3.44%MAFE 13.9819⁎⁎⁎ 13.5746⁎⁎⁎ 2.5843⁎⁎⁎ 2.91%MAFE-ln 36.3315⁎⁎⁎ 35.3924⁎⁎⁎ 2.7984⁎⁎⁎ 2.58%MAFE-sd 10.9924⁎⁎⁎ 10.6876⁎⁎⁎ 2.7071⁎⁎⁎ 2.77%MAFE-prop 37.5597 36.7094 2.1354⁎⁎ 2.26%QLIKE ARMAX(2,1,1) 11.1212 10.6704 2.2254⁎⁎ 4.05%MSFE 3.7165⁎⁎ 3.5449⁎⁎⁎ 1.8973⁎ 4.62%MSFE-ln 21.7804⁎⁎⁎ 20.7413⁎⁎⁎ 3.0392⁎⁎⁎ 4.77%MSFE-sd 2.0012⁎⁎⁎ 1.9048⁎⁎⁎ 2.4526⁎⁎ 4.82%MSFE-prop 31.3437 30.2525 0.9635 3.48%MAFE 13.9727⁎⁎⁎ 13.5545⁎⁎⁎ 2.4634⁎⁎ 2.99%MAFE-ln 36.3390⁎⁎⁎ 35.3840⁎⁎⁎ 2.6564⁎⁎⁎ 2.63%MAFE-sd 10.9898⁎⁎⁎ 10.6775⁎⁎⁎ 2.5881⁎⁎⁎ 2.84%MAFE-prop 37.5281 36.6623 2.0026⁎⁎ 2.31%

This table contains out-of-sample measures of forecast quality (given by the mean value of a loss function) for market volatility models thatexclude and include idiosyncratic volatility within their specification, and Diebold–Mariano (DM) statistics associated with the null hypothesis ofno change in mean loss when idiosyncratic volatility is added to each model (under this null the DM statistic is (asymptotically) normallydistributed with zero mean and unit variance), and the percentage improvement in mean loss achieved by using idiosyncratic volatility. Mean lossis measured using the following loss functions: mean loss implied by a Gaussian likelihood function (QLIKE); mean square forecast error basedon squared volatility forecast/realisation measures (MSFE); mean square forecast error based on log squared volatility forecast/realisationmeasures (MSFE-ln); mean square forecast error based on volatility forecast/realisation measures (MSFE-sd); mean square forecast error based onproportional squared volatility forecast/realisation measures (MSFE-prop); and the mean absolute counterparts of these given by the MAFEprefix. Each volatility model is estimated over the period January 2, 2000 to December 31, 2004, and out-of-sample 1-step-ahead forecasts ofmarket volatility are generated over the period January 2, 2005 to December 31, 2005. Realisations of squared market volatility are given byintegrated intraday squared market returns. Finally, in addition to the DM-test null hypothesis, we indicate (in the usual way) rejections of the nullhypothesis that the mean loss associated with each volatility model equals the mean loss associated with a EWMAvolatility model (with the decayparameter set equal to 0.94), against the alternative that the mean loss is lower than this EWMA mean loss.


Fig. 2. Relative forecasting performance under asymmetric loss. This figure contains plots of the mean percentage differences between the lossesassociated with volatility models that include idiosyncratic volatility within their specification and volatility models that exclude idiosyncraticvolatility from their specification. The mean percentage loss differences are calculated by estimating each volatility model over the period January2, 2000 to December 31, 2004 with out-of-sample 1-step-ahead forecasts of market volatility generated over the period January 2, 2005 toDecember 31, 2005. Positive (negative) mean percentage loss differences indicate superior (inferior) performance by models that include(exclude) idiosyncratic volatility within their specifications. The family of loss functions used in these plots is given by

L r̂M ;t ; hM ;t ; b� � ¼

r̂2M ;t

hM ;t� ln

r̂2M ;t

hM ;t� 1; if b ¼ �2:

hM ;t � r̂2M ;t þ r̂2M ;t lnr̂2M ;t

hM ;t; if b ¼ �1;

1bþ 1ð Þ bþ 2ð Þ r̂2bþ4

M ;t � hbþ2M ;t

� �� 1bþ 1

hbþ1M ;t r̂2M ;t � hM ;t

� �; otherwise;

8>>>>>>><>>>>>>>:

where σ̂M,t2 is the realised market variance; hM,t is the 1-step-ahead model-based forecast of realised market variance; and b governs the shape of

the loss function, with bb0 indicating heavier penalisation of under-prediction. Realisations of (squared) market volatility are given by integratedintraday market returns.


b=0 corresponding to the MSFE loss function, and b=−2 corresponding to the QLIKE loss function.23

To assess the impact on the results of using theabove loss functions, we calculate the mean losses

23 Patton (2007) derives the necessary and sufficient conditionsunder which a loss function will yield rankings of volatilityforecasts that are robust to noise in the proxy used to measurerealised volatility.

obtained when using ARMAX realised volatilitymodels with idiosyncratic volatility excluded as anexplanatory variable, relative to the mean lossesobtained when using models with idiosyncraticvolatility included as an explanatory variable. Thesevalues are calculated for each model with respect to theasymmetry parameter b, and are presented diagram-matically in Fig. 2. The results indicate that theinclusion of idiosyncratic volatility leads to universaldecreases in the mean losses. Moreover, these mean


loss differences tend to remain fairly high even whenunder-prediction is more heavily penalised than over-prediction, that is, when bb0.24

5. Conclusion

The dynamics of market volatility appear to befunctionally dependent on past idiosyncratic volatility.Utilising this dependence in the daily DJIA indexmarket leads to significant improvements in the qualityof the in-sample fit of market volatility models, and thequality of market volatility forecasts. These qualityimprovements are particularly pronounced when thestrong dependencies within the data are appropriatelymodelled. Furthermore, the benefits of using idiosyn-cratic volatility as an explanatory variable within marketvolatility models is demonstrated under a number ofdifferent market volatility models, and under realisticloss function assumptions. Regarding the latter set ofassumptions, non-trivial improvements in mean loss areavailable to those forecasters who incorporate idiosyn-cratic volatility within the specification of their volatilitymodel — a finding that is largely unaffected by theassumption that under-prediction of volatility is pena-lisedmore heavily than over-prediction. In the context ofthe original motivation for this paper, these results lendsupport to the former of Roll's (1988) conjectures, thatidiosyncratic volatility is (positively) associated withprivate information flow rather than irrational behaviouron the part of investors. That said, we cannot rule out thepossibility that the results obtained in this paper are dueto inefficiencies inherent in the DJIA index market.

References

Andersen, T. (1996). Return volatility and trading volume: Aninformation flow interpretation of stochastic volatility. Journalof Finance, 51, 169−204.

Andersen, T., & Bollerslev, T. (1998). DM-Dollar volatility: Intradayactivity patterns, macroeconomic announcements and longer rundependencies. Journal of Finance, 53, 219−265.

Andersen, T., Bollerslev, T., Diebold, F., & Labys, P. (2001). Thedistribution of realized exchange rate volatility. Journal of theAmerican Statistical Association, 96, 42−55.

Andersen, T., Bollerslev, T., Diebold, F., & Labys, P. (2003). Modelingand forecasting realized volatility. Econometrica, 71, 579−625.

24 Exceptions to this finding occur when the ARMAX(1,0,1) andARMAX(2,1,2) models are used, and when under-prediction carriesan extreme excess relative weighting.

Bandi, F., & Russell, J. (2008). Volatility Estimation. Chapter in J.Birge, & V. Linetsky (Eds.), Handbooks in Operations Researchand Management Science: Financial Engineering Volume 15Amsterdam: Elsevier.

Barclay, M., Litzenberger, R., & Warner, J. (1990). Privateinformation, trading volume, and stock-return variances. Re-view of Financial Studies, 3, 233−253.

Barndorff-Nielson, O., & Shephard, N. (2001). Non-Gaussian Ornstein–Uhlenbeck-based models and some of their uses in financial eco-nomics. Journal of the Royal Statistical Society, Series B,63, 167−241.

Barndorff-Nielson, O., & Shephard, N. (2002). Econometric analysisof realized volatility and its use in stochastic volatility models.Journal of the Royal Statistical Society, Series B, 64, 253−280.

Berry, T., & Howe, K. (1993). Public information arrival. Journal ofFinance, 49, 1331−1346.

Bessembinder, H., & Seguin, P. (1993). Price volatility, tradingvolume and market depth: Evidence from futures markets.Journal of Financial and Quantitative Analysis, 28, 21−39.

Bollerslev, T. (1986). Generalized autoregressive conditionalheteroskedasticity. Journal of Econometrics, 31(3), 307−327.

Bollerslev, T., & Jubinski, D. (1999). Equity trading volume andvolatility: Latent information arrivals and common long-run depen-dencies. Journal of Business and Economic Statistics, 17, 9−21.

Bollerslev, T., & Wright, J. (2001). High-frequency data, frequencydomain inference, and volatility forecasting. Review of Economicsand Statistics, 83, 596−602.

Brailsford, J., & Faff, R. (1996). An evaluation of volatility forecastingtechniques. Journal of Banking and Finance, 20, 419−438.

Brooks, C. (1998). Predicting stock market volatility: Can marketvolume help? Journal of Forecasting, 17, 59−80.

Bystrom, H. (2004). Orthogonal GARCH and covariance matrixforecasting in a stress scenario: The Nordic stock markets duringthe Asian financial crisis 1997–1998. European Journal ofFinance, 10, 44−67.

Campbell, J., Lettau, M., Malkiel, B., & Xu, Y. (2001). Haveindividual stocks become more volatile? An empirical explora-tion of idiosyncratic risk. Journal of Finance, 56, 1−43.

Chan, K., & Fong, W. (2006). Realized volatility and transactions.Journal of Banking and Finance, 30, 2063−2085.

Chong, Y., & Hendry, D. (1986). Econometric evaluation of linearmacroeconomicmodels.Review of Economic Studies, 53, 671−690.

Chua, C., Goh, J., & Zhang, J. (2007). Expected volatility,unexpected volatility, and the cross-section of stock returns.Singapore Management University Working Paper.

Clark, P. (1973). A subordinated stochastic process model with finitevariance for speculative prices. Econometrica, 41, 135−155.

Copeland, T. (1976). A model of asset trading under the assumptionof sequential information arrival. Journal of Finance, 31,1149−1168.

Corsi, F., Zumbach, G., Müller, U., & Dacorogna, M. (2001).Consistent high-precision volatility from high-frequency data.Economic Notes, 30, 183−204.

Darrat, A., Zhong, M., & Cheng, L. (2005). Trading without publicnews: Another look at the intraday volume-volatility stockrelations. Louisiana Tech University Working Paper.

Diebold, F., & Mariano, R. (1995). Comparing predictive accuracy.Journal of Business and Economic Statistics, 13, 253−265.


Donaldson, R., & Kamstra, M. (2005). Volatility forecasts, tradingvolume, and the ARCH vs option-implied volatility tradeoff.Journal of Financial Research, 27, 519−538.

Duffee, G. (2001). Asymmetric cross-sectional dispersion in stockreturns: Evidence and implications.U.C. BerkeleyWorking Paper.

Durnev, A., Morck, R., & Yeung, B. (2004). Value-enhancing capitalbudgeting and firm-specific stock return variation. Journal ofFinance, 25, 65−105.

Durnev, A., Morck, R., Yeung, B., & Zarowin, P. (2003). Doesgreater firm-specific return variation mean more or less informedstock pricing? Journal of Accounting Research, 41, 797−836.

Engle, R. F. (1982). Autoregressive conditional heteroscedasticitywith estimates of the variance of United Kingdom inflation.Econometrica, 50(4), 987−1007.

Epps, T., & Epps, M. (1976). The stochastic dependence of securityprice changes and transaction volumes: Implications for themixture-of-distributions hypothesis. Econometrica, 44, 305−321.

Ericsson, N. (1992). Parameter constancy, mean square forecast errors,and measuring forecast performance: An exposition, extensions,and illustration. Journal of Policy Modeling, 14, 465−495.

Evans, J., & Archer, S. (1968). Diversification and the reduction ofdispersion: An empirical analysis. Journal of Finance, 23, 761−767.

Fama, E., & French, K. (1993). Common risk factors in the returnson stocks and bonds. Journal of Financial Economics, 33, 3−56.

Ferreira, M., & Laux, P. (2007). Corporate governance, idiosyncraticrisk, and information flow. Journal of Finance, 62, 951−989.

Flannery, M., & Protopapadakis, A. (2002). Macroeconomic factorsdo influence aggregate stock returns. Review of FinancialStudies, 15, 23−49.

Fleming, J., Kirby, C., & Ostdiek, B. (2001). The economic value ofvolatility timing. Journal of Finance, 56, 329−352.

Foster, F., & Viswanathan, S. (1993). Variations in trading volume,return volatility, and trading costs: Evidence from recent priceformation models. Journal of Finance, 48, 187−211.

Foster, F., & Viswanathan, S. (1995). Can speculative tradingexplain the volume-volatility relation? Journal of Business andEconomic Statistics, 13, 379−396.

Franses, P. H., & McAleer, M. (2002). Financial volatility: Anintroduction. Journal of Applied Econometrics, 17, 419−424.

French, K., & Roll, R. (1986). Stock return variances: The arrival ofinformation and the reaction of traders. Journal of FinancialEconomics, 17, 5−26.

Galbraith, J., & Kisinbay, T. (2005). Content horizons for conditionalvariance forecasts. International Journal of Forecasting, 21, 249−260.

Gallant, R., Hsu, C., & Tauchen, G. (1999). Using daily range data tocalibrate volatility diffusions and extract the forward integratedvariance. Review of Economics and Statistics, 81, 617−631.

Giot, P., & Laurent, S. (2004). Modelling daily Value-at-Risk usingrealized volatility and ARCH type models. Journal of EmpiricalFinance, 11, 379−398.

Goyal, A., & Santa-Clara, P. (2003). Idiosyncratic risk matters.Journal of Finance, 58, 975−1007.

Granger, C., & Newbold, P. (1976). Forecasting economic timeseries. New York: Academic Press.

Guo, H., & Savickas, R. (2006). Idiosyncratic volatility, stockmarket volatility, and expected stock returns. Journal of Businessand Economic Statistics, 24, 43−56.

Hansen, P., & Lunde, A. (2005). A forecast comparison of volatilitymodels: Does anything beat a GARCH(1,1)? Journal of AppliedEconometrics, 20, 873−889.

Hansen, P., & Lunde, A. (2006). Realized variance and marketmicrostructure noise. Journal of Business and Economic Statistics,24, 127−161.

Harris, L. (1987). Transaction data tests of the mixture ofdistributions hypothesis. Journal of Financial and QuantitativeAnalysis, 22, 127−141.

Harvey, D., Leybourne, S., & Newbold, P. (1998). Tests for forecastencompassing. Journal of Business and Economic Statistics, 16,254−259.

Heimstra, C., & Jones, J. (1994). Testing for linear and nonlinearGranger causality in the stock price-volume relation. Journal ofFinance, 49, 1639−1664.

Hobijn, B., Franses, P., & Ooms, M. (2004). Generalizations of theKPSS-test for stationarity. Statistica Neerlandica, 58, 483−502.

Jennings, R., Starks, L., & Fellingham, J. (1981). An equilibriummodel of asset trading with sequential information arrival.Journal of Finance, 36, 143−161.

Jin, L., & Myers, S. (2006). R2 around the world: New theory andnew tests. Journal of Financial Economics, 25, 257−292.

Jones, C., Kaul, G., & Lipson, M. (1994). Information, trading, andvolatility. Journal of Financial Economics, 36, 127−154.

Jones, C., Lamont, O., & Lumsdaine, R. (1998). Macroeconomic newsand bond market volatility. Journal of Financial Economics, 47,315−337.

Kalev, P., Liu, W., Pham, P., & Jarnecic, E. (2004). Publicinformation arrival and volatility of intraday stock returns.Journal of Banking and Finance, 28, 1441−1467.

Karpoff, J. (1987). The relation between price changes and tradingvolume: A survey. Journal of Financial and Quantitative Analysis,22, 109−126.

Kauermann, G., & Carroll, R. (2001). A note on the efficiency ofsandwich covariance matrix estimation. Journal of the AmericanStatistical Association, 96, 1387−1396.

Kelly, P. (2007). Information efficiency and firm-specific returnvariation. University of South Florida Working Paper.

Kwiatkowski, B., Phillips, P., Schmidt, P., & Shin, Y. (1992). Testingthe null of stationarity against the alternative of a unit root: Howsure are we that economic time series have a unit root? Journalof Econometrics, 54, 159−178.

Lamoureux, C., & Lastrapes, W. (1990). Heteroskedasticity in stockreturn data: Volume versus GARCH effects. Journal of Finance,45, 221−229.

Lamoureux, C., & Lastrapes, W. (1994). Endogenous tradingvolume and momentum in stock return volatility. Journal ofBusiness and Economic Statistics, 12, 253−260.

Lee, D., & Schmidt, P. (1996). On the power of the KPSS test ofstationarity against fractionally integrated alternatives. Journalof Econometrics, 73, 285−302.

Liesenfeld, R. (2001). A generalized bivariate mixture model forstock price volatility and trading volume. Journal of Econo-metrics, 104, 141−178.

Luu, J., & Martens, M. (2003). Testing the mixture of distributionshypothesis using realized volatility. Journal of Futures Markets,23, 661−679.


Martens, M., & Zein, J. (2004). Predicting financial volatility: High-frequency time-series forecast vis-à-vis implied volatility. Jour-nal of Futures Markets, 24, 1005−1028.

Meddahi, N. (2003). ARMA representations of integrated andrealized variances. Econometrics Journal, 6, 334−355.

Melvin, M., & Yin, X. (2000). Public information arrival, exchange ratevolatility, and quote frequency. Economic Journal, 110, 644−661.

Mina, J., & Xiao, J. (2001). Return to RiskMetrics: the evolution of astandard. RiskMetrics Technical Document.

Mincer, J., & Zarnowitz, V. (1969). The evaluation of economicforecasts. In J. Mincer (Ed.), Economic Forecasts and Expecta-tions. New York: National Bureau of Economic Research.

Mitchell, M., & Mulherin, J. (1994). The impact of public informationon the stock market. Journal of Finance, 49, 923−950.

Morck, R., Yeung, B., & Yu, W. (2000). The information content ofstock markets: Why do emerging markets have synchronous stockprice movements? Journal of Financial Economics, 58, 215−260.

O'Hara, M. (1995). Market microstructure theory. Cambridge, MA:Blackwell Publishers.

Owens, J., & Steigerwald, D. (2006). Noise reduced realized volatility:A Kalman filter approach. In T. Fomby, & D. Terrell (Eds.), Ad-vances in Econometrics: Econometric Analysis of Economic andFinancial Time Series (Volume 20). Amsterdam: Elsevier.

Patton, A. (2007). Volatility forecast evaluation and comparisonusing imperfect volatility proxies. London School of EconomicsWorking Paper.

Pong, S., Shackleton, M., Taylor, S., & Xu, X. (2004). Forecastingcurrency volatility: A comparison of implied volatilities and AR(FI)MAmodels. Journal of Banking and Finance, 28, 2541−2563.

Poon, S., & Granger, C. (2003). Forecasting volatility in financialmarkets. Journal of Economics Literature, 41, 478−539.

Richardson, M., & Smith, T. (1994). A direct test of the mixture ofdistributions hypothesis: Measuring the daily flow of information.Journal of Financial and Quantitative Analysis, 29, 101−116.

Roll, R. (1984). A simple measure of the effective bid-ask spread inan efficient market. Journal of Finance, 39, 1127−1139.

Roll, R. (1988). R2. Journal of Finance, 43, 541−566.Ross, S. (1989). Information and volatility: The no-arbitrage

martingale approach to timing and resolution irrelevancy. Jour-nal of Finance, 44, 1−18.

Statman, M. (1987). How many stocks make a diversified portfolio?Journal of Financial and Quantitative Analysis, 22, 353−363.

Stivers, C. (2003). Firm-level return dispersion and the futurevolatility of aggregate stock market returns. Journal of FinancialMarkets, 6, 389−411.

Tauchen, G., & Pitts, M. (1983). The price variability–volumerelationship on speculative markets. Econometrica, 51, 485−505.

Tole, T. (1982). You can't diversify without diversifying. Journal ofPortfolio Management, 8, 5−11.

Zhou, B. (1996). High-frequency data and volatility in foreign exchangerates. Journal of Business and Economic Statistics, 14, 45−52.

can idiosyncratic volatility help forecast stock market volatility?

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