Modeling discontinuous periodic conditional volatility: Evidence from the commodity futures market

MODELING DISCONTINUOUS

PERIODIC CONDITIONAL

VOLATILITY: EVIDENCE

FROM THE COMMODITY

FUTURES MARKET

NICHOLAS TAYLOR

This paper examines a wide variety of models that allow for complex anddiscontinuous periodic variation in conditional volatility. The value ofthese models (including augmented versions of existing models) is demon-strated with an application to high frequency commodity futures returndata. Their use is necessary, in this context, because commodity futuresreturns exhibit discontinuous intraday and interday periodicities in condi-tional volatility. The former of these effects is well documented for variousasset returns; however, the latter is unique amongst commodity futuresreturns, where contract delivery and climate are driving forces. Using sixyears of high-frequency cocoa futures data, the results show that thesecharacteristics of conditional return volatility are most adequately cap-tured by a spline-version of the periodic generalized autoregressive con-ditional heteroscedastic (PGARCH) model. This model also providessuperior forecasts of future return volatility that are robust to variation in

I wish to thank the anonymous referee for useful comments and suggestions, and Cardiff BusinessSchool for funding received while undertaking this paper.For correspondence, Department of Accounting and Finance, Cardiff University, Cardiff CF103EU, United Kingdom; e-mail: [email protected]

Received June 2003; Accepted November 2003

� Nicholas Taylor is a Distinguished Senior Research Fellow in the Department ofAccounting and Finance at Cardiff Business School at Cardiff University in Cardiff,United Kingdom.

The Journal of Futures Markets, Vol. 24, No. 9, 805–834 (2004) © 2004 Wiley Periodicals, Inc.Published online in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/fut.20114

806 Taylor

1See French, Schwert, and Stambaugh (1987) for an example of how models of conditional returnvolatility can be successfully embedded into a mean-variance asset pricing model.2See Jorion (2000) for an overview of VaR models, and Christoffersen and Diebold (2000) for a spe-cific description of the role of models of conditional return volatility, and the associated forecasts,within the VaR framework.3See Szegö (2002) for a concise definition of market risk.4See Chew (1994) and Frye (1998) for a discussion of the importance of intraday VaR models, andGiot (in press) for an investigation into the quality of various intraday VaR models.5A series is said to be periodic if observations in certain periods display different features (e.g., mean,variance, etc.) to those in alternate periods. As these periods do not necessarily have to correspondto seasons, the more generic term periodicity is used instead of the term seasonality, though the latterterm is often used in the literature.

the loss function assumed by the user, and are shown to be beneficial tousers of Value-at-Risk (VaR) models. © 2004 Wiley Periodicals, Inc. JrlFut Mark 24:805–834, 2004

INTRODUCTION

A great deal of emphasis is placed on appropriate ways of modeling theconditional volatility of asset returns. The traditional motivation for thishas been the crucial role of risk played in the pricing of financial assets.1

More recently, changes in the financial regulation of risk management,via the introduction of various Value-at-Risk (VaR) models, has con-tributed to the widespread interest in conditional return volatilitymodels.2 Following the publication of the G-30 Report by the Group of30 in 1993, and the Basle Capital Accord introduced by the BasleCommittee on Banking Supervision in 1996, these models have prima-rily focused on market risk measured over daily frequencies.3 However,for particular users it may be more appropriate to quantify this risk overhigher frequencies. For instance, intraday market risk is important totraders and market makers seeking to manage the risk of the tradingdesk.4 It is the unique features of such risk, and problems of modelingthis risk therein, that are considered in this paper. In particular, condi-tional volatility models that take explicit account of various market riskperiodicities are developed and examined.5 As these periodicities are par-ticularly acute for high-frequency commodity futures returns, it is thesedata that are used to demonstrate the usefulness of these models.

Several models of conditional volatility have been utilized within theareas of asset pricing and risk management. Of these, the class of econo-metric model that has received the most attention has been the autore-gressive conditional heteroscedastic (ARCH) model (Engle, 1982), itsgeneralized (GARCH) version (Bollerslev, 1986; Taylor, 1986), and themany subsequent modifications to these models; see Bollerslev, Chou,

Discontinuous Periodic Conditional Volatility 807

6The main competition to the (econometric) GARCH model comes in the form of an economicmodel; namely, the implied volatility model. Estimates based on this model of return volatility havebeen shown to be useful in explaining the dynamics of conditional return volatility (see, e.g., Day &Lewis, 1992; Xu & Taylor, 1995; Blair, Poon, & Taylor, 2001; Giot, 2003; Mayhew & Stivers, 2003).However, this explanatory power has not been shown to be greatly superior to that provided bystandard GARCH models. It is for this reason that we do not consider implied volatility as anexplanatory variable in this paper.

and Kroner (1992) for a review of this vast literature.6 While thesemodels have been successfully applied to daily, weekly, and monthly assetreturn data, applications to higher frequency financial data have beenless fruitful. Bollerslev and Ghysels (1996) and Andersen and Bollerslev(1997, 1998a) argue that this failure is due to the deterministic patternof return volatility over the trading day. Such observations have led to thedevelopment of GARCH models that explicitly incorporate this periodic-ity into the parameters of the model. The purpose of the current paper isto examine the performance of augmented versions of these periodicGARCH(PGARCH) models, in comparison to competing periodic condi-tional volatility models.

Several applications of existing conditional volatility models havedocumented their usefulness. Using intraday data from a variety of mar-kets, Martens, Chang, and Taylor (2002), Giot (in press), and Clementsand Taylor (2003) assess the relative performance of various conditionalvolatility (including PGARCH) models. However, all of these studies usedata that are characterized by a U-shaped intraday return volatility pat-tern. When return volatility does not follow this pattern, the relativeperformance of these models has not been examined. This issue isaddressed in this paper. In particular, various conditional volatility mod-els (including augmented versions of existing models) are applied to datathat exhibit intraday volatility patterns (referred to as intraday perio-dicity) that do not conform to the U-shaped pattern.

The commonly observed U-shaped intraday return volatility patternis the result of systematic trading behavior around the beginning and endof the trading day; see Admati and Pfleiderer (1988) and Gerety andMulherin (1992) for detailed descriptions of this behavior. Moreover,such behavior will occur in any financial market where a deterministicbreak in trading occurs, irrespective of the timing of the closure. Forexample, when the trading session closes over the lunch time period, asin many Asian markets, a double U-shape intraday periodicity in returnvolatility is observed (Chang, Fukuda, Rhee, & Takano, 1993; Cheung,Ho, Pope, & Draper, 1994). Moreover, as lunchtime information is com-pounded into prices at the opening of the afternoon session, there may

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7Traders with trading horizons of a few minutes, referred to as scalpers, are generally assumed toprovide the vast majority of prices (and liquidity) in futures markets (Tse, 1999).

be an abrupt (and discontinuous) increase in return volatility at thispoint in time.

An alternative example of an intraday trading close is observed inthe U.K. cocoa futures market. In addition to being the largest com-modity futures market in Europe, studying this market is particularlyinteresting because of two additional complications to return volatilitydynamics. First, return volatility in European asset markets is sensitive tothe release of U.S. macroeconomic announcements (see, e.g., Harvey &Huang, 1991; Scalia, 1998; Tse, 1999). As this information is alwaysreleased at 13:30 (GMT), it seems likely that the intraday periodicity ofreturn volatility in the U.K. cocoa futures market is likely to be affected.Second, discontinuous periodicities over lower frequency cycles arelikely to be observed in futures markets. This periodicity (referred to asinterday periodicity) was first hypothesized by Samuelson (1965), anddescribes how return volatility increases as the time-to-maturitydecreases. This hypothesis also implies that there will be an abruptchange in return volatility if there is a sudden change in the maturity ofthe futures contract. As contracts are often rolled over at, or just before,maturity, it follows that such abrupt changes are likely to be observed atthis point in time. While most studies reject the Samuelson hypothesis,the empirical evidence suggests the existence of periodicities in com-modity futures return volatility that are, in some way, related to thematurity cycle (see, e.g., Sørensen, 2002; Richter & Sørensen, 2002;Allen & Cruickshank, 2002), a result affected, to some extent, byclimatic effects.

It is surprising, given the importance of appropriate models ofmarket risk, that few, if any, studies have attempted to incorporate theabove intraday and interday periodicities into a single conditional volatil-ity model. Such a model is likely to be extremely useful as there is a highproportion of short-term traders operating in commodity futuresmarkets.7 Specifically, the use of market risk models of this type is likelyto significantly improve the quality of the risk management undertakenby these traders. It is this gap in the literature that this paper addresses.In particular, this paper considers highly flexible conditional volatilitymodels designed to account for the complex periodicities in market riskfound in the U.K. cocoa futures market. To anticipate the results, wefind that the return volatility forecasts produced by a spline-version ofthe PGARCH model are more accurate than the forecasts produced by


competing conditional volatility models. In addition, the economicimportance of various return volatility forecasts is demonstrated byshowing that VaR models based on a spline-version of the PGARCHmodel are more accurate than currently used VaR models.

This paper is organized as follows. The next section describes vari-ous econometric models of periodic conditional volatility, and introducesaugmented versions of these models designed to incorporate both intra-day and interday periodicity in return volatility. The third section con-tains results pertaining to various aspects of the estimated conditionalvolatility models considered, and the final section concludes.

PERIODIC CONDITIONAL VOLATILITYMODELS

This section contains descriptions of a variety of periodic conditionalvolatility models. Throughout this description, the models consideredwill reflect the fact that commodity futures returns are used in theempirical section. In doing this, we concede that the set of modelsdescribed is not universal; rather, only models that are relevant to theapplication are considered.

Periodic GARCH Models

All of the models described in this subsection are simultaneous modelsof the time-dependency and periodicity in the conditional volatility ofreturns within a GARCH framework; however, they differentiate them-selves by allowing different periodic components in the specification ofthe conditional volatility equation.

Microstructure-Based PGARCH Models

One of the advantages of using a GARCH-type specification for condi-tional volatility is that (periodic) explanatory variables can be introducedinto the volatility equation of a standard GARCH(p, q) model withoutsubstantially complicating the estimation process. In allowing such vari-ables, this version of the PGARCH model is given by

(1)

where Rt denotes (log) returns, denotes theinformation set available at is a continuoust � 1, �t ƒ �t�1 � D(0, ht), D

�t�1�t � Rt � E[Rt 0 �t�1],

ht � v � a(L)�2t � b(L)ht � G(L)�Zt

810 Taylor

8Formally, and s and t are related by the function s(t) defined such that

(s, t) � 5(1, 1), (2, 2), p , (S � 1, S � 1), (S, S), (1, S � 1), (2, S � 2), p , (S, T)6s � 5s � �� : 1 � s � S6,

distribution with support over and mean equal to zero and vari-ance equal to ht, and Zt is a ( ) vector of (periodic) explanatory variables containingelements with associated parameter vector contain-ing elements This specification forms the basis forthe first periodic model considered in this paper. In the current applica-tion, the explanatory variables consist of various (periodic) marketmicrostructure variables, ergo this model is, henceforth, referred to asthe microstructure-version of the PGARCH model.

Flexible Fourier form PGARCH Models

In the simplest version of the PGARCH model of Bollerslev and Ghysels(1996), periodic variation in conditional volatility can be achieved byincluding a set of periodic intercept dummy variables in the volatilityequation of a standard GARCH model, where each of these variablestakes a value of unity if the current observation is in the sth stage of theperiodic cycle S, and a value of zero otherwise.8 A potential problem withsuch a model is that a large number of coefficients are required if thereare many time periods within each periodic cycle (i.e., S is large). It ispossible to overcome this problem by selecting dummy variables thatspan more than one time period. However, this assumes that conditionalvolatility is constant within the time period covered by the dummy vari-able and then changes abruptly whenever a new time period is entered.To overcome such problems, Andersen and Bollerslev (1997, 1998a)propose use of the flexible Fourier form (FFF) to model periodic condi-tional return volatility. This form can be used within a PGARCH frame-work as follows:

(2)

where is a vector of coefficients, and

This model is, henceforth, referred to as the FFF-version of thePGARCH model.

The FFF-version of the PGARCH model is somewhat restrictive inthat it assumes equality in conditional volatility at the beginning and end

i � 51, p , I6.ut � 2ps(t)�S,f�i,t, p , f�I,t], f�i,t � [sin(iut), cos(iut)],F�t � [1, f�1,t, p ,((2I � 1) � 1)V

ht � V�Ft � a(L)�2t � b(L)ht

[g1(L), p , gr(L)]�.G(L)[Z1,t, p , Zr,t]�,

r � 1a(L) � a1L � p � aq L

q, b(L) � b1L � p � bp Lp,

(��, � )


9To demonstrate this problem, note that at these respective points in time, i.e., s(t) � 0 and s(t) � S,the value of equals zero and unity, respectively, and the periodic components in (2) are

respectively. While this is not a problem when using these models in applications tocontinuously traded assets like foreign currency (see, e.g., Martens et al., 2002), it is a problemwhen using data from financial markets that are closed overnight. It is possible to overcome thisproblem by using an alternative version of the FFF that includes linear and quadratic s(t) terms; seeAndersen and Bollerslev (1997, 1998a). However, as will be demonstrated, the alternative versionproposed in this paper does not require any additional parameters.10To clarify the definitions of time used in the augmented FFF-version of the PGARCH model,consider the following example of how intraday periodicity can be modeled. Assume that we areusing 5-minute frequency returns over the trading day, and that trading starts at 9:00 and finishes at17:00, with an hour break in trading between 12:00 and 13:00. This means that the business andcalendar times of the last observations of the periodic cycle, S and Sc, will be 84 (�7 � 12) and 288(�24 � 12), respectively. Therefore, at the opening at 9:00, equals and equals

, and at the lunchtime close in trading, s(t) equals and equals .However, at the opening of trading at 13:00, still equals , but has increased to

. It is this difference in the values at the close of morning trading and opening ofafternoon trading that allows conditional return volatility to be different at these points in time.Similarly, at the close of trading at 17:00, equals unity while equals 96�288.Therefore, only use of the latter ratio will enable the periodic components to differ and, hence, willallow conditional return volatility to differ over these points in time. Moreover, this is achievedwithout the need for additional parameters in the model.

sc(t)�Scs(t)�S

sc(t)�Sc48�288sc(t)�Sc36�84s(t)�S

36�288sc(t)�Sc36�841�288sc(t)�Sc1�84s(t)�S

i � 51, p , I6,f�i,t:s(t)�S � [i sin(2p), i cos(2p)] � [0, i]

f�i,t:s(t)�0 � [i sin(0), i cos(0)] � [0, i]

s(t)�S

of the periodic cycle.9 An additional related problem exists in the currentapplication as some U.K. commodity futures markets have, until veryrecently, also closed over the lunchtime period. Such trading breaks areproblematic as the above model is incapable of modeling the resultantdiscontinuity in conditional return volatility. This is because the periodiccomponents in (2) are assumed to be measured with respect to businesstime, with As such, when markets are closed during thetrading day, this time does not increase, and the periodic components in(2) do not change. This imposes the restriction that conditional returnvolatility before and after the close are equal.

To overcome these problems we introduce an alternative measure oftime into the FFF-version of the PGARCH model. In particular, the peri-odic components in this model are now assumed to be a function of timemeasured with respect to the actual timing of events, referred to ascalendar time, rather than business time. To achieve this, we replace s(t)and S in (2) with sc(t) and Sc, respectively, where sc(t) is a function thatgives the calendar time of the tth observation within the periodic cycle,and Sc is the calendar time of the last observation of the periodic cycle.This model is, henceforth, referred to as the augmented FFF-version ofthe PGARCH model.10

t � 51, 2, . . . , T6.

812 Taylor

11Though few authors have used this functional form in the context of a PGARCH model, it hasbeen used extensively in alternative volatility models. Most notably, cubic splines have been suc-cessfully incorporated into a variety of autoregressive conditional duration (ACD) models (see, e.g.,Engle & Russell, 1998; Zhang et al., 2001; Giot, in press; Taylor, 2004).

Spline-Based PGARCH Models

Though the FFF-version of the PGARCH model is parsimonious andallows for smooth volatility dynamics, it is somewhat rigid in functionalform and, therefore, may not be able to capture complex periodic volatil-ity dynamics. To overcome this shortcoming, a spline-version of thePGARCH model is considered that allows different cubic spline func-tions to be estimated between selected points (referred to as knots) with-in the periodic cycle.11 In particular, letting kj denote the jth knot, with

and this version ofthe PGARCH model is given by

(3)

where is a ((3( J � 1) � 1) � 1) vector of coefficients,and

Dj equals unity if and zero otherwise. This model is, henceforth,referred to as the spline-version of the PGARCH model.

The problem of overnight market closure is (partially) solved by usingthe spline-version of the PGARCH model. In particular, the periodiccomponents of conditional return volatility are allowed to vary betweenthe opening and closing of the trading day. However, this model cannotallow differences in conditional return volatility when intraday tradingbreaks occur. To remedy this shortcoming, this model can also be aug-mented by using calendar time instead of business time, that is, s(t) and Sin (3) are replaced by sc(t) and Sc, respectively. This model is, henceforth,referred to as the augmented spline-version of the PGARCH model.

In addition to being a more flexible PGARCH model, the augmentedspline-version of the PGARCH model has an additional advantage overthe augmented FFF-version of the PGARCH model. The specificationsgiven by (3) are such that the estimated conditional return volatilitiesequate at each of the pre-selected knots. While this is a desirable featurefor most periods within the trading day, it may not be when consideringthe interday periodicity observed in this paper. In particular, theSamuelson hypothesis predicts that there will be sharp changes in returnvolatility when there is a large change in the maturity of the contract.This discontinuous interday periodicity can easily be incorporated intothe spline-version of the PGARCH model by carrying out the following

s(t) kj,[1, g�0,t, . . . , g�j,t, . . . , g�J,t], g�j,t � Dj[(s(t) � kj), (s(t) � kj)

2, (s(t) � kj)3],G�t �V

ht � V�Gt � a(L)�2t � b(L)ht

k0 � 0,kj � 5kj � ��: 0 � kj � S6, j � 50, 1, . . . , J6,


12This effect is, however, smaller than that predicted by standard mathematical models of returnvolatility; see French and Roll (1986) for an investigation into this effect.

steps: (1) select knots to coincide with expected (large) changes in matu-rity, and (2) relax the assumption that the conditional return volatilityequates at these knots. The second of these is achieved by assuming thatconditional return volatility evolves as follows:

(4)

where is a (4( J � 1) � 1) vector of coefficients,

and Dj equals unity if and zero otherwise. Thismodel is, henceforth, referred to as the unrestricted spline-version of thePGARCH model.

It also worth noting that the use of calendar time is also advised wheninterday periodicities are present. If business time is used in this context,then trading days are normally used as the unit of time. Using this meas-urement of time in the FFF-version and the spline-version of thePGARCH model will imply a gradually changing conditional return volatil-ity over the year that makes no allowance for days when no trading occurs.However, a literature exists suggesting that return volatility is higher afternontrading days than after trading days.12 To allow for such effects, the useof calendar time in the interday periodic components of the FFF-versionand the spline-version of the PGARCH model permits differential changesin conditional return volatility after nontrading and trading days.

Alternative Periodic Volatility Models

In addition to the above PGARCH models, this paper also considers theperformance of two competing models of conditional volatility employedin the literature. The first proposes separation in the modeling of condi-tional volatility and periodicity, while the second advocates a periodicmodel of an alternative definition of return volatility. Both of theseapproaches are described below.

Two-Step Filtration Models

Two-step procedures have been employed by several authors; see, e.g.,Andersen and Bollerslev (1997, 1998a), Bauwens and Giot (2001),Martens et al. (2002), and Giot (in press), for applications to intradayfinancial data. The initial step of these procedures involves estimatingthe periodic components of the data (henceforth referred to as the fitted

s(t) kj,50, . . . , J6,j �gu�j,t, . . . , gu�J,t], g

u�0,t � [s(t), s(t)2, s(t)3], gu�j,t � Dj[1, s(t), s(t)2, s(t)3],Gu

t � � [1, gu�0,t, . . . ,V

ht � V�Gut � a(L)�2

t � b(L)ht

814 Taylor

13Price duration intensity is defined as the number of trades occurring in the interval t � 1 to t, con-taining a price that is different to that contained in the previous trade. This definition is maintainedthroughout the text.

periodic components) and removing them from the data, with the filtra-tion element of this step achieved by dividing returns by the fittedperiodic components. The second step models these filtered data via aconventional conditional volatility model, which in the current applica-tion is assumed to be a standard GARCH(p, q) model.

The periodic components used in the first of these steps are oftenbased on simple intraday means of squared returns. However, in order tocapture the intraday and interday periodicities present in the currentapplication, we consider models based on the fitted values from an ordi-nary least squares (OLS) regression of squared returns on FFF-basedand spline-based periodic components, both of which are defined overthe intraday and interday periodicities.

Price Duration Intensity Models

A growing number of authors have modeled alternative definitions ofreturn volatility. Most notably, Engle and Russell (1998) introduce theautoregressive conditional duration (ACD) model of price duration—ameasure closely related to conditional return volatility—that includesperiodic components in its specification. Such models, and variousextended versions, have been successfully applied to transaction datafrom a variety of markets (see, e.g., Zhang, Russell, & Tsay, 2001; Giot,in press; Taylor, 2004). However, the quality of the return volatility fore-casts generated by such models, in comparison to those based onPGARCH models, is questionable. Indeed, in a recent study of high-frequency stock return volatility, Giot (in press) finds that intraday VaRmeasures based on PGARCH return volatility forecasts are far superiorto those based on ACD return volatility forecasts. Given this evidence,ACD models are not considered in this paper. However, similarly moti-vated models are considered.

As with ACD models, the motivation for the final conditional volatil-ity model considered in this paper is based on a direct linkage betweenreturn volatility and trading activity. In particular, conditional returnvolatility can be expressed as a linear function of the conditional expec-tation of price duration intensity, denoted 13

(5)ht � a cAt�1b2

lt

lt,


14This is a discretized version of equation (32) in Engle and Russell (1998).

where c is the minimum allowable nonzero price change, and At�1 is theprice of the asset.14 However, rather than modeling the reciprocal ofprice duration intensity, as in the ACD models, we model the intensitydirectly via the Poisson count model.

The Poisson model specifies that price duration intensity at time t,denoted Yt, is drawn from a Poisson distribution with intensity parameter

, which, in turn, is related to the regressors by the log-linear model,

(6)

where is one of the periodiccomponent vectors defined in the previous subsection, and is avector of explanatory variables including past values of Yt. To assess thequality of this model compared to the periodic models of return volatilitydescribed previously, the estimated count measure of return volatility, is converted to the estimated conditional return volatility measure, via (5).

EMPIRICAL RESULTS

This section contains a description of the data used, presents evidence ofvarious periodicities in these data, describes the specifications of theestimated models, and evaluates these models in terms of model fit andforecasting ability.

Data

We make use of various pieces of information concerning trades incocoa futures contracts traded on the Euronext.liffe exchange. In partic-ular, returns, defined as the log change in transaction prices; effectivebid-ask spreads, defined as the absolute difference between the transac-tion price and the quote mid-price, divided by the quote mid-price; priceduration intensity, defined as the number of “price-moving” tradesoccurring within a preselected time interval; and trading volume,defined as the number of contracts traded within this interval. Thisinformation was collected for every trade in the nearest futures contractcarried out between January 2, 1997 and December 31, 2002. Thesedata were obtained from the London International Financial Futuresand Options Exchange (LIFFE). Use of such a relatively long span of

ht,lt,

Zt�1

F� � [V�, G�], X�t�1 � [H�t�1, Z�t�1], Ht�1

ln lt � f�Xt�1

Xtlt

816 Taylor

high-frequency data allows simultaneous examination of both intradayand interday periodicity.

Cocoa futures contracts specify delivery of the underlying, 11 busi-ness days prior to the last business day of March, May, July, September,and December. As such, we assume that futures contracts with the near-est maturity are replaced (through trading) by contracts with the nextnearest maturity—a practice referred to as rolling over—on the last busi-ness day of the month preceding delivery. Though selection of this date issomewhat ad hoc, it is based on the observation that the trading volumesof the contracts with the next nearest maturity begin to exceed the con-tracts with the nearest maturity around this time.

The above transaction data are then converted to 5-minute fre-quency data. This frequency is deemed to be sufficiently low enough toavoid stale data, and high enough to avoid loss of information. Thesedata are then partitioned over various sample periods because of differ-ences in the hours in which trading occurs. The first partition is made atthe point where trading is changed from an open-outcry quote-driventrading structure to an electronic order-driven trading platform. Thisoccurred on November 24, 2000, with the introduction of the Connecttrading platform. Amongst the many changes to the trading process thatthis brought about, the most relevant to the present study of periodicconditional return volatility relates to the changes in the trading hours.In the pre-Connect period, cocoa futures were traded from 9:30 to17:00, and in the post-Connect period, the trading hours were 9:30 to16:50. In addition, an intraday close in cocoa trading occurred between12:30 and 14:00 during the period January 2, 1997 to November 2, 1998(henceforth referred to as period 1), and between 12:00 and 13:30 dur-ing the period November 3, 1998 to November 24, 2000 (henceforthreferred to as period 2). In the post-Connect period (henceforth referredto as period 3), the cocoa futures market did not close over thelunchtime period.

Periodicity Tests

Before estimating specific models of conditional volatility, it is worthconsidering the nature and extent of the intraday and interday periodici-ties in variables closely related to return volatility. To this end, resultspertaining to periodicity tests applied to 5-minute frequency returns,absolute returns, price duration intensity, effective bid-ask spreads, andtrading volume, are given in Table I. These tests are based on an OLSregression of each of these variables on a set of periodic components,


TABLE I

Testing for Periodicity

Periodicity test

Sample period Variable

Period 1 Rt 2.8444*** 0.4155 2.2372**�Rt � 549.5541*** 17.6727*** 194.9665***Yt 507.8095*** 106.2684*** 240.1154***Bt 18.4456*** 40.5594*** 33.1881***Vt 18.4728*** 8.2771*** 12.1005***

Period 2 Rt 4.7248*** 0.0735 3.5620***�Rt � 508.4521*** 14.5274*** 199.7491***Yt 510.8045*** 71.1144*** 217.6778***Bt 21.8751*** 94.1847*** 69.8751***Vt 23.3838*** 6.2302*** 12.6628***

Period 3 Rt 6.4060*** 1.4709 5.1722***�Rt � 611.5232*** 48.4201*** 236.1211***Yt 914.0288*** 171.5560*** 419.0469***Bt 61.4678*** 146.4039*** 118.0919***Vt 46.5204*** 9.9266*** 22.1245***

Note. This table contains the F-test statistics associated with spline-based tests for intraday periodicityinterday periodicity and joint tests for intraday and interday periodicity

These tests are based on 5-minute frequency futures returns (Rt), absolute futures returns, price duration intensity (Yt), effective bid-ask spreads (Bt), and trading volume (Vt). The tests are con-

ducted using three sample periods: January 2, 1997 to November 2, 1998 (period 1), November 3, 1998 toNovember 24, 2000 (period 2), and November 25, 2000 to December 31, 2002 (period 3).

*10% significance.**5% significance.***1% significance.

( ƒRt ƒ )(F(��d � ��y � 0)),

(F(��y � 0)),(F(��d � 0)),

F(V~d � V~d � 0)F(V~ y � 0)F(V~d � 0)

15The periodic components used in these regressions are those used in the augmented and unre-stricted spline-versions of the PGARCH model, defined over the intraday and interday periodicities.

with an F-test conducted, whereby the coefficients on these periodiccomponents are restricted to equal zero.15 The results indicate that mostvariables exhibit strong intraday and interday periodicities. The excep-tion to this result occurs when returns are tested for periodicity.

The intraday periodicity in return volatility is also apparent whenone considers the plots in Figure 1A–C. These present the mean priceduration intensity measure of return volatility for each 5-minute intervalwithin the trading day. Separate plots are given for each of the sampleperiods described above. A clear periodicity is apparent from these plots.In particular, return volatility appears high at the opening of trading,both at the beginning of the trading day and at the beginning of tradingafter the lunchtime period. This pattern is repeated at the close of trad-ing, both at the end of the trading day and just before the lunchtime

818 Taylor

16Information asymmetry measures the amount of informed trading versus non-informed trading ina market at any point in time.

period. It is also interesting to note that these effects are far less pro-nounced when no lunchtime close occurs.

Model Estimation

The first model considered is based on the microstructure-version of thePGARCH model (henceforth denoted M1) given by (1). The explanatoryvariables considered in this paper are motivated by the following argu-ments. A key implication of market microstructure theory (for a review,see O’Hara, 1995) is that conditional return volatility is positively relatedto the extent of information asymmetry.16 It follows that a measure ofinformation asymmetry would seem to be a useful explanatory variable.The problem here is that information asymmetry cannot be readilyobserved in financial markets. The approach taken in this paper is to

FIGURE 1Periodic return volatility.

Note. This figure shows the mean realized and estimated intraday return volatility perio-dicity using data from the period January 2, 1997 to November 2, 1998 (A), using datafrom the period November 3, 1998 to November 24, 2000 (B), using data from the periodNovember 25, 2000 to December 31, 2002 (C), and the mean realized and estimatedinterday return volatility periodicity using data from the period January 2, 1997 toDecember 31, 2002 (D). The vertical arrows in D indicate the maturity dates of thefutures contracts.


17There remains two features of M1 to be discussed. First, we conduct a grid search over thenumber of lagged values of the explanatory variables, in addition to the inclusion of Sd lags of allexplanatory variables. In doing this, we find that one lagged value of all explanatory variables pro-vides the best fit (as implied by the value of the AIC). Second, as trading hours vary over periods 1,2, and 3, Sd is allowed to take a different value in each of these periods.

include variables that are theoretically related to the extent of informa-tion asymmetry. One such variable is the bid-ask spread. Glosten andMilgrom (1985) and Easley and O’Hara (1992b) argue that marketmakers will widen spreads when the extent of information asymmetryincreases. Such action is taken by market makers to lessen the proba-bility of trading with an informed agent. The second variable included istrading volume. Inclusion of this variable is motivated by the results ofEasley and O’Hara (1987, 1992a) and Lee, Mucklow, and Ready (1993).They argue that, for a given price, informed traders have an incentive totrade a larger quantity of shares than noninformed traders.

The strong intraday and interday periodicities found in the abovemarket microstructure variables (see Table I) suggests that their usewithin a conditional volatility model may be an effective way of modelingthe complex return volatility periodicities. To this end, the first and Sd

lags of these variables are included, where Sd is given by the number of5-minutes intervals within each trading day. Moreover, we also includethe first and Sd lagged values of absolute returns in the specification.Using these variables, this model selects the lag structure by grid searchover p and q. For the allowable parameter space (p, d)we find that delivers the minimum value of the AkaikeInformation Criterion (AIC). Using this lag structure, M1 is given by

(7a)

(7b)

where D is assumed to be a normal distribution, denotes

the effective bid-ask spread, and Vt denotes trading volume.17

A summary of the results is given in Table II. In addition, the esti-mated coefficients on the above explanatory variables are given by thefollowing vector:

with associated Bollerslev-Wooldridge robust standard error vector,

[0.7656, 0.5939, 0.5747, 0.2559, 0.0085, 0.0164]�

g(L) � [1.5509, 6.3191, 7.7806, 1.2209, 0.0256, 0.0366]�

g1,Sd LSd, g2,1 L � g2,Sd

LSd, g3,1 L � g3,Sd LSd], Z�t � [ ƒRt ƒ , Bt, Vt], Bt

G(L)� � [g1,1 L �

ht � v � aP2t�1 � bht�1 � g(L)�Zt

Rt � m � Pt, Pt ƒ�t�1 � D(0, ht)

p � q � 1� 51, 2, 36,

820 Taylor

TABLE II

Volatility Model Estimates

Model

M1(�) M2(�) M3a(�) M3b(�)

Panel A: GARCH models

0.0008 0.0004 �0.0116 0.0003(0.0005) (0.0005) (0.0101) (0.0005)

0.0007 1.2612 0.3698 0.0007(0.0003) (0.3157) (0.0955) (0.0260)

0.1838 0.0861 0.0946 0.0947(0.0246) (0.0068) (0.8653) (0.0070)

0.3626 0.7409 0.8653 0.7012(0.0246) (0.0123) (0.0029) (0.0147)

LR(G(L) � 0) 25,949.2334*** — — —— 23,608.6485*** — 27,690.7810***— 302.2237*** — 410.1421***— 24,748.8846*** — 29,433.9400***

LL 36,707.7815 36,107.6014 30,183.4054 38,450.1348AIC �0.6851 �0.6729 �0.5616 �0.7158SIC �0.6842 �0.6698 �0.5550 �0.7094

Panel B: Poisson count models (fit only)

LL/100 �1,311.3084 �1,275.2213 — �1,262.5804AIC 2.7080 2.6325 — 2.6160SIC 2.7092 2.6355 — 2.6207Pseudo-R2 0.1044 0.1295 — 0.1350

Note. This table contains the parameter estimates of the models described and denoted in the text as M1, M2,M3a, and M3b (Panel A) and M1�, M2�, and M3b� (Panel B). The numbers in parentheses are Bollerslev-Wooldridge (1992) robust standard errors. The joint significance of the microstructure coefficients, and the intra-day and the interday periodic coefficients, are given by the x2-test version of the likelihood ratio test statisticsand presented in the rows, , , , and respectively. Thelog likelihood (LL), the Akaike Information Criterion (AIC), the Schwarz Information Criterion (SIC), andthe Pseudo-R2 (also referred to as the LR-index) are also given (where appropriate). All numbers in Panel A arebased on all variables being multiplied by 100.

*10% significance.**5% significance.***1% significance.

LR(��d � ��y � 0),LR(��y � 0)LR(��d � 0)LR(G(L) � 0)

LR(V�d � V�y � 0)LR(V�y � 0)LR(V�d � 0)

b

a

v

m

These results are in accordance with the above market microstructuretheories. In particular, conditional return volatility is positively (andsignificantly) related to effective bid-ask spreads and trading volume atall lags. Hence, given the above arguments, there would appear to be asignificant relationship between conditional return volatility and infor-mation asymmetry, a result revealed by proxies for information asymme-try. The results also indicate that the lagged values of absolute returnsare highly significant. Moreover, there appears to be significant intradayperiodicity in the data, as evinced by the significance of all coefficientson variables lagged by Sd.


18Some experimentation concerning the selection of these knots yielded results that were similar innature to those presented in the paper. Details of these results are available upon request.

The next model (henceforth denoted M2) attempts to explicitlymodel the periodicities in the data by combining (7a) with theaugmented FFF-version of the PGARCH model described in the previ-ous section. However, a number of issues require discussion beforegiving the exact specification of the model. First, two different periodici-ties are modeled, namely, intraday periodicity and interday periodicity.Second, variation in trading hours means that intraday periodicity willvary over the periods 1, 2, and 3. This effect is controlled for by theinclusion of dummy variables that equal unity if the tth observationoccurs in period m, and zero otherwise. Third, we have no reason, a pri-ori, to believe that interday periodicity will vary over these sample peri-ods. Therefore, the coefficients determining this periodicity are assumedto remain fixed over all periods. Consideration of all these issues givesrise to the PGARCH model given by (A.2) in the Appendix.

Using a grid search over the parameter space, and we find that the optimal fit (according to theAIC) is achieved when and Estimation detailsof this model are given in Table II. Space limitations prevent us from giv-ing the estimated coefficients on the periodic components. However, theresults of likelihood ratio tests performed on the intraday and interdayperiodicity coefficients, both individually and jointly, are given. Theresults indicate that both periodicities are significant, with intradayperiodicity appearing to be the stronger of the two effects. Despite thesignificance of these coefficients, the fit of this model is inferior to the fitof M1, as indicated by the values of the log likelihood function and vari-ous information criteria.

The next two models considered combine the periodic componentsof the augmented and unrestricted spline-versions of the PGARCHmodel, and the mean equation given by (7a). Intraday periodicity is mod-eled using the periodic components of the augmented spine-version ofthe PGARCH model, but with additional consideration given to the vari-ation in intraday periodicity over periods 1, 2, and 3. In using these peri-odic components, one must select the number and position of the knots.For simplicity, we assume that four equally spaced intraday knots occurat 10:00, 12:00, 14:00, and 16:00.18 By contrast, the modeling of inter-day periodicity is based on using the periodic components of the unre-stricted spline-version of the PGARCH model given by (4), with discon-tinuities occurring when contracts are rolled over to the next nearestcontract. Moreover, the periodic components in (4) assume that thesepoints coincide with the knots selected over the interday period.

Iy � 6.Id � 3,p � q � 1,(Id, Iy) � 51, . . . , 76,

(p, d) � 51, 2, 36,

822 Taylor

19Two-step procedures involving the microstructure and FFF-based periodic components, as used inM1 and M2, are also considered. However, the results were not reported as they were found to beinferior to those obtained using the spline-based periodic components. Details of these results areavailable upon request.20The model fit metrics are adjusted by multiplying the filtered returns by the fitted periodic compo-nents, and the estimated conditional volatility by the squared fitted periodic components.21Space limitations prevent presentation of all estimated parameters. However, these estimates (andtheir associated standard errors) are available upon request. Moreover, to allow for “overdispersion”in price duration intensity errors, the negative binomial model was also used to estimate the processfollowed by price duration intensity. These results (and all subsequent analysis) were very similar tothose presented in this paper and are, therefore, not presented. Details of these results are alsoavailable upon request.

Therefore, given the assumptions made concerning these roll-over dates(see Data above), interday knots are assumed to occur on March 1,May 1, July 1, September 1, and December 1. As these interday knots arespread over the year, one can also capture any climatic effects upon con-ditional return volatility that may be present. The resultant spline-basedperiodic components form the basis of two GARCH-based models. Thefirst model (henceforth denoted M3a) adopts a two-step procedure,whereby the fitted periodic components (obtained via an OLS regressionof squared returns on the spline-based periodic components) are firstremoved from return volatility, and then a standard GARCH(p, q) modelis fitted to the filtered returns.19 By contrast, the second model (hence-forth denoted M3b) assumes that the time-dependency and periodicityin return volatility are jointly modeled via the PGARCH model given by(A.3) in the Appendix.

The optimal fits of M3a and M3b over the parameter space,are obtained (according to the AIC) when p � q � 1.

The results associated with these estimated models are given in Table II.It is apparent from the results that it is far more efficient to jointly esti-mate the time-dependency and periodicity in return volatility. This resultis based on a comparison of the adjusted model fit metrics associatedwith M3a and the model fit metrics associated with M3b.20 It is alsonoticeable that, unlike M2, the significance of the periodicities (asevinced by the results of the likelihood ratio tests) is reflected in thesuperior fit of M3b compared to the other models.

The Poisson count models are also estimated using analogous spec-ifications to those given above, and are denoted M1�, M2�, and M3b�. Toallow for time-dependency in price duration intensity, lagged values ofthe dependent variable are included in the each of these models. A gridsearch indicates that, for each model, six lagged values of this variablegives the optimal fit (according to the AIC). These fits are given inTable II, and indicate that the spline-version of the Poisson count modelof return volatility (i.e., M3b�) has the superior fit of all models.21

(p, q) � 51, 2, 36,


22These plots are based on the fitted values from a Poisson count model with only the periodic com-ponents included. To isolate each periodicity, intraday and interday periodic components enterseparately.

The plots given in Figure 1A–C give an indication of the relativestrengths of the FFF-version and the spline-version of the Poisson countmodels.22 From these plots, one can see that intraday periodicity is moreadequately captured by the spline-version of the Poisson count model,a result that is particularly apparent around the opening and close oftrading. These plots also show that, even when no intraday close occurs,the spline-version of the Poisson count model appears more able tocapture localized return volatility dynamics. In particular, note how thismodel captures the increase in return volatility when U.S. macroeco-nomic announcements are made at 13:30 (GMT), and when the majorU.K. financial markets close between 16:00 and 16:30. This latter peakin return volatility coincides with the 16:10 close in financial futurestrading on LIFFE (in the preconnect period) and the 16:30 close inequity trading on the London Stock Exchange. It is, therefore, conjec-tured that this peak at the end of trading day is the result of day traderssimultaneously closing out positions in a variety of assets at the samepoint in time.

The ability of the models to capture features of interday periodicitycan be examined by inspection of Figure 1D. This plot shows that thespline-version of the Poisson count model implies that return volatilityand the maturity cycle are related, with discontinuous changes in returnvolatility occurring when contracts are rolled over. However, contrary tothe Samuelson hypothesis, return volatility does not appear to (uniformly)increase as the time to maturity decreases; rather, sharp increases inreturn volatility occur when the contract is rolled over. After this date,return volatility appears to gently increase up to the (nearest) maturitydate (indicated in Fig. 1D by the vertical arrows) and then decrease untilthe next roll-over date. Similar implications are found when the FFF-version of the Poisson count is used, but, crucially, the discontinuities inreturn volatility cannot be captured.

Model Adequacy

Given the importance of accurate forecasts of the volatility of assetreturns in areas such as risk management and option pricing, it seemsreasonable to assess model performance via a comparison of forecastingability. This approach will give some indication as to which model mostaccurately represents the data and, hence, is of most importance to usersof financial markets.

824 Taylor

Forecasting Ability

The estimated coefficients associated with the PGARCH and two-stepfiltration models, M1, M2, M3a, and M3b, considered in the previoussection, are used to generate time-consistent 1-step-ahead forecasts ofconditional return volatility.23 For the purpose of benchmark provision,we also consider the forecasts generated by a standard GARCH(1,1)model (henceforth denoted M0). These forecasts are then compared torealized conditional return volatility (given by the squared residual fromthe mean Equation (7a)). The mean squared forecast error (MSFE) asso-ciated with these forecasts is given in Table III.24 The results indicatethat the M0 and M3a forecasts are the poorest of the forecasts consid-ered, while the best forecasts appear to be those generated by M3b, fol-lowed by the M1 forecasts.

To formally test the comparative accuracy of the model-based fore-casts, we make use of the asymptotic test introduced by Diebold andMariano (1995). This test allows use of an arbitrary loss function instead ofthe usual squared forecast error loss, and is robust to nonzero mean fore-cast errors, nonnormally distributed forecast errors, and serially correlatedforecast errors. In the current application, it is the robustness of theDiebold and Mariano test statistic to the nonnormality assumption that ismost attractive. Indeed, when the forecast errors are tested for normality,the null is rejected at the 1% significance level on every occasion.25

A summary of the results obtained when the Diebold-Mariano testis performed on all available 1-step-ahead forecasts is given in Table III.We report the differences in the MSFE for each model and the associat-ed (one-sided Bonferroni) significance of the Diebold-Mariano test,where each model-based set of forecasts is compared with each other.26

In addition, we report the number of times one particular model-basedset of forecasts is significantly better (at the one-sided 5% Bonferronisignificance level) than the other model-based forecasts. The resultsindicate that M3b produces forecasts that are significantly more accu-rate than those produced by all the other models.

In using a symmetric loss function, we are explicitly assuming thatpositive and negative forecast errors (of the same absolute magnitude)

23The forecasts generated by M3a are multiplied by the fitted periodic components in order that theycan be compared with the forecasts generated by the other PGARCH models.24Similar results are obtained when the mean absolute forecast error (MAFE) is considered. Theseresults are available upon request.25These results are available upon request.26To account for the fact that four tests are conducted for each model, the significance of these testsis assessed using appropriate one-sided Bonferroni significance levels.


TABLE III

Testing the Statistical Significance of Return Volatility Forecasts

Model

M0(�) M1(�) M2(�) M3a(�) M3b(�)

Panel A: GARCH models

MSFE 0.0548 0.0457 0.0461 0.0592 0.0448

MSFE differences M0 — 0.0090*** 0.0087*** �0.0044 0.0100***M1 �0.0090 — �0.0004 �0.0135 0.0010***M2 �0.0087 0.0004* — �0.0131 0.0014***M3a 0.0044*** 0.0135*** 0.0131*** — 0.0145***M3b �0.0100 �0.0010 �0.0014 �0.0145 —

DM Test Score r� 0 1 2 2 0 4r� �0.02 1 2 2 0 4r� �0.04 1 2 2 0 4r� �0.06 1 2 2 0 4r� �0.08 1 3 2 0 4r� �0.10 1 3 2 0 4

Panel B: Poisson count models

MSFE 2.3168 2.1862 1.5757 — 1.6008MSFE differences M0� — 0.1306 0.7411 — 0.7160

M1� �0.1306 — 0.6105 — 0.5853*M2� �0.7411 �0.6105 — — �0.0251M3b� �0.7160 �0.5853 0.0251 — —

DM Test Score r� 0 0 0 0 — 0r� �0.02 0 0 2 — 2r� �0.04 0 0 2 — 3r� �0.06 0 1 2 — 3r� �0.08 0 1 2 — 3r� �0.10 0 1 2 — 3

Note. This table gives the MSFE, and the differences between the mean squared forecast error MSFE valuesacross models M0, M1, M2, M3a, and M3b (Panel A) and M0�, M1�, M2�, and M3b� (Panel B). This table alsogives the number of successful forecast performance comparisons under the (asymmetric) Linex loss function(with asymmetry parameter r) using a one-sided 5% Bonferroni significance level.

*10% significance, **5% significance, ***1% significance; the significance of the MSFE differences is assessedusing the Diebold-Mariano (DM) test based on one-sided Bonferroni significance levels.

receive equal weight in the loss function. In the current context, thismay not be appropriate. Both Brailsford and Faff (1996) and Bystrom(2000) argue that underprediction of return volatility should be moreheavily penalized than overprediction. This need to protect againstunderprediction can be demonstrated in the context of the VaR method-ology. Specifically, it is obvious in this context that underprediction ofreturn volatility is to be avoided if the solvency of the portfolio owner isto be maintained. Therefore, some form of asymmetric loss functionis required.

826 Taylor

27See Engle, Hong, Kane, & Noh (1993), and West, Edison, & Cho (1993), for alternative methodsof avoiding this problem.

A commonly used asymmetric loss function is the Linex loss func-tion introduced by Varian (1974) and used by Zellner (1986). It is givenby the following expression:

(8)

where et is the 1-step-ahead forecast error. When posi-tive (negative) forecast errors are weighted more heavily than negative(positive) forecast errors. Another important feature of this loss functionis that the function becomes asymptotically equivalent to the symmetricquadratic loss function (i.e., the MSFE) when Given the abovearguments, we assume that

The Diebold-Mariano test is applied using the Linex loss functionfor all combinations of possible comparisons. The results are given inTable III. Not surprisingly, when is close to zero, the number of suc-cesses closely resembles the results obtained using the symmetric quad-ratic loss function. Moreover, as becomes large (in absolute terms)and, hence, the degree of asymmetry increases, the number of successesfor most models remains unchanged. Most notably, M3b continues to bethe most successful model.

A drawback of the above analysis is that the forecasting quality of thereturn volatility models is assessed using the realized value of conditionalreturn volatility. However, the true value of this quantity cannot beobserved. Consequently, to make the evaluation criteria operational, thevalue is proxied by the squared value of the residual from the meanEquation (7a). However, using this proxy creates problems of assessment.Specifically, Andersen and Bollerslev (1998b) and Christodoulakis andSatchell (1998) show that use of this proxy necessarily leads to apparentlypoor forecasting performance. This problem is circumvented in thecurrent paper by using an alternative measure of conditional returnvolatility.27 In particular, we consider the forecasts of price durationintensity (a measure of return volatility) generated by M0�, M1�, M2�, andM3b�, and compare these with realized price duration intensity. Using thesame evaluation criteria as before, the results in Table III indicate thatM3b� becomes the dominant model when differs from zero.

VaR Analysis

An alternative method of assessing the quality of the above models is toconsider the accuracy of VaR measures generated by these models. In

r

r

r

r � 5�0.02,�0.04,�0.06,�0.08,�0.106.rS 0.

r � 0 (r � 0),

g(et) � exp[ret] � ret � 1


28Similar results were obtained for other values of � within this range. These results are availableupon request.

particular, we consider the failure rates for long and short positionsimplied by the models. These failure rates are simply the proportions oftime the observed return is outside of the VaR measures. In turn, theVaR measures are given by the dynamic interval forecasts generated bythe above models (except M0 and M0�, which generate static intervalforecasts).

In addition to the models previously described, we consider theRiskMetrics VaR model (henceforth denoted RM), which adopts anexponentially weighted moving average approach to return volatilitymodeling. In particular, it assumes that return volatility evolves asfollows:

(9)

where and � is given a value between 0.94 and 0.99. In thecurrent application, we assume that � � 0.94.28 Moreover, to accountfor the periodicity in the data, this model is applied to the same filtereddata as used in M3a. The resultant conditional volatility estimatesobtained using RM (and M3a) are then multiplied by the squared fittedperiodic components to obtain comparable measures of conditionalvolatility. In addition to this model, we also consider VaR models basedon the Poisson count models. The estimated return volatility from thesemodels is converted to the standard definition of conditional returnvolatility for use within the VaR framework, by using the result givenin (5).

Following Granger, White, and Kamstra (1989), we assume that thelower and upper limits of the interval forecasts are given by

(10)

for an interval with a nominal coverage, where is the empirical -percentile of the standardized residuals It follows thata long position VaR measure is deemed to “failure”when . (The corresponding short position failure condition is given by

By construction, the VaR measures in (10) will fail the correct num-ber of times. However, these failure rates may differ at certain pointswithin the sample. In particular, if return volatility periodicity is notadequately modeled, these empirical failure rates will differ from their

Rt � m� Qch12t).

Rt � m� Q1�c h12t

(1 � c) � 100%Pt�h

12t.°

Q°c � 100%

5Q1 �c2

h12t, Q1 �c

2 h 12t6

�t � Rt � m,

ht � (1 � b)�2t�1 � bht�1

828 Taylor

preselected values at certain points within the periodic cycle. Given thestrength of intraday periodicity observed in the previous section, weconsider the failure rates during each hour of the trading day. The empir-ical failure rates for various long position 5% VaR models are given inFigure 2. One can see from this plot that empirical failure rates for VaRmodels based on M1 and M2 exceed the VaR failure rates during theopening of trading. However, the M3b VaR model mitigates this effect.Indeed, the empirical failure rate for this model never lies outside the 4to 6% range. It is also noticeable (and rather surprising given previousresults) that the VaR models based on two-step filtration, i.e., RM andM3a, perform reasonably well over the trading day.

To give an idea of the overall accuracy of the VaR models, the meansof the absolute differences between the empirical and VaR failure ratesfor each hour of the trading day are presented in Table IV. The mostaccurate model of those considered is the M3b VaR model. Moreover,this result is robust to the type of position held (i.e., long or short) andthe magnitude of the VaR failure rate. Of the other models considered,the M2, M3a, and RM VaR models are the next most successful. Ratherunsurprisingly, the M0 VaR model is generally the least accurate of themodels considered. This is to be expected given that this model does notconsider periodicities within the data. It is also noticeable that the VaR

FIGURE 2Intraday VaR failure rates.

Note. This figure shows the empirical failure rates for various longposition 5% VaR models during each hour of the trading day.


TABLE IV

Intraday VaR Performance

Long position VaR Short position VaR

Model 5% 2.5% 1% 0.5% 0.25% 5% 2.5% 1% 0.5% 0.25%

Panel A: Accuracy

RiskMetrics RM 0.32 0.37 0.30 0.18 0.19 0.64 0.54 0.35 0.24 0.13

GARCH M0 2.09 1.35 0.74 0.45 0.27 1.80 1.12 0.70 0.44 0.28M1 1.57 1.03 0.53 0.30 0.19 1.32 0.88 0.51 0.33 0.20M2 1.12 0.57 0.30 0.20 0.11 0.95 0.57 0.35 0.22 0.12M3a 0.31 0.38 0.30 0.21 0.14 0.59 0.59 0.40 0.24 0.14M3b 0.66 0.26 0.15 0.11 0.04 0.44 0.21 0.16 0.11 0.07

Count M0� 2.36 1.45 0.74 0.43 0.23 2.02 1.26 0.67 0.42 0.28M1� 2.29 1.38 0.67 0.40 0.20 1.95 1.18 0.63 0.39 0.23M2� 1.51 1.03 0.54 0.34 0.18 1.31 0.86 0.45 0.30 0.21M3b� 1.54 0.94 0.46 0.28 0.15 1.22 0.75 0.40 0.25 0.17

Panel B: Test rejections

RiskMetrics RM 1 5 5 5 4 4 6 5 5 3

GARCH M0 7 8 7 6 4 6 6 6 7 6 M1 7 7 7 4 4 6 6 6 4 4 M2 7 4 4 3 3 6 6 5 3 2 M3a 1 5 5 4 3 4 6 6 5 3 M3b 5 2 0 1 0 3 2 2 2 1

Count M0� 8 7 7 6 4 7 6 6 5 4 M1� 8 7 6 6 4 7 6 6 5 4M2� 6 6 4 5 3 6 5 6 4 2M3b� 6 5 4 4 2 6 5 3 2 2

Note. This table gives the mean of the absolute value of the differences between the empirical failure rate andthe VaR failure rate over each hour of the trading day (Panel A), and the number of times the Kupiec test isrejected using a one-sided 5% Bonferroni significance level (Panel B), for a variety of models. A maximum ofeight rejections is possible (equal to the number of hours in the trading day). All values in Panel A are inpercentages.

29This test is equivalent to the unconditional coverage test of interval forecast evaluation introducedby Christoffersen (1998).30As eight tests are performed for each model (one for each hour in the trading day), an appropriate5% Bonferroni significance level is used.

models based on the Poisson count models, M0�, M1�, M2�, and M3b�,are almost always less accurate than their PGARCH counterparts.

Finally, differences between the empirical and VaR failure rates aresubjected to a statistical test. In particular, we test this difference usingthe test introduced by Kupiec (1995).29 This test is conducted for eachhour of trading day. The results in Table IV confirm the earlier resultsthat M3b is the superior model of return volatility.30 Most notably, theM3b VaR model cannot be rejected for low values of the long positionVaR failure rate at any point during the day.

830 Taylor

CONCLUDING REMARKS

Periodicity in conditional return volatility is particularly acute when con-sidering high-frequency commodity futures returns. This is because, inaddition to the discontinuous intraday periodicity observed in the data, adiscontinuous interday periodicity exists. The former periodicity is theresult of an intraday trading close, while the latter periodicity is driven bya relationship between trading behavior and the maturity of the futurescontract held. Given the extreme nature of these periodicities, it isimportant to control for these periodicities when using such data. Ademonstration of this importance is provided by considering the relativeperformance of various periodic conditional volatility models, and6 years of 5-minute frequency cocoa futures data.

The results from this paper indicate that failure to adequately incor-porate the above periodicities leads to inferior forecasts of future returnvolatility. This inferiority is both statistically and economically significant.Regarding the former issue, return volatility forecasts generated by thespline-version of the PGARCH model considered in this paper are shownto be less accurate than those generated by competing periodic conditionalvolatility models. On the latter issue, when used in a VaR framework, mostconditional volatility models imply failure rates that systematically divergefrom the VaR failure rates. By contrast, use of the spline-version of thePGARCH model produces accurate and consistent VaR measures.

APPENDIX: MODEL SPECIFICATIONSAND DEFINITIONS

This appendix contains the exact specifications of conditional volatilityused in the PGARCH models.

(A.1)

where Sd denotes the number of 5-minute periods

within the trading day, denotes the absolutereturn, denotes the effective bid-ask spread, and denotes tradingvolume.

(A.2)

where is a (6Id � 1) vector of coefficients, is a (2Iy � 1) vector of coefficients,

f cId� id,t� , p ,[P1,t1�2Id

, P2,t1�2Id, P3,t1�2Id

], F�cd,t� � [f c

1,t� , p , f cid,t� , p , f c

Id,t� , f c

Id �1,t� , p ,P�t �Fc

t� � [1, diag(PtF~ c

d,t� )�, F~c�y,t],

V�yV� � [v, V��d, V��y ], V�d

M2��ht � V�Fct � aP2

t�1 � bht�1

VtBt

Z�t � [ ƒRt ƒ , Bt, Vt], ƒRt ƒg2,Sd

LSd, g3,1 L � g3,Sd LSd],

Pt � Rt � m, Pt ƒ�t�1 �D(0, ht), g(L)� � [g1,1 L �g1,Sd LSd, g2,1 L �

M1��ht � v � aP2t�1 � bht�1 � g(L)�Zt


equalsunity if the tth observation occurs in period m, and zero otherwise,

is a (2Id � 1) vector of ones, and are the calen-dar times of the last observations of the intraday and interday periodiccycles, respectively, and and are functions that give the calen-dar times of the tth observations within the intraday and interdayperiodic cycles, respectively.

(A.3)

where is a vector of coefficients, isa vector of coefficients,

is a vector ofones, is the calendar time of the jdth ( jyth) knot, and equal zero, equals unity if and zero otherwise, and equals unity if and zero otherwise.

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Modeling discontinuous periodic conditional volatility: Evidence from the commodity futures market

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