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International Journal of Forecasting 29 (2013) 244–257 Contents lists available at SciVerse ScienceDirect International Journal of Forecasting journal homepage: www.elsevier.com/locate/ijforecast Robust forecasting of dynamic conditional correlation GARCH models Kris Boudt a , Jón Daníelsson b , Sébastien Laurent c,d,a KU Leuven/Lessius and V.U. University Amsterdam, Belgium b London School of Economics, United Kingdom c Maastricht University, Department of Quantitative Economics, The Netherlands d CORE, Université catholique de Louvain, Belgium article info Keywords: Jumps Conditional covariance Forecasting abstract Large one-off events cause large changes in prices, but may not affect the volatility and correlation dynamics as much as smaller events. In such cases, standard volatility models may deliver biased covariance forecasts. We propose a multivariate volatility forecasting model that is accurate in the presence of large one-off events. The model is an extension of the dynamic conditional correlation (DCC) model. In our empirical application to forecasting the covariance matrix of the daily EUR/USD and Yen/USD return series, we find that our method produces more precise out-of-sample covariance forecasts than the DCC model. Furthermore, when used in portfolio allocation, it leads to portfolios with similar return characteristics but lower turnovers, and hence higher profits. © 2012 International Institute of Forecasters. Published by Elsevier B.V. All rights reserved. 1. Introduction Prices of financial assets sometimes exhibit large jumps, caused by one-off events such as news announcements, and such extreme returns are often found to affect the volatility less than a standard GARCH model would predict. 1 Therefore, in such cases, not only does the use of standard GARCH models lead to an overestimation of the volatility for the days following the event, but also, since the unconditional volatility forecast is upward biased, all volatility forecasts tend to be larger than they would be otherwise. A similar argument applies to correlation Correspondence to: Department of Quantitative Economics, Maas- tricht University, School of Business and Economics, P.O. Box 616, 6200 MD Maastricht, The Netherlands. Tel.: +31 43 388 38 43; fax: +31 43 388 48 74. E-mail addresses: [email protected] (K. Boudt), [email protected] (J. Daníelsson), [email protected] (S. Laurent). 1 See e.g. Andersen, Bollerslev, and Diebold (2007), Bauwens and Storti (2009) and Carnero, Peña, and Ruiz (2012). estimates. If only one of the stocks is subject to a large jump in prices, that biases the correlation estimates towards zero. In the case of co-jumps of the same (opposite) signs, the correlations are biased toward (minus) one. Our objective in this paper is to develop a multivariate volatility forecasting model that is accurate in the presence of one-off events which cause large changes in prices whilst not affecting the volatility dynamics. There are two main directions one could take in the development of such a model: either modeling a jump process explicitly within a standard volatility model, or employing a robust estimation procedure for a standard volatility model. The former approach is necessary in applications where the properties of the jumps are of interest. However, jumps in daily returns are rare events, and the estimates of the jump process have large confidence bands. Therefore, if the ultimate objective is to forecast the volatility, a robust approach may be a better choice, and that is what we provide in this paper. Our starting point is the univariate procedure proposed by Muler and Yohai (2008) for the estimation of GARCH models, whereby the impact of returns on volatility 0169-2070/$ – see front matter © 2012 International Institute of Forecasters. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.ijforecast.2012.06.003

Robust forecasting of dynamic conditional correlation GARCH models

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Page 1: Robust forecasting of dynamic conditional correlation GARCH models

International Journal of Forecasting 29 (2013) 244–257

Contents lists available at SciVerse ScienceDirect

International Journal of Forecasting

journal homepage: www.elsevier.com/locate/ijforecast

Robust forecasting of dynamic conditional correlation GARCHmodelsKris Boudt a, Jón Daníelsson b, Sébastien Laurent c,d,∗a KU Leuven/Lessius and V.U. University Amsterdam, Belgiumb London School of Economics, United Kingdomc Maastricht University, Department of Quantitative Economics, The Netherlandsd CORE, Université catholique de Louvain, Belgium

a r t i c l e i n f o

Keywords:JumpsConditional covarianceForecasting

a b s t r a c t

Large one-off events cause large changes in prices, but may not affect the volatility andcorrelation dynamics as much as smaller events. In such cases, standard volatility modelsmay deliver biased covariance forecasts. We propose a multivariate volatility forecastingmodel that is accurate in the presence of large one-off events. The model is an extensionof the dynamic conditional correlation (DCC) model. In our empirical application toforecasting the covariance matrix of the daily EUR/USD and Yen/USD return series, we findthat our method produces more precise out-of-sample covariance forecasts than the DCCmodel. Furthermore, when used in portfolio allocation, it leads to portfolios with similarreturn characteristics but lower turnovers, and hence higher profits.© 2012 International Institute of Forecasters. Published by Elsevier B.V. All rights reserved.

s. P

1. Introduction

Prices of financial assets sometimes exhibit large jumps,caused by one-off events such as news announcements,and such extreme returns are often found to affectthe volatility less than a standard GARCH model wouldpredict.1 Therefore, in such cases, not only does the use ofstandard GARCH models lead to an overestimation of thevolatility for the days following the event, but also, sincethe unconditional volatility forecast is upward biased, allvolatility forecasts tend to be larger than they wouldbe otherwise. A similar argument applies to correlation

∗ Correspondence to: Department of Quantitative Economics, Maas-tricht University, School of Business and Economics, P.O. Box 616, 6200MD Maastricht, The Netherlands. Tel.: +31 43 388 38 43; fax: +31 43 38848 74.

E-mail addresses: [email protected] (K. Boudt),[email protected] (J. Daníelsson), [email protected](S. Laurent).1 See e.g. Andersen, Bollerslev, and Diebold (2007), Bauwens and Storti

(2009) and Carnero, Peña, and Ruiz (2012).

0169-2070/$ – see front matter© 2012 International Institute of Forecasterdoi:10.1016/j.ijforecast.2012.06.003

estimates. If only one of the stocks is subject to a large jumpin prices, that biases the correlation estimates towardszero. In the case of co-jumps of the same (opposite) signs,the correlations are biased toward (minus) one.

Our objective in this paper is to develop a multivariatevolatility forecastingmodel that is accurate in the presenceof one-off events which cause large changes in priceswhilst not affecting the volatility dynamics. There are twomain directions one could take in the development ofsuch a model: either modeling a jump process explicitlywithin a standard volatility model, or employing a robustestimation procedure for a standard volatility model. Theformer approach is necessary in applications where theproperties of the jumps are of interest. However, jumpsin daily returns are rare events, and the estimates of thejump process have large confidence bands. Therefore, ifthe ultimate objective is to forecast the volatility, a robustapproach may be a better choice, and that is what weprovide in this paper.

Our starting point is the univariate procedure proposedby Muler and Yohai (2008) for the estimation of GARCHmodels, whereby the impact of returns on volatility

ublished by Elsevier B.V. All rights reserved.

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K. Boudt et al. / International Journal of Forecasting 29 (2013) 244–257 245

forecasts is bounded. They term this procedure a ‘‘boundedinnovation propagation’’ (BIP) GARCH. We adjust theirprocedure to make it suitable for multivariate volatilityforecasting when the underlying assets are subject to one-off shocks. This is the main contribution of our paper.

The most widely used models for forecasting condi-tional covariances and correlations are the BEKK modelof Engle and Kroner (1995) and the dynamic conditionalcorrelation model (DCC) of Engle (2002). Boudt and Croux(2010) proposed a robust estimation method for the BEKKmodel. Their robust model bounds the impact of jumpson the conditional variance of each asset based on theextremeness of the complete lagged return vector. Thismakes their model unsuitable for volatility forecasting,since it underestimates the volatility of the assets that havenot jumped on the days following a jump in one of theassets. To avoid this and preserve the positive definite-ness of the covariance forecasts, we choose to disentanglethe robust forecasting of univariate volatilities and correla-tions, and take theAielli (2009) version of the dynamic con-ditional correlation model (cDCC) of Engle (2002) as ourbaseline model.

We provide three extensions of the cDCC model. First,we use a BIP-GARCH model for the univariate volatilitiesinstead of a standard GARCH. Second, we bound theimpact of large innovations on the correlation matrix,through a BIP procedure in the update equation of theconditional correlation, along with a robust procedurefor estimating the unconditional correlation. Finally, wepropose a robust M-estimator for the parameters of thecorrelation dynamics. These three extensions each ensurethat extreme one-off events have little influence on thecovariance predictions made by the proposed BIP-cDCCmodel.

We compare our model with the baseline models(standard GARCH and cDCC) in various ways. First, we runa Monte Carlo experiment to study the effect of jumps onthe cDCCparameter estimates. The results indicate that theestimates from the baseline model can have a large biaswhen the data has additive jumps, while the BIP procedureprovides accurate estimates of the parameters.

Second, the conditional covariance forecasts arecompared with ex post covariance estimates based onhigh-frequency data. Using the model confidence setmethodology, proposed by Hansen, Lunde, and Nason(2011),we find that the BIPmodels always belong to the setof superior forecastingmodels.Moreover, formost forecasthorizons, their covariance forecasts are significantly betterthan those of all other models considered.

Our key empirical conclusion follows from applying theBIP model to the problem of optimal portfolio allocation,where we study an investor who adopts a volatilitytiming strategy where the changes in ex ante optimalportfolio weights are determined solely by the forecastsof the conditional covariance matrix. We find that theportfolio returns have similar unconditional first andsecond moments when using the BIP method as whenthe baseline model is employed. However, by using theBIP procedure, we increase the profits, because the BIPconditional covariance matrices are more stable, resultingin a lower portfolio churn, and thus lower transaction

costs. Therefore, even if both procedures have the samemean return and standard deviation (net of transactioncosts), the BIP procedure is more profitable overall.

The structure of the rest of the paper is as follows. Sec-tion 2 describes the univariate BIP-GARCH volatility fore-cast model. Section 3 then proposes the robust conditionalcorrelation forecasting method. These forecasts rely on ro-bust M-estimates of the dependence parameters, a BIP-cDCC conditional correlation model, and a robust estimateof the unconditional correlation. Section 4 reports the re-sults of the Monte Carlo study on the effect of jumps onparameter estimates of the cDCC and BIP-cDCC models.Section 5 evaluates the forecasting precision of themodels,using high-frequency data covariance estimates as proxiesfor the true covariance. Section 6 analyzes the economicconsequences of using the BIP-cDCCmodel for a minimumvariance investor. Finally, Section 7 summarizes our mainfindings and provides some directions for future research.

2. Univariate volatility forecasting in the presence ofextremes

Many volatility models, such as GARCH models, arebased on the assumption that each return observation hasthe same relative impact on future volatility, regardlessof the magnitude of the return. This assumption is atodds with an increasing body of evidence indicating thatthe largest return observations have a relatively smallereffect on the future volatility than smaller shocks (see forinstance Andersen et al., 2007).

One reason for this is the occurrence of extremelylarge shocks, caused by one-off events that cannot beexpected to influence the future volatilitymuch. In thewebappendix, this is illustrated using the example of Apple’sstock price, which fell 52% on September 29, 2000, after awarning that its fourth-quarter profit would fall well shortof the Wall Street forecasts.

Several proposals have been made for explicitly ad-dressing the way in which extreme returns affect volatil-ity, e.g. Andersen et al. (2007) and Corsi, Pirino, and Renó(2008), who use a simple restricted autoregressive modelto forecast the realized volatility. They show that decom-posing the volatility into a jump component and a con-tinuous component results in the jump component beingconsiderably less persistent than the continuous compo-nent.

Carnero, Peña, and Ruiz (2006), Franses and Ghijsels(1999), and Muler and Yohai (2008), among others,propose new methods which are designed to estimate theparameters of a GARCH(1, 1) model in the presence ofadditive, but one-off, jumps. After subtracting the mean µ,the observed return series s∗t has a standard normal GARCHcomponent yt and a jump component at , i.e.

st = s∗t − µ = yt + at (2.1)

yt =

htzt where zt

i.i.d.∼ N(0, 1) (2.2)

ht = ω + α1y2t−1 + β1ht−1. (2.3)

The assumption of normal innovations could be replacedwith another distributional hypothesis if appropriate.

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246 K. Boudt et al. / International Journal of Forecasting 29 (2013) 244–257

In the economic literature, the occurrence of additivejumps is generally modeled by means of a Poissondistribution, as did Chan and Maheu (2002) and Vlaar andPalm (1993). Such a parametric approach requires one tofurther specify and estimate a model which governs boththe time-varying jump intensity and the jump size. Thisleads to an increased complexity of the estimationmethod(especially in a multivariate framework), and a potentialspecification bias. Moreover, because of the low frequencyof extremes in the sample, these estimates often havewideconfidence bands.

If the ultimate objective is to forecast the volatility, in-ference on the jump process is not needed in order toproduce accurate volatility forecasts. In the ‘‘robust’’ ap-proach, jumps are detected automatically in the estimationstep, and their effect on parameter estimation and volatil-ity forecasts is bounded.

In the absence of jumps (i.e., when at = 0∀t), themodelin Eqs. (2.1)–(2.3) reduces to a standard GARCH(1, 1) withnormal innovations. This model is usually estimated by(Q)ML. When at = 0 for some t ∈ {1, . . . , T }, yt andat are not directly distinguishable from s∗t . In this case,the Gaussian QML is not appropriate because at−1 hasno impact on ht , while assuming a GARCH(1, 1) for s∗twould imply ht = ω + α1(yt−1 + at−1)

2+ β1ht−1, i.e., a

large and slowly decaying effect of at−1 on future volatilitypredictions.

Furthermore, if E(at) = 0, µ is no longer theunconditional mean of s∗t , and thus both its QML estimateand the empirical mean are expected to be strongly biased.

2.1. BIP-GARCH and the robust M-estimator

Muler and Yohai (2008) (MY) show that one can limitthe effect of at on the estimation of the parameters of theGARCH model by using a modified GARCH specificationwhich downweights the effect of past jumps (at−1) onthe time t conditional variance. Their approach estimatesthe GARCH parameters and detects jumps jointly, byidentifying returns as jumps when a return is an extremeoutlier under the estimated GARCH model. Becauseof the time-varying volatility, extremes need to beidentified by the squared devolatilized return s2t−1/ht−1rather than the squared return itself. Otherwise, jumpswould be overdetected on days with high volatility andunderdetected on days with low volatility. This leads to anauxiliary GARCH(1, 1) model which places weights on theextremes:

ht = ω + α1w

s2t−1

ht−1

s2t−1 + β1ht−1, (2.4)

where w(·) is a weight function. Since s2t−1/ht−1 is chi-square distributed with one degree of freedom if thereis no jump at time t − 1, it is natural to detect a jumpoccurrence in s2t−1 if s2t−1/ht−1 exceeds kδ,1, the δ quantileof the chi-square distribution with one degree of freedom.The weight function used by MY is given by

wMYkδ,1(u) = min

1,

kδ,1

u

. (2.5)

Table 1Correction factor cδ,N for the weighted variance estimator, the BIP-GARCHmodel and theM-estimator of (c)DCCmodelswithN-dimensionalGaussian innovations.

N/δ cδ,N σN,4

1 0.99 0.975 0.95 0.90

1 1 1.0185 1.0465 1.0953 1.2030 0.82602 1 1.0101 1.0257 1.0526 1.1111 0.82585 1 1.0050 1.0122 1.0255 1.0542 0.8467

10 1 1.0028 1.0073 1.0154 1.0330 0.883550 1 1.0009 1.0025 1.0053 1.0118 0.9644

Model (2.4) with the weight function in Eq. (2.5) is calledthe Bounded Innovation Propagation (BIP)-GARCH, since theeffect of past shocks on future volatility is bounded. MYshow that the combination of a BIP-GARCHwith an outlierrobust M-estimator reduces the root mean squared error(RMSE) of the parameter estimates considerably in thepresence of additive jumps.2

For the estimation of the BIP-GARCH model, MYrecommend using an M-estimator that minimizes theaverage value of an objective functionρ(·), evaluated at thelog-transform of squared devolatilized returns, i.e.

θM= argmin

θ∈Θ

1T

Tt=1

ρ

log

s2tht

. (2.6)

For robustness, this ρ-function needs to downweight theextreme observations, and hence the jumps. The choiceof ρ(·) is a trade-off between robustness and efficiency.Based on a comparison of several candidate ρ-functions intheweb appendix (Boudt, Daníelsson, & Laurent, 2012), werecommend the one associatedwith the Student-t4 densityfunction:

ρ2(z) = −z + σ1,4ρt1,4(exp(z)),

where

ρtN,ν(u) = (N + ν) log

1 +

uν − 2

(2.7)

and

σN,ν =N

E[ρ ′tN,ν

(u)u], (2.8)

where u is a chi-squared random variable with N degreesof freedom. The values of σN,4 for N = 1, 2, 5, 10 and50 are reported in Table 1; it is needed to ensure Fisherconsistency, as was shown by Boudt and Croux (2010).Carnero et al. (2012) document the good properties of theMY procedure in forecasting the volatility in the presenceof additive outliers. We propose two modifications of theMY procedure that improve the accuracy of the volatilityforecasts further.

2.2. A modified MY procedure

In the BIP-GARCH model of Eq. (2.4), returns thatare not affected by jumps are also downweighted. Our

2 Harvey and Chakravarty (2008) propose an alternative weightfunction based on the score of the t distribution with ν degrees offreedom, wHC

ν (u) =ν+1

ν−2+u .

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K. Boudt et al. / International Journal of Forecasting 29 (2013) 244–257 247

simulations (which are available upon request) show thatsmaller biases in the GARCH parameters based on the BIPspecification are obtained when theMYweighting schemeis modified to ensure that the conditional expectationof the weighted squared unexpected shocks is still theconditional variance in the absence of jumps, i.e.

wkδ,1(u) = cδ,1wMYkδ,1(u), (2.9)

where

cδ,N =E[u]

E[wMYkδ,N

(u)u]

=1

Fχ2N+2

(χ2N(δ)) + (1 − δ)χ2

N(δ), (2.10)

with u being a chi-square random variable with N degreesof freedom.3 Table 1 reports the correction factors cδ,1 forvarious values of δ.

A second modification of the MY procedure is that weintegrate reweighted estimates of the mean and varianceinto the forecasting procedure. The above definitions ofthe BIP-GARCH model and the M-estimators are for st =

s∗t − µ. MY assume that µ = 0, and thus only consider theconditional variance. Unfortunately, this assumption maynot hold in practice, and therefore a jump-robust estimatorof µ is needed. Furthermore, MY estimate the interceptω jointly with the parameters α and β . As was noted byEngle and Mezrich (1996), this is especially difficult if αand β add up to a number which is very close to one, asthe intercept will be very small but must remain positive.

Engle and Mezrich (1996) propose variance targetingas an estimation procedure, where ω is reparameterizedas h(1 − α1 − β1) (with h being a consistent estimatorof the unconditional variance h) before the remainingparameters are estimated. Francq, Horvath, and Zakoian(2011) show that when the model is misspecified, thevariance targeting estimator can be superior to the QMLEfor long-term prediction or Value-at-Risk calculations.

In the absence of outliers, natural choices of µ and hare the sample mean and sample variance of the returns.However, these estimators are known to be very sensitiveto outliers (e.g., outliers cause a large upward bias in thesample variance). We therefore propose using the robustreweighted mean and variance estimators suggested byBoudt, Croux, and Laurent (2011a) and described in thenext subsection. In the web appendix (Boudt et al., 2012),we verify the accuracy of the BIP M-estimator with aStudent-t4 loss function, and target the robust reweighedmean and variance, relative to the QML estimator and theestimators considered in MY.

2.3. Reweighted mean and variance estimators

In a high-frequency data setting, Boudt et al. (2011a)propose to estimate the unconditional mean µ and

3 Themultivariate notation is used here, since the correction factor willalso be useful for the robust estimation of the correlation parameters inSection 3.

variance h through a mean and variance estimate in whichlocal outliers receive a weight of zero. The locality of theoutlier detection method is needed in order to avoid anoverdetection of outliers at times of high volatility andan underdetection when volatility is low (Boudt, Croux,& Laurent, 2011b). For each observation, this methodfirst estimates the median absolute deviation (MAD)t ofthe returns in a window around that observation.4 Thereweighted sample mean and variance are then

µ =

Tt=1

s∗t It

Tt=1

It

and

h = 1.318 ·

Tt=1

(s∗t − µ)2Jt

Tt=1

Jt

,

(2.11)

with

It = I

(s∗t − mediant(s∗t ))2

MAD2t

≤ χ21 (95%)

and

Jt = I

(s∗t − µ)2

MAD2t

≤ χ21 (95%)

,

(2.12)

where I is the indicator function andχ21 (δ) is the δ quantile

of the χ2 distribution with 1 degree of freedom. Thecorrection factor 1.318 is a constant which adjusts for thebias due to the thresholding.

Practically, the local window around every observations∗t is the one that spans the interval [t−K/2, t+K/2]. At theborders, when t < K/2, the interval is given by [1, K + 1],orwhen t > T−K/2, the interval is given by [T−K , T ]. Thechoice of K must be such that the local window containsas many observations as possible while still satisfying thecondition that, approximately, the returns in the localwindow that are not affected by outliers come from thesame normal distribution. Thus, the choice of K shouldideally depend on the persistence of the underlying GARCHmodel. Through simulation, we tried several values of Kfor common GARCHmodels. We found that, since K affectsonly the weights in Eq. (2.12), it is not a critical tuningparameter. Throughout the paper, we set K = 30. Futureresearch could focus on creating a data-driven method foroptimally selecting the length of the local window, as inMercurio and Spokoiny (2004), for example.

2.4. Forecasting with univariate BIP-GARCH models

In the GARCH(1, 1) model in Eq. (2.3), the optimal r-step-ahead forecast of the conditional variance, ht+r|t ≡

Et(ht+r), is given by:

ht+r|t = ω + α1y2t+r−1|t + β1ht+r−1|t , (2.13)

4 The MAD of a sequence of observations x1, . . . , xn is defined as1.486 · mediani(|xi − medianj(xj)|), where 1.486 is a correction factorto guarantee that the MAD is a consistent scale estimator for the normaldistribution.

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248 K. Boudt et al. / International Journal of Forecasting 29 (2013) 244–257

where y2t+r−1|t ≡ Et [y2t+r−1], which is ht+r−1|t for r =

2, 3, . . . and y2t for r = 1. Similarly, r-step-ahead forecastsof the conditional variance of the BIP-GARCH(1, 1) areobtained as follows:

ht+r|t = ω + α1wy2t+r−1|t + β1ht+r−1|t , (2.14)

where wy2t+r−1|t ≡ Et [wkδ,1(y2t+r−1/ht+r−1)y2t+r−1], which

is ht+r−1|t for r = 2, 3, . . . and wkδ,1

y2t /ht

y2t for r = 1,

becausewkδ,1 in Eq. (2.9) is chosen such that E[wkδ,1(u)u] =

1 for u, a χ21 random variable (which does not hold for the

original specification of MY).Note that Eqs. (2.13) and (2.14) are based on expecta-

tions of future squared returns under the assumption ofa model without price jumps. In the presence of jumps,yt is not observed, and is naturally replaced by st in theabove equations. Thus, extremes affect the forecasts notonly through a potential bias in the parameter estimation,but also because lagged returns are used to forecast futurevariances. The effect of these outliers on future variancesis unbounded under the GARCH model, but limited underthe BIP-GARCH model.

3. Extremes and multivariate volatility forecasting

The effect of extremes on univariate volatility forecast-ing can equally be expected to be present in the forecastingof correlations. In contrast with the large body of literatureon the robust estimation of univariate GARCHmodels, dis-cussed above, little work exists on the estimation of mul-tivariate GARCH models in the presence of one-off jumps.Indeed, we are aware of only one paper, that of Boudt andCroux (2010), which proposes a method for the robust es-timation of the BEKK covariance matrix (Engle & Kroner,1995).

3.1. Baseline model

Our baseline model is a multivariate version ofEqs. (2.1)–(2.3), where the vector of demeaned observedreturns St = (s1,t , . . . , sN,t)

′ is composed of two non-observable components, a GARCH(1, 1)-cDCC process Yt =

(y1,t , . . . , yN,t)′ and an N-dimensional additive jump pro-

cess At :St = Yt + At (3.1)

Yt = H1/2t Zt , where Zt

i.i.d.∼ N(0, IN). (3.2)

Let Rt be the conditional correlation matrix, with Rij,t asits (i, j)th element, and define Dt as the diagonal matrixcontaining the conditional variances hii,t , i.e.,

Dt = diag (h1/211,t · · · h

1/2NN,t). (3.3)

The cDCC conditional covariance matrix Ht is given by:

Ht = DtRtDt =

Rij,t

hii,thjj,t

. (3.4)

The conditional correlation Rt is based on the matrix pro-cessQt , whose time-variation is driven by the devolatilizedreturns Yt = (y1,t , . . . , yN,t)

′= D−1

t Yt , i.e.

Qt = (1 − α − β)Q + αPt−1Yt−1Y ′

t−1Pt−1

+ βQt−1, (3.5)

with Pt = diag (q1/211,t · · · q1/2NN,t), Q being the unconditional

correlation matrix of Pt Yt , and α and β being nonnegativescalar parameters satisfyingα+β < 1.5 Thismatrix is thenlinked to the conditional correlation matrix as follows:Rt = diag (q−1/2

11,t · · · q−1/2NN,t ) Qt

× diag (q−1/211,t · · · q−1/2

NN,t ). (3.6)

Under the cDCC model, the estimation of the matrix Qand the parameters α and β are intertwined, since Q isestimated sequentially as the correlation matrix of Pt Yt ,where Pt also depends on α and β . However, since Ptonly involves the diagonal elements of Qt , the diagonalelements of which do not depend on Q (because Q ii = 1for i = 1, . . . ,N), Aielli (2009) shows that, for given valuesof α and β ,qii,t = (1 − α − β) + αqii,t−1y2i,t−1 + βqii,t−1,

i = 1, . . . ,N. (3.7)The remainder of the estimation procedure of Aielli (2009)is an iteration until convergence of (a) estimation of Qas the sample correlation of Pt Yt , and (b) multivariateGaussian QML estimation of α and β using the cDCCspecification.

3.2. A cDCC model with weights on extremes

In the presence of additive jumps, i.e., At in Eq. (3.2), thecDCC procedure in Section 3.1 is likely to deliver biasedparameter estimates, and hence covariance forecasts. Toremedy this,we propose threemodifications of the originalcDCC, i.e.,(a) the replacement of GARCH with the BIP-GARCHmodel

on St = (s∗1,t − µ1, . . . , s∗N,t − µN), as is describedin Section 2.2 and Appendix A, to compute thedevolatilized returns St = D−1

t St ;(b) estimation of Q with a robust correlation estimator;

and(c) replacement of the cDCC model with a BIP-cDCC

specification.

In the BIP-cDCC model, the effect of St−1S ′

t−1 on Qt isbounded. The BIP-cDCC model can only bound the mostextreme returns and must still yield positive semidefinitecovariance matrices. The latter condition is verified ifthe weights are positive and identical for all elements inSt−1S ′

t−1. This calls for a scalar measure of the extremenessof St−1, and, to apply the bounding, the distribution of thisstatistic should be known.6

5 The extension of the scalar-cDCC model (and its robust versiondiscussed in the next section) to the more general cases where α and β

are N × N matrices (Engle, 2002) or are block-diagonal (Billio, Caporin, &Gobbo, 2006) is straightforward, but is not investigated here for the sakeof simplicity.6 The use of thresholds for modeling the conditional correlations of

the financial return series is also considered by Audrino and Trojani(2011). However, their threshold is based on the average cross-productof the components of St−1 . In the presence of time-varying conditionalcorrelations, the distribution of this statistic is unknown. Audrino andTrojani (2011) use a data-driven method to estimate a fixed threshold.Because of the time-variation in the distribution of their statistic, thisapproach may not be optimal.

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K. Boudt et al. / International Journal of Forecasting 29 (2013) 244–257 249

Under the standard cDCC model without jumps, St−1is conditionally normally distributed with mean zero andcovariance matrix Rt−1. Cox (1968) and Healy (1968)proposed to use the squared Mahalanobis Distance (MD)to detect extremes in multivariate normal data, i.e.,

dt−1 = S ′

t−1R−1t−1St−1. (3.8)

The MD is conditionally distributed as a chi-squarerandom variable with N degrees of freedom. If any of thecomponents in St−1 is an extreme or St−1 is a correlationoutlier, the MD will be inflated. Hence, if dt−1 exceeds ahigh quantile of the χ2(N) distribution (denoted kδ,N ), itis likely that St−1 will be an extreme return and should bedownweighted. Thus, we can use a similar weight functionas in the univariate BIP-GARCH model. The BIP-cDCC thentakes the form

Qt = (1 − α − β)Q + αwkδ,N (dt−1)Pt−1St−1S ′

t−1Pt−1

+ βQt−1, (3.9)

where wkδ,N (u) = cδ,N min1, kδ,N

u

is as in Eq. (2.9),

but the threshold kδ,N and correction factor cδ,N are nowcomputed under the χ2(N) distribution (see Table 1). Thecorrection factor cδ,N is such that E[wkδ,N (Z ′Z)ZZ ′

] = IN

if Zi.i.d.∼ N(0, IN). The choice of δ is based on an efficiency

versus robustness trade-off. If all return observationsfollow the cDCC model, then the BIP-cDCC model inducesa bias and larger mean squared errors in the estimatedparameters. The higher the values of kδ,N , the closer theBIP-cDCC model is to the cDCC model, and hence, thesmaller the misspecification bias is. However, large valuesof kδ,N also imply that the effect of extremes on correlationsbecomes larger. In the remainder of the paper, we takeδ = 0.975.

Note that, in the univariate case, the MD is justthe squared devolatilized return. Thus, our multivariatetechnique for bounding is analogous with the approachproposed in Section 2.2.

3.3. Reweighted unconditional correlation estimator

In the cDCC model, the intercept of Qt is an explicitfunction of Q , the long run correlation matrix of St , whichis typically estimated by the sample correlation of St .7 Thepresence of additive jumps is likely to bias this estimate,just as it does for volatilities. Thus, it is desirable to replacethe simple correlation matrix with an estimator that ismore robust in the presence of such extremes.

We propose the use of a robustly reweighted correla-tion estimator that is proportional to the sample correla-tion of the observations for which no extremes have beendetected using a multivariate test statistic based on lo-cal correlation estimates. It is analogous to the reweightedmean and variance estimators of Section 2.3.

7 We assume that N/T is not close to one. If N/T is close to one,the correlation matrix is ill-conditioned, and other methods, such asshrinkage estimators, are preferred (see Hafner & Reznikova, 2012).

As for the reweighed variance estimator in Eq. (2.12),the reweighted correlation estimator proceeds in twosteps. First, the Spearman correlation matrix RCt is com-puted in a local window around every observation St . Moreprecisely, write St:1, . . . , St:K+1 as time-ordered observa-tions in the window t; then rank each component serieswithin the local window. Denote the vector series con-taining these ranks by Lt:1, . . . , Lt:K+1. The raw Spearmancorrelation matrix Ct for the window around St is thesample correlationmatrix of Lt:1, . . . , Lt:K+1. Moran (1948)showed that this correlation matrix needs to be correctedas follows to ensure consistency:

SCt = 2 sin16πCt

. (3.10)

The advantages of the Spearman correlation matrix withrespect to other robust correlation estimators includeits computational simplicity, high efficiency and outlierrobustness, as was shown by Croux and Dehon (2010).The locality is needed, such that S ′

tSC−1t St is approximately

chi-square distributed with N degrees of freedom (seeSection 3.2).

The second step is then to compute the robustreweighted correlation estimator:

RC =c0.95,NT

t=1Lt

Tt=1

St S ′

tLt , (3.11)

with weights Lt = I[S ′tRC

−1t St ≤ χ2

N(0.95)]. The scalarc0.95,N is as defined in Eq. (2.10). The reweighted (RW)

correlation estimator of Q , denoted ˆQ RW, is given by

ˆQ RW = diag (RC−1/211 · · · RC−1/2

NN ) RC

× diag (RC−1/211 · · · RC−1/2

NN ). (3.12)

This correlation estimate is positive semidefinite andinherits good robustness properties from the first stepcorrelation estimate which is used to compute the weights(Lopuhaä, 1999).

3.4. Estimation of the BIP-cDCC model

The estimation of the cDCC model is straightforward,even for moderately large values of N , and we aim to keepthe estimation of the proposed model straightforward aswell.

As in the univariate approach, we use M-estimatorsto estimate α and β . We define the M-estimators forconditional correlation models as the minimizers of thesum of the average value of a ρ-function, evaluated at thesquared Mahalanobis distances and the average value ofthe log of the determinant of the correlation matrices, i.e.,

θM= argminMT (θ, ρ)

≡1T

Tt=1

log det Rt + σρ

S ′

tR−1t St

, (3.13)

where σ is a correction factor. If ρ(z) = z and σ = 1, weobtain the Gaussian QML estimator as a special case, which

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corresponds to the original estimation method advocatedby Engle (2002). From the first order condition of the M-estimator, it is clear that the influence of extremes on theM-estimate depends strongly on the derivative of the ρ-function used:∂MT (θ, ρ)

∂θj

=1T

Tt=1

TrIN − σρ ′(S ′

tR−1t St)St S ′

tR−1t

∂Rt

∂θjR−1t

= 0, (3.14)

where Tr is the trace operator. Since each return St isweighted by the derivative of the ρ-function, evaluatedat the squared Mahalanobis distance of St in terms ofRt , the extreme bias on the estimate will be lower forM-estimators with decreasing ρ-functions. The GaussianQML is very sensitive to additive jumps, because in thiscase ρ ′(S ′

tR−1t St) = 1∀t , irrespective of the Mahalanobis

distance. Conversely, theM-estimatorwith ρtN,4 as definedin Eq. (2.7) is less sensitive to extremes, since its derivativeρ ′tN,4

(z) =N+42+z decreases to zero at the rate 1/z. The

correction factor σN,4 in Eq. (2.8) makes this estimatorconsistent in the absence of extremes. For typical valuesof N , we tabulate the correction factor σN,4 in Table 1.

The M-estimation procedure of the BIP-cDCC modelis similar to the QML estimation of the cDCC model,since Q , α and β are estimated iteratively. It starts withthe estimation of Q as the reweighted unconditionalcorrelation matrix of P1S1, . . . , PT ST , where the diagonalelements of Pt are obtained using the BIP version ofEq. (3.7), i.e.,qii,t = (1 − α − β) + αw(s2i,t−1)qii,t−1s2i,t−1

+ βqii,t−1. (3.15)The estimator then iterates until convergence throughthe estimation of Q as the reweighted correlation ofP1S1, . . . , PT ST , and Student t4 M-estimation of α and βusing the BIP-cDCC specification.

3.5. Forecasting with BIP-cDCC models

We now have all of the building blocks necessary toconstruct robust multivariate volatility forecasts. Unfortu-nately, multistep forecasts of the covariance matrix can-not be made analytically, because the model is not linearin squares and crossproducts of the data. As an approxima-tion, we follow Engle and Sheppard (2001) by constructingthe BIP-cDCC r-step-ahead volatility forecasts asHt+r|t = Dt+r|tRt+r|tDt+r|t , (3.16)where Dt+r|t is the diagonal matrix holding the r-step-ahead conditional variance forecasts, as describedin Section 2.4. The correlation forecast Rt+r|t is thestandardized version of Qt+r|t , where the 1-step-aheadforecast is obtained by projecting Eq. (3.9) one step into thefuture, and for r > 1

Qt+r|t = (1 − α − β) ˆQ + (α + β)Qt+r−1|t , (3.17)

since E[wkδ,N (Z ′Z)ZZ ′] = IN if Z

i.i.d.∼ N(0, IN).

4. Simulation study of estimation precision

Robust estimation of the model parameters is a keyfeature of the proposed robust covariance forecastingmethod. In the web appendix we confirm the good finitesample properties (bias and RMSE) of the estimator usingaMonte Carlo study, and show that, in contrast to the QMLestimator, it is not influenced much by jumps in the data.

As was noted by a referee, M-estimation of the BIP-cDCC model may become unfeasible if the dimension isvery large, due to the inversion of thematrix Rt in Eqs. (3.8)and (3.13), and/or because the unconditional correlationestimate used in the correlation targeting is ill-conditionedwhen the cross-section size is close to the sample size.Similar observations were made by Engle, Shephard, andSheppard (2008) and Hafner and Reznikova (2012) forthe QML-estimator of the cDCC model. In such cases, werecommend the use of a robust version of the MacGyvermethod, as proposed by Engle (2009), in order to obtainaccurate estimates of α and β . When the cross-section sizeis close to the sample size, shrinkage methods should beused to estimate Q .

Themain idea of theMacGyvermethod is that, since thedependence structure of the cDCC model is the same forany submatrix of the covariancematrix, accurate estimatesof α and β in the N-dimensional cDCC model can beobtained as the median of a large number of pairwiseestimates of the cDCC-model.8 In Table 2, we compare theMacGyver QML and BIP M-estimates for N = 2, 10 and50. The cDCC parameters α and β are 0.1 and 0.8, andthe intercept Q was chosen to match the unconditionalcorrelation of the daily returns of 50 stocks belonging tothe S&P 500 index for the period 1991–2011.

Since we are only interested in the estimation precisionof the correlation parameters, we follow Engle andSheppard (2001) and fix the conditional variances toone and only do the second step estimation of thecDCC parameters. Additive jumps are generated as acombination of 0.4εT equispaced cojumps and a Bernoulliprocess generating an average of 0.6εT individual jumps.We simulated either 1% or 5% of jumps of size d = 3 or4 (recall that the conditional variances are set to 1), andtheir sign is set equal to the sign of Yt . For each parameter,we compute the estimation bias and RMSE over 10,000replications. For the BIP-cDCC, we consider a thresholdgiven by δ = 0.975.9

Consider first the panel with the results for the no-jumps case (ε = 0%) and N = 2. We see that theRMSE of the robust estimator is only slightly higher thanthe RMSE of the maximum likelihood estimator, for allsettings. In the next panels, it is shown that this smallloss of efficiency in the absence of jumps is compensatedfor by a smaller bias and lower RMSE in the presence ofjumps. Additive jumps lead to a downward bias in the

8 When the total number of possible combinations of two assetsexceeds 200, we take the median of 200 randomly chosen subsets.9 We repeated the simulation for δ = 0.95, but the resulting bias and

RMSE were similar to that for δ = 0.975. The results reported in thispaper are based on programs written by the authors using Ox version 6.0(Doornik, 2009) and G@RCH version 6.0 (Laurent, 2009).

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Table 2Bias and RMSE of the MacGyver Gaussian QML and robust estimator for the dependence parameters α and β of the N-dimensional cDCC model in thepresence of ε jumps of size d times the conditional standard deviation, with δ = 0.975 and T = 2000. For Q , the average bias and RMSE of the lowerdiagonal elements are reported.

N = 2 N = 10 N = 50Q α β Q α β Q α β

ε = 0% QML Bias −0.003 0.001 −0.008 0.001 −0.001 −0.004 0.000 0.000 −0.004RMSE 0.039 0.018 0.043 0.042 0.005 0.012 0.043 0.003 0.007

Robust Bias −0.010 0.010 −0.009 −0.013 0.001 −0.005 −0.010 0.001 0.004RMSE 0.038 0.021 0.051 0.042 0.006 0.014 0.045 0.004 0.009

ε = 1% QML Bias −0.019 −0.011 −0.014 −0.001 −0.013 −0.001 −0.008 −0.013 0.000d = 3 RMSE 0.045 0.029 0.084 0.045 0.016 0.018 0.046 0.015 0.010

Robust Bias −0.012 −0.004 −0.008 −0.013 −0.005 −0.004 −0.014 −0.005 −0.004RMSE 0.038 0.021 0.051 0.042 0.008 0.013 0.048 0.006 0.008

ε = 1% QML Bias −0.031 −0.019 −0.030 −0.017 −0.019 −0.001 −0.014 −0.020 0.004d = 4 RMSE 0.054 0.039 0.130 0.051 0.022 0.026 0.050 0.022 0.013

Robust Bias −0.012 −0.004 0.010 −0.013 −0.005 −0.003 −0.017 −0.005 −0.004RMSE 0.039 0.020 0.051 0.042 0.008 0.013 0.052 0.006 0.008

ε = 5% QML Bias −0.058 −0.033 −0.029 −0.035 −0.037 0.017 −0.028 −0.038 0.020d = 3 RMSE 0.072 0.046 0.150 0.058 0.038 0.031 0.055 0.038 0.024

Robust Bias −0.017 −0.021 −0.005 −0.016 −0.023 0.003 −0.026 −0.023 0.002RMSE 0.042 0.028 0.056 0.044 0.024 0.015 0.058 0.023 0.008

ε = 5% QML Bias −0.084 −0.042 −0.054 −0.052 −0.049 0.014 −0.041 −0.049 0.024d = 4 RMSE 0.097 0.057 0.190 0.074 0.050 0.043 0.066 0.050 0.028

Robust Bias −0.017 −0.022 −0.006 −0.015 −0.024 0.004 −0.036 −0.023 0.003RMSE 0.042 0.028 0.055 0.044 0.024 0.015 0.067 0.024 0.009

QML estimates of the unconditional correlation Q and thedependence parameters α and β . The larger the jumps andthe higher the percentage of jumps, the larger the bias is.Jumps have almost no impact on the bias and RMSE of theBIPM-estimator.

For N > 2, the estimates of α and β are the median ofpairwise estimates of the cDCC-model. We see that, whenN increases, the RMSE of the QML and BIP M-estimatorsof α and β decreases substantially. The numbers of pairsare 45 and 200, for N = 10 and N = 50, respectively.This explains why the decrease in RMSE is relatively moreimportant for N = 10 vs. N = 2, than for N = 50 vs.N = 10. Interestingly, since jumps have different impactson each pairwise estimate, and because of the aggregationusing the median, the bias induced by the jumps is alsoreducedwhen the cross-section becomes larger. In all casesof jump contamination, the RMSE of the robust estimatorof the dependence parameters is substantially smaller thanthe RMSE of the QML estimator.

For N > 2, Q is estimated as the (reweighted)unconditional correlation matrix of P1S1, . . . , PT ST , withP1, . . . , PT constructed using the MacGyver estimatesof α and β (see Sections 3.2–3.4). In Table 2 wereport the average bias and RMSE computed on thelower diagonal elements of the estimated unconditionalcorrelationmatrix. Since an increase inN reduces theRMSEof the QML estimates of α and β , we also find a reductionin the RMSE of the classical estimator of Q in the presenceof jumps. For N = 50, the RMSEs of the robust andclassical estimators are similar, due to the fact that in thissimulation study, 60% of the jumps are component-wise,leading to the propagation of outliers when the dimensionis high. This is consistent with the finding of Alqallaf,Van Aelst, Yohai, and Zamar (2009) that the reweightedcorrelation estimators have poor breakdown properties in

the case of independent contamination. In such cases, werecommend the use of cell-wise weights, as proposed byVan Aelst and Willems (2011).

The results obtained above are for a specific set ofdependence parameters, namely α = 0.1 and β = 0.8.In Figs. 1 and 2 we report the bias and RMSE of the QMLand BIP M-estimators of the bivariate cDCC model whenα ranges between 0.03 and 1 and α + β is fixed at 0.95.The solid lines show the bias and RMSE in the case ofno jumps. The dotted and dashed lines correspond to thecase of 5% of additive jumps of sizes d = 3 and d = 4,respectively. In the left panel, we see that jumps inducelarge increases in the bias and RMSE of the QML estimator,and these increases tend to be larger as the jump sizeincreases. The bias and RMSE of the QML estimate of α (β)increases for larger values of α (β). Comparing the left andright panels of Figs. 1 and 2, we see that, for all parametercombinations considered, the robust BIPM-estimator has asmaller bias andRMSE than theGaussianQML-estimator inthe presence of jumps.We repeat the same analysis forα+

β = 0.98 in the web appendix and obtain similar results.

5. Forecast evaluation using the model confidence set

Our first application is to forecasting the r-step-aheaddaily conditional covariance matrix of the EUR/USD andYen/USD exchange rates over the period 2004–2009.The model confidence set (MCS) approach of Hansenet al. (2011) is used to compare the forecasts. Given auniverse of model-based forecasts, the MCS allows usto identify the subset of models that are equivalent interms of their forecasting ability, but outperform all ofthe other competing models. The accuracy of the forecastsis evaluated by comparing the forecasts with a high-frequency based ex post measure of the daily covariancematrix and a robust loss function.

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Fig. 1. Bias of the Gaussian QML and robust BIPM-estimator for unconditional correlationQ and the dependence parameters α and β of the 2-dimensionalcDCC model in the presence of ε jumps of size d times the conditional standard deviation, with δ = 0.975, T = 2000 and α + β = 0.95.

Fig. 2. RMSEof theGaussianQMLand robust BIPM-estimator for unconditional correlationQ and thedependenceparametersα andβ of the 2-dimensionalcDCC model in the presence of ε jumps of size d times the conditional standard deviation, with δ = 0.975, T = 2000 and α + β = 0.95.

The next subsections present the set of competingmodels, the proxy of the true but unobserved covariance,the data, the loss function, and, finally, the results of ourapplication. Following Hansen et al. (2011), we set thesignificance level for the MCS to α = 0.25, and 10,000bootstrap samples were used to obtain the distributionunder the null of equal forecasting performance.10

10 This test has been implemented using the Ox software packageMULCOM 2.0 of Hansen and Lunde (2011).

5.1. Set of competing models

We consider eight MGARCH models with a constantconditional mean. The first three models belong to theclass of BEKK models proposed by Engle and Kroner(1995). We consider the diagonal BEKK(1, 1), the scalarBEKK(1, 1) and the multivariate exponentially weightedmoving average (EWMA) model.11 The first two models

11 The EWMA model has been popularised by Riskmetrics (1996) andis used widely by practitioners. See for example Bauwens, Laurent, andRombouts (2006) for the exact specification of these models.

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are estimated by Gaussian QML, while the EWMA modeldoes not require any parameter estimation (apart fromthe mean, set here to the empirical mean). We alsoconsider the constant conditional correlation (CCC) modelof Bollerslev (1990) and the (consistent) DCC model, withGARCH(1, 1) specifications for the conditional variances.The CCC and cDCC models are estimated by Gaussian QMLusing the three-step approach described in Section 3.1 (orthe two-step approach for the CCC). The last two modelsare the BIP versions of the CCC and cDCC models. Thefirst step is common to all three models and consists ofthe estimation ofN BIP-GARCH(1, 1)models with variancetargeting. The second step corresponds to the estimationof the correlation matrix, as described in Section 3.3. Wechoose δ = 0.975 for the BIP weight functions.

5.2. Proxy

We judge the performances of the competing modelsthrough the use of a statistical loss function. The evalu-ation of the forecasting performance of volatility modelsis challenging, since the variable of interest (i.e., the co-variance) is unobservable, and therefore the evaluation ofthe loss function has to rely on a proxy. We consider threeproxies based on the theory of quadratic variation of Brow-nian semimartingale with jumps processes, i.e., the real-ized covariance (RCov) of Andersen, Bollerslev, Diebold,and Labys (2003), the realized bipower covariation (RBP-Cov) of Barndorff-Nielsen and Shephard (2004), and therealized outlyingness weighted covariation (ROWCov) ofBoudt et al. (2011a). The motivation for the choice of theseproxies is that RCov estimates the total quadratic variation,while RBPCov and ROWCov only estimate the continuouscomponent of the quadratic variation.

5.3. Data

Our data consist of daily and 30-min log-returnscomputed from the indicative quotes provided by Olsen& Associates, on the Euro (Deutsche Mark before 1999)and Yen exchange rates, expressed in US dollars (EUR andYEN).12 The sample period goes from January 3, 1995, toDecember 31, 2009. We removed days with too manymissing values and/or constant prices, as well as weekendsand holidays, when trading is infrequent. The cleaneddataset includes 3819 trading days. One trading dayextends from21.00GMTonday t−1 to 21.00GMTonday t .For equity data, Laurent, Rombouts, and Violante (in press)find that the relative performance of MGARCH modelsdepends strongly on the state of the market. We thereforedistinguish between the calmmarket period of 2004–2006and the turbulent period of the credit crisis in 2007–2009.The standard deviations of the daily EUR and YPY returns

12 The 30-min returns are needed to compute the proxies discussedabove. Muthuswamy, Sarkar, Low, and Terry (2001), among others, showthat, because of non-synchronicity of trading, the correlations are biasedtoward zero if returns on foreign exchange prices are calculated at ultra-high frequencies such as five minutes. For our dataset, the correlations inthe 5, 10, 15 and 30min, and daily EUR and YEN returns are 26%, 30%, 31%,32% and 34%, respectively.

are 59% and 56% in the calm period, but increase to73% and 87% in the turbulent period. From the dailyreturns, rolling estimation samples of 2303 observationsare used to produce the out-of-sample r-step-ahead dailycovariance forecasts, with r = 1, . . . , 10. We report thetime series and scatter plots of the forecasted cDCC andBIP-cDCC variances and covariances in the web appendix.Overall, we find that the differences in the cDCC and BIP-cDCC variances and covariances tend to be small, exceptfor the period October–November 2008, where the cDCCvariance forecast for the Yen/USD is almost double the BIP-cDCC forecast. Carnero et al. (2012) document the fact thatrobust variances of stock returns tend to be smaller thanstandard GARCH(1, 1) variance forecasts. We find a similarresult for the exchange rate data: the 1- (10-)day-aheadGARCH variance forecast is higher than the BIP-GARCHvariance forecast on 63% (58%) and 94% (98%) of the daysfor the EUR/USD and Yen/USD, respectively.

5.4. Loss function

In the presence of outliers (or jumps), Preminger andFranck (2007) recommend using forecast performanceevaluation criteria that are less sensitive to extremeobservations. For this reason, we rely on the Entrywise 1(matrix) norm, defined as follows:

Lm,t =

1≤i,j≤N

σi,j,t − hm,i,j,t , (5.1)

where Lm,t is the Entrywise 1 (matrix) norm of model m(form = 1, . . . , 9), and day t; and σi,j,t and hm,i,j,t , indexedby i, j = 1, . . . ,N , refer respectively to the elements ofthe covariance matrix proxy for day t (i.e., Σt ) and thecovariance forecast of modelm (i.e., Hm,t ).13

5.5. Results

Table 3 indicates which models belong to the set ofsuperior forecasting models according to the MCS test.In both the calm and turbulent market regimes, the BIP-cDCC model is selected as having a superior forecastingperformance for all forecast horizons. Moreover, it alwayshas the smallest value for the loss function. Interestingly,in the calm period, a BIP-model without dynamics in theconditional correlations (i.e., the BIP-CCC) often belongsto the set of superior forecasting models. In the turbulentmarket regime, the RCov and RBPCov are clearly lessinformative about the best 1-day-ahead forecast, since notonly the BIP-cDCC, but also the cDCC, DCC, scalar-BEKK and

13 Hansen and Lunde (2006), Laurent, Rombouts, and Violante (2009)and Patton (2011) show that substituting a proxy for the underlyingvolatility may induce a distortion in the ranking, i.e., the evaluation basedon the proxymight differ from the ranking thatwould have been obtainedif the true target had been used. However, such distortions can be avoidedif the loss function has a particular functional form or if the proxy isaccurate enough. TheMonte Carlo simulation results reported by Laurentet al. (2009) suggest that when the proxy of the daily covariance iscomputed from high frequency returns (for example, 30-min returns, asin our case), all loss functions deliver the expected ranking (i.e., the onebased on the true covariance), which justifies our choice.

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Table 3Models that have a superior forecasting performance for the RCov, RPBCov and ROWCov of EUR/USD and Yen/USD returns in 2004–2006 and 2007–2009,as indicated by the model confidence set approach.

r Calm period: 2004–2006 Turbulent period: 2007–20091 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10

RCov

BIP-cDCC + + + + + + + + + + + + + + + + + + + +

BIP-CCC + + + + + + +

cDCC + +

DCC + +

CCCRM

Scalar-BEKK + +

Diag-BEKK + +

RBPCov

BIP-cDCC + + + + + + + + + + + + + + + + + + + +

BIP-CCC + + + + + + +

cDCC +

DCCCCCRM +

Scalar-BEKK +

Diag-BEKK +

ROWCov

BIP-cDCC + + + + + + + + + + + + + + + + + + + +

BIP-CCC + + +

cDCCDCCCCCRM +

Scalar-BEKKDiag-BEKK

diag-BEKK belong to the model confidence set. However,for the ROWCov and for longer horizons (RCov: horizonsof at least three days; RBPCov: horizons of at least twodays), the BIP-cDCC is the best model according the theMCS procedure.

6. Economic gains in portfolio allocation

Our key empirical application follows fromapplying theBIP model to the problem of optimal portfolio allocation.We study an investor who adopts a volatility timingstrategy where the changes in the ex ante optimalportfolio weights are determined solely by the forecastsof the conditional covariance matrix. The optimal portfolioallocation is supposed to be the minimum varianceportfolio, with the minimum variance portfolio being theonly portfolio on the efficient frontier that is independentof themean forecast. Tomake the portfolio allocationmorerealistic, portfolio weights are required to be nonnegativeand less than 10%.14

The portfolios are fully invested in equity belonging tothe same sector. The initial investment universe consistsof all S&P 500 stocks on July 31, 2010. Stocks with missingdata at the start of the estimation period and sectorswith fewer than ten stocks in the investment universe

14 In the web appendix we verify that relaxing such bound constraintson the portfolioweights has little impact on the conclusions regarding therelative profitability of using the BIP-cDCC vs. cDCC covariance forecasts.

are removed.15 We study the portfolio performance gainsobtained by allocating the sector portfolios based on theBIP-cDCC covariance forecasts instead of the forecastsfrom the baseline cDCC model, over the period January2004–July 2009.16

Because of the high dimensionality of the covarianceestimates, the dependence parameters in the correlationprocess are estimated using the MacGyver method. Thesummary statistics of the rolling estimates of α and β arereported in Table 4. We see that the range of estimatesof α and β is much narrower for the BIP-cDCC methodthan for the cDCC method, in all cases. Furthermore, theaverage estimate of β is higher when using the BIP-cDCC method than when using the cDCC estimator for allsectors. Finally, note the higher stability of the BIP-cDCCcovariance and variance forecasts compared to the cDCCforecasts, as indicated by the first order autocorrelation ofthe forecasted series.

We split up the evaluation sample into the calm periodof 2004–2006 and the turbulent years 2007–2009. Theforecasts of the conditional covariance matrix at time tare based on the data from January 1994 until t − 1. Wemeasure the portfolio performance in two ways. First, welook at the mean and standard deviation of the out-of-sample daily gross returns. The net return for an investor

15 See the web appendix for the list with tickers.16 We repeated the analysis for the BIP-CCC vs. CCC covariance forecastsand obtained similar conclusions.

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Table 4Summary statistics on rolling estimates of the correlation dependence parameters, α and β , obtained using the cDCC and BIP-cDCC models. The last twocolumns report the average first order autocorrelations of the covariance (hi,j) and variance (hii) forecasts.

α β Mean AC(1)Min Mean Max Min Mean Max hi,j hi,i

Cons. discret. cDCC 0.005 0.009 0.018 0.933 0.980 0.994 0.933 0.877BIP-cDCC 0.006 0.008 0.028 0.978 0.990 0.997 0.978 0.990

Cons. staples cDCC 0.006 0.009 0.012 0.852 0.931 0.989 0.929 0.844BIP-cDCC 0.006 0.008 0.010 0.989 0.991 0.993 0.983 0.989

Energy cDCC 0.007 0.013 0.025 0.959 0.983 0.990 0.956 0.944BIP-cDCC 0.010 0.015 0.020 0.979 0.984 0.990 0.986 0.992

Financials cDCC 0.011 0.017 0.029 0.900 0.948 0.980 0.917 0.912BIP-cDCC 0.012 0.015 0.021 0.974 0.981 0.985 0.976 0.987

Healthcare cDCC 0.006 0.010 0.014 0.899 0.941 0.993 0.901 0.770BIP-cDCC 0.006 0.007 0.009 0.990 0.992 0.994 0.976 0.985

Industrials cDCC 0.005 0.009 0.027 0.933 0.985 0.994 0.933 0.877BIP-cDCC 0.009 0.012 0.020 0.978 0.987 0.990 0.978 0.990

IT cDCC 0.005 0.008 0.017 0.952 0.983 0.995 0.953 0.885BIP-cDCC 0.010 0.012 0.016 0.983 0.987 0.989 0.981 0.993

Materials cDCC 0.007 0.013 0.025 0.959 0.983 0.990 0.955 0.925BIP-cDCC 0.009 0.011 0.014 0.986 0.989 0.991 0.987 0.990

For all sectors, the cDCC and BIP-cDCC estimates of α and β are significantly different at the 99% confidence level, using the t-test and the Wilcoxon test.

corresponds to these gross return performance measures,from which the transaction costs are deducted. As anindicator of the transaction costs, we report the averageportfolio turnover in Table 5 as well. The daily portfolioturnover is defined as the percentage of wealth traded thatday: κt = |wt − wt+|

′ι, where wt is the vector of weightsat the rebalancing period t; wt+ is the vector of weightsbefore rebalancing at t; and ι is a vector of ones. Finally, inTable 5 we report the differences in Sharpe ratios betweenthe minimum variance portfolios constructed using theproposed robust method and the classical method in thepresence of a proportional transaction cost f . Then, thedaily evolution of the wealth invested is given by

Wt+1 = Wt(1 − f · κt)(1 + pt), (6.1)

with pt being the portfolio return on trading day t andW0 = 1000 USD. Following Balduzzi and Lynch (1999), thetransaction cost per dollar of portfolio value f is set to 0,1e − 4 or 1e − 3 USD.

Similarly to Engle and Colacito (2005), we test forsignificant differences between the daily portfolio returns,squared returns and turnover using a Diebold andMariano(1995) type test. It regresses the daily differences betweenthe performance measures of two portfolio methods ona constant, and tests whether the estimated constantis significantly different from zero using a Newey–Weststandard error. The significance of the difference in Sharperatios is evaluated using the test of Jobson and Korkie(1981) and Memmel (2003), which also uses Newey–Weststandard errors. Similar results were obtained whentesting using the bootstrap procedure of Ledoit and Wolf(2008).

In Table 5, note first that the use of the BIP method hasno significant impact on the average gross portfolio return,but it reduces the portfolio standard deviation significantlyfor most sectors. For 11 of the 16 portfolios considered,the BIP method has a positive impact on the Sharpe ratio,but the impact is not statistically significant. This does notmean that there is no profit for the investor in using the BIPprocedure, since the BIP conditional covariance matrices

are more stable, resulting in a lower portfolio churn, andthus lower transaction costs. Indeed, we see that, exceptfor the portfolio invested in the IT sector in 2004–2006,the portfolio turnover is always significantly larger for theclassical portfolio than for the robust portfolio. Therefore,even if the two procedures have similar out-of-samplemean returns and standard deviations, net of transactioncosts, the BIP procedure will potential yield higher profitsoverall. This is illustrated in the last two columns of Table 5.When the Sharpe ratio is computed on the returns netof a 0.1% proportion transaction costs, the difference inSharpe ratios between the robust and classical allocationmethods is always positive (except for the IT sector), andeven significantly positive for some sectors.

7. Conclusion

We propose the BIP-cDCC model for multivariatevolatility forecasting in the presence of one-off eventswhich cause large changes in prices whilst not affectingthe volatility dynamics. Under this model, extremes have abounded impact on both the parameter estimates and thevolatility forecasts. In an application to forecasting the co-variance matrix of the daily returns of the EUR/USD andYen/USD exchange rates, the BIP-cDCC model always be-longs to the set of superior forecasting models. Further-more, it is identified as the best model for most forecasthorizons by the model confidence set approach of Hansenet al. (2011). We also show that, for the minimum vari-ance allocation of sector portfolios on US stocks, the BIPcovariance forecasts reduce the portfolio turnover signifi-cantly, without the out-of-sample portfolio return perfor-mance deteriorating.

Throughout the paper, we focused on the use of the BIP-GARCH(1, 1) model for univariate volatility forecasting. Anatural extension would be to consider volatility modelswith leverage effects, such as the asymmetric power ARCHmodel of Ding, Granger, and Engle (1993) or the GJRmodelof Glosten, Jagannathan, and Runkle (1993), provided that

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Table 5Summary statistics on the out-of-sample performances of minimum variance portfolios based on the BIP-cDCC model vs. the cDCC model: gross returns(annualized mean, standard deviation), portfolio turnover and difference in annualized Sharpe ratio when the proportional trading cost is κ .

cDCC BIP-cDCC ∆ SR BIP-cDCC vs cDCCMean SD Turn Mean SD Turn κ = 0 1e − 4 1e − 3

2004–2006

Cons. discret. 0.101 0.098∗∗∗ 0.151∗∗∗ 0.114 0.102∗∗∗ 0.244∗∗∗ 0.047 0.063 0.237Cons. staples 0.090 0.089∗ 0.289∗∗∗ 0.089 0.086∗ 0.156∗∗∗ 0.032 0.063 0.379∗

Energy 0.325 0.233∗∗∗ 0.334∗∗∗ 0.283 0.223∗∗∗ 0.173∗∗∗−0.126 −0.111 0.032

Financials 0.209 0.118∗∗∗ 0.474∗∗∗ 0.135 0.104∗∗∗ 0.321∗∗∗−0.474 −0.459 −0.237

Healthcare 0.076 0.114 0.330∗∗∗ 0.090 0.111 0.205∗∗∗ 0.142 0.174 0.395Industrials 0.129 0.106∗∗∗ 0.346∗∗∗ 0.151 0.102∗∗∗ 0.195∗∗∗ 0.269 0.316 0.601∗∗∗

IT 0.111 0.132 0.165∗∗∗ 0.123 0.132 0.180∗∗∗ 0.095 0.079 0.063Materials 0.165 0.146∗∗∗ 0.270∗∗∗ 0.131 0.134∗∗∗ 0.169∗∗∗

−0.142 −0.126 0.000

2007–2009

Cons. discret. −0.179 0.262∗∗∗ 0.186∗∗∗−0.149 0.251∗∗∗ 0.215∗∗∗ 0.032 0.063 0.379∗

Cons. staples −0.018 0.188∗∗∗ 0.314∗∗∗ 0.005 0.176∗∗∗ 0.171∗∗∗ 0.126 0.142 0.300∗∗

Energy −0.036 0.439∗∗∗ 0.305∗∗∗ 0.001 0.429∗∗ 0.187∗∗∗ 0.079 0.095 0.142∗

Financials −0.124 0.346∗ 0.359∗∗∗−0.108 0.339∗ 0.210∗∗∗ 0.047 0.047 0.142

Healthcare 0.009 0.214∗∗∗ 0.350∗∗∗ 0.016 0.201∗∗∗ 0.193∗∗∗ 0.032 0.047 0.206Industrials −0.068 0.264∗∗∗ 0.373∗∗∗

−0.059 0.251∗∗∗ 0.220∗∗∗ 0.016 0.032 0.158IT −0.028 0.282∗∗∗ 0.264∗∗∗

−0.048 0.273∗∗∗ 0.257∗∗∗−0.079 −0.079 −0.079

Materials −0.045 0.311∗∗∗ 0.243∗∗∗−0.017 0.303∗∗∗ 0.150∗∗∗ 0.095 0.095 0.158∗

∗∗∗ , ∗∗ and ∗ indicate significant differences between BIP-cDCC and cDCC at the 1%, 5% and 10% levels, respectively.

the models are adapted such that lagged returns have abounded impact on the future volatility.

A further limitation of the paper is that we have onlyconsidered one type of outlier, namely those correspond-ing to price jumps caused by large one-off events. As wasnoted by Tsay, Peña, and Pankratz (2000), for example,other definitions of outliers in multivariate time series doexist.

We leave further work along these lines for futureresearch.

Acknowledgments

The authors thank the Dutch Science Foundation forfinancial support, as well as Esther Ruiz (the Editor) andtwo anonymous referees for their helpful comments andadvice.

References

Aielli, G. (2009). Dynamic conditional correlations: on properties andestimation. Unpublished paper: Department of Statistics. Universityof Florence.

Alqallaf, F., Van Aelst, S., Yohai, V., & Zamar, R. (2009). Propagation ofoutliers in multivariate data. Annals of Statistics, 37, 311–331.

Andersen, T. G., Bollerslev, T., & Diebold, F. (2007). Roughing it up:including jump components in the measurement, modelling andforecasting of return volatility. Review of Economics and Statistics, 89,701–720.

Andersen, T., Bollerslev, T., Diebold, F., & Labys, P. (2003). Modeling andforecasting realized volatility. Econometrica, 71, 579–625.

Audrino, F., & Trojani, F. (2011). A general multivariate threshold GARCHmodel with dynamic conditional correlations. Journal of Business andEconomic Statistics, 29, 138–149.

Balduzzi, P., & Lynch, A. (1999). Transaction costs and predictability: someutility cost calculations. Journal of Financial Economics, 52, 47–78.

Barndorff-Nielsen, O.E., & Shephard, N. (2004). Measuring the impactof jumps in multivariate price processes using bipower covariation.Discussion paper. Nuffield College. Oxford University.

Bauwens, L., Laurent, S., & Rombouts, J. (2006). Multivariate GARCHmodels: a survey. Journal of Applied Econometrics, 21, 79–109.

Bauwens, L., & Storti, G. (2009). A component GARCH model with timevarying weights. Studies in Nonlinear Dynamics and Econometrics, 13,article 1.

Billio, M., Caporin, M., & Gobbo, M. (2006). Flexible dynamic conditionalcorrelation multivariate GARCH models for asset allocation. AppliedFinancial Economics Letters, 2, 123–130.

Bollerslev, T. (1990). Modeling the coherence in short-run nominalexchange rates: a multivariate generalized ARCH model. Review ofEconomics and Statistics, 72, 498–505.

Boudt, K., & Croux, C. (2010). RobustM-estimation ofmultivariate GARCHmodels. Computational Statistics and Data Analysis, 54, 2459–2469.

Boudt, K., Croux, C., & Laurent, S. (2011a). Outlyingness weightedcovariation. Journal of Financial Econometrics, 9, 657–684.

Boudt, K., Croux, C., & Laurent, S. (2011b). Robust estimation of theperiodicity in intraday volatility and intraday jump detection. Journalof Empirical Finance, 18, 353–367.

Boudt, K., Daníelsson, J., & Laurent, S. (2012). Appendix to Robustforecasting of dynamic conditional correlation GARCH models.Available at www.econ.kuleuven.be/kris.boudt/public.

Carnero, M., Peña, D., & Ruiz, E. (2012). Estimating GARCH volatility in thepresence of outliers. Economics Letters, 114, 86–90.

Carnero, M., Peña, D., & Ruiz, E. (2006). Effects of outliers on theidentification and estimation of GARCH models. Journal of Time SeriesAnalysis, 28, 471–497.

Chan, W. H., & Maheu, J. M. (2002). Conditional jump dynamics instock market returns. Journal of Business and Economic Statistics, 20,377–389.

Corsi, F., Pirino, D., & Renó, R. (2008). Threshold bipower variation and theimpact of jumps on volatility forecasting. Journal of Econometrics, 159,276–288.

Cox, D. (1968). Notes on some aspects of regression analysis. Journal ofRoyal Statistical Society A, 131, 265–279.

Croux, C., & Dehon, C. (2010). Influence functions of the Spearman andKendall correlation measures. Statistical Methods and Applications, 19,497–515.

Diebold, F., & Mariano, R. (1995). Comparing predictive accuracy. Journalof Business and Economic Statistics, 13, 253–263.

Ding, Z., Granger, C. W. J., & Engle, R. F. (1993). A longmemory property ofstock market returns and a newmodel. Journal of Empirical Finance, 1,83–106.

Doornik, J. (2009). Object-oriented matrix programming using Ox. Timber-lake Consultants Press.

Engle, R. (2002). Dynamic conditional correlation—a simple class ofmultivariate GARCH models. Journal of Business and EconomicStatistics, 20, 339–350.

Engle, R. (2009). High dimension dynamic correlations. In J. Castle, &N. Shepard (Eds.), The methodology and practice of econometrics: afestschrift for David Hendry (pp. 122–148). Oxford University Press.

Page 14: Robust forecasting of dynamic conditional correlation GARCH models

K. Boudt et al. / International Journal of Forecasting 29 (2013) 244–257 257

Engle, R. F., & Colacito, R. (2005). Testing and valuing dynamic correlationsfor asset allocation. Journal of Business and Economic Statistics, 24,238–253.

Engle, R., & Kroner, F. (1995). Multivariate simultaneous generalizedARCH. Econometric Theory, 11, 122–150.

Engle, R., & Mezrich, J. (1996). GARCH for groups. Risk, 9, 36–40.Engle, R., Shephard, N., & Sheppard, K. (2008). Fitting vast dimensional

time-varying covariance models. Mimeo.Engle, R., & Sheppard, K. (2001). Theoretical and empirical properties of

dynamic conditional correlation multivariate GARCH. UCSD: Mimeo.Francq, C., Horvath, L., & Zakoian, J.-M. (2011). Merits and drawbacks of

variance targeting in GARCHmodels. Journal of Financial Econometrics,9, 619–656.

Franses, P., & Ghijsels, H. (1999). Additive outliers, GARCH and forecastingvolatility. International Journal of Forecasting , 15, 1–9.

Glosten, L., Jagannathan, R., & Runkle, D. (1993). On the relation betweenexpected value and the volatility of the nominal excess return onstocks. Journal of Finance, 48, 1779–1801.

Hafner, C., & Reznikova, O. (2012). On the estimation of dynamicconditional correlation models. Computational Statistics and DataAnalysis, 56, 3533–3545.

Hansen, P., & Lunde, A. (2011). MULCOM 2.00. Econometric toolkit formultiple comparisons.http://www.hha.dk/~alunde/MULCOM/MULCOM.HTM.

Hansen, P., & Lunde, A. (2006). Consistent ranking of volatility models.Journal of Econometrics, 131, 97–121.

Hansen, P., Lunde, A., & Nason, J. (2011). Model confidence sets.Econometrica, 79, 453–497.

Harvey, A., & Chakravarty, T. (2008). Beta-t-(E)GARCH. Working paperseries. University of Cambridge.

Healy, M. (1968). Multivariate normal plotting. Applied Statistics, 17,157–161.

Jobson, J., & Korkie, B. (1981). Performance hypothesis testing with theSharpe and Treynor measures. Journal of Finance, 36, 889–908.

Laurent, S. (2009). G@RCH 6: estimating and forecasting GARCH models.Timberlake Consultants Ltd.

Laurent, S., Rombouts, J., & Violante, F. (in press). On the forecasting accu-racy of multivariate GARCH models. Journal of Applied Econometrics.

Laurent, S., Rombouts, J., & Violante, F. (2009). On loss functions andranking forecasting performances of multivariate volatility models.Cirano discussion paper 2009-45.

Ledoit, O., &Wolf, M. (2008). Robust performance hypothesis testingwiththe Sharpe ratio. Journal of Empirical Finance, 15, 850–859.

Lopuhaä, H. (1999). Asymptotics of reweighted estimators ofmultivariatelocation and scatter. Annals of Statistics, 27, 1638–1665.

Memmel, C. (2003). Performance hypothesis testing with the SharpeRatio. Finance Letters, 1, 21–23.

Mercurio, D., & Spokoiny, V. (2004). Statistical inference for time-inhomogeneous volatility models. Annals of Statistics, 32, 577–602.

Moran, P. (1948). Rank correlation and permutation distributions.Proceedings of the Cambridge Philosophical Society, 44, 142–144.

Muler, N., & Yohai, V. (2008). Robust estimates for GARCHmodels. Journalof Statistical Planning and Inference, 138, 2918–2940.

Muthuswamy, J., Sarkar, S., Low, A., & Terry, E. (2001). Time variation inthe correlation structure of exchange rates: high frequency analyses.Journal of Futures Markets, 21, 127–144.

Patton, A. (2011). Volatility forecast comparison using imperfect volatilityproxies. Journal of Econometrics, 160, 246–256.

Preminger, A., & Franck, R. (2007). Forecasting exchange rates: a robustregression approach. International Journal of Forecasting , 23, 71–84.

Riskmetrics (1996). Riskmetrics technical document (4th ed.). New York:J.P. Morgan.

Tsay, R., Peña, D., & Pankratz, A. (2000). Outliers in multivariate timeseries. Biometrika, 87, 789–804.

Van Aelst, S. V. E., & Willems, G. (2011). Stahel–Donoho estimators withcellwise weights. Journal of Statistical Computation and Simulation, 81,1–27.

Vlaar, P., & Palm, F. (1993). The message in weekly exchange ratesin the European monetary system: mean reversion, conditionalheteroskedasticity and jumps. Journal of Business and EconomicStatistics, 11, 351–360.