A Multivariate Generalized Autoregressive Conditional Heteroscedasticity Model with TimeVarying Correlations Author(s): Y. K. Tse and Albert K. C. Tsui Source: Journal of Business & Economic Statistics, Vol. 20, No. 3 (Jul., 2002), pp. 351-362 Published by: American Statistical Association Stable URL: http://www.jstor.org/stable/1392122 Accessed: 06/10/2009 10:08 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/action/showPublisher?publisherCode=astata. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact
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A
MultivariateGeneralized Autoregressive
Conditional
With Model Heteroscedasticity
Correlations Time-Varying Y. K. TSE School of Business,SingaporeManagementUniversity, Singapore259756 (
[email protected]) Albert K. C. Tsui of Economics,NationalUniversity of Singapore,Singapore119260(
[email protected]) Department In this article we propose a new multivariategeneralized autoregressiveconditional heteroscedasticity (MGARCH)model with time-varyingcorrelations.We adopt the vech representationbased on the conditional variancesand the conditionalcorrelations.Whereas each conditional-varianceterm is assumed to follow a univariateGARCH formulation,the conditional-correlationmatrix is postulatedto follow an autoregressivemoving average type of analog. Our new model retains the intuition and interpretation of the univariateGARCH model and yet satisfies the positive-definitecondition as found in the constant-correlationand Baba-Engle-Kraft-Kronermodels. We reportsome Monte Carloresults on the finite-sample distributionsof the maximum likelihood estimate of the varying-correlationMGARCH model. The new model is applied to some real data sets. KEY WORDS: BEKK model; Constant correlation; Maximum likelihood estimate; Monte Carlo method; MultivariateGARCH model; Varyingcorrelation.
1. INTRODUCTION After the success of the autoregressive conditional heteroscedasticity (ARCH) model and the generalized ARCH (GARCH) model in describing the time-varyingvariances of economic data in the univariatecase, many researchershave extendedthese models to multivariatedimension.Applications of the multivariateGARCH (MGARCH) models to financial data have been numerous.For example, Bollerslev (1990) studied the changing variance structureof the exchange rate regime in the EuropeanMonetary System, assuming the correlations to be time invariant.Kroner and Claessens (1991) applied the models to calculate the optimal debt portfolio in multiple currencies. Lien and Luo (1994) evaluated the multiperiodhedge ratios of currency futures in a MGARCH framework.Karolyi (1995) examined the internationaltransmission of stock returnsand volatility,using differentversions of MGARCHmodels. Baillie and Myers (1991) estimatedthe optimal hedge ratios of commodity futures and argued that these ratios are nonstationary.Gourieroux(1997, chap. 6) presented a survey of several versions of MGARCHmodels. See also Bollerslev et al. (1992) and Bera and Higgins (1993) for surveys on the methodology and applicationsof GARCH and MGARCH models. Bollerslev et al. (1988) provided the basic frameworkfor a MGARCH model. They extended the GARCH representation in the univariatecase to the vectorized conditional-variance matrix. Their specification follows the traditionalautoregressive moving averagetime series analog. This vech representation is very general, and it involves a large numberof parameters. Empirical applications require further restrictions and simplifications. A useful member of the vech-representation family is the diagonal form. Under the diagonal form, each variance-covarianceterm is postulated to follow a GARCHtype equation with the lagged variance-covarianceterm and
the productof the correspondinglagged residualsas the rightside variablesin the conditional-(co)varianceequation. It is often difficult to verify the condition that the conditional-variancematrixof an estimatedMGARCH model is positive definite. Engle et al. (1984) presented the necessary conditions for the conditional-variancematrixto be positive definite for a bivariateARCH model. Extensions of these results to more general models are, however,intractable.Furthermore, such conditions are often very difficult to impose during the optimizationof the log-likelihood function.Bollerslev (1990) suggested a constant-correlationMGARCH (CCMGARCH) model that can overcome these difficulties. He pointed out thatunderthe assumptionof constantcorrelations, the maximum likelihood estimate (MLE) of the correlation matrixis equal to the sample correlationmatrix.As the sample correlationmatrixis always positive definite, the optimization will not fail as long as the conditional variances are positive. In addition,when the correlationmatrixis concentratedout of the log-likelihood function further simplification is achieved in the optimization. Because of its computationalsimplicity,the CC-MGARCH model is widely used in empiricalresearch.However,although the constant-correlationassumption provides a convenient MGARCH model for estimation, many studies find that this assumption is not supported by some financial data. Thus, there is a need to extend the MGARCH models to incorporate time-varyingcorrelationsand yet retain the appealing feature of satisfying the positive-definitecondition during the optimization.
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? 2002 American Statistical Association Journal of Business & Economic Statistics July 2002, Vol. 20, No. 3 DOI 10.1198/073500102288618496
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Engle and Kroner (1995) proposed a class of MGARCH the notations.Additionalparameterswould be requiredto repmodel called the BEKK (named after Baba, Engle, Kraft, resent the conditional-meanequationin the complete model if and Kroner) model. The motivation is to ensure the condi- the mean were unknown.Under certain conditions, the MLE tion of a positive-definite conditional-variancematrix in the of the parametersin the conditional-meanequationis asympprocess of optimization. Engle and Kroner provided some totically uncorrelatedwith the MLE of the parametersof the theoretical analysis of the BEKK model and related it to conditional-varianceequation. Under such circumstances,we the vech-representationform. Another approachexamines the may treat y, as pre-filteredobservations [see Bera and Higconditionalvarianceas a factormodel. The works by Diebold gins (1993) for furtherdiscussions]. Otherwise,the parameter and Nerlove (1989), Engel and Rodrigues (1989), and Engle vector has to be augmentedto take account of the parameters et al. (1990) are along this line. One disadvantage of the in the unknownconditionalmean. BEKK and factor models is that the parameterscannotbe easThe conditional variance of y, is assumed to follow the ily interpreted,and theirnet effects on the futurevariancesand time-varyingstructuregiven by covariances are not readily seen. Bera et al. (1997) reported that the BEKK model does not perform well in the estima(1) Var(ytf1_) = ••, tion of the optimal hedge ratios. Lien et al. (2001) reported difficulties in getting convergencewhen they used the BEKK where (t is the informationset at time t. We denote the varimodel to estimate the conditional-variancestructureof spot ance elements of f, by it, for i = 1 .... K, and the covariance elements by oit, where 1 < i < j < K. Denoting D, as and futuresprices. the K x K diagonal matrix where the ith diagonal element is In this article we propose a new MGARCH model with it,, we let Et = Dt'lt. Thus, Et is the standardizedresidual time-varying correlations.Basically we adopt the vech representation.The variables of interest are, however, the con- and is assumed to be serially independentlydistributedwith ditional variances and conditional correlations. We assume mean zero and variance matrix Ft = {Pijt}. Of course, Ft is a vech-diagonal structurein which each conditional-variance also the correlationmatrix of y,. Furthermore,ft = DtFtDt. To specify the conditional variance of y, we adopt the term follows a univariateGARCH formulation.The remainformulationinitiatedby Bollerslev et al. (1988). vech-diagonal ing task is to specify the conditional-correlationstructure.We Thus, each conditional-variance term follows a univariate apply an autoregressive moving average type of analog to GARCH model (p, the q) given by equation the conditional-correlationmatrix.By imposing some suitable P restrictions on the conditional-correlation-matrixequation, q we construct a MGARCH model in which the conditionalih t-h t-h9 "`i2ti 2i, +E ihi, i= 1 ..... K, (2) h=1 h=1 correlationmatrix is guaranteedto be positive definite during the optimization.Thus, our new model retainsthe intuitionand where coi, aih, and fih are nonnegative, and EP a +ih + interpretationof the univariateGARCH model and yet satis1 i = for K. Note fih 1, that we may allow .. fies the positive-definitecondition as found in the constant- Eh=I (p, q) to vary with i so that (p, q) should be regardedas the correlationand BEKK models. generic order of the univariateGARCH process. Researchers The planof the restof the articleis as follows. In Section2 we adopting the vech-diagonal form typically assume that the describethe constructionof the varying-correlationMGARCH above equation also applies to the conditional-covariance model. As in other MGARCH models, the new model can terms in which o-i2is replaced by o'ijt and y2 replaced by be estimated by use of the MLE method. Some Monte for < j < K. We shall deviate from this approach, i 1 < YitYjt Carlo results on the finite-sample distributionsof the MLE however. Specifically, we shall focus on the conditionalof the varying-correlationMGARCH model are reportedin correlationmatrix and adopt an autoregressive moving averSection 3. Section 4 describes some illustrativeexamples of age analog on this matrix. Thus, we assume that the timethe new model that use some real data sets. These are the varyingconditional-correlation matrixFt is generatedfrom the exchange rate data, national stock marketprice data, and sec- recursion toral stock price data. The new model is compared against the CC-MGARCHmodel and the BEKK model. It is found (3) Ft = (1 - 0,-02) -?-or,_l-t, that the new model compares favorably against the BEKK model. Extending the constant-correlationmodel to allow for where r = {pqi}is a (time-invariant)K x K positive definite time-varyingcorrelationsprovides some interestingempirical parametermatrix with unit diagonal elements and Pt is a results. The estimated conditional-correlationpath provides a K x K matrix whose elements are functions of the lagged time historythatwould be lost in a constant-correlationmodel. observationsof y,. The functionalform of P-_, will be specified below. The parameters01 and 02 are assumed to be Finally, we give some concluding remarksin Section 5. non-negative with the additional constraintthat ,01+ 02 < 1. Thus, F, is a weighted average of F, Ft_1, and 4,t-. Hence, if -_, and F0 are well-defined correlationmatrices(i.e., pos2. A VARYING-CORRELATION MGARCHMODEL itive definite with unit diagonal elements), F, will also be a Considera multivariatetime series of observations{yt}, t = well-defined correlationmatrix. 1,..., T, with K elements each, so that yt = (ylt .... YKt)'. It can be observed that P-_, is analogous to yi, _, in the We assume that the observationsare of zero (or known) mean. univariateGARCH(1, 1) model. However,as is a standardF, This assumption simplifies the discussions without straining ized measure,we also require P>-1to depend on the (lagged)
GARCHModel Tse and Tsui:A Multivariate
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The conditions 0
