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1、Tsay-Driver-12013/10/2814:00page399#1CHAPTER7MultivariateVolatilityModelsInpreviouschapters,weassumetheinnovationsatofamultivariatetimeseriesztareseriallyuncorrelatedandhavezeromeanandpositive-definitecovariancematrix.Wealsoassumethatthecovariancematrixofatistime-invariant.LetFt−1de
2、notetheσ-fieldgeneratedbythepastdata{zt−i
3、i=1,2,...}.TheseassumptionsimplythatE(a
4、F)=0andE(aa
5、F)=Σ>0,whichisaconstantmatrix.Ontt−1ttt−1atheotherhand,mostfinancialtimeserieshaveconditionalheteroscedasticity.LetΣt=Cov(at
6、Ft−1)betheconditionalcovariancematrixofztgivenFt−1.Condi-tionalheter
7、oscedasticitymeansthatΣtistime-dependent.ThedynamicdependenceofΣtisthesubjectofmultivariatevolatilitymodelingandthefocuspointofthischapter.Foreaseinreference,weshallrefertoΣtasthevolatilitymatrixofzt.Thevolatilitymatrixhasmanyfinancialapplications.Forinstance,itiswidelyusedinassetalloca
8、tionandriskmanagement.ModelingΣt,however,facestwomajordifficulties.Thefirstdifficultyisthecurseofdimensionality.Forak-dimensionaltimeserieszt,thevolatilitymatrixΣtconsistsofkconditionalvariancesandk(k−1)/2conditionalcovariances.Inotherwords,Σtconsistsofk(k+1)/2differenttime-varyingelement
9、s.Fork=30,Σtcontains465differentelements.ThedimensionofΣtthusincreasesquadraticallywithk.Theseconddif-ficultyismaintainingthepositive-definiteconstraint.ThevolatilitymatrixΣtmustbepositive-definiteforallt.Specialattentionisneededtomaintainthisconstraintwhenkislarge.Manymultivariatevolatil
10、itymodelshavebeenproposedintheliterature,includ-ingmultivariatestochasticvolatilityandmultivariategeneralizationsofGARCHmodels.SeereviewarticlesAsai,McAleer,andYu(2006)andBauwens,Laurent,andRombouts(2006),andthereferencestherein.SeealsothehandbookofvolatilitybyBauwens,Hafner,andLaurent
11、(2012).Thegoalofthischapteristointroducesomemultivariatevolatilitymodelsthatareapplicableinpractice,yetrelativelyMultivariateTimeSeriesAnalysis:WithRandFinancialApplications,FirstEdition.RueyS.Tsay.c2014JohnWiley&Sons,Inc.Published2014byJohnWiley&Sons,Inc.399Tsay-Driver-12013/
