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1、Tsay-Driver-12013/10/2813:01page199#1CHAPTER4StructuralSpecificationofVARMAModelsStructuralspecificationseekstofindtheunderlyingstructureofamultivariatelineartimeseriessothatawell-definedVARMAmodelcanbeidentified.Thespecifiedmodelovercomesthedifficultyofide
2、ntifiabilitymentionedinChapter3andcanrevealthehiddenstructureofthesystem.Toacertaindegree,structuralspecifica-tioniscloselyrelatedtodimensionreductionandvariableselectioninmultivariateanalysis.Itisaneffectivewaytomodelingmultivariatelineartimeseries.Forsim
3、plicity,wefocusonak-dimensionalzero-meanstationarymultivariatelineartimeseries∞zt=ψiat−i,(4.1)i=0whereψ0=Ikand{at}isaniidsequenceofrandomvectorswithmeanzeroandpositive-definitecovariancematrixΣa.Intheory,themethodsdiscussedcon-tinuetoapplyforunit-rootnon
4、stationarytimeseries,butthelimitingdistributionsofthestatisticsinvolvedwillbedifferent.Thestationarityassumptionimpliesthat∞2i=0ψi<∞,whereAisamatrixnormsuchasthelargestsingularvalueofthematrixA.Therearetwomethodsavailabletoperformstructuralspecificat
5、ionformultivariatelineartimeseries.ThefirstmethodusestheideaofKroneckerindexandthesecondmethodappliestheconceptofscalarcomponentmodel(SCM).See,forinstance,Tsay(1991)andthereferencestherein.Inrealapplications,bothmethodscanbeimplementedusingthecanonicalcor
6、relationanalysis,whichisastatisticalmethodusefulinmultivariateanalysis.Forak-dimensionalVARMAtimeserieszt,theKroneckerindexapproachseekstospecifythemaximumorderoftheARandMApolynomialsforeachcom-ponent.Specifically,theKroneckerindexapproachspecifiesanindexf
7、oreachMultivariateTimeSeriesAnalysis:WithRandFinancialApplications,FirstEdition.RueyS.Tsay.c2014JohnWiley&Sons,Inc.Published2014byJohnWiley&Sons,Inc.199Tsay-Driver-12013/10/2813:01page200#2200structuralspecificationofvarmamodelscomponentzit.These
8、KroneckerindicesjointlyidentifyaVARMAmodelforzt.Theorderforcomponentzitisintheform(pi,pi).Therefore,theKroneckerindexapproachspecifiesthemaximumorderoftheARandMApolynomialsforeachcom-ponentzit.Thescalarcomponentapproach,ont
