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1、Tsay-Driver-12013/10/2812:18page27#1CHAPTER2StationaryVectorAutoregressiveTimeSeries2.1INTRODUCTIONThemostcommonlyusedmultivariatetimeseriesmodelisthevectorautoregressive(VAR)model,particularlysointheeconometricliteratureforgoodreasons.First,themodelisrelativelyea
2、sytoestimate.Onecanusetheleast-squares(LS)method,themaximumlikelihood(ML)method,orBayesianmethod.Allthreeestimationmeth-odshaveclosed-formsolutions.ForaVARmodel,theleast-squaresestimatesareasymptoticallyequivalenttotheMLestimatesandtheordinaryleast-squares(OLS)estimat
3、esarethesameasthegeneralizedleast-squares(GLS)estimates.Second,thepropertiesofVARmodelshavebeenstudiedextensivelyintheliterature.Finally,VARmodelsaresimilartothemultivariatemultiplelinearregressionswidelyusedinmultivariatestatisticalanalysis.Manymethodsformakinginfere
4、nceinmultivariatemultiplelinearregressionapplytotheVARmodel.ThemultivariatetimeseriesztfollowsaVARmodeloforderp,VAR(p),ifpzt=φ0+φizt?i+at,(2.1)i=1whereφ0isak-dimensionalconstantvectorandφiarek×kmatricesfori>0,φp=0,andatisasequenceofindependentandidenticallydistribut
5、ed(iid)ran-domvectorswithmeanzeroandcovariancematrixΣa,whichispositive-de?nite.ThisisaspecialcaseoftheVARMA(p,q)modelofChapter1withq=0.Withtheback-shiftoperator,themodelbecomesφ(B)zt=φ0+at,whereφ(B)=Ik?pφBiisamatrixpolynomialofdegreep.SeeEquation(1.21).Weshallreferi=
6、1itoφ=[φ,ij]asthelagARcoef?cientmatrix.MultivariateTimeSeriesAnalysis:WithRandFinancialApplications,FirstEdition.RueyS.Tsay.c2014JohnWiley&Sons,Inc.Published2014byJohnWiley&Sons,Inc.27Tsay-Driver-12013/10/2812:18page28#228stationaryvectorautoregressivetime
7、seriesTostudythepropertiesofVAR(p)models,westartwiththesimpleVAR(1)andVAR(2)models.Inmanycases,weusebivariatetimeseriesinourdiscussion,buttheresultscontinuetoholdforthek-dimensionalseries.2.2VAR(1)MODELSTobegin,considerthebivariateVAR(1)modelzt=φ0+φ1zt?1+at.Thismodelc
8、anbewrittenexplicitlyasz1tφ10φ1,11φ1,12z1,t?1a1t=++,(2.2)z2tφ20φ1,21φ1,22z2,t?1a2torequivalently,z1t=φ10+φ1,11z1,t
