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1、c10-VectorAutoregressPage343Thursday,October26,20062:08PMCHAPTER10VectorAutoregressiveModelsIInthischapterwediscussestimationmethodsforvectorautoregressiveI(VAR)models.Wefirstconsiderestimationofstablesystems.ThekeyresulthereisthatstableVARsystemscanbeconvenientlyestimatedwit
2、hleastsquaresmethods.Wethenproceedtotheestimationofunsta-blesystems.ESTIMATIONOFSTABLEVARMODELSWhendiscussingtheestimationofregressionmodelsinChapter3,weintroducedtwomainmethodsforestimatinglinearregressions:theleastsquaresmethodandthemaximumlikelihoodmethod.Thesemethodsappl
3、yimmediatelytounrestrictedstableVARmodels.Notethatmodelsaresaidtobe“unrestricted”iftheestimationprocessisallowedtodetermineanypossibleoutcome,and“restricted”iftheestimationpro-cedurerestrictsparametersinsomeway.Supposethatatimeseriesisgivenandthatthedatageneratingpro-cess(DG
4、P)oftheseriesistheVAR(p)model:xt=A1xt–1++A1xt–1++A1xt–1v+εtwherext=(x1,t,...,xN,t)′isaN–dimensionalstochastictimeseriesinvectornotation;AiaredeterministicN×Nmatrices;εt=(ε1,t,...,εN,t)′isamultivariatewhitenoisewithvariance-covariancematrixΣ;andv=(v1,...,vN)′isavectorofconst
5、ants.Let’sfirstassumethatstabilityconditiondet(A(z))≠0forz≤1343c10-VectorAutoregressPage344Thursday,October26,20062:08PM344FINANCIALECONOMETRICSholds,thatis,therootsofthereversecharacteristicequationarestrictlyoutsideoftheunitcircle.TheresultisthattheVAR(p)modelisstableandthe
6、correspondingprocessstationary.Wewillconsiderprocessesthatstartatt=1,assumingthatpinitialconditions:x–p+1,...,x0aregiven.Inthiscase,stableVARmodelsyieldasymptoticallystationaryprocesses.RecallthattheaboveN-dimensionalVAR(p)modelisequivalenttothefollowingNp-dimensionalVAR(1)m
7、odel:Xt=AXt–1+V+UtwhereA1A2Ap–1ApxtIN000X=xt–1,,A=t0IN0000xtp–+100IN0vεt00V=,Ut=00MatrixAiscalledthecompanionmatrixoftheVAR(p)system.GiventhattheVAR(p)modelisunrestricted,itcanbeestimatedasanylinearregressionmodel.Asweconsideronlyconsistentestimators,theestimatedpa
8、rameters(inthelimitofaninfinitesample)satisfythestabilitycondition.Howeveron
