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1、c09-VectorAutoregressModelsPage321Thursday,October26,20062:07PMCHAPTER9VectorAutoregressiveModelsInthisandthefollowingtwochapterswediscussvectorautoregressiveImodels.HereweprovidetheformalbackgroundofVARmodelsanddiscussingtheirstatisticalproperties.Thenextchapteraddressesth
2、eestimationofVARmodels.VARMODELSDEFINEDVectorautoregressive(VAR)modelsare,assuggestedbytheirname,modelsofvectorsofvariablesasautoregressiveprocesses,whereeachvariabledependslinearlyonitsownlaggedvaluesandthoseoftheothervariablesinthevector.Thismeansthatthefuturevaluesofthep
3、rocessareaweightedsumofpastandpresentvaluesplussomenoise(and,possibly,exogenousvariables).Forexample,itisknownthatthereareequityprice“l(fā)eaders”andequityprice“l(fā)aggards”inthesensethatthereturnsofsomeportfoliosoflarge-capstocksanticipatethereturnsoflargeportfoliosofsmall-capsto
4、cks.1Ananalystwhowantstoexploitthisrelationshipforaspeci?cpairofleader-laggardportfoliosmight?tabivariateVARtomodelthereturnsoftheleaderandlaggardportfolios.Suppose,forexample,thatportfolioAisaleaderportfolioandportfolioBalaggardportfolio.Theanalystcanwritethefollowingmodel
5、forreturns:1SeeJohnY.Campbell,AndrewW.Lo,andA.CraigMacKinlay,TheEconometricsofFinancialMarkets(Princeton,NJ:PrincetonUniversityPress,1997);andAngelosKanasandGeorgeP.Kouretas,“ACointegrationApproachtotheLead-LagEffectAmongSized-SortedEquityPortfolios,”WorkingPaper,Department
6、ofEconomics,UniversityofCrete,2001.321c09-VectorAutoregressModelsPage322Thursday,October26,20062:07PM322FINANCIALECONOMETRICSRA()t+1=RA()εt+A()t+1RB()t+1=aRA()εt+B()t+1whereRA(t)andRB(t)arethereturnsofthetwoportfolios,respectively,andεAandεBareindependentwhite-noiseterms(II
7、Dzero-meanvari-ables).The?rstequationstatesthattheprice-leaderportfoliofollowsarandomwalk;thesecondequationstatesthatthelaggardportfoliotendstofollowtheleaderwithadelayofoneperiod.Theaboveisasimpleexampleofamultivariateextensionoftheautoregressive(AR)model.Avectorautoregres
8、sivemodeloforderp[VAR(p)]hasthefollowinggeneralform:xt=A1xt–1++A2xt–2…++Apxtp–st+ε
