cointegration and state space models

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1、c11-CointegrationPage373Thursday,October26,20062:08PMCHAPTER11CointegrationandStateSpaceModelsnthischapter,weintroducetheconceptsofcointegratedprocessesandIstatespacemodels,aswellastherelativeestimationmethods.Statespacemodelswereintroducedintheengineeringliteratureinthe1960sespeciallythroughthewor

2、kofRudolfE.Kalman.Cointegrationanaly-sisisamorerecenteconometrictool.Thefirstarticlestointroducecoin-tegratedmodelswerepennedbyEngleandGrangerinthesecondhalfofthe1980s.Thoughvectorautoregressive(VAR)processesandstatespacemod-elsareequivalentrepresentationsofthesameprocesses,deeperinsightintotherelat

3、ionshipbetweenstatespacemodelsandcointegrationwasgainedmorerecentlywhenitwasunderstoodthatcointegrationimpliesareducednumberofcommonstochastictrends.Theideabehindcointe-grationthattherearefeedbackmechanismsthatforceprocessestostayclosetogetheristhereforeintimatelyrelatedtotheideathatthebehav-iorofl

4、argesetsofprocessesisdrivenbythedynamicsofasmallernum-berofvariables.COINTEGRATIONCointegrationisoneofthekeyconceptsofmoderneconometrics.Let’sstartbygivinganintuitiveexplanationofcointegrationanditsproper-ties.Twoormoreprocessesaresaidtobecointegratediftheystayclosetoeachotherevenifthey“driftabout”

5、asindividualprocesses.Acolor-fulillustrationisthatofthedrunkenmanandhisdog:Bothstumbleaboutaimlesslybutneverdrifttoofarapart.Cointegrationisanimpor-373c11-CointegrationPage374Thursday,October26,20062:08PM374FINANCIALECONOMETRICStantconceptbothforeconomicsandfinancialmodeling.Itimplementsthenotiontha

6、ttherearefeedbacksthatkeepvariablesmutuallyaligned.Tointroducethenotionofcointegration,recalltheconceptsofstation-aryprocessesandintegratedprocesses.KeyFeaturesofCointegrationLet’sfirstgiveanintuitivecharacterizationtotheconceptofcointegra-tioninthecaseoftwostochasticprocesses.Cointegrationcanbeunde

7、r-stoodintermsofitsthreekeyfeatures:■Reductionoforderofintegration■Regression■CommontrendsFirst,considerreductionoforderofintegration.Twoormorestochasticprocessesthatareintegratedoforderoneorhigheraresaidto