时间序列分析及var模型

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2、14Lecture66.Timeseriesanalysis:Multivariatemodels6.1Learningoutcomes·Vectorautoregression(VAR)·Cointegration·Vectorerrorcorrectionmodel(VECM)·Application:pairstrading6.2Vectorautoregression(VAR)向量自回归Theclassicallinearregressionmodelassumesstrictexog

3、eneity;hence,thereisnoserialcorrelationbetweenerrortermsandanyrealisationofanyindependentvariable(leadorlag).Aswediscovered,serialcorrelation(orautocorrelation)isverycommoninfinancialtimeseriesandpaneldata.Furthermore,weassumedapre-definedrelationofcausa

4、lity:explanatoryvariableaffectthedependentvariable.传统的线性回归模型假设严格的外生性，误差项与可实现的独立变量之间没有序列相关性。金融时间序列及面板数据往往都有很强的自相关性，假定解释变量影响因变量。WenowrelaxbothassumptionsusingaVARmodel.VARmodelscanberegardedasageneralisationofAR(p)processesbyaddingadditionaltimeseries.Henc

5、e,weenterthefieldofmultivariatetimeseriesanalysis.VAR模型可以当作是在一般的自回归过程中加入时间序列。Let’slookatastandardAR(p)processfortwovariables(ytandxt).(1)yt=α1+i=1pβ1iyt-i+ε1t(2)xt=α2+i=1pβ2ixt-i+ε2tThenextstepistoallowthatlaggedvaluesofxtcanaffectytandviceversa.Thismean

6、sthatweobtainasystemofequationsfortwodependentvariables(ytandxt).Bothdependentvariablesareinfluencedbypastrealisationsofytandxt.Bydoingthat,weviolatestrictexogeneity(seeLecture2);however,wecanuseamorerelaxedconcept,namelyweakexogeneity.Asweuselaggedvalue

7、sofbothdependentvariables,wecanarguethattheselaggedvaluesareknowntous,asweobservedtheminthepreviousperiod.Wecallthesevariablespredetermined.Predetermined(lagged)variablesfulfilweakexogeneityinthesensethattheyhavetobeuncorrelatedwiththecontemporaneouserro

8、rtermint.WecanstilluseOLStoestimatethefollowingsystemofequations,whichiscalledaVARinreducedform.(3)yt=α1+i=1pβ11iyt-i+i=1pβ12ixt-i+ε1t(4)xt=α2+i=1pβ21iyt-i+i=1pβ22ixt-i+ε2t14DrGerhardKling,QuantitativeResearchMethodsinFinance,UniversityofSouthamptonPage

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10、14Thebeautyofthismodelisthatwedon’tneedtopredefinewhetherxoryareendogenous(thedependentvariable).Infact,wecantestwhetherx(y)isendogenousorexogenoususingGrangercausalitytests.TheideaofGrangercausalityisthatpastobservations(