时间序列分析报告及VAR模型.docx
《时间序列分析报告及VAR模型.docx》由会员分享,可在线阅读,更多相关《时间序列分析报告及VAR模型.docx(13页珍藏版)》请在冰豆网上搜索。
时间序列分析报告及VAR模型
Lecture6
6.Timeseriesanalysis:
Multivariatemodels
6.1Learningoutcomes
∙Vectorautoregression(VAR)
∙Cointegration
∙Vectorerrorcorrectionmodel(VECM)
∙Application:
pairstrading
6.2Vectorautoregression(VAR)向量自回归
Theclassicallinearregressionmodelassumesstrictexogeneity;hence,thereisnoserialcorrelationbetweenerrortermsandanyrealisationofanyindependentvariable(leadorlag).Aswediscovered,serialcorrelation(orautocorrelation)isverycommoninfinancialtimeseriesandpaneldata.Furthermore,weassumedapre-definedrelationofcausality:
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.Hence,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)
(2)
Thenextstepistoallowthatlaggedvaluesofxtcanaffectytandviceversa.Thismeansthatweobtainasystemofequationsfortwodependentvariables(ytandxt).Bothdependentvariablesareinfluencedbypastrealisationsofytandxt.Bydoingthat,weviolatestrictexogeneity(seeLecture2);however,wecanuseamorerelaxedconcept,namelyweakexogeneity.Asweuselaggedvaluesofbothdependentvariables,wecanarguethattheselaggedvaluesareknowntous,asweobservedtheminthepreviousperiod.Wecallthesevariablespredetermined.Predetermined(lagged)variablesfulfilweakexogeneityinthesensethattheyhavetobeuncorrelatedwiththecontemporaneouserrortermint.WecanstilluseOLStoestimatethefollowingsystemofequations,whichiscalledaVARinreducedform.
(3)
(4)
Thebeautyofthismodelisthatwedon’tneedtopredefinewhetherxoryareendogenous(thedependentvariable).Infact,wecantestwhetherx(y)isendogenousorexogenoususingGrangercausalitytests.TheideaofGrangercausalityisthatpastobservations(laggeddependentvariables)caninfluencecurrentobservations–butnotviceversa.Sotheideaisrathersimple:
thepastaffectsthepresent,andthepresentdoesnotaffectthepast.STATAprovidesGrangercausalitytestsafterconductingaVARanalysis,whichisbasedontestingthejointhypothesisthatpastrealisationsdonotGrangercausethepresentrealisationofthedependentvariable.
Inmanyapplications,VARmodelsmakealotofsense,asacleardirectionofcausalitycannotbepredefined.Forinstance,thereisasubstantialliteratureonthebenefitsofinternationalisation(e.g.enteringforeignmarketthroughcross-borderM&A).Thereisevidencethatmultinationalsoutperformlocalpeersduetothebenefitsofoperatinginmanycountries.Atthesametime,weknowthathigh-performingcompaniesaremorelikelytoenterforeignmarketsduetotheirownershipspecificadvantages.ThisargumentisbasedontheResource-basedViewandtheOLSframeworkdevelopedbyDunningandRugman(ReadingSchoolofInternationalBusiness).TheVARmodelallowsyoutoincorporatebotheffects:
infactyoucantestwhetherperformancedrivesinternationalisationorinternationalisationdrivesperformance.
BeforeyoustartusingaVARmodel,youhavetomakesurethatthetimeseriesarestationary.SothefirststepistocheckwhetherthetimeseriesisstationaryusingDickey-FullertestsandKPSStests.Thesecondstepistospecifytheoptimallaglength(p)ofthemodel.Thisisdonebycomparingdifferentmodelspecificationsusinginformationcriteria.ApartfromusingAkaike(AIC)andBayesianSchwarz(BIC),theHannan-Quinn(HQIC)iscommonlyused.MostappliedeconometriciansfavourtheHannan-Quinn(HQIC)criterion.STATAwillhelpyoutomakeagoodchoice.Afterspecifyingyourmodel,youneedtocheckstabilityconditions.ThecoefficientmatrixofthereducedformVARhastoensurethattheiterationsequenceconvergestoalong-termvalue.STATAwillhelpyouincheckingstability.
Tobeprecise,youneedtoshowthattheeigenvaluesofthecoefficientmatrixliewithintheunitcircle.Thereasonbehinditcanbeonlyunderstoodwhenyouunderstandthemethodofdiagonalizingamatrix.
VARmodelsofferanothernicefeature:
impulseresponsefunctions.VARmodelscapturethedynamicsoftwo(ormore)stationarytimeseries;hence,wecanassessthedynamicimpactofamarginalchangeofonevariableonanother.ThestandardOLSregressionprovidescoefficients,andcoefficientsrefertothepartialimpactofanexplanatoryvariableonthedependentvariable.InthecaseofVARmodels,therelationshipbecomesdynamic,asachangeofonevariable(sayx)intcanaffectxandyint+1.Theimpactonxandyint+1inturnaffectsxandyint+2andsoonuntilt