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

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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