时间序列分析及VAR模型.docx

时间序列分析及VAR模型.docx

《时间序列分析及VAR模型.docx》由会员分享，可在线阅读，更多相关《时间序列分析及VAR模型.docx（12页珍藏版）》请在冰豆网上搜索。

时间序列分析及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。

金融时间序列与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+2andsoonuntiltheimpactdiesout.Impulseresponsefunctionsareveryusefulinillustratingtheshort-termdynamicsinamodel.

Let’slookatanexampletoseehowVARmodellingworks.InLecture5,wetriedveryhardtounderstandgoldprices.Weextendourunivariatemodelbyexploringtherelationshipsbetweengoldandsilverprices.Linkingtwo（similar）assetsorsecuritiesisaverycommontradingstrategy,whichiscalledpairs-trading.

Beforewedoanysophisticatedmodelling,itisalwaysbeneficialtolookatsomelinecharts.Figure1showstheindexedtimeseriesofnominalgoldandsilverpricesfrom1900to2010.

Figure1:

Nominalgoldandsilverprices,indexed,1900-2010

Wecanseethatthereisacertaindegreeofco-movement,whichwemightbeabletoexploitforourtradingstrategy.BeforewecanuseVAR,weneedtoensurethatbothtimeseriesarestationary.ItisobviousfromFigure1thatgoldandsilverpricesarenotstationary.However,aftertakingafirst-differencewecanshowthatpricechangesarestationary.SobothtimeseriesareI

（1）.

Thenextstepistodeterminetheoptimallaglengthusinginformationcriteria.Table1showsdifferentspecificationsusingthevarsoccommand.

Table1:

Determiningtheoptimallaglengthusinginformationcriteria

BasedontheAICandHQIC,twolagsareoptimal;however,the（S）BICprefersonlyonelag.IwouldpreferHQICandtrytwolagsfirst.Ifthesecondlagdoesnotexhibitsignificantcoefficient,wecouldtrytoreducethelaglengthinlinewith（S）BIC.

WerunaVARwithtwolagstoexplaincurrentpricechangesingoldandsilver.Table2providestheOLSestimates.

Table2:

VARmodelwithtwolags

Weseethatsilverprices（lag2）affectcurrentgoldprices,andwecanestablishautocorrelationinbothtimeseries.TotestwhethergoldGrangercausessilverorviceversa,werunGrangercausalitytestsreportedinTable3.

Table3:

Grangercausalitytests

Hence,weconfirmthatpastchangesinsilverpricescanpredictfuturegoldpricechanges.Thisisveryinteresting,asitcanbeusedtodevelopatradingstrategy.Finally,weneedtoshowthattheVARisstable（seeTable4）.

Table4:

StabilityconditionoftheVAR

Finally,wecanillustratetheimpactofsilverpricechangesonfuturegoldpricechangesusinganimpulseresponsefunction.Figure2showstheimpulseresponsefunctionandconfidenceintervalsderivedfrombootstrapping.Ifsilverpricesincreasetodayby1%,weshouldexpectasignificantdeclineingoldpricesintwoyearsby0.2%.

Figure2:

Impulseresponsefunction

6.3Cointegration

WhenweexploreFigure1abitmorecarefully,wecanseethatsilverandgoldpricesexhibitacertaindegreeofco-movement.Wecouldalmostarguethattheyshareacommonstochastictrend.ThelimitationofARIMAandVARmodelsisthattheycanbeonlyusedifthetimeseriesarestationary.Inourcase,wehadtofirst-differenceyourtimeseriestoensurestationarity.First-differencingeliminatesalotofinformationinthetimeseries.Istherenobetterwaytoanalysegoldandsilverprices.

Longbeforethedevelopmentofmultivariatetimeserieseconometrics,peoplerealisedthatgoldandsilverseemtohaveacommonmovementaroundalong-termequilibrium（gold-silverpriceratio）.Moreover,theideaofequilibriumconditionsineconomicsandtheavailabilityofmacroeconomictimeseriesledtothedevelopmentofcointegrationanalysis.

Theideaisverysimple.Eveniftwo（ormore）timeseriesarenon-stationaryandhencehavestochastictrends,theymightbestilldrivenbythesameunderlyingfactorsthatleadtotheirstochasticbehaviour.Therefore,weanalysethetimeseriesinlevelsandseewhetherwecanfindalong-termequilibrium–aso-calledcointegratingvector.

BeforeweexploretheJohansenprocedure,let’slookatthegold-silverratioovertimeshowninFigure3.

Figure3:

Thegold-silverratio,1900-2010

Theratiolookslikeamean-revertingprocess;thus,inthelongrunittendstogobacktoitslong-termequilibrium（mean）.Basedontheratio,wecouldarguethatgoldseemstobeovervaluedcomparedtosilveratthemoment.

Ofcourse,takingtheratiosuggestsaverysimplecointegratingvector–infactweassumeaone-to-onerelationship.BeforewecanusetheJohansenprocedure,wehavetomakesurethatthetimeserieshavethesameorderofintegrationI（p）.WealreadyknowthatgoldandsilverpricesarebothI

（1）timeseries.Table5showstheresultsoftheJohansentestforcointegration.InlinewiththeVARmodel,weusetwolags.

Table5:

Johansentest

Thenullhypothesisthatthereisnocointegration（r=0）canberejectedifweusethetracestatistic.However,thenullhypothesisthatwehaveonecointegratingvector（r=1）cannotberejected.Theproblemisthatthemax-lambdastatisticdoesnotsupportcointegration.Ialsotriedlog-pricesinstead,wh