Lag Length Selection in Vector Autoregressive Mode

LagLengthSelectioninVectorAutoregressiveModels:SymmetricandAsymmetricLagsOmerOzcicekBatonRouge,LA70803Phone:[phone]Fax:[phone]email:[e-mail]tofEconomicsLouisianaStateUniversityBatonRouge,LA70603-6306Phone:[phone]Fax:[phone]email:[e-mail]gressive(VAR)modelsarewidelyusedinforecastingandinanalysisoftheeffectsofstructuralshocks.AcriticalelementinthespecificationofVARmodelsisthedeterminationofthelaglengthoftheVAR.TheimportanceoflaglengthdeterminationisdemonstratedbyBraunandMittnik(1993)whoshowthatestimatesofaVARwhoselaglengthdiffersfromthetruelaglengthareinconsistentasaretheimpulseresponsefunctionsandvariancedecompositionsderivedfromtheestimatedVAR.Lütkepohl(1993)indicatesthatoverfitting(selectingahigherorderlaglengththanthetruelaglength)causesanincreaseinthemean-square-forecasterrorsoftheVARandthatunderfittingthelaglengthoftengeneratesautocorrelatederrors.HaferandSheehan(1989)findthattheaccuracyofforecastsfromVARmodelsvariessubstantiallyforalternativelaglengths.MostVARmodelsareestimatedusingsymmetriclags,i.e.thesamelaglengthisusedforallvariablesinallequationsofthemodel.ThislaglengthisfrequentlyselectedusinganexplicitstatisticalcriterionsuchastheAICorSIC.SymmetriclagVARmodelsareeasilyestimated;sincethespecificationofallequationsofthemodelisthesame,estimationbyordinaryleastsquaresyieldsefficientparameterestimates.However,thereisnocompellingreasonfromeconomictheorythatlaglengthsshouldbethesameforallvariablesinallequations.Infact,Hsaio(1981)suggestedestimatingVARsinwhichthelaglengthoneachvariableineachequationcoulddiffer.TheissueofwhetherVARsshouldbeestimatedusingsymmetricorasymmetriclagshasrecentlybeenresurrectedbyKeating(1993;1995).HesuggestedestimatingasymmetriclagVARmodelsinwhichthelaglengthpotentiallydiffersacrossthevariablesinthemodelbutisthesameforaparticularvariableineachequationofthemodel.HedemonstratedthatthispatternofasymmetrycanbederivedfromastructuralrepresentationoftheVARmodelthathasasymmetriclags.IntheKeating-typeasymmetriclagmodel,thespecificationofeachequationisthesame,sotheVARcanbeestimatedusingordinaryleastsquares.Withinthecontextofasmallstructural3VARmodel,Keating(1995)foundthatestimatesoflong-runstructuralparameterstypicallyhadsmallerstandarderrorsforhisasymmetriclagVARthanforasymmetriclagVARandthattheconfidenceintervalsforimpulseresponsefunctionsandvariancedecompositionsforanasymmetriclagVARweresmallerthanforasymmetriclagVAR.GiventhedemonstratedimportanceoflaglengthselectionforVARmodels,theaimofthispaperistoexaminetheperformanceofalternativelagselectioncriteriaforVARmodelsusingMonteCarlosimulations.AlthoughtheperformanceofalternativestatisticalcriterionforlaglengthselectionofsymmetriclagVARshasbeenstudiedby,amongothers,Lütkepohl(1993),theperformanceofstatisticallagselectioncriteriainselectinglaglengthsforasymmetriclagVARmodelshasnotbeenstudied.ThelagselectioncriteriaconsideredincludeAkaike’sinformationcriterion(AIC),Schwarz’sinformationcriterion(SIC),Phillips’posteriorinformationcriterion(PIC),andKeating’s(1995)’sapplicationoftheAICandSICcriterion(KAICandKSIC).1Fourdifferentbivariatelagmodelsareexamined;twoaresymmetriclagmodelsandtwoareasymmetriclagmodels.InthespiritofKennedyandSimons(1991),theparametersofthelagmodelsareestimatedusingtendifferentbivariatemodelsconstructedfromactualeconomicdata.Theremainderofthestudyisnowoutlined.SectionIIofthepaperdescribestheempiricalmethodologywhiletheempiricalresultsarereportedinSectionIII.SectionIVprovidesasummaryandconclusion.II.MethodologyMonteCarlosimulationsof1000drawsareusedtoevaluatetheperformanceofthealternativelagselectioncriterioninbivariateVARmodels.TheVARsaresimulatedusingprespecifiedmodelparametersandlaglengthandarandomnumbergenerator.Thealternativelagselectioncriteriaareevaluatedbycomputingthefrequencydistributionsoflaglengthsselectedbyeachlagselectioncriterion.Fourdifferentbivariatelagmodelsareconsidered.Theout-of-sampleforecastingperformanceofthemodelsselectedbyeachlagselectioncriterionarealso4examined,asistheabilityofeachlagselectioncriteriontogenerateimpulsereponsefunctionsthatmimicthetrueimpulseresponsefunction.ThelagselectioncriterionaretheAIC,SIC,PIC,andKeating’s(1995)’sprocedureusing,alternatively,theAICandSICcriteriontoselectthelagorder.TheAICandSICarewell-knownandthedefinitionofeachispresentedwithoutdiscussion:AIC=ln~∑+2T(numberoffreelyestimatedparameters),andSIC=ln~∑+lnTT(numberoffreelyestimatedparameters)where~∑=estimatedcovariancematrixandT=numberofobservations.Phillips(1994)hasrecentlyproposedanotherlagspecificationcriterioncalledtheposteriorinformationcriterion(PIC).PhillipsderivedthePICfromBayesiananalysisinwhichflatpriorswereimposedontheparameterstofindtheposteriordistributions.ThePICisdefinedas:PIC=ln~~∑+∑⊗′-11TXX,where⊗isthekroneckerproductoperatorandX=matrixofexplanatoryvariables.Keating’sinsightaboutasymmetriclagscanbedevelopedinthefollowingway.Let(1)Φ(L)yCtt=+εbeastructuralmodelwhereyt=Nx1vectorofendogenousvariables,