Local power of likelihood ratio tests for the coin

July14,1997LocalPowerofLikelihoodRatioTestsfortheCointegratingRankofaVARProcessbyPenttiSaikkonenandHelmutLutkepohlDepartmentofStatisticsInstitutfurStatistikundOkonometrieUniversityofHelsinkiWirtschaftswissenschaftlicheFakultatP.O.Box54Humboldt{UniversitySF-00014UniversityofHelsinkiSpandauerStr.1FINLAND10178BerlinGERMANYTel.:[phone]Tel.:[phone]Fax:[phone]Fax:[phone]AbstractLikelihoodratio(LR)testsforthecointegratingrankofavectorautoregressive(VAR)processhavebeendevelopedunderdierentassumptionsregardingdeterministicterms.Forinstance,nonzeromeantermsandlineartrendshavebeenaccountedforinsomeofthetests.Inthispaperweprovideageneralframeworkforderivingthelocalpowerpropertiesofthesetests.Therebyitispossibletoassessthevirtueofutilizingvaryingamountsofpriorinformationbymakingassumptionsregardingthedeterministicterms.OneinterestingresultfromthisanalysisisthatifnoassumptionsregardingthespecicformofthemeantermaremadewhilealineartrendisexcludedthenatestisavailablewhichhasthesamelocalpowerasanLRtestderivedunderazeromeanassumption.WethankRalfBruggemannforperformingthecomputationsandtheDeutscheForschungsgemeinschaft,SFB373,fornancialsupport.01IntroductionFollowingthederivationofafullmaximumlikelihood(ML)analysisofcointegratedGaussianvectorautoregressive(VAR)processesbyJohansen(1988,1991a),likelihoodratio(LR)testsforthecointegratingrankhavebeendevelopedundervarioussetsofassumptions.Themaindierencesintheseassumptionsrelatetothedeterministictermssuchasinterceptandmeantermsaswellaspolynomialtrends.Inparticular,LRtestsforthecointegratingrankhavebeenderivedunderthefollowingconditions:(1)thereisnodeterministictermatall,(2)anintercepttermispresentonlyinthecointegrationrelationsandthereisnolineartrendterm,(3)alineartrendmaybeinthevariablesbutnotinthecointegrationrelations,(4)alineartrendispresentinboththecointegrationrelationsandinthevariables,(5)anadditivelineartrendwithoutanyrestrictionsisaddedtothezeromeancointegratedstochasticpartoftheprocess.AllthesedierentassumptionsresultindierentasymptoticnulldistributionsoftheLRtests.InthisstudywewillderivethecorrespondinglocalpowerpropertiesoftheLRtests.Theseresultsenableustoassessthevalueofincorporatingvaryingamountsofpriorinformationincludedinthedierentsetsofassumptions.Moreover,itisseenwhichfactorsarethecrucialdeterminantsofthelocalpowerofthetests.Animportantresultisalsothatifanintercepttermispresentonlyinthecointegrationrelationsandnolineartrendispresentintheprocessthenatestcanbeconstructedwithidenticallocalpowertoatestderivedunderscenario(1)wherenodeterministictermispresentatall.Forsomeofthescenariosconsideredinthisstudy,Johansen(1991b,1995),Rahbek(1994)andHorvath&Watson(1995)haveperformedlocalpoweranalyses.Ourapproachdiersfromthatusedinthesearticles,however.WewilldevelopageneralframeworkrstinwhichthelocalpoweroftheLRtestscanbereadilyestablished.Thisstudyisstructuredasfollows.Inthenextsectionthemodelset-upisdescribedandtheLRtestsareconsideredinSection3.Sinceallthesetestsmaybeviewedasbeingobtainedfromareducedrank(RR)regressionageneralresultforsuchmodelsisderivedinSection4.InSection5thisresultisusedtoobtainthelocalpoweroftheLRtestsforthecointegratingrankofaVARprocess.ConclusionsaregiveninSection6andproofsarecontainedintheAppendix.Thefollowingnotationisusedthroughout.Thevectoryt=(y1t;...;ynt)0denotesanobservablen-dimensionalsetoftimeseriesvariables.Thelaganddierencingoperatorsare1denotedbyLand,respectively,thatis,Lyt=yt1andyt=ytyt1.ThesymbolI(d)isusedtodenoteaprocesswhichisintegratedoforderd,thatis,itisstationaryafterdierencingdtimeswhileitisstillnonstationaryafterdierencingjustd1times.Thesymbolsd!andp!signifyconvergenceindistributionandprobability,respectively,anda.s.isshortforalmostsurely.O(),o(),Op()andop()aretheusualsymbolsfortheorderofconvergenceandconvergenceinprobability,respectively,ofasequence.Thenormaldistributionwithmean(vector)andvariance(covariancematrix)isdenotedbyN(;).Thesymbolsmax(A),rk(A)andtr(A)signifythemaximaleigenvalue,therankandthetraceofthematrixA,respectively.IfAisan(nm)matrixoffullcolumnrank(nm)weletA?standforan(n(nm))matrixoffullcolumnrankandsuchthatA0A?=0.Foran(mn)matrixAandan(ms)matrixB,[A:B]isthe(m(n+s))matrixwhoserstncolumnsarethecolumnsofAandwhoselastscolumnsarethecolumnsofB.ForasymmetricmatrixAwewriteA0toindicatethatAispositivedenite.The(nn)identitymatrixisdenotedbyIn.LSisshortforleastsquaresandDGPabbreviatesdatagenerationprocess.RRmeansreducedrank.Asageneralconvention,asumisdenedtobezeroifthelowerboundofthesummationindexexceedstheupperbound.2PreliminariesOurpointofdepartureistheDGPofann-dimensionalmultipletimeseriesyt=(y1t;...;ynt)0denedbyyt=0+1t+xt;t=1;2...;(2:1)where0and1areunknown,xed(n1)parametervectorsandxtisanunobservableerrorprocesswithVAR(1)representationinerrorcorrection(EC)formxt=xt1+t;(2:2)wheretiidN(0;),x0=0andisan(nn)matrixofreducedrankr(0rn).Ofcourse,thismodelset-upissimplerthaninmostappliedstudieswithrespecttotheorderoftheprocessandthedistributiono