Fitting an error distribution in some heteroscedas

arXiv:math/0607040v1[math.ST]3Jul2006TheAnnalsofStatistics2006,Vol.34,No.2,994–1012DOI:10.1214/009053606000000191cInstituteofMathematicalStatistics,2006FITTINGANERRORDISTRIBUTIONINSOMEHETEROSCEDASTICTIMESERIESMODELS1ByHiraL.KoulandShiqingLingMichiganStateUniversityandHongKongUniversityofScienceandTechnologyThispaperaddressestheproblemoffittingaknowndistributiontotheinnovationdistributioninaclassofstationaryandergodictimeseriesmodels.TheasymptoticnulldistributionoftheusualKolmogorov–Smirnovtestbasedontheresidualsgenerallydependsontheunderlyingmodelparametersandtheerrordistribution.Toovercomethedependenceontheunderlyingmodelparameters,weproposethattestsbebasedonavectorofcertainweightedresid-ualempiricalprocesses.Underthenullhypothesisandundermini-malmomentconditions,thisvectorofprocessesisshowntoconvergeweaklytoavectorofindependentcopiesofaGaussianprocesswhosecovariancefunctiondependsonlyonthefitteddistributionandnotonthemodel.Undercertainlocalalternatives,theproposedtestisshowntohavenontrivialasymptoticpower.TheMonteCarlocriti-calvaluesofthistestaretabulatedwhenfittingstandardnormalanddoubleexponentialdistributions.TheresultsobtainedareshowntobeapplicabletoGARCHandARMA–GARCHmodels,theoftenusedmodelsineconometricsandfinance.Asimulationstudyshowsthatthetesthassatisfactorysizeandpowerforfinitesamplesatthesemodels.Thepaperalsocontainsanasymptoticuniformexpan-sionresultforageneralweightedresidualempiricalprocessusefulinheteroscedasticmodelsunderminimalmomentconditions,aresultofindependentinterest.1.Introduction.Let{yi:i∈Z:=0,±1,±2,...}beastrictlystationaryandergodicrealtimeseries.Oftenthefinite-dimensionaldistributionsofsuchseriesarecharacterizedbythestationarydistributionandthecondi-tionaldistributionofyi,giventhepast.OneproblemofinterestistofitthisReceivedNovember2003;revisedFebruary2005.1SupportedbytheHongKongRGCGrantsHKUST4765/03HandHKUST6022/05P.AMS2000subjectclassifications.Primary62F05,62M10;secondary60G10.Keywordsandphrases.Nonlineartimeseriesmodels,goodness-of-fittest,weightedempiricalprocess.ThisisanelectronicreprintoftheoriginalarticlepublishedbytheInstituteofMathematicalStatisticsinTheAnnalsofStatistics,2006,Vol.34,No.2,994–1012.Thisreprintdiffersfromtheoriginalinpaginationandtypographicdetail.12H.L.KOULANDS.LINGconditionaldistribution.Ingeneral,thisisadifficultproblem.However,insomespecialtimeseriesmodelswherethisconditionaldistributionisde-terminedbytheinnovationdistribution,itispossibletoobtainreasonableanswers.Inparticular,inthispaperweshallfocusonthegeneralizedau-toregressiveconditionallyheteroscedastic(GARCH)andARMA–GARCHmodels.Todescribethesemodels,letq,rbeknownpositiveintegersandΘ1(Θ2)beasubsetoftheq(r)-dimensionalEuclideanspaceRq(Rr),andletΘ=Θ1×Θ2.Inthemodelsofinterestoneobservestheprocessyisuchthat,forsomesequencesofpastmeasurablefunctionsμifromΘ1toRandhifromΘtoR+:=(0,∞),andforsomeθ′=(θ′1,θ′2),θ1∈Θ1,θ2∈Θ2,ηi:=yi−μi(θ1)phi(θ),i∈Z,(1.1)areindependentlyandidenticallydistributed(i.i.d.)standardizedr.v.’s.Here,“pastmeasurable”meansthat,foreverys:=(s1,s2)∈Θ,s1∈Θ1,thefunctionsμi(s1)andhi(s)shouldbeFi−1measurable,whereFiistheσ-fieldgeneratedby{ηi,ηi−1,...,y0,y−1,...},i∈Z.LetFdenotethecom-mondistributionfunction(d.f.)oftheerrors{ηi},andF0beaknownd.f.Theproblemoffittingtheconditionaldistributionofyi,givenFi−1,inthemodel(1.1)isnowequivalenttotestingthegoodness-of-fithypothesisH0:F=F0vs.H1:F6=F0.Theknowledgeoftheerrordistributionisimportantinstatistics,inpar-ticular,invalueatrisk(VaR).Ineconomicsandfinance,VaRisasinglenumbermeasuringtheriskofafinancialposition.Forexample,whenyiisaprocessofdailyreturns,theVaRforaone-dayhorizonofaportfolioisthe95thconditionalquantileofthedistributionofyi+1,giventheinformationavailableattimei.Afterestimatingtheparameterθ,theVaRforaone-daypositionofyiandprobability0.05isμi(θ1)−1.6449phi(θ),providedtheηi’shavethestandardnormaldistributionandtheestimatedparame-terθiscorrect.Thismeansthat,withprobability0.95,thepotentiallossofholdingthatpositionthenextdayisμi(θ1)−1.6449×phi(θ).Clearly,theknowledgeoftheerrordistributionplaysacrucialroleindeterminingthisprobability,and,hence,inevaluatingVaRviamodel(1.1).Thus,itisimportanttotestthehypothesisH0inpractice.FormoreonVaR,see,forexample,[29].Thegoodness-of-fittestingproblemunderthei.i.d.setuphasalonghis-tory;see,forexample,acollectionofpapersin[7],andreferencestherein.AcommonlyusedtestisbasedontheKolmogorov–Smirnovstatistic.Theprimaryreasonforthisisthatthistestisdistributionfree,thatis,thenulldistributionofthisstatisticdoesnotdependonF0.However,whenthei.i.d.FITTINGERRORD.F.INGARCHMODELS3sequencesuchasηiinthemodel(1.1)isnotobservedandhastobeesti-matedfromaspecialmodel,thispropertyofbeingdistributionfreeislostevenasymptotically.Thesekindsofproblemshavebeenextensivelyinvesti-gatedintheliteratureinvariousmodels,includingregressionmodels;see,forexample,Durbin[9],Loynes[28]andKoul[14],amongothers.Inthecontextoftimeseriesmodels,Boldin[2]observedthatforthezeromeanlinearautoregressive(AR)modelsthetestsbasedontheresidualempiricalprocessareasymptoticallydistributi