Bayesian Methods for Change-point Detection in Lon

BayesianMethodsforChange-pointDetectioninLong-rangeDependentProcessesBonnieK.Ray∗DepartmentofMathematicalSciencesandCenterforAppliedMathandStatisticsNewJerseyInstituteofTechnologyRueyS.TsayGraduateSchoolofBusinessUniversityofChicagoApril12,2001AbstractWedescribeaBayesianmethodfordetectingstructuralchangesinalong-rangedependentprocess.Inparticular,wefocusonchangesinthelong-rangedependenceparameter,d,andchangesintheprocesslevel,µ.MarkovchainMonteCarlomethodsareusedtoestimatetheposteriorprobabilityandsizeofachangeattimet,alongwithothermodelparameters.Atime-dependentKalmanfilterapproachisusedtoevaluatethelikelihoodofthefractionallyintegratedARMAmodelcharacterizingthelong-rangedependence.Themethodallowsformultiplechangepointsandcanbeextendedtothelong-memorystochasticvolatilitycase.WeapplythemethodtoinvestigateachangeinpersistenceoftheyearlyNileRiverminima.WealsousethemethodtoinvestigatestructuralchangesintheseriesofdurationsbetweenintradaytradesofIBMstockontheNewYorkStockExchangeandtodetectstructuralbreaksindailystockreturnsfortheCocaColaCompanyduringthe1990’s.Keywords:Gibbssampler;Kalmanfilter;longmemory;structuralchange1IntroductionStationaryprocessesexhibitinglong-term,persistentfluctuationshavebeenobservedinmanyareas,includinghydrology,meteorology,economicsandfinance,andtelecommunications;see,forexample,Beran(1994)andBaillie(1996).Acommonlyusedmodelforsuchprocessesistheautoregressivefractionallyintegratedmoving-average(ARFIMA)model,introducedbyGrangerandJoyeaux(1990)andHosking(1981).AccurateestimationofanARFIMAmodeloftenrequiresalargesampleofdatatakenoveralongperiodoftime,whichinturnincreasesthechanceofstructural∗CorrespondingAuthor:BonnieK.Ray,DepartmentofMathematicalSciences,NJIT,Newark,NJ07102;Phone:[phone];Fax:[phone];e-mail:[e-mail]sedbyachangeinthephysicalmechanismthatgeneratesthedataorbyachangeinthewaythatobservationsarecollectedovertime.DuetotheslowlydecayingcorrelationstructureofanARFIMAprocess,teststatisticscommonlyusedforassessingthestabilityofthemodelovertimemayencountersomedifficulties.KuanandHsu(1998)showthat,althoughtheleast-squaresestimatorofachange-pointinmeanisconsistentfordetectingachangeifoneexists,itislikelytosuggestaspuriouschangeifthereisnosuchchange.Wright(1998)showsthattheusualsup-WaldandCUSUMtestsforstructuralstabilitywithunknownpotentialbreakdatefalselyindicateabreakalmostsurelywhenappliedtoapolynomialregressionmodelwithlong-rangedependenterrors.EvenstructuralchangetestsdesignedspecificallyforARFIMAdatawithknownpotentialbreakdatemaystillhavelargesizedistortionsinsmallsamples;seeHidalgoandRobinson(1996).BeranandTerrin(1996)deriveatestthatcanbeusedtodetectasinglechangeinoneoftheparametersofanARFIMAmodel,buttheirmethodisnotapplicablefordetectingachangeinmean.Furthermore,mostcommonlyusedtestsaredesignedtodetectasinglechangepoint,althoughforalongseries,severalstructuralbreaksmaybepresent.McCullochandTsay(1993)giveaBayesianmethodforestimatingrandomlevelandvarianceshiftsinanARtimeseries.Theirmethodisbasedonestimatingtheprobabilityandsizeofashiftateachtimepoint,togetherwithothermodelparameters,usingMarkovchainMonteCarlo(MCMC)techniques.Althoughonlychangesinlevelandvariancearediscussedintheirpaper,themethodisbroadlyapplicable.Forexample,itcanbeusedtotestforachangeinothermodelparameters,suchasanARparameter.Inthispaper,weextendthemethodofMcCullochandTsay(1993)todetectchangesintheparametersofanARFIMAprocess.Weconcentrateondetectingchangesinthelongmemoryparameter,d,andinthemean,µ,althoughasstatedabove,themethodcanbeusedtodetectchangesinanymodelparameter.Themethodislikelihood-basedandcandetectmultiplechangepoints.ArecentpaperbyLiuandKao(1999)proposesanalternativeBayesianmethodforestimatingmultiplechangesinthelong-rangedependentparameterofanARFIMAprocessbasedonareversiblejumpMarkovchainMonteCarloalgorithm.Theirmethodrequiresthatthemaximumnumberofchangesbespecifiedapriori.Theremainderofthepaperisorganizedasfollows.Section2outlinesthemethodanddiscussesitsimplementation.Section3presentstheresultsofasmallsimulationstudy.Section4appliesthemethodtotheseriesofyearlyminimaoftheNilerivertoinvestigatethestabilityofthelong-rangedependentparameter,andtotwofinancialseries,thedurationbetweentradesofIBMstockontheNewYorkStockExchangeandtheabsolutereturnsonCocaColastock,toinvestigatethepresence2oflevelshiftsand/orchangesinpersistence.Section5discussesextensionstothelongmemorystochasticvolatilitymodel.Section6concludes.2DescriptionofMethodLet{yt}beazero-meanlong-rangedependentprocess.Wemodel{yt}asanARFIMA(p,d,q)process(1−φ1B−...−φpBp)(1−B)dyt=(1−θ1B−...−θqBq)at,(1)whereatisaGaussianwhitenoiseprocesswithvarianceσ2aandBdenotesthebackshiftoperator.Thedparameterisarealvaluegoverningtheamountofpersistenceintheprocess.Theprocessisstationaryandinvertibleifallrootsof(1−φ1z−···−φpzp)and(1−θ1B−...−θqBq)lieoutsidetheunitcircleand−0.5d0.5.Itislong-rangedependentwhen0d0.5.AnARFIMAprocesswithmeanµhasaninfiniteordermoving-average(MA)representationoftheformyt=µ+at+∞
i=1ψi(