Estimation of continuous-time models with an

JournalofFinancialEconomics82(2006)227–249Estimationofcontinuous-timemodelswithanapplicationtoequityvolatilitydynamics$GurdipBakshia,NengjiuJub,
,HuiOu-YangcaDepartmentofFinance,SmithSchoolofBusiness,UniversityofMaryland,CollegePark,MD20742,USAbSchoolofBusinessandManagement,HongKongUniversityofScienceandTechnology,ClearWaterBay,HongKongcFuquaSchoolofBusiness,DukeUniversity,Durham,NC27708,USAReceived18August2004;receivedinrevisedform17August2005;accepted23September2005Availableonline9June2006AbstractThetreatmentofthisarticlerendersclosed-formdensityapproximationfeasibleforunivariatecontinuous-timemodels.Implementationmethodologydependsdirectlyontheparametric-formofthedriftandthediffusionoftheprimitiveprocessandnotonitstransformationtoaunit-varianceprocess.Offeringmethodologicalconvenience,theapproximationmethodreliesonnumericallyevaluatingone-dimensionalintegralsandcircumventsexistingdependenceonintractablemultidimensionalintegrals.Density-basedinferencescannowbedrawnforabroadersetofmodelsofequityvolatility.Ourempiricalresultsprovideinsightsoncrucialoutstandingissuesrelatedtotherank-orderingofcontinuous-timestochasticvolatilitymodels,theabsenceorpresenceofnonlinearitiesinthedriftfunction,andthedesirabilityofpursuingmoreflexiblediffusionfunctionspecifications.r2006ElsevierB.V.Allrightsreserved.JELclassification:G11;G12;G13;C15;C32;C52Keywords:Continuous-timemodels;Maximum-likelihoodestimation;Densityapproximation;Equityvolatility;MarketvolatilitydynamicsARTICLEINPRESS:10.1016/j.jfineco.2005.09.005$Wearegratefulforthefeedbackoftherefereewhosedetailedcommentshavesubstantiallyimprovedthispaper.WeacknowledgeusefulcommentsbyYacineAı¨t-Sahalia,DoronAvramov,CharlesCao,PeterCarr,Ren-RawChen,BobDittmar,SteveHeston,GautamKaul,MarkLoewenstein,DilipMadan,BertrandMelenberg,JunPan,MattPritsker,BillSchwert,TylerShumway,GuojunWu,andLiurenWu.SeminarparticipantsatMichiganBusinessSchool,RutgersUniversity,2004EuropeanFinanceAssociationmeetings(Maastricht),and2005ChinaInternationalConferenceinFinanceprovidedseveralconstructivesuggestions.Computercodesusedtoimplementthecontinuous-timemodelsareavailablefromtheauthors.
Correspondingauthor.Tel.:[phone];fax:[phone].E-mailaddress:[e-mail](N.Ju).1.IntroductionWhethertheend-goalismartingalepricingormaximum-likelihoodestimation,theoryinvariablyrequirestheknowledgeofthetransitiondensityoftheeconomicvariable,whichisgenerallyunamenabletoclosed-formcharacterization.Inthissensethelackofanalyticityofthedensityfunctionhashamperedempiricaltestingandthevalidationofalternativehypothesesaboutcontinuous-timemodels.Toremedythisdeficiency,Aı¨t-Sahalia(1999,2002)proposesamethodtoapproximatethetransitiondensityinaone-dimensionaldiffusionsetting.Givenitsutilitytotheresearcherinappliedandtheoreticalwork,thepurposeofourarticleistoexpandontheanalyticaldensityapproachofAı¨t-Sahalia(1999,2002),andourtreatmentrenderstheoriginalmethodfeasibleforasubstantiallylargerclassofone-dimensionalmodels.Basedonthismodificationweempiricallyimplementthedensityapproximationmethodtostudytheplausibilityofgeneralmodelsofequityvolatility.Density-basedinferencesallowustodisentangleissuesconnectedwiththerank-orderingofcontinuous-timevolatilitymodels,thepresenceofnonlinearitiesinthedriftfunction,andthedesirabilityofadoptingmoreflexiblediffusionspecifications.ThemotivationforouranalysistoexpandonAı¨t-Sahalia(1999,2002)derivesfromtwoconsiderations.First,inthecontextofone-dimensionaldiffusionsdXt¼m½Xt
dtþs½Xt
dWt,foreconomic-variableXt,extantdensityapproximationshingeontransform-ingXttoaunit-varianceprocessviaRXdu=s½u
andthenoninvertingRXdu=s½u
.Thisrequirementhasprovedanalyticallychallengingforsomeeconomicmodels(seeBakshiandJu,2005).Whiledevelopingthelikelihoodfunctionofarbitrarymultivariatediffusions,Aı¨t-Sahalia(2003)constructsaTaylorseriessolutionoftheexpansioncoefficients.Suchaprocedureiscapableofdeliveringaclosed-formdensityapproximationevenwhenthemultivariatediffusionisnotreducible.Apotentialtrade-offexistsbetweenthefullyclosed-formirreduciblemethodandourapproach.Second,intheenhanced-methodofAı¨t-Sahalia(1999,2002),therecursivelydefinedcoefficientsthatfulfilltheforwardandbackwardequationhaveamultidimensionalintegraldependenceandareseldomtractableoutsideoftheconstantelasticityofvariancediffusionclass.TheframeworkofourpaperovercomesbothhurdlesassociatedwithimplementingAı¨t-Sahalia(1999,2002).Broadeningtheappealofthemethodologyweshowthatthedensityapproximationcanbederivedwithoutreducingtheprimitiveprocesstoaunit-varianceprocessandwithoutanalyticallyintegratingandinvertingRXdu=s½u
.Thecontributionofourapproachalsoliesindeterminingtherecursivelydefinedexpansioncoefficientsthatexhibitatmostasingleintegraldependenceandconsequentlyaffordstractability.Anadvantageofthisnewapproachisthatitcausesthedensityapproximationtobevirtuallyanalyticalforcontinuous-timemodelswithnonlineardriftanddiffusionfunctionsofthegeneraltypeanalyzedinAı¨t-Sahalia(1996).M