long-run equity

Howusefularehistoricaldataforforecastingthelong-runequityreturndistribution?JohnM.MaheuandThomasH.McCurdy¤Forthcoming,JournalofBusinessandEconomicStatisticsAbstractWeprovideanapproachtoforecastingthelong-run(unconditional)distrib-utionofequityreturnsmakingoptimaluseofhistoricaldatainthepresenceofstructuralbreaks.Ourfocusisonlearningaboutbreaksinrealtimeandassess-ingtheirimpactonout-of-sampledensityforecasts.Forecastsuseaprobability-weightedaverageofsubmodels,eachofwhichisestimatedoveradi®erenthistoryofdata.Theempiricalresultsstronglyrejectignoringstructuralchangeorusinga¯xed-lengthmovingwindow.Theshapeofthelong-rundistributionisa®ectedbybreakswhichhasimplicationsforriskmanagementandlong-runinvestmentdecisions.keywords:densityforecasts,structuralchange,modelrisk,parameteruncertainty,Bayesianlearning,marketreturns¤Maheu([e-mail]),DepartmentofEconomics,UniversityofTorontoandRCEA;McCurdy([e-mail]),JosephL.RotmanSchoolofManagement,UniversityofToronto,andAssociateFellow,CIRANO.WearegratefultotheEditor,ArthurLewbel,anAssociateEditorandtwoanonymousrefereesformanyhelpfulandconstructivesuggestions.WethankBillSchwertforprovidingequityreturndataforthe1885-1926period,andGregBauer,RobEngle,DavidGoldreich,StephenGordon,EricJacquier,MarkKamstra,LisaKramer,JanMahrt-Smith,LubosPastor,NickPolson,LukaszPomorski,JeroenRombouts,MikeVeall,BenjaminVerschuere,KevinWang,aswellasseminarparticipantsattheCIREQ-CIRANOFinancialEconometricsconference,the(EC)2conferenceIstanbul,theNorthernFinanceAssociationannualmeetings,theBankofCanada,HECMontreal,McMasterUniversityandYorkUniversityformanyhelpfulcomments.LoisChanprovidedexcellentresearchassistance.WearealsogratefultotheSSHRCfor¯nancialsupport.11IntroductionForecastsofthelong-rundistributionofexcessreturnsareanimportantinputintomany¯nancialdecisions.Forexample,Barberis(2000)andJacquier,Kane,andMarcus(2005)discusstheimportanceofaccurateestimatesforlong-horizonportfoliochoice.Ourpapermodelsandforecaststhelong-run(unconditional)distributionofexcessreturnsusinga°exibleparametricdensityinthepresenceofpotentialstructuralbreaks.Ourfocusisonlearningaboutbreaksinrealtimeandassessingtheirimpactonout-of-sampledensityforecasts.Weillustratetheimportanceofuncertaintyaboutstructuralbreaksandthevalueofmodelinghigher-ordermomentsofexcessreturnswhenforecastingthereturndistributionanditsmoments.Theshapeofthelong-rundistributionandthedynam-icsofthehigher-ordermomentsarequitedi®erentfromthosegeneratedbyforecastswhichcannotcapturestructuralbreaks.Theempiricalresultsstronglyrejectignoringstructuralchangeinfavorofourforecastswhichweighthistoricaldatatoaccommodateuncertaintyaboutstructuralbreaks.Wealsostronglyrejectthecommonpracticeofusinga¯xed-lengthmovingwindow.Thesedi®erencesinlong-runforecastshaveim-plicationsformany¯nancialdecisions,particularlyforriskmanagementandlong-runinvestmentdecisionssuchasthosebyapensionfundmanager.Existingworkonstructuralbreakswithrespecttomarketexcessreturnshasfocusedonconditionalreturndynamicsandtheequitypremium.ApplicationstotheequitypremiumincludePastorandStambaugh(2001)andKim,Morley,andNelson(2005)whoprovidesmoothedestimatesoftheequitypremiuminthepresenceofstructuralbreaksusingadynamicrisk-returnmodel.Inthisenvironment,modelestimatesarederivedconditionalonamaintainednumberofbreaksin-sample.Thesepapersfocusontheposteriordistributionofmodelparametersforestimatingtheequitypremium.LettauandvanNieuwerburgh(2007)analyzetheimplicationsofstructuralbreaksinthemeanofthedividendpriceratioforconditionalreturnpredictability;Viceira(1997)investigatesshiftsintheslopeparameterassociatedwiththelogdividendyield.PayeandTimmermann(2006)andRapachandWohar(2006)presentevidenceofinstabilityinmodelsofpredictablereturnsbasedonstructuralbreaksinregressioncoe±cientsassociatedwithseveral¯nancialvariables,includingthelaggeddividendyield,shortinterestrate,termspreadanddefaultpremium.Additionalworkonstructuralbreaksin¯nanceincludesPesaranandTimmermann(2002)whoinvestigatewindowestimationinthepresenceofbreaks,PettenuzzoandTimmermann(2005)whoanalyzethee®ectsofmodelinstabilityonoptimalassetallo-cation,Lettau,Ludvigson,andWachter(2007)whofocusonaregimechangeinmacro-economicrisk,AndreouandGhysels(2002)whoanalyzebreaksinvolatilitydynamics,andPesaran,Pettenuzzo,andTimmermann(2006b)whoexplorethee®ectsofstructuralinstabilityonpricing.Toourknowledge,noneoftheexistingapplicationsstudythee®ectsofstructural2changeonforecastsoftheunconditionaldistributionofreturns.Anadvantagetoworkingwiththelong-rundistributionisthatitmaybelesssusceptibletomodelmisspeci¯ca-tionthanshort-runconditionalmodels.Forexample,anunconditionaldistributionofexcessreturnscanbeconsistentwithdi®erentunderlyingmodelsofrisk,allowingustominimizemodelmisspeci¯cationwhilefocusingontheimplicationsofstructuralchange.Wepostulatethatthelong-runorunconditionaldistributionofreturnsisgeneratedbyadiscretemixtureofnormalssubjecttooccasionalbreaksthataregovernedbyani.i.d.Bernoullidistribution.Thisimpliesthatthelong-rundistributionistim