版权说明:本文档由用户提供并上传,收益归属内容提供方,若内容存在侵权,请进行举报或认领
文档简介
FinanceandEconomicsDiscussionSeries
FederalReserveBoard,Washington,D.C.
ISSN1936-2854(Print)
ISSN2767-3898(Online)
FactorSelectionandStructuralBreaks
SiddharthaChibandSimonC.Smith
2024-037
Pleasecitethispaperas:
Chib,Siddhartha,andSimonC.Smith(2024).“FactorSelectionandStructuralBreaks,”FinanceandEconomicsDiscussionSeries2024-037.Washington:BoardofGovernorsoftheFederalReserveSystem,
/10.17016/FEDS.2024.037
.
NOTE:StafworkingpapersintheFinanceandEconomicsDiscussionSeries(FEDS)arepreliminarymaterialscirculatedtostimulatediscussionandcriticalcomment.TheanalysisandconclusionssetfortharethoseoftheauthorsanddonotindicateconcurrencebyothermembersoftheresearchstafortheBoardofGovernors.ReferencesinpublicationstotheFinanceandEconomicsDiscussionSeries(otherthanacknowledgement)shouldbeclearedwiththeauthor(s)toprotectthetentativecharacterofthesepapers.
FactorSelectionandStructuralBreaks
SiddharthaChiba,SimonC.Smithb
aOlinSchoolofBusiness,WashingtonUniversityinSt.LouisbFederalReserveBoard
Draft:May31,2024
Abstract
Wedevelopanewapproachtoselectriskfactorsinanassetpricingmodelthatallowsthesettochangeatmultipleunknownbreakdates.UsingthesixfactorsdisplayedinTable1since1963,wedocumentamarkedshifttowardsparsimoniousmodelsinthelasttwodecades.Priorto2005,fiveorsixfactorsareselected,butjusttwoareselectedthereafter.Thisfindingoffersasimpleimplicationforthefactorzooliterature:ignoringbreaksdetectsadditionalfactorsthatarenolongerrelevant.Moreover,allomittedfactorsarepricedbytheselectedfactorsineveryregime.Finally,theselectedfactorsoutperformpopularfactormodelsasaninvestmentstrategy.
Keywords:Modelcomparison,Factormodels,Structuralbreaks,Anomaly,Bayesiananal-ysis,Discountfactor,Portfolioanalysis,Sparsity.
JELclassifications:G12,C11,C12,C52,C58
Emailaddresses:chib@wustl.edu(SiddharthaChib),simon.c.smith@(SimonC.Smith)
WethankDanieleBianchiandAndyNeuhierl.Anyremainingerrorsareourown.TheviewsexpressedinthispaperarethoseoftheauthorsanddonotnecessarilyreflecttheviewsandpoliciesoftheBoardofGovernorsortheFederalReserveSystem.CenterforResearchinSecurityPricesdatawereobtainedbySiddharthaChibunderthepurviewofWashingtonUniversitylicenses.
1
1.Introduction
“USsmall-capstocksaresufferingtheirworstrunofperformancerelativetolargecompaniesinmorethan20years[...]TheRussell2000indexhasrisen24%sincethebeginningof2020,laggingtheS&P500’smorethan60%gainoverthesameperiod.Thegapinperformanceupendsalong-termhistoricalnorminwhichfast-growingsmall-capshavetendedtodeliverpunchierreturnsforinvestorswhocanstomachthehighervolatility.
”1
(FinancialTimes,2024)
Theempiricalliteratureonassetpricinghasproposedahugenumberoffactorsthatclaimtoexplainthecross-sectionofexpectedstockreturns(
Cochrane
2011
).Morerecently,thefieldhasbeendealingwithhowtohandlethisproliferationoffactors.Variouspotentialsolutionshavebeenoffered(
Fengetal.
2020
).
Thispaperpresentsanintuitivelysimplepointofviewthathassomehowbeenoverlookedintheliterature.Ifthesetoffactorsthatexplainthecrosssectionofexpectedreturnsisvaryingovertime,itiscriticaltoaccountforthisfeaturewhenevaluatingwhichfactorsarerelevantatanygiventime
.2
Otherwise,usingallavailablehistoricaldatawilltendtopickupfactorsthatwereimportantatsomepointinthepastbutarenotriskfactorsatpresent.Asasimpleexample,imaginethatonlytwofactorsarerelevantforthefirsthalfofthesampleandthattwodifferentfactorsarerelevantinthesecondhalf.Thecommonapproachintheliteratureofusingallthehistoricaldatawilltendtosuggestthatallfourfactorsarerelevantfortheentiresample,wheninfactnomorethantwoarerelevantatanygiventime.Thismaypartlyexplaintheproblemofthe“factorzoo”(
Harveyetal.
2016
;
1ThisquoteisfromaMarch27,2024FinancialTimesarticleentitled‘USsmall-capssufferworstrunagainstlargerstocksinmorethan20years.’
2Forexample,thepublicationeffectof
Schwert
(2003),and/ortheadaptiveefficientmarkethypothesis
of
Lo
(2004),maycausethesetofriskfactorstochange
.Thesetofriskfactorsmayalsochangedue,forexample,tothetechnologicalrevolutioninfinancialmarketstowardstheendofthetwentiethcentury,shiftingmonetarypolicyregimesthatledtotheanchoringofinflationexpectations,orregulatorychanges.
2
Houetal.
2020
),aswellasthedecliningperformanceofriskfactorsinacomprehensivesetofanomalies(
McLeanandPontiff
2016
).Therefore,itisimportanttoconsidertimevariationwhenselectingfactors.
Ifoneknewthetimeatwhichthesetoffactorschanges,onecoulddiscardtheoldirrel-evantdatawithasubsamplesplit.Inreality,however,thisdateisnotknownandthereforemustbeestimated
.3
Furthermore,thelongerthesampleperiodunderconsideration,themorelikelyitisthattheremaybemultipletimesatwhichthesetchanges,whichfurthercomplicatestheproblem.Thissettingistechnicallychallengingbecauseoneneedstoes-timateboththetimesatwhichthesetofrelevantfactorschangesandthesetofrelevantfactorswithineachsubperiod.Inotherwords,boththeassetpricingmodelandtheparame-tersofthatmodelchange
.4
Inthispaper,weproposeasolutiontothischallengingproblembydevisingthefirstmethod(Bayesianorfrequentist)thatcansimultaneouslyestimateboththetimesatwhichthemodelchangesandhowtheparametersofthemodelchange,takingtheguessworkoutofhowtodeterminethesubsamplesplits(orregimes).
Ourmethodologygeneralizestheframeworkof
ChibandZeng
(2020)
–whodevelopedaBayesianmodelselectionapproachfortime-invariantfactorselection–byblendingitwiththeBayesianbreakpointapproachinthecontextofmodeluncertaintydevelopedby
Chib
(2024),producingasingleunifiedframeworkwhichestimatestheselectedriskfactorsand
allowsthisselectedsettochangeatmultipleunknownbreakdates.NotethataBayesianapproachiswellsuitedtothisproblembecauseitcanallowforbothabruptandgradualchanges,dependingontheuncertaintysurroundingthebreakdate.ABayesianapproach
3
Greenetal.
(2017),forexample,imposeapredeterminedsubsamplesplitintheearly2000sandfind
thatthenumberofrelevantcharacteristicshasdeclinedovertime.
4Thissettingismorecomplexthanstandardbreakpointproblemsinwhichthemodelparametersshiftafterabreakbutthemodelitself(i.e.theselectedfactors)remainsunchanged.Awidelyappliedapproachforthissettingwasdevelopedin
Chib
(1998),firstappliedinthefinancesettingby
P´astorandStambaugh
(2001)andsubsequentlyinmanyotherpapers
.Standardbreakpointproblemshavebeenappliedtoarange
ofissuesinempiricalassetpricing,suchasreturnpredictability(Viceira
1997;
LettauandVanNieuwerburgh
2008;
Rapachetal.
2010;
SmithandTimmermann
2021
),estimatingtime-varyingriskpremia(
P´astorand
Stambaugh
2001;
SmithandTimmermann
2022),anddatingtheintegrationofworldequitymarkets(Bekaert
etal.
2002
).
3
alsoinherentlyprotectsagainstproblemsassociatedwithmultipletests(
Kozaketal.
2020
;
Jensenetal.
2023
;
Bryzgalovaetal.
2023
).Weperformanexhaustivesearchacrossallpossibleassetpricingmodelsimpliedbythestartingsetofriskfactorsandallpossiblebreakdatesforagivennumberofbreaks,identifyingtheoptimalsubsetofpotentialfactorsthatcanpricemost(ifnotall)oftheremainingfactorsineachregime
.5
Ourexhaustivesearchcircumventstheriskofgettingstuckatlocalmaximiathatisassociatedwithstochasticsearchalgorithms
.6
Inourempiricalanalysis,wefocusonthesix-factormodelof
FamaandFrench
(
2018
)
.7
UsingmonthlydatafromJuly1963throughDecember2023,ourmethodidentifiesthreebreakscorrespondingtoaregimelastingfor15yearsonaverage
.8
ThebreaksoccurinMarch1975,October1995,andSeptember2005
.9
Thesetofriskfactorschangesaftereachofthesebreaks.Atleastfivefactorsareselectedinthefirstthreeregimes(upto2005),whileonlytwofactors(marketandprofitability)areselectedinthefinalregime(post-2005)
.10
Incontrast,thepreferredmodelwhenusingallhistoricaldataisafour-factormodelthatexcludessizeandvaluewhichshowsthatfailingtodiscardpre-breakdatacanleadtoariskfactorsetbeingselectedthatisnottherelevantoneforpricinginthecurrentregime
.11
5Ourmethodalsoperformsinferenceoverthenumberofbreaks.
6Whiletheconventionalapproachtotestthepricingabilityofrisk-factorsistousevarioustestassetsorportfolios,following
ChibandZeng
(2020)weleveragetheintuitionthatifasubsetoftheavailablefactorsare
foundtoberiskfactors,thenthosefactors,byvirtueofbeingriskfactors,shouldpricethecomplementarysetofnonriskfactors.
7Themodelscanisthereforeover63models,includingthepopularrisk-factorcollectionssuchasthe3-and5-factorFama-Frenchmodels,butitalsoincludesallothercombinationsofrisk-factorsthathavenotpreviouslybeenconsidered.
8Weconsiderothernumbersofbreaks,butfindthreetobeoptimal.
9Thebreakin1975correspondstotheoilpriceshocksofthe1970sandthecorrespondinghighinflation-aryperiodthatwasonlystoppedwhenasharpcontractionarymonetarypolicyregimewassubsequentlyimplemented.October1995coincideswiththeInternetrevolutionandthetechboomontheNASDAQ
(Griffinetal.
2011
).Thisbreakalsocoincideswithaperiodofdramaticchangesinmarketefficiencythathasbeendocumentedby
Chordiaetal.
(2011)
.TheSeptember2005breakcorrespondstoalittlebeforetheonsetoftheGlobalFinancialCrisis.
10Allbutthesizefactorareselectedinthefirstregime(1963-1975),allsixfactorsareselectedinthesecond(1975-1995),andallbutthevaluefactorareselectedinthethird(1995-2005).
11Thisselectedmodelisunabletopriceoneoftheomittedfactors–size–usingthewholesampleofavailabledata,highlightingitsshortcomings.Furthermore,usingtheentiredatasample,ourapproach
4
Moreover,themediannumberoffactorsselectedinthebestperformingtenmodelsin
thethreeregimesupto2005isfive,butthisfallsto2.5inthefinalregime.Thisclearlyindicatesashifttomoreparsimoniousmodelsinthemostrecenttwodecades
.12
Infact,theCapitalAssetPricingModel(CAPM)–whichperformspoorlyupto2005–isinthetoptenmodelsafter2005andoutperformsthe3-and5-factormodelsofFama-French(andbothofthosemodelsplusmomentum).
Ineveryregime,eachoftheomittedfactorsispricedbytheselectedfactors,suggestingthattheyarespannedbythesmallersubsetofselectedfactorsandcanthereforebeconfi-dentlyexcluded.Post-2005,constructingthetangencyportfoliothatconsistsoftheselectedfactorsandtheindividualstocksthatarenotpricedbythosefactorsgeneratesaSharperatioof2.74.ThisismuchhigherthanthecorrespondingSharperatios(whichrangefrom0.87to1.82)generatedfromthe3-and5-factormodelsofFama-French,thesametwomodelsplusmomentum,andtheCAPM.Thetworiskfactorsthatourprocedurehasisolatedsince2005–marketandprofitability–captureimportantsystematicrisks.Theroleofthemarketfactorasasystematicriskfactorisarguablyunquestioned.Theprofitabilityfactorcapturesthepartofthecrosssectionofexpectedreturnsthatcovarieswithprofitability.Inaddition,ourmethodologywouldbeusefulfordetectinganychangeinthecurrentsetofriskfactorsinthefuture.
Finally,ourmethodologyprovidesregime-specificestimatesoffactorriskpremiaandtheirpriceofrisk
.13
Mountingempiricalevidenceofsizeableriskpremiaassociatedwiththesefactorshasimportantimplicationsforinvestmentstrategiesandhasmarkedlychangedtheinvestmentlandscape,leadingtotheproliferationofmutualfundsspecializingincertaininvestmentstylessuchassmallcapsorvaluestocks.Theappealofsuchstrategiesisnot
revealsthatthemomentumfactorisnotpricedbytheFama-French5-factormodel;andthemomentum,investment,andprofitabilityfactorsarenotpricedbytheFama-French3-factormodel.
12
Kellyetal.
(2019)useInstrumentedPrincipalComponentsAnalysistodocumentthatjustfivelatent
factorscanoutperformexistingfactormodels.
13Asmallsubsetofstudiesthatestimatetime-varyingriskpremiainclude
FersonandHarvey
(1991);
Freybergeretal.
(2020),
Guetal.
(2020),
Gagliardinietal.
(2016),
AngandKristensen
(2012),and
Adrian
etal.
(2015)
.
5
onlydependentonthemagnitudeoftheassociatedriskpremia,butalsoonthestabilityof
theirriskpremiaovertime
.14
Wefindcleartime-variationintheriskpremiaforallsixfactorssince1963.Forexample,thevaluepremiumwas5.6%from1963to1975butdecreasedto4.3%from1975to1995(
FamaandFrench
2021
).Since1995,thevaluefactorhasnotbeenselectedasariskfactor.TheimpliedweightsonthevaluefactorinthemaximumSharperatioportfoliothereforedeclinedfrom18percent(1963-1975)to15percent(1975-1995)andhavebeenzerosince.Thisindicatesthathighallocationstovaluestockshavebecomenotablylessattractiveovertime.
Bessembinderetal.
(2021)estimatefactorriskpremiausingafixed60-monthrolling
windowanddocumentcleartime-variationinthenumberoffactorsselectedovertime.However,asweshowinourempiricalanalysis,arollingwindowleadstofactorsenteringandexitingtheSDFveryfrequently,sometimesonamonthlybasis.Theeconomicmotivationforthisbehavior,however,isdifficulttojustify.Thisiswhyaformalmethodisneededtoidentifythesetofriskfactorsthatisstablewithinaregime,butisallowedtoshiftoccasionallyovertime.Wepresentthefirstapproach(eitherBayesianorfrequentist)todoso
.15
Therestofthepaperisorganizedasfollows.InSection2wedetailourmethodology.InSection3wepresentevidenceofbreaksandtheregime-specificselectedfactorsandtheirriskpremiaestimates.Section4hasthepricingperformanceandinvestmentimplicationsofourselectedfactorcollection,andSection5concludes.
14Factorpremiamaytime-varyduetoinvestorsdifferinginsophisticationorinvestmentobjectives,en-ablingthemarginalinvestortodifferacrossstocksandovertimeforagivenstock.Individualinvestorscanformmean-varianceportfolios,whileothersmaypursueverylargepayoffs.Someinvestorsmaypursue“buy-and-hold”strategies,andothersmayperiodicallyrebalancetotargetcertainweights.
15
Bianchietal.
(2019)alsodocumentevidenceoftime-varyingsparsityinfactormodels
.
6
2.Methodology
Wenowsetouttheeconomicmotivationforbreaksintheriskfactormodel.Then,tobuildintuition,weexplainhowthemethodologyworksfortheno-breakandsingle-breakcases,beforeexplainingourmethodologyforthemostgeneralcaseinwhichthesubsetofriskfactorscanshiftacrossanunknownnumberofbreaksthatoccuratunknowntimes.Finally,wedetailourpriorspecification.
2.1.EconomicSourcesofBreaksintheFactorModel
Formally,supposethatforatimeseriessamplefromt=1,...,T,wehavedata{ft},t≤TonasetofK(potential)riskfactors.Supposethatthestochasticdiscountfactor(SDF)attimetisgivenby
Mt=1−b′(ft−λ)
wherebisthevectorofmarketpricesoffactorrisksandλisthevectoroffactorrisk-premia.Inanenvironmentwheretheunderlyingfirm-levelproductionfunctionissubjecttobreaks,duetotechnologicalinnovations,itismoreappropriatetoassumethatfirm-levelprofitabilitywoulddependonatime-varyingsetoffirm-levellaggedcharacteristics.Inthissituation,theSDFwouldbemoreappropriatelycharacterizedbyatime-varyingSDF
Mt=1−b(ft−λt)
wherethemarketpricesandfactorrisk-premiaaretime-varying.Ifweimaginethatsomeofthelaggedcharacteristicsthatdeterminefirm-levelprofitabilityceasetobesignificantforperiodsoftimeduetochangesinpersistentshocks(innovations)toproduction,thiswouldimplythatsomeoftheelementsinthemarketpricevectorbtwouldbezeroandthecorrespondingelementsofftwoulddropoutoftheSDF,i.e.,ceasetoberiskfactors.
7
Todescribethissituation,letxt⊆ftdenoteasubsetofftwithnon-zeromarketpricesoffactorrisks.Supposethatthemarketpricesbtchangeatunknownbreakdates
1<t<t<···<t<T(1)
wherem(thenumberofbreaks)isalsoanunknownparameter.Inparticular,adifferentsetofriskfactorsenterstheSDFineachregimeandthusthereare(m+1)riskfactorsets
'''
(''xt≤t
''xt−1<t≤t
''
('x+1t<t≤T.
Theobjectivesoftheanalysisaretofind
•thenumberofbreaksm∈{0,1,2,...,M}
•thetimingofthebreaks,t,...,t
•andtheriskfactorsineachregimex,...,x+1.
Wenowoutlinetheframeworkdevelopedby
ChibandZeng
(2020)tofindriskfactorsin
theabsenceofbreaks.Wethengeneralizetheirframeworktofindriskfactorswithasinglebreakinthemarketpricevector(tohelpbuildintuition)andthenconsidertheextensiontomultiplebreaks(whichwesubsequentlytaketothedata).
2.2.Nobreaks
ChibandZeng
(2020)developaBayesianmodelscanningapproachtodeterminewhich
subsetofpotentialriskfactorsenterstheSDF.Todothis,theyexploitthefactthatasset
8
pricingtheoryplacesrestrictionsonthejointdistributionoffactorsthatentertheSDFand
thosethatdonot.Onekeyrestrictionisthatthenon-riskfactorsshouldbepricedbytheriskfactors.Onecanthereforeconstructallpossibledecompositionsofthejointdistributionoffactorsintermsofamarginaldistributionoftheriskfactorsandaconditionaldistributionofthenon-riskfactors(imposingthepricingrestrictiononthelatter)anddeterminebyBayesianmarginallikelihoodswhichsuchdecompositionisthebest
.16
Theriskfactorsinthatbestdecompositionarethentakentobetheriskfactorsbestsupportedbythedata.
Toisolatethebestsetofriskfactors,considerallpossiblesplitsofftintoxt,theriskfactors,andyt,thenon-riskfactors.ThesesplitsproducemodelsthatweindicatebyMj,forj=1,...,J=2K−1.Attimet,thedatageneratingprocessunderMjisgivenby
xj,t=λj+uj,t
yj,t=Γjxj,t+εj,t,t=1,...,T,(3)
wheretheerrorsaredistributedasmultivariateGaussian
uj,t~N(0,Ωj),εj,t~N(0,Σj).(4)
Lettheunknownparametersinthismodelbedenotedby
θj=(λj,Ωj,Γj,Σj).(5)
Notethateachofthesemodelshasadistinctsetofriskfactorsandadistinctsetofparam-eters.
Apartfromλj,theprioroftheparametersΩj,Γj,Σjarederivedbychange-of-variablefromasingleinverseWishartpriorplacedonthematrixΩjinthemodelwhereallfactors
16MarginallikelihoodsareBayesianobjectsthatarecalculatedbyintegratingouttheparametersfromthesamplingdensitywithrespecttotheprioroftheparameters.
9
arerisk-factors.ThehyperparametersofthissingleinverseWishartdistribution,andthoseofthemodel-specificλj,arecalculatedfromatrainingsample(whichwetaketobethefirst15%ofthesampledata).Thetrainingsampledataaresubsequentlydiscarded,whichmeansthatitisnotusedforestimationormodelcomparisonpurposes.
Letπ(θj)denotetheprioronθj.Then,themarginallikelihoodofMjisgivenby
marglik(f|Mj)=∫N(xj|λj,Ωj)N(yj|Γjxj,Σj)dπ(θj),j≤J.(6)
Theseareclosedformasshownin
Chibetal.
(2020)
.However,theirapproachassumesthatthesetofriskfactorsistime-invariant.
2.3.Singlebreak
Assumefornowthecaseofasinglebreak.Thisbreakoccursatanunknownlocationtthat
separatesthesampledataintoregimess∈{1,2}.Asetofriskfactors(x)enterstheSDF
inthefirstregime(fromtimeperiodst=1,...,t)andanotherset(x)entersinthesecond
regime(fromtimeperiodst=t+1,...,T)
.17
Theobjectiveistoestimatethetimingof
thebreak(t)andtheidentitiesoftheriskfactorsinthefirstregime(x)andthesecond
(x)regime.
Toinferthebreakdate,wefocusonthequantity
marglik(f1,t,ft+1,T|t)(7)
whichisthemarginallikelihoodofthedatasegmentedbythebreakdate.Wecalculatethisquantityonalargegridofpossiblebreakdatesandchoosethebreakdatewiththelargestvalueofthismarginallikelihood.
17Theriskfactorsetisstablewithineachregime.
10
Theproblemincalculatingtheprecedingquantityisthatwedonothavethedata-
generatingprocess(DGP)oneithersideofthesplit.Inotherwords,wedonotknowtheidentityofriskfactorsbeforeandafterthesplit.Todealwiththistwo-waymodeluncertainty,weconsiderallpossibledivisionsofftintoxtandyt,oneithersideoft.Ontheleft,wedenotethemodelsbyMj,1andontherightbyMk,1,for(j,k)=1,...,J=2K−1.Whenj=kthesplitsareidenticalbuttheparametersofthemodelaredifferent.JustaswedidinEquation(
3
),thejthmodelinregimes,s=1,2takestheform
xj,t,s=λj,s+uj,t,s
yj,t,s=Γj,sxj,t,s+εj,t,suj,t,s∼N(0,Ωj,s)
εj,t,s∼N(0,Σj,s),t∈Ts,1,s=1,2,(8)
whereT1,1=(1,2,...,t)andT2,1=(t+1,...,T).Wedenotetheunknownparametersinthesemodelsbyθj,s=(λj.s,Ωj,s,Γj,s,Σj,s).Notethateachofthesemodelshasadifferentsetofriskfactorsandadistinctsetofparameters,andbecausewehaveabreak,theseparametersdifferbetweenregimes.
Lettingπ(θj,s)denotetheprioronθj,s,themarginallikelihoodofMj,sisgivenby
marglik(fs,m|Mj,s,t)
=∫N(xj,t,s|λj,s,Ωj,s)N(yj,t,s|Γj,sxj,t,s,Σj,s)dπ(θj,s),j≤J,s=1,2(9)
whichwecalculatebythemethodof
Chib
(1995a)
.
NowbyextendingtheargumentandmarginalizationthemarginallikelihoodinEquation
11
(7)canbewrittenas
marglik(f1,t,ft+1,T|t)=marglik(f1,t,ft+1,T|Mj,1,Mk,1,t)Pr(Mj,1)Pr(Mk,1)
(10)
=marglik(f1,t|Mj,1,t)marglik(ft+1,T|Mk,2,t)(11)
whereinthesecondlinewehaveassumedequalpriorprobabilitiesofmodelsandthefactthatthejointfactorsintoindependentcomponentsgiventhemodels.Ineffect,whatwedoispaireachoftheJpossiblemodelsinthefirstregimewitheachpossiblemodelinthesecondandthenmarginalizeoverallpossiblesuchpairings.
Werepeattheabovecalculationforeverypossiblebreakdate.Thebreakdateandtwocollectionsofregime-specificriskfactorsbestsupportedbythedataarethosewiththehighestmarginallikelihood.
2.4.Multiplebreaks
Withmultiplebreaks,weperformthesamemarginallikelihoodcalculationasinthesinglebreakapproach,butthistime,givenmbreaks,wecalculatethemarginallikelihoodofthedatasegmentedbythembreaks:
marglik(f1,m,...,fm+1,m|t1,...,tm).(12)
WecalculatethisquantityforeverypossiblecombinationofthembreaksandhenceeverypossiblecombinationoftheJmodelsineachofthem+1regimes.
Letthetimepointsinthe(m+1)regimesof[1,T]inducedbythesembreakdatesbe
12
denotedbythesets
Ts,m={t:ts−1<t≤ts},s=1,...,m+1.(13)
LetthedataonthefactorsinTs,mbegivenby
fs,m={ft:ts−1<t≤ts},s=1,...,m+1.(14)
Onceagain,weconsiderallpossiblesplitsofftintoxtandyt,ineachofthem+1regimes.Forregimess=1,...,m+1,thesesplitsproducemodelsthatweindicatebyMj,sforj=1,...,J=2K−1.Attimet,inregimes,thedatageneratingprocessunderMj,sisgivenby
xj,t,s=λj,s+uj,t,s
yj,t,s=Γj,sxj,t,s+εj,t,suj,t,s∼N(0,Ωj,s)
εj,t,s∼N(0,Σj,s),t∈Ts,m.(15)
Denotingtheunknownparametersinthesemodelsbyθj,s=(λj,s,Ωj,s,Γj,s,Σj,s),themarginallikelihoodofMj,sisgivenby
=∫N(xj,t,s|λj,s,Ωj,s)N(yj,t,s|Γj,sxj,t,s,Σj,s)dπ(θj,s),j≤J,s=1,...,m+1(16)
whichwecalculatebythemethodof
Chib
(1995a)
.
Thenextstepistocalculatethemarginallikelihoodofallthedataforgivenpairingsofmodelsfromeachofthem+1regimes.ThereareJ(m+1)suchpairingsinallregimes.The
13
marginallikelihoodinEquation(
12
)canbewrittenas
marglik(f1,m,...,fm+1,m|Mj1,1,Mj2,1,...,MjJ,1,...,Mj1,m+1,Mj2,m+1...,MjJ,m+1,t1,...,tm)
Wecangetthedesiredmarginallikelihoodbysummingtherighthandsideoverallpossiblepairingsofmodels.Ifm=3andJ=63,asinoneofourcasesweconsider,therearemorethan15millionsuchmodelcombinations.Thus,
marglik···jmarglik
(18)
Thisisthemarginallikelihoodforthebreakdatest1,...,tm
.18
ThecalculationisrepeatedforallpossiblelocationsofthembreaksandallpossiblecombinationsoftheJmodelsacrossthecorrespondingm+1regimes.Forthisassumednumberofmbreaks,theoptimalbreak
datest,...,tandthem+1collectionofregime-specificriskfactorsarethosethathavethe
highestmarginallikelihood.
Finally,werepeatthiscalculationfordifferentnumbersofbreaksm∈{0,1,2,...,M}.
Theopt
温馨提示
- 1. 本站所有资源如无特殊说明,都需要本地电脑安装OFFICE2007和PDF阅读器。图纸软件为CAD,CAXA,PROE,UG,SolidWorks等.压缩文件请下载最新的WinRAR软件解压。
- 2. 本站的文档不包含任何第三方提供的附件图纸等,如果需要附件,请联系上传者。文件的所有权益归上传用户所有。
- 3. 本站RAR压缩包中若带图纸,网页内容里面会有图纸预览,若没有图纸预览就没有图纸。
- 4. 未经权益所有人同意不得将文件中的内容挪作商业或盈利用途。
- 5. 人人文库网仅提供信息存储空间,仅对用户上传内容的表现方式做保护处理,对用户上传分享的文档内容本身不做任何修改或编辑,并不能对任何下载内容负责。
- 6. 下载文件中如有侵权或不适当内容,请与我们联系,我们立即纠正。
- 7. 本站不保证下载资源的准确性、安全性和完整性, 同时也不承担用户因使用这些下载资源对自己和他人造成任何形式的伤害或损失。
最新文档
- 商业建筑用空调装置项目评价分析报告
- 日期印戳相关项目实施方案
- 挖泥船相关项目实施方案
- 婴儿车车罩市场环境与对策分析
- 开箱刀项目可行性实施报告
- 医疗器械箱相关项目实施方案
- 家用篮项目评价分析报告
- 宝鸡文理学院《公共事业管理》2022-2023学年第一学期期末试卷
- 小便池项目可行性实施报告
- 木蜡石膏或塑料制半身像市场环境与对策分析
- 篮球突分技术与配合-教学设计
- 设计开发记录表及设计开发各过程表单
- 新概念英语第一册Lesson5-6练习题
- 养老行业的法律体系与风险控制
- ISO11898-3中文翻译完整
- 血液透析相关感染检测课件
- 航空公司行政招聘笔试真题
- 低碳环保知识竞赛试题
- 2024年南京铁道职业技术学院高职单招(英语/数学/语文)笔试历年参考题库含答案解析
- 2024年浙江省公务员考试《行测》真题及答案解析
- 食品仓储业食品安全从业人员培训
评论
0/150
提交评论