美联储-因素选择与结构断裂（英）-2024.5

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