美聯(lián)儲-線性因子模型與期望收益估計(jì)（英）

FinanceandEconomicsDiscussionSeriesFederalReserveBoard,Washington,LinearFactorModelsandtheEstimationofExpectedReturnsCisilSarisoy,PeterdeGoeij,andBasJ.M.Werker2024-014Sarisoy,Cisil,PeterdeGoeij,andBasJ.M.Werker(2024).“LinearFactorMod-elsandtheEstimationofExpectedReturns,”FinanceandEconomicsDiscussionSe-/10.17016/FEDS.2024.014.NOTE:StafworkingpapersintheFinanceandEconomicsDiscussionSeries(FEDS)arepreliminarymaterialscirculatedtostimulatediscussionandcriticalcomment.TheanalysisandconclusionssetfortharethoseoftheauthorsanddonotindicateconcurrencebyothermembersoftheresearchstafortheBoardofGovernors.ReferencesinpublicationstotheFinanceandEconomicsDiscussionSeries(otherthanacknowledgement)shouldbeclearedwiththeauthor(s)toprotectthetentativecharacterofthesepapers.1LinearFactorModelsandtheEstimationofExpectedReturnsTilburgUniversityTilburgUniversityAbstractThispaperanalyzesthepropertiesofexpectedreturnestimatorsonindividualassetsimpliedbythelinearfactormodelsofassetpricing,i.e.,theproductofβandλ.Weprovidetheasymptoticpropertiesoffactor–model–basedexpectedreturnestimators,whichyieldthestandarderrorsforriskpremiumestimatorsforindividualassets.Weshowthatusingfactor-model-basedriskpremiumestimatesleadstosizableprecisiongainscomparedtousinghistoricalaverages.Finally,inferenceafactor–model–basedestimatesofexpectedreturnstranslainout–of–sampleperformanceofoptimalportfolios.*WethankTorbenG.Andersen,BertilleAntoine,SvetlanaBryzgalova,FrankdeJong,JoostDriessen,StefanoGiglio,BryanKelly,FrankKleibergen,YinyingLi,PauloMaio,AdamMcCloskey,DinoPalazzo,AndrewPatton,EricRenault,EnriqueSentana,GeorgeTauchen,ViktorTodorov,BrianWeller,DachengXiu,andGuofuZhouforhelpfulcommentsanddiscussionsatBlackRock,FederalReserveBoard,NorthwesternUniversityKelloggSchoolofManagement,ErasmusUniversityRotterdam,TilburgUniversity,andCIREQMontrealEconometricsConferenceinhonorofEricthoseoftheauthorsandshouldnotbeinterpretedasrelectingtheviewsoftheBoardofGovernorsoftheauthor:CisilSarisoy,FederalReserveBoard,Washington,D.C.20551U.S.A.E-mail:cisil.sarisoy@.21Introduction f 31SeealsoKanandZhang(1999)Gospodinov,Kan,andRobotti(2014),Bryzgalo(2015),Gospodinov,Kan,andRobotti(2017,2019),Giglio,Xiu,andZhang(2021)ontheroleofspuriousorweaklyidentiiedfactorsforinferenceaboutthepricesofrisk.42.MonteCarlosimulationresultsdocumentthatapproximationsoftheinite-samplevariancesofthefactor-model-basedesestimatesofexpectedirsttwomomentsofassetreturns.Constructingoptimalportfolioswiththeimprecisesuchasthe1/Nstrategyortheglobalminimumvariansubjecttoestimationriskonratiosofoptimalportfolioswhenconportfoliosconstructedwiththefactor–model–basedrisk–premiumestimatesperformbetterthanboththeGMVportfolioandthe1/Nstrategyportfolio.presentsthelinearfactormodelwiththeassumptionsthatformthebasisofourstatisticalanalysis.Next,weintroducefactor–mimickingportfoliosandclarifythelinkbetw2See,forexample,FrostandSavarinoGrauer(1991).3Severalstudiesprovidesolutionsonimprovingthecovariancematrixestimates(see,e.g.Wolf,2003,DeMigueletal.,2009amongothers).However,theestimationerrorinassetreturnmeansimprecisioninestimatesoftheexpectedreturnsafectstheoptimalportfolioweightsmoredrasticallycomparedtotheimprecisionincovarianceestimat52ModelandAssumptionsE[MR]=0.withK<Nsatisfythefollowingconditions:2.ThecovariancematrixoffactorsΣFFhasfullrankK,e,F],hasfullrankK.6E[Re]=βλ,,F地]Σ,andλ=-E]ΣFFb.fnitefourthmoment.Assumption3.Letεt=R-α-βFt.AssumethatE[εtjFt]=0andVar[εtjFt]=Σεε.2.1Factor–MimickingPortfolios4Wefocusonconstantparameterfactormodels.Gagliardini,Ossola,andScaillet(2016)andKelly,Pruitt,andSu(2019)allowfortimevaryingriskexposuresandtimevaryingfactorriskpremiabyincor-poratinginformationfromstockcharacteristicsandmacroeconomicvariables.Ourresultscanbeextendedtosuchasetting,atthecostofadditionalassumptions.Asconstantparametermodelsarestillwidelyusedinempiricalapplications,we’vedecidedtofocusonthatsetting.70E[ut]=0K1,andEutR\=0KN.8Φ=ΣFFβ地ΣRe;andβm=β(β地ΣReβ,-1Σ:Theorem2.1.UnderAssumption1,wehaveβλ=βmλmwithλm=E[Ftm].theoryimpliesthatthepriceofriskattachedtoit,λm,equalsitsexpectation.Thisaddi-mayhopethattheexpecte3Estimationresulting“errors-in-variables”problemwhencalculatinglimitingdistributions.Inaddition,weimmediatelyobtainthejointlimitingdistributionofestimatesforβandλwhichisWethenincorporatet93.1MomentConditions-GeneralCaseE[ht(α,β,λ)]=EFt[R-α-βFt]R-βλt(α,β,λ)]=0withA=IN(1+K)0K根N(1+K)0N(1+K)根NΘK根N.TΣ1.3.2MomentConditions-TradedFactorCaseAssetpricingtheoryprThestandardtwo–passestimationprocedurecommonlyfouintheirTable1,bothOLSandGLSriskpriceestimatesofthemardiferentfromthesampleaverageoftheexcessmarketreturn.Itisimportanttopointconditionswiththeconditionthattheirexpectationofthevectoroffactorsequalsλ.E[ht(α;β;λ)]=EFt[R-α-βFt]Fte-λ=0;fparametersequalsthenumberofmomentconditions.Notethatifthefactoristraded,butAlternatively,wecouldincorporatethetheoreticalrestrictiononfactorpricesintotheRe-βλ.Thissetofmomentconditionswouldbesimilartothegeneralcase,withtheonlydiferencebeingthattheliRadditiontotheoriginalsetoftestassets,i.e.,wedeineR=.Underthissetting,FtE[ht(α,β,λ)]=E[R-α-[R-α-βFt]FtR-βF,Rλ=0,IKmatrixandsetΘ=β,RΣRF.BecauseweindthatGMMbasedon(3.3)leadstothe3.3MomentConditions-Factor–MimickingPortfoliosE[ht(αm,βm,Φ0,Φ,λm)]=ER1Ftm[Ft-Φ0-ΦR][R-αm-βmFtm]ΦR-λm4PrecisionofRisk–PremiumEstimators56Then,5ExamplesincludeShanken(1992),JagannathanandWang(1998),Kleibergen(2009),Lewellen,NandShanken(2010),KanandRobotti(2011),andKan,Robotti,andShanken(2013).6Wealsoprovideresultsforinferenceonriskpremiumswhenthemodelismisspeciiedintheappendix.tions.Thefollowingtheoremprovidesthelimitingdistributionofthehistoricalaveragesesti-wehavepT(e-E[Re],Ⅵ(0;ΣReRe).toticdistributionsofexpected(excess)returnestimatorsgiventhelineaimpliedbyassetpricingmodels.Notethatthejointdistributionsofλandβarediferentforeachsetofmomentconditions,whichleadstodiferentasymptoticdistributionsfortherisk4.1PrecisionwithGeneralMomentConditions7WethroughoutassumethatstandardassumptionsformartingaleCLTshold.ΣReRe-(1-λ地Σλ,(Σεε-β(β地Σ1β)-1β地,.precisiongainsforestimatingtheriskpremiumsfromincorporatinginformationaboutthethenaiverisk–premiumestimatoranofriskλ,theexposuresβ,andtheidiosrisk–premiumestimatoris(1-λ地Σλ,(Σεε-β(β地Σ1β)-1β地).Thefollowingcorollaryformalizesthisrelation.Corollary4.1.SupposetreturnestimatorisatmostΣReRe.Σλ,the4.2PrecisionwithMomentConditionsforTradedFactorsΣReRe-(1-λ地Σλ,Σεε:1.Ifλ地isatmostΣReRe.λ地4.3PrecisionwithMomentConditionsUsingFactor–MimickingPortfoliosΣReRe-(μe{ΣRe-ΣReβ(β地ΣReβ)-1β地ΣRe}μRe)根(ΣReRe-β(β地ΣReβ)-1Σ(βΣReβ)-1β地)-（1-μeΣReμRe）(ΣReRe-β(β地ΣReβ)-1β地);withμRe=E[R].μe{ΣRe-ΣReβ(β地ΣReβ)-1β地ΣRe}μRe根(ΣReRe-β(β地ΣReβ)-1Σ(βΣReβ)-1β地)+（1-μeΣReμRe）(ΣReRe-β(β地ΣReβ)-1β地):mandλm,5InferenceaboutRiskPremiumswhentheβ’sarefinKleibergen(2009)fortLudvigson(2001).8Accordingly,Kleibergen(2009)providesidentiication-robuststatisticsfactors.Recallthatthefocusofthisstudyi9casewhereβ=BforafxedfullrankNKmatrixB.Then,thebehavioroftherisk(-βλ)（B+YΣFF-1）l（B+YΣFF-1）、ΣEE-1（B+YΣFF-1）]-1、（B+YΣFF-1）ΣEE-1[Bλ+U]-βλ,(5.5)8Kleibergen(2009)documentsthat%95percentconidenceboundsofthepricesofriskonthescaledconsumptiongrowthcoincideswiththewholerealline.9Theresultsinthissectionhavebeenrevisedinthecurrentversionofthepaper.10Thelimitingvarianceoftheriskpriceestimatorinthepresenceofweak–factorsisunboundedforFama(-βλ)N（0;λ、ΣFF-1λΣEE）:(5.6)fouranalysisregardingtheissueofsmalfconditions(3.1),thelimitingbehaviouroftheriskpremiumestimatorisnon-standardanddifersfromnormality.ThisresultisinlinewiththeliteraturedocumentingunreliablewithGMMbasedon(3.2),thenthelimitingvariancesoftherinotafectedbytheβhavingasmallvalueornot.Ob6MonteCarloSimulationsInthissection,weconductaMonteCarloexperimenttostudytheinite-samplepropertiesWecalibratetheparametersofthetandKleibergenandZhan(2015).empiricalresearchwithquarterlyandmonthlydata.VaryingTisusefulforunderstandingf6.1UnderthenullthatassetpricingrestrictionsholdInthiscase,themodeliscalibratedtomimicthmultivariatenormaldistribution.12Figure1reportsthesimulationresultsfmodelbasedestimatorsofexpectedreturns(GMM–Gen,GMM–Tr,GMM–Mim).Inpar-basedestimatorsofexpectedreturns(GMM–Gen,GMM–Tr,GMM–Mim)aresmallerthan11Resultsforthree-factormodelisavailableuponrequest.12Inthethree-factormodelsetting,theparametersofthemodelarecalibratedtomimictheFama–Frenchthree-factormodel.6.2Presenceofweakfactors,t.Thedatat7PortfolioChoicewithParameterUncertaintyfportfolioweightswsuchthatE[U]=wμe-2w地ΣRRw,(7.8)tothismaximizationproblemiswewopt=1Σμe.Weconsiderfourportfoliosconstructedwithdiferentrisk–premiumestimators:theop-timalportfolioconstructedwithhistoricalaverages,andtheoptimalportfoliosconstructedwiththethreefactormodel–basedGMMrisk–premiumestimateswitusingthetraditionalsamplecounterpart.13(GMVthereafter)portfoliotowhichwecomparetheperformanceoftheportfoliosbasedontherisk–premiumestimates.Notethattheimplementationofthisportfolioonlyre-performanceofthe1/Nportfolio.131/(T-1)：(Rt-t)(Rt-t)、,wheretisthesampleaverageofreturns.14Thisportfolioisobtainedbyminimizingtheportfoliovariancewithrespecttotheweightswiththeonlyconstraintthatweightssumupto1andtheN–vectorofportfolioweightsisgivenbywgmv=15SeePe?narandaandSentana(2011)foranasamplemeanvariancefrontiersbasedonassetpricingmodelrestrictions,tangencyorspanningconstraints.French3–factormodelwith25FFporfrommultivariatenormaldistribution.WesimulateindependentsetsofZ=20,000return line),thebiasasthepercentageofthepopulationSharperatios,(SR-SR)/SR(secondline)andtheroot–mean–squareerror(RMSE),thesquarerootof：(Ss-SR)/Z,(thirdline),whereZ=20,000.resultswithbothtrueandestimatedΣRR.First,notethatthetrueSharperatiooftheop-timalportfolioissuperiortotheportfoliosbasedonestimatematrixofassetreturns.Comparingtheaveragweseethatthebiasisstrikinglylargearatiossubstantiallytoabout-12%whenthetruecoviariancematrixisused,andtoabout-9%whenthecovariancematrixisestimated.Inparticular,withGMM–Genestimates,areminimalforalloptimalportfoliosconstructedwithfactor–modelbasedrisk–premiumfareminorforalloptimalportfoliosconstructedwithfactor–modelbasedrisk–premiumrisk–premiumestimateswithGMVand1/N,weseethatopnaiveestimatorperformsworsethanboththe1/NstrategyandtheGMVportfolio.More-over,bothGMVand1/NhavesubstantiallylowerRMSEs.ThisresultistheindingsintheliteraturethatGMVportfolioaswellas1/Nstrategyhasbetterout–of–estimatesarelargerthanboththeGMVand1/Nporfolios.Moreover,theiroutofSharpestrategy.manceofoptimalportfoliossubstantiallyovertheoptimalportfoliosbasedontheplugintotheoptimalportfolioswithhistoricalaverages,theseportfolbetterthantheglobalminimumvarianceportfolio.Inpractice,whilethereislittlehopeinknowingtheuniversallytruefactormodel,weknowthatparticularfactormodelsworkLedoit,O.,andWolf,M.(2003).Improvedestimationofthereturnswithanapplicationtoportfolioselection.JournalofEmpiricalFinance,10(5),Merton,R.C.(1973).Anintertemporalcapitalassetpricingmodel.Econometrica:JournalMerton,R.C.(1980).Onestimatingtheexpectedreturnonthemarket:AnexplorMichaud,R.O.(1989).Themarkowitzoptimizationenigma:Is‘optimized’optimal?Fi-Shanken,J.(1992).Ontheestimationofbeta-pricingmodels.ThereviewofSharpe,W.F.(1964).CapitalassetprAProofsIntherestofthepaper,thecovariancematrixofthefactor–mimickingportfoliosisdenotedbyΣFmFm.A.1EquivalenceoffactorpricingusingmimickingportfoliosTheorem2.1.DeineMmastheprojectifMm=r(Mj1,Re)(A.1)=E[Mm],(A.2)Cov[M,Re]=Cov[Mm,Re].(A.3)βλ=Cov[Re,F地]Σ(-E[]ΣFFb)(A.4)=-Cov[Re,F地]b=-CovlRe,Fm地]bm]CovlRe,Fm地]ΣFmΣFmFmbA.2PrecisionofParameterEstimatorsGivenaFactorModelunderthespeciiedlinearfactormodel.ThelemmaA.1belowillustratestheasymptoticdistributionoftheGMMestimatorswithagivensetofmomentconditionsprovidedthatapre–speciiedmatrixA,thatessentiallydeterminestheweigthsoftheoveridentifyingthefollowingresult.GivenaprespecifedmatrixA2Rp根q,itsconsistentestimatorpT(-θ)Ⅵ(0,[AJ]-1ASA\[J\A\]-1),(A.5)J=E,(A.6)S=E[ht(θ)ht(θ)\].(A.7)GMMcontext.Inthesubsequentlemmas,weprovidethelimitingdistributionsfortherpT(-θ)Ⅵ(0,V),(A.8)withV=1+μΣμF-ΣμF-μΣΣΣεεVccV\c(1+λ\Σλ)(β\Σ1β)-1+ΣFF1+μΣλwhereμF=E[Ft]andVc=-ΣλProof.Theprooffollowsfrompluggingtheappropriatematricesforthemomatrixmultiplications.Below,weprovidethelimitingvariancecovariancematrix(S)andtheJacobian(J)forthisspeciicsetofmomentconditions,ΣεεμΣεεΣεεS=μFΣεε[ΣFF+μFμ]ΣεεμFΣεεS=ΣεεμΣεεβΣFFβ\+ΣεεJ(θ)=E=FurthermoreA=-1μμFΣFF+μFμ[0NXN-λ\IN]IN(K+1)0KXN(K+1)0N(K+1)XNβ\Σ1IN0N(K+1)XK-β..sothatthelimitingvarianceofGMMestimatorforθisobtainedbyperformingthematrixmultiplications[AJ]—1ASA\[J\A\]—1.pT(-θ)Ⅵ(0,V),(A.9)withV=1+μΣμF-ΣμF-μΣΣΣεε0N(K+1)XK.0KXN(K+1)ΣFFProof.Theprooffollowsftions(3.2)intothevariancecovarianceformulain(A.5)andperformitiplications.Below,weprovidethelimitingvariancecovariancematrix(S),Jacobian(J)forthisspeciicsetofmomΣεεμΣεε0NXKS=μFΣεε[ΣFF+μFμ]Σεε0NKXKS=0KXN0KXNKΣFF-1μJ(θ)=μFΣFF+μFμ0KXN(K+1)IN0N(K+1)XKIK.Thus,thelimitingvarianceoftheGMMestimatorforθisobtainedbyperformingthematrixmultiplicationsJ—1S[J\]—1sinceA=IN(K+1)+K.ThenextlemmaprovidestheasymptoticpropertiesoftheGMMestimatiorwithfactor–mimickingportfolios.pT(-θ)Ⅵ(0,V),(A.10)withΣFmΦΣReReΦ\ΣFmβmΣuuβm\+ΣFmΣεmεm-ΣFmμFmβmΣuuV=.-μmΣFmΣuuβm\μeΣReμReΣuu+ΣFmFmconditions(3.4)intothevariancecovarianceformulmultiplications.Now,observethatfrom(A.7),wehaveeΣuu0K(1+N)xN(K+1)0K(1+N)xKμReΣReRe+μReμeS=1μm,0N(K+1)xK(1+N)Σεmεm0N(K+1)xKμFmΣFmFm+μFmμFm\0KxK(1+N)0KxN(K+1)ΣFmFmJ(θ)=E-1R\IKRRR\-0R\βm0Kx1Φ(RR\)0KR\IK0K(1+N)xN(K+1)-1Ftm\FtmFtmFtm\0KxN(K+1)0K(1+N)xKIN0N(K+1)xK-IK,withA=IK(1+N)+N(K+1)+K.Thus,thelimitingvarianceoftheGMMestimatorforθ=(vec(βm)\,λm\)\isobtainedbyperformingthematrixmultiplicationsJ-1S[J\]-1.Here,itisworthstressingthatthelimitingvariancecovariancematrixobtainedbyperformingthematrixmultiplicationscorrespondstotheparametervectorrightKN+KbyKN+Ksub-matrixofthelargervariancecovariancematrix.LemmasA.2–A.4allowustosmiumestimators.Itisworthmentioningthatthelower–leftNK+KdimensionalsquareProofofTheorem4.1.ThisfollowsfromadirectapplicationoftheCentralLimitTheorem.βλ.Given)Ⅵ(0,Vβ;),pT（g(,)-g(β,λ))Ⅵ(0,、Vβ;),with=[λ、INβ].RememberthatLemmaA.2andA.3givetheasymptoticdistributionsofpT(-θ)isthelowerNK+KblockdiagonalmatrixofthevariancecovariancematricesprovidedProofofTheorem4.4.Weareinteresteding(βm,λm)=βmλm.GivenpT(g(β,)-g(βm,λm))Ⅵ(0,地Vβm,m)INβmi地Vβm,m=λm地ΣFmλmΣεmεm+βmΣFmFmβm地(A.11)+(μeΣμRe-λm地ΣFmFm-1λm)βmΣuuβm地LemmaA.5.LetK=K11K21K12K22andK11-K12K1K21≥0.ΣReRe-(ΣReRe-（1-λ地Σλ)[Σεε-β(β地Σ1β)-1β地])=（1-λ地Σλ)[Σεε-β(β地Σ1β)-1β地].InordertoshowthatΣεε-β(β地Σ1β)-1β地ispositivesemi–deinite,wewilluseLemmaA.5.Now,letK1=Σ2andK2=β地Σ1/2.Then,K=[KK]=K=Then,LemmaA.5yieldsthatΣεε地β地Σ""-1β Σεε-β(β地Σ1β)-1β地≥0K1KK2KInordertoproveCorollary4.2–1,weneedtostudythediferencebetweenΣReRe-(ΣReRe-(1-λ地Σλ)Σεε)=(1-λ地Σλ)Σεεispositivesemi–deinite.SinceΣεεispositivesemi-deinite,CorollaryInordertoproveCorollary4.2–2,weneedtostudythediferencebetweenthat(ΣReRe-（1-λ\Σλ)[Σεε-β(β\Σ1β)-1β\])-(ΣReRe-（1-λ\Σλ)Σεε)=（1-λ\Σλ)β(β\Σ1β)-1β\ispositivesemi–deinite.ThisfollowsimmediatelyfromΣεεbeingpositivesemi–deinite.smallβcasewhereβ=BforaixedfullrankN根KmatrixB.Thenstudy(-βλ)=ReF(、1)-1(、1e)-βλ.(A.12)Note,usingRte=α+βFt+εt,ReF=（R-e)（Ft-)、(A.13)=[β（Ft-)+(εt-)]（Ft-)、=βFF+εF,=β+εF.(A.14)、=(β+εF)εε(α+β+ε),、=(β+εF)εε(β+εF).Consideringβ=B/pTandassumingthat^(T)εFY,andpTB+YΣ）、Σεε（B+YΣ）.ER=βλ=α+βEF,α=β(λ-EF)=B/pT(λ-EF).TB+YΣFF-1）、Σεε(Bλ+U).(A.15)(A.16)(A.17)(A.18)(A.19)(A.20)(-βλ)（B+YΣFF-1）l（B+