,Vol.75,No.4(July,2007),1175–1189NOTESANDCOMMENTSLEASTSQUARESMODE - PDF document

,Vol.75,No.4(July,2007),1175–1189NOTESANDCOMMENTSLEASTSQUARESMODELAVERAGINGE.HThispaperconsiderstheproblemofselectionofweightsforaveragingacrossleast BRUCEE.HANSENtionsineconometricsincludeworksbySala-i-Martin,Doppelhofer,andMiller),BrockandDurlauf(),Avramov(),Fernandez,Ley,andSteel),Garratt,Lee,Pesaran,andShin(),Brock,Durlauf,andWest(),andWright().Inthefrequentistliterature,Buck-land,Burnham,andAugustin()andBurnhamandAnderson()sug-gestedexponentialAICweights.Theriskpropertiesofasimilarclassofesti-matorswasexaminedbyLeungandBarron().Yang()andYuanandYang()proposedamixingestimator.HjortandClaeskens()pro-videdanasymptoticanalysisofmodelaverageestimatorsinlikelihood-basedShrinkageandparameterpenalizationareotheralternativestomodelselec-tionandaveraging.Somerecentcontributionsincludethelasso-typeestima-torsofKnightandFu(),thepenalizedlikelihoodestimatorsofFanandLi)andFanandPeng(),andtheempiricalBayesestimatorofKnox,Stock,andWatson(Thereisalsoalargeliteraturethatdiscussestheeffectsofmodelselectiononinference.Potscher()showedthatAICselectionresultsindistortedin-ference.Kabaila()examinedtheimpactoncondenceregions.Buhlmann)presentedconditionsunderwhichpost-model-selection(PMS)estima-torsareadaptive.LeebandPotscher()examinedtheuncon-ditionalandconditionaldistributionofPMSestimatorsandarguedthattheycannotbeuniformlyestimated.Theapproachwetakeinthispaperissimilartothatofselectingthenumberoftermsinaseriesexpansion.Andrews()andNewey()studiedtheconvergenceratesforseriesestimatorsandgiveconditionsforasymptoticnor-mality,butdidnotgiverulesforselection.Shibata()demon-stratedtheasymptoticoptimalityofAICselectioninthecontextofGaussianregressions.Shibata’sanalysiswasextendedtonon-GaussianautoregressionsbyLeeandKaragrigoriou().Li()demonstratedtheasymptoticop-timalityofmodelselectioninhomoskedasticlinearregressionusingMallows’criterion,cross-validation,andgeneralizedcross-validation.Andrews(extendedLi’sresultstothecaseofheteroskedasticerrors.AthoroughreviewoftheasymptoticpropertiesofmodelselectioncriteriahasbeenprovidedbyShao().TheoptimalitycriterionusedinthesepaperswascritiquedbyKabaila(WeproposeamodelaverageestimatorwithweightsselectedbyminimizingaMallowscriterion.OurmaincontributionisademonstrationthattheMal-lowscriterionisasymptoticallyequivalenttothesquarederror,andthusourMallowsmodelaverage(MMA)estimatorasymptoticallyachievesthelowestpossiblesquarederrorintheclassofmodelaverageestimators.OurproofisanapplicationofTheorem2.1ofLi(Therearetwoimportantlimitationsofourresults.First,werestrictatten-tiontoregressionswithconditionallyhomoskedasticerrors.Andrews(showedthatmodelselectionbyMallows’criterionisnotoptimalunderhet-eroskedasticity.TheoptimalityofMMAwillsimilarlyfailunderheteroskedas- LEASTSQUARESMODELAVERAGINGticity.Second,ourasymptotictheoryrestrictsthemodelaverageweightstoadiscretesetduetothedifcultyofestablishinguniformityoveraweightvec-torwhosedimensionisunbounded.Developingweightselectionmethodsthatallowforheteroskedasticityandextendingtheprooftechniquetoallowforcontinuousweightsareimportanttopicsforfutureresearch.discussestheestimationframeworkandmodelaverageestimators.calculatestheaveragesquarederrorofthemodelaverageestimator.introducestheMallowscriterionforthemodelaverageestimatoranditssamplingproperties.SectionpresentssimulationevidenceinsupportofthenewMMAestimator.Proofsoftheresultsarepresentedinthe.AGaussprogramthatcalculatestheMMAestimatorisavailableontheauthor’swebpage,MODELAVERAGEESTIMATORLet

nbearandomsample,whereisreal-valuedwhile
)iscountablyinnite.ThemodelisthehomoskedasticlinearE(eE(eWeassumeandthat()convergesinmeansquare.Thelinearityof()isnotessentialtotheideaofmodelaveraging,butitgreatlysimpliesthealgebraiccalculations.Becausetheelementsofmaybetermsinaseriesexpansion,()includesnonparametricregression.Considerasequenceofapproximatingmodels

wherethemodelusestherstelementsof,where0.Thethap-proximatingmodeliswheretheapproximationerroris.Inmatrixnotation,,whereisthematrixwith

b,andLurkingbehind()isanexplicitorderingoftheregressors.Insomecases(suchasaseriesexpansion)thismaynotbetroubling,butinothercasesanaturalorderingoftheregressorsmaynotbeobvious.Inpractice,itmaybe BRUCEE.HANSENfeasibletoordertheregressorsbygroups,andthismaybeacommonapplica-tionofmodelaveraging.Letbeanintegerforwhichisinvertible.Forall,theleastsquaresestimateof.Letbeaweightvectorintheunitsimplexinin0
1]M:M
m=1wm=1(6)AmodelaverageestimatorofAmodelaverageestimatorbearssomeresemblancetoashrinkageestima-tor.Thiscanbeseenmostplainlywhentheregressorsareorthogonal.Inthiscase,thethelementofthemodelaverageestimatoristhethelementoftheunconstrainedestimatormultipliedby.Thusthecoefcientestimatesshrinktowardzero,withthedegreeofshrinkageincreasingwithHowever,inthestandardcasewheretheregressorsarenotorthogonal,suchasimplerepresentationisnotpossible.Inthethapproximatingmodel(),letsothat.Theestimateofinthethapproximatingmodel,where.Themodelaverageesti-mateofµ(W)P(W)Y,whereP(W)istheim-plied“hat”matrix.BecausethematrixP(W)playsanimportantroleinthealgebraicstructureofthemodelaverageestimator,wediscussheresomeofitsproperties.NoteP(W)issymmetricbutgenerallynotidempotent.LetdenotethelargesteigenvalueofanddeneWehave(i)tr(P(W))k(W)(ii)tr(P(W)P(W))(P(W)) LEASTSQUARESMODELAVERAGINGSQUAREDERRORDenetheaveragesquarederror(W)µ(W)µ(W)conditionalsquarederror(W)E(L(W),whereWehave(W))W
isdeÞnedinFurthermoreLemmashowsthattheconditionalsquarederror(W)isaquadraticfunctionintheweightvector,anellipsoidincenteredatthezerovector.Itisinterestingtoobservethattheoptimalweightvector,whichminimizes(W),necessarilyputsnon-zeroweightonatleasttwomodels,exceptinthespecialcasethat.Toseethis,supposethat2,inwhichcase(W)isuniquelyminimizedby,whichisinTHEMALLOWSCRITERIONTheMallowscriterionforthemodelaverageestimatoris(W)k(W)
k(W)denedinLemmaistheeffectivenumberofparameters.De-nition()dependsontheunknown.Wediscussbelowthereplacementofwithanestimate.TheMallowscriterionmaybeusedtoselecttheweightvector.Deneargmin(W)
theempiricalMallowsselectedweightvector.Becausethereisnoclosed-formsolutionto(),theweightvectormustbefoundnumerically.Forthiscalcu-lation,itisconvenienttowrite()inthefollowingform.Letbetheresidualvectorfromthethmodel,letbethe BRUCEE.HANSENcollectionoftheseresiduals,andletbethe1vectorofthenumberofparametersinthemodels.Then()equals(W)whichislinear-quadraticin.Thesolution()minimizes()subjecttothenonnegativityandsummationconstraints().Thisisaclassicquadraticprogrammingproblemforwhichnumericalalgorithmsarereadilyavailable.(Forexample,intheGaussprogramminglanguage,theprocedureQPROGisappropriate.)Thesolutionmaybeaunitvectororaninteriorvalue.Ifmoderatelylarge,atypicalsolutioncanputzeroweightonmanyofthein-dividualmodels.TheMallowsmodelaverageestimatoris()usingtheweightWepresenttwojusticationsfortheMallowscriterion.Ourrstistheclassicobservationthat(W)isanunbiasedestimateoftheexpectedsquarederrorplusaconstant.Wehave(W)(W)Oursecondjusticationisthatiftheweightsarerestrictedtoadiscreteset,theempiricalMallowsweightvectorasymptoticallyminimizesthesquareder-ror.Specically,forsomeinteger,lettheweightsberestrictedtothe N
2 andlet(N)bethesubsetofrestrictedtothissetofweights.Letargmin(N)(W)betheMallowsweightvector,thechoiceobtainedbyminimizingtheMallowscriterionoverthediscreteweightset(N)ThefollowingresultisanapplicationofTheorem2.1ofLi(),whoshowedtheasymptoticoptimalityofMallows’criterionformodelselection.(W)almostsurelyandforsomeÞxedinteger (N)(W) LEASTSQUARESMODELAVERAGINGNotethatthetheoremplacesnorestrictionon,thelargestmodelincludedinthemodelaverage(otherthantherequirementthatisinvertible).maybexedasmaydivergetoinnity.showsthatthesquarederrorobtainedusingtheMallowsweightisasymptoticallyequivalenttotheinfeasibleoptimalweightvector.ThismeansthattheMMAestimatorisasymptoticallyoptimalintheclassofmodelaverageestimators()wheretheweightvectorisrestrictedtotheset(N)Therestrictionof(N)canbemadelessbindingbypickingwhichcanbedoneaslongastheconditionalmomentbound()holds.Thisrestrictionisimposedbecausetheproofof()requiresthat(W)isas-ymptoticallyequivalentto(W)uniformlyover.Thetroubleisthatthedimensionofthesetisunboundedwhen,renderingconventionalproofmethodsinapplicable.requirescondition(),whichspeciesthatthereisnoniteapproximatingmodelforwhichthebiasiszero.Thisassumptionisconven-tionalfornonparametricregression.Forexample,if,thenwehavetheexplicitrate.If()fails,thenMMAwillnotsatisfytheopti-mality(Inpractice,isunknown,so()needstobecomputedwithasampleesti-mate.Onechoiceis,wherecorrespondstoa“large”approximatingmodel.Otherestimatorsforbeenproposedinthenonparametricregressionliterature.Lemmatoholdifisunbiasedfor,whichholdsif0,sothethapproximat-ingmodelhasnobias.Theoremholdsasstatedaslongasisconsistent,whichisvalidasshownnext.K/nFINITESAMPLEINVESTIGATIONWenowinvestigatethenitesamplemeansquarederroroftheourmodelaverageestimatorinasimplesimulationexperiment.Thesettingistheinnite-orderregression.Weset1tobetheintercept;theareindependentandidenticallydistributed.Theer-andindependentof.(Otherexperiments,notreported,showedthattheresultsarenotsensitivetoalternativedistributionsfortheregressorsandregressionerror.)Theparametersaredeterminedbytherule .Thepopulationiscontrolledbythepara-Thesamplesizeisvariedbetween50,150,400,and1000.Theparame-isvariedbetween05,1.0,and15.Thelargerimpliesthatthecoef-declinemorequicklywith.Thenumberofmodelsisdeterminedbytherule11,16,22,and30forthefoursamplesizes). BRUCEE.HANSENThecoefcientwasselectedtocontrolthepopulationtovaryonagridbetween0.1and0Weconsiderveestimators:(1)AICmodelselection(AIC),(2)Mallows’modelselection(Mallows),(3)smoothedAIC(S-AIC),(4)smoothedBIC(S-BIC),and(5)Mallows’modelaveraging(MMA).TheAICcriterionforisAIC.TheAICmodelselectionestimatorisminimizesAIC.S-AICwasintroducedbyBuckland,Burnham,andAugustin()andembracedbyBurnhamandAnderson()andHjortandClaeskens().Itistheleastsquaresmodelaverageestimator()withtheweights 2AICm)/Mj=1exp(Š1 .S-BICisasimpliedformofBayesianmodelaveraging.Itistheleastsquaresmodelaveragees-timator()withtheweights 2BICm)/Mj=1exp(Š1 ,whereToevaluatetheestimators,wecomputetherisk(expectedsquarederror).Wedothisbycomputingaveragesacross100,000simulationdraws.Foreachparameterization,wenormalizetheriskbydividingbytheriskoftheinfeasibleoptimalleastsquaresestimator(theriskofthebest-ttingmodelTheriskcalculationsaredisplayedinFigures5,1.0,and1.5,respectively.Ineachgure,thefourpanelsdisplaysamplesizes.Ineachpanel,risk(expectedsquarederror)isdisplayedontheaxisandthepopulationdisplayedontheaxis.ThetwodottedlinescorrespondtoAICandMallowsselection.Thedashed,dash-dotted,andsolidlinescorrespondtoS-AIC,S-BIC,andMMA,respectively.Ineachpanel,theAICandMallowsselectionmethodshavequitesimilarrisk.ThesmoothedAICestimatorachievesalowerriskthanAICmodelselec-tion,whichisconsistentwiththendingsintheearlierliterature.TheS-AICandMMAestimatorsarenearlyequivalentforthecase5andlargeotherwise,MMAachievesalowerriskthanS-AIC.Inmanycases,itsnormal-izedriskislessthan1,meaningthatitislowerthanthatofinfeasibleoptimalmodelselection.ItisalsoinstructivetocontrasttheperformanceoftheMMAandS-BICestimators.TheMMAestimatorachieveslowerriskinmostcases,butS-BIChaslowerriskwhenaresmall,anditsrelativeperformanceimprovesislarge.Inparticular,S-BIChasmuchlowerriskwhen5and50.Theirrelativeperformancedependsstronglyonsamplesize,withtheS-BICestimatorshowingincreasingrelativeriskandtheMMAshowingde-creasingrelativerisk,asincreases.Inmanycases,however,theriskoftheS-BICestimatorisquitepoorrelativetotheothermethods.Dept.ofEconomics,UniversityofWisconsin,1180ObservatoryDrive,Madison,WI53706,U.S.A.;ManuscriptreceivedJanuary,2006;ÞnalrevisionreceivedAugust,2006. LEASTSQUARESMODELAVERAGING ROOFOF:Parts(i)and(ii)followfromthefactsthattr,tr,andsimplealgebra.Part(iii)usesthefactthatisidempotentsothat(P(W))P(W)\f \f\fM
m=1wmmax\f\fPm\f Q.E.D.ROOFOF:Notethatµ(W)P(W))µP(W)e(W)P(W))(IP(W))µP(W)BP(W)P(W)eLemmaandassumption()implythatE(eP(W)P(W)e(P(W)P(W)) BRUCEE.HANSEN Takingconditionalexpectationsof(),weobtainE(L(W)P(W))(IP(W))µ))µb\n1

b\nM].ThenP(W))µNotethatfor.Thenandthus.ItfollowsthatP(W))(IP(W))µandweobtain().Furthermore,fornotethatandthusasclaimed. LEASTSQUARESMODELAVERAGING Wenowshowthat0,whichholdsif,forall)�0.Ifistherstunitvector,then)0.Otherwise,if,notethat0,andbythedenitionandsomealgebraicmanipulations,+···+)�0asrequired.Q.E.D.ROOFOF:Bystraightforwardalgebra,(W)(W)P(W))µP(W)ek(W))Lemmaandassumption()implythatE(eP(W)e(P(W))k(W) BRUCEE.HANSENTakingexpectationsof(),Equation()followsdirectly.Q.E.D.ROOFOF:Theorem2.1ofLi()established()forabroadclassoflinearestimators.Itissufcienttoverifythathisequations(A.1),(A.2),and(A.3)holdalmostsurely,conditionalon.Indeed,(A.1)isimpliedbypart(iii)ofLemma,and(A.2)holdsby().Itremainstoshow(A.3),whichinournotationis(N)(W)almostsurelyasForintegers1···,let

jbetheweightvectorthatsets

N,andtheremainderzero.Wecanwrite(N)

j···Therestrictionoftheweightstotheform1iswithoutlossofgenerality,becausetheweakorderingoftheintegersallowsties.Wethenhave(N)(W)

jNowbreakthesumintotwogroupsbasedonwhether.Fortherstgroup(whichhaslessthanelements),usethebound(W)from()andforthesecondgroup,usethesimplebound

j

j

j N2kjN2 wheretherstinequalityisimpliedby()andthesecondusesthedenitions

jUsingthesebounds,

j N2jNŠ(N+1)\bŠ1n+2 N2Š(N+1)

jN=\bnjŠ2N LEASTSQUARESMODELAVERAGING almostsurelyas.Togetherwith(),thisestablishes()asde-Q.E.D.ROOFOF:Sinceweseethat nŠKe(IŠPK)e+1 nŠKbK(IŠPK)bK+21 Weexaminethetermsontherightsideof().First,because,byTheorem2ofWhittle(K)Thusforany0,byMarkov’sinequality, nŠKe(IŠPK)eŠ2�E|e(IŠPK)eŠ2(nŠK)|2 2(nŠK)2C2\t1/(N+) .Second, E(b andthesquareintegrabilityof0as.Thisimplies0.Similarly,thethirdtermontherightsideof()isandweconcludethatQ.E.D.KAIKE,H.(1973):“InformationTheoryandanExtensionoftheMaximumLikelihoodPrinci-ple,”inSecondInternationalSymposiumonInformationTheory,ed.byB.PetrocandF.Csake.Budapest:AkademiaiKiado,267–281.267–281.1175]ANDREWS,D.W.K.(1991a):“AsymptoticNormalityofSeriesEstimatorsforNonparametricandSemiparametricRegressionModels,”,59,307–345. (1991b):“AsymptoticOptimalityofGeneralized,Cross-Validation,andGeneral-izedCross-ValidationinRegressionwithHeteroskedasticErrors,”JournalofEconometrics,47, BRUCEE.HANSEN,D.W.K.,B.L(2001):“ConsistentModelandMomentSelectionProceduresforGMMEstimationwithApplicationtoDynamicPanelDataModels,”JournalofEconometrics101,123–164.123–164.1175]AVRAMOV,D.(2002):“StockReturnPredictabilityandModelUncertainty,”JournalofFinancial,64,423–458.423–458.1176]BROCK,W.,S.DURLAUF(2001):“GrowthEmpiricsandReality,”WorldBankEconomicReview,15,229–272.229–272.1176]BROCK,W.,S.DURLAUFK.D.W(2003):“PolicyAnalysisinUncertainEconomicEn-BrookingsPapersonEconomicActivity,1,235–322.235–322.1176]BUCKLAND,S.T.,K.P.BN.H.A(1997):“ModelSelection:AnIntegralPartofInference,”,53,603–618.603–618.1176,1182]BUHLMANN,P.(1999):“EfcientandAdaptivePost-Model-SelectionEstimators,”JournalofSta-tisticalPlanningandInference,79,1–9.1–9.1176]BURNHAM,K.P.,D.R.AModelSelectionandMultimodelInference:APracticalInformationÑTheoreticApproach.Berlin:Springer-Verlag.rlag.1176,1182]CLAESKENS,G.,N.L.HJORT(2003):“TheFocusedInformationCriterion,”JournaloftheAmericanStatisticalAssociation,98,900–916.900–916.1175]CRAVEN,P.,G.W(1979):“SmoothingNoisyDatawithSplineFunctions:EstimatingtheCorrectDegreeofSmoothingbytheMethodofGeneralizedCross-Validation,”,31,377–403.377–403.1175]DRAPER,D.(1995):“AssessmentandPropagationofModelUncertainty,”JournaloftheRoyalStatisticalSociety,Ser.B,57,45–70.45–70.1175]FAN,J.,R.L(2001):“VariableSelectionviaNonconcavePenalizedLikelihoodandItsOracleProperties,”JournaloftheAmericanStatisticalAssociation,96,1348–1360.1348–1360.1176]FAN,J.,H.P(2004):“NonconcavePenalizedLikelihoodwithaDivergingNumberofParameters,”TheAnnalsofStatistics,32,928–961.928–961.1176]FERNANDEZ,C.,E.LM.F.J.S(2001a):“BenchmarkPriorsforBayesianModelAveraging,”JournalofEconometrics,100,381–427. (2001b):“ModelUncertaintyinCross-CountryGrowthRegressions,”JournalofApplied,16,563–576.563–576.1176]GARRATT,A.,K.L,M.H.PESARANY.S(2003):“ForecastingUncertaintiesinMacroeconomicModelling:AnApplicationtotheUKEconomy,”JournaloftheAmericanStatisticalAssociation,98,829–838.829–838.1176]HALL,A.R.,A.I,K.JC.S(2007):“InformationinGeneralizedMethodofMomentsEstimationandEntropy-BasedMomentSelection,”JournalofEconometrics,138,138,1175]HJORT,N.L.,G.C(2003):“FrequentistModelAverageEstimators,”JournaloftheAmericanStatisticalAssociation,98,879–899.879–899.1176,1182]HOETING,J.A.,D.MADIGAN,A.E.RAFTERYC.T.VOLINSKY(1999):“BayesianModelAveraging:ATutorial,”StatisticalScience,14,382–417.382–417.1175]HONG,H.,B.PM.S(2003):“GeneralizedEmpiricalLikelihood-BasedModelSelectionCriteriaforMomentConditionModels,”EconometricTheory,19,923–943.923–943.1175]KABAILA,P.(1995):“TheEffectofModelSelectiononCondenceRegionsandPredictionRe-EconometricTheory,11,537–549. (2002):“OnVariableSelectioninLinearRegression,”EconometricTheory,18,913–925.913–925.1176]KNIGHT,K.,W.F(2000):“AsymptoticsforLasso-TypeEstimators,”TheAnnalsofStatistics28,1356–1378.1356–1378.1176]KNOX,T.,J.H.SM.W.WATSON(2004):“EmpiricalBayesRegressionwithManyRegressors,”WorkingPaper,HarvardUniversity.[,S.,A.KARAGRIGORIOU(2001):“AnAsymptoticallyOptimalSelectionoftheOrderofaLinearProcess,”,Ser.A,63,93–106. LEASTSQUARESMODELAVERAGING,H.,B.M.P(2003):“TheFinite-SampleDistributionofPost-Model-SelectionEstimatorsandUniformversusNon-UniformApproximations,”EconometricTheory19,100–142. (2005):“ModelSelectionandInference:FactsandFiction,”EconometricTheory,21, (2006):“CanOneEstimatetheConditionalDistributionofPost-Model-SelectionEsti-TheAnnalsofStatistics,34,2554–2591.2554–2591.1176]LEUNG,G.,A.R.B(2006):“InformationTheoryandMixingLeast-SquaresRegres-IEEETransactionsonInformationTheory,52,3396–3410.3396–3410.1176]LI,K.-C.(1987):“AsymptoticOptimalityfor,Cross-ValidationandGeneralizedCross-Validation:DiscreteIndexSet,”TheAnnalsofStatistics,15,958–975.958–975.1176,1180,1186]MALLOWS,C.L.(1973):“SomeCommentsonTechnometrics,15,661–675.661–675.1175]NEWEY,W.K.(1997):“ConvergenceRatesandAsymptoticNormalityforSeriesEstimators,”JournalofEconometrics,79,147–168.147–168.1176]POTSCHER,B.M.(1991):“EffectsofModelSelectiononInference,”EconometricTheory,7,,1176]RAFTERY,A.E.,D.MADIGANJ.A.H(1997):“BayesianModelAveragingforRe-gressionModels,”JournaloftheAmericanStatisticalAssociation,92,179–191.179–191.1175]RAFTERY,A.E.,Y.Z(2003):“Long-RunPerformanceofBayesianModelAveraging,”WorkingPaper,UniversityofWashington.[ARTIN,X.,G.DR.M(2004):“DeterminantsofLong-TermGrowth:ABayesianAveragingofClassicalEstimates(BACE)Approach,”AmericanEco-nomicReview,94,813–835.813–835.1176]SCHWARZ,G.(1978):“EstimatingtheDimensionofaModel,”TheAnnalsofStatistics,6,,1175]SHAO,J.(1997):“AnAsymptoticTheoryforLinearModelSelection,”StatisticaSinica,7,,1176]SHIBATA,R.(1980):“AsymptoticallyEfcientSelectionoftheOrderoftheModelforEstimatingParametersofaLinearProcess,”TheAnnalsofStatistics,8,147–164. (1981):“AnOptimalSelectionofRegressionVariables,”,68,45–54. (1983):“AsymptoticMeanEfciencyofaSelectionofRegressionVariables,”AnnalsoftheInstituteofStatisticalMathematics,35,415–423.415–423.1176]STONE,M.(1974):“Cross-ValidatoryChoiceandAssessmentofStatisticalPredictions,”oftheRoyalStatisticalSociety,Ser.B,36,276–278.276–278.1175]WHITTLE,P.(1960):“BoundsfortheMomentsofLinearandQuadraticFormsinIndependentVariables,”TheoryofProbabilityandItsApplications,5,302–305.302–305.1187]WRIGHT,J.H.(2003a):“BayesianModelAveragingandExchangeRateForecasts,”FinanceDiscussionPapers,779,BoardofGovernors,FederalReserveBoard. (2003b):“ForecastingUSInationbyBayesianModelAveraging,”InternationalFinanceDiscussionPapers,780,BoardofGovernors,FederalReserveBoard.Board.1176]YANG,Y.(2001):“AdaptiveRegressionbyMixing,”JournaloftheAmericanStatisticalAssociation96,574–586.574–586.1176]YUAN,Z.,Y.Y(2005):“CombiningLinearRegressionModels:WhenandHow?”naloftheAmericanStatisticalAssociation,100,1202–1214.