2 Fra Equity premium prediction the role of economic and ... Paper: “Equity premium prediction:...

Paper:“Equitypremiumprediction:theroleofeconomicandstatistical

constraints”byJiahan LiandIlias Tsiakas

July 2016

Student:FrancescaCaturano

AdvancedFinancialEconometricsIII

February 2020

Introduction

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ThispapershowsthattheequitypremiumispredictableOOSforshortpredictablehorizonwhenweuseapredictiveregression thatconditionsonalargesetofeconomicfundamentals,subjectto:1. economicconstraintsonthesignofcoefficientsandreturnforecasts2. statisticalconstraintsimposedbyshrinkageestimation

Paper:“Equity premiumprediction:therole ofeconomic andstatistical constraints”

§ Thekeytoestablishingequitypremiumpredictabilityisimplementingapredictiveframeworkbasedon3aspects:

i. usingasinglepredictiveregressionthatconditiononalargenumberofpredictorsà kitchen-sinkregression

ii. imposingeconomicconstraintsonthesign ofcoefficientsandreturnforecasts

iii. usingashrinkageestimatordesignatedtoimproveperformancebyreducingtheeffectsoflessinformativepredictorsinOOSforecasting

Introduction

3

ThispapershowsthattheequitypremiumispredictableOOSforshortpredictablehorizonwhenweuseapredictiveregressionthatconditionsonalargesetofeconomicfundamentals,subjectto:1. economicconstraintsonthesignofcoefficientsandreturnforecasts2. statisticalconstraintsimposedbyshrinkageestimation


§ Thekeytoestablishingequitypremiumpredictabilityisimplementingapredictiveframeworkbasedon3aspects:

i. usingasinglepredictiveregressionthatconditiononalargenumberofpredictorsà kitchen-sinkregression

ii. imposingeconomicconstraintsonthesign ofcoefficientsandreturnforecasts

iii. usingashrinkageestimatordesignatedtoimproveperformancebyreducingtheeffectsoflessinformativepredictorsinOOSforecasting

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Introduction:the3aspectsmoreindetail


i. Specifyakitchen-sinkregressionà providesadirectwayofpoolinginformationbyconstructiona“super”modelwhichnestseachofthemodelsthatconditiononasinglepredictor

ii. Imposingeconomicconstraintsonthesignofcoefficientsandreturnforecastsà imposeeconomictheoryonthepredictiveregressionsandusuallyimproveperformance

iii. Implementashrinkageestimationofthekitchen-sinkregressionà kitchen-sinkregressiontransformsfrombeingtheworstmodelwhenestimatedwithOLStobeingthebestmodelwhenestimatedwithashrinkageestimator.Shrinkageestimationproducesbiasedparameterestimatesbyshrinkingallestimatetoward0,whichisthevalueimpliedbythebmk historicalmeanmodel.

iv. Authorsuses3shrinkageestimators:o ridgeregressiono lassoregressiono elasticnet

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Introduction:the3aspectsmoreindetail


i. Specifyakitchen-sinkregressionà providesadirectwayofpoolinginformationbyconstructionamodelwhichnestseachofthemodelsthatconditiononasinglepredictor

ii. Imposingeconomicconstraintsonthesignofcoefficientsandreturnforecastsà imposeeconomictheoryonthepredictiveregressionsandusuallyimproveperformance

iii. Implementashrinkageestimationofthekitchen-sinkregressionà kitchen-sinkregressiontransformsfrombeingtheworstmodelwhenestimatedwithOLStobeingthebestmodelwhenestimatedwithashrinkageestimator.Shrinkageestimationproducesbiasedparameterestimatesbyshrinkingallestimatetoward0,whichisthevalueimpliedbythebmk historicalmeanmodel.

iv. Authorsuses3shrinkageestimators:o ridgeregressiono lassoregressiono elasticnet

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Predictingequitypremium:kitchen-sinkregression


§ Equitypremiumà continuouslycompoundedreturnontheS&P500indexincludingdividendminus theT-billrate

§ MonthlyEconomicfundamentalsusedforpredictingthemonthlyequitypremiumfortheperiodJan.1927toDec.2014

§ à use11monthlypredictorsprimarilybasedonstockcharacteristicsandinterestrates:dividendyield,earnings-priceratio,book-to-marketratio,netequityexpansion,stockvariance,T-Billrate,termspread,long-termrateofreturn,defaultyieldspread,defaultratespread,inflation

§ Themodelusedforpredictingtheequitypremiumisbasedonthekitchensinkregression:

𝑟"#$% = 𝛼 + ∑*+$, 𝛽*𝑥*," + 𝜀"#$,where§ 𝒓𝒕#𝟏𝒆 = 𝒓𝒕#𝟏 − 𝒓𝒇 isequitypremiumatt+1,𝑟"#$istotalreturnontheS&P500at

t+1,𝑟7 istheT-billrate;𝒙𝒋,𝒕 à isthe𝑗 ≤ 𝑁 predictorattimet;𝜺𝐭#𝟏à normalerrorterm;𝛂 and𝜷 = 𝛽* areconstantparameterstobeestimated

§ KSregressionallowstocaptureallavailableinfoinasingleregressionàwayofpoolinginfointoasingleforecast

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Predictingequitypremium:kitchen-sinkregression


§ Equitypremiumà continuouslycompoundedreturnontheS&P500indexincludingdividendminus theT-billrate

§ MonthlyEconomicfundamentalsusedforpredictingthemonthlyequitypremiumfortheperiodJan.1927toDec.2014

§ à use11monthlypredictorsprimarilybasedonstockcharacteristicsandinterestrates:dividendyield,earnings-priceratio,book-to-marketratio,netequityexpansion,stockvariance,T-Billrate,termspread,long-termrateofreturn,defaultyieldspread,defaultratespread,inflation

§ Themodelusedforpredictingtheequitypremiumisbasedonthekitchensinkregression:

𝑟"#$% = 𝛼 + ∑*+$, 𝛽*𝑥*," + 𝜀"#$,where§ 𝒓𝒕#𝟏𝒆 = 𝒓𝒕#𝟏 − 𝒓𝒇 isequitypremiumatt+1,𝑟"#$istotalreturnontheS&P500at

t+1,𝑟7 istheT-billrate;𝒙𝒋,𝒕 à isthe𝑗 ≤ 𝑁 predictorattimet;𝜺𝐭#𝟏à normalerrorterm;𝛂 and𝜷 = 𝛽* areconstantparameterstobeestimated

§ KSregressionallowstocaptureallavailableinfoinasingleregressionàwayofpoolinginfointoasingleforecast

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§ Implementtheshrinkageestimationà shrinktheregressioncoefficientstowards0(i.e.thevalueimpliedbythehistoricalmeanbmk)inawaythatdirectlyminimizestheOOSmeansquarederroro IncontrasttotheOLSestimatorthatisunbiased,ashrinkageestimatorisbiasedbutmayhavelowervarianceandlowerMSEthanOLS

§ ShrinktheregressioncoefficientsbyestimatingtheKSregressionwiththeelastic-net estimator,whichsolves:

§ 𝒎𝒊𝒏𝜷𝟏𝟐∑𝒕+𝟏𝑻H𝟏(𝒓𝒕#𝟏𝒆 − 𝜶 − ∑𝒋+𝟏𝑵 𝜷𝒋𝒙𝒋,𝒕)𝟐 s.t. ∑*+$, 𝛽* < 𝑠$,∑*+$, 𝛽*O < 𝑠O

§ 𝑠$,𝑠O positiveconstants,estimatedinawaythatminimizesMSEofforecasts

§ Specialcaseoftheelastic-netestimator:o ridgeregressionà 𝑠$ = ∞ (firstconstraintisunbounded)o lassoregressionà 𝑠O = ∞ (second constraintisunbounded)

Statisticalconstraintsduetoshrinkageestimation


Authorsestimate theKSregressionwithashrinkageestimatorbecausetheOOSperformanceoftheKSregressionestimatedwithOLSisverypoorrelativetothehistoricalmeanbmk

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AuthorsestimatetheKSregressionwithashrinkageestimatorbecausetheOOSperformanceoftheKSregressionestimatedwithOLSisverypoorrelativetothehistoricalmeanbmk

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AuthorsestimatetheKSregressionwithashrinkageestimatorbecausetheOOSperformanceoftheKSregressionestimatedwithOLSisverypoorrelativetothehistoricalmeanbmk

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§ 1st constraintà equitypremium>0ineveryperiod§ How? Replacenegativeforecastswithzero§ CampbellandThompson(2008)arguethatareasonableinvestorwouldnothaveusedamodeltoforecastanegativeequitypremium

§ 2nd constraintà constraintsignoftheslopecoefficientstobeconsistentwitheconomictheory

§ How? Setvalueof0foracoefficientthatdoesnothavethetheoreticallymotivatedsigno Inthid study:slopecoefficientispositiveforallpredictorsexceptnetequityexpansions,T-billrateandinflation

§ CampbellandThompson(2008)explainthat«aregressionestimatedoverashortsampleperiodcaneasilygenerateperverseresults,suchasanegativecoefficientwhentheorysuggeststhatthecoefficientsshouldbepositive...»

Economicconstraints


Authorsimposealso2constraints motivatedbyeconomictheory

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§ 1st constraintà equitypremium>0ineveryperiod§ How? Replacenegativeforecastswithzero§ CampbellandThompson(2008)arguethatareasonableinvestorwouldnothaveusedamodeltoforecastanegativeequitypremium

§ 2nd constraintà constraintsignoftheslopecoefficientstobeconsistentwitheconomictheory

§ How? Setvalueof0foracoefficientthatdoesnothavethetheoreticallymotivatedsigno Inthisstudy:slopecoefficientispositiveforallpredictorsexceptnetequityexpansions,T-billrateandinflation

§ CampbellandThompson(2008)explainthat«aregressionestimatedoverashortsampleperiodcaneasilygenerateperverseresults,suchasanegativecoefficientwhentheorysuggeststhatthecoefficientsshouldbepositive...»

Economicconstraints


Authorsimposealso2constraints motivatedbyeconomictheory

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§ TheBmkà historicalsamplemeanfortheequitypremium:caseof𝛽* = 0§ Individualpredictorsà predictiveregressionsthatconditiononasinglepredictor(inthiscase,11regressionseachconditioningon1predictor)

§ Combinationsofpredictorso “Modelselection”(MS)approach:estimateregressionswithallpossiblecombinationsofpredictorsandateachpointintimeselecttheoneforecastthathasperformedthebest(lowestcumulativeMSEuptothatpoint)à chooseoneamong11O models

o PrincipalComponentAnalysis(PCA):estimateasetofPCsthatparsimoniouslyincorporateinfofromthe11predictors

§ Estimatekitchen-sinkregressionwithOLS§ Forecastscombinationà combinetheforecastsofseveralpredictiveregressionsthatconditionononepredictor.Implement2approaches:o Meancombination:computeequally-weightedaverageofallforecastsateachpointintime

o MSEcombination:weightedaverageoftheindividualforecastateachpointintimeusingasweightstheinverseofthediscountedMSEofeachmodeluptothatpoint

Otherpredictiveregressions


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§ Grandcombinationsofpoolinginformationandpoolingforecastso Authorspredictiveframeworkprovidesawayofpoolinginformationsinceitisbasedonakitchen-sinkregressionthatdirectlyconditionsonalargenumberofpredictors

o incontrast,combinedforecastsaredesignedtopoolforecastsratherthanpoolinformation

§ Poolingforecastsinvolvestwostages:1. estimateseveralpredictiveregressionseachconditioningonone

predictor2. combinetheindividualforecastsintooneforecastcombination.

Intheory,thistwo-stageprocessintroducesanefficiencylossandignoresthecorrelationsbetweenthepredictors.

Therefore,itisoftenarguedthatpoolinginformationisoptimalrelativetopoolingforecasts



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§ Grandcombinationsofpoolinginformationandpoolingforecastso inadditiontojustpoolinginformationorjustpoolingforecasts,authorsalsoforma“grand”combination ofthetwoapproaches.

o Thisprovidesaframeworkforassessingwhetherpoolingforecastsaddstothepredictiveabilityofpoolinginformationandviceversa.

o Thegrandcombinationisaninterestingadditiontothemodelsetbecause,thecombinedforecastsaretheclosestcompetitortoourpredictiveframework.

o Authorsformequally-weightedgrandcombinationsofforecastsbycombining:(i)thee-netKSforecastswiththemeancombinedforecasts;(ii)thee-netKSforecastswiththeMSEcombinedforecasts;(iii)thelassoKSforecastswiththemeancombinedforecasts;(iv)thelassoKSforecastswiththeMSEcombinedforecasts.



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§ ThemainstatisticalcriterionforevaluatingtheOOSpredictiveabilityofthemodelisOOSR2statistic

𝑅ZZ[O = 1 −𝑀𝑆𝐸 �̂�"#$|"

%

𝑀𝑆𝐸 �̅�"#$|"% = 1 −

∑"+$bH$(𝑟"#$% − �̂�"#$|"% )O

∑"+$bH$(𝑟"#$% − �̅�"#$|"% )O

§ Itcompares:theunconditional1-monthaheadforecastc𝒓𝒕#𝟏|𝒕

𝒆 ofthehistoricalmeanbmktotheconditionalforecastd𝒓𝒕#𝟏|𝒕

𝒆 ofthealternativemodel§ 𝑅ZZ[O > 0à alternativemodeloutperformsthebmk bymeansoflowerMSE

AllempiricalmodelsareevaluatedOOSrelativetothehistoricalmeanbmk.AuthorsgenerateOOSforecastwithrollingpredictiveregressionsusinga20yestimationwindow(1st forecastisforJan1947,lastforDec.2014)

Out-of-sampleanalysis


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AccesstheOOSperformanceoftheempiricalmodelsbyreportingthe 𝑅ZZ[OEmpiricalResults


OLSpredictiveregressionsthat

conditionon1predictorattime

§ Tabledisplaysthe𝑅ZZ[O forpredictivemodelsofthemonthlyequitypremiumagainstthenullofthehistoricalmean.The𝑅ZZ[O isformodelsthatimposeasignconstraintontheslopecoefficientsandapositivityconstraintontheforecasts.

§ Inadditiontothefullsample,arereportedalsoresultsforthe2subsamplesofexpansions andrecessions

3KSregressionusing≠ shrinkage

estimation

Forecastcombination

4grandcombinations

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The3shrinkageestimatorsoftheKSregressiondeliveran𝑅ZZ[O thatispositive,significantandhigher thanallothermodels.Indeed,theshrinkagemodelsaretheonlymodelsthathaveapositiveandsignificant𝑅ZZ[O inbothexpansionsandrecessions

EmpiricalResults


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EmpiricalResults


§ Theclosestcompetitorstotheshrinkagemodelsarethecombinedforecasts.However,theshrinkagemodelsexhibitamuchhigher𝑅ZZ[O forthefullsample,whichbecomesevenhigherinrecessions

§ à Evidence stronglyfavorsauthors’approachofpoolinginfotothestandardapproachofpoolingforecasts

§ TheOLSKSmodelisbyfartheworstperformingmodel,whichindicatesthatthewayauthorsestimatetheKSregressionisofcriticalimportance

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EmpiricalResults


§ Turningtothegrandcombinationsofpoolinginfoandpoolingforecast,authorsfindthattheydeliversimilarperformancetotheplainshrinkagemodel

§ à grandcombinationsdonotaddtothepredictiveabilityofauthors’approach

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EmpiricalResults


§ Amongthesinglepredictors,thebestperformingmodelisdividendyield(dy).However,thedy hasan𝑅ZZ[O thatislessthanhalfofthevalueofthee-netorlassoanditisonlypositiveinexpansions,whilebeingnegativeinrecession

§ Noneofthe11predictorshaveasignificantlypositive𝑅ZZ[O inbothexpansionsandrecessions.

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Tosummarize,evidencebasedon𝑅ZZ[O indicatesthatthebestmodelforpredicting theequitypremiumOOS isakitchen-sinkregressionthatconditionsonalargesetofpredictorsandimposesstatisticalconstraintsthroughshrinkageestimationtogetherwitheconomicconstraintsonslopecoefficientsandforecasts.

EmpiricalResults


EmpiricalResults

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§ Effectofeconomicconstraintsonperformanceo Thetabledemonstratestheeffectsofconstraintsonthe𝑅ZZ[O ofmodels

o Firstà isolate theeffect ofstatisticalconstraintsintheabsenceofeconomicconstraints


o StatisticalconstraintduetoshrinkageestimationdeliveramassiveimprovementonpredictabilityrelativetoOLS,butalonetheyfailshortofproducing apositive𝑅ZZ[O

TheunconstrainedOLSKSdeliversan𝑅ZZ[O of–11.86%(insignificant)butwhenimposingthelassoconstraintthe𝑅ZZ[O risesto-0.63%(significant)

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§ Effectofeconomicconstraintsonperformanceo Itisthecombination ofeconomic andstatisticalconstraintsinthecontextofkitchen-sinkregressionthatdeliversthemostpowerfulresults


Imposingtheeconomicconstraintsraisesthelasso𝑅ZZ[O

from-0.63%to1,77%whichisnowpositive,significantat1%andthehighestofallmodels

§ Theeconomicconstraintsseemtobeveryeffectivewhenappliedtomanypredictorsinasingleregressionestimatedwithshrinkage.

EmpiricalResults

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§ Effectofeconomicconstraintsonperformance


§ Itisthecombination ofeconomic andstatisticalconstraintsinthecontextofthekitchen-sinkregressionthatdeliversthemostpowerfulresults.

EmpiricalResults

Predictabilityandassetallocation

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§ Thestrategyinvolvesmonthlyrebalancingofaptf investedintheS&P500index(theriskyasset)andtheT-bill(risklessasset)

§ Authorsconsideramean-varianceinvestorwitha1-monthaheadhorizon,whodeterminesoptimalweightsbyimplementingamaximumexpectedutilityrule:

§ 𝐦𝐚𝐱𝒘𝒕

𝑬𝒕 𝑼(𝒓𝒑,𝒕#𝟏) = 𝒓𝒑,𝒕#𝟏|𝒕 −𝜸𝟐𝝈𝒑,𝒕#𝟏|𝒕𝟐

§ s.t. 𝒓𝒑,𝒕#𝟏|𝒕 = 𝑤"𝑟"#$|" + (1 − 𝑤")𝑟7 and𝝈𝒑,𝒕#𝟏|𝒕𝟐 = 𝑤"O𝜎"#$|"

O

§ 𝒓𝒑,𝒕#𝟏|𝒕 isthet+1forecastofptf returnconditionalontimet information,𝜸 istheinvestor’sdegreeofrelativeriskaversion,𝝈𝒑,𝒕#𝟏|𝒕

𝟐 isthet+1forecastofptf variancemadeattimet,𝒓𝒕#𝟏|𝒕 isthet+1forecastoftheS&P500indexreturnmadeattimet,𝒓𝒇 istherisk-freerateofreturnand𝝈𝒕#𝟏|𝒕

𝟐 isthet+1forecastofthevarianceoftheS&P500indexreturnmadeattimet

§ Thesolutiongivestheriskyassetweight:𝒘𝒕 =𝟏𝜸𝒓𝒕r𝟏|𝒕H𝒓𝒇𝝈𝒕r𝟏|𝒕𝟐 ,𝑤" ∈ 0, 1.5 (ie.

shortsellingnotallowedandleveragingislimitedtonomorethan50%

Authorsassesstheeconomicvalueofequitypremiumpredictabilityusingadynamicassetallocationstrategy



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O









31






O









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§ SRà averageexcessreturnofaptf dividedbythestandarddeviationofptf returns

§ CERà 𝐶𝐸𝑅 = (�̅�w −xOc𝜎wO)where�̅�w =meanptf returnand c𝜎wO =ptf

varianceovertheforecastevaluationperiod§ TheCERcanbeinterpretedastheperformancefeetherisk-averseinvestoriswillingtopayforswitchingfromtherisklessassettotheriskyptf.

§ Focuson:∆𝑪𝑬𝑹= 𝐶𝐸𝑅|"7 }%~%�"%� �� 7��"� �7 ��"%�~�"��% ��%�− 𝐶𝐸𝑅|"7 }%~��"%� �� "�% ��"�� %�~ ��

§ ∆𝑪𝑬𝑹measurestheperformancefeetherisk-averseinvestoriswillingtopayforswitchingfromtheriskyptf generatedbythebmkmodeltotheriskyptf generatedbythealternativemodel

Authorsevaluate theperformance oftheptf generatedbyagivensetofequitypremiumforecastsusingtheSharperatio(SR)andthecertaintyequivalentreturn (CER)



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§ Focuson:∆𝑪𝑬𝑹= 𝐶𝐸𝑅|"7 }%~%�"%� �� 7��%��"� �7 ��"%�~�"��% ��%�− 𝐶𝐸𝑅|"7 }%~��"%� �� "�% ��"�� %�~ ��





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§ ∆𝑪𝑬𝑹measurestheperformancefeetherisk-averseinvestoriswillingtopayforswitching fromtheriskyptf generatedbythebmkmodeltotheriskyptf generatedbythealternativemodel



Portfolioperformance

36Paper:“Equity premiumprediction:therole ofeconomic andstatistical constraints”

Assesstheperformanceofdynamicallyrebalancedptfs generatedbythemonthlyforecastsofthepredictivemodels§ TableshowstheOOSptf performanceforamean-varianceinvestor,whoeachmonth

rebalancestheptf byinvestinginoneriskyasset(S&P500)andtherisklessasset(T-bill).Theinvestorhasadegreeofriskaversionequalto5andfollowsamaximumutilitystrategy.TheOOSmonthlyforecastsareobtainedusinga20yrollingwindowforthesampleperiodJan.1927toDec.2014.Forthehistoricalmeanbmk,thelevelofCERisreported

§ ∆𝑪𝑬𝑹à isthegaininpercentannualizedCERforswitchingfromtheforecastsofthebmk totheforecastsgeneratedbythealternativemodel



§ Thehistoricalmeanbmk deliversaCERof5,82%peryearrelativetotherisklessinvesting.TheCERrisesto8.33%inexpansionsbutfallsto-6.81%inrecessions.

§ à Historicalmeanisapoorpredictorofequitypremiuminrecessions

§ Anyofthe3shrinkageKSregressionsperformsbetterthatanyoftheothermodelso Ex:lassoregressiondeliversa∆𝐶𝐸𝑅

relativetothehistoricalmeanbmk thatof2.71%peryear,whichbecomes12.71%inrecessions



§ AlsoconsideringtheannualizedSharperatio,shrinkagestrategiessubstantiallyoutperformstheothermodels

§ Overall,thisisevidencethattheequitypremiumispredictableOOSand,inthecontextofadynamicmean-variancestrategy,thereishigheconomicvalueinusingaKSregressionwithbothstatistical andeconomicconstraints

o Forexample,theeconomicgainsofthelassoapproachcanbesummarizedintoaperformancefeeof2.71%peryearbeforetransactioncosts,togetherwithanincreaseintheSharperatiofrom0.45to0.66.

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Conclusion


§ Authorsinvestigatethepredictabilityoftheequitypremiumusingakitchen-sinkregressionthatconditionsonalargesetofeconomicfundamentals.Theregressionisestimatedwithashrinkagemethodologydesignatedtomaximizepredictiveperformance

§ Authorsimplementthisframeworkusingalongsampleofmonthlydataandarriveattheempiricalfindingsthat:equitypremiumispredictableOOS:authors’predicativeframeworkconsistentlyoutperformsboththehistoricalmeanbmk andcompetingmodels

§ Equitypremiumforecastsbasedonauthors’predictiveframeworkconsistentlyoutperformthehistoricalmeanbmk,especiallyduringrecessions

§ Thesuperiorperformanceoftheshrinkageestimatorsinrecessionsisveryimportantbecausethepredictiveinformationofeconomicfundamentalsismorevaluabletoaninvestorduringrecessions.Thisistruebecauseduringrecessionstheequitypremiumisonaveragenegativewithhighvolatility,whichmakesthehistoricalmeanapoorforecast.

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Conclusion




§ Equitypremiumforecastsbasedonauthors’predictiveframeworkconsistentlyoutperformthehistoricalmeanbmk,especiallyduringrecessions

§ Thesuperiorperformanceoftheshrinkageestimatorsinrecessionsisveryimportantbecausethepredictiveinformationofeconomicfundamentalsismorevaluabletoaninvestorduringrecessions.Thisistruebecauseduringrecessionstheequitypremiumisonaveragenegativewithhighvolatility,whichmakesthehistoricalmeanapoorforecast.

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Conclusion




§ Equitypremiumforecastsbasedonauthors’predictiveframeworkconsistentlyoutperform thehistoricalmeanbmk,especiallyduringrecessions

§ Thesuperiorperformanceoftheshrinkageestimatorsinrecessions isveryimportant becausethepredictiveinformationofeconomicfundamentalsismorevaluabletoaninvestorduringrecessions.Thisistruebecauseduringrecessionstheequitypremiumisonaveragenegativewithhighvolatility,whichmakesthehistoricalmeanapoorforecast.

Assessingperformanceover

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§ Figuresplotthedifferenceofthecumulativesquarederrorofthenull(historicalmean)minusthecumulativesquarederrorofalternative.

FiguresplottheOOSperformanceofeachmodelovertime.TheyshowthatshrinkageestimatorsdisplaythemostpronouncedupwardtrendintheirOOSperformanceovertime


Assessingperformanceover

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Thegoodperformanceofshrinkageestimatorsisnotduetoaparticularsubsamplebutissystematicoveralongsamplespanningthefullpostwarperiod


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