Post on 25-Jul-2020
Paper:“Equitypremiumprediction:theroleofeconomicandstatistical
constraints”byJiahan LiandIlias Tsiakas
July 2016
Student:FrancescaCaturano
AdvancedFinancialEconometricsIII
February 2020
Introduction
2
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
Paper:“Equity premiumprediction:therole ofeconomic andstatistical constraints”
§ Thekeytoestablishingequitypremiumpredictabilityisimplementingapredictiveframeworkbasedon3aspects:
i. usingasinglepredictiveregressionthatconditiononalargenumberofpredictorsà kitchen-sinkregression
ii. imposingeconomicconstraintsonthesign ofcoefficientsandreturnforecasts
iii. usingashrinkageestimatordesignatedtoimproveperformancebyreducingtheeffectsoflessinformativepredictorsinOOSforecasting
4
Introduction:the3aspectsmoreindetail
Paper:“Equity premiumprediction:therole ofeconomic andstatistical constraints”
i. Specifyakitchen-sinkregressionà providesadirectwayofpoolinginformationbyconstructiona“super”modelwhichnestseachofthemodelsthatconditiononasinglepredictor
ii. Imposingeconomicconstraintsonthesignofcoefficientsandreturnforecastsà imposeeconomictheoryonthepredictiveregressionsandusuallyimproveperformance
iii. Implementashrinkageestimationofthekitchen-sinkregressionà kitchen-sinkregressiontransformsfrombeingtheworstmodelwhenestimatedwithOLStobeingthebestmodelwhenestimatedwithashrinkageestimator.Shrinkageestimationproducesbiasedparameterestimatesbyshrinkingallestimatetoward0,whichisthevalueimpliedbythebmk historicalmeanmodel.
iv. Authorsuses3shrinkageestimators:o ridgeregressiono lassoregressiono elasticnet
5
Introduction:the3aspectsmoreindetail
Paper:“Equity premiumprediction:therole ofeconomic andstatistical constraints”
i. Specifyakitchen-sinkregressionà providesadirectwayofpoolinginformationbyconstructionamodelwhichnestseachofthemodelsthatconditiononasinglepredictor
ii. Imposingeconomicconstraintsonthesignofcoefficientsandreturnforecastsà imposeeconomictheoryonthepredictiveregressionsandusuallyimproveperformance
iii. Implementashrinkageestimationofthekitchen-sinkregressionà kitchen-sinkregressiontransformsfrombeingtheworstmodelwhenestimatedwithOLStobeingthebestmodelwhenestimatedwithashrinkageestimator.Shrinkageestimationproducesbiasedparameterestimatesbyshrinkingallestimatetoward0,whichisthevalueimpliedbythebmk historicalmeanmodel.
iv. Authorsuses3shrinkageestimators:o ridgeregressiono lassoregressiono elasticnet
6
Predictingequitypremium:kitchen-sinkregression
Paper:“Equity premiumprediction:therole ofeconomic andstatistical constraints”
§ 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
7
Predictingequitypremium:kitchen-sinkregression
Paper:“Equity premiumprediction:therole ofeconomic andstatistical constraints”
§ 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
Paper:“Equity premiumprediction:therole ofeconomic andstatistical constraints”
Authorsestimate theKSregressionwithashrinkageestimatorbecausetheOOSperformanceoftheKSregressionestimatedwithOLSisverypoorrelativetothehistoricalmeanbmk
9
§ 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
Paper:“Equity premiumprediction:therole ofeconomic andstatistical constraints”
AuthorsestimatetheKSregressionwithashrinkageestimatorbecausetheOOSperformanceoftheKSregressionestimatedwithOLSisverypoorrelativetothehistoricalmeanbmk
10
§ 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
Paper:“Equity premiumprediction:therole ofeconomic andstatistical constraints”
AuthorsestimatetheKSregressionwithashrinkageestimatorbecausetheOOSperformanceoftheKSregressionestimatedwithOLSisverypoorrelativetothehistoricalmeanbmk
11
§ 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
Paper:“Equity premiumprediction:therole ofeconomic andstatistical constraints”
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
Paper:“Equity premiumprediction:therole ofeconomic andstatistical constraints”
Authorsimposealso2constraints motivatedbyeconomictheory
13
§ 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
Paper:“Equity premiumprediction:therole ofeconomic andstatistical constraints”
14
§ 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
Paper:“Equity premiumprediction:therole ofeconomic andstatistical constraints”
15
§ 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
Paper:“Equity premiumprediction:therole ofeconomic andstatistical constraints”
16
§ 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
Paper:“Equity premiumprediction:therole ofeconomic andstatistical constraints”
17
§ Grandcombinationsofpoolinginformationandpoolingforecastso Authorspredictiveframeworkprovidesawayofpoolinginformationsinceitisbasedonakitchen-sinkregressionthatdirectlyconditionsonalargenumberofpredictors
o incontrast,combinedforecastsaredesignedtopoolforecastsratherthanpoolinformation
§ Poolingforecastsinvolvestwostages:1. estimateseveralpredictiveregressionseachconditioningonone
predictor2. combinetheindividualforecastsintooneforecastcombination.
Intheory,thistwo-stageprocessintroducesanefficiencylossandignoresthecorrelationsbetweenthepredictors.
Therefore,itisoftenarguedthatpoolinginformationisoptimalrelativetopoolingforecasts
Otherpredictiveregressions
Paper:“Equity premiumprediction:therole ofeconomic andstatistical constraints”
18
§ Grandcombinationsofpoolinginformationandpoolingforecastso inadditiontojustpoolinginformationorjustpoolingforecasts,authorsalsoforma“grand”combination ofthetwoapproaches.
o Thisprovidesaframeworkforassessingwhetherpoolingforecastsaddstothepredictiveabilityofpoolinginformationandviceversa.
o Thegrandcombinationisaninterestingadditiontothemodelsetbecause,thecombinedforecastsaretheclosestcompetitortoourpredictiveframework.
o Authorsformequally-weightedgrandcombinationsofforecastsbycombining:(i)thee-netKSforecastswiththemeancombinedforecasts;(ii)thee-netKSforecastswiththeMSEcombinedforecasts;(iii)thelassoKSforecastswiththemeancombinedforecasts;(iv)thelassoKSforecastswiththeMSEcombinedforecasts.
Otherpredictiveregressions
Paper:“Equity premiumprediction:therole ofeconomic andstatistical constraints”
19
§ 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
Paper:“Equity premiumprediction:therole ofeconomic andstatistical constraints”
20
AccesstheOOSperformanceoftheempiricalmodelsbyreportingthe 𝑅ZZ[OEmpiricalResults
Paper:“Equity premiumprediction:therole ofeconomic andstatistical constraints”
OLSpredictiveregressionsthat
conditionon1predictorattime
§ Tabledisplaysthe𝑅ZZ[O forpredictivemodelsofthemonthlyequitypremiumagainstthenullofthehistoricalmean.The𝑅ZZ[O isformodelsthatimposeasignconstraintontheslopecoefficientsandapositivityconstraintontheforecasts.
§ Inadditiontothefullsample,arereportedalsoresultsforthe2subsamplesofexpansions andrecessions
3KSregressionusing≠ shrinkage
estimation
Forecastcombination
4grandcombinations
21
The3shrinkageestimatorsoftheKSregressiondeliveran𝑅ZZ[O thatispositive,significantandhigher thanallothermodels.Indeed,theshrinkagemodelsaretheonlymodelsthathaveapositiveandsignificant𝑅ZZ[O inbothexpansionsandrecessions
EmpiricalResults
Paper:“Equity premiumprediction:therole ofeconomic andstatistical constraints”
22
EmpiricalResults
Paper:“Equity premiumprediction:therole ofeconomic andstatistical constraints”
§ Theclosestcompetitorstotheshrinkagemodelsarethecombinedforecasts.However,theshrinkagemodelsexhibitamuchhigher𝑅ZZ[O forthefullsample,whichbecomesevenhigherinrecessions
§ à Evidence stronglyfavorsauthors’approachofpoolinginfotothestandardapproachofpoolingforecasts
§ TheOLSKSmodelisbyfartheworstperformingmodel,whichindicatesthatthewayauthorsestimatetheKSregressionisofcriticalimportance
23
EmpiricalResults
Paper:“Equity premiumprediction:therole ofeconomic andstatistical constraints”
§ Turningtothegrandcombinationsofpoolinginfoandpoolingforecast,authorsfindthattheydeliversimilarperformancetotheplainshrinkagemodel
§ à grandcombinationsdonotaddtothepredictiveabilityofauthors’approach
24
EmpiricalResults
Paper:“Equity premiumprediction:therole ofeconomic andstatistical constraints”
§ Amongthesinglepredictors,thebestperformingmodelisdividendyield(dy).However,thedy hasan𝑅ZZ[O thatislessthanhalfofthevalueofthee-netorlassoanditisonlypositiveinexpansions,whilebeingnegativeinrecession
§ Noneofthe11predictorshaveasignificantlypositive𝑅ZZ[O inbothexpansionsandrecessions.
25
Tosummarize,evidencebasedon𝑅ZZ[O indicatesthatthebestmodelforpredicting theequitypremiumOOS isakitchen-sinkregressionthatconditionsonalargesetofpredictorsandimposesstatisticalconstraintsthroughshrinkageestimationtogetherwitheconomicconstraintsonslopecoefficientsandforecasts.
EmpiricalResults
Paper:“Equity premiumprediction:therole ofeconomic andstatistical constraints”
EmpiricalResults
26
§ Effectofeconomicconstraintsonperformanceo Thetabledemonstratestheeffectsofconstraintsonthe𝑅ZZ[O ofmodels
o Firstà isolate theeffect ofstatisticalconstraintsintheabsenceofeconomicconstraints
Paper:“Equity premiumprediction:therole ofeconomic andstatistical constraints”
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
Paper:“Equity premiumprediction:therole ofeconomic andstatistical constraints”
Imposingtheeconomicconstraintsraisesthelasso𝑅ZZ[O
from-0.63%to1,77%whichisnowpositive,significantat1%andthehighestofallmodels
§ Theeconomicconstraintsseemtobeveryeffectivewhenappliedtomanypredictorsinasingleregressionestimatedwithshrinkage.
EmpiricalResults
28
§ Effectofeconomicconstraintsonperformance
Paper:“Equity premiumprediction:therole ofeconomic andstatistical constraints”
§ Itisthecombination ofeconomic andstatisticalconstraintsinthecontextofthekitchen-sinkregressionthatdeliversthemostpowerfulresults.
EmpiricalResults
Predictabilityandassetallocation
29
§ 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
Paper:“Equity premiumprediction:therole ofeconomic andstatistical constraints”
Predictabilityandassetallocation
30
§ 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
Paper:“Equity premiumprediction:therole ofeconomic andstatistical constraints”
Predictabilityandassetallocation
31
§ 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
Paper:“Equity premiumprediction:therole ofeconomic andstatistical constraints”
Predictabilityandassetallocation
32
§ 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)
Paper:“Equity premiumprediction:therole ofeconomic andstatistical constraints”
Predictabilityandassetallocation
33
§ 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)
Paper:“Equity premiumprediction:therole ofeconomic andstatistical constraints”
Predictabilityandassetallocation
34
§ 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)
Paper:“Equity premiumprediction:therole ofeconomic andstatistical constraints”
Predictabilityandassetallocation
35
§ 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-averseinvestoriswillingtopayforswitching fromtheriskyptf generatedbythebmkmodeltotheriskyptf generatedbythealternativemodel
Authorsevaluate theperformance oftheptf generatedbyagivensetofequitypremiumforecastsusingtheSharperatio(SR)andthecertaintyequivalentreturn (CER)
Paper:“Equity premiumprediction:therole ofeconomic andstatistical constraints”
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
Portfolioperformance
37Paper:“Equity premiumprediction:therole ofeconomic andstatistical constraints”
§ Thehistoricalmeanbmk deliversaCERof5,82%peryearrelativetotherisklessinvesting.TheCERrisesto8.33%inexpansionsbutfallsto-6.81%inrecessions.
§ à Historicalmeanisapoorpredictorofequitypremiuminrecessions
§ Anyofthe3shrinkageKSregressionsperformsbetterthatanyoftheothermodelso Ex:lassoregressiondeliversa∆𝐶𝐸𝑅
relativetothehistoricalmeanbmk thatof2.71%peryear,whichbecomes12.71%inrecessions
Portfolioperformance
38Paper:“Equity premiumprediction:therole ofeconomic andstatistical constraints”
§ AlsoconsideringtheannualizedSharperatio,shrinkagestrategiessubstantiallyoutperformstheothermodels
§ Overall,thisisevidencethattheequitypremiumispredictableOOSand,inthecontextofadynamicmean-variancestrategy,thereishigheconomicvalueinusingaKSregressionwithbothstatistical andeconomicconstraints
o Forexample,theeconomicgainsofthelassoapproachcanbesummarizedintoaperformancefeeof2.71%peryearbeforetransactioncosts,togetherwithanincreaseintheSharperatiofrom0.45to0.66.
39
Conclusion
Paper:“Equity premiumprediction:therole ofeconomic andstatistical constraints”
§ Authorsinvestigatethepredictabilityoftheequitypremiumusingakitchen-sinkregressionthatconditionsonalargesetofeconomicfundamentals.Theregressionisestimatedwithashrinkagemethodologydesignatedtomaximizepredictiveperformance
§ Authorsimplementthisframeworkusingalongsampleofmonthlydataandarriveattheempiricalfindingsthat:equitypremiumispredictableOOS:authors’predicativeframeworkconsistentlyoutperformsboththehistoricalmeanbmk andcompetingmodels
§ Equitypremiumforecastsbasedonauthors’predictiveframeworkconsistentlyoutperformthehistoricalmeanbmk,especiallyduringrecessions
§ Thesuperiorperformanceoftheshrinkageestimatorsinrecessionsisveryimportantbecausethepredictiveinformationofeconomicfundamentalsismorevaluabletoaninvestorduringrecessions.Thisistruebecauseduringrecessionstheequitypremiumisonaveragenegativewithhighvolatility,whichmakesthehistoricalmeanapoorforecast.
40
Conclusion
Paper:“Equity premiumprediction:therole ofeconomic andstatistical constraints”
§ Authorsinvestigatethepredictabilityoftheequitypremiumusingakitchen-sinkregressionthatconditionsonalargesetofeconomicfundamentals.Theregressionisestimatedwithashrinkagemethodologydesignatedtomaximizepredictiveperformance
§ Authorsimplementthisframeworkusingalongsampleofmonthlydataandarriveattheempiricalfindingsthat:equitypremiumispredictableOOS:authors’predicativeframeworkconsistentlyoutperformsboththehistoricalmeanbmk andcompetingmodels
§ Equitypremiumforecastsbasedonauthors’predictiveframeworkconsistentlyoutperformthehistoricalmeanbmk,especiallyduringrecessions
§ Thesuperiorperformanceoftheshrinkageestimatorsinrecessionsisveryimportantbecausethepredictiveinformationofeconomicfundamentalsismorevaluabletoaninvestorduringrecessions.Thisistruebecauseduringrecessionstheequitypremiumisonaveragenegativewithhighvolatility,whichmakesthehistoricalmeanapoorforecast.
41
Conclusion
Paper:“Equity premiumprediction:therole ofeconomic andstatistical constraints”
§ Authorsinvestigatethepredictabilityoftheequitypremiumusingakitchen-sinkregressionthatconditionsonalargesetofeconomicfundamentals.Theregressionisestimatedwithashrinkagemethodologydesignatedtomaximizepredictiveperformance
§ Authorsimplementthisframeworkusingalongsampleofmonthlydataandarriveattheempiricalfindingsthat:equitypremiumispredictableOOS:authors’predicativeframeworkconsistentlyoutperformsboththehistoricalmeanbmk andcompetingmodels
§ Equitypremiumforecastsbasedonauthors’predictiveframeworkconsistentlyoutperform thehistoricalmeanbmk,especiallyduringrecessions
§ Thesuperiorperformanceoftheshrinkageestimatorsinrecessions isveryimportant becausethepredictiveinformationofeconomicfundamentalsismorevaluabletoaninvestorduringrecessions.Thisistruebecauseduringrecessionstheequitypremiumisonaveragenegativewithhighvolatility,whichmakesthehistoricalmeanapoorforecast.
Assessingperformanceover
42
§ Figuresplotthedifferenceofthecumulativesquarederrorofthenull(historicalmean)minusthecumulativesquarederrorofalternative.
FiguresplottheOOSperformanceofeachmodelovertime.TheyshowthatshrinkageestimatorsdisplaythemostpronouncedupwardtrendintheirOOSperformanceovertime
Paper:“Equity premiumprediction:therole ofeconomic andstatistical constraints”
Assessingperformanceover
43
Thegoodperformanceofshrinkageestimatorsisnotduetoaparticularsubsamplebutissystematicoveralongsamplespanningthefullpostwarperiod
Paper:“Equity premiumprediction:therole ofeconomic andstatistical constraints”