美聯(lián)儲-光譜回溯測試無界和折疊 Spectral backtests unbounded and folded 2024

FinanceandEconomicsDiscussionSeries

FederalReserveBoard,Washington,D.C.

ISSN1936-2854(Print)

ISSN2767-3898(Online)

Spectralbacktestsunboundedandfolded

MichaelB.GordyandAlexanderJ.McNeil

2024-060

Pleasecitethispaperas:

Gordy,MichaelB.,andAlexanderJ.McNeil(2024).“Spectralbacktestsunboundedandfolded,”FinanceandEconomicsDiscussionSeries2024-060.Washington:BoardofGover-norsoftheFederalReserveSystem,

/10.17016/FEDS.2024.060

.

NOTE:StafworkingpapersintheFinanceandEconomicsDiscussionSeries(FEDS)arepreliminarymaterialscirculatedtostimulatediscussionandcriticalcomment.TheanalysisandconclusionssetfortharethoseoftheauthorsanddonotindicateconcurrencebyothermembersoftheresearchstafortheBoardofGovernors.ReferencesinpublicationstotheFinanceandEconomicsDiscussionSeries(otherthanacknowledgement)shouldbeclearedwiththeauthor(s)toprotectthetentativecharacterofthesepapers.

1

Spectralbacktestsunboundedandfolded*

MichaelB.Gordy

FederalReserveBoard,WashingtonDC

AlexanderJ.McNeil

SchoolforBusinessandSociety,UniversityofYork

15July2024

Abstract

Inthespectralbacktestingframeworkof

GordyandMcNeil

(2020)aprobability

measureontheunitintervalisusedtoweightthequantilesofgreatestinterestinthevalidationofforecastmodelsusingprobability-integraltransform(PIT)data.WeextendthisframeworktoallowgeneralLebesgue-Stieltjeskernelmeasureswithun-boundeddistributionfunctions,whichbringspowerfulnewtestsbasedontruncatedlocation-scalefamiliesintothespectralclass.Moreover,byconsideringuniformdistri-butionpreservingtransformationsofPITvaluesthetestframeworkisgeneralizedtoallowteststhatarefocusedonbothtailsoftheforecastdistribution.

JELCodes:C52;G21;G32

Keywords:Backtesting;Volatility;Riskmanagement

*Theopinionsexpressedhereareourown,anddonotreflecttheviewsoftheBoardofGovernorsoritsstaff.AddresscorrespondencetoAlexanderJ.McNeil,UniversityofYork,alexander.mcneil@york.ac.uk.

2

1Introduction

GordyandMcNeil

(2020)studyaclassofbacktestsforforecastdistributionsinwhichthetest

statisticdependsonaspectraltransformationofaquantileexceedanceindicatorfunction.Thespectraltransformationweightsquantileexceedanceeventsusingakernelmeasurewhichischosenbythevalidatortoreflectthevalidator’sprioritiesformodelperformance.Thepresentpaperextendstheoriginaltreatmentintwodirections.First,whereas

Gordyand

McNeil

(2020)restrictthekernelmeasuretotheclassofprobabilitymeasures,inthispaper

weallowthekernelmeasuretobeunbounded,subjecttoanintegrabilitycondition.Weshowthatunboundedkernelsdelivertestsmateriallymorepowerfulthantestsbasedontheboundedkernelsstudiedby

GordyandMcNeil

(2020)

.Second,weintroduceapre-processingofthedatabyafoldingtransformationthatleavesthesizeofthebacktestunalteredbutincreasesitspoweragainstmisspecificationsofforecastvolatilitythatareextremelycommoninpractice.

Ourextensionstothespectralbacktestingframeworkaregermanetoanyvalidationexerciseinwhichperformancethroughoutoneorbothtailsoftheforecastdistributionisofspecialinterest.Ourinvestigationismotivatedbyrecentdevelopmentsinthecapitalregulationofthetradingoperationsoflargebanks.

UnderthecurrentBaselIIIrules(Basel

CommitteeonBankSupervision,

2019),minimumcapitalrequirementsforabank’strading

bookaredeterminedbythebank’sself-reporteddailyExpectedShortfall(ES)attheα1=97.5%confidencelevel.TheadoptionofESdepartsfromearlierBaselregimestiedtoValue-at-Risk(VaR)attheα*=99%confidencelevel.LeftunchangedinBaselIIIistheroleoftheregulatorinvalidationofthebank’smodelthroughbacktesting.Forthispurpose,banksintheUnitedStatesreporttoregulatorsforeachtradingdaytheprobabilityassociatedwiththerealizedprofit-and-loss(P&L)inthepriorday’sforecastdistribution,i.e.,theprobabilityintegraltransform(PIT)associatedwithrealizedP&L.ObservingthePITvaluesisequivalenttoobservingVaRexceedancesateverylevelα∈[0,1].Besides

GordyandMcNeil

(2020),bank-reportedPITdatahavebeenstudiedby

Lynchetal.

(2023)

3

and

Iercosanetal.

(2023)

.

UnderaVaR-basedregime,theregulatorwouldhaveparticularinterestintestingmodelperformanceoversomerangeofconfidencelevelsintheneighborhoodofα*.Accordingly,

GordyandMcNeil

(2020)illustratetheirmethodswithkernelsplacingmassinawindow

[α1,α2]with0<α1<α*<α2<1,e.g.,[0.985,0.995].Insuchasetting,onlyboundedmeasuresproducefiniteteststatistics,andsinceourteststatisticisinvarianttothemeasureofthewindow,withoutlossofgeneralitywecanrestrictattentiontoprobabilitymeasures.BecauseESisanintegralofVaRaboveathresholdlevel,itisnaturalunderthenewregimetoconsidercontinuouskernelsthatweightoneveryαabovesomethreshold,e.g.,α1=0.975intheBaselcontext.Inthissetting,evensomeunboundedmeasurescanbeguaranteedtoyieldvalidteststatistics.Further,becausebankmodelstendtobreakdownunderextrememarketeventsandunboundedmeasuresweightmostheavilyonsuchtailevents,weexpectunboundedmeasurestodelivermorepowerfultests.Weconfirmthisintuitioninsimulationexercisesandshowaswellthatthispowerdoesnotcomeattheexpenseofsizedistortions.

Thetopicofbacktestingexpectedshortfallhasledtoalivelydebateaboutwhether

ornotESisamenabletobacktesting(Gneiting,

2011;

AcerbiandSzékely,

2014;

Fissleret

al.

,

2016;

AcerbiandSzékely,

2023)

.Agrowingliterature,including

Pattonetal.

(2019)

and

Barendseetal.

(2023),employselicitabilitytheorytodevelopjointbacktestsofVaR

andES.Forregulatoryuse,thismethodologywouldgenerallyrequirethatbankssubmittimeseriesofbothVaRandESestimates,althougharecentpaperof

BayerandDimitriadis

(2022)suggestsaworkaroundtoobtainatestofESestimatesonly,atthepossibleexpense

ofsomemodelmisspecification.

IssuesrelatedtobacktestingestimatesofriskmeasuressuchasESaresidesteppedinourframeworkbecausewetesttheforecastdistributionsfromwhichriskmeasuresareestimated,ratherthantheestimatesthemselves.ItmaybenotedthatanumberofrecentpapersproposePIT-basedapproachestobacktestingexpectedshortfalland,inparticular,exploitthecumulativeviolationprocessof

DuandEscanciano

(2017),whichcanbeviewedasa

4

particularchoiceofspectraltransformation.Theseinclude

Duetal.

(2023),whoproposean

improvedconditionalESbacktest,

HogaandDemetrescu

(2023),whoproposeareal-time

monitoringprocedureforESforecastsand

Huéetal.

(2024)whouseorthogonalpolynomials

tojointlytestmomentconditionsforthecumulativeviolationprocessandtheprocessofdurationsbetweenVaRexceedances.

EveniftheregulatorisinterestedexclusivelyintheuppertailofthePITdistribution,itisoftenthecasethatmodelsthataremisspecifiedintheuppertailmaybesimilarlymisspecifiedinthelowertail.Forexample,inariskmanagementsetting,afailuretocapturestochasticvolatilityinthedistributionoffinancialreturnsleadstounderestimationofextremegainsaswellasextremelosses.

Berkowitzetal.

(2011)and

O’BrienandSzerszen

(2017)provideevidenceofneglectedstochasticvolatilityinthebankingcontextbyshowing

thatsimpleGARCHmodelsfittedtobankP&Loftenoutperformbankinternalmodels.ExpressedintermsoftheobservedPITvalues,suchamisspecificationwouldproducetoofewmiddlingPITandtoomanylowandhighPIT.Thus,eveniftheregulatorisconcernedonlywithlargelosses,akernelthatassignsnoweighttothelowertailofthePITdistributionfailstocapturedatathatmayberelevanttodetectingmisspecificationintheuppertail.

Weshowhowtheregulatorcanpre-processthePITvalues,byanoperationwedescribeasfolding,sothattailvaluesfromleftandrightintheoriginalPITdistributionaremappedtotheuppertailofthepre-processeddistributionwithoutalteringthedistributionoftheteststatisticunderthenullhypothesis.

Asimpleexampleofasuitablepre-processorwouldapplythev-shapedmappingT(u)=|1?2u|tothePITvalues.Underthismapping,theeventthatapre-processedPITvalueisintheuppertail,T(PIT)∈[v,1],isequivalenttotheeventthatthePITliesinaunionofintervalsinbothtails,{PIT∈[0,(1?v)/2]∪[(1+v)/2,1]}.ItisstraightforwardtoseethatifthePITareinfactuniformlydistributed(asunderthenullhypothesisofthebacktest)thenthetransformedPITareuniformlydistributedaswell.ThelinearsymmetricmappingT(u)=|1?2u|isonlyasingleexampleofaverylargeclassofuniformdistributionpreserving

5

(u.d.p.)transformations.Acommonfindingintheempiricalliteratureisthatthedistribu-

tionofmarketreturnsisasymmetricsuchthatthetailoflargelossesisheavierthanthetailoflargegains,aphenomenonthatledtothedevelopmentofasymmetricGARCH-typemodelsincorporatingbothleverageeffectsandskewedinnovationdistributions,including

AGARCH(Engle,

1990),EGARCH(Nelson,

1991)andGJR-GARCH(Glostenetal.,

1993)

.Weshowthatasymmetricmembersoftheu.d.p.classcanbechosentohighlightmodelskewnessaswellaskurtosis.

InSection

2,weextendthebacktestingframeworkof

GordyandMcNeil

(2020)toal

-lowforunboundedkernelsandu.d.p.foldingtransformations.Akeyresultdemonstratesthatfoldingisnotredundant,i.e.,pre-processingdeliversbackteststhatcannototherwisebeobtained.InSection

3

weintroducetwonovelfamiliesofunboundedkernels.MonteCarlosimulationsdemonstratethatthesekernelsdeliverbackteststhatarewell-sizedandhighlysensitivetounmodelledkurtosis.Aparsimoniousbutflexiblefamilyofv-shapedpre-processorsisintroducedinSection

4.

MonteCarlosimulationsshowhowpre-processingfurtherhighlightsunmodelledkurtosis.Pre-processorscanbeeffectiveaswellinthepres-enceofunmodelledskewness.However,intheabsenceofmaterialexcesskurtosis,apoorlychosenpre-processorcanmaskratherthanenhancethesignatureofmodelmisspecificaton.Section

5

offersguidanceonimplementationinpracticalsettings.

2Extendedspectralbacktesting

2.1Backtestingset-up

Weassumethataforecastermodelsportfoliolosses(Lt)onafilteredprobabilityspace(?,F,(Ft)t∈N0,P)whereFtrepresentstheinformationavailabletotheforecasterattimet,N0=N∪{0}andNdenotesthenon-zeronaturalnumbers

.1

Foranytimet∈N,thelossLtisanFt-measurablerandomvariablewithconditionaldistributionfunction(df)givenby

1LtisthenegativevalueofP&L,solargelossesareassociatedwiththerighttailofthedistribution.

6

Ft(x)=P(Lt≤x|Ft?1).Inmostapplicationsthisdistributionisnottime-invariant,duetoserialdependenciesin(Lt)andchangesinthecompositionoftheportfolioovertime.

AttimettheforecasterbuildsamodelF-tofFtbasedontheinformationFt?1.PIT-valuesaretherandomvariablesobtainedbysettingPt=.Ifthemodelsformasequenceofidealprobabilisticforecastsinthesenseof

Gneitingetal.

(2007),i.e.,coinciding

withtheconditionallawsFtofLtforeveryt,thentheresultof

Rosenblatt

(1952)implies

thattheprocess(Pt)isasequenceofiidstandarduniformvariables.PIT-valuescontaininformationaboutexceedancesofquantileestimatesatanylevelu:ifu,t=denotestheestimateoftheu-quantileofFtcalculatedusingthegeneralizedinverseofF-tatprobabilitylevelu,thenPt≥u??Lt≥u,t.

Weadoptthepositionofanexternalmodelvalidator,suchasaregulator,whousesthePIT-values(Pt)totakeadecisiononthequalityoftheforecastingmethodology.Forthepurposesofthispaper,weassumethatthevalidatorhasaccessonlytothesePITvaluesalthoughthisrestrictioncouldberelaxedconsiderably.WhatisessentialisthatthevalidatordoesnotobservetheentiredistributionF-twhichreflectstherealityofmostregulatoryregimes.Further,forbrevity,weconsideronlytestsofunconditionalcoverage.Applicationofunboundedmeasuresandfoldingpre-processorswouldapplywithoutcomplicationtothetestsofconditionalcoveragedescribedin

GordyandMcNeil

(2020)

.

2.2Spectralbacktests

ThemodelvalidatoremploysaspectraltransformationofthePITvaluesoftheform

Wt={T(Pt)≥u}dν(u)(1)

where(i)νisaLebesgue-Stieltjesmeasurereferredtoasthekernelmeasureand(ii)T:I→Iisauniformdistributionpreserving(u.d.p.)transformation;ifU～U(0,1)isastandarduniformrandomvariableandTau.d.p.transformation,thenT(U)～U(0,1).Throughout

7

thepaper,Idenotestheunitinterval[0,1].

In

GordyandMcNeil

(2020)themeasure

νwasrestrictedtobeaprobabilitymeasureandthetransformationTwassimplytheidentitytransformationT(v)=v.Thisset-upwasappropriateforafocusontherighttailoftheforecastdistribution.BylookingatPITexceedancesoflevelsuandusingtheprobabilitymeasureνtoselectandweightlevelsofinterestuattheupperendoftheunitinterval,teststatisticswerederivedthatthatweresensitivetoforecastmodelspecificationatarangeofquantilesintherighttail.

WithanyLebesgue-StieltjesmeasureνondomainI,thereisanassociatedincreasingright-continuousfunctionGν,referredtoasadistributionfunction(df),suchthatν([0,u])=Gν(u).

Itiseasilyseenthat(1)isequivalenttotheclosed-formexpression

Wt=ν([0,T(Pt)])=Gν(T(Pt))(2)

whichshowsthatWtisincreasinginT(Pt).Notethatweemploydfinageneralizedsense,sinceGνisaprobabilitydfonlyiflimu→1Gν(u)=1.Tostreamlinethepresentation,wewillhenceforthimposethefollowingmildregularityconditiononν.

Assumption1.Gνhasatmostafinitesetofdiscontinuitiesandisotherwiseabsolutelycontinuous.

Theunivariatetransformationextendsnaturallytothemultivariatecaseinwhichasetofdistinctkernelmeasuresν1,...,νmisappliedtoPIT-valuestoobtainthevector-valuedvariablesW1...,Wnwhere

Wt=(Wt,1,...,Wt,m)′,Wt,j=νj([0,T(Pt)])=Gj(T(Pt)),j=1,...,m.(3)

WerefertoanybacktestbasedonW1...,Wnasaspectralbacktest.Thenullhypothesisaddressedbyanunconditionalspectralbacktestis

H0:Wt~F(4)

8

whereFdenotesthedfofWtwhenPtisuniform.In

GordyandMcNeil

(2020)twotypes

oftestswereconsidered:spectralZ-testsbasedoncentrallimittheoremargumentsandspectrallikelihood-ratiotests(LR-tests).Theresultsshowedanumberofadvantagesoftheformeroverthelatter,includingbettercontrolofsizeforsimilarorsuperiorpower,easeofimplementationandspeedofexecution.InthispaperwefocusonZ-testsandprovidethenecessaryextensionofthetheorytotheLebesgue-Stieltjescase

.2

WhendimWt=maspectralZ-testisbasedonthefactthatunderthemultivariate

CLT√wheren=n?1Σ1WtandμWandΣWarethe

meanvectorandcovariancematrixofthenulldistributionF.Henceatestcanbebasedonassumingforlargeenoughnthat

Tn=n(Wn?μW)′Σ1(Wn?μW)~χ,(5)

wherewerefertoTnasanm-spectralZ-teststatistic.Whenm=1thechi-squaredtestisequivalenttoatwo-sidedtestbasedon

whereμW=E(Wt)andσ=var(Wt)arethemomentsinthenullmodelFforWt.

Bydefinition,theu.d.p.transformationT(P)doesnotalterthemomentsofPunderthenullhypothesis.Thus,asin

GordyandMcNeil

(2020),thefirstmoment

μWofthetransformedPIT-valuesWtiseasilyobtainedas

μW=(1?u)dν(u)(7)

ThevarianceσofWtandthecross-momentsinthecovariancematrixΣWofWtare

obtainedusingasimpleproductruleforspectrallytransformedPITvalues.

2ThetheoryofspectralLR-testspresentedin

GordyandMcNeil

(2020)carriesthroughinthemore

generalcasewithouttheneedforanysignificantmodification.

9

Theorem2.1.ThesetofspectrallytransformedPITvaluesdefinedbyWt,j=Gj(T(Pt))isclosedundermultiplication.TheproductWt*=Wt,1Wt,2isgivenbyWt*=G*(T(Pt))whereν*isaLebesgue-StieltjesmeasureandtheassociatedfunctionG*satisfies

Itfollowsthatσ=μW*?μ,whereμW*

isfoundbyapplying(7)underthemeasure

ν*obtainedwhenν1=ν2=ν.Thisyields

μW*=(1?u)(2Gν(u)?ν({u}))dν(u).(8)

ThecentrallimittheoremunderpinningtheZ-testrequiresfinitesecondmoments.Fortheunivariatecase,thefollowingpropositionprovidesasufficientconditiononthetailbehaviorofGν.

Proposition2.2.IfGν(u)=O((1?u)?0.5+?)asu→1forsome?>0,thenσisfinite.

Inthemultivariatesetting,theasymptoticdistributionin(5)holdsiftheconditioninPropo

-sition

2.2

issatisfiedforeachνj,j=1,...,m.

2.3Uniformdistributionpreservingtransformations

Weareinterestedinu.d.p.transformationsTthatcanextendourtestingframeworktouncoverdeficienciesinforecastmodelsthatarenotrevealedbytheidentitytransformation(generaltheoryforu.d.p.transformationscanbefoundin

Porubskyetal.,

1988,among

others).Sincethechoiceofkernelmeasureinourframeworkisquiteflexible,onemightaskwhethertheinsertionofanygivenu.d.p.transformationT

in(1)deliversanewZ-testthat

couldnotbeobtainedbychangingthekernelmeasure.

Tothisendweintroducetheconceptofredundancyinthetestframework.Letμandσbethemomentsassociatedwithkernelν.Wesaythatau.d.p,transformationTisredundant

10

forkernelνifthereexistsanotherkernelwithmomentsandthatalwaysdeliversthe

samemagnitude|Zn|

forthetest-statisticin(6)

.Thatis,let{P1,...,Pn}beanarbitrarysampleofPITvalues,andletThenTisredundantif

almostsurely.

Asasimpleexample,considertheu.d.p.transformationT(v)=1?v.

Lemma2.3.Theu.d.p.transformationT(v)=1?visredundantforanyboundedmeasureνandnotredundantforanyunboundedmeasureν.

Furtherexamplesofnon-redundanttransformationsareobtainedbyconsideringu.d.p.transformationsthatarefolding.BythiswemeantransformationsTforwhichalmostallvaluesu∈Iareassociatedwithmultiplevaluesinthepreimageof{u}underT.Tomakethispreciseweintroducesomeadditionalnotationandgiveadefinition.Foragenericfunctionf:D→YandforasetD1?Dwewritef[D1]tomeantheimageofD1underf;similarly,forasetY1?Y,f?1[Y1]isthepreimageofY1underf.

Definition2.4.Forau.d.p.transformationT:I→IletIT?IbethesetdefinedbyIT={u|card(T?1[{u}])≥2}.TisfoldingifIThasLebesguemeasureone.

Forexample,theu.d.p.transformationT(v)=|1?2v|hasIT=I\{0.5}andisclearlyfolding.Thefoldingclassincludesv-shaped,m-shaped,w-shapedandmoregeneralsaw-shapedfunctions.Ourgeneralresultforthefoldingclassis

Proposition2.5.LetνbeameasureforwhichGνisstrictlyincreasingonasub-intervalofI.IfTisafoldingu.d.p.transformationthenitisnotredundantforν.

Theintuitionisthatanduppertailobservationscontributeinoppositesigns(therebyoffsettingoneanother)inthesampleteststatistic.Bycontrast,whenwepre-processthePITvalueswithafolding

11

u.d.p.transformation,Gν(T(P))cannotbemonotonicinP.PITvaluesfromnon-contiguousregionsoftheunitintervalwillmaptothesamevalueofGν(T(P)).WhichPITobservations

contributeinthesamesignandwhichoffseteachotherdependsontheshapeofthepre-processor.

3TwofamiliesofLebesgue-Stieltjeskernels

Weconsidersomepossibilitiesfornovelkernelswhicharenotnecessarilyprobabilitymea-sures.FornotationalsimplicitywepresentthetheoryforthecasewhereTistheidentitytransformation.

Fortheremainderofthepaper,thetestsweconsiderarebasedondfsGνwithdensitiesgνsatisfyinggν(u)>0forα1<u<α2andgν(u)=0foru<α1andu>α2.Incertaincasesweallowformassattheboundaries,i.e.,ν({αi})≥0.Werefertotheinterval[α1,α2]asthekernelwindow.

Remark3.1.Forunboundedmeasures,wehaveGν(u)→∞asu→α2.Insuchcases,onlyα2=1isadmissible.Werewetochooseα2<1,wewouldhavePr(α2<Pt≤1)=(1?α2)>0underthenullhypothesis,sothefirstmomentμWwouldbeinfinite.

3.1Simplekernelsofpowerform

GordyandMcNeil

(2020)observethatthebeta-typedensity

(u?α1)a?1(α2?u)b?1providesaflexibleyetparsimoniousandtractableformforthedensityofGν.Sincethatpaperrestrictedνtothesetofprobabilitymeasures,itwasnecessarytorestricta>0,b>0andtoregularizethekernelbythebetafunctionB(a,b).Herewerelaxtherestrictiononbanddiscardtheregularization.

Foruintheunitinterval,letB(u;a,b)denotethe(unregularized)incompletebetafunc-tion

B(u;a,b)=xa?1(1?x)b?1dx.(9)

12

Wedefinethebetakernelνviathedf

Thiskernelispurelycontinuous,i.e.,ν({α1})=ν({α2})=0.Whenb>0,B(u;a,b)isboundedfromabovebyB(a,b)soWt=Gν(Pt)certainlyhasfinitemoments.However,whenb≤0,B(u;a,b)isunboundedasu→1and,byRemark

3.1,wehavetoset

α2=1.Moreover,theexistenceofmomentshastobecheckedwiththehelpofProposition

2.2

andthefollowingresult.

Proposition3.2.Asu→1,

(

IncombinationwithProposition

2.2,itfollowsthat

Wt=Gν(Pt)hasfinitefirstandsecondmomentsifandonlyifb>?1/2.Inthecaseb=0wenotethat

?ln(1?u)=O((1?u)k),u→1,(10)

foranyk<0,afactthatisusedanumberoftimesinthefollowingsections.Theb=0caseisparticularlyimportantforpracticalapplication.Forsmall|b|,standardalgorithmsforB(u;a,b)maybenumericallyunstableforunear1.However,forb=0,

González-Santander

(2021

,Theorem1)providesafiniteseriesexpansioninelementaryfunctions.

WeperformMonteCarloanalysestoexplorehowthesizeandpowerofspectralbacktestswithbeta-typekernelsdependonthebetaparameters(a,b).WeconsiderfourdifferentchoicesforthedfFofthetruemodelofLt:thestandardnormal,andthescaledt10,scaledt5andscaledt3.TheStudenttdistributionsarescaledtohavevarianceonesodifferencesstemfromdifferenttailshapesratherthandifferentvariances.Wetaketheforecaster’s

13

modelF-tobethestandardnormal,i.e.,wetransformthesampledLttoPIT-valuesasPt=Φ(Lt).Therefore,whenthesamplesofLtaredrawnfromthestandardnormal,thePIT-valuesareuniformlydistributedandareusedtoevaluatethesizeofthetests.ThePITsamplesarisingfromtheStudenttdistributionsshowthekindofdeparturesfromuniformitythatareobservedwhentheforecaster’smodelistoothin-tailed.

Wefixakernelwindowof[α1,1]forα1=0.975.Oursamplesizeisfixedton=500correspondingtotwo-yearsamplesoftradingdayreturns.Ourtablesreportthepercentageofrejectionsofthenullhypothesisatthe5%confidencelevelbasedon216=65,536replications.Allreportedp-valuesarebasedontwo-sidedtests.

Parameters

(1,1)

(2,1)

(1,1/4)

(1,1/8)

(1,0)

(2,0)

(5,0)

Normal

4.7%

4.6%

4.6%

4.5%

4.4%

4.3%

4.9%

Scaledt10

13.7%

19.4%

24.1%

28.6%

34.2%

40.8%

45.1%

Scaledt5

21.2%

34.0%

45.7%

55.0%

64.6%

72.2%

76.4%

Scaledt3

13.1%

28.7%

46.5%

61.3%

75.0%

82.2%

86.5%

Table1:Sizeandpoweroftestsbasedonbetamonokernels.

Kernelwindowis[0.975,1].2^16trialswith500observationspertrial.

ResultsforunivariatebetakernelsarereportedinTable

1.

Forallsetsofbetaparameters,testsarewell-sized.ForeachalternativetruemodelF,wefindthatpowerincreasesasbdeclinesandaincreases.Themagnitudeoftheeffectisextremelylarge.Forexample,againstthescaledt3alternative,therejectionrateincreasesfrom13.1%fortheuniform(beta(1,1))kernelto75.0%forthebeta(1,0)to86.5%forthebeta(5,0).ThispatternisconfirmedacrossafinergridofbetaparametersforthecaseoftheStudentt5inFigure

1.

Tounderstandthispattern,observethatforanytwoPITvaluesα1<p1<p2<1,theratioofthebetakernelsgν(p2)/gν(p1)decreasesinbandincreasesina.Thehigherthisratio,thegreatertheweightinthetestonPITintheneighborhoodofp2relativetoPITintheneighborhoodofp1.AsshowninFigure

2,withinthekernelwindowof[0.975,1],the

distributionsofPITunderthescaledStudenttalternativesdiffermostfromthedistributionunderthenull(greensolidline)aswemovedeeperintothetail.Thus,wegenerallyexpect

14

RejectionRate(logscale)

0.5

0.3

0.1

02468

a

b

0

0.51

248

Figure1:Poweroftestsbasedonbetamonokernelsagainstscaledt5alternative.Kernelwindowis[0.975,1].2^16trialswith500observationspertrial.

teststhatweightmoreheavilyontheright-handtailtodeliverhigherpower.

ResultsforbivariatebetakernelsarereportedinTable

2.

GordyandMcNeil

(2020)

demonstratedthatbikerneltestsaregenerallymorepowerfulthanmonokerneltestswhenthecomponentkernelsofthebivariatetestemphasizeoppositeendsofthekernelsupport.Putanotherway,thelowerthec