韩国央行-利用风险增长密度预测改进韩国GDP增长平均预测（英）

No.2024-10

UsingDensityForecastfor

Growth-at-RisktoImproveMeanForecastofGDPGrowthinKorea

YoosoonChang,Yong-gunKim,BoreumKwak,JoonY.Park

2024.9

ChangyongRhee

04531,Korea

JaeWonLee

(DirectorGeneraloftheInstitute)

2024

BOKWPNo.2024-10

UsingDensityForecastforGrowth-at-

RisktoImproveMeanForecastof

GDPGrowthinKorea

YoosoonChang*,Yong-gunKim**,BoreumKwak***,JoonY.Park†

September,2024

Theviewsexpressedhereinarethoseoftheauthors,anddonotnecessarilyreflecttheofficialviewsoftheBankofKorea.Whenreportingorcitingthispaper,theauthors’namesshouldalwaysbeexplicitlystated.

∗DepartmentofEconomics,IndianaUniversity,Email:yoosoon@.∗∗BankofKorea,Email:ygkim@bok.or.kr.

∗∗∗BankofKorea,Email:br.kwak@bok.or.kr.

†DepartmentofEconomics,IndianaUniversity,Email:joon@.

TheresearchreportedinthispaperwassuggestedbyChangyongRhee,thegovernoroftheBankofKorea.WearegratefultoDomenicoGiannone,MichaelMcCracken,TatevikSekh-posyan,NamKangLee,RaffaellaGiacomini,andtoTaeyoungDohfortheirhelpfulcommentsanddiscussions.WealsothankDowanKimandJihyunKimfortheircarefulandconstructivereviewsandtheparticipantsoftheBOKseminarfortheirfeedback.ThisresearchisfinanciallysupportedbytheBankofKorea.

Contents

I.Introduction 1

II.ForecastsofKoreanGDPGrowth 5

III.TheModelandEconometricMethodology 12

IV.EmpiricalResults 18

V.Conclusion 41

A.ConstructingRealGDPGapandFinancialCondition

Index 45

B.RestrictedModelswithSingleandDoubleFactors 49

C.AdditionalResultsforRestrictedModelwithMean

Factor 53

UsingDensityForecastforGrowth-at-Riskto

ImproveMeanForecastofGDPGrowthinKorea

Inthispaper,westudyhowwemayusedensityforecaststoimprovepointfore-castsfortheKoreanGDPgrowthratesduringtheperiodfrom2013:Q3to2022:Q1.

Althoughthetimespanunderinvestigationismuchshorterthandesired,ourcon-clusionsareclear.Densityforecastsimprovepointforecasts,aslongastheyareeffectivelyapproximatedandrepresentedasfinitedimensionalvectorsbyappro-priatelychosenfunctionalbases.However,theymayonlybeusedtoadjustpointforecasts.Combiningthemwithpointforecaststodefineweightedmeanforecastsdoesnotyieldanymeaningfulimprovement.Thefunctionalbasesweuseforourbaselineapproacharetheleadingfunctionalprincipalcomponents,whichbycon-structionmostefficientlyextractthevariationsindensityforecastsovertime.Todisentangletheeffectsofthemeanandotheraspectsofdensityforecasts,however,wealsousethefunctionalbasis,whichdesignates,astheleadingfactor,themeanfactorthatcapturesthetemporalchangesinthemeanofdensityforecasts.Especiallywiththeuseofthisfunctionalbasis,weseeadrasticincreaseintheprecisionofpointforecastsfortheKoreanGDPgrowthrates.Infact,themeansquarederrorofpointforecastsdecreasesbymorethan33%,iftheyareadjustedbydensityforecastswithourfunctionalbasisincludingthemeanfactor.

Keywords:GDPgrowthrate,pointforecast,growth-at-riskdensityforecast,func-tionalregression,functionalbasis,functionalprincipalcomponentanalysis

JELClassification:C53,E17,E37

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BOKWorkingPaperNo.2024-10

I.Introduction

TheBankofEnglandintroduceditsfamousfanchartsin1996tohelppolicy-makersandthepublicbetterunderstandtherisksanduncertaintiessurroundingtheircentralinflationprojections.Sincethen,centralbanksandmajorresearchinstitutionsworldwidehavebeenprovidingmoreinformationregardingtheun-certaintiesaroundtheirmeanorpointforecastsofkeyeconomicindicators,no-tablyforGDPgrowthandinflation.Thistrendhasledtothepublicationofdensityforecastsbyvariousinstitutionsincluding,amongothers,theBankofCanada,theNorgesBank,theFederalReserveBoardofGovernors,theNYFED,andtheIMF,eachofferingitsownestimatesfortheprobabilitydistribu-tionusingvariouseconometricapproaches,inadditiontotheconventionalpointforecasts,ofthesevariables.TheBankofKorea(BOK)isinternallyexaminingdensityforecastsaswellasannouncingpointforecastsfortheGDPgrowthinKoreatoprovidetheirassessmentsofthegrowth-at-risk(GaR),i.e.,therisksanduncertaintiesassociatedwiththefutureeconomicgrowth,inKorea.

ThetwoforecastsforGDPgrowthrates,pointforecasts,anddensityfore-castsaretypicallypreparedbydistinctiveworkinggroupsfordifferentpurposesrelyingonnotentirelyoverlappingsetsofinformation.However,pointforecastsareconsideredbymostpeopleasmeanforecasts,i.e.,forecastsofthemeanofGDPgrowthrates,1)whicharealsoprovidedbydensityforecastsasoneoftheircharacteristics.Publishingdensityforecastsaswellaspointforecaststhereforenecessarilycreatesaproblemofdiscordance,sincethemeanofadensityforecastwouldnotagreewiththecorrespondingpointforecastunlesstheyarealignedwitheachotherbeforetheirpublication.Therefore,raisedarethreeimportantissuesregardingthejointpublicationofpointanddensityforecasts:(i)whetherdensityforecastsprovideanyusefulinformationforpointforecasts,(ii)howtocombinetheadditionalinformationindensityforecastswiththatinpointfore-

1)PointforecastsmayalsobeinterpretedastheforecastsforotherdistributionalcharacteristicsofGDPgrowthratessuchasmedian,modeorevenaparticularquantilebyspecificallylookingatanappropriatelossfunctionfortheforecastingerror.Inthepaper,weusethemeansquaredlossfunctionfortheforecastingerrorassumingthatpointforecastsareregardedastheforecastsforthemeanofGDPgrowthrates.

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casts,andfinally(iii)howtoalignthemeanofadensityforecastwiththecor-respondingpointforecast.

Inthispaper,weaddressandfindasolutiontoeachoftheseissuesfortheforecastsofGDPgrowthratesinKorea.Aspointforecasts,weusetheBOKofficialone-year-aheadforecastsduringtheperiodfrom2013:Q3to2022:Q1.Overthesameperiod,densityforecastsareobtainedbasedonacopulabasedapproachasproposedinLee(2020).2)Weconstructakeyvariable,theFinancialConditionIndex(FCI),usingvariouseconomicvariablesthatarebelievedtoreflectmacroandfinancialmarketconditionsinKorea.Ascovariates,weincludetherealGDPgap,theU.S.federalfundsrate(FFR),thespreadbetweentheU.S.FFRandtheKoreancallrate,aswellastheFCI.3)Foreachtimeperiod,wefollowAdrianetal.(2019)andcomputefourconditionalquantilevalues,atthelevels5%,25%,75%,and95%ofGDPgrowthratesconditionalonthesetofourcovariates,anddefineaskewedt-densitywithfourparametersthatmostcloselymatchesthecomputedconditionalquantilevaluesasourdensityforecast.

OurstudyemploysafunctionalregressionofthefutureGDPgrowthrateonitsdensityforecastasanadditionalfunctionalcovariate,aswellasitspointforecastasausualscalarcovariate.Thoughsimple,thisfunctionalregressionisexpectedtoprovidedirectanswerstoourquestions.Ifthedensityforecastisasinformativeasthepointforecast,thenitwouldcertainlyimprovetheprecisionofthepredictionifwecombinethedensityforecastwiththepointforecasttocomeupwithanewpredictor.Ourfunctionalregressioncanbeveryusefulinthiscontext,sincewemayjustrunthefunctionalregressionandeasilydefineanewpredictorasalinearcombinationofthetwocovariates:thepointforecastandthedensityforecast.Evenifthedensityforecastisnotasinformativeasthepointforecast,wemaystillusethedensityforecasttoadjustthepointforecastandimprovetheprecisionoftheforecast.Inthiscase,wemaysimplyconsiderthefunctionalregressionofthepredictionerrormadebythepointforecastonthedensityforecastasafunctionalcovariate.

2)LeewasinchargeofproducingdensityforecastsforinternaluseattheBOK,andwasworkingattheBOKatthetimethisprojectwasstartedin2022.

3)Adrianetal.(2019)useasconditioningvariablesGDPgrowthrateandFCI.

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Toestimatethefunctionalregressionrequiredforourstudy,weneedtocon-vertthedensityforecastintoafinitedimensionalvector.Forthis,weapproximatethedensityforecastasafinitelinearcombinationofanappropriatelychosenfunctionalbasis,whichisrepresentedasafinitedimensionalvectorofthecoeffi-cientsappearinginthelinearcombinationofthebasisusedtoapproximatethedensityforecast.Thetransformationtomaptheapproximatedensityforecasttoafinitedimensionalvectorisone-to-oneandpreservesthedistance.Therefore,onceweapproximateandrepresentthedensityforecastasafinitedimensionalvectorusingafunctionalbasis,ourfunctionalregressionessentiallyreducestothestandardregressionthatmaybeestimatedbytheusualOLSprocedure.Anestimateforthefunctionalcoefficientforthedensityforecastcanbeeasilyob-tainedbymappingtheOLSestimatorobtainedfromthecorrespondingstandardregressionsbacktoafunctionalestimatebyapplyingtheinversetransformation.

Forthebaselinefunctionalregressions,weusetheleadingfunctionalprin-cipalcomponents(FPCs)ofobserveddensityforecastsasourfunctionalbasis.Byconstruction,theleadingFPCsmostefficientlyextractthevariationinanyfunctionalobservationsand,forthisreason,itismostwidelyusedasafunc-tionalbasisinawiderangeofapplicationsinfunctionaldataanalysis.Indeed,Changetal.(2022)showthatusingtheleadingFPCsasafunctionalbasistoapproximateandrepresentfunctionalobservationsasfinitedimensionalvectorsentailssomeoptimalpropertiesinestimatinggeneralfunctionalregressions.Inthepaper,however,wealsoemployanotherfunctionalbasistodisentangletheeffectofthemeanofthedensityforecast,fromtheeffectsofanyotheraspectsofthedensityforecast,onthepredictionofactualgrowthrates.Forthispurpose,weusethefunctionalbasisconsistingofthefirstelementdesignatedasthemeanfactor,whichcapturesthetemporalchangesinthemeanofthedensityforecast,andotherelementsgivenbytheleadingFPCsofthecentereddensityforecasts,i.e.,thedensityforecastswiththeirmeansshiftedtozero.

Themostseriouslimitationofourstudyisthatthereareonly35quarterlyobservationsavailablefortheofficialBOKpointforecasts.Wearefullyawareofthefactthatoursamplesizeismuchsmallerthandesiredandthatthecon-sequenceofthislimitedavailabilityofdatacanbeseverelydetrimentaltoour

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study.Thismakesithardforustorelyonthesophisticatedfunctionaldataanalysisthatweneedtoadopttoinvestigateourproblemsinfullgenerality.4)Fortunately,however,weareabletodrawasetofclearandcoherentconclusionsonallofourthreemainquestions:whethertheuseofdensityforecastsishelp-fulatallforimprovingpointforecasts,howtousetheinformationondensityforecaststoprovidebetterpointforecasts,andhowtomakethemeanofden-sityforecastsaccordwithpointforecasts.Ourresultsareconsistentandrobustacrossdifferentchoicesoffunctionalbasesandvariousothertuningparameters,andtheyseemtobequitereliable.

Theuseofdensityforecasts,inadditiontopointforecasts,appearstogener-allyimprovetheprecisionoftheforecastforfutureKoreanGDPgrowthrates.Ifdensityforecastsarecombinedwithpointforecastsbasedonourfunctionalregressiontodefineweightedmeanforecasts,however,theforecastprecisiondoesnotimprovesignificantly.Foramoremeaningfulimprovement,densityforecastsshouldbeusedonlytoadjustpointforecasts.TheseareconclusionsthatwedrawfromourfunctionalregressionestimatedwiththefunctionalbasisconsistingoftheleadingFPCsofobserveddensityforecasts.Todisentangletheeffectsofthemeanandotherdistributionalaspectsofdensityforecasts,wealsouseanotherfunctionalbasisincludingthemeanfactor,whichcapturesthetemporalchangesinthemeanofdensityforecasts,andtwootherfactorsextractedastheFPCsofthecentereddensityforecasts.Withtheuseofthisfunctionalbasis,weseeamostdrasticincreaseintheprecisionoftheforecastforfutureGDPgrowthrates.Infact,themeansquarederrorofpointforecastsdecreasesbymorethan33%,iftheyareadjustedbydensityforecastswiththefunctionalbasisincludingthemeanfactor.

ThedetailsofourempiricalresultshavefurtherimplicationsonhowbestdensityforecastscanbeusedtoadjustpointforecaststoimprovetheprecisionofthepointpredictionofthefutureKoreanGDPgrowthrates.First,ourresultsshowthathistoricallythepointforecastsforKoreangrowthratestendtobelow

4)Duetothelimitedavailabilityofdata,wedidn’tperformanyformalback-testing.Wedrewourconclusionsmostlybasedonthebiasandvariancecomputedsimplyfromtheforecastingregressionmodel.

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whenpessimisticfuturescenariosareprevailingwithpotentiallyhighdownsiderisks.Althoughthetendencytooverreactalsoexistswhenfuturesarehighlyoptimistic,itisnotassignificantasinthepessimisticcase.Second,accordingtoourresults,themeanofadensityforecastisimportantandshouldbeexploitedtoproduceamoreprecisepointforecast.Asaconsequence,itisnotrecommendedtoshiftthedensityforecasttomakeitsmeanalignedwiththatofthepointforecast.Thelossofinformationincurredbysuchapracticecanbesubstantial.Finally,wemayalsouseourresultstodealwiththediscrepancybetweenapointforecastandthemeanofadensityforecast.Thebestwaytodealwiththisproblemisfirsttoadjustthepointforecastusingourfunctionalregressionwiththedensityforecast,andthenrefitthedensityforecastwithitsmeanalignedwiththeadjustedpointforecast.

Therestofthepaperisorganizedasfollows.SectionIIdescribeshowweconstructdensityforecastsofGDPgrowthrates.SectionIIIprovidesabriefin-troductiontothefunctionalregressionweusetopredictgrowthrateusingdensityforecastsalongwiththeBOK’sofficialpointforecasts.SectionIVpresentsourempiricalresults.Itprovidesestimatesoftheweightsonpointforecasts,anddensityforecastsusedtoconstructanewpredictor,computestheadjustmentfactorfromdensityforecaststoimprovethepointforecast,andinvestigateshowthedensityforecastsimprovethepointforecast,especiallyinwhichwaythemeanfactorofthedensityforecastscontributestoimprovingthepointforecast.SectionIValsoprovidesdiscussionsonourfindingsandtheirimplications.Sec-tionVconcludes,andtheAppendixprovidesdetailsoftheanalysesprovidedinthemaintextandsomerobustnesschecks.

II.ForecastsofKoreanGDPGrowth

1.PointForecastofGDPGrowth

ThepointforecastsanalyzedinourstudyaretheforecastsofKoreanGDPgrowthratesconstructedbytheBankofKorea(BOK).TheBOKproducesGDPforecastsforthecurrentyearandthenextyeareveryquarterandreleasestheir

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forecastsfourtimesperyearinFebruary,May,August,andNovember.5)TheBOK’sofficialpointforecastsarefixed-eventforecaststhatareproducedeachquarterforthesametargetvariables,theGDPgrowthratesforthecurrentcal-endaryearandthenextcalendaryear,withdecreasingforecasthorizonsastimeprogressestowardtheendoftherespectivecalendaryear.Underthisfixed-eventforecastingscheme,forexample,theBOK’spointforecastsmadeinMay2022andAugust2022bothprovideaforecastforthesametargetvariable,i.e.,thecurrentcalendaryear2022,butwithashorterforecasthorizonfortheforecastmadelaterinAugust2022.Thefixed-eventforecaststhereforereflectdecreasinguncertaintiesineconomicconditionsastheforecasthorizonshortens,and,conse-quently,theresultingforecasterrorvariancesshowsuchseasonalcharacteristics.

Forourstudy,however,weuseanalternateforecastingschemethatpro-ducesfixed-horizonforecasts.Thefixed-horizonpointforecastsaremoresuit-ableforourempiricalanalysesfortworeasons.First,unliketheBOK’sofficialfixed-eventpointforecasts,fixed-horizonforecastsarelesssusceptibletoseasonalcharacteristicsofforecasterrorvariances.Second,withfixed-horizonforecasts,itisstraightforwardtomatchtheforecasthorizonwiththatofdensityfore-caststhatarecommonlyusedbypolicymakerstocharacterizetheuncertaintyoffuturegrowthratesaftersomefixedamountoftime.Detailsonhowwecon-structdensityforecastsfortheKoreangrowthratesareprovidedinthefollowingsubsection.

Morespecifically,weanalyzepointforecastsforone-year-aheadGDPgrowthrates.Inthisscheme,theforecasthorizonisfixedatfourquarters,whilethefore-casttargetvariablevariestorepresentthefuturegrowthratefourquartersaftertheforecastismade.Incontrasttothefixed-eventpointforecastsillustratedabove,thefixed-horizonone-year-aheadpointforecastsmadeinMay2022andAugust2022provideforecastsfortwodistincttargetvariables,thegrowthratein2023Q1andthegrowthratein2023Q2,respectively,withthesamefore-casthorizonatfourquarters,andconsequentlywithnoaforementionedseasonal

5)InNovember,theforecastsforthecurrentyearandthefollowingtwoyearsaremade.TheBOK’scurrentreportingschedulewasadoptedin2020.Priorto2020,theBOKreleaseditsforecastsinJanuary,April,July,andOctober.

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patternsinforecasterrorvariances.

ToobtaintheBOK’sone-year-aheadpointforecast,wecollectthevintagequarterlypointforecastpathsconstructedbytheBOK.6)Vintagepointforecastpathsforthefirst,second,third,andfourthquartersofeachyearincludepointforecastsofGDPgrowthratesuptoseven-,six-,five-,andeight-quarters-ahead,respectively,fromthequarterswhentheofficialforecastsweremade.Eachvin-tagepointforecastpathwasconstructedusingthesameforecastingprocedureusedtoproducetheBOK’sofficialpointforecastpublishedeachquarterinoursampleperiod.Sincethesepointforecastpathswereconstructedwiththesamedataset,thesameforecastingmodel,andthesameexpertjudgmentsasthoseusedtoproducetheofficialpointforecasts,weassumethepointforecastsusedinourstudyasofficialBOKone-year-aheadpointforecasts.

Forourstudy,weconstructaquarterlytimeseriesofone-year-aheadpointforecastsusing,foreachquarter,thefour-quarter-aheadforecastfromthevintagepointforecastpathfortheperiod2013:Q3to2022:Q1.Oursampleperiodisdeterminedbythedataavailability.Itstartsfrom2013:Q3sincethevintageBOK’spointforecastpathsforGDPgrowthratesareavailableonlyfromthen,anditendsin2022:Q1sinceweneedactualone-year-aheadGDPgrowthratesastheforecasttargetvariableforouranalysis.7)Oursamplesizeisrathershortwithonly35quarterlyobservations,butthisisthelongestwecanstretchifwewanttouseonlyvintagedata.Wemay,ofcourse,extendthesampleperiodifweusemodel-basedpointforecastsconstructedex-postfortheearlieryears.

2.DensityForecastofGDPGrowth

Therearemanydifferentwaystoconstructdensityforecasts.Oneofthemostcommonlyusedapproaches,duelargelytoitssimplicity,istofollowAdrianetal.

6)Quarterlypointforecastpathsareconfidentialdataandconstructedforinternalanalysisandjudgmentsonly.

7)Toconstructthetargetannualgrowthrateateachquarter,sayat2020:Q4,weneedfourquarterlyGDPleveldatafor2020:Q4,2020:Q3,2020:Q2,and2020:Q1,alongwithfouryear-over-yearGDPchanges,2021:Q4-2020:Q4,2021:Q3-2020:Q3,2021:Q2-2020:Q2,and2021:Q1-2020:Q1.

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(2019)andobtaindensityforecastsfromthelinearquantileregressionoffutureGDPgrowthratesonthecurrentfinancialconditionindex(FCI),aswellasGDPgrowthrate,andfitaskewedt-densitywithfourparametersthatmostcloselymatchesthecomputedquantilevalues.Asanalternative,Carrieroetal.(2020)proposemodel-implieddensityforecastsconstructedfromaBayesianVARmodel.Linearityofthequantileregressionandnormalityoftheerrordistributionhavebeenchallenged,andtherearemoreflexiblemodelsandmethodsallowingfornonlinearityandnonparametricerrordistributions.Theyincludeanonpara-metricestimationofnonlinearVARbyAdrianetal.(2021),aMarkovswitchingmodelbyCaldaraetal.(2021),useofGARCH-typevolatilitybyBrownleesandSouza(2021),andasemi-parametricestimationusingsurveyforecastsbyClarketal.(2020).

FollowingLee(2020),weestimateconditionalquantilesoffutureGDPgrowthbyutilizingaD-vinecopulabasedquantileregressionmethodasintroducedbyKrausandCzado(2017).Givencovariates(Xt),Xt=(X1t,...,XMt),thefunc-

tionQYt+h∣X(τ∣x)representingtheτ-thconditionalquantilevalueofthevariable(Yt+h)ofinterestisdefinedastheinverseoftheconditionaldistributionfunc-

tionFY−th∣Xt(r∣xt)of(Yt+h)on(Xt).Incontrasttothelinearquantileregression

modelusedbyAdrianetal.(2019),assumingalinearrelationshipbetweentheconditionalquantilesandexplanatoryvariables,thecopulabasedquantilere-gressionmethodallowsustoinvestigateanonlinearrelationshipbetweentheconditionalquantilesandcovariatesbymodelingtheconditionaldistributionfunction(FYt+h∣Xt)asacopulafunction.8)

Specifically,wesupposethatthevariable(Yt+h)ofinterestandcovariates

spectively.Thenwedefine(Vt+h)and

1followunivariatemarginaldistributionfunctionsFYt+hand(FXj)1,re-

8)LinearquantileregressionmodelssusceptibletothepotentialissueofquantilecrossingasconditionalquantilesareaffinefunctionsofcovariatesXt.Underthisspecification,theslopeparametersdependontheprobabilityindexτ,whichmaycausequantilesatdifferentvaluesofτtocrosseachother,and,therefore,theconditionalquantilescannotbelinearincovariatesXt.Thisissuepossiblyleadstobiasintheestimation,which,inturn,mayresultinover-orunderestimatingrisks.Ontheotherhand,thecopula-basedconditionalquantilefunctionnaturallysatisfiesmonotonicitywithoutsuchanissue.

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FXj(Xj)bytheprobabilityintegraltransformation(PIT)of(Yt+h)and(Xjt),re-spectively,whichareuniformlydistributedontheinterval[0,1],andsetthejointdistributionof(Yt+h)and(Xt)as

F(yt+h,x1t,...,xMt)=C(vt+h,u1t,...,uMt),

whereCdenotesacopulathatisa(M+1)-dimensionaldistributionfunctiononthehypercube[0,1]M+1withuniformlydistributedmarginals.Theconditionaldistributionfunction(FYt+h∣Xt)of(Yt+h)on(Xt)cannowbewrittenasthatoftheirPITcounterparts,i.e.,

FYt+h∣Xt(r∣xt)=CVt+h∣Ut(r∣ut),

forr∈R.Asaresult,theconditionalquantilefunctionof(Yt+h)on(Xt)canbeobtainedfromtheconditionalcopulaquantilefunctionCVt+h∣Utof(Vt+h)on(Ut)

as

QYt+h∣Xt(τ∣xt)=FY−th∣Xt(CVt+h∣Ut(τ∣ut)),(II.1)

forτ∈(0,1).AD-vine,asasubclassofregularvinecopulas,allowsustomodelmultivariatecopulasusingtheblocksofbivariatecopulas,aso-calledpair-copulaconstruction.RefertoAasetal.(2009)foradetailedexaminationofbivariatepair-copulas.KrausandCzado(2017)implementaD-vinecopulatomodelquan-tileregressionsandshowthattheproposedmethodworksfastandaccuratelyeveninhighdimensions.

Inpractice,usingaD-vinecopulabasedquantileregression,weestimateconditionalquantilesofh-quarter-aheadrealGDPgapforh=1,...,4.Toobtain

adensityforecastofone-year-aheadrealGDPgrowth,wetransformestimatesoftheh-quarter-aheadrealGDPgaptotheconditionalquantilesoftheone-year-aheadrealGDPgrowthrateandthenfitthoseestimatestotheskewedt-distribution.Toestimatethenon-linearquantileregressionmodel(II.1),weconsidertheh-quarter-aheadrealGDPgap,whichisthecyclicalcomponentofrealGDPastheresponsevariable.FollowingHamilton(2018),wedecomposetherealGDPintotrendandcyclicalcomponentsbyregressingtheh-quarter-ahead

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logrealGDPontheconstant,current,andlaggedvaluesoflogrealGDP.Forexplanatoryvariables,weusetherealGDPgap,theFCI,theU.S.FFR,andthedifferencebetweentheU.S.FFRandthecallrateinKoreaavailableattimet.DetailsforconstructingrealGDPgapandFCIareavailableinAppendixA.Thedatasetusedintheestimationofdensityforecastisavailablefrom1991:Q2to2022:Q4.AllvariablesusedfordataconstructionareobtainedfromtheBOKEconomicStatisticsSystem(ECOS).

WeobtainapproximateestimatesoftheconditionalquantilefunctionfortheGDPgapforuptofourquartersaheadbyestimatinganonlinearquantileregression(II.1).Foreachquantile,weconverttheh-quarter-aheadGDPgapintothelevelofGDPbyaddingbackthetrendcomponentestimatedusingHamilton’sregression-basedfilter.Werandomlydrawh-period-aheadrealGDP30,000timesfromtheconditionalquantilesoftherealGDPlevelandcalculatetheone-year-aheadGDPgrowthrateforeachsimulationoverthepreviousfourquartersofrealGDPobservationstoobtainconditionalquantilesoftheone-year-aheadrealGDPgrowthrate.

Inthesubsequentstep,asinAdrianetal.(2019),wefittheskewedt-densitytosmooththeconditionalquantilevaluesandrecoveraprobabilitydensityfunctionof