纽约联储-新冠肺炎疫情后自然利率的测算（英）

NO.1063

JUNE2023

MeasuringtheNaturalRateofInterestafterCOVID-19

KathrynHolston|ThomasLaubach|JohnC.Williams

MeasuringtheNaturalRateofInterestafterCOVID-19

KathrynHolstonThomasLaubach,andJohnC.Williams

FederalReserveBankofNewYorkStaffReports,no.1063

June2023

JELclassification:C32,E43,E52,O40

Abstract

WemodifytheLaubach-WilliamsandHolston-Laubach-Williamsmodelsofthenaturalrateofinteresttoaccountfortime-varyingvolatilityandapersistentCOVIDsupplyshockduringthepandemic.ResultingestimatesofthenaturalrateofinterestintheUnitedStates,Canada,andtheEuroAreaattheendof2022areclosetotheirrespectivelevelsestimateddirectlybeforethepandemic;thatis,wedonotfindevidencethattheeraofhistoricallylowestimatednaturalratesofinteresthasended.Incontrast,estimatesofthenaturalrateofoutputhavedeclinedrelativetothoseprojectedbeforethepandemic.

Keywords:naturalrateofoutput,time-varyingvolatility,Kalmanfilter,trendgrowth,COVID-19pandemic

_________________

Williams:FederalReserveBankofNewYork(email:john.c.williams@).Holston:Harvard

University(email:kathrynaholston@).Thispaperbuildsuponandextendstheworkofthethreeauthorsthatwasinitiallycontainedinthenote“AdaptingtheLaubachandWilliamsandHolston,Laubach,andWilliamsModelstotheCOVID-19Pandemic,”May27,2020.TheauthorsthankparticipantsattheThomasLaubachResearchConferenceforvaluablecomments.TheyalsothankLoganCaseyforoutstandingresearchassistance.Thispaperpresentspreliminaryfindingsandisbeingdistributedtoeconomistsandotherinterestedreaderssolelytostimulatediscussionandelicitcomments.Theviewsexpressedinthispaperarethoseoftheauthor(s)anddonotnecessarilyreflectthepositionoftheFederalReserveBankofNewYork,theFederalOpenMarketCommittee,ortheFederalReserveSystem.Anyerrorsoromissionsaretheresponsibilityoftheauthor(s).Toviewtheauthors’disclosurestatements,visit

/research/staff_reports/sr1063.html

.

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1Introduction

Thedownwardtrendinestimatesofthenaturalrateofinteresttohistoricallylowlev-elsobservedinmanycountrieshasgarneredconsiderableattentionanddebateabouttheirsourcesandconsequences(LaubachandWilliams,2016,GourinchasandRey,2019,RachelandSummers,2019).1Theeventsofthepastfewyears,includingtheCOVID-19pandemicandsubsequentpolicyactions,haverenewedthedebateoverwhetherhistoricallylownaturalratesofinterestwillpersistinthepost-pandemicera(InternationalMonetaryFund,2023,Obstfeld,2023).Answeringthisquestionusingempiricalmodelsofthenaturalrateofinteresthasbeenchallengingowingtotheunprecedentedmacroeconomicvolatilityacrosstheglobeduringthepandemic.Thispaperdevelopsandimplementsadata-drivenapproachthataddressestheextraordi-naryeffectsofthepandemicusingtheHolston,Laubach,andWilliams(HLW,2017)andLaubachandWilliams(LW,2003)modelsofthenaturalrateofinterest.Ourapproachpreservestothegreatestextentthebasicstructureandflexibilityoftheoriginalmodels,whileprovidingconsistentmodelestimatesofnaturalratesbefore,during,andafterthepandemicperiod.Italsohasbroaderapplicationtomodelsthatestimatelatentvariablesusingfrequentist(asinHLWandLW)andBayesianmethods.

TheHLWandLWmodelsapplytheKalmanfiltertotranslatemovementsinrealGDP,inflation,andshort-terminterestratesintoestimatesoftrendgrowth,thenaturalrateofoutput,andthenaturalrateofinterest.Themodel’sstructureisflexibleandincorporatestransitoryandpermanentshockstosupplyanddemandanddynamicendogenousbehaviorofinflationandoutput.However,likeothermodelsthatusetheKalmanorotherstatisticalfilters,identifyingassumptionsregardingthenatureoftheshockprocessesareimposed.Inparticular,theshocksareassumedtobeseriallyuncorrelatedanddescribedbytime-invariantGaussiandistributions.TheCOVID-19pandemicgeneratedextraordinaryswingsinmacroeconomicdatathataredramaticallyatoddswithbothoftheseassumptions.

First,theassumptionofatime-invariantGaussiandistributionfortheshockpro-cessesisclearlycontradictedbythedata.Relativetothehistoricalexperienceofthepriorhalfcentury,theCOVID-19pandemicisanextremetaileventintermsofits

1Thereisarelatedliteratureonthehistoryofrealinterestratesoveraverylongtimespan.See,forexample,Rogoff,Rossi,andSchmelzing(2022)andreferencestherein.

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demonstratethatthesetwomodificationseffectivelyaddressthetwoeconometricissuesassociatedwiththepandemic.3Theestimationprocedureyieldsparameterestimatesconsistentwiththemodelstructure,andthepre-pandemicestimatesofthenaturalratesofoutputandinterestandthetrendgrowthrateareverysimilartothoseestimatedondataendingin2019.

ThepatternofhistoricallylowestimatesoftrendGDPgrowthandthenaturalrateofinterestexperiencedbeforethepandemicpersistaftertheCOVID-19pandemic.Inallthreeeconomies,theestimatesoftrendgrowthandthenaturalrateofinterestin2022arewithinafewtenthsofapercentagepointofthecorrespondingestimatesfor2019.Inparticular,theseestimatesprovidenoevidenceofareversalofthetrenddeclineinestimatesofthenaturalrateofinterestbasedondatathrough2022.

Inallthreeeconomies,estimatesoftheCOVID-adjustednaturalrateofoutputin2022aresignificantlylowerthanwhatthemodelpredictsbasedonpre-pandemicdata.ThesedeclinesreflectboththeestimatedeffectsofCOVID-relatedrestrictionsandpermanentnegativeshockstothenaturalrateofoutput.Accordingtothemodel,thesedeclinesinthenaturalrateofoutputarethemosteconomicallysignificantlastingeffectsoftheCOVIDera.

Thispaperisorganizedasfollows.Section2describestheHLWmodelandtheevidenceofsignificantdeparturesfromthemodel’sassumptionsbroughtonbytheCOVID-19pandemic.Section3describesthemodificationstothemodeltoaddressthepandemic-relatedeffects.Section4reportsestimationresults.Section5reportsresultsfromrobustnessexercises.Section6concludes.

2ResultsfromthePre-PandemicModel

Inthissection,weprovideashortdescriptionoftheoriginalHLWmodel.Weshowthatthedataduringthepandemicgeneratelargeoutliersthatareinconsistentwiththeassumptionsofthemodel.

ceduresforbothmodels.Specifically,consistentwiththespecificationoftheLWmodel,wenowestimatetherelationshipbetweentrendgrowthandthenaturalrateofinterestintheHLWmodel,insteadofrestrictingittounity.Wealsomakeminortechnicaladjustmentsthataligntheassump-tionsusedinthevariousstagesofthemodelestimationprocedure,describedinAppendixA1.

3InHLW(2017),themodelwasalsoestimatedusingdatafromtheUnitedKingdom.ExtendingthesampletoincludethemostrecentyearshasweakenedtheestimatedrelationshipbetweentheoutputgapandrealinterestratesintheUKdata,makingestimatesofthenaturalrateofinteresthighlyunreliable.Forthatreason,wenolongerestimatethemodelfortheUnitedKingdom.

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2.1TheOriginalHLWModel

IntheHLWmodel,thenaturalrateofinterest,r,istherealinterestrateconsistentwithoutputequalingitsnaturalrate,y,andstableinflation.Asisstandardinthisliterature(e.g.,seeWoodford,2003),wemodeltheoutputgapandinflationdynamicsasafunctionoftherealinterestrategap,rt−r,usinganintertemporalISequationandPhillipscurverelationship,inlinewiththeNewKeynesianframework:

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t=ay,1t−1+ay,2t−2+工(rt−j−r−j)+ϵ,t

j=1

πt=bππt−1+(1−bπ)πt−2,4+byt−1+ϵπ,t

(1)

(2)

Theoutputgapisdefinedast=100·(yt−y),whereytandyarelogarithmsofrealGDPandtheunobservednaturalrateofoutput,respectively,rtistherealshort-terminterestrate,πtdenotesconsumerpriceinflation,andπt−2,4istheaverageofthesecondtofourthlagsoftheinflationrate.4Thestochasticdisturbancesϵ,tandϵπ,taretransitoryshockstotheoutputgapandinflationequations,respectively.

WeusetheKalmanfiltertoestimatethelatentvariables,whicharethenaturalrateofoutput,itstrendgrowthrate,andaprocesscapturingotherlow-frequencydeterminantsofthenaturalrateofinterest.InkeepingwiththestandardKalmanfilterapproach,thestochasticinnovationstothemeasurementequations–theISandPhillipscurveequations–areassumedtofollowaGaussiandistributionwithstandarddeviationsσy~andσπ,respectively,andtobemutuallyandseriallyuncorrelated.

Incontrasttothetransitoryshockstotheoutputgapandinflationequations,movementsinrreflecthighlypersistent,orpermanent,shiftsintherelationshipbetweentherealshort-terminterestrateandtheoutputgap(Williams,2003).Thelawofmotionforthenaturalrateofinterestisgivenby

r=c·gt+zt(3)

wheregtisthetrendgrowthrateofthenaturalrateofoutput,andztcapturesotherdeterminantsofrWespecifythethreelatentvariablesinourstate-spacemodel

4SeeHLW(2017)Section2andAppendixAfordetailsofthemodelspecification.Wetakeasastartingpointtheopen-economyNewKeynesianmodelspecificationasinGal´ı(2008)andrelaxtwostandardrestrictionstoworkwithreduced-formISandPhillipscurveequations.

5Notethat,consistentwithLW(2003),werelaxtheassumptioninHLW(2017)ofaone-for-one

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asfollows.Thelogarithmofthenaturalrateofoutputfollowsarandomwalkwithastochasticdrift,gt,thatitselffollowsarandomwalk,

y=y−1+gt−1+ϵy*,t

gt=gt−1+ϵg,t

(4)

(5)

andthecomponentztcapturingotherdeterminantsofr,whichisassumedtofollowarandomwalkaswell,

zt=zt−1+ϵz,t(6)

Weassumethatthedisturbancesϵy*,t,ϵg,t,andϵz,tarenormallydistributedwithstan-darddeviationsσy*,σg,andσz,respectively,andareseriallyandcontemporaneouslyuncorrelatedwithallotherdisturbances.

Equations1and2makeupthemeasurementequationsinourstate-spacemodelandcanbeexpressedas

yt=A\·xt+H\·ξt+ϵt(7)

withstochasticinnovationsϵt.Equations4,5,and6makeupthestateequationsinourstate-spacemodel,writtenas

ξt=F·ξt−1+ηt(8)

whereξtisthestatevectoroflatentvariablesandηtisthevectorofstochasticinnovations.SeeAppendixA1forthefullstate-spacerepresentationofthemodel.

2.2OutliersinEstimationwith2019:Q4ModelParameters

WenowanalyzehowtheextrememovementsinGDPandinflationduringtheCOVID-19pandemicyieldlargeoutliersinthestandardHLWmodel.Wethenshowthatesti-matesofthelatentvariablesareheavilyaffectedbythesesizableoutliers,evenwhenweconstrainthemodelparametersattheirpre-pandemicvalues,anddemonstratethatmodificationstotheHLWandLWmodelsarenecessary.

relationshipbetweentrendgrowthandthenaturalrateofinterestandestimatethisrelationship.SeeAppendixA1fordetailsonchangestotheHLW(2017)model.

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Beforemakinganyadjustmentstothemodels,webeginbyestimatingthestan-dardHLWmodelinSection2.1withdatathrough2019:Q4,priortotheonsetoftheCOVID-19pandemic.6Forreference,theupperthreepanelsofFigure2showthefullsamplesofdata.Wefixthemodelparametersattheirestimatedvaluesandre-estimatethelatentvariablesthrough2022:Q4usingtheKalmanfilter,takingallparametervaluesasgivenfromthe2019:Q4estimatedmodel.Wealsofixtheinitialvectorofunobservedstatesanditscovariancematrixatthe2019:Q4values.7Thisexerciseisequivalenttodroppingobservationsbeginningin2020:Q1throughtheendofthesampleduringthemaximumlikelihoodestimationofmodelparameters,whileallowingtheKalmanupdatingproceduretocontinuewithoutmodificationthroughtheendofthesample.Inotherwords,wemakenomodificationstothestate-spacemodel,exceptthatthemodelcoefficientmatricesandcovariancematricesintheKalmanfilteringprocedurearefixedattheir2019:Q4values.Thiswouldbeasuit-ableapproachifwetaketheviewthatthepandemicperiodisnotinformativeforthemodelparameters,suchastheslopesoftheISandPhillipscurveequations,butisinformativeforthelatentvariables.

ThefinalstepoftheKalmanfilteringproceduretoestimatethevectorofunob-servedstatevariablesattimet(denotedasξˆt|t,andconditionalontheinformationsetattimet)isgivenbytheKalmanupdatingequation,

ξˆt|t=ξˆt|t−1+Kt·(yt−A′·xt−H′·ξˆt|t−1)

山、

尸使

one-step-aheadpredictionerror

(9)

whereξˆt|t−1istheinitialestimateofthestatevectorduringtheperiod,conditionalontheinformationsetattimet−1,andKtistheKalmangainmatrix.Thefinaltermcontainstheone-step-aheadpredictionerrors(orforecasterrors)correspondingtothemeasurementequationsinthemodel.Theseone-step-aheadpredictionerrorsaretheresidualstotheISandPhillipscurveequations,usingtheforecastofyt(thevectorofcontemporaneousendogenousvariables,thatis,theoutputgapandinflation)based

6Throughoutthepaper,weusethecurrentdatavintageatthetimeofpublication,regardlessofthesampleperiod.

7Westoretheestimatedparametervectorθfromthefinal(stage3)modelaswellasthesignal-to-noiseratiosλgandλXfromthemedianunbiasedestimationproceduresfollowingstages1and2,respectively.SeeHLW(2017)foradescriptionoftheestimationprocedureandfootnote6fortheinitializationprocessofthevectorofunobservedstates,itsconditionalexpectationξ1|0inthefirstperiod,andthecovariancematrixP1|0.

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onthedataattimetandinformationattimet−1,correspondingtothestatevector

ξˆt|t−1:

yt−E[yt|xt,ζt−1]=yt−(A′·xt+H′·ξˆt|t−1)

(10)

InEquation9,theKalmangainmatrixdictatestheweightplacedontheone-step-aheadpredictionerrorsduringthelatentvariableestimation.AlargerKalmangainKtindicatesthatthefinalestimatesofthelatentvariablesaremoreheavilyinfluencedbythegapbetweentherealizeddataandthemodel’sprediction,relativetoξˆt|t−1,

thepriorestimateofξtconditionaloninformationintheprecedingperiod.

Inthisinitialexercise,thecoefficientmatricesH′(onthestatevectorξˆt|t−1)andA′(onthedataxt)arefixedattheir2019:Q4values.Theresultingone-step-aheadpredictionerrorsareverylargeduringmuchofthepandemicperiod.Theselargeforecasterrorstranslatedirectlytotheestimatedvectorofunobservedlatentvariables(y,gt,zt),sothatthedataduringthisperiodhaslargeeffectsonestimatesoftheselatentvariables.

BecausetheHLWmodelisanunobserved-componentsmodel,thereexistsan-othersetofmodelresidualsinadditiontotheone-step-aheadpredictionerrorsthatarecommonlyusedfordiagnostictesting.Theseauxiliaryresidualsaresmoothedestimatorsofthedisturbancestothemeasurementequations,ϵt,andtothestateequations,ηt,meaningthattheyincorporateallavailableinformationoverthefullsampleperiodandprovideadifferentinterpretationofthestochasticinnovations(HarveyandKoopman,1992;Harveyetal.1999).Theyhavetheadvantageinapplyingatestforoutliers:undertheassumptionthatthestochasticinnovationsarefromaGaussiandistribution,standardizedauxiliaryresidualsgreaterthan2(inabsolutevalue)indicateeitherthepresenceofoutliersorstructuralchange.

WeusethealgorithmfromKoopmanandDurbin(2000)toobtainthestandard-izedauxiliaryresidualstothemeasurementequations,t/e,t,andtothestateequa-tions,t/η,t.ThegoldlinesinFigure3showsthatthestandardizedauxiliaryresid-ualstothemeasurementequations,givenby

te,t

=

E[ϵt|yT,xT,ζT]

SD[ϵt|yT,xT,ζT],

(11)

indicateextremeoutlierstotheoutputgapequationinalleconomiesinoursample,underthestandardHLWmodelusing2019:Q4parametervalues.IntheUnitedStatesandCanada,standardizedauxiliaryresidualstotheISequationare9to10timesthe

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outlierthresholdforaGaussianmodelinthesecondquarterof2020,and17timesthethresholdintheEuroArea.Whilenotasextreme,themodelresidualsalsodetectoutlierstotheinflationequationineachofthethreeeconomies,withstandardizedauxiliaryresidualstothePhillipscurveequationreachingdoubletheoutlierthresholdintheUnitedStates.

Standardizedauxiliaryresidualstotheunobservedstateequationsaregivenby

tη,t

=

E[ηt|yT,xT,ζT]

SD[ηt|yT,xT,ζT]

(12)

AsshownbythegoldlinesinFigure4,theseresidualsdemonstratethepresenceofextremeoutlierstothenaturalrateofoutputinthestandardHLWmodelacrossalloftheeconomiesinoursample.

ThisobservationisnotuniquetotheHLWmodel.Figure1showsthatGDPreal-izationsduringthepandemicareoutlierswithrespecttohistoricaldata.Macroeco-nomicfluctuationsofthismagnitudewouldresultinoutliersinanystandardmacroe-conomicmodel.Asexpected,evenwhenpandemic-eraGDPandinflationdataareexcludedduringparameterestimation,usingtheKalmanfilterwiththeseextremeoutlierspresentinthesamplesignificantlydistortsestimatesofthelatentvariablesduringthepandemicperiod.ThegoldlinesinFigure5displayslargeswingsintheestimatesofthelatentvariablesusingthemodelwith2019:Q4parametervalues.Theextremevolatilityintheseestimatesisinconflictwiththespecificationoftheselatentvariablesasreflectinglower-frequencymovements.

3COVID-adjustedModel

TheobjectiveofthispaperistoestimatethenaturalrateofinterestfollowingtheCOVID-19pandemicinawaythatisconsistentwiththeHLWmodeloutlinedinSec-tion2.1.Thelargemovementsineconomicactivityandthepersistentsupplyshocksduringthepandemicperiodviolatetwostandard,butimportant,modelassumptionsinHLW.AsweshowinSection2.2,droppingtheobservationsfromthisperiodduringthemaximumlikelihoodestimationofmodelparametersisinsufficienttoovercometheseviolatedmodelassumptions,withmodelresidualsindicatingthepresenceoflargeoutliers.Theresultingestimatesofthenaturalratesofoutputandinterestareextremelyvolatileandinconsistentwithourspecificationofrasamedium-run

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conceptthatisdrivenbylow-frequencymovements.

ThissectiondetailstwoadjustmentstotheHLWmodelthat,takentogether,addresstheviolationstotheoriginalmodel.Thefirstistheintroductionoftime-varyingvolatilityinthemodel,whichweimplementbyallowingthevariancesofthestochasticinnovationstotheoutputgapandinflationequationstobehigherintheCOVIDerarelativetothenon-pandemicperiod.Thesecondistheintroductionofapersistent,butultimatelytemporary,COVIDsupplyshock,inadditiontothetransitoryandpermanentdemandandsupplyshocksthatarealreadypresentinthemodel.Eachofthesemodificationsaloneisinsufficienttoovercometheestimationchallengesposedbythepandemic,butinconjunctionwitheachothertheyallowforcontinuedestimationofthenaturalrateofinterest.

Importantly,becauseweareestimatinglatentvariablesthatarespecifiedasran-domwalks,simplydroppingtheobservationsduringthepandemicperiodwouldnotonlyunderstatethetrueuncertaintyassociatedwiththisperiod,butwouldalsone-cessitateinterpolatingtheselatentvariablesfromtheirpre-pandemicvalues.Instead,ratherthanimposingthatthesevariablesdonotchangeasaresultoftheCOVID-19pandemic,weareabletoletthedatainformtheestimatednaturalratesofinterestandoutputafterthepandemichasabated.Whiletheintroductionoftime-varyingvolatilityhasasimilareffectonthelatentvariablesasdroppingtheobservationsfrom2020,ourapproachprovidesmoreflexibilityinthelateryearsofthepandemicanddoesnotrequirethestrongassumptionthatseveralyearsofdatahasnoeffectonthelatentvariables.Additionally,theHLWmodelwithtime-varyingvolatilitybutwithoutseriallycorrelatedshockstosupplywouldconstrainhowthenaturalrateofoutputcouldevolveinresponsetothepandemicdata.ModelingthepersistentCOVIDsupplyshockisnecessaryinordertocapturetheeffectsofthepandemiconthenaturalrateofoutput.

3.1Time-VaryingVolatilityduringCOVID-19

COVID-19representsanextremetaileventrelativetotheassumptionofGaussiandisturbances.AsweshowinSection2.2,theresultingoutlierscontaminateestimatesofthelatentvariablesevenwhenweexcludethemfromtheestimationofmodelparameters.Thisisastatisticalproblemthatisnotuniquetoourmodelortoestimationofthenaturalrateofinterest.Wepresentastraightforwardapproachto

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accountforthesubstantialincreaseinvolatilityduringthisperiodbyintroducingtime-varyingvolatilityinthemodelduringthewindowoftimeassociatedwiththeCOVID-19pandemic.WebuildonaninsightfromLenzaandPrimiceri(2022):ifthetimingofincreasedvolatilityisknown–asisthecasefortheCOVIDpandemic–wecanintroducetime-varyingvolatilityinthemodeldirectlybyapplyingascalefactortotheinnovationvariancesduringtheperiodofincreasedvolatility.Weapplythisinsighttoourunobserved-componentsmodelinordertoestimatethenaturalrateofinterest,butourapproachcangeneralizetoanystate-spacemodelwithlatentvariables.

Inparticular,weintroducethreenewmodelparameters,κ2020,κ2021,andκ2022.Thesearethevariancescaleparametersfor2020,2021,and2022,respectively,whichmultiplythevariancesoftheinnovationstotheoutputgapandinflationequations.Wedefinethevectorκtofvariancescaleparametersattimet,thattakesthevalues

κ2020

κ2022

κt=〈κ2021

(1

2020:Q2≤t≤2020:Q4

2021:Q1≤t≤2021:Q4

2022:Q1≤t≤2022:Q4

otherwise

(13)

Weestimatethethreevariancescaleparametersbymaximumlikelihoodtogetherwiththeothermodelparameters,withtheconstraintsκ2020≥1,κ2021≥1,andκ2022≥18.κttakesthevalueof1beforethepandemicperiodandin2023andbeyond.Section5.1considersalternativespecificationsoftime-varyingvolatility.

Thecovariancematrixofthestochasticinnovationstotheoutputgapandinflationequationsisnowtime-varyingandisgivenby

Rt=κ·R=[(κt]

(14)

withtime-varyinginnovationvariancestotheIScurveandPhillipscurveequationsof(κtσy~)2and(κtσπ)2,respectively.

8Therestrictionthatκt≥1isnecessarytoensurethatthelikelihoodestimationcannotdown-weightthevarianceofcertainobservations,whichwouldineffectallowittoplacemoreweightonfavorableobservations.Instead,theestimatedκtfactorscanonlyincreasetheinnovationvariancesduringthepandemicperiod.

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Outsideofthepandemicperiod,theinnovationvariancesarespecifiedexactlyasinHLW(2017).Thatis,theinnovationvariancestotheoutputgapequation,σ,andinflationequation,σ,areconstantoverthesamplepriorto2020andafter2022.Duringtheyear2021,forexample,theinnovationvariancestakethevalues(κ2021·σy~)2and(κ2021·σπ)2,respectively.Thereforeκtisaratioofthestandarddeviationsofthedisturbancestothemeasurementequations(theoutputgapandinflationequations)attimetrelativetothestandarddeviationsinthenon-pandemicsample.Whenκt>1,aswefindfor2020through2022inalleconomiesinoursample,theinnovationvariancesaregreaterthaninthenon-pandemicsample.

Introducingtime-varyingvolatilitytothestochasticinnovationsviathevariancescalefactorsinκthastheeffectofdown-weightingextremeoutlierobservationsinthemaximumlikelihoodestimationofmodelparametersaswellasinestimationofthelatentvariablesviatheKalmanfilter.Whenκt>1,thediagonalcovariancematrixRtofthedisturbancestotheoutputgapandinflationequationsislargerrelativetothecasewhereκt=1,andtheresultingKalmangainissmaller.AsshowninEquation9,theKalmangaindictatestheweightplacedontheone-step-aheadpredictionerrors–thedifferencebetweenrealizedvaluesoftheoutputgapandinflationinagivenperiodandthemodel’spredictedvaluesbasedoninformationinthepriorperiod–inupdatingthefilteredestimatesofthelatentvariables.Astheinnovationvariancesinagivenperiodbecomelarge,theKalmangainshrinks,sothattheKalmanfilterplacesrelativelylittleweightonthesenewobservationsand

theestimatesofthelatentvariablesinthestatevector(thatis,y,gt,ztandthereforer)remainclosetotheestimatesfromthepriorperiod.

Inthelimitastheinnovationvariancestendtowardinfinity,theKalmangainapproacheszero,sothatnoweightisplacedonthetime-tobservationsinestimatingthestatevector.Ineffect,themodeldoesnotmakeuseoftime-tinformation,sothattheforecastofthestatevectorattimetgiventhetime-tinformationsetisunchangedfromtheforecastgiventheinformationsetattimet−1.ThislimitingcaseisequivalenttodroppingtheCOVID-19observationswhenestimatingthelatentvariables.Thesameholdsforparameterestimation:whenκtislarge,themodelforecasterrorinthisperiodisdown-weightedwhencomputingtheloglikelihoodfu