美联储-关于通货膨胀尾巴的故事 Tale About In

FinanceandEconomicsDiscussionSeries

FederalReserveBoard,Washington,D.C.

ISSN1936-2854(Print)

ISSN2767-3898(Online)

TaleAboutInlationTails

OlesyaV.GrishchenkoandLauraWilcox

2024-028

Pleasecitethispaperas:

Grishchenko,OlesyaV.,andLauraWilcox(2024).“TaleAboutInlationTails,”FinanceandEconomicsDiscussionSeries2024-028.Washington:BoardofGovernorsoftheFederalReserveSystem,

/10.17016/FEDS.2024.028

.

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

TaleaboutInflationTails*

OlesyaV.GrishchenkotLauraWilcox‡

May7,2024

Abstract

WestudyprobabilitiesofextremeinflationeventsintheUnitedStatesandtheeuroarea.Usingastate-spacemodelthatincorporatesinformationfromalargesetofprofessionalforecasters,wegeneratethetermstructureofinflationforecastsaswellasprobabilitiesoffutureinflationforanyrangeofinflationoutcomesinclosedformatanyhorizon.SincetheonsetoftheCOVID-19pandemic,inflationexpectationsincreasedmateriallyamidheighteneduncertaintyaboutfutureinflation.Likelihoodofsignificantdeparturesofinflationtargetsinthelongertermreachedabout15percentinthemiddleof2022,increasingfromnearzerolevelsin2020.Suchanincreaseintherighttailoftheprobabilitydistributionoverfutureinflationoutcomesdrivesanincreaseininflationexpectationsandinflationriskpremiums.Severalpopularexternaluncertaintymeasuresareassociatedwithvariationintailprobabilities.

JELClassification:G12,G13,G14

Keywords:Inflationforecasts,inflationstate-spacemodel,probabilityofrareinflationevents,infla-tionanchoring

*TheopinionsexpressedinthispaperarethoseoftheauthorsanddonotnecessarilyreflecttheviewsoftheFederalReserveSystem.WeappreciatehelpfulcommentsfromAnthonyDierks,DonH.Kim,AndrewMeldrum,Jean-PaulRenne,seminarparticipantsfromtheMonetaryandFinancialMarketAnalysissectionattheFederalReserveBoard,andconference

parcgti0,o.

‡JPMorganChaseInstitute,Washington,DC20006;Laura.F.Wilcox@

1

1Introduction

In2022,consumerpriceinflationintheUnitedStatesandtheeuroareareachedthehighestlevelsseenindecades,driveninlargepartbyCOVID-19-inducedsupplychainimbalancesandgeopoliticaldevelopments.Centralbanksrespondedbyraisingtheirpolicyratesandcommunicatingtheirstrongcommitmenttobringinflationbacktotheirrespectiveinflationobjectives.Inparticular,theEuropeanCentralBank(ECB)statedinitsFebruary2,2023MonetaryPolicyStatement(

EuropeanCentralBank,

2023

):“TheGoverningCouncilwillstaythecourseinraisinginterestratessignificantlyatasteadypaceandinkeepingthematlevelsthataresufficientlyrestrictivetoensureatimelyreturnofinflationtoits2%medium-termtarget.”Also,theFederalReserveBoardChair,JeromePowell,statedintheFederalReserve’stestimonybeforetheCommitteeonBanking,Housing,andUrbanAffairsattheU.S.Senate

(Powell,

2023

):“Althoughinflationhasbeenmoderatinginrecentmonths,theprocessofgettinginflationbackdownto2percenthasalongwaytogoandislikelytobebumpy.AsImentioned,thelatesteconomicdatahavecomeinstrongerthanexpected,whichsuggeststhattheultimatelevelofinterestratesislikely

tobehigherthanpreviouslyanticipated.”

Giventheimportanceofatimelyreturntocentralbanks’inflationtargets,gaugingandanalyzingdevelopmentsininflationexpectationsisamajorfocusofmacroeconomicsandmonetarypolicyliterature.Oneofthetoolsthatcentralbankersaroundtheworldusetomeasureinflationexpectationsisso-calledinflationcompensation,orthedifferencebetweenthenominalandinflation-adjusted(real)ratesofcomparablematurities.Inflationcompensationmeasureshowmuchcompensationinvestorsinnominalbondsrequireoverholdinginflation-adjusteddebt,suchasTreasuryInflation-ProtectedSecurities(TIPS).ForstudiesrelatedtotheU.S.,see,forexample,

G¨urkaynak,Sack,andWright

(2007),

G¨urkaynak,Sack,

andWright

(2010),

Christensen,Lopez,andRudebusch

(2010),

GrishchenkoandHuang

(2013),

D’Amico,

Kim,andWei

(2018),and

Chang

(2019),tonamejustafew.ForstudiesintheU.K.,Japan,andtheeuro

area,see,forexample,

BarrandCampbell

(1997),

Evans

(1998),and

KitaandTortorice

(2018).However,

market-basedinflationcompensationmeasuresareaffectedbyinflationriskpremiums—compensationforriskrequiredbythemarketparticipantsinthenominalbondmarketsincetherealvalueofnominalbondsdeclineswithincreasinginflation—thataretimevaryingandcanbelargeand,therefore,candistortinflationexpectationsreadingsbasedsolelyoninflationcompensation(See,forexample,

Campbell

andViceira,

2001;

BuraschiandJiltsov,

2005;

Ang,Bekaert,andWei,

2008;

H¨ordahlandTristani,

2010;

Ajello,Benzoni,andChyruk,

2020;

ChernovandMueller,

2012;

Haubrich,Pennacchi,andRitchken,

2012;

2

Abrahams,Adrian,Crump,andMoench,

2013;

GrishchenkoandHuang,

2013;

Fleckenstein,Longstaff,

andLustig,

2013;

Crump,Eusepi,andMoench,

2016;

D’Amico,Kim,andWei,

2018;

Breach,D’Amico,

andOrphanides,

2020)

.

Thispaperprovidesinflationexpectationsreadingsbasedonthewidesetofinflationforecastscon-tainedinsurveysofprofessionalforecasters.Surveyshavebeendocumentedtobesuccessfulinforecastinginflationrelativetovarioustime-seriesmodels(forexample,

Ang,Bekaert,andWei,

2007;

Aruoba,

2020)

.Theadvantageofusingsurveysofprofessionalforecastersisthat,unlikemarket-basedreadingsofin-flationexpectations,theyarefreefrominflationriskpremiums.Usingsurveys,weachieveadualgoal.First,weprovideatermstructureofinflationexpectations.Second,wecomputeinflationriskpremi-ums,bycomparingmarket-basedinflationcompensationrates(eitherTIPS-orinflationswap-based)and

model-impliedinflationexpectationsofcomparablematurities.

Webuildourframeworkonthestate-spacemodelof

Grishchenko,Mouabbi,andRenne

(2019,GMR)

.Thismodelpresentsasophisticatedwaytojointlyaggregatesurvey-basedinflationexpectationsandsurvey-baseddistributionsoverfutureinflationoutcomesavailableintheU.S.andtheeuro-areasurveysofprofessionalforecasters.Themodelproducesthetermstructuresofinflationexpectations(inflationexpectationscurve)inrespectiveeconomicareas.Themodelalsousesthesecondmomentsofsurvey-baseddistributionsoffutureinflationratesandaggregatesthemintothetermstructuresofinflationuncertainty(inflationuncertaintycurve)inrespectiveeconomicareas

.1

Inaddition,themodelextractssignalsaboutinflationforecastsfromseveralsurveysintwoeconomicareas(theU.S.andtheeuroarea).Inadditiontomodelinginflationforecastsatdifferenthorizons,importantly,weattempttoanswerquestionsrelatedtothebalanceofriskssurroundinginflationexpectations.Weachieveitinthreeways.First,weprovideameasureofuncertaintyaboutfutureinflationbecausetheGMRmodelallowsthevolatilityofinflationtovaryovertime

.2

Weestimatethestochasticvolatilityofinflationusingthesecondmomentsoftheprobabilitydistributionsaboutfutureinflationavailableinseveralsurveysof

professionalforecasters.

Second,wecomputetheGMRanchoringmeasureofinflationexpectations—theprobabilityof

1Thefactthattheinflationstate-spacemodelmatchessecond-ordermomentspresentsanimportantdistinctionbetweenourapproachandtheonedevelopedby

Aruoba

(2020)

.Thelatterstudyalsousesastatisticalmodeltoaggregatevarioussurveystoproduceatermstructureofinflationexpectations.However,

Aruoba

(2020)’sapproachimplicitlyassumesthat

inflationuncertaintyisconstantovertime.Inaddition,

Aruoba

(2020)focusesoninflationforecastsintheU.S

.only,whilewefocusonmodelinginflationexpectationsanduncertaintycurvesjointlyfortheU.S.andeuroarea.

2

Engle

(1982

)wasthefirstwhoemphasizedtime-varyinginflationuncertaintyinthecontextofaneconometricmodelbyspecifyinganewclassofstochasticprocesses—autoregressiveconditionalheteroscedastic(ARCH)processes.

Zarnowitz

andLambros

(1987

)emphasizedtime-varyinginflationuncertaintyinthecontextofthesecondmomentofsurvey-basedinflationdistributions;theconceptthatweuseinourmodeltoproxyforinflationuncertainty.

3

futureinflationbeinginacertainrangearoundthecentralbankingtarget.Thereareseveralproposed

measuresofanchoring,orstabilityofinflationexpectations,suchasaresponseofmarket-basedinflationcompensationmeasuresorinterestratestoincomingmacroeconomicnews(

G¨urkaynak,Levin,Marder,

andSwanson,

2007;

Mishkin,

2007;

Beechey,Johannsen,andLevin,

2011;

DePooter,Robitaille,Walker,

andZdinak,

2014;

Speck,

2016),aresponseof(changesin)long-terminflationexpectationsto(changes

in)short-termones(BuonoandFormai,

2016;

Gerlach,Moessner,andRosenblatt,

2017),theprecision

aroundestimatesofthelevelofinflation(

MehrotraandYetman,

2014

),thevolatilityofshockstotrend

inflation(Mertens,

2016),andtheclosenessofaveragebeliefstothecentralbank’sinflationtarget(

Kumar,

Afrouzi,Coibion,andGorodnichenko,

2015;

LyziakandPaloviita,

2016)

.ThedifferencebetweenthesemeasuresandtheGMRanchoringmeasureisthatmostofthemaremainlyrelatedtothestabilityoftheconditionalmeanofinflationanddonotcapturetheconditionalvarianceofinflationthatcanbe

relativelyhigheventhoughtheconditionalmeanisclosetothetarget

.3

Third,wecomputeprobabilitiesoffutureinflationbeinghigherthanacertainthreshold(tailproba-bilities)inclosedform.WecanaccomplishthisbecausetheGMRmodelishighlytractable—itoffersclosed-formsolutionsforfirstandsecondconditionalmomentsoffutureinflationratesatanyhorizon—duetothefactthatthefactorsinoureconometricmodel,includingthosedrivinginflationuncertainty,

followso-calledaffineprocesses

.4

Anadditionaladvantageofourmodelisthatitusessurveyinflationforecastsratherthanmarketquotes.

KitsulandWright

(2013)and

Hilscher,Raviv,andReis

(2022)useinflationoptionstoextract

informationaboutprobabilitiesofextremeinflationevents.However,recently,tradinginthemarketforinflationoptionshasbeenverylow,ifnotvirtuallynonexistent,meaningthatitisnotclearwhethermarketparticipantscouldactuallytradeattheprovidedquotes,orwhetherthosequotesrepresenttheviewsofmarketparticipants.Instead,weuseamodelthatreliesonthemostup-to-datesurveysthatprovidedistributionsoverfutureinflationoutcomesandthusreflectinformationaboutextremeinflation

outcomes.

Ourfindingsareasfollows.First,ourmodelimpliesthatbothshort-andlong-terminflationexpec-

tationsincreaseddramaticallyaftertheonsetoftheCOVID-19pandemicbutmoderatedsomewhatat

3Consider,forinstance,asituationwhereamacroeconomicsurpriseresultsinasubstantialincreaseinthelong-termconditionalvariancebuthasnoeffectontheconditionalmean.Thatis,supposewefaceequalincreasesinbothdownsideandupsiderisks.Inthissituation,whilelong-terminflationexpectationsremainstable,theprobabilityofhavingveryhighorverylowfutureinflationratesincreasessubstantially,whichisatoddswiththeconceptofanchoring.

4Theaffinepropertyofourfactorsimpliesthatthemodelcanbeeasilycastinstate-spaceformandsubsequentlyestimatedusingKalmanfilteringtechniques.Inparticular,thesetechniqueshandlemissingobservations,whichisparticularlyusefulinourcase,becausevarioussurveysarereleasedatdifferentpointsintime.

4

theendofoursample,inlate2022andbeginningof2023.Nevertheless,short-termexpectationsremain

notablyhigherthanlonger-terminflationexpectations,bothintheU.S.andtheeuroarea.Second,ourmodelimpliesthatuncertaintyaboutfutureinflation,asmeasuredbythesecondmomentofthefittedsurveyprobabilitydistributionoverfutureinflationoutcomes,increasedsignificantlyin2021-2022buthasdeclinedsincethen.WefindthatinflationuncertaintyiscurrentlyaroundthelevelslastseenjustbeforetheGlobalFinancialCrisis(GFC).Third,theprobabilityofU.S.five-yearaverageinflationexceeding3percentincreasedsubstantiallyandtheprobabilityofU.S.five-yearaverageinflationfallingbelow1percentdeclinedsubstantially.Namely,ourmodelimpliesthattheprobabilityofhigher-than-3-percentaverageinflationoverthenextfiveyearswasaround25percentagepointsinJanuary2023,comparedwithtenpercentagepointsinearly2020.Likewise,higher-than-3-percentprobabilityofeuro-areainflationinthenextfiveyearsreachedabout12percentagepointsinlate2022comparedwithonly4percentagepointsinthewakeoftheCOVIDcrisis.Last,inflationriskpremiumsincreasedin2021butdeclinedsincethen.Inflationriskpremiumsforlonger-horizonsbecamepositivein2022,whichisnotablebecauseestimatesforinflationriskpremiumsusuallyhoveraroundzerolevels(see,forexample,

Grishchenkoand

Huang,

2013,andreferencesabove).Wehavealsoexploredrelationshipbetweenmodel-impliedinflation

tailprobabilitiesandpopularexternaluncertaintymeasures,suchasmacroeconomic,realeconomic,andfinancialuncertaintymeasuresdevelopedin

Jurado,Ludvigson,andNg

(2015),economicpolicyuncer

-taintymeasuresdevelopedin

Baker,Bloom,andDavis

(2016),andageopoliticalriskmeasuredeveloped

in

CaldaraandIacoviello

(2022).Ingeneral,wefoundthatvariationininflationtailprobabilitiesinboth

economicareasappearstobeassociatedwithvariationintheseuncertaintymeasures,dependingontail

probabilitiesandconsideredhorizon.

Therestofthepaperisorganizedasfollows.Section

2

describesthesurveysofprofessionalforecastersweusetoaggregateinflationforecastsoffutureinflationatdifferenthorizonsandestimateinflationexpectationsandinflationuncertaintycurvesfortheU.S.andtheeuroarea.Section

3

describestheinflationstate-spacemodelandthesurveys’fittothemodel.Section

4

describesourempiricalresults

andSection

5

concludes.

2Data

Section

2.1

and

2.2

brieflydiscussavailabledataintheU.S.andeuro-areasurveys,respectively.Ourdata

issincetheonsetoftheeuroareainJanuary1999untilFebruary2023,withdifferentsurveysavailable

5

indifferentfrequenciesandpublishingforecastsfordifferenthorizons.

2.1SurveysofinflationforecastsintheUnitedStates

SurveysintheUSusedinourstudyincludethefollowingfoursurveys.PanelAofTable

1

summarizes

thesurveysofprofessionalforecastersintheUnitedStates.

TheSurveyofProfessionalForecasters(USSPF)publishedbytheFederalReserveBankofPhiladel-

phiaisconductedquarterlyandprovidesforecastsonawiderangeofmacroeconomicandfinancial

variablessince1968:Q4.

5

Forthepurposeofthisstudy,weuseseveralinflationforecastsfromthe

US-SPF.

First,weusedensityforecasts—availableintheformofhistograms—forthepricechangeintheGDPpricedeflator(surveyvariablePRPGDP)forthecurrentandthefollowingcalendaryear

.6

Thedensityfunctionsareavailableonanindividualforecasterbasisandweaggregatethisinfor-mationbyusingtheaverageforecastdensityfunctions.TheUS-SPFdefinesapricechangeastheannual-averageoverannual-averagepercentchangeintheleveloftheGDPpriceindexthatisavailablequarterly.Notethatforecastdensityfunctionsarethefixedeventforecasts(theytargetthecurrentandthenextcalendaryears),therefore,theforecasthorizonchangeswiththesurvey’s

timing.Oursampleforthedensityfunctionsisfrom1999:Q1to2022:Q4

.7

Second,weusetheUSSPFfive-yearaverageheadlineCPIinflationconsensusforecasts(surveyvariable:CPI5YR)inordertoidentifylonger-horizoninflationforecasts.Thisprojectionisdefinedastheannualaverageinflationrateoverthenextfiveyears.The“nextfiveyears”includestheyearinwhichthesurveyisconductedandthefollowingfouryears.Oursampleforthisvariablespans

from2005:Q3(itsstartingpointintheUS-SPF)to2022:Q4.

TheBlueChipsurveys—TheBlueChipFinancialForecasts(BCFF)andBlueChipEconomicIndicators(BCEI)surveys—arepublishedmonthly.BothBlueChipsurveysprovideindividual

pointestimatesofinflationforecasts,fromwhichconsensusanddisagreementmeasurescanbe

5TheUSSPFsurveywasformerlyconductedbytheAmericanStatisticalAssociationandtheNationalBureauofEconomicResearchandwastakenoverbythePhiladelphiaFedin1990:Q2.

6US-SPFstartedprovidingdensityprojectionsofthecoreConsumerPriceIndex(surveyvariablePRCCPI)andofthecorePersonalConsumerExpendituresIndex(surveyvariable:PRCPCE)onlyin2007:Q1.ThereforeweconcentrateonthedensityprojectionsoftheGDPpricedeflator(despitesmallleveldifferenceswiththeheadlineCPIindex)inordertohaveinformationaboutthesecondmomentsofthefutureU.S.inflationratesstartingfromthebeginningofoursample,1999:Q1.TheUS-SPFdoesnotprovideanydensityprojectionsaboutheadlineCPIinflation.

7Thebeginningofoursampleismotivatedbytheonsetoftheeuro-zoneandavailabilityoftheeuro-areasurveys.

6

obtained.Monthlysurveysprovideinflationforecastsuptosixquartersout.Inadditiontothose,

BCFFandBCEIsurveyspublishlong-rangeforecaststwiceayear

.8

Theselong-rangeforecastscontainaverageannualforecastsusuallyfiveyearsoutfromthesurveypublicationyearandtheaveragefive-yearforecastfiveyearsahead.Weusefive-year,five-years-aheadconsensusinflation

forecastsinourmodelestimation.

TheSurveyofPrimaryDealers(USSPD)ispublishedbytheFederalReserveBankofNewYorksince2004.InadvanceofeachFederalOpenMarketCommittee(FOMC)meeting,thesurveypro-videsprimarydealers’macroeconomicforecastsincludinginflationforecasts.

9

RelativetoinflationforecastsavailableintheUSSPF,theUSSPDprovidesinflationforecastsforthelongerhorizons:Namely,probabilitydistributionsoftheaverageannualfive-yearaheadCPIinflationandofthefive-yearfive-years-aheadaverageannualCPIinflation.ThesurveyispublishedattheFOMC

frequency.

TheConsensusEconomicsSurvey(CES)providesconsensusinflationforecastsforarangeofde-velopedcountries,onamonthlybasis.USaverageannualinflationforecastsareavailableforthecurrentandthenextcalendaryear,fromwhichweextrapolateafixedhorizonone-year-ahead

inflationforecast.

2.2SurveysofinflationforecastsintheEuroArea

Surveysintheeuroareausedinourstudyincludethefollowingtwosurveys.PanelBofTable

1

summarizestheeuro-areadatasetdescribedbelow.

TheEuropeanCentralBankSurveyofProfessionalForecasters(ECBSPF)providesvariousmacroeconomicforecastsintheeuroareaincludinginflationforecastssince1999,theonsetoftheeuroarea.Inparticular,thesurveyprovides,asinthecaseoftheUSSPFandtheUSSPD,theprobabilitydistributionsofinflationforthecurrentandthenextcalendaryear,andforalongerhorizon(five-year-ahead).InflationisdefinedasthechangeintheHarmonizedIndexofConsumer

Prices(HICP).

8BCFFpublishlong-rangeforecastsinJuneandDecember,andBCEIpublishtheminMarchandOctober,sotheseforecastsarenotevenlyspacedoutthroughtheyear.

9Thesurveyquestionssometimesvarydependingontheeconomicenvironment.SeepostedquestionsonthewebsiteoftheFederalReserveBankofNewYork:

/markets/primarydealer_survey_questions.html.

Nonetheless,certainquestionssuchasthedensityforecastsforheadlineCPIinflationareroutinelyasked.

7

TheCESprovideslong-termeuro-areaconsensusinflationforecastsonasemi-annualbasis(inAprilandOctober),inwhichfive-yearfive-year-aheadinflationforecastsareavailable.Theseforecasts

areavailablesince1999,theonsetoftheeuroarea.

3State-SpaceModelofSurveysofProfessionalForecasters

Inthissection,webrieflyoutlinetheGMRstate-spacemodelaccordingtowhichwecomputeinflationexpectation,inflationuncertainty,andprobabilityofinflationbeinginacertainrangeintheanalyticalform.WeestimatethemodelusingtheKalmanfiltermethodology—analgorithmthatisusuallyusedforestimatingofstate-spacemodels.Astate-spacemodelconsistsoftwotypesofequations:transitionequationsandmeasurementequations.Transitionequationsdescribethedynamicsofthelatentfactors,discussedbelowinSection

3.1.

Measurementequationsspecifytherelationshipbetweentheobservedvariablesandthelatentfactors,discussedfurtherinSection

3.2.

Conditionalonthemodelparameter-izationandonobservedvariables,theKalmanfiltercomputesthedistributionofthelatentvariables.Besides,aby-productofthealgorithmisthelikelihoodfunction.Modelparameterscanthereforebeestimatedbynumericallymaximizingthisfunction.Oncethisisdone,thelastpassofthealgorithmpro-videsestimatesofthelatentvariables.

10

Wediscussthefittingofsurveydatatomodel-impliedmoments

ofdistributioninSection

3.3.

3.1Transitionequationsofinflationanditsdrivingfactors

Weassumethattheannualinflationrate,π

vectorYt=(Y1,t,...,Yn,t)′.Asspecifiedbelow,thedynamicsofYtissuchthatthemarginalmeanofYt

iszero.Importantly,Yj,tfactors,wherej∈{1,...,n},maybecommontodifferenteconomies:

11

π12,t=π(i)+δ(i)′Yt.

(1)

WeassumethatthedistributionofYtisGaussianconditionallyonitspastrealizationYt−1={Yt−1,Yt−2,...}

andonanotherq×1exogenousvectorzt=(z1,t,...,zq,t)′thataffectsthevarianceofYt

.12

Specifically,

10WereferthereadertotheGMRpaperformorespecificdetailsonmodelestimation.

11U.S.andeuro-areainflationratesareweightedaveragesofstate-levelandcountry-levelinflationrates,respectively.Modellingsuchdisaggregatedinflationratesmayhelpinvestigatequestionssuchastheextenttowhichextentde-anchoringinonearearelatestocross-regionheterogeneity.Thisishoweverbeyondthescopeofthispaper.

12NotethatthisdoesnotimplythatthemarginaldistributionofYtisGaussian(asitisinGARCHmodels).

8

Ytisgivenby:

Yt=ΦYYt−1+diag(4ΓY,0+Γ1zt)εY,t,εY,t∼N(0,I),(2)

whereΓY,0isann×1vectorandΓY,1isaq×nmatrix.Accordingtoeq.(

2

),ztaffectstheconditionalvarianceofYt.Giventhatthevectorztisessentialformodellingthetime-varyingvarianceofinflation,

werefertoitastheuncertaintyvector(andtothezj,t’sastheuncertaintyfactors)hereinafter.

Thespecificationoftheconditionalvarianceineq.(

2

)impliesthattheentriesofΓY,0+Γ1zthave

tobenon-negativeforallt.Tothatend,weassumethatallelementsofΓYvectorsarenon-negativeandthatztfollowsamultivariateauto-regressivegammaprocess.AsshowninGMRAppendixA.4,the

dynamicsofztadmitsthefollowingsemi-strongVARrepresentation:

zt=µz+Φzzt−1+diag(4Γz,0+Γ1zt−1)εz,t,

(3)

where,conditionalonzt−1,εz,thasazeromeanandaunitdiagonalcovariancematrix,andwhereΓz,0is

aq×1vectorandΓz,1isaq×qmatrix.

GiventhedynamicsofYtandzt,thesemi-strongVARformofthedynamicsfollowedbyXt=(Yt′,z)′

is:

lYt」lYt−1」

Xt=zt=µX+ΦXzt−1+ΣX(zt−1)εX,t,(4)

whereεX,tisa(n+q)-dimensionalunit-variancemartingaledifferencesequence,µX=(01×n,µ)′,ΦXisa

block-diagonalmatrixwithΦYandΦzonitsdiagonalandΣX(zt−1)ΣX(zt−1)′—thatis,theconditionalcovariancematrixofXt(givenitsownpast)isadiagonalmatrixwhosediagonalentriesarelinearin

zt−1

.13

AnimportantpropertyofXtisthatitisaffine.Thisimpliesthat,conditionallyonXt={Xt,Xt−1,...},

thefirstandsecondconditionalmomentsofanylinearcombinationoffuturevaluesofXtareaffinefunc-

tionsofXt.Inparticular,sincetherealizedlogannualgrowthrateofthepriceindexπ12,tisanaffine

transformationofXt(eq.(

1

)),itsfirstandsecondmomentscanbewrittenasaffinefunctionsoftheXt

13Specifically,thefirstndiagonalentriesarethecomponentsofΓY,0+Γ1(μz+Φzzt−1)andthelastqarethoseofΓz,0+Γ1zt−1.

9

factorsaswell:

Et(πh−12,t+h)=π(i)+a)+b)′Xt

Vart(πh−12,t+h)=α)+βi)′Xt,

(5)

(6)

whereEt(.)andVart(.),respectively,denotetheexpectationsandvariancesconditionalonXt.AsexplainedinSection

2

,wehavetoconsiderothermeasuresofinflationbecauseofthenatureofthe

differentsurveyswefit

.14

3.2Measurementequations

Thestate-spacemodelinvolvesthreetypesofmeasurementequations:

(a)Thefirstsetofequationsstatesthat,foreacheconomicareai,arealisedinflationrateisequaltoalinearcombinationoffactorsYt,asstatedbyeq.(

1

),witharea-specificloadingsδ(i)’s,measured

withou