Disagreement About the Term Structure of Inflation Expectations

Finance and Economics Discussion Series Federal Reserve Board, Washington, D.C. ISSN 1936-2854 (Print) ISSN 2767-3898 (Online) Disagreement About the Term Structure of Inflation Expectations Hie Joo Ahn and Leland E. Farmer 2024-084 Please cite this paper as: Ahn, Hie Joo, and Leland E. Farmer (2024). “Disagreement About the Term Structure of Inflation Expectations,” Finance and Economics Discussion Series 2024-084. Washington: Board of Governors of the Federal Reserve System, https://doi.org/10.17016/FEDS.2024.084. NOTE: Staff working papers in the Finance and Economics Discussion Series (FEDS) are preliminary materials circulated to stimulate discussion and critical comment. The analysis and conclusions set forth are those of the authors and do not indicate concurrence by other members of the research staff or the Board of Governors. References in publications to the Finance and Economics Discussion Series (other than acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers.

Disagreement About the Term Structure of Inflation Expectations* HieJooAhn† LelandE.Farmer‡ FederalReserveBoard UniversityofVirginia September16,2024 Abstract Wedevelopamodeloftheindividualtermstructureofinflationexpectationsacrossforecastinghorizons.UsingtheSurveyofProfessionalForecasters,wedecomposedisagreement aboutinflationexpectationsintoindividuals’long-termbeliefs,privateinformation,and publicinformation. Wefindthatinnormaltimes,long-horizondisagreementispredominantlydrivenbyindividuals’long-termbeliefs,whileshort-horizondisagreementstems fromprivateinformation.Duringeconomicdownturns,heterogeneousreactionstopublic informationbecomeakeydriverofdisagreementatallhorizons.Whenforecastersdisagree aboutpublicinformation,monetarypolicyexhibitsadelayedresponseandapricepuzzle emerges,underscoringtheimportanceofanchoringinflationexpectations. JELclassification:E17,E31,E37,E52,E58,E65. Keywords:InflationExpectations,TermStructure,Disagreement,MonetaryPolicy *WethankHassanAfrouzi,BoragˇanAruoba,MichaelBauer,ThomasDrechsel,RogerE.A.Farmer,Andrew Figura, JimHamilton, EdwardHerbst, ElmarMertens, EmiNakamura, JeremyRudd, JónSteinsson, Luminita Stevens,EricSwanson,AllanTimmermann,FabianWinkler,andChristianWolfforhelpfulcomments,andTravis Bergeforthesupportofresources. Disclaimer: The views expressed in this paper are those of the authors and do not necessarily reflect the viewsandpoliciesoftheBoardofGovernorsortheFederalReserveSystem. †FederalReserveBoardofGovernors,20thStreetandConstitutionAvenueNW,Washington,DC20551,U.S.A. Email:hiejoo.ahn@frb.gov ‡UniversityofVirginia,Email:lef2u@virginia.edu

“Ofcourse,anextendedperiodofhighgoodsandservicesinflationresultingfroma seriesofdemandandsupplyshocksassociatedwiththepandemicandthewarcould leadtoariseininflationexpectations,whichwouldmakeitmuchmoredifficultto bringinflationdown. Thatiswhyithasbeenimportantformonetarypolicytotakea risk-managementposturetodefendtheexpectationsanchor. Andtheevidencefrom market-andsurvey-basedmeasuressuggeststhatlonger-terminflationexpectations are well anchored, while year-ahead measures have recently declined but remain elevated."—LaelBrainard(January19,2023) 1 Introduction Households,firms,financialmarketparticipants,andpolicymakersdisagreeonfutureeconomic conditions(CornandandHubert,2022). Thisdisagreementhassignificantimplicationsforthe effectivenessofmonetarypolicyandtheanchoringofinflationexpectation(Falcketal.,2021; Reis,2020;FofanaandReis,2024). Heterogeneityinagents’expectationshasalsobeencentral tomacroeconomicmodeling, servingasakeydriverofbusinesscycledynamics(Lorenzoni, 2009;AngeletosandLa’O,2013;IlutandSchneider,2014),inflationdynamics(Michael,2002; MackowiakandWiederholt,2009),andassetpricing(ScheinkmanandXiong,2003;Burnside etal.,2016;BarillasandNimark,2017). Agrowingbodyofliteratureonexpectationformation hasfocusedondisagreementanditstimevariationtoexplaindeparturesfromrationalexpectationsandtoexplorethestructuralmechanismsunderlyingagents’expectationformation(e.g., Andradeetal.,2016;Mac´kowiaketal.,2023;FofanaandReis,2024). Despitethegrowinginterestindisagreementamongeconomicagents,noresearchhasyet fullycharacterizedthecross-sectionaldistributionofindividualinflationexpectationsacross theentirepathofforecastinghorizons. AsFofanaandReis(2024)note,mostacademicstudies onexpectationformationhavefocusedprimarilyonshort-terminflationexpectations,typically withahorizonofoneyearorless.However,centralbankscloselymonitorlong-termexpectations to assess how well expectations are anchored. Thus, a comprehensive understanding of the termstructureofinflationexpectationsisessential. Unfortunately,expectationsdataareoften aggregatedintobroadcategoriesofforecastinghorizons,leadingtoanincompletepictureofthe termstructure. Whenevaluatingtheextenttowhichmonetarypolicyanchorsinflationexpectations,itis crucialtoidentifytheinformationsourcesdrivingdisagreementovertimeandacrossforecasting horizons. Ifshort-termexpectationsareprimarilybasedonforecasters’privateinformationor 1

long-termbeliefs,theeffectivenessofmonetarypolicycommunicationinreducingdisagreement aboutnear-terminflationprojectionsmaybelimited. Conversely, iflong-termexpectations aremainlyinfluencedbypublicinformation,effectivemonetarypolicycommunicationcould significantly reduce long-term disagreementand aid in anchoring inflation expectations. In thiscontext,itisimportanttounderstandtheextenttowhicheachinformationsourceshapes forecasters’projectionsandtheirdisagreementacrossforecasthorizons. Thispapermakesthreeoriginalcontributions.First,wedevelopanewindividual-levelmodel ofinflationexpectationsacrossforecastinghorizons,whichwecalltheindividualterm-structure of inflation expectations. This model describes a forecaster’s trajectory of inflation forecasts acrossdifferenthorizonsusingtwofactors: levelandslope. ThisapproachisinspiredbyNelson andSiegel’stermstructureofinterestrates(NelsonandSiegel,1987). Thelevelreflectslong-term inflationprojections,whichwerefertoasthelongend. Theslopecapturestheoveralldifference between the long end and the current quarter’s inflation nowcast.1 We estimate the model using Bayesian methods, applying it to forecaster-level data from the Survey of Professional Forecasters (SPF). Although the SPF provides only a partial snapshot of a forecaster’s term structureofinflationprojections,ourmodelisabletorecoverthecross-sectionaldistributionof inflationexpectationsacrossallforecastinghorizonsateachpointintime. Theestimatedshort-andlong-terminflationexpectationsexhibitdifferentdynamics. Althoughthe6-monthand1-yearaheadconsensusforecastscloselytrackrealizedCPIinflation, the10-yearaheadconsensusexpectationtrendsdownwardduringthe1990sandstabilizesjust below2.5%from2000onward. EvenduringtheCOVID-19pandemic,the10-yearconsensus shows limited variation, rising slightly before returning to its pre-pandemic level. Although theestimatedconsensusforecastssuggestwell-anchoredlong-terminflationexpectations,the estimateddisagreementacrossforecastinghorizonsrevealsamorenuancedpicture. Forecastersexhibitedgreaterdisagreementaboutlong-runinflationduringtheGreatRecessionandits recoverythanintheearly1990s,wheninflationwasestimatedtohavebeennonstationary.Inparticular,duringthepandemic,boththevarianceandskewnessoflong-termexpectationsspiked abovelevelsseenintheearly1990s,suggestingthatexpectationswereweaklyanchoredwhen measuredbyhighermoments. Ourmodelshowsthattheconsensusforecastanddisagreement oftenyielddistinctinsightsintotheanchoringofagents’inflationexpectations. 1Forthesakeofparsimony,weconsiderthecurvatureonlyinanappendix(AppendixH),asitplaysalimitedrole intheobservedindividual-leveltermstructureofinflationexpectations. 2

Second, we develop a novel dynamic factor model where the individual term structure elements are characterized by a common component, an idiosyncratic component, and an individualfixedeffect. Weprovideaneconomicinterpretationtoeachelementbybuildinga noisyinformationmodel. Inthisstructuralmodel,aforecasterwhohasownlong-runbeliefs observesbothpublicandprivatesignalswhenupdatinginflationprojections. Wedemonstrate thatthisstructuralmodelnaturallyalignswiththedynamicfactorcharacterization,andcaptures thekeyempiricalresultsofthestatisticalmodelwithasimulationexercise. Thus,weinterpret thecommonandidiosyncraticcomponentsasreflectingthecontributionsofpublicandprivate informationtotheforecaster’sinflationexpectations,whiletheindividualfixedeffectcaptures theforecaster’slong-termbeliefs.2 Unlikepreviousstudies(e.g.,HerbstandWinkler,2021),we allowforheterogeneousreactionstopublicinformation,whicharecapturedbytheindividual factorloadingsonthecommoncomponent. Tothebestofourknowledge,thispaperisthefirst torecoverthedistributionofsensitivitytopublicinformationacrossforecastersandidentifythe roleofpublicandprivateinformationindisagreementbasedonaformalstatisticalmodel. Weobservedistinctrolesforthethreesourcesofinformationincontributingtodisagreement acrossforecastinghorizons. Long-termbeliefsandprivateinformationaccountforthemajority ofdisagreementinlong-runandshort-runexpectationsrespectively. Wealsofindthatforecastersexhibitheterogeneousreactionstopublicinformation,asevidencedbythenon-degenerate distribution of individual-level loadings on the common component. The role of public information in disagreement is small on average.3 However, during economic downturns and periodsofhighinflationuncertainty,publicinformationbecomesakeydriverofdisagreement. AtthepeakoftheGreatRecession,publicinformationaccountedforabouthalfoflong-term disagreement. Similarly,duringthedeflationattheonsetofthepandemic,publicinformation drovethemajorityofdisagreement,anditcontributedtoaboutone-thirdofdisagreementduring thesubsequentperiodofhighinflation. Finally,weinvestigatetheeffectsofeachcomponentofdisagreementontheeffectivenessof monetarypolicy. First,weshowthattheFed’sresponsestorecentdatareleases(BauerandSwanson,2023)reducetheportionofdisagreementabout2-year-aheadinflationattributabletopublic 2Throughoutthepaperwewillusethetermlong-termbelief torefertoanunconditionalbiasinanindividual’s forecastrelativetorationalexpectations.Thisbiascanbeinterpretedasarisingduetosomecognitivelimitation/ behavioralbias,orfromthepersistenteffectofpriorsaboutlong-runinflationdynamicsasinFarmeretal.(2021). Weremainagnosticastothesourceofthisbiasthroughoutouranalysis. 3ThisresultisinlinewithpreviousstudiessuchasPattonandTimmermann(2010),Farmeretal.(2021),and LahiriandSheng(2008). 3

information,butnotthecomponentsattributabletoprivateinformationorindividuallong-run means. Next,weexplorehowdisagreementattributabletopublicinformationinfluencesthe effectivenessofmonetarypolicy. Forthisanalysis,weconstructanewmeasure,thesensitivityof disagreementtopublicinformation,anduseittoestimatethenonlineareffectsofmonetary policyshocks. Wefindthatwhenpublicinformationisthemainsourceofdisagreement,the economy’s responses to monetary policy shock are delayed and a price puzzle emerges. In contrast,whennon-publicinformationistheprimarysourceofdisagreement,monetarypolicy hasrapidandstatisticallysignificantstabilizingeffectsonthemacroeconomy. Our empirical findings have important implications for the conduct of monetary policy. First,ourapproachidentifiesateachpointintimewhyprofessionalforecastersdisagreeabout future inflation. Second, our results suggest that at important junctures (such as the Covid inflation),publicinformationisakeysourceofdisagreementand,thus,apotentialcontributor tothede-anchoringofinflation. Clearcommunicationfromthemonetaryauthorityaboutthe macroeconomiclandscapecanhelpanchortheexpectationsofeconomicagentsbyreducing theirdisagreementaboutfuturemacroeconomicconditions. Lastly,ourfindingsofferanew perspectiveonthesourceofthepricepuzzle. Ourresultssuggestthatthesensitivityofdisagreementtopublicinformationcontributestodeliveringapricepuzzle,andthatthispricepuzzle can be mitigated through effective expectation management by the monetary authority. We leavefurtherexplorationoftheseimplicationsforfutureresearch. RelatedLiteratureThispapermakescontributionstoseveralstrandsoftheliterature. Thefirstistheliteratureontheaggregatetermstructureofinflationexpectations. Aruoba (2020) models the term structure of inflation expectations using a Nelson-Siegel yield curve (NelsonandSiegel,1987). Aruobatreatsthelevel,slope,andcurvaturefactorsaslatentstates andestimatesthemwithalinearstate-spacemodelusingconsensusSPFandBlueChipforecasts. Clarketal.(2022)constructthetermstructureofinflationexpectationsanduncertaintyusinga statespacemodelinwhichstochasticvolatilityandpersistentbiasesinforecastsareallowed. Dieboldetal.(2008)employaNelson-SiegelmodelfromDieboldandLi(2006)topredictgovernmentbondyieldsintheinternationalcontext. Tomodelglobalbond-yielddynamics,Diebold etal.(2008)consideradynamicfactorstructure,similartoourmodelingscheme,butdifferent inthatourmaininterestisforecaster-specificinflationexpectations. Recentstudiesfocusonmodelingindividuals’inflationexpectations(e.g.,HerbstandWinkler, 4

2021;Crumpetal.,2023;Fisheretal.,2022).4 ThispaperisclosesttoHerbstandWinkler(2021)in thatinflationforecastsacrossforecastinghorizonsaremodeledandestimatedattheindividual level. However, there are some key differences. First, our focus is on the term structure of inflationexpectationsattheindividuallevel,andhencewemodelthecompletetermstructure ofinflationexpectationsfromthecurrentquarterthroughtenyearsoutwithflexibleLaguerre polynomialsfromtheNelson-Siegelmodel. HerbstandWinkleronlymodelindividual-level inflationexpectationsoverhorizonsuptooneyearout,sincetheirgoalistocharacterizethe jointdynamicsofvariousmacroeconomicvariables.5 Second,wecomputetheextenttowhich public and private information account for changing disagreement about future inflation at eachforecastinghorizon,withaparticularemphasisonforecasters’heterogeneousresponses topublicinformation. TheseheterogeneousresponsesarenotexplicitlyconsideredinHerbst and Winkler (2021). Lastly, we provide a unique decomposition that summarizes the role of long-termbeliefs,public,andprivateinformationindisagreementateachpointintime. Ourresearchdirectlyspeakstoavastliteratureondisagreement. Studiesonlearningmodels havefocusedonthesourceofdisagreement. Forexample,LahiriandSheng(2008)developa theoreticalBayesianlearningmodelinwhichexperts’forecastsareshapedbytheirlong-term beliefsandtheirinterpretationsofpublicinformation. Thismodelaccountsfortheevolutionof bothwithin-forecastervariabilityandbetween-forecasterdisagreementinGDPprojectionsover variousforecasthorizons.LahiriandShengestimatethemodelparametersusingforecaster-level datafromsevendifferentcountriesprovidedbyConsensusEconomics. Second,PattonandTimmermann(2010)findthatthekeysourceofpersistentdisagreementstemsfromheterogeneity inpriorsandshowthatthedifferencesinopinionmovecountercyclically. Farmeretal.(2021) documenttheimportanceoflong-termbeliefsintheformationofmacroeconomicexpectations byprofessionalforecasters. Comparedtopreviousstudies,ourpaperoffersthreedistinctcontributions. First,weincorporatelong-termbeliefs,publicinformation,andprivateinformation intoacoherentempiricalframework,allowingforheterogeneousresponsestopublicinformation. Second,weestimatethecontributionofeachinformationsourcetoindividualforecasters’ 4Crumpetal.(2021)estimateamultivariatetrend-cyclemodelusingtheuniverseofprofessionalforecastsfor theU.S.,andshowthatthemultivariatemodelbetterfitsthedatathanestimatingindividualunivariatemodelson eachtimeseries.Fisheretal.(2022)modelpeople’sexpectationsofinflationusingatrend-cycledecompositionand estimatethetermstructureofexpectationsusingthefullpanelstructureoftheSPFassumingagentsreceiveprivate andpublicsignals. 5Theyadoptthreecommonfactorsandanidiosyncraticcomponent,andestimatethemodelwithBayesian methods. 5

inflationexpectationsusingaformalstatisticalmodelwithminimalaprioristructure. Third,we explicitlydemonstratehoweachinformationsourcecontributestothedisagreementininflation expectationsacrossvariousforecastinghorizonsandateachpointintime. Our paper also relates to recent studies on the interaction between monetary policy and disagreementofeconomicagents. Andradeetal.(2016)provideempiricalevidenceforheterogeneousbeliefsaboutforwardguidanceandanalyzetheeffectofmonetarypolicyinthecontextof anewKeynesianmodel. GlasandHartmann(2016)distinguishindividualinflationuncertainty anddisagreementbetweenforecasters,andshowthatoveralldisagreementincreasesduring periodsofcontractionarypolicy. Ehrmannetal.(2019)showslong-horizontime-contingentand state-contingentforwardguidanceeffectivelyreducesdisagreement,whileshort-horizontimecontingentforwardguidancedoesnot. Falcketal.(2021)showthatapricepuzzlearisesinstates withhighdisagreementbutdisappearsinstateswithlowdisagreementusingastate-dependent localprojection. Dongetal.(2024)empiricallyshowthatinflationdisagreementweakensthe powerofforwardguidanceandconventionalmonetarypolicyandprovideastructuralmodel wherehouseholdshaveheterogeneousbeliefsabouttheinflationtargetofcentralbanks. Our researchdiffersfromrecentstudiesinthatwefurtheridentifythesourceofdisagreementand explicitly show that disagreement attributable to public information is the component that delaysmonetary-policyeffectsandcreatesthepricepuzzle. Lastly,wecontributetothediscussiononhowtoassesstheanchoringofinflationexpectations. Clarida(2021)mentionsthattheassessmentofanchoredlong-runinflationexpectations ispivotaltooutcome-basedforwardguidance. BundickandSmith(2020)analyzetheeffectof monetarypolicyoninflationexpectationsmeasuredfromtheMichiganSurveyofConsumers andTIPSbreak-evenrates,andfindthatinflationbecamemoreanchoredaftertheannouncementoftheinflationtargetin2012. Incontrast,Reis(2020)developaparsimoniousstructural modelthatcancharacterizediscrepanciesininflationexpectationsbetweenfinancialmarkets andhouseholds,andfindthatinflationbecamegraduallymoreunanchoredfrom2014onwards, whichposesatrade-offintheconductofmonetarypolicy. Ournovelempiricalapproachcontributestotheliteratureintwokeyways: First,itdemonstratesthatdisagreementcanserveas anindependentmetricforassessinganchoredexplanations;Second,itintroducesanempirical methodtoquantifytheextenttowhichcommunicationaboutpublicinformation,including monetarypolicyannouncements,canreduceinflationexpectationsanddisagreementabout futureinflation. 6

The paper is organized as follows. Section 2 discusses the data on inflation expectations fromtheSurveyofProfessionalForecasters. Section3introducestheindividual-levelparametric modelusedforthetermstructureofinflationexpectations. Section4presentstheestimation results. Section 5 introduces a noisy information model that decomposes each forecaster’s inflationprojectionintocontributionsfromlong-termbeliefs,privateinformation,andpublic information. Section6showshowtodecomposedisagreementaboutinflationexpectationsinto contributionsfromthethreeinformationsources. Section7analyzestheeffectsofdisagreement attributable to public and non-public information on the effectiveness of monetary policy. Section8concludes. 2 Data: The Survey of Professional Forecasters ThissectiondiscussesinflationexpectationsdatafromtheSurveyofProfessionalForecasters. 2.1 Notation First, wedefinesomenotationthatwewillusethroughouttherestofthepaper. Denotethe pricelevelattimet byP t (inourcasethiswillrefertotheconsumerpriceindex). Letπ s→t be thecontinuouslycompoundedinflationratebetweentimes andtimet: π s→t ≡log(P t )−log(P s ). (1) Throughoutwewillworkwithcontinuouslycompoundedinflationratesbecauseoftheirtimeadditiveproperties. Definetheforecast,madeattimet,oftheinflationratebetweentimesr and s,asπ r→s|t . Finally,letq a−1 (t)denotethefinalquarteroftheyearpriortotheyeartimet isin. 2.2 DefinitionofForecastedQuantities WecollectdataonCPIinflationforecastsfromtheSPF,conductedbytheFederalReserveBank of Philadelphia. The survey is sent out in the first month of each quarter and responses are collectedaroundthemiddleofthequarter,e.g. mid-FebruaryinQ1. Surveyparticipantsare askedtoforecasttheaveragequarterlyleveloftheCPI(ortransformationsofthisquantity)at varioushorizons. TheSPFCPIInflationForecastscanbebrokeninto4categories: 7

• 1-periodbackcaststo4-quarteraheadforecastsofannualizedquarter-over-quarterCPIinflation: 100× (cid:183)(cid:181) P t+h (cid:182)4 −1 (cid:184) P t+h−1 forh=−1,...,4. • 1to3-yearaheadforecastsofQ4overQ4CPIinflation: 100× (cid:183) P qa−1(t)+4j −1 (cid:184) P qa−1(t)+4(j−1) for j =1,...,3. • ForecastsofaverageQ4overQ4CPIinflationoverthenext5years:  (cid:195) (cid:33)1  100×  (cid:89) 5 P qa−1(t)+4j 5 −1 j=1 P qa−1(t)+4(j−1) • ForecastsofaverageQ4overQ4CPIinflationoverthenext10years:  (cid:195) (cid:33)1  100×  (cid:89) 10 P qa−1(t)+4j 10 −1 j=1 P qa−1(t)+4(j−1) Thefirsttypeofforecastiswhat’sknownasafixedhorizonforecastandtheotherthreetypes areknownasfixedevent forecasts. Weassumethattheseforecastscorrespondtoforecastsof continuouslycompoundedinflationsothattheylineupwithourmodelspecificationdirectly.6 Thatis,weassume 100×E t (cid:183)(cid:181) P t+h (cid:182)4 −1 (cid:184) ≈400×π t+h−1→t+h|t P t+h−1 100×E t (cid:183) P q P a q − a 1 − (t 1 ) ( + t) 4 + ( 4 j− j 1) −1 (cid:184) ≈100×π qa−1(t)+4(j−1)→qa−1(t)+4j|t  (cid:195) (cid:33)1  100×E t j (cid:89) = 5 1 P q P a q − a 1 − (t 1 ) ( + t) 4 + ( 4 j− j 1) 5 −1 ≈20×π qa−1(t)→qa−1(t)+19|t  (cid:195) (cid:33)1  100×E t j (cid:89) 1 = 0 1 P q P a q − a 1 − (t 1 ) ( + t) 4 + ( 4 j− j 1) 10 −1 ≈10×π qa−1(t)→qa−1(t)+39|t . 6Thisturnsouttoberelativelyinnocuousassumption,seeAruoba(2020)foradiscussion. 8

Oursamplebeginsin1991Q4,whichisthefirsttimethatforecastsofaverageQ4overQ4 inflationover thesubsequenttenyearsbecome available, andrunsthrough2023Q3. Thisis necessarytobeabletoestimatethelong-endofthetermstructure. Three-yearaheadforecasts ofQ4ofQ4inflationandforecastsofaverageQ4overQ4inflationoverthesubsequentfiveyears firstbecomeavailablein2005Q3. Werestrictoursampletoforecasterswhoreportnowcasts tofour-quarteraheadforecastsandeithera5-yearor10-yearaverageforecastinatleastone quartertoensurethatwecanidentifylong-runforecasts. 2.3 PropertiesofSPFCPIForecasts Thereare172uniqueforecastersinthedatasetduringoursampleperiod. Inanygivenquarter, therearebetween28and53forecasterswhoreportaforecastandamedianof37. Forecasters remaininthedatasetforbetween1and112quarterswithamediantenureof14quarters(3and ahalfyears). Figure1displaysthedistributionof1-quarteraheadforecastsand10-yearaheadforecasts. Theupperpanelsdisplaythemean(whichwewillrefertoasconsensus)oftheforecastsandthe projectionsoftwoindividualforecasters. Theupper-leftpanelshows1-quarteraheadforecasts and the upper-right panel reports 10-year ahead forecasts. The gray bars represent NBER recessions. Inthefirsttentofifteenyearsofthesamplethereisadownwardtrendinbothshortandlong-runinflationforecastsfromaround4%to2%. Short-runforecaststendtoexhibitmore timeseriesvolatilitythanlong-runforecastsastheyreactmorestronglytotransitoryshocks. Thebottompanelsalsoshowthecross-sectionalstandarddeviation(whichwewillrefertoas dispersionordisagreement)ofprojectionsonequarteraheadanda10yearsahead. Short-run forecaststypicallyexhibithigherdispersionthanlong-runforecasts,withanotableexception beingtheearly1990s. InadditiontoFigure1whichcapturestheaggregatepropertiesoftheSPFforecasts,wealso wish to explore how the expectations of the individual forecasters evolve over time. To this end,Figure2plotsthereportedforecastsoverallhorizonsoftwodifferentforecastersforthree differenttimeperiods:1)2007Q3,2)2009Q2and3)2022Q3. Thesolidblacklineistheconsensus, andthedashedlinesarethe5thand95thpercentilesofthedistribution.Thethreedatesreported inthefigurehelpillustratethewidevarietyofshapesthetermstructureofinflationexpectations cantakeandhowdifferentanindividualforecastercanberelativetotheconsensus. Thelevels offorecasts,aswellasthetrajectoriesovertheforecastinghorizons,arealldifferent. 9

Figure1: SPF FORECAST SUMMARY STATISTICS AND EXAMPLES Notes:Theblacklinesintheuppertwopanelsshowtheaverageofinflationforecasts1quarter(leftpanel)and10 years(rightpanel)ahead.ThecoloredlinesaretheforecastsoftwoforecasterswhoseIDsare107and122inthe survey.Thebottomtwopanelsreportthecross-sectionalstandarddeviationoftheforecasts. Sources:SPFandauthors’calculation. Figure2: THREE TERM STRUCTURES OF OBSERVED SPF FORECASTS Short-Run Forecasts Long-Run Forecasts 4 4 3 3.5 3 2 2.5 1 2 0 Q3-2007 Q4-2007 Q1-2008 Q2-2008 Q3-2008 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 4 4 3 3 2 2 1 1 0 0 Q2-2009 Q3-2009 Q4-2009 Q1-2010 Q2-2010 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 10 10 Mean 8 Forecaster 107 Forecaster 122 6 5 4 2 0 Q3-2022 Q4-2022 Q1-2023 Q2-2023 Q3-2023 2022 2023 2024 2025 2026 2027 2028 2029 2030 2031 2032 Notes:Theleftpaneldisplaysshort-horizonforecasts,whiletherightpaneldisplayslong-horizonforecasts.Thetop panelspresentasnapshotofSPFforecastsfor2007Q3,themiddlepanelsshowthesnapshotfor2009Q2,andthe bottompanelsdisplaythesnapshotfor2022Q3.Theblacklinescaptureconsensusforecasts.Thecoloredlines capturetheforecastsoftwoforecasterswhoseIDsare107and122inthesurvey.Thedashedlinesaretheforecasts atthe5thand95thpercentilesofthedistribution. Sources:SPFandauthors’calculation. 10

FollowingPattonandTimmermann(2010),weonlykeepforecasterswhosubmit12ormore forecasts. This allows us to have a higher degree of confidence inour dynamic factor model decomposition. However, information loss from this treatment is not substantial, as it only lowersthenumberofforecastersinagivenquarterbyafewpeopleonaverage. 3 An Individual Term Structure of Inflation Forecasts Inthissection,wespecifyandestimateamodeltorecoverthecompletepathofinflationforecasts overa10-yearhorizonateachpointintime,forallforecasters. 3.1 Model FollowingAruoba(2020), wesetupaNelson-Siegelmodelforthetermstructureofinflation expectations: (cid:181) 1−e −λ ih(cid:182) π i,t→t+h|t =L i,t − λ h S i,t , (2) i whereL andS areforecaster-specificlevelandslopefactors,andtheλ areforecaster-specific i,t i,t i shapeparameters. Giventhisrepresentation,theforecastofinflationbetweenanytwohorizons h andh isgivenby 1 2 (cid:181) e −λ ih1 −e −λ ih2 (cid:182) π i,t+h1 →t+h2 |t =L i,t − λ (h −h ) S i,t . i 2 1 FollowingDieboldetal.(2008),wespecifythefollowingdecompositionforthefactors L =αL+βLL +εL (3) i,t i i t i,t S =αS+βSS +εS (4) i,t i i t i,t whereL and,S arelevelandslopefactorswhicharecommontoallforecasters,αL,andαS are t t i i forecaster-specificconstanttermswhichgoverntheoverallleveloftheindividualfactors,βL and i βS areforecaster-specificloadingsonthecommonfactors,andεL andεS capturethepurely i i,t i,t idiosyncraticcomponentsoftheindividualfactors.7 7Inthisdecomposition,weassumethattheloadingsaretime-invariant. Inthedynamicfactormodel,what isidentifiedisthecommoncomponent,whichistheproductoftheloadingandthetime-varyingfactor. This commoncomponentissufficienttocaptureforecasteri’sheterogeneousreactionstocommonshocks.Therefore, 11

Ourgoalistospecifyamodelwhichcancapturerichheterogeneityacrossforecasterswhile remainingparsimoniousandinterpretable. Forthisreason,weomitthecurvaturefactor. We findthatcurvatureplaysalimitedroleintheobservedindividual-leveltrajectoriesofinflation expectations.8 Wealsoassumeλ =λ. Sinceλ primarilydeterminesthepeakofthecurvature, i i this simplification does not have material effects in the estimation given the absence of the curvaturefactor. WeassumethatthecommonfactorsfollowindependentAR(1)processes: L t =ρ L L t−1 +u t L S t =ρ S S t−1 +u t S. (5) Sincethescaleofthecommonfactorsandthefactorloadingsarenotseparatelyidentified,we normalizetheshockstothecommonfactorsuL anduS tohaveunitvariance,andweassume t t theshocksareuncorrelated.9       uL 0 1 0  t  ∼N ,  (6) uS 0 0 1 t Inaddition,weassumethattheidiosyncraticcomponentsevolveaccordingtoAR(1)processes whichareindependentacrossforecasters.10 εL =ρL εL +uL i,t i,ε i,t−1 i,t εS =ρS εS +uS (7) i,t i,ε i,t−1 i,t Wealsoassumethatthecovariancematrixisdiagonal,sothatthefactorsevolveindepeninprinciple,ourmodeldoesnotprecludetime-varyingloadings. Wecouldallowforslowtimevariationinthe loadingparameters,butthiswouldrequireidentifyingassumptionsaboutthedynamicprocessofboththeloading andthefactortoexplicitlyestimatethetime-varyingloadingandfactor.Inthispaper,wefollowthemoststandard approach—constantloadingsandtime-varyingfactors—inthedynamicfactormodelliterature. 8Weconsiderthecurvatureandamoreflexiblefactordynamics(AR(3))inAppendixH.Ourempiricalresults remainrobust. 9Thistreatmentisastandardapproachintheliteratureofdynamicfactormodel(e.g.,Dieboldetal.,2008). 10ThedynamicsofthecommonandidiosyncraticcomponentscanbegeneralizedtofollowVARswithadditional lagsatminimalcomputationalcost.Thismakeslittledifferenceintheestimatedfactorsandloadingswhichiswhy westickwithAR(1)processesinourbaselinespecification. 12

dently:       uL 0 σ2 0  i,t  ∼N , i,L . (8) uS 0 0 σ2 i,t i,S Forparsimony,wemakethefurthersimplifyingassumptionthatρL ,ρS ,σ2 ,andσ2 arethe i,ε i,ε i,L i,S sameforallforecastersi andequaltoρL,ρS,σ2,andσ2 respectively.11 ε ε L S Since the model is specified for continuously compounded inflation at the quarterly frequency,mappingthemodelpredictionstoobservedSPFforecastsisstraightforward. Ourmodel canbecastasalinearGaussianstatespacemodeloftheform x t =Fx t−1 +u t , u t ∼N(0,Q) (9) y =µ +Hx +v , v ∼N(0,R) (10) t y t t t wherex isavectorofstatescontainingthecommonandidiosyncraticfactors,Fcapturesthe t dynamicsofthestatesovertime,µ isavectorofforecaster-specificfixedeffects,Hdetailsthe y mappingfromthestatestotheobservedforecasts,andQandRarethecovariancematricesof theinnovationstothestateequationandthemeasurementerrorsrespectively. Thedetailsofthe state-spacerepresentationarepresentedinAppendixA. 3.2 Estimation Ourbaselinemodelhasatotalof431parametersconsistingof • Forecaster-specificmeans (cid:169)αL,αS(cid:170)n i i i=1 • Forecaster-specificfactorloadings (cid:169)βL,βS(cid:170)n i i i=1 • Factorautocorrelationparametersρ ,ρ ,ρL,andρS L S ε ε • Idiosyncraticfactorconditionalvariancesσ2 andσ2 L S • Shapeparameterλ 11SeeAppendixG.1formorediscussionsonthemodelingassumptionsandtheirimplication.Thesimplifyingassumptionsdonotimplythattherealizedidiosyncraticcomponentsareidentical;rather,theestimatedidiosyncratic componentscanstilldiffersubstantiallyacrossindividuals.Evenwiththesesimplifyingassumptions,wehave431 parameterstoestimate.Whileitisinprinciplepossibletoallowtheseparameterstovaryacrossindividuals,this moreflexibleapproachwouldaddover200additionalparameters,likelyincreasingtheuncertaintyoftheestimates. Therefore,ourcurrentapproachstrikesagoodbalancebetweenflexibilityandparsimony. 13

• Measurementerrorvariancesσ2 ,...,σ2 .12 v,1 v,20 Theparametervectorisdenotedas θ=(cid:163)αL,...,αS,βL,...,βS,ρ ,ρ ,ρL,ρS,σ2,σ2,λ,σ2 ,...,σ2 (cid:164)′ . 1 n 1 n L S ε ε L S v,1 v,20 SinceourmodelisalinearGaussianstate-spacemodel,weemploytheKalmanfiltertoconductinferenceonthelatentvariablesandformthelikelihoodfunction. Themodelisestimated withaGibbssampler,detailedinAppendixB.13 4 Estimation Results Thissectionreportsanddiscussesourresultsfromtheestimationoftheterm-structuremodel. 4.1 ParameterEstimates Table1reportsthemedian,5th,and95thpercentilesoftheposteriordistributionsforourmodel parameters. In the case of the forecaster fixed-effects and factor loadings αL,αS,βL, and βS, i i i i we report the median, 5th, and 95th percentiles across posterior draws of the average value across forecasters. The average value of αL is consistent with the Fed’s 2% inflation anchor, i sinceCPIinflationisknowntobeslightlyhigher—byabouthalfapercentagepointonaverage— thanPCEinflationwhichtheFedexplicitlytargets. Onaverage,thetermstructureofinflation expectationsisupwardslopingasindicatedbyapositivevalueofαS. Boththecommonand i idiosyncraticfactorsareestimatedtobehighlypersistent,withautocorrelationcoefficientsof between0.73and0.95atthequarterlyfrequency.Finally,measurementerrorstandarddeviations areestimatedtobebetween10basispoints(forthreeyearforwardexpectations)and67basis points(foronequarteraheadexpectations). 12SeeAppendix??formoredetailsonthemeasurementequation.Insummary,weuseone-quartertofour-quarter aheadfixed-horizonforecasts,alongwithtwo-yearforward,three-yearforward,five-yearaverage,andten-year averagefixed-eventforecasts.Foreachquarteroftheyear,thefixed-eventforecastscoverdifferentforecasthorizons, resultinginfourdistinctmeasurements.Consequently,themeasurementmodelcomprisestwentyequations:four fromthefixed-horizonforecastsandsixteenfromthefixed-eventforecasts. 13Alternatively,wealsoconsideredaFrequentistapproachforrobustnesschecks,whichisoutlinedinAppendix G.2.Theoverallresults,availableuponrequest,remainqualitativelyrobust. 14

Table1: POSTERIOR PARAMETER DISTRIBUTION STATISTICS Parameter Median 95%CI Parameter Median 95%CI αL 2.482 [2.422,2.548] σ 0.261 [0.247,0.276] i v,6 αS 0.192 [0.071,0.319] σ 0.216 [0.202,0.232] i v,7 βL 0.011 [-0.027,0.051] σ 0.095 [0.079,0.112] i v,8 βS 0.375 [0.315,0.447] σ 0.304 [0.285,0.326] i v,9 ρ 0.904 [0.786,0.996] σ 0.323 [0.303,0.346] L v,10 ρ 0.835 [0.685,0.979] σ 0.297 [0.279,0.318] S v,11 ρL 0.947 [0.928,0.964] σ 0.331 [0.311,0.352] ε v,12 ρS 0.732 [0.696,0.767] σ 0.185 [0.172,0.199] ε v,13 σ 0.125 [0.117,0.134] σ 0.197 [0.183,0.211] L v,14 σ 0.440 [0.421,0.458] σ 0.237 [0.222,0.254] S v,15 λ 0.174 [0.166,0.182] σ 0.258 [0.242,0.275] v,16 σ 0.672 [0.656,0.688] σ 0.193 [0.181,0.206] v,1 v,17 σ 0.469 [0.458,0.480] σ 0.188 [0.176,0.201] v,2 v,18 σ 0.444 [0.434,0.454] σ 0.181 [0.169,0.194] v,3 v,19 σ 0.443 [0.433,0.454] σ 0.221 [0.208,0.235] v,4 v,20 σ 0.270 [0.256,0.285] v,5 Notes:The“Median"columnreportstheposteriormedianofthecorrespondingparameterandthe“95%CI" columnreportstheparameter’s95-percentcredibleinterval.FortheparametersαL,αS,βL,andβS,thetable i i i i reportsstatisticsfortheaveragevalueacrossforecasters. Sources:Authors’calculation 4.2 Consensus Figure3plotsthemedianoftheposteriordistributionforthesmoothedcommonfactors,L t|T andS t|T ,alongwith95percentcredibleintervals. Notethatbothseriesarenormalizedsothat theirconditionalvariancesare1. Thetoppanelplotsthesmoothedcommonlevelfactor. The estimatecloselytracksvariationsinlong-runinflationexpectations. Theestimateexhibitsa sharpdownwardtrendinthe1990s. Inthelaterpartofthesample,theestimatedropssharply aftertheGreatRecessionandstaysdepressedforseveralyearsafterwards. DuringtheCOVID-19 pandemic,theestimatedipsintheearlyphaseofthepandemic,quicklyrecovers,anddoesnot exhibitanynotablechangesthereafter. Thebottompanelplotsthesmoothedcommonslopefactor. Theestimatetracksexpected changesininflationbetweenthecurrentquarterandthelongrunateachpointintime.Theslope istypicallypositive,indicatingthatthetermstructureofinflationexpectationsisupwardsloping onaverage. Thesteepestpositiveslopesoccurintheyearsfollowingthedotcombubbleand intheGreatRecession. Theslopesystematicallydeclinedoverthe2010s,reflectingthedecade 15

Figure3: SMOOTHED COMMON FACTORS Notes:Theupperpanelplotstheposteriormedianofthesmoothedcommonlevelfactor(blueline)alongwith95 percentcredibleintervals(blackdashedline).Thebottompanelplotstheposteriormedianofthesmoothed commonslopefactor(blueline)alongwiththe95percentcredibleintervals(blackdashedline).Theshadedareas denoteNBERrecessions. Sources:Authors’calculation oflowandstableinflationaftertheGreatRecession. Theslopeincreaseswiththeonsetofthe COVID-19pandemic.Thisreflectsthatforecastersexpectinflationtoincreaserelativetoinflation inthecurrentquarter,giventhatinflationplungedintheearlyphaseofthepandemicandlongrunexpectationsdidnotchangemuch. From2021onward,wheninflationpickeduprapidly, theslopeestimatesharplydeclinedandstayednegative,reflectingforecasters’expectationthat inflationwouldeventuallydecline. Figure4plotsthemedianand95-percentcredibleintervalforthemeanoftheforecasting distribution(acrossforecasters)atfourdifferentforecasthorizons. Thetoppanelsplotmean 6-monthand1-yearaheadinflationexpectations,whichtracktheconsensusinflationnowcast andthusrealizedinflationtoalargeextent. Thebottompanelsdisplaymean5-yearand10year ahead inflation expectations, which are significantly less variable than the short-term expectationsastheyincorporatelessofthevariationinrealizedinflation. The10-year-ahead inflationconsensusforecastlargelytrackschangesinthecommonlevelfactor. 16

Figure4: SMOOTHED CONSENSUS FORECASTS Notes:Thisfigureplotstheposteriormedianoftheaverageinflationforecastposteriordistribution(blueline) recoveredfromtheindividual-leveltermstructuremodelalongwith95-percentcredibleintervals(blackdashed line).Thefourpanelscorrespondtodifferentforecastinghorizons,6-month,1-year,5-year,and10-year respectively.TheshadedareasdenoteNBERrecessions. Sources:Authors’calculation 4.3 Dispersion: AProxyforDisagreement Thissectiondiscussesthedistributionofindividualterm-structurecomponentsandforecasts, withaparticularemphasisondispersion,asourprimaryinterestliesinunderstandingdisagreement. Figure5plotsthedistributionsofforecasters’smoothedlevelandslopefactors. Specifically, weplotposteriormediansofthe5th,25th,50th,75th,and95thpercentilesofforecastsacross forecasters. Thefactorsareinunitsofannualizedpercentagepoints. Thetoppaneldisplays thedistributionofindividualsmoothedlevelfactors. Althoughconsensuslong-runinflation expectationsremainlowandstableafter2005,thedispersionofestimateschangessubstantially overtime. Inparticular,thedispersionseenaftertheGreatRecessionislargerthanthatseen intheearly1990s,whenconsensuslong-runinflationexpectationswerearoundfourpercent andtrendingdown. DuringtheCOVID-19pandemic,themedianlevelfactoredgedupslightly, but the distribution dramatically skewed to the right, reflecting the perceived upside risk in long-runinflation. Thebottompaneldisplaysthedistributionofindividualsmoothedslope factors.Similartothelevelestimates,theslopedispersionincreasessubstantiallyoverthecourse 17

Figure5: SMOOTHED FACTOR DISTRIBUTIONS Notes:Thefigureshowsthecross-sectionaldistributionsoftheindividuallevelfactors(upperpanel)andindividual slopefactors(bottompanel).Thesolidbluelineistheposteriormedianofthemedianfactoracrossforecasters.The dashed-dottedlinesdepicttheposteriormediansofthe25thand75thpercentiles.Thedashedlinesdepictthe posteriormediansofthe5thand95thpercentiles.TheshadedareasdenoteNBERrecessions. Sources:Authors’calculation oftheGreatRecessionandtheCOVID-19pandemic. Inparticular,thedistributionbecomes skewedtotherightattheonsetofthepandemic,butitbecomesskeweddramaticallytotheleft followingarapidriseininflationin2021. Figure6plotsthecross-sectionaldistributionoftheconstants(αL andαS)andthefactor i i loadings(βL andβS).Weplotthedistribubtionacrossforecastersoftheposteriormedianofeach i i parameter. Theupperpanelsshowthatforecastersexhibitsignificantdisagreementaboutthe long-runmeansofthelevelandslopefactors. Althoughmostforecastersexpectthelong-term meanleveltobebetween2%and2.5%,asignificantfractionanticipatesittobeabove2.5%(the upperrightgraph). Forthelong-runslope,themajorityofforecastershavealong-runmean close to zero, though the distribution shows a large dispersion ranging from -0.7 to 1.4. The bottom panels display the distributions of factor loadings. For both the level factor and the slopefactor,theloadingsshowsignificantdispersion. Specifically,theloadingforthelevelfactor rangesfrom-1to2,whiletheloadingfortheslopefactorspansfrom-1.5to1.5. Thedistributions oflong-runmeansshowthatforecastersexhibitconsiderableheterogeneityintheirreactionsto commonshockswhenformingexpectationsaboutfutureinflation. 18

Figure6: DISTRIBUTIONS OF POSTERIOR MEDIANS: CONSTANTS AND FACTOR LOADINGS Notes:Thefigureshowsthecross-sectionaldistributionofposteriormedianconstanttermsandfactorloadingsfor boththelevelandslopefactors. Sources:Authors’calculation Figure7displaysthedistributionofinflationforecastsatfourdifferenthorizons: 6months,1 year,5years,and10yearsfromtoplefttobottomright. Thisfigurerevealsstarkchangesinthe forecastdistributionsthatareobscuredintheconsensusexpectations,offeringinsightsintothe anchoringofinflationexpectationsthatdiffersignificantlyfromwhattheconsensussuggests. Forexample,whenthemeaninflationprojectionsarelowandstable,from2005onward,the dispersionsbecomelarger than thoseobservedinthe 1990s(whenthe levelofinflationwas higherandmorevolatile). Itisalsonotablethatrightskewnessincreasedsignificantlyduringthe COVID-19pandemic,indicatingthatforecastershaddifferingopinionsabouttheupsideriskof inflation,evenasconsensusexpectationsappearedrelativelystable. ToensurethatourconclusionsfromFigure7arenotpurelydrivenbyoutliers,weexamine thestandarddeviation(disagreement)andskewnessoftheforecastingdistribution,displayed inFigure8. ConsistentwithourobservationinFigure7,disagreementincreaseddramatically overthecourseoftheGreatRecessionandstayedelevatedforafewyearsaftertheendofthe recession. Duringthistime,disagreementismuchlargerthaninthe1990sacrossforecasting horizons. Inaddition,skewnessalsoincreasedduringthepandemicandreacheditshighest 19

Figure7: DISTRIBUTION OF FORECASTS Notes:Thefigureshowsthecross-sectionaldistributionofindividualinflationforecastsatfourdifferentforecast horizons.Thesolidbluelineistheposteriormedianofthemeanforecastacrossforecasters.Thedottedlinesdepict theposteriormediansofthe25thand75thpercentiles.Thedashedlinesdepicttheposteriormediansofthe5th and95thpercentiles.TheshadedareasdenoteNBERrecessions. Sources:Authors’calculation levelsinthesampleperiod. Bothdisagreementandskewnessdeclinedafter2022butremained elevatedrelativetothepre-pandemiclevelsattheendof2023. Thisobservationsuggeststhat disagreementcarriesinformationaboutthedegreeofanchoringininflationexpectationsquite differentfromwhatiscapturedbytheconsensus. 4.4 IndividualDynamics Next, we examine the individual-level term structure of inflation expectations. The purpose of this section is to show that the individual term structures of inflation expectations differ significantlyfromtheconsensustermstructure,revealingthatforecastershavesubstantively differentviewsabouttheoverallpathofinflationinthefuture. Thisdisagreementregarding the term structure of inflation expectations cannot be captured by a model based solely on theconsensustermstructure. Meanwhile,theindividualtermstructuresnotonlycomovebut alsoexhibitdistinctvariationsacrossindividuals,whichourdynamicfactormodelsuccessfully captures. Itisimportanttonotethatourparsimoniousindividual-levelmodelisflexibleenough toaccommodatethevariouspatternsofdisagreementacrossdifferentforecastinghorizons. 20

Figure8: DISAGREEMENT ABOUT AND SKEWNESS OF FORECASTS Notes:Thefigureshowsthestandarddeviationandskewnessofindividualinflationforecastsatfourdifferent forecasthorizons.Thesolidbluelineistheposteriormedianofthedisagreementacrossforecasters.Thesolidred lineistheposteriormedianoftheskewnessacrossforecasters.Thedashedlinesdepicttheposterior5thand95th percentilesofthedisagreementandskewness.TheshadedareasdenoteNBERrecessions. Sources:Authors’calculation Figure9showsthetermstructureofinflationforecastsoftwoindividualforecastersalong withtheconsensus(leftpanels)andthedisagreementacrossforecastinghorizons(rightpanels). First, we consider 2007Q3 (upper panels). This was the final survey conducted before the startoftheGreatRecession. Theconsensustermstructureisrelativelyflat,withthenowcast and10-yearaheadinflationexpectationsbeingaround2.25%. Meanwhile,thetwoforecasters disagreeaboutthenowcastsandthedirectionofslopes,asshowninthedownwardslopingterm structureofforecaster535(redline)andtheupwardslopingtermstructureofforecaster518 (blueline). Overall,disagreementislargestintheshortrunat70basispointsbutdecreasesover theforecastinghorizontoabout30basispoints,yieldingadownward-slopingtermstructureof disagreement(upperrightpanel). Averydifferentpatternisobservedin2009Q2–thepeakoftheGreatRecession. Theterm structuresareupward-slopingforbothindividualforecastersandtheconsensus. Althoughthe twoforecasterssignificantlydisagreeaboutnear-terminflation,theyarestillexpectingarisein inflationovertheforecasthorizon,inlinewiththeconsensus. Thedisagreementishighinthe shortrunat1%,smallestatabouttheone-yearhorizonat85basispoints,andthenincreases again in the long run to about 1.05% (middle right panel). This non-linearity signals higher 21

Figure9: TERM STRUCTURE OF INFLATION EXPECTATIONS AT THREE DATES Forecasts Disagreement 0.7 2.5 0.6 0.5 2 0.4 1.5 0.3 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 Forecasts Disagreement 3 1.1 2 1 0.9 1 0.8 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 Forecasts Disagreement 4 2 3.5 1.5 3 1 2.5 2023 2024 2025 2026 2027 2028 2029 2030 2031 2032 2023 2024 2025 2026 2027 2028 2029 2030 2031 2032 Notes:Theleftpanelsdisplayestimatedposteriormediansofthetermstructureofinflationexpectationsforthe consensus(blackline)andtwoindividualforecasters(forecaster518inblueandforecaster535inred).Theright panelsdisplayposteriormediansofthetermstructureofdisagreement(solidblueline)alongwith90%credible intervals(dashedblacklines)overthecorrespondingforecasthorizon.Thethreerowscorrespondtothedates 2007Q3,2009Q2,and2022Q2,respectively. Sources:Authors’calculation uncertaintyintheimmediatefutureandinthelongrunthanintheshorttomediumrun. Finally, we consider 2022Q2 (bottom panels). Inflation expectations are monotonically decreasingforbothforecastersandintheconsensus. Thedisagreementislargestinthecurrent quarterbutreducesovertheforecastinghorizon,producingadownward-slopingtermstructure ofdisagreement(lowerrightpanel). Althoughthetermstructureofdisagreementissimilarto thatpriortotheGreatRecession,themagnitudeofdisagreementisthreetimesaslargeasthat seenin2007intheshortrun. 5 A Noisy Information Model of Disagreement Thissectionintroducesanoisyinformationmodelthatprovideseconomicinterpretationsofthe threecomponentsoftheindividual-levelandslopefactors: theindividual-specificconstants, common components, and idiosyncratic components. The structural model also helps to motivatethedisagreementdecompositionwewillpresentinSection6. Inthenoisyinformation model, aforecasterwithindividuallong-termbeliefsupdatesinflationprojectionsbasedon 22

bothpublicandprivatesignals. Wedemonstratethatthisstructuralmodelnaturallyalignswith the dynamic factor characterization presented in Section 3. We show that the common and idiosyncraticcomponentsofthefactormodelreflectthecontributionsofpublicandprivate informationtoaforecaster’sinflationexpectations,whiletheindividualfixedeffectcapturesa forecaster’slong-runbelief.14 FollowingStockandWatson(2016),wedescribeinflationattimet (π )asthesumofitslowt frequencycomponent(τ )andtransitorycomponent(c ). Considerthefollowingtrend-cycle t t state-spacemodelforinflation: π =τ +c t t t (cid:179) (cid:180) τ t =τ t−1 +ετ t , ετ t ∼N 0, (cid:161)στ ε (cid:162)2 (cid:179) (cid:180) c t =ρ c c t−1 +εc t , εc t ∼N 0, (cid:161)σc ε (cid:162)2 . AsinBeveridgeandNelson(1981),thetrendcomponentofavariablecapturesitslong-runforecast(π t→t+∞|t ),whereastheforecastforashorterhorizonrepresentstransitorydeviationsfrom thislong-runforecast. Deviationsofcurrentinflationfromitstrendcomponentarecapturedby thetransitorycomponent. ThisnotionofBeveridgeandNelsondecompositionnaturallymaps thetrendtothelevelelementoftheterm-structuremodelandthetransitorycomponenttothe slopeelementofthemodel.15 We assume that forecasters receive signals about both the trend and cycle components separately16,whicharesubjecttoprivatenoiseshocksv τ andvc ,andpublicnoiseshocksu τ i,t i,t t anduc: t (cid:179) (cid:180) y τ=τ +u τ , u τ∼N 0, (cid:161)στ(cid:162)2 t t t t u (cid:181) (cid:182) z τ =τ +v τ , v τ ∼N 0, (cid:179) στ (cid:180)2 i,t t i,t i,t i,v 14Ourstructuralcharacterizationcombineselementsfrombothnoisyinformationmodels(e.g.,Coibionand Gorodnichenko,2012a)andlearningmodels(e.g.,LahiriandSheng,2008).Thenoisyinformationmodelincludes both public and private information but lacks a long-run belief component. In contrast, the learning model incorporatespublicinformationandalong-runbeliefcomponentbutdoesnotaccountforprivateinformation. 15Thedeviationofh-period-aheadinflation(forh<10years)formedattimet (denotedπ t→t+h|t )fromitstrend (π t→t+∞|t )isalsoconsideredtransitory,withthemagnitudeofthedeviationbeingcapturedbytheslopeelementat timet withintheterm-structuremodel. 16Thisassumptionisnotnecessarybutservestosimplifytheexposition.Ifonlysignalsofinflationareobserved, thenthedynamicsoftheslopeandlevelfactorsinourstatisticalmodelbecomeavectorautoregressioninsteadof independentunivariateautoregressiveprocesses. 23

(cid:179) (cid:180) yc =c +uc, uc ∼N 0, (cid:161)σc(cid:162)2 t t t t u (cid:181) (cid:182) (cid:179) (cid:180)2 zc =c +vc , vc ∼N 0, σc i,t t i,t i,t i,v y τ and yc arepublicsignalsofthetrendandcyclecomponentsrespectively,whilez τ andzc t t i,t i,t areprivatesignalsofthetrendandcyclecomponentsrespectively. Weallowforthepossibility thatthevarianceoftheprivatesignal,σ ,isdifferentacrossforecastersforboththetrendand i,v cyclecomponents. Forecastersmaintaintime-invariantlong-runbeliefsaboutthetrendandcyclecomponents ofinflation. Forsimplicity,weassumethattheselong-runbeliefsarezeroforallforecasters. This assumptioncanbeeasilyrelaxedtoallowforindividual-specificnon-zeroconstants. Although thetransitorycomponentismodeledasamean-zerostationaryprocess,empirically,forecaster i’srealizedmeanpredictionofthetransitorycomponentcanbenon-zero. Infact, whilethe medianoftheestimatedα isclosetozero,thedistributionofα isnon-degenerate.17 Furtheri,s i,s more,wecharacterizethetrendcomponentasanon-stationaryprocesswithoutawell-defined first moment. Nevertheless, we can still estimate the empirical average of each forecaster’s predictionforthetrend.18 Theseempiricalestimatesofindividualforecasters’meantrendand transitorycomponentsreflecttheirtime-invariantlong-runbeliefs. 19 Definethegainsthatforecasteri placesonerrorsmadeinforecastingthepublicandprivate signalsofthetrendcomponentasg τ andg τ respectively. Wedefinegc andgc analogously i,y i,z i,y i,z forthecyclicalcomponent. Ifallagentsarerational,thegainsplacedonforecastsofthepublic signalswillbethesameforallagents. Similarly, ifthevarianceofprivatesignalsisthesame acrossallagents,thenthegainsplacedonforecastsofprivatesignalswillbethesameacross allagents. Inourmodel, agentscanhavedifferentforecastseitherbecausetheyarerational buthaveheterogeneousprivatesignalprecisionsorbecausetheyhavedifferentgainsdueto behavioralreasonsorcognitivelimitations. LetF betheexpectationofagenti formedwithtimet information. Insteady-state,agenti i,t 17SeetheupperrightpanelofFigure6. 18SeetheupperleftpanelofFigure6. 19Theselong-runbeliefscanbethoughtofascomingfromsometypeofbehavioralbiasorcognitivelimitation,or ascapturingdeviationsfromthetruthwhichresultfromlearningaboutthelongrunstartingfromaninformative butbiasedpriorbelief(Farmeretal.,2021). 24

updatestheirbeliefsaccordingtothefollowingequations (cid:179) (cid:180) F i,t τ t =F i,t−1 τ t +g i τ ,y (cid:161) y t τ−F i,t−1 τ t (cid:162)+g i τ ,z z i τ ,t −F i,t−1 τ t (cid:179) (cid:180) F i,t c t =F i,t−1 c t +g i c ,y (cid:161) y t c−F i,t−1 c t (cid:162)+g i c ,z z i c ,t −F i,t−1 c t . Aftersomesimplificationtheaboveequationscanberewrittenas F i,t τ t =(cid:161) 1−g i τ(cid:162) F i,t−1 τ t−1 +g i ττ t +g i τ ,y u t τ+g i τ ,z v i τ ,t F i,t c t =(cid:161) 1−g i c(cid:162)ρ c F i,t−1 c t−1 +g i cc t +g i c ,y u t c+g i c ,z v i c ,t , whereg τ :=g τ +g τ andgc :=gc +gc . i i,y i,z i i,y i,z Wearenowreadytostateourmainproposition: Proposition1. Giventhedynamicfactormodelcharacterizedbyequations(3)–(8),thereisan exactequivalencewiththenoisyinformationmodelfortheparameters ρ =ρ ρ =1 S c L ρS =(cid:161) 1−gc(cid:162)ρ ρL =1−g τ i,ε i c i,ε i αS =0 αL=0 i i βS =   (cid:161) g i c(cid:162)2 (σc ε )2+ (cid:179) g i c ,y (cid:180)2 (σc u )2(cid:161) 1−ρ2 c (cid:162)   1/2 βL= (cid:183) (cid:161) g τ(cid:162)2 (στ )2+ (cid:179) g τ (cid:180)2 (στ )2 (cid:184)1/2 i  1+(cid:161)ρ g¯c (cid:162)2−ρ2  i i ε i,y u c c σS = (cid:34) (cid:179) gc σc (cid:180)2 − (cid:161)βS i (cid:162)2(cid:163)(cid:161) g i c−g¯c(cid:162)ρ c (cid:164)2(cid:35)1/2 σL =g τ στ i,v i,z i,v 1−ρ2 i,v i,z i,v c whereg¯c isthecross-sectionalpopulationaverageofgc. i SeeAppendixCfortheproof. Ourindividual-leveldynamicfactormodelalignswiththeeconomicinterpretationsofthenoisyinformationmodel. Specifically,thecommoncomponentsL t andS capturetheinfluenceofpublicinformation,whiletheidiosyncraticcomponentsϵL and t i,t ϵS capturetheinfluenceofprivateinformation. Notethatreactionstopublicinformation—βS i,t i andβL—differacrossforecasters. Finally,wesetthelong-runmeanparameters—αS andαL—to i i i zeroforsimplicity,toclearlydemonstratethemappingbetweenourstatisticalmodelandthe standardnoisyinformationmodel. Asmentionedearlier,theseparameterscantakenon-zero 25

valueswithoutcompromisingthestructuralcharacterizationorthemappingtothestatistical model. Inotherwords,ifweassumethatagentsbelievethepermanentcomponenthasadrift andthecyclicalcomponenthasanon-zeromean,thesebeliefswillbecapturedbytheempirical fixedeffectsαL andαS,respectively.20 Thiswillintroduceanadditionalsourceofdisagreement, i i whichwecharacterizeasthe‘long-termbeliefs’componentofdisagreement. 6 Empirical Decomposition of Disagreement Inthissection,weshowhowourstatisticalmodeldecomposesindividualinflationforecastsat eachpointintimeintothreedistinctcomponents: (1)individuallong-termbeliefs(orindividual fixedeffects);(2)heterogeneousresponsestopublicinformation(thecommoncomponent);and (3)privateinformation(theidiosyncraticcomponent). Section6.1illustratesthedecomposition ofindividualforecasts. Section6.2providesanewmeasureforthesensitivityofdisagreement toeachsourceofinformation. Section6.3discussestheresultsofthedecomposition. Section 6.4providesasimulationexercisebasedonthenoisyinformationmodeltofurtherjustifyour economicinterpretationsofthestatisticalresults. Thesimulationconfirmsthattheempirical findings align with the theoretical predictions, even in the absence of individual long-term beliefs. 6.1 DecomposingDisagreement Weproceedintwosteps. First,wedecomposeforecasteri’sinflationforecastateachforecasting horizonintothethreecomponentsoutlinedabove. Second,wedecomposethecross-sectional varianceintocontributionsfromthethreesourcesateachpointintime. Equations(3)and(4)decomposeindividuallevelandslopefactorsintothethreeinformation sources. Here,wegiveeconomicinterpretationstoeachcomponentusingthelevelfactorasan example. L = αL + βLL + εL (11) i,t i i t i,t (cid:124)(cid:123)(cid:122)(cid:125) (cid:124)(cid:123)(cid:122)(cid:125) (cid:124)(cid:123)(cid:122)(cid:125) long-termbelief publicinfo. privateinfo. The level factor of forecaster i, L , is decomposed into portions representing long-term i,t 20Inthiscase,thetime-varyingelementsinthenoisyinformationmodelwillbedemeanedvalues,analogousto thoseinthedynamicfactormodel. 26

beliefs(αL),publicinformation(βLL ),andprivateinformation(εL ).21 NotethatαL isestimated i i t i,t i withindividualfixedeffects,andβLL andεL arethecommoncomponentandtheidiosyncratic i t i,t component,respectively,inthedynamicfactormodelforthelevelfactor. Definethelong-termbeliefcomponentofL tobeLltb=αL,thecommoncomponentof i,t i i L tobeL pub=βLL ,andtheidiosyncraticcomponentofL tobeL priv =εL . Wethenrewrite i,t i,t i t i,t i,t i,t L as i,t L =Lltb+L pub+L priv . (12) i,t i i,t i,t Likewise,wegivesimilareconomicinterpretationstoeachcomponentofS inEquation(4).22 i,t Thus,S isrewrittenas: i,t S =Sltb+S pub+S priv . (13) i,t i i,t i,t UsingthedecompositionsinEquations(12)and(13),forecasteri’sh-quarter-aheadinflation forecastattime t, π i,t→t+h|t , isexpressedasthreecomponentsrepresentinglong-runbeliefs (πltb ),publicinformation(πpub ),andprivateinformation(πpriv ): i,t→t+h|t i,t→t+h|t i,t→t+h|t (cid:181) 1−e −λh(cid:182) π i,t→t+h|t =L i,t − λh S i,t (cid:181) 1−e −λh(cid:182) (cid:179) (cid:180) =αL+βLL +εL − αS+βSS +εS i i t i,t λh i i t i,t (cid:181) 1−e −λh(cid:182) (cid:181) 1−e −λh(cid:182) (cid:181) 1−e −λh(cid:182) =αL− αS+βLL − βSS +εL − εS i λh i i t λh i t i,t λh i,t (cid:181) 1−e −λh(cid:182) (cid:181) 1−e −λh(cid:182) (cid:181) 1−e −λh(cid:182) =Lltb− Sltb+L pub− S pub+L priv− S priv i λh i i,t λh i,t i,t λh i,t (cid:124) (cid:123)(cid:122) (cid:125) (cid:124) (cid:123)(cid:122) (cid:125) (cid:124) (cid:123)(cid:122) (cid:125) long-termbelief publicinformation privateinformation =πltb +πpub +πpriv i,t→t+h|t i,t→t+h|t i,t→t+h|t Thefactorstructureleadsustodecomposethecross-sectionaldispersionofπ i,t→t+h|t into thedispersioncomponentsdrivenbythethreeinformationsources. Var i (π i,t→t+h|t )≈Var i (πl i, t t b →t+h|t )+Var i (π i p , u t→ b t+h|t )+Var i (π i p , r t→ iv t+h|t ) (14) 21Weusepublic(private)informationandcommon(idiosyncratic)informationinterchangeably,butwilluse public(private)informationinthissectiongiventheeconomicinterpretationfromthenoisyinformationmodel. 22Analogoustothelevelfactor,αSisestimatedwiththeindividualfixedeffects,andβSS andεS arethecommon i i t i,t componentandtheidiosyncraticcomponent,respectively,inthedynamicfactormodelfortheslopefactor. 27

Note that the variance of individual long-term beliefs (the first term in (14)) will not create changes in the dispersion, if the pool of forecasters does not change over time. Since each forecasterhasdifferentaverageforecasts, thecompositionalchangeinforecastersovertime createsvariationinlong-termbeliefdispersion. Inthissense,dispersionduetolong-termbeliefs isinterpretedasthenaturallevelofdisagreementorfundamentaldisagreement(Andradeetal., 2016). Thisvariancedecompositionispossiblebecauseinnovationstothecommonandidiosyncraticcomponentsareassumedtobeindependentofeachotherinthedynamicfactormodel. However,realizedshockstothecommonandidiosyncraticcomponentscanexhibitfinitesample comovement. Thiscomovementcreatesawedgebetweenthetwosidesof(14). Nonetheless,the finite-samplecomovementwillbeclosetozeroinlargesamples. 6.2 DisagreementShares Thissectionprovidesanalternativedecompositionofdisagreementaboutinflationh periods aheadintothecomponentsofthreeinformationsources—whatwecallthedisagreementshares. Ourgoalistoassesstheextenttowhicheachinformationsourceinfluencestheoverallcrosssectional variance in forecasts of inflation h periods ahead at each point in time. This new measureavoidsthecaveatof(14)thatobscuresthecontributionofeachinformationsourcein thedisagreement. Conceptually,ourdisagreementsharesresembletheideaofbetainfinance,whichgaugesthe sensitivityofstockstoacommonfactor.Asanalternativetosimplylookingatthecross-sectional varianceofeachcomponent,weconstructtheinformationshareβfordisagreementusingan approachproposedinFujitaandRamey(2009).23 Thisnewmeasureovercomestheissuethat finite-samplecomovementsamongcomponentscausetheindividualvariancesharesnotadd up to one, which is a well-known issue in the literature on dynamic factor models (e.g., Ahn andLuciani,2024). FujitaandRamey(2009)showwhenavariableisexpressedasthesumof differentsub-components,theratioofthecovariancebetweenthesumandeachcomponent dividedbythevarianceofthesum—thebeta—adduptoone. Notethattheh-periodahead inflationforecastofindividuali attimet iscomposedoftheportionaccountedforbylong-term 23FujitaandRamey(2009)provideadecompositionforthecontributionofinflowstounemploymentandthat ofoutflowsfromunemploymenttothevarianceofunemploymentrate, asimilarproblemintheliteratureof unemploymentdynamics. 28

beliefs,publicinformation,andprivateinformation. Therefore,wecanapplyFujitaandRamey’s decompositiontoourcase. Restatingourdecompositionofinflationprojectionswehavethat: π i,t→t+h|t =πl i, t t b →t+h|t +π i p , u t→ b t+h|t +π i p , r t→ iv t+h|t . (15) Taking the covariance of both sides with π i,t→t+h|t and dividing through by the variance of π i,t→t+h|t ,weobtainthefollowingexpression: 1=βltb+βpub+βpriv , (16) h,t h,t h,t where (cid:179) (cid:180) βltb = Cov i π i,t→t+h|t ,πl i, t t b →t+h|t h,t Var i (π i,t→t+h|t ) (cid:179) (cid:180) βpub = Cov i π i,t→t+h|t ,π i p , u t→ b t+h|t h,t Var i (π i,t→t+h|t ) (cid:179) (cid:180) βpriv = Cov i π i,t→t+h|t ,π i p , r t→ iv t+h|t . h,t Var i (π i,t→t+h|t ) Notethatβltb,βpub ,andβpriv cantechnicallybenegativeorgoabove1. h,t h,t h,t 6.3 DecompositionResults Figure10presentstheresultsofestimatingEquation(16)foreachtimeperiodandforecasting horizon. For clarity, we will focus on discussing the results for the 1-year and 10-year ahead forecast horizons.24 The upper panels show the level of disagreement about 1-year and 10yearaheadinflation: total(blackline),theportionattributabletolong-termbeliefs(blueline), the portion attributable to public information (orange line), and the portion attributable to privateinformation(yellowline). Thebottompanelsdisplaythedisagreementsharesofpublic informationfor1-yearand10-yearaheadinflationprojections. Wemakeafewkeyobservations. First,privateinformationistheprimarysourceofshort-run (1-yearahead)disagreement,explainingapproximately60percentofthedispersion. Second, 24ResultsforadditionalforecastinghorizonscanbefoundinSectionAppendixDoftheappendix. 29

individuallong-termbeliefsarethemainsourceofdisagreementforlong-runinflationforecasts, accountingforabout55percentofthedispersion. Finally,publicinformationcontributesthe smallestshareofdisagreementacrossallforecastinghorizons,makinguparound10percentfor bothshort-runandlong-runforecasts. However,publicinformationplaysasignificantlylarger roleinexplainingdisagreementduringthreemajorperiods: theearlyandmid-1990s,theGreat Recession,andtheCOVID-19pandemic. Theseperiodscorrespondtoeconomicrecessionsor episodesofheightenedinflationuncertainty. Figure10: FORECAST VARIANCE DECOMPOSITION Notes:Thetoptwopanelsshowthedecompositionofthecross-sectionalvarianceofinflationforecasts(blackline) intothecomponentsdrivenbyindividuallong-termbeliefs(denotedbyltb,blueline),heterogeneousresponsesto publicinformation(denotedbypub,redline),andprivateinformation(denotedbypriv,yellowline).Eachline correspondstotheposteriormedian.Thebottompanelsshowthevarianceshareofpublicinformation,βpub .The h,t solidbluelinecorrespondstotheposteriormedianandthedottedblacklinescorrespondtopointwise95%credible intervals.Theleftandrightcolumnscorrespondto1-and10-yearforecastinghorizonsrespectively.Theshaded areasdenoteNBERrecessions.Thebottompanelsshowthevarianceshareofpublicinformation. Sources:Authors’calculation Tobetterunderstandthistime-varyingroleofpublicinformationasadriverofdisagreement, weexamineβpub inthebottompanelsofFigure10moreclosely.25 Thecontributionofpublic h,t informationtodisagreementchangesdramaticallyovertime. Typicallyitfluctuatesatvalues under10percentforbothshort-andlong-horizonforecasts. However,theshareexhibitsstrong countercyclicality,increasingintimesoflargeeconomicshocksacrossforecastinghorizons. For 25SectionAppendixDoftheappendixprovidesthedisagreementsharesofpublicinformationforadditional forecastinghorizons. 30

Figure11: TERM STRUCTURE OF VARIANCE SHARES AT THREE DATES 2007 Q3 0.1 0.05 0 0 5 10 15 20 25 30 35 40 2009 Q2 0.6 0.4 0.2 0 5 10 15 20 25 30 35 40 2022 Q3 0.6 0.4 0.2 0 5 10 15 20 25 30 35 40 Notes:Thefigureshowsthefractionofoveralldisagreementaboutinflationexpectationsdrivenbyheterogeneous responsestocommoninformationoveraten-yearforecasthorizonatthreedifferentpointsintimeasmeasuredby theβmeasureproposedinFujitaandRamey(2009).Thetoppanelreportstheestimatesasof2007:Q3,themiddle panelasof2009:Q2,andthebottompanelasof2022:Q3.TheshadedareasdenoteNBERrecessions. Sources:Authors’calculation example,itspikestoapproximately65percentforshort-horizonforecastsand90percentfor long-horizonforecastsduringthepandemic.Thisobservationsuggeststhatforecasterspaymore attention to public information during economic downturns and periods of higher inflation uncertainty,buttranslatethepublicinformationintotheirforecastsindifferentways. Furthermore, the term structure of common disagreement varies dramatically over time as well. Figure 11 shows the share of public information in disagreement across forecasting horizonsforthreeparticulartimeperiods. In2007Q3,justbeforetheGreatRecession,therole of common information in disagreement is low, near 2%, across all forecasting horizons. In contrast,attheheightoftheGreatRecessionin2009Q2,publicinformationplaysamuchlarger role. Theshareislargerforlong-runforecaststhanforshort-runforecasts(middlepanel),about 50%comparedto20%. TheCOVID-19pandemicisuniqueinthatpublicinformationwasthe primarydriverofincreaseddisagreementacrossallforecastinghorizons(bottompanel). The shareisgreaterthan30percentacrossallforecastinghorizons,decliningfromabout60%inthe shortruntojustbelow35%inthelongrun. Inthepresenceofshocksthatareunprecedentedin natureandmagnitudeandtheconsequentpolicyresponse,forecasterspaidattentiontopublic 31

informationduringthepandemic. However,duetohigheconomicuncertainty,forecastershad verydifferentinterpretationsaboutthesamepublicinformationandproducedquitedifferent inflationexpectationsacrossforecastinghorizons. Theimportanceofpublicinformationinlong-rundisagreementintimesofhigheconomic uncertaintysuggeststhatmonetarypolicymaybeabletoanchorlong-horizonexpectations effectivelywithclearercommunication. 6.4 RelationtoNoisyInformationModel Next, we conduct a simulation exercise based on the noisy information model described in Section5,andshowthatourempiricaldisagreementdecompositionresultsareconsistentwith ourtheoreticalmodel’spredictions. Ourmodelallowsforecasterstorespondheterogeneouslytothesamepublicnews. These heterogeneousresponsesdrivethesignificantincreaseindisagreementattributabletopublic information during periods of large shocks, even though we do not account for individuallevel uncertainty. Moreover, by allowing for varying sensitivity to public information across differentforecastinghorizonsforeachindividual,ourmodelcapturesthedistinctrolesofpublic informationindrivingdisagreementintheshortrunversusthelongrun. Incontrast,previous literatureonthenoisyinformationmodelhastypicallyassumeduniformreactionstopublic informationacrossindividualsandhasnotaccountedfordifferingreactionsacrossforecasting horizons.26 Usingsimulateddata,wedemonstratehowthenoisyinformationmodelpresentedinSection 5canaccountforthestylizedfactsontime-varyingdisagreementandtime-varyingsharesof disagreement. We assume that gc =g and σc =σ for all agents i, and we assume that the i i,v v trendcomponentτ isknowntobeconstantovertimeandequalto0. Thus,alltime-variation t in inflation comes from the cyclical component c . These simplifications are for illustrative t purposes only, to help isolate the time variation in the relative importance public vs private information. Byallowingforanon-zerotrendcomponent,wecangeneratevariationsinthe sourceofdisagreementacrossforecastinghorizonsovertime. Similarly,weabstractfromlongtermbeliefs,becausetheydonotcontributetotime-variationindisagreementunlesswealso 26Afewpreviousstudieshaveconsideredanoisyinformationmodelwhereagentsupdatetheirinformationat differentfrequencies,resultinginlimiteddispersionofforecastsinresponsetopublicinformation. Adetailed literaturereviewonthistopicisprovidedinSectionAppendixFoftheappendix. 32

modelforecasterturnover,asisobservedintheSPF. Giventheabovesimplifications,beliefsofagenti aboutinflationfollowthestochasticprocess F i,t π t =(cid:161) 1−g (cid:162)ρ c F i,t−1 π t−1 +gπ t +g i c ,y u t c+g i c ,z v i c ,t Letgc denotethecross-sectionalpopulationaverageofgc . Furthermore,letσ2 andσ2 y i,y g,y g,z denotethecross-sectionalpopulationvariancesofgc andgc respectively. Thecross-sectional i,y i,z meanofbeliefsfollows F¯ i,t π t =(cid:161) 1−g (cid:162)ρ c F¯ i,t−1 π t−1 +gπ t +g¯ y cu t c (17) andthecross-sectionalvariancefollows Var i (cid:161) F i,t π t (cid:162)=(cid:163)(cid:161) 1−g (cid:162)ρ c (cid:164)2 Var i (cid:161) F i,t−1 π t−1 (cid:162)+σ2 g,y (cid:161) u t c(cid:162)2+σ2 g,z σ2 v (18) Equation (18) implies that time-variation in disagreement is driven entirely by public noise shocks uc. The magnitude of the impact on cross-sectional disagreement is driven by the t magnitudeoftheshockandthevarianceofthegains gc ,σ2 ,acrossagents. Largershocks i,y g,y (in magnitude) and a larger dispersion of gains on the public signal lead to larger and more persistentswingsindisagreement. Toillustratetheconnectionbetweenthenoisyinformationmodelandourstatisticalmodel, weconsiderasimulationexperiment. Weparameterizetheprocessforbeliefsusingρ =0.95, c σc =0.83, g =0.4, σ =2.1, σ =2.1, g¯c =0.2, and σ =0.1. The volatility of shocks to the ε u v y g,y cyclicalcomponent,σc,ischosensothattheunconditionalvarianceofc isthesameasthe ε t unconditionalvarianceofinflationoveroursampleperiod.Thevarianceofthepublicandprivate noiseshocksarebothchosentobeequaltotwicethevarianceofthecyclicalcomponent.27 The overallgainandaveragepublicsignalgaing andg¯c aresettobethegainsthatwouldbechosen y byarationalagentfacingthespecifiedsignalextractionproblem. Thegc arerandomlydrawn i,y fromaNormaldistributionwithmeang¯c andstandarddeviationσ . Thegc aresetequalto y g,y i,z g−gc foreachi. i,y Wesimulateabalancedpaneloffortyforecastersforthree-hundredtimeperiods. Wethen construct common and idiosyncratic components by decomposing the simulated nowcasts usingourstatisticaldynamicfactormodelinSection3assumingnomeasurementerror. The 27This implies a signal to noise ratio of one third for both the public and private signals. This is chosen to approximatelymatchtheobservedcross-sectionaldispersionininflationnowcastsfromtheSPF. 33

Figure12: NOISY INFORMATION MODEL SIMULATIONS Variance of Beliefs Total pub priv 1 0.5 0 50 100 150 200 250 300 Variance Share of Public Information 1 0.5 0 50 100 150 200 250 300 Notes:Thefigureshowsthedynamicsofdisagreementanddisagreementsharesinasimulationexperimentfrom ournoisyinformationmodel.Thetoppanelplotstheoverallcross-sectionalvarianceofbeliefsaboutinflationin black,thevarianceattributabletopublicinformationinred,andthevarianceattributabletoprivateinformationin yellow.Thebottompanelplotstheshareofthetotalvarianceattributabletopublicinformation. Sources:Authors’calculation disagreementandvariancesharemeasuresareconstructedexactlyasinSection6. Figure12presentstheresultsfordisagreement. Thetoppanelplotsthetotalvarianceof beliefsinblackalongwiththeportionsofvariancedrivenbypublicandprivateinformationin redandyellowrespectively. Thebottompanelplotstheshareofthevariancedrivenbypublic information. These panels broadly match the patterns observed in Figure 7. Disagreement exhibitsconditionalheteroskedasticitycorrespondingtoperiodsoflargepublicnoiseshocks. In periodsoflargepublicnoiseshocks,theshareofdisagreementexplainedbypublicinformation rises. There are also spikes in disagreement which arise from a large realized dispersion of privatesignals,andintheseperiodsthereisnocorrespondingriseinthevarianceshareofpublic information,aspredictedbyourtheory. Thestructuralmodeleffectivelycapturesourkeyempiricalfinding—theincreasedimportanceofpublicinformationindrivingdisagreementduringperiodsoflargeinflationaryshocks.It isimportanttonotethatthisresultwasachievedevenwithoutallowingforchangesinindividualleveluncertainty. Thisobservationfurthervalidatestheeconomicinterpretationsofthethree elementsofthedynamicfactormodelaslong-termbeliefs,heterogeneousreactionstopublic information,andprivateinformation. 34

7 Implications for Monetary Policy Thissectionexploresthelinkbetweenthesourceofdisagreementandtheeffectivenessofmonetarypolicy. Section7.1examineswhetherthenewscomponentofmonetarypolicysurprises reduces the disagreement attributable to public information and discusses implications for anchoringinflationexpectations.Section7.2investigatestheeffectofdisagreementaboutpublic informationonthetransmissionofmonetarypolicyshocks. 7.1 EffectoftheFed’sResponsetoNewsonDisagreement Sofar,wehaveimplicitlyassumedthatdisagreementaboutpublicinformationcanbemitigated throughmonetarypolicycommunication. Wenowexaminewhetherthisisthecaseempirically. Forthisanalysis,weconsideralocalprojection(LP)withanexternallyidentifiedshock. Weuse theFed’sresponsetoeconomicnewsfromBauerandSwanson(2022),astheexternallyidentified shock. Thisnewscomponentofmonetarypolicysurprises,reflectingtheFed’sinterpretation ofrecentdatareleases,ismeasuredasthedifferencebetweenhigh-frequencymonetarypolicy surprisesandtheorthogonalizedmonetarypolicyshock. IfforecasterspayattentiontotheFed’s reactionstodatareleases,thisnewscomponentshouldreducethedisagreementaboutpublic informationamongforecasters. p Let y denote the disagreement about 8-quarter-ahead inflation attributable to public t+h information. Similarly, let yo denote the disagreement about 8-quarter-ahead inflation att+h tributabletonon-publicinformation,whichincludesbothprivateinformationandlong-term beliefs. Wemeasuredisagreementusingthestandarddeviationoftheportionofindividual-level forecastsattributabletoeachinformationsource. Wefocusonan8-quarterhorizontoaccount forpolicylags. Thelocalprojectionmodelisspecifiedasfollows: y t j +h =α h j +β h j z t +Γ h j X t−1 +e t j +h for j ∈[p,o] h=0,1,···,H, (19) whereαj isaconstant, z isthenews-componentshock,βj capturesthemagnitudeofpassh t h j troughoftheshockh quartersafterimpact,ande t+h isanerrorterm. ThenotationX t−1 denotes asetofmacroeconomiccontrols,alllaggedbyoneperiod. Thecontrolsincludefourlagsofthe two-yearTreasuryyield,thefirstdifferencedlogofindustrialproduction(IP),thefirstdifferenced 35

logoftheconsumerpriceindex(CPI),theunemploymentrate,theexcessbondpremiumfrom GilchristandZakrajšek(2012),thelevelofdisagreementattributabletopublicinformationand thatattributabletonon-publicinformation.Wealsoincludetwolagsofz inthecontrols.28 Note t thattheparameterdrawsfromtheGibbssamplerareusedtoconstruct y p and yo . Foreach t+h t+h draw,wecomputethecorrespondingimpulseresponse. Wethenobtainposteriordistributions oftheimpulseresponses. Thesampleperiodis1991:Q4-2019:Q4.29 Figure13displaystheresponsesoftwodisagreementmeasurestothenewscomponentofa monetarypolicyshock. Wereporttheeffectsofaone-standard-deviationinnovationinthenews component. AsshowninPanelA,theFed’spositivereactiontoeconomicnewsimmediately andsignificantlyreducesdisagreementabout8-quarter-aheadinflationattributabletopublic information. However,thesameshockhasasmallerandstatisticallyinsignificanteffectonthe correspondingdisagreementattributabletonon-publicinformation. Thisresultconfirmsthat disagreementaboutpublicinformationistheportionthatcanbereducedbymonetarypolicy communicationandisthereforerelevantforanchoringinflationexpectations. WefurtherinvestigatewhethertheeffectsoftheFed’sreactionstonewschangewhenforecastershaverecentlyexperiencedsubstantialdisagreementaboutpublicinformation.Toexplore thispossibility,weconsidertworegimes: inRegime1,non-publicinformationisthesourceof disagreement,whileinRegime2, publicinformationisthesource. Theseregimesaredistinguishedbythedisagreementshareofpublicinformation,asintroducedinSection6,withafocus ondisagreementregarding8-quarter-aheadinflation. Considerthefollowingnonlinearlocal projectionmodel. y t j +h =α h j +β 1 j ,h (1−s t−1 )z t +β 2 j ,h s t−1 z t +Γ h j X t−1 +e t j +h for j ∈[p,o], h=0,1,···,H (20) In this model, s t−1 is the indicator of Regime 2 and is measured using βc 8,t−1 from Equation (16). TheindicatorofRegime1isthus(1−s t−1 ),measuredwith(1−βc 8,t−1 ). Theparameterβ 1 j ,h capturesthemagnitudeofpass-throughwhendisagreementisdrivenbynon-publicinformation (Regime1),whileβj reflectsthemagnitudewhenpublicinformationisthesourceofdisagree- 2,h ment(Regime2). Weuseaone-quarterlagofs toavoidcontemporaneousfeedbackfrompolicy t actions(AuerbachandGorodnichenko,2013). TheparameterdrawsfromtheGibbssamplerare 28WeusethecontrolsofmacroeconomicvariablesasconsideredbyBauerandSwanson(2022). 29ThesampleperiodofBauerandSwanson’sorthogonalizedshockendsinFebruary2020.Forthisreason,weend thesampleperiodofthislocalprojectionanalysisin2019:Q4. 36

Figure13: PROPAGATION OF FED’S REACTIONS TO NEWS PanelA.AverageResponses Public Private 0.04 0.04 0.02 0.02 0 0 -0.02 -0.02 -0.04 -0.04 -0.06 -0.06 -0.08 -0.08 -0.1 -0.1 0 2 4 6 8 0 2 4 6 8 PanelB.NonlinearResponses Notes:Thefigureshowstheresponsesofdisagreementabout8-quarter-aheadinflationexpectationsattributableto publicinformationandnon-publicinformationfollowingtheFed’sresponsetonewsfromBauerandSwanson (2023).HACstandarderrorsarereported.PanelApresentstheresultsfromthelinearmodelinEquation.(19),while PanelBshowstheresultsfromthenonlinearmodelinEquation(20).InPanelB,thebluelinesrepresentthe responsesinRegime1,whilethemagentalinesrepresentthoseinRegime2.Non-publicinformationistheprimary sourceofdisagreementinRegime1,whereaspublicinformationisthemainsourceinRegime2.Theaverage disagreementsharesofnon-publicandpublicinformationare0.9and0.1,respectively.InPanelB,theestimated responsesforRegimes1and2arescaledby0.9and0.1,respectively,sothattheirsumcloselyapproximatesthe averageimpulseresponsesinPanelA.Thedashedlinesindicatethe95%posteriorintervals. Source:Authors’calculation usedtoestimateβc 8,t−1 andhences t−1 ,whicharethenusedtocomputeimpulseresponses. PanelBofFigure13showstheimpulseresponsesforthetworegimes. Wereporttheeffects ofaone-standard-deviationinnovationinthenewscomponent. Theresponsesofregimes1 and2arescaledby0.9and0.1,correspondingtotheaveragedisagreementsharesofnon-public informationandpublicinformation,respectively. Thisapproachensuresthatthesumofthetwo impulseresponsescloselyapproximatestheaverageresponsesreportedinPanelA.Notabledif- 37

ferencesemergeacrossthetworegimes. Whenpublicinformationisthesourceofdisagreement (magentalines),thenewscomponentdoesnotstatisticallysignificantlyreducedisagreement. In contrast,whennon-publicinformationisthesource,thereductionofdisagreementisstatistically significantonimpact. Thisfindingsuggeststhatincreaseddisagreementaboutpublicinformationmayreflectsituationswhereforecastersdisagreeaboutmonetarypolicycommunication, giventhattheFed’scommunicationisalsopublicinformation. Thus,heighteneddisagreement attributabletopublicinformationmaysuggestaweakenedeffectivenessofmonetarypolicy communicationinanchoringinflationexpectations. This result has a few important implications. First, our empirical model effectively identifies the sources of information contributing to disagreement, highlighting that the portion attributabletopublicinformationistheamountthatisreduciblebymonetarypolicycommunicationandisrelevantinanchoringinflationexpectations. Second,theFed’sinterpretation of economic news plays a crucial role in shaping public information essential for inflation forecasting. In this sense, clear communication of the monetary authority can help anchor economicagents’expectationsbyreducingtheirdisagreementaboutfuturemacroeconomic conditions. Finally, theextentofdisagreementdrivenbypublicinformationcanserveasan auxiliaryindicatorofhowwell-anchoredinflationexpectationsare. InappendixE.1,weprovideextensiverobustnesschecksfortheanalysispresentedinthis section. First,wefurtherconsiderdisagreementabout10-year-aheadinflationinsteadofabout 8-quarter ahead inflation projections. Second, we consider expanding the macroeconomic controlstoincludeameasureofindividualuncertaintyfromBinder(2017)andtheconsensus inflation expectations about next year.30 Third, we consider not including lags for the news componentofthemonetarypolicyshock. Fourth,wealsoconsideralternativemethodologies suchasthetwo-stagelocalprojectionwithexternalinstrumentalvariable(LP-IV).Theresults arerobust. 7.2 DisagreementandMonetaryPolicyEffectiveness Thissectionexplorestherelationshipbetweentheroleofpublicinformationindisagreement andthestabilizingeffectsofmonetarypolicyshocks. Specifically,weassesstheextenttowhich thedisagreementshareofpublicinformationaffectstheeffectivenessofmonetarypolicy. 30Forthisrobustnesscheck,wealsoincludeanexternalconsensusmeasure,ratherthanusingthemodelestimates, toensurethatourresultsarerobusttotheinclusionofexternaldata. 38

Figure14:PROPAGATIONOFMONETARYPOLICYSHOCKSOFTHETWOREGIMES(8-QUARTERAHEAD) Notes:Thefigurereportstheresponsesoffourmacroeconomicvariablestotheorthogonalizedmonetarypolicy shockfromBauerandSwanson(2023).ThefigureshowstheimpulseresponsesofRegimes1and2scaledby0.9 and0.1,respectively.Thebluelinesshowtheresponseswhennon-publicinformationisthesourceofdisagreement (Regime1),whilethemagentalinesrepresenttheresponseswhenpublicinformationisthesourceofdisagreement (Regime2).Theupperleftfigureshowsthecumulativeresponseofpercentchangesinindustrialproduction;the upperrightfigureshowsthecumulativeresponsesofpercentchangesintheCPI;thebottomleftfigureshowsthe responsesoftheunemploymentrate;andthebottomrightfiguredisplaystheresponseoftheexcessbondpremium (EBP).Thedashedlinesrepresentthe90%posteriorintervals. Sources:Authors’calculation Forthisanalysis,weutilizeanonlinearlocalprojectionmodelsimilartotheoneinequation (20), with some modifications. First, the dependent variables are macroeconomic variables, namelythegrowthrateofindustrialproduction,CPIinflation,theunemploymentrate,andthe excessbondpremium. Second,weusetheorthogonalizedmonetarypolicyshockfromBauer andSwanson(2022)astheexternallyidentifiedshock. Finally, tomaintainconsistencywith previousstudies,ourlocalprojectionmodelisspecifiedatamonthlyfrequency. Weconsider12 lagsforthemacroeconomiccontrolsandtwolagsforthemonetarypolicyshocks.31 31Asarobustnesscheck,weincludetheuncertaintymeasurefromBinder(2017)andtheconsensusinflation expectationsfromtheSPF(long-runandnextyear)asmacrocontrols.Fortheconsensusexpectations,weusethe dataastheyareinsteadofusingthemodel-impliedconsensustoavoidpotentialissuesspecifictoourmodel.The estimationresultsarerobusttotheinclusionofadditionalmacrocontrols.Allrobustnesschecksarereportedinthe appendix. 39

Thenonlinearmodelhastworegimeswhicharethesameasthoseinequation(20). NonpublicinformationisthesourceofthedisagreementinRegime1,andpublicinformationin Regime2. Eachregimeisalsodistinguishedbythedisagreementshareofpublicinformation8 quartersahead. Again,toavoidcontemporaneousfeedbackfrompolicyactions,weconsider aone-periodlagfortheregimeindicator. Sincethedisagreementshareisquarterly,weassign thesamequarterlyvaluetothethreemonthsofthecorrespondingquarterandusetheprevious quarter’sshareastheregimeindicatorformontht.32 ThesampleperiodisfromOctober1991to December2019. Figure 14 presents the impulse responses for the two regimes. We report the effects of a one-standard-deviationinnovationintheorthogonalizedmonetarypolicyshock. InPanelA,the responsesofregimes1and2arescaledby0.9and0.1,respectively,againreflectingtheaverage disagreementsharesofnon-publicinformationandpublicinformation.33 Fortheresponses ofIPgrowthandCPIinflation,wereportthecumulativeeffectsofthemonetarypolicyshock, providingestimatesthatdirectlyreflectchangesinlevels(StockandWatson,2018). Significant differencesintheeffectsofmonetarypolicyareobservedbetweenthetworegimes. Inregime 1 (blue lines), where non-public information is the source of disagreement, contractionary monetarypolicyhasrapidandstatisticallysignificanteffectsonmacroeconomicvariables. In contrast,inregime2(magentalines),wherepublicinformationisthesourceofdisagreement, theoverallmonetarypolicyeffectsbecomeweakerandmacroeconomicvariablesshowdelayed responses. Notably,apricepuzzleemergesinthesecondregime. Ourempiricalfindingshaveimportantimplicationsfortheeffectivenessofmonetarypolicy. First, the sensitivity of disagreement to public information—captured by the disagreement share of public information—is an important determinant of monetary policy effectiveness. Whenforecastersdisagreeaboutpublicinformation,includingthestanceofmonetarypolicy, thestabilizingeffectsofmonetarypolicyweakensignificantly.34 Second,ourempiricalresult also offers a new perspective on the source of the price puzzle. Our result suggests that the 32Asarobustnesscheck,wealsoconsideraone-monthlagfors . SincetheSPFisconductedduringthefirst t weekinthesecondmonthofeachquarter,thesubmittedforecastslargelyreflecttheinformationsetthroughthe previousquarter.Forthisreason,theone-monthlagfors islesslikelytocreateafeedbackeffectinthecurrent t period.Theestimationresultsremainrobustregardless. 33Thisapproachensuresthatthesumofthetwoimpulseresponsescloselyapproximatestheaverageresponses. 34OurfindingalignswithDongetal.(2024), whoreportthathouseholds’disagreement, asmeasuredinthe Michigansurvey,weakenstheeffectivenessofmonetarypolicy.However,beyondthedifferencedatasource,we furtheridentifythatthekeycomponentofdisagreementthatweakenstheeffectivenessofmonetarypolicyisthe portionofthatdisagreementwhichisattributabletopublicinformation. 40

contributionofpublicinformationtodisagreementisanimportantchanneldrivingtheprice puzzle. Thisimpliesthatthepricepuzzlemayberelatedtotheeffectivenessofmonetarypolicy communicationorthecredibilityofthecentralbank.35 Overall,ourempiricalresultssuggest that clear communication of monetary policy can enhance policy effectiveness by reducing disagreementattributabletopublicinformation. AppendixE.2providesextensiverobustnesschecksfortheanalysispresentedinthissection. First,weconsiderthedisagreementsharefor10-year-aheadinflationinsteadof8-quarter-ahead inflationprojections.Second,weexploretheuseofregimeindicatorsbasedonbothaone-month lagandthecurrentquarter toassessthesensitivityoftheresultstothetimingoftheregime indicator. Third,weexpandthemacroeconomiccontrolstoincludetheuncertaintymeasure fromBinder(2017),themonetarypolicyuncertaintyfromHustedetal.(2020),andtheconsensus inflationexpectationsforthenextyear. Notethatwecontrolforindividual-leveluncertainty anduncertaintyregardingmonetarypolicytoisolatetheeffectsofdisagreementaboutpublic informationontheeffectivenessofmonetarypolicy. Fourth,wepresenttheimpulseresponses withtheaveragedisagreementsharesseenduringeconomicrecessions. Theempiricalresults remainrobustacrossthesealternativespecifications.36 8 Conclusion This paper makes three key contributions. First, we develop a parametric model which we callthe‘individualterm-structureofinflationexpectations,’whichusestwofactors—leveland slope—todescribeforecasters’inflationpredictionsacrossdifferenttimehorizons. Second,we extendthismodeltoadynamicfactorframework,decomposingindividual-levelelementsinto 35Falcketal.(2021)alsoobservethatthepricepuzzlebecomesmorepronouncedwhenprofessionalforecasters disagree,basedonSPFdata.OurfindingdiffersfromFalcketal.(2021)inthatweidentifypublicinformationas thekeydriver. Thesensitivityofdisagreementtopublicinformationincreaseswhenoveralldisagreementrises, explainingtheobservedconsistencybetweenthetwofindings. 36Additionally, weexaminealternativeregimesbasedonhigh-andlow-leveldisagreementwithoutfurther decomposingbyinformationsource.SeeAppendixE.2formoredetails.Inthisanalysis,theregimesaredetermined bywhethertotal4-quarter-aheaddisagreement—theforecasthorizonfrequentlyoftenconsideredbyprevious studies. Similartotheearlierfindings,wefindlargerandmorestatisticallysignificantcontractionaryeffectsof monetarypolicyintheregimeoflowdisagreement,whiletheeffectsaremutedandlessstatisticallysignificantin theregimeofhighdisagreement.However,thepricepuzzleisobservedinthelowdisagreementregimenotinthe highdisagreementregime,whichissomewhatdifferentfromtheearlierstudies. Thedifferencemaystemfrom differencesindataandthesampleperiod.Alltold,thedifferenceinpolicyeffectivenessbetweenthetworegimesis lesspronouncedwhendistinguishingtheregimesbasedonthelevelofdisagreementcomparedtoourbaseline. Thisobservationreinforcestheimportanceoftheinformationsourceinevaluatingtheeffectivenessofmonetary policy. 41

common and idiosyncratic components and an individual constant term. We build a noisy informationmodelwhereforecastersexhibitheterogeneousreactionstopublicinformation. Weshowthatourstructuralmodelmapsdirectlyintotheestimatesfromourdynamicfactor model,whichallowstointerpretthecommonandidiosyncraticcomponentsasresponsesto publicandprivateinformation,respectively,andtheindividualfixedeffectsaslong-termbeliefs. Thisuniquedecompositionallowsustoseparatedisagreementaboutinflationprojectionsinto thesethreesourcesateachpointintimeandacrossforecastinghorizons. Third,weinvestigate howthesensitivityofdisagreementtopublicinformationimpactstheeffectivenessofmonetary policyandtheanchoringofinflationexpectations. Allofthesecontributionsareentirelynovel totheliterature. Ourresearchhighlightstheimportanceofconsideringdisagreementwhenevaluatingthe anchoringofinflationexpectations. Althoughconsensusforecastssuggestwell-anchoredlongtermexpectations,inflationforecastsacrossforecastinghorizonsshowgreaterdisagreementand increasedskewness,particularlyduringtheGreatRecessionandtheCOVID-19pandemic. This observationindicatesthatexpectationswerelikelylesswell-anchoredthanpreviouslythought. Ourmodelshowsthattheconsensusforecastanddisagreementoftenyielddistinctinsightsinto theanchoringofagents’inflationexpectations. We find distinct roles for the three sources of disagreement across forecasting horizons. Long-termbeliefsandprivateinformationaccountforthemajorityofdisagreementinlong-run andshort-runexpectationsrespectively. Theroleofpublicinformationindisagreementissmall onaverage. However,duringeconomicdownturnsandperiodsofhighinflationuncertainty, public information becomes a key driver of disagreement. A noisy information model with heterogeneousreactionstopublicinformationpredictsthatthisvaryingimportanceofpublic informationinthedisagreementarisesinresponsetolargepublicnewsshocks. Finally,wefindthatwhenpublicinformationisthemainsourceofdisagreement,theeconomy’sresponsestomonetarypolicyshockaredelayedsignificantlyandapricepuzzleemerges. When public information is not important in disagreement, monetary policy has rapid and statisticallysignificantstabilizingeffects. Theseresultssuggeststhatdisagreementaboutpublic informationisanimportantdeterminantoftheeffectivenessofmonetarypolicy,underscoring theimportanceofanchoringinflationexpectations. Central bank communication about the macroeconomic outlook plays a crucial role in managing inflation expectations, especially during times when economic agents are highly 42

attentivetomonetarypolicyandmacroeconomicnews. Clearcommunicationbypolicymakers canreducedisagreementandprovideastrongeranchorforinflationexpectationsduringperiods ofheighteneduncertainty. Ourfindingsofferanewperspectiveonthesourceofthepricepuzzle anditsrelationshipwithcentralbanks’expectationsmanagement. Weleavethistopictofuture research. 43

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Appendix A State-Space Representation of Nelson-Siegel Model This section provides a full characterization of the state-space representation of the Nelson- Siegel model’s equations, along with detailed definitions of all of the coefficient vectors and matrices. A.1 StateEquation Westartwiththestateequation. Letthe(2(n+1)×1)statevectorx bedefinedas t (cid:183) (cid:184)′ x t := L t ,S t ,εL 1,t ,εS 1,t ,···,εL n,t ,εS n,t (A1) DefinethetransitionmatrixFas   ρ 0 0 0 ... 0 0 L      0 ρ 0 0 ... 0 0   S       0 0 ρL 0 0... 0 0  ε     F:=  0 0 0 ρS ε ... 0 0   . (A2)     . . . . . . . . . . . . ... . . . . . .          0 0 0 0 ... ρL ε 0      0 0 0 0 ... 0 ρS ε Letu bethevectorofshockstothestatevectordefinedasfollows: t (cid:183) (cid:184)′ u t := uL,uS,uL ,uS ,···,uL ,uS . (A3) t t 1,t 1,t n,t n,t 48

Thecovariancematrixoftheshocksisgivenby:   1 0 0 0 ... 0 0      0 1 0 0 ... 0 0         0 0 σ2 0 0... 0 0   L    Q=  0 0 0 σ2 ... 0 0   . (A4) S     . . . . . . . . . . . . ... . . . . . .          0 0 0 0 ... σ2 0   L    0 0 0 0 ... 0 σ2 S Finally,wearriveatthestateequation: x t =Fx t−1 +u t , u t ∼N(0,Q). (A5) A.2 MeasurementEquation Themeasurementequationisoftheform: y =µ +Hx +v , v ∼N(0,R), (A6) t y t t t whereµ isavectorofforecasterfixedeffects,Hisamatrixoffactorloadingsontheaggregate y andidiosyncraticlevelandslopefactors,andv isavectorofmeasurementerrors. Therestof t thesectiondetailseachcomponentinEquation(10). A.2.1 TheVectorofObservations: y t Forestimation,weuseone-quartertofour-quarteraheadfixed-horizonforecastsandtwo-year forward,three-yearforward,five-yearaverage,andten-yearaveragefixedeventforecasts.Forthe five-yearandten-yearaverageforecasts,weuseobservednowcastsandone-quarterbackcasts 49

whenavailable,andrealizedinflationtwoquartersandthreequarterspriorfromthemostrecent CPIvintageatthetimethesurveywasconductedtocapturerealizedinflation. Theobservationvectorforanyperiody isgivenby t  ′ π 1,t→t+1|t , π 1,t+1→t+2|t , π 1,t+2→t+3|t , π 1,t+3→t+4|t , ···       π 1,t+3→t+7|t , π 1,t+2→t+6|t , π 1,t+1→t+5|t , π 1,t→t+4|t , ···         π 1,t+7→t+11|t , π 1,t+6→t+10|t , π 1,t+5→t+9|t , π 1,t+4→t+8|t , ···     y t =  π 1,t→t+19|t , π 1,t→t+18|t , π 1,t→t+17|t , π 1,t→t+16|t , ···   . (A7)       π 1,t→t+39|t , π 1,t→t+38|t , π 1,t→t+37|t , π 1,t→t+36|t , ···       ······       ··· π n,t→t+37|t , π n,t→t+36|t Thefirstfourelementsofy correspondtofixedhorizonforecastsofonetofourquartersahead t andaretypicallyobservedeveryperiod. Onlyfourofthefinalsixteenelementsofy areobserved t inanygivenquarter. Thesefinalsixteenelementscorrespondtofixedeventforecasts,where eachgroupoffourcorrespondtothefixedeventcorrectlymappedtothequarterinwhichthe surveywasconducted. Forthefinaleightelements,whichcorrespondtoforecastsofaverageinflationoverfiveand tenyearperiodsincludingthecurrentcalendaryear,wemustadjustthemtoaccountforthefact thattheyincluderealizedinflationoverpreviousquarters. Specifically, • InQ1,wedefine (cid:181) (cid:182) 4 1 π i,t→t+19|t = 5π i,t−1→t+19|t − π i,t−1→t|t 19 4 (cid:181) (cid:182) 4 1 π i,t→t+39|t = 10π i,t−1→t+19|t − π i,t−1→t|t 39 4 • InQ2,wedefine (cid:181) (cid:182) 4 1 1 π i,t→t+18|t = 5π i,t−1→t+19|t − π i,t−1→t|t − π i,t−2→t−1|t 18 4 4 50

(cid:181) (cid:182) 4 1 1 π i,t→t+38|t = 10π i,t−1→t+19|t − π i,t−1→t|t − π i,t−2→t−1|t 38 4 4 • InQ3,wedefine (cid:181) (cid:182) 4 1 1 1 π i,t→t+17|t = 5π i,t−1→t+19|t − π i,t−1→t|t − π i,t−2→t−1|t − π i,t−3→t−2|t 17 4 4 4 (cid:181) (cid:182) 4 1 1 1 π i,t→t+37|t = 10π i,t−1→t+19|t − π i,t−1→t|t − π i,t−2→t−1|t − π i,t−3→t−2|t 37 4 4 4 • InQ4,wedefine (cid:181) (cid:182) 4 1 1 1 1 π i,t→t+16|t = 5π i,t−1→t+19|t − π i,t−1→t|t − π i,t−2→t−1|t − π i,t−3→t−2|t − π i,t−4→t−3|t 16 4 4 4 4 (cid:181) (cid:182) 4 1 1 1 1 π i,t→t+36|t = 10π i,t−1→t+19|t − π i,t−1→t|t − π i,t−2→t−1|t − π i,t−3→t−2|t − π i,t−4→t−3|t 36 4 4 4 4 Forthenowcastsπ i,t−1→t|t andbackcastsπ i,t−2→t−1|t ,weusethereportedvaluesfromtheSPF. Forthetwoandthreeperiodbackcastsπ i,t−3→t−2|t andπ i,t−4→t−3|t ,weusethemostrecently availablevintageoftheCPIatthetimethattheforecastwasmade. µ A.2.2 TheVectorofForecasterFixedEffects: y Wedefinetheloadingfunctionontheslopefactorforforecastsofinflationbetweenhorizonsat twodatest+h andt+h as 1 2 e −λh1 −e −λh2 f (h ,h )= (A8) S 1 2 λ(h −h ) 2 1 51

Thisexpressionisusedinµ andH. Definetheconstantvectorinthemeasurementequation, y µ ,as y  ′ α −α f (0,1), α −α f (1,2), α −α f (2,3), α −α f (3,4), L S S L S S L S S L S S      α −α f (3,7), α −α f (2,6), α −α f (1,5), α −α f (0,4),   L S S L S S L S S L S S       α −α f (7,11), α −α f (6,10), α −α f (5,9), α −α f (4,8)   L S S L S S L S S L S S    µ y :=  α L −α S f S (0,19), α L −α S f S (0,18), α L −α S f S (0,17), α L −α S f S (0,16),   . (A9)      α −α f (0,39), α −α f (0,38), α −α f (0,37), α −α f (0,36),   L S S L S S L S S L S S      ······       ······α −α f (0,37), α −α f (0,36) L S S L S S Definetheerrorterminthemeasurementequation,v ,as t  ′ v , v , v , v , 1,1,t 1,2,t 1,3,t 1,4,t      v , v , v , v ,   1,5,t 1,6,t 1,7,t 1,8,t       v , v , v , v ,   1,9,t 1,10,t 1,11,t 1,12,t    v t :=  v 1,13,t , v 1,14,t , v 1,15,t , v 1,16,t ,   . (A10)      v , v , v , v ,   1,17,t 1,18,t 1,19,t 1,20,t      ······       ······v , v n,19,t n,20,t The covariance matrix of the measurement error vector (v ), R, is given by the following t 52

diagonalmatrix.  ′ σ2 , σ2 , σ2 , σ2 ,  v,1 v,2 v,3 v,4     σ2 , σ2 , σ2 , σ2 ,   v,5 v,6 v,7 v,8       σ2 , σ2 , σ2 , σ2 ,   v,9 v,10 v,11 v,12    R:=diag    σ2 , σ2 , σ2 , σ2     (A11) v,13 v,14 v,15 v,16      σ2 , σ2 , σ2 , σ2   v,17 v,18 v,19 v,20      ······       ······,σ2 , σ2 v,19 v,20 Notethattheargumentinthesquarebracketisavector. 53

Finally,wedefinethemeasurementequationmappingmatrixHas   βL −βSf (0,1) 1 f (0,1) ... 0 0  1 1 S S     βL −βSf (1,2) 1 f (1,2) ... 0 0   1 1 S S       βL −βSf (2,3) 1 f (2,3) ... 0 0   1 1 S S      βL 1 −βS 1 f S (3,4) 1 f S (3,4) ... 0 0        βL −βSf (3,7) 1 f (3,7) ... 0 0   1 1 S S       βL −βSf (2,6) 1 f (2,6) ... 0 0   1 1 S S     βL −βSf (1,5) 1 f (1,5) ... 0 0   1 1 S S       βL −βSf (0,4) 1 f (0,4) ... 0 0   1 1 S S       βL 1 −βS 1 f S (7,11) 1 f S (7,11) ... 0 0       βL −βSf (6,10) 1 f (6,10) ... 0 0   1 1 S S       βL −βSf (5,9) 1 f (5,9) ... 0 0   1 1 S S    H:=  βL 1 −βS 1 f S (4,8) 1 f S (4,8) ... 0 0   . (A12)      βL −βSf (0,19) 1 f (0,19) ... 0 0   1 1 S S        βL 1 −βS 1 f S (0,18) 1 f S (0,18) ... 0 0      βL −βSf (0,17) 1 f (0,17) ... 0 0   1 1 S S       βL −βSf (0,16) 1 f (0,16) ... 0 0   1 1 S S       βL 1 −βS 1 f S (0,39) 1 f S (0,39) ... 0 0       βL −βSf (0,38) 1 f (0,38) ... 0 0   1 1 S S       βL −βSf (0,37) 1 f (0,37) ... 0 0   1 1 S S      βL 1 −βS 1 f S (0,36) 1 f S (0,36) ... 0 0       . . . . . . . . . . . . ... . . . . . .          βL n −βS n f S (0,37) 0 0 ... 1 f S (0,37)      βL −βSf (0,36) 0 0 ... 1 f (0,36) n n S S 54

A.2.3 Remark Inthemeasurementequation,eachseriesiny isassumedtobeobservedwithmeasurement t error. Thefixedeventforecastsaretreatedseparatelyineachquarterthroughoutthecalendar yeartoreflectthefactthattheforecastinghorizonshrinksasthecalendaryearprogresses. This leavesuswithatotalof20observablesforeachforecasterineachquarter,12ofwhicharemissing byconstruction. 55

Appendix B Gibbs Sampler 1. Sampleθ =(cid:169)αL,αS,βL,βS(cid:170)N conditionalonremainingparameters. 1 i i i i i=1 Sincealloftheshocksareassumedtobeindependent,thiscanbetreatedasaseparate regression model for each forecaster i. We assume independent, multivariate normal priorsforeachgroupoffourparameters (cid:163)αL,αS,βL,βS(cid:164)′ acrosseachforacasteri,where i i i i   αL  i     αS    i   ∼N (cid:161)µ i ,Σ i (cid:162)    βL   i    βS i 2. Sampleθ =vec(A)conditionalonremainingparameters. SinceL andS areobserved, 2 t t thisisastandardmultivariateregressionmodel. 3. Sampleθ =vec(B)conditionalonremainingparameters. SinceεL andεS areobserved 3 i,t i,t foreveryforecasteri =1,...,n,thisisastandardmultivariateregressionmodelwithknown covariancematrixwherewepoolthedataacrossforecasters. 4. Sampleθ =(cid:163)σ2,σ2(cid:164)′ conditionalonremainingparameters.SinceεL andεS areobserved 4 L S i,t i,t foreveryforecasteri =1,...,n,thisisastandardvarianceestimationproblemwithknown regressioncoefficientswherewepoolthedataacrossforecasters. 5. Sampleθ =λwithaMetropolisHastingsstepconditionalonremainingparameters. It 5 boilsdowntoanonlinearregressionproblem. (cid:104) (cid:105)′ 6. Sampleθ = σ2 ,...,σ2 conditionalonremainingparameters. Givenotherparame- 6 v,1 v,20 ters,v isdirectlyobserved. t 7. Sampleθ ={x }T conditionalonremainingparametersusingasimulationsmoother. 7 t t=1 56

Appendix C Noisy Information Model Proofs ProofofProposition1. Aftersomealgebraicmanipulationonecanshowthat (cid:34)(cid:195) ρL (cid:33) ρL (cid:35) L i,t =ρL i,ε L i,t−1 +βL i 1− ρ i,ε L t + ρ i,ε u t L +v i L ,t L L (cid:34)(cid:195) ρS (cid:33) ρS (cid:35) S i,t =ρS i,ε S i,t−1 +βS i 1− ρ i,ε S t + ρ i,ε u t S +v i S ,t S S Takingthecross-sectionalmeanacrossforecastersofthecommonterm(involvingβ )and i addingandsubtractingweget L i,t =ρL i,ε L i,t−1 +βL i (cid:183)(cid:181) 1− ρ ρ ¯L ε (cid:182) L t + ρ ρ¯L ε u t L (cid:184) +v i L ,t −βL i ρL i,ε ρ −ρ¯L ε (cid:161) L t −u t L(cid:162) L L L S i,t =ρS i,ε S i,t−1 +βS i (cid:183)(cid:181) 1− ρ ρ¯S ε (cid:182) S t + ρ ρ ¯S ε u t S (cid:184) +v i S ,t −βS i ρS i,ε ρ −ρ¯S ε (cid:161) S t −u t S(cid:162) S S S Matchingtermstothoseintheequationswhichgoverntheevolutionofbeliefsinthenoisy informationmodel,wecansolveforthestatisticalmodelparametersasfunctionsofthenoisy informationmodelparameters. Startingwiththeslopedynamicswehave ρS i,ε S i,t−1 =(cid:161) 1−g i c(cid:162)ρ c F i,t−1 c t−1 (C13) (cid:183)(cid:181) ρ¯S(cid:182) ρ¯S (cid:184) βS 1− ε S + ε uS =gcc +gc uc (C14) i ρ t ρ t i t i,y t S S ρS −ρ¯S vS −βS i,ε ε (cid:161) S −uS(cid:162)=gcvc (C15) i,t i ρ t t i i,t S WestartbyrecognizingthatS =F c ,andthusfromthefirstequationweimmediately i,t i,t t obtain ρS =(cid:161) 1−gc(cid:162)ρ (C16) i,ε i c (cid:179) (cid:180) Wenowassumethatgc ∼i.i.d. g¯c,(σc)2 acrossforecasters(thiscanbejustifiedbyagents i g havingdifferentvariancesofprivatesignals). 57

Workingwithequation(C14)andassumingthatρ =ρ , S c (cid:183)(cid:181) ρ¯S(cid:182) ρ¯S (cid:184) βS 1− ε S + ε uS i ρ t ρ t S S (cid:183)(cid:181) (cid:161) 1−g¯c(cid:162)ρ (cid:182) (cid:161) 1−g¯c(cid:162)ρ (cid:184) =βS 1− c S + c uS i ρ t ρ t S S =βS(cid:163) g¯cS +(cid:161) 1−g¯c(cid:162) uS(cid:164) i t t =βS i (cid:163) g¯c(cid:161)ρ S S t−1 +u t S(cid:162)+(cid:161) 1−g¯c(cid:162) u t S(cid:164) =βS i (cid:163)ρ c g¯cS t−1 +u t S(cid:164) =gcc +gc uc i t i,y t Weknowthatthelasttwoexpressionsmusthavethesametime-seriesvariance,thus (cid:161)ρ c βS i g¯c(cid:162)2 +(cid:161)βS(cid:162)2=(cid:161) gc(cid:162)2 (σc ε )2 + (cid:179) gc (cid:180)2 (σc)2 1−ρ2 i i 1−ρ2 i,y u c c SolvingforβS gives i  (cid:161) gc(cid:162)2 (σc)2+ (cid:179) gc (cid:180)2 (σc)2(cid:161) 1−ρ2(cid:162) 1/2 βS = i ε i,y u c  (C17) i  1+(cid:161)ρ g¯c (cid:162)2−ρ2  c c Nextweworkwithequation(C15), ρS −ρ¯S vS −βS i,ε ε (cid:161) S −uS(cid:162) i,t i ρ t t S =v i S ,t +βS i (cid:161) g i c−g¯c(cid:162)ρ c S t−1 =gc vc i,z i,t Asbefore,weknowthatthelasttwoexpressionsmusthavethesametime-seriesvariance, thus (σS )2+ (cid:161)βS i (cid:162)2(cid:163)(cid:161) g i c−g¯c(cid:162)ρ c (cid:164)2 = (cid:179) gc σc (cid:180)2 i,v 1−ρ2 i,z i,v c 58

Solvingfor(σS )2gives i,v (σS )2= (cid:179) gc σc (cid:180)2 − (cid:161)βS i (cid:162)2(cid:163)(cid:161) g i c−g¯c(cid:162)ρ c (cid:164)2 (C18) i,v i,z i,v 1−ρ2 c Using similar arguments for the level factor / permanent component (but matching the conditionalvarianceinsteadoftheunconditionalvariance),weobtain ρL =1−g τ (C19) i,ε i βL= (cid:183) (cid:161) g τ(cid:162)2 (στ )2+ (cid:179) g τ (cid:180)2 (στ )2 (cid:184)1/2 (C20) i i ε i,y u σL =g τ στ (C21) i,v i,z i,v 59

Appendix D Additional Distributional Results Thissectionshowsfiguresonmomentsofthefactordistributions,theforecastvariancedecomposition,andtheforecastvarianceshareofpublicinformation,thatarenotincludedinthemain text. FigureD1: SMOOTHED IDIOSYNCRATIC FACTOR DISPERSION Notes:TheshadedareasdenotetheNBERrecessions. Sources:Authors’calculation FigureD2: SMOOTHED IDIOSYNCRATIC FACTOR SKEWNESS Notes:TheshadedareasdenotetheNBERrecessions. Sources:Authors’calculation 60

FigureD3: SMOOTHED IDIOSYNCRATIC FACTOR KURTOSIS 5 Notes:TheshadedareasdenotetheNBERrecessions. Sources:Authors’calculation FigureD4: FORECAST VARIANCE DECOMPOSITION Notes:Thefigureshowsthedecompositionofthecross-sectionalvarianceofinflationforecasts(blackline)intothe componentsdrivenbyindividuallong-termbeliefs(denotedbyltb,blueline),heterogeneousresponsestopublic information(denotedbypub,redline),andprivateinformation(denotedbypriv,yellowline).Eachline correspondstotheposteriormedian.TheshadedareasdenoteNBERrecessions. Sources:Authors’calculation 61

FigureD5: FORECAST VARIANCE SHARE OF COMMON COMPONENT Notes:Thefigureshowsthefractionofoveralldisagreementaboutfoursetsofinflationexpectationsdrivenby heterogeneousresponsestocommoninformationovertimeasmeasuredbytheβmeasureproposedinFujitaand Ramey(2009).TheshadedareasdenoteNBERrecessions. Sources:Authors’calculation 62

Appendix E Additional Results on Monetary Policy Effectiveness E.1 EffectsofNewsComponentonDisagreement FigureE6: (ROBUSTNESS CHECK 1) PROPAGATION OF FED’S REACTIONS TO NEWS: 10-YEAR AHEAD PanelA.AverageResponses Public Private 0.04 0.04 0.02 0.02 0 0 -0.02 -0.02 -0.04 -0.04 -0.06 -0.06 -0.08 -0.08 -0.1 -0.1 0 2 4 6 8 0 2 4 6 8 PanelB.NonlinearResponses Public Private 0.04 0.04 0.02 0.02 0 0 -0.02 -0.02 -0.04 -0.04 -0.06 -0.06 -0.08 -0.08 Private (+ long-run beliefs) Public -0.1 -0.1 0 2 4 6 8 0 2 4 6 8 Notes:Thefigureshowstheresponsesofdisagreementabout40-quarter-aheadinflationexpectationsattributable topublicinformationandnon-publicinformationfollowingtheFed’sresponsetonewsfromBauerandSwanson (2022).Thealternativemodelalsoincludestheindividual-leveluncertaintyfromBinder(2017)andtheconsensus inflationexpectationforthenextyearastheadditionalcontrols.Inaddition,theregimesaredeterminedbythe disagreementsharesfor40-quarterahead.HACstandarderrorsarereported.PanelApresentstheresultsfromthe linearmodelinEquation.(19),whilePanelBshowstheresultsfromthenonlinearmodelinEquation(20).InPanel B,themagentalinesrepresenttheresponseswhenthepreviousperiod’sdisagreementisattributabletopublic information,andthebluelinesrepresenttheresponseswhenthepreviousperiod’sdisagreementisattributableto non-publicinformation.Estimatesforregimes1and2arescaledby0.9and0.1,respectively,correspondingtothe averagevaluesofthedisagreementsharesforpublicandnon-publicinformationfor10-yearahead.Thedashed linesindicatethe95%posteriorintervals. Source:Authors’calculation 63

FigureE7: (ROBUSTNESS CHECK 2) PROPAGATION OF FED’S REACTIONS TO NEWS: ALTERNATIVE CONTROLS PanelA.AverageResponses Public Private 0.04 0.04 0.02 0.02 0 0 -0.02 -0.02 -0.04 -0.04 -0.06 -0.06 -0.08 -0.08 -0.1 -0.1 0 2 4 6 8 0 2 4 6 8 PanelB.NonlinearResponses Public Private 0.04 0.04 0.02 0.02 0 0 -0.02 -0.02 -0.04 -0.04 -0.06 -0.06 -0.08 -0.08 Private (+ long-run beliefs) Public -0.1 -0.1 0 2 4 6 8 0 2 4 6 8 Notes:Thefigureshowstheresponsesofdisagreementabout8-quarter-aheadinflationexpectationsattributableto publicinformationandnon-publicinformationfollowingtheFed’sresponsetonewsfromBauerandSwanson (2022).Thealternativemodelwiththeindividual-leveluncertaintymeasurefromBinder(2017)andtheconsensus inflationexpectationsforthenextyearareadditionallyconsideredasthecontrols.HACstandarderrorsare reported.PanelApresentstheresultsfromthelinearmodelinEquation.(19),whilePanelBshowstheresultsfrom thenonlinearmodelinEquation(20).InPanelB,themagentalinesrepresenttheresponseswhentheprevious period’sdisagreementisattributabletopublicinformation,andthebluelinesrepresenttheresponseswhenthe previousperiod’sdisagreementisattributabletonon-publicinformation.Estimatesforregimes1and2arescaled by0.9and0.1,respectively,correspondingtotheaveragevaluesofthedisagreementsharesforpublicand non-publicinformationfor8-quarterahead.Thedashedlinesindicatethe95%posteriorintervals. Source:Authors’calculation 64

FigureE8: (ROBUSTNESS CHECK 3) PROPAGATION OF FED’S REACTIONS TO NEWS: NO LAGS OF z t PanelA.AverageResponses Public Private 0.04 0.04 0.02 0.02 0 0 -0.02 -0.02 -0.04 -0.04 -0.06 -0.06 -0.08 -0.08 -0.1 -0.1 0 2 4 6 8 0 2 4 6 8 PanelB.NonlinearResponses Public Private 0.04 0.04 0.02 0.02 0 0 -0.02 -0.02 -0.04 -0.04 -0.06 -0.06 -0.08 -0.08 Private (+ long-run beliefs) Public -0.1 -0.1 0 2 4 6 8 0 2 4 6 8 Notes:Thefigureshowstheresponsesofdisagreementabout8-quarter-aheadinflationexpectationsattributableto publicinformationandnon-publicinformationfollowingtheFed’sresponsetonewsfromBauerandSwanson (2022).Thealternativemodelwithzerolagsforz isconsidered.HACstandarderrorsarereported.PanelApresents t theresultsfromthelinearmodelinEquation.(19),whilePanelBshowstheresultsfromthenonlinearmodelin Equation(20).InPanelB,themagentalinesrepresenttheresponseswhenthepreviousperiod’sdisagreementis attributabletopublicinformation,andthebluelinesrepresenttheresponseswhenthepreviousperiod’s disagreementisattributabletonon-publicinformation.Estimatesforregimes1and2arescaledby0.9and0.1, respectively,correspondingtotheaveragevaluesofthedisagreementsharesforpublicandnon-publicinformation 8-quarterahead.Thedashedlinesindicatethe95%posteriorintervals. Source:Authors’calculation 65

FigureE9: (ROBUSTNESS CHECK 4) PROPAGATION OF FED’S REACTIONS TO NEWS: LP-IV Impulse response (LP with controls) Treasury IP 0.6 2 1.5 0.4 1 0.2 0.5 0 0 0 2 4 6 8 0 2 4 6 8 CPI EBP 0.6 0.2 0.1 0.4 0 -0.1 0.2 -0.2 0 0 2 4 6 8 0 2 4 6 8 Public (8-quarter) Private (8-quarter) 0.02 0 0.01 0 -0.02 -0.01 -0.04 -0.02 -0.06 0 2 4 6 8 0 2 4 6 8 Notes:Thefigureshowstheresponsesoffourmacroeconomicvariablesandthedisagreementabout 8-quarter-aheadinflationexpectationsdrivenbypublicandprivateinformation(includinglong-termbeliefs)to Fed’sresponsetonewsfromBauerandSwanson(2023).TheLP-IVisemployedfortheestimation.TheLP-IV includesfourlagsofdependentvariablesandfourlagsofexternalshocksinthefirststageregression.The F-statisticsissignificantlylargerthan10.ThecumulativeresponsesarereportedfortheIPgrowthandCPIinflation. Thedashedlinecapturesthe90percentposteriorintervals. Source:Authors’calculation 66

E.2 NonlinearEffectsofaTraditionalMonetaryPolicyShock Figure E10: (ROBUSTNESS CHECK 1) PROPAGATION OF MONETARY POLICY SHOCKS: 10-YEAR AHEAD IP CPI 1 0.2 0 0 -1 -0.2 -2 -0.4 0 5 10 15 20 25 0 5 10 15 20 25 Unemployment Rate EBP 0.3 0.2 0.2 0.1 0.1 0 0 -0.1 -0.2 -0.1 0 5 10 15 20 25 0 5 10 15 20 25 Notes:Thefigurereportstheresponsesoffourmacroeconomicvariablestotheorthogonalizedmonetarypolicy shockfromBauerandSwanson(2022).Fortheregimeindicator,10-yearaheaddisagreementsharesoftheprevious quarterareconsidered.PanelAshowstheimpulseresponsesofregimes1and2scaledby0.9and0.1,respectively. Thebluelinesrepresenttheresponseswhendisagreementisattributablenon-publicinformation(regime1),while themagentalinesshowtheresponseswhendisagreementisattributabletopublicinformation(regime2).Ineach panel,theupperleftfigureshowsthecumulativeresponseofpercentchangesinindustrialproduction;theupper rightfigureshowsthecumulativeresponsesofpercentchangesintheCPI;thebottomleftfigureshowsthe responsesoftheunemploymentrate;andthebottomrightfiguredisplaystheresponseoftheexcessbondpremium (EBP).Thedashedlinesrepresentthe90%posteriorintervals. Sources:Authors’calculation 67

FigureE11: (ROBUSTNESS CHECK 2) PROPAGATION OF MONETARY POLICY SHOCKS OF THE TWO REGIMES: 1-MONTH LAG FOR THE REGIME INDICATOR) IP CPI 0.5 0.2 0.1 0 0 -0.1 -0.5 -0.2 -1 -0.3 0 5 10 15 20 25 0 5 10 15 20 25 Unemployment Rate EBP 0.2 0.2 Private (+ long-run beliefs) Public 0.1 0.1 0 0 -0.1 -0.1 0 5 10 15 20 25 0 5 10 15 20 25 Notes:Thefigurereportstheresponsesoffourmacroeconomicvariablestotheorthogonalizedmonetarypolicy shockfromBauerandSwanson(2022).Fortheregimeindicator,theone-monthlagsof8-quarter-ahead disagreementsharesareconsidered.Thepanelsshowtheimpulseresponsesofregimes1and2scaledby0.9and 0.1,respectively.Thebluelinesrepresenttheresponseswhendisagreementisattributablenon-publicinformation (regime1),whilethemagentalinesshowtheresponseswhendisagreementisattributabletopublicinformation (regime2).Ineachpanel,theupperleftfigureshowsthecumulativeresponseofpercentchangesinindustrial production;theupperrightfigureshowsthecumulativeresponsesofpercentchangesintheCPI;thebottomleft figureshowstheresponsesoftheunemploymentrate;andthebottomrightfiguredisplaystheresponseofthe excessbondpremium(EBP).Thedashedlinesrepresentthe90%posteriorintervals. Sources:Authors’calculation 68

FigureE12: (ROBUSTNESS CHECK 3) PROPAGATION OF MONETARY POLICY SHOCKS OF THE TWO REGIMES: NO LAG FOR THE REGIME INDICATOR) IP CPI 0.2 0.1 0 0 -0.2 -0.1 -0.4 -0.6 -0.2 0 5 10 15 20 25 0 5 10 15 20 25 Unemployment Rate EBP 0.1 0.1 Regime 1 Regime 2 0.05 0.05 0 0 -0.05 0 5 10 15 20 25 0 5 10 15 20 25 Notes:Thefigurereportstheresponsesoffourmacroeconomicvariablestotheorthogonalizedmonetarypolicy shockfromBauerandSwanson(2022).Fortheregimeindicator,8-quarter-aheaddisagreementsharesare consideredwithoutalag.Thepanelsshowtheimpulseresponsesofregimes1and2scaledby0.9and0.1, respectively.Thebluelinesrepresenttheresponseswhendisagreementisattributablenon-publicinformation (regime1),whilethemagentalinesshowtheresponseswhendisagreementisattributabletopublicinformation (regime2).Ineachpanel,theupperleftfigureshowsthecumulativeresponseofpercentchangesinindustrial production;theupperrightfigureshowsthecumulativeresponsesofpercentchangesintheCPI;thebottomleft figureshowstheresponsesoftheunemploymentrate;andthebottomrightfiguredisplaystheresponseofthe excessbondpremium(EBP).Thedashedlinesrepresentthe90%posteriorintervals. Sources:Authors’calculation 69

FigureE13: (ROBUSTNESS CHECK 4) PROPAGATION OF MONETARY POLICY SHOCKS OF THE TWO REGIMES: ALTERNATIVE CONTROLS (1) IP CPI 0.2 0.1 0 0.05 -0.2 0 -0.4 -0.05 -0.6 -0.1 -0.8 -0.15 0 5 10 15 20 25 0 5 10 15 20 25 Unemployment Rate EBP 0.15 0.1 Private (+ long-run beliefs) Public 0.1 0.05 0.05 0 0 -0.05 -0.05 0 5 10 15 20 25 0 5 10 15 20 25 Notes:Thefigurereportstheresponsesoffourmacroeconomicvariablestotheorthogonalizedmonetarypolicy shockfromBauerandSwanson(2022).Fortheregimeindicator,8-quarteraheaddisagreementsharesofthe previousquarterareconsidered.Asthecontrols,theuncertaintymeasurefromBinder(2017)andtheconsensus expectationsforthenextyearareadditionallyincluded.Thepanelsshowtheimpulseresponsesofregimes1and2 scaledby0.9and0.1,respectively.Thebluelinesrepresenttheresponseswhendisagreementisattributable non-publicinformation(regime1),whilethemagentalinesshowtheresponseswhendisagreementisattributable topublicinformation(regime2).Ineachpanel,theupperleftfigureshowsthecumulativeresponseofpercent changesinindustrialproduction;theupperrightfigureshowsthecumulativeresponsesofpercentchangesinthe CPI;thebottomleftfigureshowstheresponsesoftheunemploymentrate;andthebottomrightfiguredisplaysthe responseoftheexcessbondpremium(EBP).Thedashedlinesrepresentthe90%posteriorintervals. Sources:Authors’calculation 70

FigureE14: (ROBUSTNESS CHECK 4) PROPAGATION OF MONETARY POLICY SHOCKS OF THE TWO REGIMES: ALTERNATIVE CONTROLS WITH MONETARY POLICY UNCERTAINTY (2) IP CPI 0.2 0.05 0 0 -0.2 -0.05 -0.4 -0.6 -0.1 0 5 10 15 20 25 0 5 10 15 20 25 Unemployment Rate EBP 0.1 0.1 Regime 1 0.08 Regime 2 0.06 0.05 0.04 0.02 0 0 -0.05 0 5 10 15 20 25 0 5 10 15 20 25 Notes:Thefigurereportstheresponsesoffourmacroeconomicvariablestotheorthogonalizedmonetarypolicy shockfromBauerandSwanson(2022).Fortheregimeindicator,8-quarteraheaddisagreementsharesofthe previousquarterareconsidered.Asthecontrols,theuncertaintymeasurefromBinder(2017),themeasureof monetarypolicyuncertaintyfromHustedetal.(2020),andtheconsensusexpectationsforthenextyearare additionallyincluded.Thepanelsshowtheimpulseresponsesofregimes1and2scaledby0.9and0.1,respectively. Thebluelinesrepresenttheresponseswhendisagreementisattributablenon-publicinformation(regime1),while themagentalinesshowtheresponseswhendisagreementisattributabletopublicinformation(regime2).Ineach panel,theupperleftfigureshowsthecumulativeresponseofpercentchangesinindustrialproduction;theupper rightfigureshowsthecumulativeresponsesofpercentchangesintheCPI;thebottomleftfigureshowsthe responsesoftheunemploymentrate;andthebottomrightfiguredisplaystheresponseoftheexcessbondpremium (EBP).Thedashedlinesrepresentthe90%posteriorintervals. Sources:Authors’calculation 71

FigureE15: (ROBUSTNESS CHECK 5) PROPAGATION OF MONETARY POLICY SHOCKS OF THE TWO REGIMES WITH RECESSION WEIGHTS (8-QUARTER AHEAD) IP CPI 1 0.5 0 0 -1 -0.5 -2 -3 -1 0 5 10 15 20 25 0 5 10 15 20 25 Unemployment Rate EBP 0.6 0.6 Private (+ long-run beliefs) Public 0.4 0.4 0.2 0.2 0 0 -0.2 -0.2 -0.4 0 5 10 15 20 25 0 5 10 15 20 25 Notes:Thefigurereportstheresponsesoffourmacroeconomicvariablestotheorthogonalizedmonetarypolicy shockfromBauerandSwanson(2022).Fortheregimeindicator,8-quarteraheaddisagreementsharesofthe previousquarterareconsidered.Thefigureshowstheimpulseresponsesofregimes1and2scaledby0.6and0.4, respectively.Theseweightsrepresentthedisagreementsharesofnon-publicandpublicinformationduringan averageeconomicdownturnoftheperiodofhighinflationuncertainty.Thebluelinesshowtheresponseswhen non-publicinformationisthesourceofdisagreement(regime1),whilethemagentalinesrepresenttheresponses whenpublicinformationisthesourceofdisagreement(regime2).Theupperleftfigureshowsthecumulative responseofpercentchangesinindustrialproduction;theupperrightfigureshowsthecumulativeresponsesof percentchangesintheCPI;thebottomleftfigureshowstheresponsesoftheunemploymentrate;andthebottom rightfiguredisplaystheresponseoftheexcessbondpremium(EBP).Thedashedlinesrepresentthe90%posterior intervals. Sources:Authors’calculation Here,wefurtherexaminetheresponsesduringperiodsofeconomicrecessionsandheightenedinflationrisks. Theresponsesofregimes1and2arescaledby0.6and0.4,reflectingthe averagedisagreementsharesofnon-publicinformationandpublicinformationduringthese periods,respectively. Comparedtonormaltimes,theresponsesinregime2becomedominant. Overall,theeffectsofmonetarypolicyarenotstatisticallysignificant. Forunemploymentrate andtheEBP,thestatisticallysignificanteffectsappearbutwithsubstantialdelays. 72

FigureE16: (ROBUSTNESSCHECK6)PROPAGATIONOFMONETARYPOLICYSHOCKS: HIGHANDLOW DISAGREEMENT REGIMES IP CPI 0.2 0.1 0 0 -0.2 -0.1 -0.4 -0.6 -0.2 0 5 10 15 20 25 0 5 10 15 20 25 Unemployment Rate EBP 0.15 0.1 High Disagreement Low Disagreement 0.1 0.05 0.05 0 0 -0.05 -0.05 0 5 10 15 20 25 0 5 10 15 20 25 Notes:Thefigurepresentstheresponsesoffourmacroeconomicvariablestotheorthogonalizedmonetarypolicy shockfromBauerandSwanson(2022).Theresponsestoaone-standard-deviationinnovationshockarereported. Thepanelsshowtheimpulseresponsesofthelow-disagreementandhigh-disagreementregimesscaledby0.35and 0.65,respectively,representingtheaverageregimeprobabilities.Eachregimeprobabilityiscomputedasfollows. Afterstandardizingthelevelof4-quarter-aheaddisagreementcomputedwiththevarianceofforecasts,weplugin thestandardizedvalueintoalogitfunctionwiththesmoothingparameter5toproducetheprobabilityof low-disagreementregime.Theprobabilityofhigh-disagreementregimeisoneminustheprobabilityof low-disagreementregime.Forthecontrolsoflocalprojection,weconsidertheindividual-leveluncertaintyfrom Binder(2017)andnextyear’sconsensusinflationexpectationsinadditiontothebaselinecontrols.Thebluelines representtheresponsesofthehigh-disagreementregime,whilethemagentalinesshowtheresponsesofthe low-disagreementregime.Ineachpanel,theupperleftfigureshowsthecumulativeresponseofpercentchangesin industrialproduction;theupperrightfigureshowsthecumulativeresponseofpercentchangesintheCPI;the bottomleftfiguredisplaystheresponseoftheunemploymentrate;andthebottomrightfigurepresentsthe responseoftheexcessbondpremium(EBP).Thedashedlinesrepresentthe95%posteriorintervals. Sources:Authors’calculation 73

Appendix F Literature Review of Related Theory Thissectiondiscussesimplicationsofourempiricalfindingsfortheoreticalmodels. Tothebestofourknowledge,wecontributetothetheoreticalliteratureinthefollowingways. First,thecoreinnovationthatwebringtotheliteratureistouncover“heterogeneousresponses" topublicinformationasanimportantsourceofdisagreement. Second,wefoundthedifferential contributionsofthreedistinctinformationsourcestochangesinthetermstructureofinflation expectations. Specifically,theindividuallong-termbeliefsisthemaincontributortothelongrundisagreementfollowedbyprivateinformation,whileprivateinformationisthemainfactor drivingshort-rundisagreement. Last,heterogeneousresponsestopublicinformationarethe keydriverofincreaseddisagreementaboutbothshort-runandlong-runinflationforecastsin timesofeconomicdownturnsorhighinflationuncertainty. Wereviewthefeaturesofexisting theoreticalmodelsandidentifythefeaturesthatcouldbeaddedtomakethemodelsaccountfor ourempiricalfindings. Wediscusstheliteratureonthestickyinformationmodel,thenoisyinformationmodel,and disagreementaboutmeansandlong-runpriors. TableF1summarizesthemainfeaturesofeach modelinthecontextofourempiricalfindings. Notethatthereisnoscopefordisagreement inthefull-informationrationalexpectation(FIRE)model,whereeconomicagentsareexante identicalandefficientlyprocessallavailableinformation. Therefore,wefocusontheremaining models. Tobeginwith,theimportanceofindividualpriorbeliefsinlong-rundisagreementaligns withthefindingsofPattonandTimmermann(2010)andFarmeretal.(2021),whichemphasize the significance of beliefs about long-run means and individual forecasters’ priors for their long-term macroeconomic forecasts. It is worth noting that our statistical model captures a forecaster-specificlong-runpriorthroughindividualfixedeffects.Incontrasttothesestudies,we alsouncoverbothprivateandpublicinformationasadditionalimportantfactorsthatcontribute todisagreementaboutthelongrun. 74

Second,disagreementdrivenbyprivateinformationcanbeaccountedforbythenoisyinformationmodel. ThenoisyinformationmodelWoodford(2001)cangeneratedisagreement amongforecasters,butislimitedincharacterizingcountercyclicaldisagreementorincreased disagreementinresponsetoalargeshock. Inthemodel,forecastersareex-anteidenticalwith time-invariantinformationprecisionwhichisthesameacrossforecasters,buttheyarefacedwith idiosyncraticsignalswhichareuncorrelatedovertime. Thisfeaturegeneratestime-varyingdisagreement,butislimitedingeneratingthecountercyclicalityseenindisagreementattributable to public information. Thus, the noisy information model can account for disagreement attributabletoprivateinformationbutnottheportionattributabletopublicinformation. Both the noisy information model with heterogeneity and the sticky information model offerinsightsintocharacterizingdisagreementdrivenbyheterogeneousresponsestopublic information. To begin with, the noisy information model with heterogeneous information precision across forecasters, as demonstrated by Coibion and Gorodnichenko (2012b), can generate increased disagreement to economic shocks compared to the model without such heterogeneity.Notably,inourstatisticalmodel,thedifferencesinfactorloadingsontheleveland slopecanbeinterpretedasreflectingheterogeneousinformationprecision,therebycontributing todisagreementarisingfrompublicinformation. Additionally, the sticky information model is capable of generating disagreement at all timesandcapturingincreaseddisagreementintimesoflargeshocksunderspecificconditions. In this model, economic agents update their information set periodically due to the cost of informationacquisition,asexplainedbyMankiwandReis(2002). Consequently,disagreement arisesbecauseonlyafractionofforecastersupdatetheirforecastsinresponsetomacroeconomic news,whileothersdonot. Whilethismodelcancaptureincreaseddisagreementinresponse tomacroeconomicnews,theincreaseddisagreementtendstodissipatequicklyovertimeas moreforecastersupdatetheirinformationset. Thisfeatureallowsthemodeltocharacterizethe limitedroleofpublicinformationinnormaltimes,butthemodelislimitedincapturingthe increasedandpersistentimportanceofpublicinformationindisagreementduringaneconomic 75

TableF1: DISAGREEMENT IN THE MODELS OF EXPECTATION FORMATION FIRE Sticky Noisyinfo.Noisyinfo.Disagreement Information (Same) (Different)aboutmeans Scopeofdisagreement X ✓ ✓ ✓ ✓ Long-termbeliefs(heterogeneity) X X X X ✓ Changingidiosyncraticdisagreement X X ✓ X X Countercyclicalcommondisagreement X ✓ X ✓ X Forecast-horizondifferences X X X X X downturn.37 Lastly,whileexistingmodelscanaccountforcertainaspectsofourempiricalfindings,they areconstrainedintheirabilitytofullycapturethetime-varyingimportanceofpublicandprivate information,aswellaslong-runbeliefs,inshort-andlong-terminflationforecasts. Notethatthe factorloadingsonthelevelandslopefactorsplayacrucialroleingeneratingdifferentialeffects ofpublicinformationondisagreementacrossforecastinghorizons,whichisthemissingpiece in the literature. The two sets ofloadings suggest a need for two-dimensional heterogeneity inreactionstonews: onerelatedtothelongrunandanotherconcerningthetransitionfrom theneartermtothelongrun. Alltold,ourempiricalfindingssuggestdirectionsforimproving existingmodelsofexpectationformationandhelpingthemcapturetheobservedpatternsof diasgreement. 37It’sworthnotingthatifthefrequencyofeconomicshocksexceedsthefrequencyofinformationupdating,the stickyinformationmodelcanalsogeneratedisagreementatalltimes. 76

Appendix G Discussion of Modeling Choices This section discusses an alternative model and methodology. Section G.1 considers a timevaryingparametermodel. SectionG.2considersanon-parametricapproachasanalternativeof ourparametricmodel. G.1 DynamicFactorModel A.FixedFactorLoadings Inourmodel,thefactorloadingsofeachforecasterarefixed. Apotentialconcernisthatour modelislimitedincapturingthechangingresponsivenessofforecasterstocommoninformation. However,anindividualforecasterstaysinthesampleforonly27quartersonaverage,whichis tooshorttoallowforregimechangesinthefactorloadingsforeachindividual. Notethatourmodelallowseachforecastertohaveuniqueloadings,althoughtheloadings areconstantforaforecaster. Therefore,inourmodel,twosimilarforecastersobservedattwo differentpointsintimehavedifferentloadings,reflectingincreasedordecreasedattentionto potentiallysimilarpolicychanges,forinstance. Ourgoalistoparseouttheportionofcross-sectionalvarianceattributedtocommoninformation. Inotherwords,aslongasthecommoncomponent—theproductofcommonfactor and factor loading—is distinguished from the idiosyncrasy, the distinction of factor loading fromcommonfactorisnotnecessary. Forinstance,ifthecommoncomponentdoesnotchange in spite of an increase in the factor loading, the portion of disagreement driven by common informationdoesnotchangeandhencetheincreaseinthefactorloadingdoesnotmatter. B.StochasticVolatility We do not allow for time-varying variances in the dynamics of the factors. However, by allowing forecaster-specific loadings on the common factor and accounting for forecasters movinginandoutofthesample,themodelcanindirectlycapturethestochasticvolatilityof 77

aggregate inflation projections. As the composition of forecasters changes, these forecasterspecificloadingscaneffectivelyrepresentslow-movingstochasticvolatilityintheaggregate. TheresilienceofthemodelestimatestotheCOVID-19shockisapracticalconcern. Ifthe pandemicobservationsdramaticallyaltertheparameterestimates,thepre-pandemicinference maydramaticallywiththeinclusionofahandfulofpandemicobservations. Inthiscase,includingstochasticvolatilitymayrobustifytheinference,asitdiscountsthepandemicobservations and largely prevents the model from carrying backward the COVID shock for pre-pandemic inference. TocheckhowreliabletheestimatesaretotheCOVIDshock,wecomparethemodel estimatesthrough2019:Q4withthosethrough2023:Q3.Inparticular,theestimatesandthemain conclusionarerobusttotheinclusionofthepandemicobservationsfortheperiodpriortothe COVIDera. Thisobservationsuggeststhatourresultisrobustevenintheabsenceofstochastic volatility. Theresultsareavailableuponrequest. G.2 EvidencefromaNon-ParametricModel Ourbaselinemodelisahighlyparameterizedmodelwithalargenumberofparameters. Since themodelisestimatedwithMCMCsampling,theestimationiscostlyandtime-consuming. One mayarguethatourconclusionsaresensitivetotheparticularparametricassumptionsthatwe imposeandthattheparameterestimatesmaybeunstablebecauseofthesizeofthemodel. Alternatively, we can consider a non-parametric two-step model that is estimable least squares and MLE. The first step is to construct a non-parametric model that describes the individualtermstructureofinflationexpectations. Inthismodel,weuseLegendrepolynomials to fit the short-term (less than 1 year ahead) inflation forecasts and a log function to fit the long-term(morethan1yearahead)inflationforecasts.Wefiteachforecaster’sobservedinflation forecasts in each quarter with least squares. This results in individual level and slope factor estimates. Inthesecondstep,weestimateadynamicfactormodelforeachfactortoparseout commonandidiosyncraticcomponentsusingthealgorithmofBanburaandModugno(2014). Finally,werecoverthefractionsofthetermstructureattributabletolong-termbeliefs,common 78

information,andidiosyncraticinformation. Relativetothebaselinemodel,thealternativemodelislesscostlytoestimate. Inaddition, wecanallowformorethanonecommonfactorforthelevelorslopewithoutmuchadditional effort. However, this convenience comes at a cost. The log function is not flexible enough tocaptureobservedforecastsbeyondoneyearoutandhenceproducesunrealisticlong-end estimates—anoticeabledeclineinthelongend—duringtheCOVID-19pandemic. Thisproblem isnotobservedinourbaselinemodel. Thatsaid,theoverallconclusionsaboutthedriversof disagreementarerobust. Furtherdetailsontheempiricalapproachandtheresultsareavailable uponrequest. 79

Appendix H Model with Curvature and AR(3) Dynamics Inthissection,wepresentsomeresultsfromamoregeneralizedmodelthatincorporatesthe curvature factor and an AR(3) process for factor dynamics. We provide the distribution of smoothedfactors,thedistributionofforecasts,thedispersionandskewnessofforecasts,andthe contributionofpublicinformationtodisagreement.38 Overall,theestimationresultsarevery similartothoseobtainedfromourbaselinetwo-factormodelwithAR(1)dynamics. FigureH17: SMOOTHED FACTOR DISTRIBUTIONS Notes:Thefigureshowsthecross-sectionaldistributionsoftheindividuallevelfactors(upperpanel)andindividual slopefactors(bottompanel).Thesolidbluelineistheposteriormedianofthemedianfactoracrossforecasters.The dashed-dottedlinesdepicttheposteriormediansofthe25thand75thpercentiles.Thedashedlinesdepictthe posteriormediansofthe5thand95thpercentiles.TheshadedareasdenoteNBERrecessions. Sources:Authors’calculation 38Additionalresultsareavailableuponrequest. 80

FigureH18: DISTRIBUTION OF FORECASTS Notes:Thefigureshowsthecross-sectionaldistributionofindividualinflationforecastsatfourdifferentforecast horizons.Thesolidbluelineistheposteriormedianofthemeanforecastacrossforecasters.Thedottedlinesdepict theposteriormediansofthe25thand75thpercentiles.Thedashedlinesdepicttheposteriormediansofthe5th and95thpercentiles.TheshadedareasdenoteNBERrecessions. Sources:Authors’calculation FigureH19: DISAGREEMENT ABOUT AND SKEWNESS OF FORECASTS Notes:Thefigureshowsthestandarddeviationandskewnessofindividualinflationforecastsatfourdifferent forecasthorizons.Thesolidbluelineistheposteriormedianofthedisagreementacrossforecasters.Thesolidred lineistheposteriormedianoftheskewnessacrossforecasters.Thedashedlinesdepicttheposterior5thand95th percentilesofthedisagreementandskewness.TheshadedareasdenoteNBERrecessions. Sources:Authors’calculation 81

FigureH20: FORECAST VARIANCE DECOMPOSITION Notes:Thetoptwopanelsshowthedecompositionofthecross-sectionalvarianceofinflationforecasts(blackline) intothecomponentsdrivenbyindividuallong-termbeliefs(denotedbyltb,blueline),heterogeneousresponsesto publicinformation(denotedbypub,redline),andprivateinformation(denotedbypriv,yellowline).Eachline correspondstotheposteriormedian.Thebottompanelsshowthevarianceshareofpublicinformation,βpub .The h,t solidbluelinecorrespondstotheposteriormedianandthedottedblacklinescorrespondtopointwise95%credible intervals.Theleftandrightcolumnscorrespondto1-and10-yearforecastinghorizonsrespectively.Theshaded areasdenoteNBERrecessions.Thebottompanelsshowthevarianceshareofpublicinformation. Sources:Authors’calculation 82