Inattention in individual expectations

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EconomiA18(2017)40–59

Inattention

in

individual

expectations

夽

Yara

de

Almeida

Campos

Cordeiro

a

,

Wagner

Piazza

Gaglianone

b

,

João

Victor

Issler

a,∗

a_Graduate_School_of_Economics_–_EPGE,_Getulio_Vargas_Foundation,_Brazil b_Research_Department,_Banco_Central_do_Brasil,_Brazil

Received3March2016;accepted9December2016

Availableonline19January2017

Abstract

Thispaperinvestigatestheexpectationsformationprocessofeconomicagentsaboutinflationrate.UsingtheMarket Expec-tationsSystem of CentralBank of Brazil, weperceive that agents do not update theirforecasts every period and that even agentswhoupdatedisagreeintheirpredictions.Wethenfocuson thetwomostpopulartypesofinattentionmodelsthathave been discussedinthe recent literature: sticky-informationand noisy-informationmodels. Estimating ahybrid modelwefind that,although formallyfitting theBrazilian data,it happensatthe costofamuchhigher degreeofinformationrigiditythan observed.

© 2017 The Authors. Production and hosting by Elsevier B.V. on behalf of National Association of Postgraduate Cen-ters in Economics, ANPEC. This is an open access article under the CC BY-NC-ND license (http://creativecommons. org/licenses/by-nc-nd/4.0/).

JELclassiﬁcation: C14;C52;D84;E37

Keywords:Expectations;Inflation;Imperfectinformation;Rationalinattention

Resumo

Este artigo investigaoprocesso deformaçãodeexpectativasde inflaçãodeagentes econômicos.UtilizandooSistemade ExpectativasdeMercadodo BancoCentraldo Brasil, percebemosque osagentes nãoatualizam suasprojeçõesem todosos períodosemesmoaquelesagentesqueofazemdiscordamsobreosvaloresprevistos.Nestesentido,investigamososdoistiposde modelosmaispopularessobreinatençãodiscutidosnaliteraturarecente:informaçãocomrigidezeinformaçãocomruído.Combase

夽 _We_are_very_thankful_to_Valdemar_Rodrigues_de_Pinho_Neto,_Marco_Antonio_César_Bonomo_and_Cecília_Machado_Berriel_for_the_helpful

suggestionsandcomments,totheInvestorRelationsandSpecialStudiesDepartment(Gerin)oftheBancoCentraldoBrasilforkindlyproviding

thedatausedinthispaperandtoPhilippeAndrade,whoprovidedhisGAUSScodestous.WethankalsotheexcellentresearchassistanceofMarcia

WaleriaMachadoandAndreaV.Machado.Theopinionsexpressedinthisarticlearethoseoftheauthorsanddonotnecessarilyreflectthoseofthe

BancoCentraldoBrasil.Anyremainingerrorsareours.

∗_{Corresponding}_author_at:_Graduate_School_of_Economics,_Getulio_Vargas_Foundation,_Praia_de_Botafogo_190,_s.1104,_Rio_de_Janeiro,_RJ

22.253-900,Brazil.

E-mailaddresses:cordeiro.ya@gmail.com(Y.d.A.C.Cordeiro),wagner.gaglianone@bcb.gov.br(W.P.Gaglianone),

joao.issler@fgv.br(J.V.Issler).

PeerreviewunderresponsibilityofNationalAssociationofPostgraduateCentersinEconomics,ANPEC.

http://dx.doi.org/10.1016/j.econ.2016.12.002

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Y.d.A.C.Cordeiroetal./EconomiA18(2017)40–59 41

naestimac¸ãodeummodelohíbrido,concluímosque,emboraformalmenteomodelosejacapazdeseajustaraosdadosbrasileiros, talresultadoocorreaocustodeumgraumuitomaiorderigidezinformacionaldoqueoobservado.

© 2017 The Authors. Production and hosting by Elsevier B.V. on behalf of National Association of Postgraduate Cen-ters in Economics, ANPEC. This is an open access article under the CC BY-NC-ND license (http://creativecommons. org/licenses/by-nc-nd/4.0/).

Palavras-chave: Expectativas;Inflação;Informaçãoimperfeita;Inatençãoracional

1. Introduction

Theexpectationsformationprocessofeconomicagentsaboutmacroeconomicvariableshaslongbeenoneofthe mostdebatedquestionsinmacroeconomics.Neverthelessremainsanopenquestionhowexpectationsareformed,and howbesttomodelit.Inmuchclassicaltheory,thereisnoroomfordisagreementinexpectations,sinceitisusually assumed thatallagentsformexpectations conditionalonacommoninformationset.Howeverifnoteveryone has thesameexpectationsandtheinformationfrictionsarelargeandeconomicallysignificant,thedegreeofinformation rigiditymayhavesignificantimplicationsformacroeconomicdynamicsandoptimalpolicy.

What we aim todo is relatedto the recentempirical work tryingto determinethe nature of the expectations formationprocess.Rationalexpectationsmodels withinformationfrictionssuchas MankiwandReis(2002),Reis (2006a,b),Sims(2003)andWoodford(2003)havebeenassociatedtoagents’inattentiontonewinformation,dueto costsofcollectingandprocessinginformation.Thesemodelshavethekeyadvantageofparsimoniouslyexplaining somepatternsofindividualexpectationsobservedinthedata–suchasdisagreementacrossforecastersandpredictable forecasterrors–thatareconflictingwiththestandardhypothesisofperfectinformation.

Thesticky-informationmodelsproposedbyMankiwandReis(2002)andReis(2006a,b)arebasedontheassumption thattheagentsdonothaveaccesstoinformationinstantly.InMankiwandReis(2002),forinstance,itisassumedthatthe acquisitionofinformationfollowsaPoissonprocessinwhich,ateachdate,agentsfaceagivenandconstantprobability

λof beingabletogetnewinformation. Nevertheless,onceagentsupdatetheirinformationset,theyobtainperfect informationandformexpectations rationally.Thuswe refertoλastheattentiondegreeforthe sticky-information modeland(1−λ)canbeseenasthedegreeofinformationrigidity.

Theinfrequentupdateimpliesthat,eachperiod,onlyafractionoftheagentshasaccesstothelatestmacroeconomic news andthe expectationsandactionsof thosewhodidnot updatetheir informationsetscontinue tobe basedon theiroldinformation.Asaresult,agentswhoupdatedtheirinformationsetsinthesameperiodmustmakethesame forecastsandagentswhodidnothaveaccesstonewinformationshouldnotrevisetheirlastprediction.

On theotherhand,inthenoisy-informationmodelsdevelopedbySims(2003)andWoodford (2003),although agentscontinuouslytrackvariablesandupdatetheirinformationset,theyonlyobservenoisysignalsaboutthetrue state.Asagentsknowtheyhaveanimperfectaccesstothenewstheygetateachperiod,theydonotcompletelypassit ontotheirforecast.Moreprecisely,forecastsareaweightedaverageofthenewandthepreviousinformationreceived, sotheweightonpreviousbeliefsistakenasthedegreeofinformationrigidity.

FollowingAndradeandLeBihan(2013),wewillfocusonthesetwomostpopulartypesofinattentionmodelsthat havebeendiscussedintherecentliterature:sticky-informationandnoisy-informationmodels.Themodel–developed bytheseauthors–isahybridone:itassumesthat,ateachdate,everyforecasterfacesagivenprobabilityofbeing abletoupdatehisinformationsetandthat,whenupdating,hegetsanoisyperceptionofthestateoftheeconomy.The modelisthenestimatedbyaMinimumDistanceEstimation(MDE)procedure,whichallowsustotestifthemodelis capableofquantitativelyfittingthedata,particularlytheforecasterrorsandthedisagreementamongforecasters.

Asfarasweknow,weareoneofthefirstauthorstouseexpectationsdataoninflationinBrazilinordertotryto modelitbasedoninattentionmodels.1Astheresponseofagentstomacroeconomicdynamicsisstronglyimpacted bythewayindividualexpectationsareformed,modelingtheexpectationsformationprocessisimportantforbetter conductof economicpolicyandbetterunderstandingitsimplications.AnadvantageoverAndrade andLeBihan’s

1 _Guillén₍₂₀₀₈₎_investigates_a_set_of_expectations_theories_using_Brazilian_data_and_concludes_that_the_median_inflation_forecast_is_more_likely_to

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(2013)studyisthatforecastsonourdatabaseareobservedinadailyfrequency–theirsareonaquarterlyfrequency. Thismeansweareabletofollowthesequenceofforecastsmadebyeachagentmoreclosely.

Theremainderofthispaperisstructuredasfollows.InSection2wemakeabriefliteraturereviewandinSection3we presentsomebasicfactsaboutforecasterrorsanddisagreementbetweenforecasters.Section4presentsthemethodology weapplyandinSection5weintroducethedatabasewewilluse,namelytheMarketExpectationsSystemofCentral BankofBrazil.Finally,Section6presentstheresultsofourestimationandSection7concludes.

2. Literaturereview

Thesticky-informationPhillipscurveproposedbyMankiwandReis(2002)wasdevelopedinordertoobtainan alternativetothe newKeynesian Phillipscurve– asticky-pricemodel – thatfailedinexplainingsomeaspects of macroeconomicfluctuations.2Inthismodel,ineachperiod,ashareλofthefirmsupdatetheirinformationset3and computeanewpathofoptimalprices.Theonesthatdidnotreceivenewinformationcontinuetosetpricesbasedon oldplans.Afirm’soptimalpriceisgivenby(allvariablesexpressedinlogs):

p∗_t =pt+αyt

wherept isoverallpricelevelandytoutputgap.Afirmthatupdateditsinformationsetjperiodsagosetstheprice

Et−j[p∗t]andaggregatepricelevelistheaverageofthepricessetbyallfirms:

pt=λ

∞

j=0

(1−λ)jEt−j(pt+αyt)

Thusinflationratecanberepresentedby:

πt = αλ 1−λ yt+λ ∞ j=0 (1−λ)jEt−1−j(πt+αyt).

Definingmoneysupply(oraggregatedemand)bym=p+yandtakingittobeexogenous,theyexaminehowoutput andinflationrespondtovariationofmunderthesticky-informationmodel,comparingittothedynamicpropertiesof thesticky-pricemodelandabackward-lookingmodel.Theresponsesofthesticky-informationmodelseemconsistent withwhathappenswheneconomiesareexposedtotheseshocks.

Inthesameline,Reis(2006b)andReis(2006a)studyrespectivelytheproblemofaproducerandtheconsumption decisionsofagentswhofacecostsofacquiringand/orinterpretingnewinformationandtrytounderstandthedynamic responseofpricestoshocksinordertoexplaininflationdynamic.In thissetup,the agentsrationallydecidetobe inattentivetonewinformationandchoosetheoptimallengthofinattentiveness.Butoncetheypayattentiontonew information,theybecomeawareofeverythingthatisrelevant.

Sims(2003),inturn, assumesthat people havelimited information-processing capacity– instead of assuming thatagentsupdatetheir informationsetonlysporadically.Here thelimitedcapacityisrepresentedbythefact that informationthatagentshaveaccessmaybecontaminatedwitharandomnoise.Moreprecisely,informationisthought ofas movingthrough achannel inwhichoneentersinputdata,andoutputdataemerges,possiblywitherror.The authortakesthenatureofthenoiseagentsfaceasexogenousandassumethatnoisechangessystematicallyaccording tochangesinthedynamicpropertiesoftheeconomy.UsingtheideaofafiniteShannoncapacity,4heanalysesthe implicationsofincludingtheseinformation-processingconstraintstothedynamicprogrammingproblemusedtomodel behavior.Theimplications–intermsofalteringpartofitsoutcomes–seeminlinewithobservedmacroeconomic behavior.

2 _For_example:_the_sticky-price_model_cannot_explain_why_inflation_is_so_persistent_or_why_shocks_to_monetary_policy_have_gradual_and_delayed

effectoninflation,andityieldsthatannouncedandcredibledisinflationleadstobooms.

3 _Each_firm_has_the_same_probability_of_being_one_of_the_firms_that_receive_new_information,_regardless_of_how_long_it_has_been_since_its_last_update.

4 _In_information_theory,_the_Shannon_capacity_is_the_maximum_amount_of_information_that_can_be_transmitted_by_a_channel_without_error._It_depends

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Y.d.A.C.Cordeiroetal./EconomiA18(2017)40–59 43 Following the approach proposed by Sims (2003), Woodford (2003), Moscarini (2004) and Mackowiak and Wiederholt(2009)modelinattentivenessbyprice-settersassumingthatagentshavelimitedcapacitytoacquireand/or processalltheinformationintheirenvironment.InMoscarini(2004),agentsalsochoosenottoupdatetheirinformation systematically.Inthissense,themodelisalsorelatedtoReis’(2006a,b).Buthere,oncetheyupdate,theydonotobtain aperfectsignalonthestateoftheeconomy.

Woodford(2003)assumesthateachprice-setteractstakingintoaccounthisownsubjectiveperceptionofthestate ofaggregatedemand,whichismodeledasanobservationofthetruevaluewithanidiosyncraticerror,andthathe formsoptimalestimatesoftheaggregatestatevariablesgiventhisimperfectinformation.Heassumesthattheestimates areupdatedinrealtimeusingaKalmanfilter.Theprice-setter(correctly)believesthattheeconomy’saggregatestate evolvesaccordingto

¯

Xt =¯c+M ¯Xt−1+mut

where¯candmarevectors,Mamatrix,utisassumedtorepresentamonetarypolicyshockand

¯ Xt = ⎡ ⎢ ⎢ ⎣ Xt ∞ k=1 ξ(1−ξ)k−1X(k)_t ⎤ ⎥ ⎥ ⎦

withX(k)t beinghigher-orderexpectations –others’ expectationsof others’expectations...–andξ∈ (0,1)being

ameasureof strategiccomplementarity.∞_k=1ξ(1−ξ)k−1X(k)_t indicatesthat weightedaverageof expectationsand higher-orderexpectationsalsomatterforthedeterminationofprices,outputandoftheirfutureevolution.

Theonlyinformationreceivedbysupplieriinperiodt(theobservationequation)takestheform

zt(i)=e1X¯t+vt(i)

wherevt(i)isamean-zeroGaussianwhitenoiseerrorterm,distributedindependentlybothofthehistoryoffundamental

disturbancesandoftheobservationerrorsofallothersuppliers.ejrefertojthunitvector.

Thus,i’soptimalestimateofthestatevectorevolvesaccordingto: ¯

Xt|t(i)= ¯Xt|t−1(i)+k[zt(i)−e1X¯t|t−1],

wherekisthevectorofKalmangains.Andthedatetstateperceivedatt−1isgivenby ¯

Xt|t−1(i)=¯c+M ¯Xt−1|t−1(i).

Woodfordanalysestheresponsesofinflationandoutputtoamonetaryshockimpliedbyhismodelandcomparesit tothoseimpliedbythesticky-pricemodel.WhentheeffectofchangesonnominalGDPgrowthpresentconsiderable persistence,thenoisy-informationmodelmatchestheempiricalevidencebetter.

Someauthorshavealsoexploitedsurveyofforecastsofmacroeconomicvariablestoproducemicro-factsthat char-acterizetheformationofexpectationsand,then,triedtoassesswhichmodelbestdescribethebehaviorofforecasters.

Mankiwetal.(2004)investigatewhethertheamplitudeofdisagreementobservedinthedataaboutinflation expecta-tions,aswellasitsevolutionovertime,canbepredictedbythesticky-informationmodel.Theyfindthatthisapproach seemscapableofaccountingformanyaspectsoftheobserveddispersionandcentraltendencyduringtheperiodunder study,butitdoesnotconsistentlygenerateenoughtimevariationindisagreement.

Branch(2007)extendedMankiwandReis’s(2002)sticky-informationmodelallowingthedistributionof informa-tionacrossagentstovaryovertimeandfoundthat thismodelprovidesabetterfittothedistributionofthesurvey datausedthanthebasic/staticversiondoes.5PattonandTimmermann(2010),ontheotherhand,focusontherelation ofthesourceofdisagreementanddifferencesinagents’beliefs.Theyexploresurveydatacross-sectionaldispersion inforecasters’predictionsatseveralforecasthorizonsandfindevidencethatdifferencesinopiniondonotstemfrom differencesininformationsets,butfromheterogeneityinpriors.

5 _In_this_sense,_models_that_consider_the_interaction_between_dispersed_and_sticky_information_also_provide_an_alternative_framework_to_the_standard

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BothCoibion and Gorodnichenko(2012a) andCoibion andGorodnichenko (2012b) find evidence against the nulloffullinformationconsistentwiththepresenceofinformationrigidities.CoibionandGorodnichenko(2012b)

useaggregatedsurveydatafromUSprofessionalforecasters, consumers,firms, andcentralbankers toexplorethe conditionalresponse of the averageforecast errorandof the disagreementacrossforecasters tovariousstructural shocks.Theyfindevidenceinfavorofthetwomodels.CoibionandGorodnichenko(2012a)proposeanapproachthat doesnotrequiretheidentificationofshocks.Theydocumentevidencethattheeconomicconditionsaffecthowmuch resourcesaredevotedtothecollectionandprocessingofinformation.Thedegreeofinformationrigidityseemstovary acrossmacroeconomicvariablesandthisvariationgoesinthemannerpredictedbynoisy-informationmodels.

AndradeandLeBihan(2013)andAndradeetal.(2013)alsousesurveydatatoanalyzeinformationrigiditymodels fortheeurozoneandtheUnitedStates.Theyfindevidenceonthedatainfavorofimperfectinformationmodelsand focusonstickyandnoisy-informationmodels.Anadvantageofbothstudiesisthattheyrelyonmultivariatemodels, whichtakeintoaccountthedynamicinteractionsacrossvariableswhenagentsformtheirexpectations.Andradeetal. (2013)useaggregatedsurveydatatostudythetermstructureofdisagreementforUSrealoutputgrowth,consumer priceindexinflationandthefederalfundsrate,encounteringthatdisagreementistimevaryingatallhorizons,including theverylongrunandthatthetermstructureofdisagreementdifferssignificantlyacrossvariables.Theirresultsindicate thatbothsticky-andnoisy-informationmodelsareabletocharacterizethedata.

AndradeandLeBihan(2013)exploitthe paneldimensionof theECBSurveyofProfessionalForecasters.The authors developa hybrid sticky-noisy-information model and testthe empirical performanceof the model. They assumethatthe economycanbesummarized byastate vectorZt made ofp lagsof 3variablesXt –yearonyear

inflationrate,changeinunemploymentrateandrealGDPgrowth–withassociatedinnovationt.Itsdynamicsare

describedby:

Zt=FZt−1+ηt,

whereZt =(XtXt−1 ...Xt−p+1),andηt =(t0...0)hasacovariancematrix η.

Letidenoteanindividualforecaster.Ateachdate,everyimayreceivenewrelevantinformationwithprobability

λ,andwhenupdating,agentsobserveanoisysignal(Yit)ofthetruestate(Zt),thatfollows

Yit=HZt+vit, vit∼i.i.d.(0, v)

whereHisassumedtobeequaltotheidentityandwith vadiagonalmatrix.Besides,becauseeveryperiodonlya

shareofthepopulationupdates,wheneveraggregatingtheforecasts–oranyothermeasure–oneshouldweighby

λ∞_j=0(1−λ)j,asinMankiwandReis(2002),wherejdenotetheforecastersthatlastupdatedint−j.Theauthors detectthatalthoughthefactsobservedfromthedataqualitativelysupportsbothexpectationmodels,whenestimating them,theycannotquantitativelyreplicatethehighpersistenceandvarianceofforecasterrortogetherwiththelowlevel andtime-varianceofdisagreementobservedinthedata.

3. Whyinattentionmodels?

Twoparticular patternsof individualexpectations observedinthe datathat areinconflictwiththe assumption ofperfectinformationandthatcanbereproducedbymodelsof rationalinattentionarerelatedtothedisagreement amongforecastersandthepredictabilityofforecasterrors.Inthissection,wepresentsomebasicfactsfocusingon bothforecasterrorsanddisagreement.

3.1. Forecasterrors

Thepredictabilityofforecasterrorsisapropertythatarisesfrombothsticky-informationandnoisy-information models.Theinfrequentupdateofthesticky-informationmodelimpliesthat,indatet−1,theexpectationsofthosewho didnotupdatetheirinformationsetscontinuetobebasedontheinformationavailableatt−2.Accordingly,theforecast erroratdatetofagentswhodidnotupdatetheirinformationsetatt−1willbepredictable,basedoninformation availableatt−1.Ontheotherhand,inthenoisy-informationmodel,althoughhavingaccesstonewinformationevery period,agentsknowtheinformationtheygeteachperiodiscontaminatedwithnoise.Thustheydonotcompletely passitontotheirforecastsandforecasterrorwillalsobepredictable.

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Fig.1.Timeseriesofrealizationofmonthlyinflation(purpledotted),averageforecasts(purplesolid)anddifference(blue).

Individualforecasterrorisdefinedas:

eit,T =xT −fit,T

wherexTdenotestherealizedinflationatdateTeventandfit,Tindicatestheforecastmadebyindividualiatdatetfor

thedateTevent.

Andtheaverageforecasterrorisgivenby:

et,T =xT −ft,T

where ft,T =(1/nt)

_n_t

i=1fit,T denotetheaverageforecastacrossagents,andntisthenumberofinstitutionstaken

intoaccount.

Fig.1exhibitsthetimeseriesofaverageBrazilianmonthlyinflation(asmeasuredbytheIPCA–TheBroadNational ConsumerPriceIndex)forecastspoolingthecurrentandthenext11monthaheadhorizons,usingmonthlyinflation realizationsfromJanuary2002toDecember2014.Onecaneasilyseethat thedifferencebetweenthetwoseries– inflationrealizationsminusforecasts–presentconsiderablypersistence.Periodsofunder-oroverestimationusually lastforsomeconsecutiveperiods,whichleadstopredictableforecasterrors.Also,thefirst-orderautocorrelationof theaverageforecasterror(e(1))is0.6366.

Allthissuggestpastforecasterrorscontaininformationthathasnotbeenexploitedwhenproducingpresentforecasts. Therefore,the predictabilityof forecast errors derivedfrom bothsticky andnoisy-information models– and also obtainedfromthehybridmodelfeaturingbothtypesofinattention–areobservedinourdata.

3.2. Disagreement

Disagreementamongforecastersisdefinedasthecross-sectionstandarddeviationofforecastsateachdate:

σt,T = (1/nt) nt i=1 (fit,T −ft,T)2

Itemergesfrominformationfrictionsandthefactthattheagentsdonotallhaveacommoninformationset,andcan alsobeexplainedbybothsticky-andnoisy-informationmodels.Fig.2exhibitsthetimeseries(fromJanuary2002to

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Fig.2.Averagedisagreement.

November2015)ofaveragedisagreementacrossforecasterspoolingthecurrentandthenext11monthaheadhorizons. Noticeably,disagreementisalwaysgreaterthanzero,whichisevidenceinfavorofinformationrigidity.

However,thesetwomodelsmakesomedifferentpredictionsaboutdisagreement.First,sticky-informationmodels, asopposedtothebasicnoisy-information–whichassumesconstantnoiseinthesignalandnoheterogeneitybetween forecasters–admittimevariationindisagreement.Sinceinsticky-informationapproachesonlyafractionoftheagents updatetheirinformationset,thesizeoftheshockshittingtheeconomywilldeterminethevariationofdisagreement overtime.Moreprecisely,thedifferencebetweentheforecastsproducedbytheagentswhoupdatedtheirinformation andtheoneswhodidnotwillbemuchlargerunderlargethanundersmallshocks.Ontheotherhand,inbasic noisy-informationmodels,disagreementisnotaffectedbythemagnitudeoftheshocks,becausethedifferentperceptions ofrealityareresponsibleforthedifferenceinforecasts.Itthusdependsonlyonthevarianceofthenoise,whichis assumedtobeconstantovertime.

AlsoinFig.2,wecanseethat,apartfromthedisagreementbeingalwaysnon-zero,itvariesovertime.Therefore, wecanmakeananalysisoftheresponseofdisagreementtothedynamicsoftheeconomytoseeiftheyarecorrelated. Apossiblemeasureforthedynamicoftheeconomy–ortheshockhittingit–isthesquaredvariationinlastperiod inflation,i.e.,(πt−1)2,orthesquaredchangeinaverageforecast,i.e.,(ft)2.So,weregressdisagreementofforecasts

onthesemeasures.TheresultisshowninTable1:botharestatisticallysignificantat1%.Thisindicatesdisagreement

Table1

Disagreementamongforecastersofmonthlyinflation.Mean(σ)standsforthemeanofthedisagreementacross

datesandStd-Dev(σ)foritsstandarderror.(πt−1)2denotesthesquaredvariationinpastinflationand(ft)2

denotesthesquaredchangeinaverageforecast.Standarderrorsareinparentheses.

Disagreement,σ MonthlyIPCA

Descriptivestatistics Mean(σ) 0.091 Std-Dev(σ) 0.039 Regressionsofσon (πt−1)2 0.055***(0.014) (ft)2 0.827***(0.078)

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Fig.3.Averagedisagreementamongrevisers.

respondtoshocks,whichisevidenceinfavorofthesticky-informationmodel.Butthisalsogoesinthemannerpredicted bythehybridmodel.Althoughmaintainingtheconstantnoiseinthesignal,inthehybridapproachwehaveashareof thepopulationthatupdatesitsinformationsetandasharethatdoesnot.Thisisenoughtogeneratetimevariancein disagreement.

Asecondpointofdivergencyregardstheanalysisofthedisagreementamongforecasterswhorevisetheirforecasts. Non-zerodisagreementinthiscaseisevidenceinfavorofthenoisy-informationapproach.Whilethesticky-information modelassumesthatagentswhoreceivenewinformationinthesameperiodhavethesameinformationset–andthus makethesamepredictions,whichshouldindicatezerodisagreementamongrevisers–thenoisy-informationmodel considersthateachagentreceivesnewinformationcontaminatedwithnoise.Therefore,evenagentswhoupdatetheir forecastsconcomitantlymaymakedifferentpredictions.Fig.3plotsdisagreementamongrevisersformonthlyIPCA forecasts.Asonecansee,disagreementisalwaysnon-zerobetween2002and2015–evidenceinfavourofthenoisy. Stillthisisalsopredictedbythehybridmodel.Asagentswhoupdatereceivenewinformationcontaminatedwithnoise, itisunlikelythatallforecastersthatupdatetheirinformationsetinagivendatemakeexactlythesamepredictions. Thisleadsdisagreementamongreviserstobenon-zero.

4. Methodology

4.1. Attention/inattentiondegree

Workingwithahybridmodelimplieswe’llderivetwomeasuresofattention/inattentiondegree,whicharerelated tothedegreeofinformationrigidityfacedbytheeconomy.AsmentionedinSection1,inthesticky-informationmodel theprobabilityofupdatingaforecastbetweentwoconsecutivedatescanbeseenasameasureofagents’attention tonewinformation.So,inordertoobtainourfirstmeasureoftheattentiondegree6–representedbyλ–wewantto estimate

P(fit,T =/ fit−1,T)

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Fig.4.Average(sticky-information)degreeofattention,λ.

Table2

Degreeofattention(λ)–probabilityofupdatingaforecastofmonthlyIPCAinflationforadateTeventbetween

twoconsecutivemonths.

Degreesofattention,λ MonthlyIPCA

Descriptivestatistics

Mean( ˆλ) 0.4999

Std-Dev( ˆλ) 0.0657

wherefit,Tdenotesagenti’sforecastfortheIPCAinflationatdatetforthedateTevent.Assumingthattheprobability

ishomogeneousacrossinstitutionsandacrosshorizons,wecanestablishsomeempiricalcounterpartstothisattention degreefromthesurveydata:

ˆ

λt,T =(1/nt) nt

i=1

I(fit,T =/ fit−1,T)

whereI(·)isaindicatorfunction,thatassumesvalue1whentheindividual’sforecastfordateeventTchangesbetween

t−1andt,and0otherwise.

Inotherwords, ˆλt,T tellsusthepercentageofourpopulation–heretheinstitutionsrespondingthesurvey–that

updatedtheirforecastsandsupposedlyhadaccesstorelevantnewinformationbetweenthedatest−1andt.Thusthe greaterλ,thelowerthedegreeofinformationrigidityfacedbytheeconomy.

Fig.4showstheaverage(acrossinstitutionsandforecasthorizons)attentiondegreeforpredictionsmadeforeach dateTevent,specifiedinthehorizontalaxis.7Ascanbeseen,theattentiondegreeneverreaches1.Itinfactremains alwaysbetween0.3and0.8duringtheperiodunderstudyanditsmean(overtime)–denotedby ˆλanddescribedin

Table2–is0.4999.

7 _Given_the_structure_of_our_database,_to_construct_a_sequence_of_forecasts_made_by_the_same_institution_for_the_same_date_T,_we_pool_the_monthly

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Y.d.A.C.Cordeiroetal./EconomiA18(2017)40–59 49 Oursecondmeasureofinattentiondegreeisobtainedfromthenoisy-informationmodelandisrelatedtothevariance ofthemeasurementerror,i.e.,theerrorassociatedtothesignalaboutthestateoftheeconomyreceivedbytheinstitutions eachperiod–denotedby v.Ittellsushownoisyistheperceptionofthestateoftheeconomy–or howdifferent

fromtheactualinflationrealizationtheforecastcanbe–andthusthegreater v,thegreateristheinformationrigidity

facedbyagents.Thisparameterwillbeexplainedinmoredetailsbelow.

4.2. Themodel

DisagreementamongforecastersandpredictableforecasterrorsobservedinsurveydataanddocumentedinSection3

indicatethatbothsticky-andnoisy-informationmodelsmaydescribeexpectationformationinasatisfactoryway.So, we will implementthe methodologyproposedbyAndrade andLeBihan (2013). Thehybrid modelfeatures both approachesandinordertoassesstheempiricalperformanceofthisexpectationsmodel,itiscomparedtosomedata characteristics throughaMinimumDistanceEstimation(MDE)procedure.Thisprocedureallowstoanalyzeifthe modelcanreplicatetheforecasterrorandthedisagreementobservedinthesurveydata.

Wesketchbelowthe structureof themodel. WeassumethatZt isthestate that summarizesthe economy.The

followingAR(p)describeitsdynamics:

A(L)Zt=wt,

withA(L)=p_k=0ρkLk,andwherewt haszeromeanandvarianceequaltoQ.

AsinMankiwandReis(2002),ateachdate,everyforecasterimayupdatehisinformationset,withagivenand constantprobabilityλ–modeledasaPoisson.Theagentswholastupdatedtheirinformationsetint−jarenamedthe generationj.Howeverwealsoconsideranoisyperceptionofthenewinformationwhenupdating,whichiscaptured bythesignalYitthatfollows

Yit=Zt+vit, vit∼i.i.d.(0, v),

wherevitisaprivateshockand vistheinattentionparameterofthenoisyapproach.

Theestimationprocedureinvolvesthreesteps.First,anAR(p)modelforinflationrate,withthevariablepreviously centered, is estimated.Second, takingthe AR(p) parameteras given, andfor a givensetof structural parameters (λ, v),fourmomentsaresimulatedusingtheKalmanfilterandthehybridmodel.Theselectedmoments–thesame

usedinAndradeandLeBihan(2013)–are:themeansquareofforecasterrors,forecasterrors’firstautocorrelation, disagreements’averagelevelanddisagreements’variance,respectivelydenotedby:E[e2],e(1),Mean[σ]andV[σ].

These momentsare relatedeither toforecast errors or disagreement, two key features that canbe reproduced by imperfectinformationmodelsandthat,inthishybridmodel,arefunctionsofthetwomeasuresofattention/inattention degree–λand v.

Next,thepreviousmodelisestimatedbytheMDEprocedure.Takethestructuralparametersvectorθ0=(λ, v),

thereducedformparametervectorμ0=(E[e2],e(1),Mean[σ],V[σ]),andh:R2→R4acontinuouslydifferentiable

function.TheMDEprocedureconsistsoffirstestimatingμ0byμ,ˆ andthenchoosinganestimatorθofθ0byminimizing thedistancebetweenμˆ andh(θ).Fromnowon,wedenoteh(θ)=μ(λ, v).Wethuscomputethedistancebetween

thevectorofdata-momentsμˆ andthecorrespondingmomentsgeneratedbythemodel,whichareafunctionofthe attention/inattentionparametersμ(λ, v).Theobjectivefunctionisminimizedoverthespaceofparameters.I.e.,one

mustminimize:

[μˆ −μ(λ, v)]ˆ−1[μˆ −μ(λ, v)],

where ˆisaconsistentestimatoroftheasymptoticvarianceofμ,ˆ i.e.,√T(μˆ −μ)→N(0,a )andμ(λ, v)takesthe

form: μ(λ, v)= ⎡ ⎢ ⎢ ⎢ ⎣ E[e2] e(1) Mean[σt] V[σt] ⎤ ⎥ ⎥ ⎥ ⎦= ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ Et{Ej[(ej_t,T) 2 ]} e(1) Et[σt,T] Vt[σt,T] ⎤ ⎥ ⎥ ⎥ ⎥ ⎦

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where8 Ej[(ejt,T) 2 ]=ρ2(T−t)(1−Gt)2λ ∞ j=0 (1−λ)jPt|t−j−1+ρ2(T−t)G2t v+ρ 2(T−t)₋₁ ρ2₋₁ Q , e(1)=

E{[λ∞_j=0(1−λ)j−1]2Ei(Zit|t−j−1)Ei(Zit|t−j−2)}

E{[λ∞_j=0(1−λ)j−1]2Ei(Zit|t−j−1)2} and σt,T = λ∞ j=0 (1−λ)j ρ2(T−t)_{V i(Zit|t−j−1|j)+G2t[Vi(Zit|t−j−1|j)+ v]} +ρT−tEi(Zit|t−j−1)+ρT−tGtZt+ρT−tGtEi(Zit|t−j−1) − ⎛ ⎝λ∞ j=0 (1−λ)j ⎞ ⎠ ρT−t_E i(Zit|t−j−1)+ρT−tGtZt+ρT−tGtEi(Zit|t−j−1)]] 2 } 5. Data

Inordertoascertainthenatureoftheexpectationsformationprocessandtoperformanempiricalassessmentof inattentionmodelswewilltakeprofessionalforecastsontheheadlinerateofinflation(π),IPCA,surveyedbytheMarket ExpectationsSystemdevelopedbytheCentralBankofBrazil.9 Professionalforecastersareparticularlyinteresting becausetheseagentsaresomeofthemostinformedintheeconomy.Asaresult,anyevidenceofinformationrigidity observedintheirexpectationsshouldbeindicativeofexistenceofinformationrigidityfacedbythewholeeconomy.

Since1999,theCentralBankofBrazilconductsadailysurveyofforecastsofthemainBrazilianmacroeconomic variables.Theso-called“Focussurvey”currentlycoversmorethan100institutions,includingcommercialbanks,asset managementfirms,consultingfirms,non-financialinstitutionsandotherlegalentities.Nonetheless,thedatasetusedin thispaperincludesforecastsmadeby254institutions,takingintoaccountthewholesampleperiodandtheinstitutions thatarecurrentlyactiveinthesystemandtheonesthatnolongerare.Wethusdealwithanunbalancedpanel.10

TheCentralBankofBrazilalsomakesavailablemonthlyrankingsincludingthe5bestforecasterswithregardto short,mediumandlongrun.Thisstimulatestheinstitutionstoupdatetheirforecastsregularlyandtoestimatethem withaccuracy,whichmakesthissurveydatamorereliable.Moreover,thesystemonlytakesintoaccountforecasts imputedinthelast30days,whichpreventsthestatisticstobeinfluencedbyforecaststhathaven’tbeenupdated.See

Marques(2013)forfurtherdetails.

Anotherimportantadvantageisthat,whilemostexpectations surveysconductedalloverare aggregated,which impliesoneisnotabletodirectlyobserveupdatesofforecastsinthemicrolevel,intheMarketExpectationsSystem wecantrackthesequenceofresponsesofaparticularsurveyparticipantovertime.11Finally,apartfrombeingableto exploitindividualdata,wehaveaccesstodailyresponses,whatmeanswecancloselyfollowthesequenceofforecasts madebyinstitutions.ThisisanadvantageoverAndradeandLeBihan’s(2013)study,sincetheECBSPFprovides forecastsonlyonaquarterlyfrequency.FortheIPCAinflation,weobserve(inadailybasis)5annualIPCAinflation forecasts,i.e.,forthecurrentcalendaryearandthenext4calendaryears,besidestheforecastforthetwelve-month

8 _See_Appendix_A_for_detailed_calculation.

9 _The_collection_and_manipulation_of_data_from_the_Focus_survey_is_conducted_exclusively_by_the_staff_of_the_Central_Bank_of_Brazil.

10 _The_survey_respondents_are_followed_throughout_time_with_a_reasonable_turnover._As_new_participants_are_often_added_to_the_survey,_and_others

dropout,thepanelofsurveyforecastsisunbalanced.Thus,fromasetof254registeredinstitutionsinthesystem(inoursample),thereisasmaller

activegroupofaround100institutionsthatfrequentlyupdatetheirforecasts.Onaverage,87institutionsaredailysurveyedinourdataset.

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Table3

Informationcriterion–LikelihoodRatio(LR),Akaike(AIC),Hannan-Quinn(HQIC)andSchwartzBayesian(SBIC)informationcriteria.

Selection-ordercriteria

Sample:May2002toDecember2014 Numberofobs=152

Lag LR AIC HQIC SBIC

0 0.961446 0.969527 0.98134

1 109.97* 0.251084* 0.267247* 0.290872*

2 0.14116 0.263313 0.287558 0.322995

3 0.07645 0.275968 0.308295 0.355544

4 0.05561 0.28876 0.329168 0.38823

*denotestheoptimallagineachcolumn.

aheadcumulatedinflationrateandmonthlyIPCAinflationforecastsforthepresentmonthandupto17monthsahead thecurrentdate.

In Brazil,theinflationrate asmeasuredbythe IPCAiscalculatedandreleasedonamonthlyfrequencybythe BrazilianInstituteofGeographyandStatistics(IBGE).Thus,inthisstudy,weusemonthlyIPCAinflationforecasts. TherangeofdatagoesfromJanuary2002toNovember2014.Theaggregationofthedailyintomonthlyresponseis donebyusingthedateofreferenceforIPCA,alsoknownasthe“criticaldate”,whichisthelastbusinessdaybefore theIPCA-15releasedate.12Wesimplytakethelastforecastmadeuntilthedateofreferenceasthemonthlyresponse, becausethisistheforecastusedbytheCentralBankofBraziltoformulatetheTop5rankingsthataremadepublicly available.So,weunderstandtheinstitutionshaveincentivestoupdatetheirforecastsclosetothisdate.

Weassumethattheprobabilityofupdatingaforecastisconstantacrosshorizonsandovertime.Havingestablished that,inordertoconstructasequenceofforecastsmadeforthesamedateTeventbythesameinstitution–sothatwe areabletocomparerevisionsofconsecutiveforecastsforthesameTandsamei–wepoolthemonthlyforecastsmade byeachinstitutionforthecurrentandthenext11months.Wethususuallyhaveasequenceof12forecasts–therefore 11revisions–foreachdateT.13

6. Estimationandresults

Asmentioned,firstwemodeltheinflationprocess.Theautoregressivemodelthatseemstobestdescribetheinflation rateistheAR(1).Thechoiceismadebytheanalysisofautocorrelationandpartialautocorrelationfunctionsandthe Schwartz’sBayesianInformationCriterion(SBIC)–althoughtheresultremainsthesamewithalternativeinformation criteria,seeTable3–consideringupto4lags.

Finally,theBreusch–Godofreytestofautocorrelationoftheresidualsdoesnotrejectthenullofnoautocorrelation (p-value=0.6855)andtheresidualsanalysiscorroboratethischoice.Thegraphicofthetimeseriesoftheresiduals canbeseeninFig.5,aswellastheoriginalIPCAandfittedIPCA.

Havingestablishedtheprocessthatrulesinflation,weproceedtothesimulationofthemoments,namelythemean squareofforecasterrors(E[e2]),forecasterrors’firstautocorrelation(e(1)),averagelevelofdisagreement(Mean[σt])

andvarianceofdisagreement(V[σt]).TheresultsobtainedfromsimulationarepresentedinTable4,Column(1).The

respectivesampledmoments–thatweideallywouldliketomatchwiththesimulatedmoments–arealsospecifiedin

Table4,Column(2).

FortheMDEprocedureweuseadiagonalmatrixasestimatorof(asymptoticcovariancematrix).Thevariance of each momentiscomputed utilizingNewey-Westestimatortoovercome autocorrelation inthe errorterms.The attention/inattentiondegreessimulatedbythehybridmodelare: ˜ v=0.1158and ˜λ=0.2256,whichisverydistant

fromtheλvalueof0.4999observedfromthedata–Table4.Infact,somethingonecanimmediatedrawfromthese

12_The_collecting_period_of_IPCA_is_the_calendar_month,_whereas_for_the_IPCA-15_it_ranges_from_the_16th_of_the_previous_month_to_the_15th_of_the

referencemonth.

13_Except_for_the_first₁₁_and_the_last₁₁_dates_T_sampled_–_i.e.,_from_January₂₀₀₂_to_November₂₀₀₂_and_from_January₂₀₁₅_to_November₂₀₁₅_–

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Fig.5.TimeseriesoftheresidualsafterfittinganAR(1)toinflationprocess.

Table4

Simulatedandsampledattention/inattentiondegrees.λdenotesmean(overtime)probabilityofupdatingaforecastfordateTbetweentwoconsecutive

monthsand vstandsforthevarianceofthenoiseinthesignal.Firstcolumnspecifiesthemeasuressimulatedbythehybridmodelandsecond

columnthemeasureofλobtainedfromthedata,usingupto12monthsaheadforecasts.

Attention/inattentiondegrees

Simulated Sampled

λ 0.2256 0.4999

v 0.1158

Testofoveridentifyingrestrictions χ2

2statistic 2.3343

***Significantat1%;**Significantat5%;*Significantat10%.

numbersisthatinordertominimizethedistancebetweentheselectedmoments,themodelgeneratesamuchhigher degreeofinformationrigiditythantheobservedinthedata.

Using the MDE procedure, we can also perform a test of over-identifying restrictions – which null is that the parameters to be estimated (λ and v) can accurately describe the 4 selected moments, or alternatively,

that the distance between the simulated moments and the observed ones is zero. The test statistic has asymp-toticChi-squaredistributionwith2 degrees of freedom,χ22.Our statisticequals2.3343 (p-value=0.3113), which leadsus to not reject the null evenunder the statistical significance level of 10%. Although the over-identifying restrictions are not rejected by the data, as mentioned, that happens at the cost of a much lower frequency of updating(λ).14

Nowweanalyzehowthemomentsrespondtosomealternativecombinationsofλand v.Thisexerciseispresented

inColumns3and4ofTable5.Oncewefixλatthevalueobservedfromdata( ˆλ =0.4999)and vatthevaluegenerated

bythehybridmodel( ˜ v=0.1158)–Column3–thetwomomentsrelatedtotheforecasterrorandtime-variance

ofdisagreementbecome lowerandholdofffurtherfromthedata,andthemeanofdisagreementgetshigher.Ifon theotherhandwefixλat0.4999andraise vto0.9for example–Column4–whencomparedtoColumn3,we

observearaiseinthe firsttwomoments,relatedtoforecasterror, whilebothmomentsassociated todisagreement fall.

14 _Looking_for_a_local_minimum,_we_could_restrict_the_interval_for_possible_values_of_λ_around_the_value_encountered_in_the_data._{Nevertheless,}

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Table5

Moments.E[e2_]_denotes_mean_square_of_forecast_errors,

e(1)forecasterrors’firstautocorrelation,Mean[σt]theaveragelevelofdisagreementand

V[σt]itstimevariance.Firstcolumnspecifiesthemomentssimulatedbythehybridmodelandsecondcolumnthemomentsobtainedfromthedata,

usingupto12monthsaheadforecasts.Columns3to5specifytheresultsunderalternativeconfigurations.

Moments (1) (2) (3) (4) (5) Simulated λ= ˜λ Data λ=0.4999 λ=0.4999 λ=0.4999 v= ˜ v v= ˜ v v=0.9 vfree(=0.4596) E[e2_] _0.0685 _0.1267 _0.0536 _0.0757 _0.0686 e(1) 0.6271 0.6366 0.5919 0.6449 0.6280 Mean[σt] 0.0912 0.0912 0.1150 0.0752 0.0908 V[σt] 0.000414 0.0015 0.000154 0.01549×10−3 0.04305×10−3

Thesepatternsarewhatwould beexpectedfromthemodel.Accordingtothemodelanincreasein v (i.e.,an

increaseinthevarianceofthenoiseassociatedtothesignalreceivedbytheagents)oradecreaseinλ(i.e.,adecreasein theprobabilityofupdatingaforecastbetweentwoconsecutivemonths)arebothindicativeofanincreaseininformation rigidity.Itshouldthusraisebothvarianceandpersistenceofforecasterrors.Theeffectonaveragedisagreement,onthe otherhand,isambiguous.Finally,anincreasein vshoulddecreasetimevarianceofdisagreement,whileadecrease

inλshouldraiseit–seeAppendixAformoredetails.Theoppositeshouldhappenwhenwefaceanincreaseinλor adecreasein v.

Finally, in Column 5 of Table 5 we fix λ at the value observed from the data ( ˆλ =0.4999) and let v

free to be simulated by the model. We obtain v=0.4596. When compared to the case where all

parame-ters are freely simulated (Column 1), the two moments related to forecast errors and the mean of disagreement do not change too much. Nevertheless the variance of disagreement gets even lower, holding off from the data value.

Theresultsindicatethat,despiteoffittingthepersistenceofforecasterrorsandthelevelofdisagreementrelatively well,themodelcannotaccountforbothhighvarianceofforecasterrorsandofdisagreement.Infact,wheneverwefit sampledvarianceofdisagreement,forexample,wemustset vatamuchlowerlevel.Butthisinturnleadstomuch

higheraveragedisagreementthanobservedandevenlowervarianceofforecasterrors.Ontheotherhand,togetcloser tothevarianceofforecasterrorsweneedalowerλandamuchhigher v.Neverthelessitleadstohigherpersistence

offorecasterrorsandmuchlowerlevelandvarianceofdisagreement.

7. Conclusion

In thisstudywe usedthe MarketExpectations Systemof CentralBankofBraziltoanalyzehowgoodcanthe expectationsformationprocessofinflationforBraziliandatabemodeledbythehybridmodelfeaturingboth sticky-informationandnoisy-informationmodels.Themodelisbasedontheassumptionthattheagentsdonothaveaccessto informationinstantly/everyperiod,andthatwhenagentsupdatetheirinformationset,theyonlyobservenoisysignals aboutthetruestateandformexpectationsrationally.

Utilizing a Minimum Distance Estimation (MDE) procedure, we were able to test if the model is capable of quantitatively fitting the Brazilian data. Estimating the model we encounter that formally the model fits the data. Nevertheless, it happens at the cost of a much higher degree of information rigidity than observed in the data.

Infutureresearch,onepossiblewayofimprovingthefitofthemodeltothedatamaybeallowingthedegreeof attentionofthesticky-informationapproach(λ)tovaryacrossinstitutions–somethingthatisobservedinthedata.Or onecouldtrytoincludethestickinessasinReis(2006a,b).Thatis,theforecasterexplicitlychoosestobeinattentive, solvingaoptimizationproblemtodecidewhentoupdatehisinformationset–insteadofbeingsubjecttoanexogenous constantprobabilityofbeingabletogetnewinformation.Otherpossibleroutesaretotestdifferentmodels,following theapproachofCoibionandGorodnichenko(2012b),ortoconsiderdispersedinformationwithstrategicinteractions, inlinewiththemethodologyofMorrisandShin(2002).

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AppendixA. Themodel

Thestatespacerepresentationisgivenby:

Yit=Zt+vit

Zt =ρZt−1+wt

(A.1)

wherevit=it+ηt∼i.i.d.(0, v)andwt∼i.i.d.(0,Q).

Thefirst(A.1)equationiscalledtheobservationequationandthesecondthestateequation. Thedatetstateoftheeconomyperceivedbyagentiatdatet−1is

Zi,t|t−1=E[Zt|Ii,t−1] (A.2)

whereIi,t−1=(Yi,t−1 ,Yi,t−2 ,...,Yi,1 )and

Pt|t−1=E[(Zt−Zi,t|t−1)(Zt−Zi,t|t−1)] (A.3)

Besides,

Yi,t|t−1=E[Yi,t|Ii,t−1] (A.4)

E[Zt]=ρE[Zt−1] (A.5)

E[Yi,t|Zt]=Zt (A.6)

Thus,

Yi,t|t−1=E[Yi,t|Ii,t−1]=E[E(Yi,t|Zt)|Ii,t−1]=E[Zt|Ii,t−1]A.2=Zi,t|t−1 (A.7)

TheoptimalforecastfordateT15eventmadebygenerations0andj–i.e.,theagentsthatupdatedtheirinformation setinthecurrentperiodandtheoneswhoupdatedjperiodsago–isrespectivelygivenby:

fit,T =E[ZT|Zi,t|t]=ρT−tZi,t|t (A.8)

fit−j,T =E[ZT|Zi,t−j|t−j]=ρT−tZi,t|t−j (A.9)

From(A.1)and(A.7)

Yi,t−Yi,t|t−1=Zt+vit−Zi,t|t−1=(Zt−Zi,t|t−1)+vit (A.10)

thus

E[(Yi,t−Yi,t|t−1)(Yi,t−Yi,t|t−1)]

=E[[(Zt−Zi,t|t−1)+vit][(Zt−Zi,t|t−1)+vit]]

=E[(Zt−Zi,t|t−1)(Zt−Zi,t|t−1)]+E[vitvit]=Pt|t−1+ v (A.11)

wherethesecondequalitycomesfromvitandZtbeingnon-correlated.

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Y.d.A.C.Cordeiroetal./EconomiA18(2017)40–59 55 TheKalmanFiltersolutiontothissignalextractionproblemisgivenby:

Zi,t|t =Zi,t|t−1+Git(Yit−Yi,t|t−1) (A.12)

whereGitistheKalmangain.Weassumeittobehomogeneousacrossagents:G_it=G_t =P_t|t−1(P_t|t−1+ _v).

Usingthissolution,wecanrewritetheoptimalforecastas:

fit,TA.8= ρT−tZi,t|t=ρT−tZi,t|t−1+ρT−tGt(Yit−Yi,t|t−1) (A.13)

Theaverageforecastwithinagenerationjis:

Ei[fit−j,T|j] =ρT−tEi[Zi,t|t−j−1]+ρT−tGtEi

×[Zt−Zi,t|t−j−1+vit]A.1= ρT−t+1Ei[Zi,t−1|t−j−1]+ρT−tGt[Zt−ρEi(Zi,t−1|t−j−1)] (A.14)

andtheforecasterrorwithinagenerationjis:

ej_t,T =ZT −Ei[fit−j,T|j]=ρT−tZt+ T−t

i=1

ρT−t−iwt+i−{ρT−tEi[Zi,t|t−j−1]+ρT−tGt[Zt−Ei(Zi,t|t−j−1)]}

=ρT−t(1−Gt){Zt−Ei[Zi,t|t−j−1]}+ T−t

i=1

ρT−t−iwt+i (A.15)

Theaverageforecastacrossgenerationsfollows:

Ej{Ei[fit−j,T|j]}=λ

∞

j=0

(1−λ)jEi[fit−j,T] (A.16)

andtheaverageforecasterrorisgivenby:

et,T =Ej[ejt,T]=ZT −Ej{Ei[fit−j,T|j]}=ρT−tZt+ T−t i=1 ρT−t−iwt+i−λ ∞ j=0 (1−λ)jEi[fit−j,T] =ρT−tZt+ T−t i=1 ρT−t−iwt+i−λ ∞ j=0 (1−λ)j{ρT−tEi[Zi,t|t−j−1]+ρT−tGt[Zt−Ei[Zi,t|t−j−1]} =ρT−t(1−Gt)λ ∞ j=0 (1−λ)j[Zt−Ei(Zit|t−j−1)]+ T−t i=1

ρT−t−iwt+i (A.17)

Wenowturntothe disagreementgenerated bythishybrid-model.Our measureofdisagreement– presentedin Section3.2–isthecross-sectionstandarddeviationateachdate.So,wederivethecorrespondingformulaobtained bythetheoreticalmodel.Inordertomakecalculationssimpler,sowedonothavetodealwithsquaredroots,wewill herelookatthecross-sectionvariance.

Notethatthetotalcross-sectionvarianceofpointforecastsacrossindividualsiindifferentgenerationsjcanbe decomposedintwoelements:firstthedifferencesofperceptionsofthestateoftheeconomywithinagivengeneration, weightedbyitsrelativeshareinthetotalpopulationandsecondthedifferencesinaverageperceptionofthestateof

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theeconomybetweendifferentgenerations:

Vij(fit−j,T) = Ej[Vi(fit−j,T|j)]+Vj[Ei(fit−j,T|j)]

= λ ∞ j=0 (1−λ)jVi(fit−j,T|j)+λ ∞ j=0 (1−λ)j{Ei(fit−j,T|j)−Ej[Ei(fit−j,T|j)]}2 A.13_{= λ}∞ j=0

(1−λ)j Vi[(ρT−tZit|t−j−1+ρT−tGt(Zt−Zit|t−j−1+vit))|j]+ {Ei(fit−j,T|j)

−Ej[Ei(fit−j,T|j)]}2 = λ ∞ j=0 (1−λ)j ⎧ ⎨ ⎩ρ2(T−t){Vi(Zit|t−j−1|j)+G2t[Vi(Zit|t−j−1|j)+ v]} + ρT−tEi(Zit|t−j−1)+ρT−tGtZt+ρT−tGtEi(Zit|t−j−1) − ⎛ ⎝λ∞ j=0 (1−λ)j ⎞ ⎠ ρT−t_E i(Zit|t−j−1)+ρT−tGtZt+ρT−tGtEi(Zit|t−j−1) ⎤_⎦2 ⎫ ⎪ ⎬ ⎪ ⎭ (A.18) So,disagreementgeneratedbythemodelisgivenby:

σt,T = Vij(fit−j,T) (A.19)

A.1. Selectedmoments

Basedonthepreviousequation,letsderivethefourmoments-themeansquareofforecasterrors,forecasterrors’ firstautocorrelation,disagreements’averagelevelanddisagreements’variance–usedintheprocedure.

First,wederivethemeanofsquaredforecasterror. Ej[(ejt,T) 2 ]=Ej{Ei[(ZT −(fit−j,T|j))2]} =Ej{Ei{[(ρT−t(1−Gt)(Zt−Zit|t−j−1))−ρT−tGtvit] 2 }+Ei _T_−t i=1 (ρT−t−iwt+i) 2 ! } =Ej{Ei[ρ2(T−t)(1−Gt)2(Zt−Zit|t−j−1)2]+ρ2(T−t)G2tEi[v2it]+ _T_−t i=1 ρ2(T−t−i) ! Ei[w2t+i]} =ρ2(T−t)(1−Gt)2Ej{Ei[(Zt−Zit|t−j−1)2]}+ρ2(T−t)G2t v+ρ 2(T−t)₋₁ ρ2₋₁ Q =ρ2(T−t)(1−Gt)2Ej[Pt|t−j−1]+ρ2(T−t)G2t v+ ρ2(T−t)−1 ρ2₋₁ Q =ρ2(T−t)(1−Gt)2λ ∞ j=0 (1−λ)jPt|t−j−1+ρ2(T−t)G2t v+ρ 2(T−t)₋₁ ρ2₋₁ Q (A.20)

Ourfirstmomentisthemean(overtime)ofmeansquaredforecasterror,i.e.,

E[e2]=Et{Ej[(ejt,T)

2

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Y.d.A.C.Cordeiroetal./EconomiA18(2017)40–59 57 Theforecasterrors’ﬁrstautocorrelationisdetailedbelow.

e(1)= γ j 1 V(ej_t,T) = E[(ej_t,T −et,T)(ejt−1,T−et−1,T)] V(ej_t,T) (A.22) where γ1j=E{[ρT−t(1−Gt)(Zt−Ei(Zit|t−j−1))+ T−t i=1 ρT−t−iwt+i−ρT−t(1−Gt)λ ∞ j=0 (1−λ)j[Zt−Ei(Zit|t−j−1)] −T−t i=1

ρT−t−iwt+i]× [ρT−t(1−Gt)(Zt−Ei(Zit|t−j−2))

+T−t i=1 ρT−t−iwt+i−ρT−t(1−Gt)λ ∞ j=0 (1−λ)j[Zt−Ei(Zit|t−j−2)]− T−t i=1 ρT−t−iwt+i]} =ρ2(T−t)(1−Gt)2E{(λ ∞ j=0 (1−λ)j−1) 2

Ei(Zit|t−j−1)Ei(Zit|t−j−2)} (A.23)

and V(ej_t,T)=E{[ρT−t(1−Gt)(λ ∞ j=0 (1−λ)j−1)Ei(Zit|t−j−1)] 2 } (A.24) Thus e(1)=

E{[λ∞_j=0(1−λ)j−1]2Ei(Zit|t−j−1)Ei(Zit|t−j−2)}

E{[λ∞_j=0(1−λ)j−1]2Ei(Zit|t−j−1)2}

(A.25) Theaveragedisagreementis:

Mean[σt]=Et[σt,T] (A.26)

anddisagreement’stimevariance:

V[σt]=Vt[σt,T] (A.27)

whereσt,Tisgivenby(A.19).

A.2. Selectedmomentsandthedegreeofinformationrigidity

Weshallnowtakealookathowthesemomentsrelatetochangesinthedegreesofattention/inattention.Recallλ

istheprobabilityofupdatingaforecastafteroneperiodofnewinformation.Thusλisinverselyproportionaltothe degreeofinformationrigidity. v,ontheotherhand,describeshowvolatileisthenoiseoftheperceptionofthestate

oftheeconomy.Thisimpliesthatthegreater v,thehigherthedegreeofinformationrigidity.

Supposeanincreaseinλoradecreasein v,whicharebothequivalenttoadecreaseininformationrigidity.This

meansagentshaveaccesstomoreaccurateinformationandshouldconsistentlymakebetterpredictions.Everything elseremainingconstant,wewillfaceadecreaseinthevarianceof theforecasterrorsandthesamehappenstoits persistence.Itmeanstheresponseofthefirsttwomomentsgoesinthesamedirectionasthechangeininformation rigidity.

Withregardtothetimevarianceofdisagreement,anincreaseinλdecreasesthedifferenceofinformationbetween generations.Thusthearrivalofanewinformationsetthatisobservedonlybygenerationj=0andthatcancontain alotoforjustafewrelevantinformationunnotedbythepreviousgenerations–whichcanbeseenasthesizeofa shockhittingtheeconomyonthatdate–willhavealowerimpactonthemagnitudeofdisagreementandthatleads

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tolowertimevarianceofdisagreement.Ontheotherhand,adecreasein vdecreasestheamountofnoiseaboutthe

stateoftheeconomyfacedbyagents.Thisimpliesnewinformationismoreinformativeandthusmoreweightisgiven tonewinformationwhenmakingaforecast.Thusitcanalsorisetimevarianceofdisagreement,sinceitwouldleadto disagreementbeinghigherwhenmorerelevantinformation(oralargershock)arisesandtolessdisagreementwhen fewrelevantinformation(orasmallershock)arises.

Finally,withrespecttotheresponseoftheaveragedisagreement,wehavetwoconcurrenteffectsforbothdegreesof (in)attention.Anincreaseinλ,forexample,decreasesthelengthbetweeninformationupdating–becausetheprobability toreceivenewinformationfacedbyeachagenteachperiodisnowgreater-,whichdecreasestheheterogeneitybetween forecastsmadebydifferentgenerationsandshouldimplyadecreaseindisagreement.Butitalsodecreasestheshareof generationsbasingtheirpredictiononoldinformation.Withinthegenerationsthathaven’tupdatedtheirinformation setfortoolong,theoptimalforecastissimplytheunconditionalmeanoftheinflationprocessanddisagreementtends tozero.Thusadecreaseintheshareofthesesgenerationscontributestoincreasedisagreement.

Considernowadecreasein v.Ononehand,itdecreasesthedifferentperceptionswithineachgenerationof

fore-casters(viathefallintheamountofnoisefacedbyagentsaboutthestateoftheeconomy)anddecreasesdisagreement. Ontheotherhand,adecreasein vimpliesagentstendtoincorporatemoreofthenewinformationintotheirforecast,

becausethesignalismoreprecise.Thisleadstoanincreaseinaveragedisagreement.

Therefore,whiletheeffectofachangeinthedegreeofinformationrigidityonthemomentsrelatedtoforecasterrors canbeeasilyanticipated,thedirectionofthemomentsrelatedtodisagreementdependonwhichforceswillprevail.

TableA.1summarizesit:

TableA.1

Responseofthemomentstochangesintheattention/inattentiondegrees.λstandsfortheprobabilityofupdatingaforecastbetweentwoconsecutive

monthsand vforthevarianceofthenoiseinthesignal.E[e2]denotesmeansquareofforecasterrors,e(1)forecasterrors’firstautocorrelation,

Mean(σt)theaveragelevelofdisagreementandV(σt)itstimevariance.

Moments

DecreaseinInformationRigidity

Increaseinλ Decreasein v

E[e2_] _↓ _↓

e(1) ↓ ↓

Mean[σt] ambiguous ambiguous

V[σt] ↓ ↑

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