Imperfect Information and Slow Recoveries in the Labor Market

Board of Governors of the Federal Reserve System International Finance Discussion Papers ISSN 1073-2500 (Print) ISSN 2767-4509 (Online) Number 1423 September 2025 Imperfect Information and Slow Recoveries in the Labor Market Anushka Mitra Please cite this paper as: Mitra, Anushka (2025). “Imperfect Information and Slow Recoveries in the Labor Market,” International Finance Discussion Papers 1423. Washington: Board of Governors of the Federal Reserve System, https://doi.org/10.17016/IFDP.2025.1423. NOTE: International Finance Discussion Papers (IFDPs) 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 International Finance Discussion Papers Series (other than acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers. Recent IFDPs are available on the Web at www.federalreserve.gov/pubs/ifdp/. This paper can be downloaded without charge from the Social Science Research Network electronic library at www.ssrn.com.

Imperfect Information and Slow Recoveries in the Labor Market AnushkaMitra* FederalReserveBoard Abstract Abstract:Theunemploymentrateremainselevatedlongafterrecessions,apersistencethatstandard search-and-matchingmodelscannotexplain.Ishowthatnoiseshocks—expectationalerrorsduetothe noiseinreceivedsignalsaboutaggregateshocks—accountformuchofthissluggishness.Usingastructural VAR,Ifindthatabsentnoiseshocksunemploymentwouldhaverecoveredtoitspre-recessionlevelsix quartersearlierover1968–2019.Tointerpretthisevidence,Idevelopasearch-and-matchingmodelwithonthe-jobsearch,endogenoussearcheffort,andwagerigidity.Embeddingimperfectinformationgenerates twochannelsofpersistence: slowlearningamplifiestheeffectsofpersistentproductivityshocks,and noiseshocksprovideanadditionalsourceofsluggishness,furthermagnifiedbystickywagesandvacancy posting. Themodelsuccessfullyreplicatesboththeslowrecoveryofunemploymentandsystematic forecasterrors,highlightingimperfectinformationasakeymechanismbehindpost-recessionlabor marketdynamics. Keywords:ImperfectInformation,LaborMarket,BusinessCycles JELCodes:E24,E32,E70 *anushka.mitra@frb.gov.IamdeeplygratefultoAyşegülŞahin,AndreasMuellerandOlivierCoibion,fortheirconstantguidance andsupport. IremainindebtedtoSarojBhattarai,AndresDrenik,StefanoEusepiandChristopherHuckfeldtfortheirinsightful discussions.IthankGadiBarlevy,JasonFaberman,BartHobijn,andGiorgioTopafortheirfeedbackandinsights.Ifurtherthank SadhikaBagga,RobertBarsky,ChrisBoehm,JoelDavid,KeshavDogra,FilippoFerroni,FrancoisGourio,GizemKosar,NiklasKroner, DonggyuLee,ThomasLubik,DavideMelcangi,LeonardoMelosi,NityaPandalai-Nayar,JaneRyngaert,PierreSarte,AndreaTambalotti, AntonellaTrigari,SuryanshUpamanyu,MarceloVeracierto,AlessandroVilla,DongchenZhaoandconferenceandseminarparticipants atSED2025,EUSaM2025,UTAustin,ChicagoFedandNewYorkFedforhelpfulcommentsandsuggestions.Theviewsexpressedhere aresolelytheresponsibilityoftheauthorandshouldnotbeinterpretedasreflectingtheviewoftheBoardofGovernorsoftheFederal ReserveSystemorofanyotherpersonassociatedwiththeFederalReserveSystem.

1 Introduction Oneofthelong-standingchallengesforconventionalbusinesscyclemodelshasbeentomatchthe persistenceintherecoveryofthelabormarket,especiallyoftheunemploymentratefollowing recessions. Inthispaper,Idocumentanewstylizedfact: professionalforecasterssystematically overestimatethetimethatittakesunemploymenttorecoverfromarecession,suggestingthat ex-antetheyexpecttherecoveriestobeevenlongerthanobserved. Thissystematicoverestimationsuggeststhepresenceofinformationfrictions,whereagentsmisperceivethepersistenceof aggregateshocksandrelyonnoisysignalstoformexpectations. Thepresenceofsuchfrictions offerapotentialmechanismtoslowjobcreationandsearchactivity,therebygeneratingapositive feedback cycle that lead to slower unemployment recoveries. To investigate such a channel, I estimatenoiseshocks,persistentTFPshocks,andtransitoryTFPshocksfromatri-variateVAR.I findthatthenoiseshocksplayanessentialroleinexplainingthepersistenceofthelabormarket. I thenincorporatethenoiseshocksintoamodeloflaborsearchwithimperfectinformation,and confirmthatthepresenceofimperfectinformationiskeyforthemodel’ssuccessinmatchingthe sluggishnessoftheunemploymentrateduringrecoveries. I begin by presenting a set of empirical facts using data on the unemployment rate and forecastsfromtheSurveyofProfessionalForecasters(SPF).Ittakesbetween5and16quarters fortheunemploymentrateintheUnitedStates,torecoverhalfofitsrecessionaryincrease,and morethan20quarterstoreturntoitspre-recessionlevel. UsingforecastdatafromtheSPF,Inext documentthatprofessionalforecastersareevenmorepessimisticabouttherecoverythanthe realizeddatasuggest. Thereisaconsistentwedgebetweenexpectedandactualunemployment ratesacrossrecessions,withforecastspredictingevenslowerrecoveries. Thispatternsuggests thatforecasters,likeagentsintheeconomy,maylackperfectinformationabouttheaggregatestate andinsteadrelyonnoisysignalstoformexpectationsaboutwhetherchangesinfundamentals arepersistentortransitory. Suchmisperceptionscaninfluencekeyeconomicdecisions,suchas hiringandinvestment,therebycontributingtoasluggishlabormarketrecovery. Tounderstandthesefacts,IbuildupontheidentificationstrategyfromChahrour,Nimark, andPitschner,2021andEnders,Kleemann,andMüller,2021,toestimatenoiseshocks(shocks thatarisefromchangesinexpectationswithoutanychangesinfundamentals), persistentTFP shocks,andtransitoryTFPshocksfromaVARinutilization-adjustedTFP,realGDPgrowth,and nowcasterrorsofprofessionalforecasters. Thenoiseshocksareidentifiedusingsignrestrictions with the assumption that the noise shocks, by definition, must affect expectations more than thefundamentals. Ifurtherrefinethisstrategybyimposingazerorestrictionthatrulesoutany contemporaneouseffectofnoiseshocksonactualTFP.Additionally,todistinguishpersistentfrom transitoryTFPshocks,Iidentifythepersistentcomponentastheshockthatmaximizesthelong-run forecasterrorvarianceofTFP.Idocumenttwokeyfindingsfromthisexercise. First,labormarket variables,includingunemployment,vacancies,andjob-findingrates,respondsignificantlytothe estimatednoiseshocks,whichaccountforroughlyone-thirdoftheirvariationatbusinesscycle frequencies. Second,intheabsenceofnoiseshocks,unemploymentwouldhaverecoveredtoits pre-recessionlevelanaverageofsixquartersearlier.1 Thesefindingssuggestacriticalrolefor 1Whileittakesonaverage17quartersforunemploymentratetorecover50%ofitsrecessionaryrisesincethe 1

thenoiseshocksintherecoveryofthelabormarketandindicatethepresenceofinformation frictions.2 To study the role of noise shocks in driving labor market persistence, I embed imperfect information(Lorenzoni,2009)intoamodelofequilibriumunemploymentfeaturingendogenous searcheffort,on-the-jobsearch,andstickywagesàlaGertler,Huckfeldt,andTrigari,2020.3 The modelincludestwofundamentalproductivityshocks,transitoryandpersistent,butagentscannot distinguishbetweenthem. Instead,theyobserveaggregateproductivityandreceiveanoisysignal aboutitspersistentcomponent,whichtheyusetoformbeliefsaboutfutureproductivity. Noise shocksaddcomplexitybymakingitharderforagentstodiscernwhetherobservedchangesreflect truepersistentshocksorsimplynoiseinthesignal. Thecalibratedmodelsuccessfullyaccounts forthesluggishlabormarketrecovery: relativetoamodelwithfullinformation,themodelwith imperfect information generates an additional six quarters (30%) of elevated unemployment followingrecessions. Therearetwochannelsforthesuccessofthemodel. First,learningunderimperfectinformationgeneratespersistenceendogenously: ittakestimeforagentstolearnwhetherashockis persistentornot,leadingtoaninitialunder-reaction.Wagestickinessfurtherdelaysthisdynamics, asinfrequentrenegotiationmakesagentshesitanttorevisewageswithoutfullyknowingtheshock’s persistence. Thisinteractionbetweenimperfectinformationandwagerigidityisnovelandcentral tothepersistencemechanism. Second,noiseshocksthemselvesmayprolongrecoveries,asagents mistakethemforactualproductivityshocksandrespondtothemaccordingly,eventhoughthe fundamentalsareunchanged.Together,thesemechanismssuccessfullyaccountforthepersistence intheunemploymentdynamicsfollowingdownturns. Further, the model successfully replicates the systematic overestimation of medium-run unemploymentbyprofessionalforecastersobservedinthedata. Thisovershootingarisesendogenouslyfromagents’misperceptionsdrivenbynoiseshocksandcannotbereplicatedinmodels withonlystructuralproductivityshocks,which,bydefinition,implythatactualoutcomesexceed expectationsandthustendtogenerateforecastunder-reaction,awell-knownlimitationofstandard macroeconomic models. This ability to match both the persistence in unemployment andthe forecasterrorsconstitutesnotonlyasuccessforthemodelbutisalsoakeyempiricalvalidationof theimperfectinformationchannel. Thispaperbuildsontheliteraturedocumentingthepersistenceofunemploymentfollowing recessionsandthedifficultyconventionalbusinesscyclemodelsfaceinexplainingthisfact,as documentedbyColeandRogerson(1999),andmorerecentlybyHallandKudlyak(2022)andFerraro (2023). Slowlabormarketrecoverieshavebeenaconsistentfeatureofpostwarrecessions, yet beginningoftherecession, itwouldbe11quartersintheabsenceofnoiseshocks. Noiseshocksalsodampened job-findingratesandvacancies. 2Iffirmsandworkershadperfectinformationabouttheshockbeingnoise,noiseshockswouldnotexistand,it wouldnotbeoptimalforagentstorespondtoit. However,initiallyagentsmisperceivethenoiseshockasanactual negativeproductivityshockandhencefirmsdecreasetheirhiring.Asaresult,therearefewerjobopportunitiesfor workersandjob-findingratedecreases.Thiscontributestoanincreaseinunemployment. 3Pissarides(2009)critiquesmodelsthatrelyonstickywagesfornewhires.Severalpapersprovideresponsestothis critique:Gertler,Huckfeldt,andTrigari(2020)showthatstaggeredNashbargainingcanreconcilewagestickinesswith dataonunemploymentfluctuations;Grigsby,Hurst,andYildirmaz(2021)andHazellandTaska(forthcoming)present empiricalevidenceofwagerigidityinadministrativeandvacancy-leveldata.AnalternativeviewisofferedbyKudlyak (2014),whoemphasizesthecyclicalityoftheusercostoflaborratherthanwagesdirectly. 2

thereremainsnounifiedconsensusontheirunderlyingdrivers. Prominentexplanationsinclude jobpolarization(JaimovichandSiu,2020),organizationalrestructuringfollowinglongexpansions (Bergeretal.,2012;Koenders,Rogerson,etal.,2005),changesinthepersistenceofbusinesscycles (Bachmann,2012;Panovska,2017),therisingimportanceoftechnologyshockssincethemid-1980s (Barnichon,2010b),extendedunemploymentinsurance(MitmanandRabinovich,2019),andthe convergenceoffemaleemployment(Fukui,Nakamura,andSteinsson,2023). Theseexplanations focusonevolvingfundamentals. Inthispaper,Iinsteaddocumentanovelrolefornoiseshocksthat arisewhenagentsfaceimperfectinformationaboutchangesinfundamentals. Thismechanism complementstheexistingliteraturebyshowinghowinformationalfrictions—whenembeddedin asearchandmatchingmodel—ratherthanchangesinfundamentalsthemselves,cangeneratethe sluggishunemploymentrecoveriesobservedinthedata. Aseparatestrandoftheliteratureemphasizesasymmetriclabormarketdynamics,where recessions trigger sharp rises in unemployment but recoveries are slow. This asymmetry has beenmodeledthroughendogenousjobseparation(Andolfatto,1997)andworkerheterogeneity in productivity (Ferraro, 2018). These models generate slow recoveries by amplifying shocks throughnon-linearitiesinthelabormarket. Mypaperinsteaddocumentsnewevidencethatthe underlyingdrivingforcesdiffersystematicallybetweenrecessionsandexpansions,pointingto acomplementarymechanism. Inparticular,Ishowthatprofessionalforecasterssystematically overestimateunemploymentduringrecoveries—astylizedfactthatconventionalmodelscannot replicate. Thispaperthereforeconnectstothelong-standingresearchagendaoninformationfrictions inmacroeconomics,datingbacktoLucas(1972,1975)anddevelopedfurtherinAngeletosandLa’O (2010,2013),Blanchard,L’Huillier,andLorenzoni(2013)andAngeletos,Collard,andDellas(2020). Thesetheoriesdemonstratehownoisysignalsaboutfundamentalscandistortbeliefsandpropagate shocks. Mycontributionistoembedsuchinformationfrictionsinasearch-and-matchingmodel ofthelabormarket,therebylinkingthemtothepersistenceofunemployment,anapplicationnot previouslyexploredinthisliterature. Inparallel,anempiricalliteraturehasdevelopedmethods for identifying noise shocks from macroeconomic data, most recently Chahrour, Nimark, and Pitschner(2021)andEnders,Kleemann,andMüller(2021). Iadvancethisempiricalagendaby refiningtheiridentificationstrategy: Iimposeadditionalrestrictionsandalsoidentifypersistent andtransitoryproductivityshocksseparately,allowingmetoquantifythedistinctroleofnoise shocksindrivinglabormarketrecoveries. Takentogether,thesecontributionsbridgetheoretical and empirical work on information frictions, offering a comprehensive account of how noisy signalsshapebothbeliefsandlabormarketdynamics. Withinthelabormarketliterature,Venkateswaran(2014)showsthatfirms’inabilitytodisentangleaggregatefromidiosyncraticshockscangeneratevolatilityfarlargerthaninstandardsearch models. Inthatframework,firmsmisattributepartofaggregateshockstoidiosyncraticfactorsand, becausetheyrespondmorestronglytothelatter,hiringfluctuationsareamplified,helpingresolve theShimer(2005)volatilitypuzzle. Mypapercomplementsthisbyfocusingonadifferentfriction: imperfectinformationaboutthepersistenceofaggregateshocks. WhereasVenkateswaran(2014) emphasizesvolatilitydrivenbymisperceptionsofthesourceofshocks,Ishowthatnoiseshocks 3

tiedtomisperceptionsofpersistencegeneratesluggishlabormarketrecoveriesandsystematic forecasterrors. MorerecentworkbyFacciniandMelosi(2022)andMorales-Jiménez(2022)quantitatively assessestheimportanceofinformationfrictionsinlabormarketdynamics. Further,Kozlowski, Veldkamp,andVenkateswaran(2020)andD’Agostino,Mendicino,andPuglisi(2022)showthatimperfectinformationaboutthedistributionofshockscanproducepersistenteffectsfromtransitory disturbances,particularlyduringtheGreatRecession. Myanalysiscomplementsthesepapersby combiningnewempiricalevidence—systematicforecasterrorsbyprofessionalforecasters—witha structuralmodelthatdemonstrateshownoiseshocksandimperfectinformationjointlyaccount forsluggishlabormarketrecoveriesacrossrecessions. Therestofthepaperisorganizedinthefollowingmanner. Section2discussestheempirical evidenceaswellastheidentificationofnoiseshocksusingastructuralVARanditsimpactonlabor marketdynamics. Section3introducesimperfectinformationstructuretoageneralequilibrium searchandmatchingmodel. Section4discussesthecalibrationandestimationstrategyforthe model parameters. Section 5 presents the results from the quantitative exercise and Section 6 concludes. 2 Unemployment Recoveries and Noise Shocks In this section, I first document the sluggish recovery of unemployment in the United States between1968-2019. Ishowthatittakesonaverage25quartersfortheunemploymentratetorecover toitspre-recessiontrough. Ithendocumentthemisperceptionabouttheunemploymentrateby professionalforecastersacrossrecessionsandshowthatforecastersconsistentlypredictmore sluggishrecoveriesinthelabormarketthanwhatactuallyoccurs. Havingestablishedthesefacts,I thenproceedtoidentifynoiseshocksusingnowcasterrorsinSection2.1,whereIdiscussindetail theidentificationstrategy. IthendocumentinSection2.2,thatthesenoiseshockshaveasignificantimpactonaggregate labormarketoutcomes,includingunemployment,vacanciesandjob-findingrates. Specifically,I documentthefollowingempiricalfacts. First,noiseshockshavepersistenteffectsonthedynamics ofthelabormarket. Aonestandarddeviationnoiseshockleadstoanincreaseintheunemployment rateof0.4percentagepointsat4quartersandrecoversbetween8-10quarters. Thisresponseresults fromanincreaseintheinflowintounemploymentandadecreaseinthejob-findingrate,influenced byadecreaseinvacanciesandhiringratesbyfirms. Wagesrespondweaklyandareslowtoadjust. Theimpulseresponsessuggestthatfirmsandworkersarelearningunderimperfectinformation. Second,aforecasterrorvariancedecompositionshowsthatnoiseshocksaccountforaboutonethirdofthefluctuationsinthekeylabormarketvariables. Third,ahistoricalshockdecomposition showsthatthenoiseshockscontributesignificantlytotherecoveryoftheunemploymentrate: absentnoiseshocks,ittakesunemploymentanadditional6quartersonaveragetorecover50%of itsrecessionaryincrease. 4

UnemploymentDynamicsDuringRecoveries. U.S.labormarketrecoveriestypicallyhavebeen slow,withtheunemploymentrateremainingelevatedevenafterthejobdestructionsubsides. To haveaconsistentmetricoflabormarketrecoveryovertime,IfollowHeise,Karahan,andŞahin, 2022whoproposeasimplemeasureoflabormarketrecovery—theunemploymentrecoverygap. I considertheshareoftheriseintheunemploymentrateduringtheprecedingrecessionthathas beenreversedduringthesubsequentexpansion. Specifically,foreachrecession,Iidentifythepeak quarterlyunemploymentrate,u andcomputetheincreaseintheunemploymentraterelative peak toitsprecedingtrough,u . Thisallowsmetoevaluatetheprogressintheunemploymentrate trough u peak –ut asafractionoftheunemploymentgapu peak –u trough byconsideringthetimefor25%, 50%,75%and100%ofthegaptorecover. Specifically, u peak –ut (1) URecoveryt = . u –u peak trough Table1andFigure1showtheunemploymentrecoverydynamicsforeachrecessionstarting in 1968. As Table 1 shows it took the unemployment rate between 5 to 16 quarters to recover halfofitsrecessionaryincreaseandlongerthan20quarterstorecoverbacktoitspre-recession level. Moreover,unemploymentrecoveriesbecameslowerovertime. Onaverage,post-recession unemploymenttakes10quarterstoreduceby50%and25quartersforfullrecovery. Before2000, 50%recoveryoccurredwithin9quarters;post-2000,thisextendsto13quarters. Figure 1: Unemployment Recovery Across Table1: UnemploymentRecoveryAcrossRe- Recessions cessions Note: Figure(1)andTable(1)reportthenumberofquarterstakentorecover25%,50%,70%and100%oftheriseinthe unemploymentratefromitspeakacrossrecessionsbetween1968-2019,excepttherecessionin1980whichwasquickly followedbythedownturnin1981-82.TheNBERCycleisthedurationforeconomicactivitytogofromtroughtoitspeak duringeachrecession. ForecastErrorsandMisperceptionabouttheEvolutionoftheUnemploymentRate. Whilethe unemploymentrateremainspersistentlyelevatedduringrecoveries,forecasterstendtobeeven morepessimisticabouttherecoveryofthelabormarket. Thisisevidentintheforecasterrors from the Survey of Professional Forecasters. The Survey of Professional Forecasters (SPF) is a quarterlysurveywhichelicitstheexpectationsofprofessionalforecastersaboutthestateofthe 5

economyintheUS.Itisoftenregardedasabenchmarkmeasureofprivatesectorexpectations. Thepatternisclearinlong-runprojectionsoftheone,twoandthreeyearaheadunemployment rate. Figure2ashowsthatforecastersconsistentlyoverestimatedtheunemploymentrateduring 1981-82intheLivingstonSurvey.4 Mostrecently,Figure2bdocumentsthatprofessionalforecasters predictedanevenslowerrecoveryaftertheGreatrecession. FigureA4plotsthemedian1yearahead unemploymentrateprojectionsfromtheSPF,whichshowthatforecasterspredicttherecoveries tobeslowerthantheyactuallywere. Theseobservationssuggestthatthereistypicallyawedge betweentheexpectedandtheactualunemploymentrateacrossrecessions. Figure2: UnemploymentRate:ProjectionsandActual (a)1981-82Recession (b)2009-07Recession Note:InPanel(a),thevariouscoloredlinesrepresentthemedian1and2yearaheadprojectionoftheunemployment ratefromtheLivingstonSurvey.Thesolidredlineistheactualunemploymentrateduringthe1981-82recession.In Panel(b)thevariouscoloredlinesrepresentthemedianlong-run(1year,2yearand3yearahead)projectionsofthe unemploymentratefromtheSurveyofProfessionalForecastersduringtheGreatRecession.Thedashedredlineisthe actualunemploymentrate. Onepotentialexplanationforthemismatchbetweenrealizedandexpectedunemployment ratescouldbetheimperfectinformationaboutwhetherthechangesintheaggregatefundamental processintheeconomyispersistentortransitory(Edge,Laubach,andWilliams,2007). Dueto imperfectinformation,agentsmustbasetheirdecisionsontheirexpectationsaboutthepersistence ofchangesinthetruefundamentalprocessbyobservingsomesignals. Undersuchaframework, agentsthenmaypredictconsistentlyhigherunemploymentrateastheycannotdistinguishthe trueshocksintheeconomyfromnoiseshocks(errorsinexpectationsduetothenoiseinreceived signals)andthesebeliefsmayinturnaffecteconomicoutcomes. Totestthishypothesis,Iproceed intwosteps. IfirstidentifynoiseshocksusingaSVARthatIdiscussinthefollowingsection. Then, Istudywhetheraggregatelabormarketoutcomesrespondtotheidentifiedshocks. Arecentapproachtoidentifyingnoiseshocksreliesonameasureofmisperceptions: the deviationofrealizedoutcomesfromexpectedoutcomes(Enders,Kleemann,andMüller,2021). I usethe’nowcasterrors’—thedifferencebetweentheactualoutcomeandthereal-timeperceived outcome—foridentifyingnoiseshocks. Thenowcasterrorscontainsignificantinformationabout thereal-timedeviationinexpectationsofprofessionalforecastersrelativetorealizedoutcomes. Since these deviations may arise due to the noise in observed signals about current economic 4TheSurveyofProfessionalForecastersstartedreportingthelong-runprojectionsonlysince2009.Therefore,Irely ontheLivingstonSurveyforlonger-rununemploymentexpectationsduringearlierrecessions. 6

activity,nowcasterrorscanbeexploitedtoidentifythesenoiseshocks. NowcastErrors Thenowcasts,whicharemedianexpectationsaboutthecurrentGDPgrowth rate,arecollectedfromtheSurveyofProfessionalForecasters. Thenowcasterrorsarecomputed asthedifferencebetweentheex-postgrowthrateofGDPforaquarterandthecontemporaneous forecastofwhatthatgrowthratewouldbefromprofessionalforecasters. Fortherestofthepaper, itisdefinedas (2) ncet = ∆yt –E t median (∆yt) whereyt isthecurrentrealGDPgrowthrate. Thetimingofthesurveyissuchthattheparticipating professionalforecastersareaskedtoreporttheirexpectationsaboutthecurrentquarteroutput growthbythesecondmonthofthequarter. Atthispoint,thecurrentoutputisnotobservable. Therefore,attimet,nowcasterrorsarenotobservableinrealtimeandarenotpartofanyagent’s information set. This gives an informational advantage to the econometrician over economic agentsasthenowcasterrorsonlybecomeavailableex-post. Further,thesenowcasterrorsplaysa keyroleintheidentificationofnoiseshocksifoneassumesthatnowcasterrorsandoutputgrowth haveoppositeresponsetonoiseshocks. 2.1 IdentificationofNoiseShocks Inthissection,IdescribetheempiricalstrategytoidentifyapersistentTFPshock,atransitoryTFP shock,andanoiseshockandthendiscusstheeffectsoftheseshocksonkeylabormarketoutcomes. Here,Itestthehypothesisthattheobservedwedgebetweenrealizedandexpectedunemployment rates,asshowninFigure2andA4,ariseduetoimperfectinformationaboutwhetherthechanges intheaggregatefundamentalprocessintheeconomyarepersistentortransitory. Iproceedintwo steps. First,Iidentifynoiseandproductivityshocksusingatri-variateSVAR.Theidentificationof noiseshocksisachievedbyimposingsignandzerorestrictions. identifypersistentandtransitory productivityshocksbymaximizingtheforecasterrorvarianceofaggregateproductivityinthelong run. Second,usinglocalprojections,Itestwhethernoiseshockshaveasignificanteffectsonthe dynamicsofkeylabormarketindicatorslikeunemploymentandvacancies. EmpiricalSpecification. Theaggregateproductivityprocessisassumedtoconsistofapersistent andatransitorycomponent. Whilethelevelofproductivityisobservable,itsunderlyingcomponentsarenot. Therefore,economicagentsmustformtheirbeliefsaboutaggregateproductivity usingpublicsignals. Noiseshocksarethechangesinthesignalnotcomingfromshockstothe actualproductivity. Theaimistonowidentifythethreeshocks,apersistentshocktoaggregateproductivity,atransitoryshocktoaggregateproductivityandanoiseshock.Theempiricalspecification consistsofavector-autoregressionoftheform p (3) A 0 Yt = a+Σ j=1 A j Y t–j +et 7

where the set of variables Yt ≡ [TFPt,GDPt,NCEt] includes the utilization-adjusted TFP from Fernald,2014,realGDPgrowthandNowcasterrors.5 Thesampleperiodrangesfrom1968q4to 2019q4. A j is the weighton past realizationsof Yt, et is a vectorof structuraleconomic shocks, andA–1 isthestructuralmatrixthattheSVARprocedureseekstoidentifyfromthesetofreduced- 0 formresiduals. Thefactthatagentscannotobservethenowcasterrorsinrealtimeprovidesthe econometrician an informational advantage over the economic participants in real time, thus makingtheSVARmodelinvertible(Blanchard,L’Huillier,andLorenzoni,2013). Itfollowsthatthereduced-formrepresentationis p (4) yt = b+Σ j=1 B j y t–j +ut Hereb = A– 0 1isann×1vectorofconstants,B j = A– 0 1A j ,ut = A– 0 1(cid:15) t. var(ut) = E(utu t (cid:48) ) = ∑ = A– 0 1(A– 0 1) (cid:48) isthen×nvariance-covariancematrixofreduced-formerrors. Letφ = (B,∑)collectthereducedform parameters. Finally, following Uhlig, 2005 , I define the set of all IRFs through an n×n orthonormalmatrixQ ∈ Θ(n)whereΘ(n)isthesetofalln×northonormalmatrices. IdentificationAssumptions Aggregatenoiseshocksinanimperfectinformationstructureare identified in the data using a combination of zero and sign restrictions as well as max share identificationinatri-variatestructuralVAR.Thesignrestrictionsidentifythenoiseshockandthe max-shareapproachidentifiesthepersistentshocksfromthetransitoryproductivityshocks. 1. Iimposethefollowingrestrictionsontheimpactmatrixtoidentifythenoiseshocks. (a) Noise shocks have zero impact on aggregate productivity. Noise is an error in the expectations of economic agents. It should not affect the underlying fundamental productivityprocessintheeconomy,whichisthetotalfactorproductivityhere. Iuse theTFPseriesfromFernald,2014andassumethatthisisanerror-freemeasureofTFP. (b) Onimpact,thepersistentandthetransitoryTFPshockscontemporaneouslyaffectTFP andGDPgrowthinthesamedirection. Theresponseofthenowcasterrortoapersistent aswellasatransitoryshockisunrestricted.6 (c) Noiseshockscontemporaneouslyaffectnowcasterrorsintheoppositedirectionasthey doGDPgrowth. Inotherwords,noiseshocksareassumedtomoveexpectationsabout realGDPmorethanrealGDPitself. GDPalsoincreasesasagentsrespondbutitincreases lessthantheexpectations. Thisimpliesthatncet = ∆yt –E t median(∆yt) < 0while∆yt > 0. ThisassumptionismadebyEnders,Kleemann,andMüller,2021andChahrour,Nimark, andPitschner,2021whoidentifybeliefshocksinabi-variateVARusingsignrestrictions. Theseidentifyingrestrictionsholdacrossabroadclassofmodelswithinformationstructures consistentwithLorenzoni,2009,Blanchard,L’Huillier,andLorenzoni,2013,andAngeletos 5ThefactthattheVARdoesnotincludeanylabormarketoutcomessuchastheunemploymentrate,allowsthe identifiedshockstobeunaffectedbyfluctuationsinthelabormarketdirectly. 6ATFPshockmaycausealargerchangeinactualGDPgrowththanitdoesinexpectationsasevidencesuggeststhat consensusforecastsunder-reactrelativetofull-informationrationalexpectations(Bordaloetal.,2020).Forrobustness, IconsideranalternatespecificationwhereIimposetherestrictionthattheTFPshocksaffectthenowcasterrorinthe samedirectionasTFPandoutput.However,theresultsfromthisexerciseareinlinewiththemainexercise. 8

andLa’O,2010. Let(cid:15) t bethepersistentshock,η t bethetransitoryshockandν t bethenoise shock. Thus,therestrictionsontheimpactmatrixcanbedemonstratedbythefollowing:        zt zt–p + + 0 (cid:15) t (5)   yt   = Σ p j B j   yt–p   +   + + +     η t   ncet ncet–p ∗ ∗ – ν t 2. Thesignrestrictionsidentifythenoiseshocksbutdonotdistinguishbetweenthepersistent and the transitory shock. To separately identify the persistent shock from the transitory shock,Iusewhatisreferredtointheliteratureasthemax-share identificationstrategy. I extractthepersistentshockastheinnovationthataccountsforthemaximumforecasterror variance(FEV)shareofutilizationadjustedTFPatalongbutfinitiehorizon. Thismethod buildsonUhligetal.,2004andhasbeenusedbyFrancisetal.,2014toidentifylongrunTFP shocks. MorerecentlythishasbeenusedbyKurmannandE.Sims,2021incontextofnews shocks. Toformalizetheidentificationstrategydescribedabove,letj ∈ {1,2,3}bethestructuralshocks, andi ∈ {1,2,3}denoteTFP,GDPgrowthandnowcasterrorrespectively. DefineI = 1,...,kasa –j subsetoftheshocksofinterest. Lets bethesignrestrictionsontheimpulseresponsevectorto jh thejth structuralshockathorizonh. Inthiscase,theimpulseresponseisgivenbythejth column vectorofIRh = C h (B)∑ tr Q. ThesignrestrictionsarerepresentedbyS j (φ)q j ≥ 0, forj ∈ I S. Let CEFEVi(H)denotethefactorerrorvariance(%contribution)athorizonH ofvariableiexplained j bythejth structuralshock. ΣH c (φ)c (cid:48) (φ) (6) CEFEV i (H) = q(cid:48) Γh i (φ)q ; Γi (φ) = h=0 ih ih j j j H ΣH c (cid:48) (φ)c (φ) h=0 ih ih whereΓi (φ)isn×npositivesemi-definitematrix. H Thus, the identificationof thethreeshocks, Q = [q ,q ,...,q ]requiresus tosolvethe 1:k 1 2 k followingproblem (7) q ∗ = argmaxq (cid:48) Γ1 (φ)q 1 1 H 1 q 1 subjectto (cid:48) (8) q (1,3) = 0 1 (9) S j (φ)q j ≥ 0, forj ∈ I S (cid:48) (10) q q = 1 1 1 HerethehorizonisassumedtobeH = 40quarters,whichisamediumrunhorizon. This isbecausetheeffectsoftransitoryandnoiseshocksarenotexpectedtopersistforaslongasa 9

decade.7 Now,thisimpliesthattheshockcanonlybeextractedtill2012. Toextendtheseries,for 2012-2019,IcalculateH asthemaximumavailablehorizonfromthatpoint. In2017,thisissetto H = 20. AsseeninAppendixFigureA7,persistentshocksexplainthemaximumvarianceofTFP evenat20quarters. Equation8istherestrictionthatnoiseshockshavezeroeffectonTFP,which followsfromthedefinitionofthenoiseshock. Equation9consistsofthesignrestrictionsdetailed inequation5. Equation10ensuresthattheidentifiedshocksaremutuallyorthogonal. Ifollowthe algorithmoutlinedbyCarrieroandVolpicella,2022tosolvethisoptimizationproblem. Iassume4 lagsassuggestedbytheAkaikeInformationCriterionanduniformpriors. Theimpulseresponseofthenowcasterrorstotheidentifiedshockssuggestthatforecasters donothavefullinformationabouttheeconomy. AppendixFigureA6showstheimpulseresponse ofTFP,GDPgrowthandnowcasterrorstotheidentifiedpersistent,transitoryandnoiseshocks. Thenowcasterrorincreasesonimpactofthepersistentshockbutdoesnotrecoverimmediatelyin thenextperiod. Furthermore,thenowcasterrorrespondsweaklytothetransitoryshockonimpact andhasadelayedpositiveresponse. Thissignifiesthatforecasterscannotdistinguishimmediately ifashockispersistent,transitoryornoiseandlearnwithsomepersistence. Noiseshockshavea negativeeffectonimpactonthenowcasterrorssincethisisarestrictionimposedbytheVAR. Predictably,thepositivepersistentshockincreasesTFPasonimpactanddeclinespersistently. GDPgrowthweaklyrespondstoapersistentproductivityshockonimpact,buthasadelayedpositive andpersistentresponse. AtransitoryshockincreasesTFPandGDPgrowthonimpactbuttheeffect isnotpersistent. Finally,TFPdoesnotrespondtonoiseshocks,inlinewiththezerorestriction imposed. TheNoiseshockhasapositiveandsomewhatpersistenteffectonGDPgrowth. 2.2 EffectofNoiseShocksonLaborMarketDynamics Key labor market variables exhibit significantly persistent impulse responses to the identified noiseshocksatthebusinesscyclefrequency(8-10quarters). Ahistoricaldecompositionshows thatnoiseshockshaveanincreasinglyimportantroletoplayintheevolutionofunemployment, vacanciesandjobfindingratesoverthebusinesscycle,whichisamotivationtointroduceimperfect informationinasearchandmatchingmodel. SmoothLocalProjections OncetheshocksareextractedfromtheVAR,Icannowstudyhow labormarketvariablesrespondtotheseshocksusingsmoothlocalprojections(SLP)(Barnichon andBrownlees,2019). Foreachshocku ,theJordà(2005)localprojectionsaregivenby j P j j j ∑ j j (11) y t+h = α h +β h u t + γ p ω t–p+µ h,t+h p=1 whereω j isthesetoflaggedvaluesofy anduj. t–p Following Barnichon and Brownlees, 2019, one can approximate β j ≈ ∑K b j B j (h) using h k=1 k k alinearB-splinesbasisfunctionexpansionintheforecasthorizonh. Thus,thecorresponding 7Theresultsarerobusttolongerhorizons,uptoH=60quarters. 10

smoothLinearProjectionscanbewriitenasEquation12. K K P K (12) y ≈ ∑ a j B (h) j + ∑ b j B j (h)u j + ∑ ∑ c j B j (h)ω j +µ j t+h k k k k t pk k t–p h,t+h k=1 k=1 p=1k=1 TheSLPisestimatedusinggeneralizedridgeestimationandfurtherdetailscanbefoundinAppendixsectionA.5andBarnichonandBrownlees(2019). Here,yt =aggregatelabormarketoutcomessuchasunemploymentrate,vacancies,rateof outflowfromunemployment(UE),job-to-jobtransitionrates(EE),hiringrateandwagegrowth. uj j arethethreeshocksrespectivelywhileµ istheresidualerrorforeachregression. Alllabormarket t dataarefromCurrentPopulationSurveyandJOLTSforvacanciesandhiringrate.8 Figures3and 4showtheimpulseresponsesofthelabormarketvariablestostandardized1standarddeviation negativenoiseandpersistentTFPshocks.9 Figure3: ImpulseResponsetoNoiseShocks Note: Thisfigureshowsthesmoothedcumulativeimpulseresponsefunctionsforkeylabormarketvariablestoanoise shock,estimatedusingequation11,whereu isthenoiseshockidentifiedusingtheSVARdescribedbytheoptimization j probleminequation7. Thesampleperiodis1968q4: 2019q4. DataforthelabormarketoutcomesarefromCPS, vacanciesfromBarnichon,2010aandwagesfromBEA’saveragehourlyearningsseries.Theshadedarearepresentsa 95%confidenceinterval. Noiseshockshaveasignificantandpersistenteffectonunemployment,vacancies,UE,EEas wellashiringrateforupto10quarters. Thenegativeeffectonwagegrowthisdelayed,although weak,indicatingthatwagesaresluggish. Unemploymentrisesby0.6percentagepointsinresponse to a one standard deviation noise shock. The number of job vacancies decreases, transitions 8TheresultsareconsistentwithusinganAutoregressiveDistributedLag(ADL)pecificationforlocalprojections. 9ResponsetothetransitoryTFPshocksisdocumentedinAppendixFigureA10. 11

fromunemploymenttoemploymentreduce,andjob-to-jobtransitionsdecline. Astherearefewer vacancies, there are fewer number of jobs to be found, dampening the job-finding rate of the workers. Furthermore, as wage growth declines, there are fewer number of workers making job-to-jobtransitions. Thisfurtherdampensjobfindingratesfortheunemployedasjobsinlower endoftheladdersremainoccupiedsincefewerworkersaremovinguptheladder,makingitharder forunemployedworkerstofindjobs. Theseultimatelyleadtounemploymentratebeinghigherfor longer. Theseresultssuggestthatagentscannotdistinguishthetypeofshocktheyfacecorrectlyand thattheylearnslowlyovertimeaboutthetrueshocks. Thehumpshapeoftheimpulseresponses suggestthatinitiallyagentsmisperceivethenoiseshockasanactualnegativeproductivityshock andhencerespondasiffacedwithanactualnegativeproductivityshock. Firmsdecreasetheir hiringandincreaselayoffs. Astherearelessjobstobematchedwithnow,thejobfindingrate decreasesandtranslatestoanincreaseinunemployment. However,asfirmsandworkerslearn aboutthetrueprocessintheeconomy,theyplacehigherweightontheshockbeinganoiseshock andgraduallystartincreasinghiring. Asaresult,outflowfromunemploymentincrease,resulting inadeclineinunemployment. Figure4: ImpulseResponsetoPersistentTFPShocks Note: Thisfigureshowsthesmoothedcumulativeimpulseresponsefunctionsforkeylabormarketvariablestoa persistentTFPshock(inblue),estimatedusingequation11,whereu isthepersistentTFPshockidentifiedusingthe j SVARdescribedbytheoptimizationprobleminequation7.The95%confidenceintervalisshadedinblueaswell.Itis superimposedontheIRFsfromthenoiseshocksinFigure3.Thesampleperiodis1968q4:2019q4.Dataforthelabor marketoutcomesarefromCPS,vacanciesfromBarnichon,2010aandwagesfromBEA’saveragehourlyearningsseries. Theshadedarearepresentsa95%confidenceinterval. Theseresultsareconsistentwithlearningwhichmotivatestheintroducingofimperfectinformationinageneralequilibriummodelofsearchandmatching. Iftherewascompleteinformation 12

intheeconomyfirmsandworkerswouldnotrespondtonoiseshocksbecausethesedon’tchange thefundamentaleconomicconditions. Moreover,thetimeittakesfortheimpulseresponsesto recover,suggeststhatthislearningprocessisquitegradual. Iflearninghappenedmorerapidly, theeconomywouldadjusttonoiseshocksmuchfaster. ForecastErrorVarianceDecomposition The forecast error variance decomposition is informative of the variance in an outcome explained by each of the shocks at a specific horizon. I use the estimator proposedby Gorodnichenko andLee, 2020 for calculatingthe forecasterror variancedecompositionwithlocalprojections. Theforecasterrorfortheh-periodaheadvalueof anendogenousvariableyt isgivenby (13) f ≡ (y –y )–P[y –y |Ω ] t+h|t–1 t+h t–1 t+h t–1 t–1 whereP[y –y |Ω ]istheprojectionofy –y ontheinformationsetΩ ≡ {∆y ,µ ,∆y ,µ ,···}. t+h t–1 t–1 t+h t–1 t–1 t–1 t–1 t–2 t–2 Theforecasterrorsduetoinnovationsinµcanbedecomposedasfollows: (14) f t+h|t–1 = ψ µ,0 µ t+h +···+ψ µ,h µ t +v t+h|t–1 wherev t+h|t–1 istheerrortermduetoinnovationsorthogonalto{µ t,µ t+1 ,··· ,z t+h }andΩ t–1 . Theshareofvariancesexplainedbythecontemporaneousandfutureinnovationsinµ t tothe totalvariationsinf canbedefinedasfollows(C.A.Sims,1980): t+h|t–1 var(ψ µ,0 µ t+h +···+ψ µ,h µ t) (15) s = h var(f ) t+h|t–1 s inequation15isestimatedusingthecoefficientofdeterminationestimatorforFEVDsasproposed h byGorodnichenkoandLee,2020. TheresultofthisexerciseissummarizedinTable2. TheFEVDanalysisrevealsthatatashort-runhorizonof0to8quarters,noiseshocksarenotably influentialinaccountingforthevariabilityinkeylabormarketmetricssuchasunemployment,job openings,inflowsandoutflowsfromunemployment,andratesoftransitionsbetweenjobs. Specifically,atan8-quarteraverage,noiseshocksaccountfor34%ofthevariationinunemployment,37% injobvacancies,35%intheoutflowratefromunemployment,27%inemployment-to-employment transitions,and14%inwagegrowth. Whilepersistentfactorsgenerallymakeupalargershare,rangingfrom38%to61%across theseindicators,andtransitoryfactorscontributebetween21%and31%,theinfluenceofnoise shocksissubstantial. Especiallyintermsofjobvacanciesandunemployment,noiseshocksaccount formorethanone-thirdoftheobservedvariability,highlightingtheirsignificantroleinshort-term fluctuationsinthelabormarket. Atalongerrunhorizonof8-16quarters,persistentshocksaretheprimarydriversofvariance acrossalllabormarketindicators. Specifically,theyaccountfor63%ofthevariationinunemployment,61%invacancies,63%inthejob-findingrate,65%injob-to-jobtransitions(EE),and92%in wagechanges. Predictably,noiseshocksshowacomparativelymodestinfluence,accountingfor 15-19%ofthevarianceinunemployment,vacancies,job-findingrate,andEEtransitions,and3%in 13

Table2: ForecastErrorVarianceDecomposition:ShorterRunHorizon ShortRun MediumRun Horizon: 0-8quarters Horizon: 9-16quarters Persistent Transitory Noise Persistent Transitory Noise Unemployment 0.43 0.23 0.34 0.63 0.21 0.16 Vacancies 0.42 0.21 0.37 0.61 0.20 0.19 UE 0.38 0.27 0.35 0.63 0.20 0.17 EE 0.42 0.31 0.27 0.65 0.16 0.19 WageGrowth 0.61 0.25 0.14 0.92 0.05 0.03 Note: ThistablereportstheaverageforecasterrorvariancedecompositionforU,V,E–E,U–Eand∆W,estimated usingequation15,overashortrun(0-8quarters)andamediumrun(8-16quarters)horizon.Eachrowaddsto1.Noise shocksexplainasignificantvariationinthelabormarketatashortrunhorizon.Thesampleperiodis1968q4:2019q4. DataforthelabormarketoutcomesarefromCPS,vacanciesfromBarnichon,2010aandwagesfromBEA’saverage hourlyearningsseries. wages. Transitoryshocksplayalesssubstantialrole,contributingtolessthan25%ofthevariance inunemployment,vacancies,UE,andEEtransitionsrespectively,andonly5%inwagegrowth. HistoricalContributionofNoiseShocks. Tounderstandtheroleofimperfectinformationover thebusinesscycle,itisusefultounderstandhowmuchofthedeviationofthekeylabormarket outcomesfromtheirpredictedpathcanbeexplainedbytheproductivityshocks. Ifnoiseshocksare notimportant,theproductivityshockswouldexplainalmostallthefluctuationsinthesevariables. Here,thedecompositionforj = {1,2,3}shockscanbewrittenasthefollowing: (16) yt –Ψ¯ t = ∑ t ∑ –t0 β h ·µ j,t–h j h=0 where,Ψ¯ t isthepuredeterministiccomponentandyt isvariouslabormarketoutcomessuch asunemploymentrate,vacancypostings,jobfindingrateandaveragehourlyearnings. Twokeyfactsemergefromthehistoricaldecomposition: first,thattheproductivityshocks alonefailtoaccountforthepersistenceofunemploymentratepost1985,andsecond,thatnoise shockshavebeenplayinganincreasinglyimportantrolesince1990s. Figure5aplotsthedeviation ofunemploymentratefromitspredictedpathduetothepersistentandtransitoryproductivity shocksalone. Thus,theremainingmovementisexplainedbythenoiseshocks. An examination of the recessions in 1990, 2001, and 2007-09 reveals that the productivity shocksareinsufficienttocompletelyexplainthefluctuationsintheunemploymentrate,vacancy postingsandjobfindingrates. Theproductivityshockspredictedafasterrecoveryacrossthese recessionsandadiminishedpeakduringtheGreatRecession. Moreover,Figure2bdemonstrates thatprofessionalforecastersduringtheGreatRecessionanticipatedunemploymentratesthatwere bothhigherandmorepersistentthantheactualunemploymentrate. Outflowfromunemploymentandvacanciesfollowsasimilarpattern,wherethefundamental shocks do not fully explain the fluctuation as well as the speed of the recovery. Noise shocks dampenedthejobfindingratesduringtheexpansioninthe90s,butamplifiedthevacancypostings. 14

Figure5: HistoricalContributionofPersistentandTransitoryTFPShocks (a)UnemploymentRate (b)OutflowfromUnemployment (c)Vacancies Note:Thisfigureshowsthehistoricaldecompositionofunemploymentrate,vacancypostingsandoutflowratefrom unemploymentfollowingequation16.Thedashedblacklineisthecumulativecontributionoftheidentifiedpersistent andtransitoryTFPshockstothemovementsinthedemeanedvacancypostings(solidredline).Theremainingmovement isexplainedbythenoiseshocks,whichcontributesignificantlytothevacancypostingsduringtherecessionsin1990-91, 2001and2007-09. 15

Thereseemstobeadisconnectbetweentheeffectofimperfectinformationduringexpansionson householdsandfirms,butduringthedownturns,noiseshocksconsistentlyamplifythedeclinein bothjobfindingratesaswellasvacancypostings. Anaggregateassessmentofthesefindingsimpliesthatduringtheserecessions,noiseshocks ledtoamis-estimationofthepersistenceoftheshockbyeconomicagents. Thismisperception ledtoanoverestimationoftheactualpersistenceoftheshock,therebyinfluencingdecisionsconcerningemploymentandproduction. Inotherwords,firmsandworkersperceivedtherecessions tobeworsethantheyactuallywere. Consequently,therewasamorepronouncedreductionin vacancypostings,accompaniedbyadecreaseinjob-findingrates. Thesefactorstogetherresulted inunemploymentlevelsthatwerenotonlyelevatedbutalsopersistent,reflectingapersistence thatwasgreaterthanoriginallyanticipated. Tounderstandthecontributionofthenoiseshockstothepersistenceofunemployment,I computeforeachrecessionbetween1968-2019,theshareoftheriseinunemploymentduringthe recessionthathasbeenreversedduringtheexpansionfollowingEquation1 Ithendefinepersistenceasthenumberofquarterstorecover50%oftheriseinunemployment duringarecession,thatisurecovery,t = 0.5. Now,fromthehistoricaldecomposition,Icancalculate what fraction of this persistence can be attributed to each of the shock by first computing the predictedunemploymentratefromeachshockandthencalculatingthepersistenceasdefined above. TheresultsaresummarizedinAppendixTableA1. Forthegreatrecession,noiseshocks accountforabout35%ofthe50%oftheriseinunemploymentandonaveragenoiseshocksaccount for32%ofthisrecoveryacrossrecessions. Thesecondobservationthatemergesfromthisanalysisisthattheroleofnoiseshocksappears tobemoreprominentposttheGreatModeration. FigureA5plotstheshockseriesretrievedfrom theVARandascanbeseen,thenoiseshocksduringthethreerecessionspost1990hadalarger negativedrawthaninthepre1990decades. Interestingly,thepersistentshockdisplaystheopposite pattern. This merely demonstrates that noise shocks have had a larger role to play post 1990, althoughthetime-seriesisnotlongenoughtoestablishifthisisasystematicpattern. Pre-2000 ittookonaverage26quartersfortheunemploymentratetorecovertoitspre-recessiontrough whileafter2000ittook32quarters. Onaverage,noiseshocksexplain19%ofthisrecoveryduration pre-2000,and36%ofthisdurationforthepost-2000recessions. The empirical results indicate that noise shocks play a significant role in explaining the dynamicsofthelabormarketoverthebusinesscycleandspecificallythesluggishrecoveryfrom recessions. Now,tounderstandthemechanismthroughwhichnoiseshocksaffectthepersistence ofthelabormarket,Iintroduceimperfectinformationinageneralequilibriummodelofsearch andmatchinginthefollowingsection. 2.3 Robustness: InclusionofUnrestrictedShocks Structuralshocksdonotsatisfythesignrestrictionsforthenoiseshocks,sinceanystructuralshock resultsinalargerchangeinactualoutputthanexpected. However,theremaybepotentiallyother shockssuchasmonetarypolicyshocksorfinancialshocks,notincludedintheSVAR,whichmay behavelikethenoiseshocks. Toaddressthisconcern,Iincludingafourthunrestrictedshockin 16

thesystem. Thekeyargumenthereisthatifthereareothershocksthatarebeingpickedupbyany ofthepersistent,transitoryornoiseshocks,inclusionofthefourthshockshouldthenaccountfor thoseshocks. Iincludeunemploymentasthefourthvariableandleaveunrestrictedtheimpact matrix. ThemodifiedVARisthusgivenbyequation17.        zt zt–p + + 0 ∗ (cid:15) t         yt  p  yt–p   + + + ∗  η t  (17)   = Σ B  +     ncet   j j  ncet–p     ∗ ∗ – ∗     ν t   ut ut–p ∗ ∗ ∗ ∗ µ t Onceagain,Iassumethat(cid:15) t maximizestheforecasterrorvarianceofTFPatalongrunhorizon. I Figure6: 4VariableVAR:HistoricalContributionofShockstoUnemploymentRate Note:Thisfigureshowsthehistoricaldecompositionofunemploymentratefollowingequation16.Theblacklineis thecumulativecontributionoftheidentifiedpersistentandtransitoryTFPshockstothemovementsindemeaned unemploymentrate(redline).ThedashedbluelineisthecontributionoftheTFPshockandthenoiseshocks,which contributesignificantlytotheunemploymentrateduringtherecessionsin1990-91,2001and2007-09.Theremaining movementhereisexplainedbythe4thshockµ t. presentheretheresultofthehistoricaldecompositionexerciseinFigure6. Hereaswesee,the noiseshocksstillexplainasignificantvariationinunemploymentrate. However,inthe1973-75,as wellasthe1980-81,1982-83recessions,thecontributionofthe4thshockisthehighestinexplaining themovementinunemploymentrate. Thisisconsistentwiththefactthattheserecessionswere mostly explained by monetary policy shocks or oil shocks, which cannot be attributed to TFP shocks. Finally,Iconductfurtherrobustnessbycontrollingforvariousshockswhenestimatingthe 17

impulse response of labor market outcomes to the noise shocks. In Section A.7, I control for contemporaneousuncertaintyshocksaswellasitslags(Bloom,2009),andfindthattheresultsare robusttoit. Theimpulseresponseoflabormarketoutcomestoanoiseshock,whencontrollingfor theuncertaintyshockremainswithinthe90percentconfidenceintervaloftheresponsewithout thecontrol. Further,theshapeoftheseresponsesremainunchangedandconsistentwithimperfect information. 3 A Search and Matching Model with Information Frictions Standardmodelsofsearch-and-matchingfailtofullyexplainthevolatilityofunemploymentand vacanciesaswellastheslowrecoveryofthelabormarketfromrecessionsColeandRogerson, 1999.10 Motivatedbytheempiricalevidencethatimperfectinformationplaysanimportantrolein explainingthesedynamics,thissectionintroducesanimperfectinformationstructuretoasearch andmatchingmodel. Themodelisbasedonarealbusinesscyclemodelwithsearchandmatchinginthelabor marketasinMerz, 1995andAndolfatto, 1996andfollowstheextensionsbyGertler, Huckfeldt, andTrigari,2020whointroducestaggeredwagecontractingandallowforon-the-jobsearchwith variableintensity. Theprimaryreasonforintroducingastaggeredwagecontractinginthecontext ofthispaperisthatwagerigidityamplifiestheroleofimperfectinformation,aswillbecomeclear inthesubsequentsubsections(ChahrourandJurado,2018;Morales-Jiménez,2022). Thereason forintroducingendogenoussearcheffortaswellason-the-jobsearch,istocapturetheresponse comingfromworkerswhenfacedwithinformationfrictions. Job-to-jobtransitionscapturenot onlythecyclicalwagegains, butalsocrowdoutunemployedworkerssearchingforajob, thus capturinganimportantmomentofthelabormarket. Inthefollowingsub-sectionsIdescribethe environmentforthemodelanddiscusstheproblemsoffirmsandhouseholds. Environment Thereisacontinuumoffirmsandworkers,eachofmeasureunity. Firmsthatpost vacanciesandworkerslookingforjobsmeetrandomly. Theaggregateproductivityintheeconomy isgivenbyzt. Idiosyncraticmatchqualityisrevealedonceaworkerandafirmmeet. Matchquality ofaworkerwithinthefirmiseithergood(g)withprobabilityξorbad(b)withprobability1–ξ. Theproductivityofabadmatchisafractionφoftheproductivityofagoodmatch,whereφ ∈ (0,1). Thefirms’effectivelaborforceis (18) lt = gt +φbt 10AsColeandRogerson,1999note,theDMPmodelcanaccountforbusinesscyclefactsonlyiftheaveragedurationof non-employmentspellisninemonthsorlonger,whichisquitehighthanobservedinthedata. 18

Thetotalnumberifunemployedworkersisgivenby: u¯ t = 1–n¯ t –b ¯ t (cid:90) n¯ t = ntdi i (cid:90) ¯ bt = btdi i wheren¯ t andb ¯ t arethetotalnumberofworkersingoodandbadmatchesrespectivelyacrossall firms,indexedbyi. Workers search for jobs when they are unemployed with endogenous search intensity ζ u. Employedworkersinabadmatchalsosearchonthejobsothattheycanmoveuptheladderand matchwithagoodjob. Theysearchwithendogenoussearchintensityζ . Workerssearchingon b thejobonlytransitiontogoodjobs. Iftheyarematchedwithanotherbadjob,theystayintheir currentbadjobsandhencelateralmoveemntstootherbadjobsareeliminated.11 Searchiscostly andthecostofsearchingischaracterizedby 1 c(ζ ) = µ(ζ )1+ω jt jt where, ζ is the search intensity of unemployed workers (j = u) and employed workers in bad jt matches(j = b). Therearetwowaysamatchcanbedissolved. First,firmsandworkersmayreceive anexogenousseparationshockwithprobability1–σ. Workerswhoreceivetheseparationshock become unemployed at the beginning of next period. Second, if the match is not destroyed, a worker in a bad match searches on the job. If she finds another job and accepts it, the worker movestothenewfirmwithintheperiodandthematchwiththecurrentemployerisdissolved. The totalefficiencyunitsofsearchisthereforegivenbythesearchintensityweightedsumofsearchers s¯ t = ζ utu¯ t +σζ bt b ¯ t The aggregate number of matches are thus a function of the efficiency weighted number of searcherss¯ t andthenumberofvacanciesv¯ t: m¯ t = Ψs¯ t α v¯ t 1–α where α is the elasticity of matches to units of search and Ψ is the matching efficiency. The probabilitythataunitofsearchleadstoamatchisgivenby m¯ t pt = s¯ t 11AsGertler,Huckfeldt,andTrigari,2020explain,theexpectedgainfromalateralmoveisquantitativelytrivialand canberuledoutwithasmallmovingcost. 19

Itfollowsthattheprobabilitythatthematchisgood(p g )orbad(pb)isasfollowsrespectively t t g p t = ξpt p t b = (1–ξ)pt Forafirm,theprobabilitythatavacancywillleadtoamatchis: m m¯ t q = t v¯ t Now,notallmatcheswillleadtohiressinceIassumethatworkersinbadmatchesacceptonly g goodjobs. Thus,theprobabilitythatavacancyleadstoagoodqualityhire(q )ortoabadquality t hire(qb)isgivenby t q g = ζq m t t q b = (1–ζ) (cid:18) 1– σζ bt b ¯ t (cid:19) q m t s¯ t t g Since all workers accept good matches, q is simply the product of the probability of a match t beinggoodconditionalonamatchandtheprobabilityofamatch. However,sinceworkersinbad matchesdonotmakelateralmovements,thefractionofsearcherswhosearchon-the-jobfrombad matches, σζ bt b ¯ t isnettedouttocalculateqb. Thus,theexpectednumberofworkersinefficiency s¯ t t unitsoflaborthatafirmcanexpecttohirefrompostingavacancyas: qt = q t g +φq t b Thus,thetotalnumberofnewhires(inefficiencyunits)isqtvt andthehiringrateχ t istheratioof newhirestotheexistingstocklt,givenby: χ t = qtvt lt Wecannowdefinesomelawofmotionsforthegoodandbadmatches. respectively. (19) g¯ t+1 = σg¯ t +ξpts¯ t (20) b ¯ t+1 = σ(1–ζ bt (1–ξ)pt)b ¯ t +(1–ξ)ptu¯ t Totalgoodmatchesnextperiodareasumofsurvivinggoodmatchesinthecurrentperiodand aninflowofsearchersintogoodmatches,whichdependsontheirprobabilityoffindingagood match. Similarly,totalnumberofbadmatchesnextperiodareasumoftwoterms. thefirstterm representsthenumberofworkersinbadmatcheswhoareunabletofindagoodmatchandthus remaininthebadmatch. Thesecondtermisthenumberofunemployedworkerswhofindbad matchesandmoveintothem. Thecurrentvaluesoflt,gt andbt arepredeterminedstatevariables. Theintra-periodtimingprotocolthatthefirm’sdecisionproblemisbaseduponis: (i)realizationofaggregateandfirm-levelshocks,(ii)wagebargainingandproduction,(iii)realizationof 20

match-levelseparationshocks,and(iv)searchandmatching. Wecannowlookattheproblems thatthefirmsandworkersfaceinthesubsequentsub-sections. Information Structure This section introduces an imperfect information structure which is analogoustothestructuralVARdeployedtorecoverthebeliefshocksaswellasthepersistentand transitoryproductivityshocks. Theinformationstructureaimstocapturethefactthatagentsdo nothavefullinformationaboutthestateoftheeconomy. However,itisimportanttonotethatagentshaverationalexpectations,giventheirinformation set. Agentsdonotperfectlyknowwhetherthecurrentaggregateproductivity,whichistheonly source of aggregate uncertainty, is persistent or transitory. They get a public signal about the persistentcomponentandformexpectationsbasedonit. Letzt = logZt. Fromnowon,alowercasevariablewilldenotethelogofthecorrespondinguppercasevariable. xt isthepermanent componentandη t isthetemporarycomponent. (21) zt = xt +η t ; η t ∼N(0,σ2 η) xt followsanAR(1)process: (22) xt = ρx t–1 +(cid:15) t ; (cid:15) t ∼N(0,σ2 (cid:15)) Eachperiod,allagentsintheeconomyobserveanoisysignalsˆ t aboutthepermanentcomponent oftheproductivityprocess,whichisgivenby (23) sˆ t = xt +at (24) at = ρ aa t–1 +ν t ; ν t ∼N(0,σ2 ν) Theshocksη t,(cid:15) t andν t aremutuallyindependent. Thenoisetermν t inthesignalat preventstheagentsfromperfectlyidentifyingpermanentinnovationstotechnologyandgenerates variationintheagents’beliefsregardingxt,independentofthefundamentals. Itisapureshockto expectationsanddoesnotaffectproductivity. Permanentshocktoproductivityis(cid:15) t whichaffects aggregateproductivityandalsoaffectsbeliefs. Thetemporaryshockη t affectsagents’beliefsand realizedproductivityinthefirstperiodandonlyaffectsbeliefsinthesubsequentperiods. PersistenceinNoise. Here,thenoiseisassumedtobepersistent. Thisservesthepurposeof makingthesignalextractionproblemmorecomplexfortheagents. Agentsnownotonlycannot discern if a shock is persistent, transitory or noise, but they also cannot discern whether the persistenceinchangeinproductivityisattributedtoatruepersistentchangeinproductivityora persistentsignal. (cid:12) Letx t|t ≡ xt (cid:12) (cid:12) It denotetheagents’expectationsregardingxt conditionalontheirinformation setatdatet. Thisimplies,x t|t ≡ E t[xt]. Agentsupdatetheirbeliefsaboutxt inaBayesianmanner, usingaKalmanfilter. 21

Thus,thedynamicsofx is. t|t (25) x t|t = ρ xx t–1|t–1 +K t–1 (st –s t|t–1 ) where,st isthevectorofsignals(st = [zt,sˆ t]),andKt istheKalmangainmatrix. Thedetailsofthe filteringprocessareinAppendixSectionB.1. Timing. Here,thetimingofthesignalandexpectationformationiskey,whichisasfollows: 1. Firmsandworkersformexpectationsatbeginningoft withinformationsetIt–1. 2. Firmsandworkersmaketheirdecisionsfortimeperiodt. 3. Publicsignalisrevealedasisthevalueofzt. However,firmsandworkersdonotlearnfrom thissignalintimeperiodt (Pre-commitment). Thus,agentsmaketheirdecisionsabouttimetoutcomesbasedonthesignalaswellastheaggregate productivitytheyobservedattheendoftimeperiodt –1. Now,akeypointtonotehereisthat technically, theagentsinthemodelcouldobservevariousrealoutcomesintheeconomysuch aszt,yt,ct,ut,vt,st,wt andlearnaboutthetrueaggregateproductivity. However,assumingthat thatsignalextractionsiscostlyandagentsonlyusethesignaltolearnandmakedecisions. This assumptionissimilartoWoodford,2001,MankiwandReis,2002andAngeletos,Iovino,andJennifer, 2020whereagentsdonotlearnendogenously. Assumption1 Firmsandworkersobservepubliclyavailablerealvariablesinthemodelattimet,butdo notincludethemintheirinformationsetIt sincethissignalextractioniscostly. Furthermore,allthe informationprocessingthattheworkersundertaketolearnthestateoftheeconomyissummarizedin thepublicsignala . Agentsformtheirexpectationsatthebeginningoftheperiodbeforethesignalis t–1 revealedfort andpre-committotheirdecisionsthattheymakeint,whicharebasedonIt–1.12 Firm’sProblem Thereisacontinuumoffirmsindexedwithamassnormalizedto1. Allfirms produceahomogeneousgoodthatissoldinacompetitivemarket. Theaggregateproductivityin theeconomyiszt withatransitorycomponentη t andapermanentcomponentxt aboutwhich theagentsreceiveanoisypublicsignal. Firmsproducewithcapitalandlabor,andtheiroutput canbeusedforconsumptionorforcapitalaccumulation. Capitalisperfectlymobileandfirms rentcapitalonaperiodbyperiodbasis. Firmsaddlaborthroughasearchandmatchingprocess ζ 1–ζ describedabove. Theproductionfunctionisyt = ztk t l t . Letthestochasticdiscountfactorbe Λ t,t+1 ,wt bethewageperefficiencyunitoflaborandrt bethecapitalrentalrate. Iassumethat laborrecruitingcostsareconvexinthehiringrateoflaborinefficiencyunits,χ t. Thefirmsdecisionproblemisthereforetochooseχ t tomaximizethevalueofthefirmswhich isthediscountedstreamofprofitsnetofrecruitingcosts,wagesandcapitalrentalexpenses,subject 12Thisassumptioncanbethoughtofaseconomicdatareleasesbeinglaggedbyoneperiod.Firmsandworkerssee thedatareleaseint,whichcontaintheinformationfromt–1.Thisassumptiontriestomimicthisaspect.Inthemodel, therefore,nowcasterrorsareanalogoustothedatanowcasterrors. 22

tothelawofmotionforlt,gt andbt,andgiventheexpectedpathsofwagesandrentalrate. Firm’s solvethefollowingproblem: (cid:40) (cid:12) (cid:41) (26) Ft = m kt, a χ x t E t ztk t ζ l t 1–ζ – (1+ κ η h ) χ ( t 1+η h ) lt –wtlt –rtkt +Λ t,t+1 F t+1 (cid:12) (cid:12) (cid:12) It–1 subject to the law of motions of lt,gt and bt given in equations 18, 19 and 20. The first order conditionsgiveustherentalrateofcapitalandafirstorderconditionforhiring. (27) k t+1 : zt ζ (cid:16)lt (cid:17)1–ζ –rt = 0 kt (28) χ t : –κ(χ t) η hlt + E t(Λ t,t+1 F t+1 ) = 0 GivenCobb-Douglasproductiontechnologyandperfectmobilityofcapital,kt doesnotvaryacross firms. Itisalsoimportanttonotethatwhilethefirmpaysthesamerecruitmentcostsforbadand goodworkers(inqualityadjustedunits),badworkershavedifferentsurvivalrateswithinthefirm duetotheirincentivetosearchon-the-job. Thefirstorderconditionforhiringratecanbesolved togetthevacanciesvt sinceχ t = qtvt/lt. Eachfirmoptimizestheirhiringrateandinequilibrium, (cid:82)1 totalvacanciesaregivenbysummingacrossallfirms,v¯ = v di. 0 i Household’sProblem Thereisaunitmeasureoffamilies,eachwithameasureoneofworkers. Thefamilypoolsallwageandunemploymentincome. Consumptionandsavingsdecisionsare madeatthehouseholdlevel,buthouseholdmembersmaketheirdecisionsbasedonthesame informationsetI. Eachfamilyownsdiversifiedstakesinfirmsthatpayoutprofitsandassigns consumptionc¯ t tomembersandsavesintheformofcapitalk ¯ t,whichisrentedtofirmsatratert anddepreciatesattherateδ. Thehouseholdsolvesthefollowingproblem, (cid:110) (cid:111) (29) Ω t = max E t log(c¯ t)+βΩ t+1 k¯ t+1,c¯ t subjectto c¯ t +k ¯ t+1 +c(s t b )b ¯ t +c(s t u )u¯ t = w¯ tn¯ t +φw¯ tb ¯ t +u¯ tu B +(1–δ+rt)k ¯ t +Tt +Π t g¯ t+1 = σg¯ t +ξpts¯ t b ¯ t+1 = σ(1–ζ bt (1–ξ)pt)b ¯ t +(1–ξ)ptu¯ t g UnemployedWorkers LetUt bethevalueofunemployment,V t thevalueofagoodmatch,and Vb thevalueofabadmatch. u istheflowbenefitfromunemployment. Anunemployedworker t B searcheswithanendogenoussearchintensityζ ut. Thevalueofunemploymentisgivenby: (cid:40) (cid:32) (cid:33)(cid:12) (cid:41) (30) Ut = m ζ a u x E t u b –c(ζ ut)+Λ t,t+1 (1–pt)ζ utU t+1 +ζ ut(1–ξ)ptV t b +1 +ζ ut ξptV t g +1 (cid:12) (cid:12) (cid:12) It–1 t 23

Here,inthecurrentperiod,anunemployedworkerreceivesu ,netofsearchcosts. Int+1,With b probability (1–pt)ζ ut, an unemployed worker does not find a job and remains unemployed in t+1.13 Unemployedworkersfinditoptimaltoaccepteitherabadoragoodmatchiftheyreceive oneifthewagesaregreaterthantheiroutsideoption. EmployedWorkers Employedworkersearnawagew whileemployedatfirmj. Theworkersina j abadmatchsearchonthejobwithendogenousintensityσ andarematchedwithanotherfirm bt withprobabilityσ bt pt. However,Iassumethatemployedworkersonlymoveuptheladder. They switchjobsonlyiftheyfindafirmthatoffersabettercontinuationvalue. Employedworkersare separatedfromtheirjobwithexogenousprobability(1–σ),inwhichcasetheyhavetospendat leastoneperiodinunemploymentbeforetheycanbematchedwithanotherfirm. Theemployed workersolvesthefollowingproblem Valueofbeingemployedforaworkerinagoodmatchisgivenbythefollowing (cid:40) (cid:32) (cid:33)(cid:12) (cid:41) (31) V t g = E t wgt +Λ t,t+1 σV t g +1 +(1–σ)U t+1 (cid:12) (cid:12) (cid:12) It–1 Aworkerinagoodmatchearnswagewgt whileemployedinagoodmatch. Sincethereis no ladder to move up, these workers do not search on-the-job. In the next period, the worker caneithergettheseparationshockinwhichcasesheflowsintounemployment. Otherwisethey continuebeinginagoodmatchinthesubsequentperiod. Now,thevalueofbeingemployedforaworkerinabadmatchisgivenby: (cid:40) (cid:16) V t b = max E t φwt –σc(ζ bt )+Λ t,t+1 σζ bt (1–ξ)ptV t b +1 ζ bt (32) (cid:12) (cid:41) +σζ bt ξptV t g +1 +(1–σ)U t+1 (cid:17)(cid:12) (cid:12) (cid:12) It–1 Aworkerinabadmatchsearcheson-the-jobandhencechoosestheirsearchintensitytooptimize theirvaluefromabadmatch. Whileinabadmatch,theworkerearnsthewagew ,andifthe bt worker survives within the firm, which occurs with probability σ, she searches with variable intensityζ ,andsincesearchiscostly,theypaythecostofsearching. Inthenextperiod,ifthey bt arehitbytheseparationshocktheyflowintounemployment. Iftheyremainemployedinthebad match,theworkermightbematchedwithagoodjobinwhichcasetheymovetothegoodjobnext period. Ifmatchedwithanotherbadmatch,theworkerchoosestostayinthecurrentbadjob. WageContracts WorkersandfirmsdividethejointmatchsurplusviastaggeredNashbargaining àlaGertlerandTrigari,2009. Thefirmbargainswithworkersingoodmatchesforawagewhile workersinbadmatchesthenreceivethefractionofthewageforgoodworkers,correspondingto theirrelativeproductivity.14 Thus,whenbargainingwithgoodworkers,firmsalsotakeaccountof 13TheaveragevalueofemploymentinthecontinuationvalueofUt shouldbethatofanewhireratherthanthe unconditionalone.However,GertlerandTrigari,2009showthatthetwoareidenticaluptoafirstorder. 14Thiswageruleforworkersinbadmatchesapproximatestheoptimumwagefromdirectbargaining. 24

theimpliedcostsofhiringbadworkers. Forthefirm,therelevantsurplusperworkeris: Ft Jt = lt Forgoodworkers,therelevantsurplusisthedifferencebetweenthevalueofagoodmatchand unemployment: g Ht = V t –Ut The expected duration of a wage contract is exogenous. At each period, a firm faces a fixed probability 1–λ of renegotiating the wage and with λ probability, the wage from the previous 1 periodisretained. Theexpecteddurationofawagecontractis . Workershiredinbetween 1–λ contractingperiodsreceivetheprevailingfirmwageperunitoflaborqualitywt. Thewagew t N is chosentomaximize: (cid:40) (cid:12) (cid:41) (33) w t N = argmax Ht(wt) η Jt(wt) (1–η) (cid:12) (cid:12) (cid:12) It–1 wt subjectto (cid:40) wt withprobabilityλ (34) w = t+1 wN withprobability1–λ t+1 wherewN isthewagechoseninthenextperiodifthereisrenegotiationandηisthehouseholds t+1 relativebargainingpower. Now,toafirstorderapproximation,theevolutionofaveragewagescan bewrittenasfollows (35) w¯ t = (1–λ)w¯ t N +λw¯ t–1 Here,theaveragewagesandtheaveragecontractwagearedefinedby (cid:90) w¯ t = wdGt(w,γ) w,γ (cid:90) w¯ t N = w t N (γ)dGt(w,γ) w,γ dGt(w,γ)denotesthetimet fractionofunitsoflaborqualityemployedatfirmswithwagelessthan orequaltow andratioofbad-to-goodworkerslessthanorequaltoγ.15 ResourceConstraint Toclosethemodel,theresourceconstraintstatesthatthetotalresource allocationtowardsconsumption,investment,vacancypostingcosts,andsearchcostsisequalto 15Undermulti–periodbargaining,theoutcomedependsonhowthenewwagesettlementaffectstherelativesurpluses offirmsandworkersinsubsequentperiodswherethecontractisexpectedtoremainineffect. AsshowninGertler andTrigari,2009,uptoafirstorderapproximation,thecontractwagewillbeanexpecteddistributedleadofthetarget wagesthatwouldariseunderperiod-by-periodNashbargaining,wheretheweightsonthetargetforperiodt+idepend onthelikelihoodthecontractremainsoperativewhichisλi. 25

aggregateoutput κ (cid:90) (36) y¯ t = c¯ t +k ¯ t+1 –(1–δ)k ¯ t + κ t 2 ltdi+c(s t b )b ¯ t +c(s t u )u¯ t 2 i Thegovernmentfundsunemploymentbenefitsthroughlump-sumtransfers: (37) Tt +(1–n¯ t –b ¯ t)b = 0 Equilibrium TheaggregatestateoftheeconomyisdefinedbyΩ = {l,g,b,k,z,x T ,n T }. Arecursiveequilibriumischaracterizedasasolutionforasetof(i)valuefunctions{Jt,V t g ,V t b,Ut},(ii) prices{rt,w t N,w t+1 ,w¯ t,w¯ t N},(iii)allocations{χ t,ζ ut,ζ bt ,k ¯ t+1 ,c¯ t,g¯ t,g¯ t},(iv)thedensityfunctionof compositionandwagesacrossworkersdGt,atransitionfunctionQt,t+1,alawofmotionforthe economyΠ t,suchthatgiventhelawofmotionforexogenousvariableszt,xt andnt: 1. householdsoptimizesuchthatct,k t+1 satisfytheoptimalityconditions; 2. optimalsearchandhiring: ζ bt ,ζ ut,χ t optimizetheBellmanequationsforV t b,Ut,Jt; 3. wagewN satisfiestheNashBargainingRuleandw isgivenby35; t t+1 4. allMarketsClear: RentalmarketofCapitalclears,householdsoptimizeconsumptionand searchintensities. Firmsoptimizeonhiringdecisionsandcapitalinvestment; 5. g¯ t and b ¯ t evolve according to their respective laws of motion and the evolution of Gt is consistentwiththetransitionfunctionQ ; t,t+1 6. at each point in time, agents’ beliefs are determined by their information set It–1, their perceivedlawofmotionfortheeconomy. Agentsupdatetheirbeliefsabouttheaggregate productivityinaBayesianmannerwiththetimingconsistentwithAssumption1. SpecialCases 1. FullInformationBenchmark. Thegoaloftheoreticalframeworkistwofold. First,toassess whetherintroducingimperfectinformationimprovesthepredictionofdurationofrecovery ofunemploymentascomparedtoafullinformationframework. Second,tounderstandthe propagationmechanismforimperfectinformation. Foreitherscenario,itisimportantto definethefullinformationbenchmark. Underfullinformation,theagentsperfectlyobserve zt andxt eachperiodalongwithothervariables. Therefore,whenmakingtheirdecisions,the agentsarefullyawareofthestateoftheeconomyandcanperfectlyobserveeachcomponent of aggregate productivity. Hence there are only two shocks in this case: persistent and transitory productivity shock. As there is no information friction, and they immediately adjusttheirexpectationsinresponsetoanychangesintheeconomy. 2. ImperfectInformationWithoutNoise. Anotherimportantconsiderationistheroleofimperfectinformation,evenwithoutnoiseshocks. Inthisframework,Iassumethattheinformation structure is the same as in the imperfect information with noise shocks framework, and 26

Table3: ParameterValues Parameters Interpretation Value Source β Discountrate 0.99 Shimer,2005 δ Depreciationrate 0.025 GertlerandTrigari,2009 ζ Productionfunctionparameter 0.33 Gertler,Huckfeldt,andTrigari,2020 ω Elasticityofsearchcost 3.60 Fabermanetal.,2022 γ Worker’sbargainingpower 0.5 Shimer,2005 λ Renegotiationfrequency 0.75 Gertler,Huckfeldt,andTrigari,2020 α Elasticityofmatchestosearchers 0.4 Gertler,Huckfeldt,andTrigari,2020 η Hiringcostconvexity 2.40 MerzandYashiv,2007 h ρ z Technologyautoregressiveparameter 0.949 Shimer,2005 Note:Thistablereportstheparametervaluesthathavebeenfixedtowidelyacceptedexternalvaluesintheliterature. agentsobservezt andasignalsˆ t aboutthepersistentcomponentoftheaggregateproductivity xt. Here,thenoiseshocksareneverrealized,butagentsbelievethatthereissomenoisein theeconomyandadjusttheirexpectationsaccordingly. Searchandmatchingwithlaborforvhrmotivntodothismaximisthevalueofthefirmsacross tthesectors 4 Parameterization and Estimation I estimate the parameters in the model at a quarterly frequency using a three-step procedure. First,Ifixtheparameters{β,δ,ζ,ω,γ,λ,α,η h ,ρ z}towidelyacceptedvaluesfromtheliterature. Then,Iestimate{Ψ,κ,µ,σ,φ,u ,ξ}bytargetingsomeunconditionalstationarymomentsusingthe b simulatedmethodofmoments. Finally,theremainingparameters{σ (cid:15),σ nu,ρ n,K}areestimated tomatchtheimpulseresponsesofunemploymentrate,vacancies,outflowfromunemployment, job-to-jobtransitionrates,hiringratesandwagegrowthtotheidentifiednoiseshocksaswellas thepersistentproductivityshockinthedata. Thisexerciseyieldsasignal-to-noiserationof0.23, whichisconsistentwiththeliterature. Table3summarizestheresultofthecalibrationstrategy. I calibratetheoutputelasticityoflaborα = 0.33,thediscountfactorβ = 0.99,anddepreciationrate δ = 0.025towidelyacceptedvaluesintheliterature. TargetingUnconditionalMoments. Asafirststep,Itargetthesteadystateunemploymentrate, unemployment-to-employmenttransitions,job-to-jobtransitionsandseparationrateinthemodel tomatchtheaveragevaluesfromtheUnitedStatesfortheperiod1968-2019. Ialsotargettheflow value of unemployment, u , to match the relative value of non-work to work activity u¯ = 0.71 B T followingHallandMilgrom,2008.16 TheefficiencyparameterΨistargetedsuchthatthesteadystateunemploymentrateinthe modelmatchestheaverageunemploymentratefrom1968-2019inthedata,andtakesavalueof0.49. Thehiringcostparameterκdeterminestheresourcesthatfirmsinvestintorecruiting,andhence, 16Therelativenon-worktoworksatisfiesEXPRESSION.Thevalueofnon-workincludessavedsearchcostsfrom on-the-jobsearchandthevalueofworkincludessavedvacancypostingcosts. 27

Table4: UnconditionalTargetedMoments Parameters Interpretation Value Target Ψ Matchefficiency 0.49 UnemploymentRate= 0.055 κ Costofhiring 7.21 U –E = 0.28 µ Scaleparameterofsearchcost 0.082 E–E = 0.025 1–σ Separationrate 0.010 E–U = 0.010 φ SSproductivityfrombadjob 0.76 AverageE-Ewageincrease= 0.045 ξ Probabilityoffindingagoodjob 0.24 Averagewage-improv. flowshare= 0.53 u Flowvalueofunemployment 2.43 Relativevalue,nonwork= 0.71 b Note:ThistablereportstheparametersestimatedusingSimulatedMethodofMomentstotargetsomekeystationary momentsinthedata.Thesemomentsare:unemploymentrate,unemploymenttoemploymenttransitionrate,job-tojobtransitionrate,employmenttounemploymentseparationrateandwagechangeofworkerswhomakejobtojob transitions.Thesemomentsarecalculatedoverthesampleperiod1968q4-2019q4usingtheCPS. influencestheprobabilitythataworkerfindsajob. Isetthesteadystatejobfindingprobabilityto matchthequarterlyUEtransitionprobability,p˜ = 0.28;andthencalibrateκtobeconsistentwith p˜. Furthermore,ahighersearchcostimpliesalowerEEprobabilityandhence,thesearchcost parameterµistargetedtomatchEEprobability,andtakesavalueµ = 0.082. Theseparationrateσ istargetedtomatchtheE-Uprobability. Thesteadystateproductivityfromabadjob,φistargeted tomatchthechangeinwageofworkerswhomakejob-to-jobtransitions. Theratioofbadjobs togoodjobsisheldconstantandiscalibratedfollowingGertler,Huckfeldt,andTrigari,2020.17 I furthercalibrateζtomatchtheaverageshareofjobtransitionsinvolvingpositivewagechangesout oftotaljobflowsandtargetthisnumbertobe0.527,followingGertler,Huckfeldt,andTrigari,2020. Thecorrespondingvalueifζ = 0.23. Alowerprobabilityoffindingagoodjobcorrespondstoa highersteadystatevalueofbad-to-goodworkers,andhenceahigheraverageshareofbad-to-good flows. Finally,thehiringcostconvexityη istargetedtomatch h InformationParameters: ImpulseResponseMatching. Iestimatetheinformationparameters bymatchingmodel-impliedresponsesfollowinganoiseshock,andTFPshocktotheircounterparts intheempiricalexercise(Christiano,Eichenbaum,andEvans,2005;RotembergandWoodford, 1997). Thetargetsaretheresponsesofunemploymentrate,UErateandEErateforhorizonsofup to20quarters. Theimpulseresponsematchingisdonebyminimizingthedistancebetweenthe model-generatedimpulseresponsefunctions(IRFs)andtheempiricalIRFs. Letf,bethecolumn vectorstackingthepointestimatesofeachoftheseimpulseresponses,wherei = 1,··· ,N indexes thedifferentIRFs,andhisthehorizonatwhichtheIRFsarebeingevaluated. Themodel-generated IRFsaredenotedasfm(Θ),whereΘisavectorofmodelparameters. Theoptimizationproblemis givenas: (cid:48) (38) min(f –fm(Θ)) W (f –fm(Θ)) θ 17 b¯ (1–λ)(pEE+pEU) = g¯ pEE+λpEU wherepEEandpEU areprobabilityofE-EtransitionsandE-Utransitionsrespectively 28

Table5: EstimatedParametersfromIRFMatching Parameters Interpretation Estimate Std. Error σ (cid:15) Std. DevofPersistentTFPshock 0.062 0.009 σ ν Std. DevofNoiseShock 0.096 0.007 ρ n NoiseAutoregressiveParameter 0.921 0.004 K Signal-to-NoiseRatio 0.23 0.003 Note:Thistablereportstheestimatedparametersfromtheimpulseresponsematchingexerciseoutlinedinequation38. Thethirdcolumnreportstheestimatedvalueswhilethefourthcolumnreportsthestandarderrorsforthesevalues.The impulseresponsesarematchedbyGMMandthestandarderrorsarecalculatedusingthedeltamethod. where,Θ = σ2 (cid:15),σ2 ν,ρ n,K,whereKisthesignal-to-noiseratio. TheweightmatrixW istheinverse ofthevariance-covariancematrixoftheempiricalIRFestimates.18 TheresultofthisestimationprocessisdocumentedinTable5,withstandarderrorscalculated usingthedeltamethod(Guerron-Quintana,Inoue,andKilian,2017). Thesignal-to-noiseratiois 0.23,whichislowasthenoiseshockshavealargevariance. Thus,theimpliedstandarddeviation ofthetransitoryproductivityshockisfoundtobeσ η = 0.192,whichisrelativelylargecompared tothestandarddeviationofthenoiseshockaswellasthatofthepersistentTFPshock,sothat learningaboutthepersistentcomponentofproductivityisgradual. Thisimpliesthattheagentsin theeconomylearnquiteslowlyaboutthetruepersistentTFPcomponent. Figures7plotstheimpulseresponsesfromtheempiricalexerciseandtheimpulseresponses impliedbytheestimatedmodelinresponsetoanoiseshockandapersistentTFPshock. Themodel fitisgood,withallthemodelimpliedimpulseresponsesfallingwithintheconfidencebandsfrom theempiricalexercise. Theimpactaswellasthedynamicsforunemploymentrate,jobfinding rateandjob-to-jobtransitionsmatchestheempiricalimpulseresponseswell. Thedynamicsfor vacanciesandhiringratearenotmatchedwellasthemodelfailstocapturethecurvaturewhich theempiricalimpulseresponsesdisplay. Thebenchmarkmodelfortherestofthepaperistheimperfectinformationmodelwithnoise shocks. Iconsiderseveralcounterfactualmodels. Thefirstisthefullinformationmodelwhichis re-calibratedasdiscussedinAppendixSectionB.2. Inthisframework,firmsandworkersperfectly observezt,xt everyperiodandimmediatelyrevisetheirexpectations. Thesecondframework, istheimperfectinformationmodelwithoutnoiseshocks. Thismodelisnotre-calibrated. Here, firmsandworkersstillhaveimperfectinformationanddonotobservext,butonlyseethesignal sˆ t eachperiod. However,thenoiseshocksareneverrealized. Thismodelservesasanimportant comparisontohighlighttheroleofimperfectinformation. BusinessCycleStatistics Tounderstandhowthemodelwithimperfectinformationperformswith respecttotheobservedbusinesscyclestatisticsinthedata,Ireportthevolatilityandcorrelation ofseverallabormarketoutcomeswithoutputinTable6. Thetablecomparesthebusinesscycle statistics obtained by simulating the benchmark model and the full information model, to the 18Theobjectivefunctionbecomesaformofthegeneralizedmethodofmomentsestimator. Inthiscase,theoptimizationproblemaimstomatchmoments(theIRFs)inawaythatisefficientgiventhevariabilityoftheempirical estimates. 29

Figure7: ImpulseResponsesfromDataandEstimatedModel (a)NoiseShocks (b)PersistentTFPShock Note:ThisfigureshowstheresultsofIRFmatchingfornoiseshocksandpersistentTFPshocks. 30

statisticsintheUSeconomyfrom1968-2019forunemploymentrate(U),jobvacancies(V),job-tp-job transitions(EE),jobtransitionsfromunemploymenttoemployment(UE),andhiringrate. Whilebothmodelsofferreasonableapproximationsofoutputdata,theimperfectinformationmodeloutperformsthefull-informationmodelacrossallothervariables,bothintermsof standarddeviationandcorrelationwithoutputandisclosertoempiricalobservations. It’sworth Table6: BusinessCycleStatistics ImperfectInfo. ImperfectInfo. Data FullInformation withoutNoise withNoise (1) (2) (3) (4) (5) (6) (7) (8) x SD corr(Y,x) SD corr(Y,x) Info corr(Y,x) SD corr(Y,x) Y 0.019 1 0.019 1 0.021 1 0.024 1 U 0.162 -0.859 0.121 -0.742 0.132 -0.768 0.151 -0.792 V 0.182 0.702 0.131 0.642 0.157 0.675 0.196 0.728 EE 0.102 0.720 0.067 0.629 0.071 0.661 0.088 0.825 UE 0.069 0.734 0.044 0.639 0.058 0.653 0.077 0.692 HiringRate 0.058 0.677 0.034 0.571 0.036 0.622 0.042 0.723 Note:Thistablereportsstandarddeviationofkeylabormarketvariablesandtheircorrelationwithoutputinthemodel. ThedataherehasbeensimulatedfromthemodelandHP-filtered(100,00). acknowledgingthatthefull-informationmodelalreadyincorporatesfeatureslikewagerigidityand on-the-jobsearch—factorsknowntoinducevolatilityinsearchmodels(Shimer,2005).19 Yet,the introductionofimperfectinformationaugmentsthevolatilityofunemploymentbyanadditional 18%relativetothefull-informationbenchmark. Similarly,theimperfectinformationframework yields higher volatility for job vacancies and transition rates. This underscores the imperfect informationmodel’senhancedefficacyincapturingthedynamicsoflabormarkets. ForecastErrorVarianceDecomposition Theidentifiednosieshocksexplainaboutathirdofthe varianceinthelabormarketatashortrunhorizon. Tounderstandhowthebenchmarkmodel comparestotheobservedmoments,Ireporttheforecasterrorvariancedecompositioncalculated bysimulatingtheimperfectinformationmodel,inTable7,for8quarters. Thebenchmarkmodel canmatchtheforecasterrorvariancesof thekeylabormarketoutcomesobservedin thedata reasonalbly well. The model predicts that the noise shocks explain 31% of the forecast error varianceinunemploymentratewhichis90%oftheforecasterrorvarianceinthedataexplained bynoiseshocks. Onaverage,themodeloverpredictstheforecasterrorvarianceby7%. 5 Role of Imperfect Information in Labor Market Dynamics In this section, I first document that the calibrated model successfully matches the observed persistenceoftheunemploymentrate. Specifically,ittakes15quartersfortheunemploymentrate torecover50%ofitsrecessionaryincreaseinthemodelwhilethecorrespondingnumberis17 19InthecalibrationinTable6,thefull-informationmodelgeneratesabout75%oftheobservedunemployment volatility,whichalignswithGertler,Huckfeldt,andTrigari,2020. 31

Table7: FEVD:DataandModel Data Model Horizon: 0-8quarters Horizon: 0-8quarters Persistent Transitory Noise Persistent Transitory Noise Unemployment 0.43 0.23 0.34 0.48 0.21 0.31 Vacancies 0.42 0.21 0.37 0.49 0.19 0.32 Job-findingRate 0.38 0.27 0.35 0.42 0.21 0.37 E-E 0.42 0.31 0.27 0.49 0.17 0.34 Wages 0.61 0.25 0.14 0.60 0.22 0.18 Note:Thistablereportstheforecasterrorvarianceinthemodelwithimperfectinformationandcomparesittothe momentsinthedata. quartersinthedata. Inthemodelwithoutimperfectinformation,itwouldtakeonly9quarters. Thisadditionalpersistenceisgeneratedthroughtwochannels. First,learningunderimperfect informationgeneratesendogenouspersistence: agentsinitiallyunder-reacttoproductivityshocks, andwithstickywages,thiscausesaninitialdelayinthedynamicsthatplayoutinthelabormarket, thus generating higher persistence. Second, noise shocks themselves may prolong recoveries, as agents mistake them for actual productivity shocks and adjust behavior accordingly—even thoughfundamentalsareunchanged. Forexample,anegativenoiseshockleadsagentstopartially attributetheperceiveddeclineinproductivitytoanactualchangeinproductivity,althoughthe trueproductivityremainsunchanged. Thiscausesfirmstoanticipatelowerreturnsfromhiring, reducingvacancies. Unemployedworkers,expectinglowerwages,reducetheirsearcheffort,and employedworkersarelessmotivatedtomoveupthejobladder. Theseresponseslowerjob-finding ratesandmatchingefficiency,resultinginhigherandmorepersistentunemployment. Together, thesemechanismsgeneratepersistenceintheunemploymentdynamicsfollowingdownturns. Afterestablishingthemechanism,Idocumentacounterfactualexercisetodemonstratethat imperfectinformationcancontributesignificantlytotheslowrecoveryofunemploymentrate inrecessions,andshowthatthemodelwithimperfectinformationcanmatchthepathofunemploymentrateacrossvariouspostwarrecoveriessuccessfully. Finally,Idiscusstheimportanceof variousotherchannelssuchasstickywagesandon-the-jobsearch,whichhavebeenproposedas possiblechannelsforgeneratinghigherpersistenceinsearchandmatchingmodels. Ifindthat whilestickywagesalonecangenerateabout45%ofthepersistence,itsinteractionwithimperfect informationaccountsforabout70%ofthepersistence,highlightingtheroleofthisinteraction. 5.1 MechanismforPropagationofShocksunderImperfectInformation Figure8illustratestheeffectofaonestandarddeviationnegativepersistentproductivityshock on key outcomes. The solid lines represent the imperfect information framework, while the dashedlinesarethefullinformationbenchmarkwherefirmsandworkerscanperfectlyobserve thepersistentandtransitorycomponentsofaggregateproductivityeachperiod. Inanenvironmentcharacterizedbyimperfectinformation,agents—bothfirmsandworkers—operateundera Bayesianlearningframeworkwheretheyassignprobabilisticweightstoshocksaseitherpersistent, 32

transitory,ormerenoise. Ifindthatincorporatingimperfectinformationabouttheunderlying persistenceofaggregateproductivityshocksincreasesthepersistenceofunemploymentrelative tothefull-informationmodel,by30percentandthroughtwochannels. Channel1. First,theresponsetoapersistentproductivityshockisevenmorepersistent. Consider anegativepersistentproductivityshock. Initiallyfirmsandworkersassignpositiveprobabilities totheshockbeingpersistent,transitoryornoise. Thiscausesanoverallunder-reactioninitially thatisamplifiedbywagerigidity. Aswagesaresticky, firmsdonotwanttoadjustwagesifthe shockisnoiseortransitory,sinceanincreaseinwageswouldnotbeoptimalwhensubsequently productivitylevelsactuallydecline,therebyreducingthefuturediscountedprofits. Now,initially relativetoafullinformationmodel,wages,vacancies,andsearcheffortdoesnotdeclinemuch, sincetheagentsbelievethattheshockcouldjustbenoiseortransitory. Thiscausesanoverall initialdelayinthedynamicresponseofthelabormarket. Figure8: ImpulseResponsetoaPositivePersistentTFPShock Note:Thisfigureshowstheimpulseresponsefunctionsforthere-calibratedfullinformationmodel(dashedblackline) andtheimperfectinformationmodel(solidgreenline)toanegative1standarddeviationpersistentTFPshock. However,asfirmsandworkersupdatetheirbeliefsaboutthechangeinproductivity,they assignmoreweighteachperiodtothechangebeingtrulypersistent. Thisadjustmentinexpectationsissignificantlysloweddownbythepersistenceinthesignalasagentsfaceamorecomplex signalextractionproblem. Asthefirmsandworkersputmoreweightontheshockbeingpersistent, thefirmsthatgetachancetorenegotiatenowofferlowerwagesandasaresulttheaveragewage decreases. Unemployedworkersdecreasetheirsearcheffortasreturnsfromemploymentdeclines. Employedworkerslookingtomoveuptheladderdecreasetheirsearcheffortasaveragewages 33

havedeclined. Further,firmspostfewervacanciesastheynowplacemoreweightontheshock beingatruenegativepersistentshock. Consequently,thisgeneratessluggishnessinthejob-finding rate. Eventually,unemploymentstartsincreasinginahump-shapedtrajectoryandweobservea morepersistentresponseinunemploymentthanthefullinformationframework. Thisillustratestheimportanceofstickywagesinpropagatingtheimperfectinformation. If wageswereflexible,theycouldkeepadjustimmediatelyasagentslearnaboutthenatureofthe shock,andrecoverywouldbefaster. Furtherpersistenceisgeneratedbytheon-the-jobsearch,as employedworkerscrowdouttheunemployedworkersinbadmatches,andalsocausefirmsto postfewervacancies. Thissignificantlydelaysthejob-findingprocessfortheunemployedworkers, thus contributing to the persistence of unemployment. With the combination of sticky wages, on-the-jobsearchandimperfectinformation,persistentproductivityshockshavesignificantly morepersistenteffectsonunemploymentthanwouldunderthefullinformationframework. Channel2. Thesecondchannelcomesfromthenoiseshocksitself,whichprovideanindependent sourceofpersistenceinthelabormarket. Whenthereisanegativenoiseshock,agentspartially attributetheperceiveddeclineinproductivitytoanactualchangeinproductivity,althoughthe trueproductivityremainsunchanged. Thisleadsfirmstoexpectlowerreturnsfromnewhires whichsubsequentlyreduceslabordemandandthenumberofjobvacancies. Atthesametime, unemployedworkersanticipatelowerwagesduetotheperceivedfallinthepersistentcomponent ofproductivity,decreasetheirsearchintensity,resultinginalowerjob-findingrate. Employed workersalsoreducetheirsearcheffortastheirincentivetomoveupthejobladderdeclinesas theyexpectlowerwages. Inequilibrium,thesefactorsleadtofewermatchesbetweenfirmsand workers,therebyincreasingtheunemploymentrate. Boththesechannelsareamplifiedthroughwagerigidityandendogenouson-the-jobsearch. During downturns, both these channels may act together to amplify the persistence of unemployment as firms and workers receive a sequence of all the shocks. Agents overestimate the persistenceofthenegativeproductivityshockduetopresenceofnoiseshocksandperceivethe negativeproductivityshocktobepersistentlyworsethanitis. Idiscussthedynamicsoftheunemploymentrateduringrecessionsinthenextsection,whereIshowthatthemodelwithimperfect informationcanexplaintheslowrecoveryofunemploymentrateduringrecessions. Duringdownturns,boththesechannelsmayacttogethertoamplifythepersistenceofunemploymentasfirmsandworkersreceiveasequenceofalltheshocks. Agentsoverestimatethe persistenceofthenegativeproductivityshockduetopresenceofnoiseshocksandperceivethe negativeproductivityshocktobepersistentlyworsethanitactuallyis. Sincetheygraduallylearn whetherashockispersistentortransitory,theyrespondasiffacingamorepersistentnegative productivityshockthanthetrueshock. Specifically,firmsanticipateproductivitytobepersistently worseandthereforeexpectlowerfuturerevenueandpostfewervacanciesforlongerthanwould beconsistentwiththetruedeclineinproductivity. Thedeclineinvacancieslowerjobofferarrival ratesforboththeunemployedandtheemployedwholowertheirsearcheffortduetothedecline inreturntosearch. Sincemostofthebadjobsremainoccupiedbytheemployedworkersandgood jobsarehardtofind,thejob-findingrateoftheunemployeddeclinesfurtherduetothecongestion 34

inthelowerranksoftheladder. Consequently,thejob-findingratedeclinesfurtherwhichkeeps theunemploymentrateelevatedlongerthanimpliedbythetruestateoftheeconomy. 5.2 UnemploymentDynamicsacrossRecessions: DatavsModel Inthissection,Isimulatethecalibratedimperfectinformationmodeltogeneratecounterfactual unemploymentrateseriesfor5recessionsbetween1970-2019. Thisexerciseshowsthatimperfect information explains the slow recovery of the unemployment rate, specially in the last three recessions. Forthisexercise,themodelisnormalizedtomatchthestartingunemploymentrate foreachoftherecessions. Whilesimulatingtheimperfectinformationmodelwithnoise,allthree identifiedshocksfromtheVARareincorporatedeachperiod. Fortheimperfectinformationmodel withoutnoise,onlythepersistentandtransitoryshocksfromtheVARareincorporatedeachperiod. Forthefullinformationmodel,Iintroducethepersistentandthetransitoryshockseachperiod. Furthermore,thefullinformationmodelisre-estimatedasdescribedintheprevioussection,to matchtheempiricalIRFstothepersistentproductivityshocks. Theestimatedparametersforthe fullinformationmodelarepresentedintheAppendix. Figure9: ModelImpliedRecoveryofUnemploymentforRecessionsPost1990s Note:Thisfigureshowsthemodelimplied,simulatedunemploymentrateforthere-calibratedfullinformationmodel (dashedblueline),theimperfectinformationmodelwithoutnoise(grayline)andtheimperfectinformationmodel (solidgreenline)fortheGreatRecession,2001recessionandthe1990-91recession. Theimperfectinformationmodelpredictsthepersistenceoftheunemploymentrateforthe threerecessionspost1990quitewell. Incontrast,thefullinformationmodelpredictsmuchfaster recoveries. FortheGreatRecession,theactualunemploymentratetook37quarterstorecovertoits pre-recessiontrough. Theimperfectinformationpredictstherecoveryat32quarterswhereasthe fullinformationmodelpredictsamuchfasterrecoveryat24quarters. Inotherwords,theimperfect informationmodelpredictsalmost33%slowerrecoverythanthefullinformationbenchmark. 35

Forthe2001recession,thedurationofrecoveryforunemploymentwas24quarters,andthe imperfectinformationmodel(20quarters)predicts25%slowerrecoverythanthefullinformation model(16quarters). In1991recession,theunemploymentrecoverytook28quarters. Theimperfect informationmodelpredictedarecoveryat22quarters,almost38%higherthanthefullinformation benchmarkwhichpredictedrecoveryat16quarters. Thishighlightsthecontributionofimperfect information to the persistence of the labor market. Larger noise shocks further dampen the economyasfirmsandworkersperceiveanegativeproductivityshocktobemorepersistentthan itactuallyis. Theslowlearningbyagentscombinedwithstickywages,translatesintoaslower recovery. Forthepre-90srecessions,thefullinformationandthetwoimperfectinformationmodels arecomparableintheirpredictionforrecoveryofunemploymentrate. Thisisbecausethenoise shocksidentifiedintheSVARplayamuchsmallerroleinexplainingthefluctuationsinthelabor market. Withsmallernoiseshocks,theagents’perceivedproductivity,whilestilllowerthanactual, wasnottoofarofffromthetrueproductivity. Thus,theimperfectinformationmodelspredict similarrecoveryrelativetothefullinformationmodel. Tobeprecise,inthe1981-82recession(14 quarters),theimperfectinformationmodelwithnoiseshockspredictedrecoveryin11quarters whilethefullinformationmodelpredictedrecoveryin9quarters. Forthe1973-78recession(23 quarters),theimperfectinformationmodelwithnoiseshockspredictedrecoveryin17quarters whilethefullinformationmodelpredictedrecoveryin14quarters.Theresultsarealsosummarized inAppendixTableB2. Figure10: ModelImpliedRecoveryofUnemploymentforRecessionsPre1990s Note:Thisfigureshowsthemodelimpliedsimulatedunemploymentrateforthere-calibratedfullinformationmodel (dashedblueline),theimperfectinformationmodelwithoutnoise(grayline)andtheimperfectinformationmodelwith noiseshocks(solidgreenline)fortheGreatRecession,2001recessionandthe1990-91recession. 36

5.2.1 ModelDecomposition: StickyWages,On-the-jobSearchandImperfectInformation Inthissection,Icomparethepersistenceofunemploymentundervariousmechanismswithand withoutimperfectinformation. Inthismodel,therearethreefactorsthataddtothepersistence of the unemployment rate: on-the-job search, sticky wages and learning. To understand the contributionofeachchannel,Icompareafullinformationbenchmarktoimperfectinformation modelunder4scenarios: a)flexiblewageswithouton-the-jobsearchb)flexiblewageswithonthe-job search, c) sticky wages without on-the-job search, and d) sticky wages with on-the-job search. The measure of persistence I use is the average number of quarters across recessions to recover50%oftheriseinunemploymentduringrecession. Icalculateforeachrecessionbetween 1968-2019, the share of the rise in unemployment during the recession that has been reversed duringtheexpansionfollowingEquation1. Foreachrecession,Icalculatetheaveragenumberof quartersittakesfromthebeginningoftherecessiontorecover50%oftheriseinunemployment. Empirically, it took 17 quarters from the beginning of the recession to recover 50% of the rise inunemploymentacrossrecessionsbetween1968-2019. Tohighlightthatlearningcontributesto persistenceundereachspecification,IplotthismeasureinFigure11. Inthestackedbargraph,I re-calibratethefullinformationmodelasdiscussedinSectionB.2. Figure11: AverageDurationtoRecover50%ofRiseinUnemploymentAcrossModels Note:Thisfigureshowsthemodelimplieddurationtorecover50%oftheriseinunemploymentfromthebeginningof therecession,averagedacrossrecessionsbetween1968-2019forvariousmodelspecification.Thepercentagesarethe percentofthedata(18quarters)thattheparticularmodelspecificationexplains,whilethex-axisistheactualnumber ofquartersexplainedbytheparticularspecification.Thegreenbarsareincrementalcontributionsbylearning,which impliesthatthetotalcontributionoftheimperfectinformationmodelisthesumoftheblueandthegreenbar.Here, thefullinformationmodelisre-calibratedasdiscussedinSectionB.2.Further,Ishutdowneachmechanismoneby oneinbothmodels. 37

Roleofstickywages. Stickywagesgeneratepersistenceinthelabormarketbymakingwage adjustmentsluggish. Aswagesareslowertoadjustdownwardsduringrecessions,incentivefor firmstohireworkersremainslow. Thus,hiringdeclinesandresultsinalowerjob-findingratewith higherunemploymentforlongerthanifthewageswereflexibletoadjust. Introductionofsticky wageswithnoon-the-jobsearchcontributessignificantlytothepersistenceofunemployment,and underfullinformation,itaccountsfor45%(7quarters)ofthe50%oftheriseinunemployment. Role of on-the-job search. During downturns, on-the-job search creates congestion for the unemployedworkersasthepoolofemployedworkersinbadmatchesincreases. Thisisbecause theincentiveforemployedworkerstomoveuptheladderislowduetolowerproductivity. Andas theprobabilityoffindingagoodjobisexogenouslylow,unemployedworkershavefewervacant lowproductivityjobstomoveintoandthus,findingajobtakeslonger,creatingpersistenceinthe unemploymentrate. Furthermore,employedworkerssearchwithlowerintensityduetowhichthe firmspostfewervacanciesandthusalsodampenstheunemployedworkersprobabilityoffinding ajob. Whenon-the-jobsearchisintroducedtoaflexwagefullinformationmodel,itpredictsabout 30%(5quarters)ofthedurationtorecover50%oftheriseinunemploymentrate. Whenonthejob searchinteractswithstickywages,thefullinformationmodelpredictsabout54%(9quarters)of thepersistenceobservedinthedata. Roleofimperfectinformation. Theprevioustwoparagraphshighlightedthatbothon-the-job searchandstickywagesendogenouslygeneratepersistence,predictingtheunemploymentrateto recover50%ofitsrisein10quarters. Tounderstandhowmuchdoesimperfectinformationadd tothepersistence,wemustlookatthecontributioncomingfromlearningacrossallthemodel specifications. Duringdownturns,duetonoiseshocks,firmsandworkersperceivetheaggregate productivityshockstobemorepersistentthantheyare. Sincetheyonlylearnovermultipleperiods whetherashockispersistentortransitory,thisgeneratesasluggishnessastheykeepbehavingasif facinganegativeproductivityshockwhichismorepersistentthanthetrueshock. Firmsanticipate lowerreturnsfromhiringwhileworkersdeclinetheirsearcheffort,leadingtolowernumberof matchesforaslongastheylearnaboutthetrueshock. Whenintroducingimperfectinformation tothetheflexiblewageswithouton-the-jobsearchframework,persistenceincreases,predicting theunemploymenttorecover50%ofit’srisein6quarters(35%ofthetotalduration). Learninginteractswithon-the-jobsearchandgeneratesanadditional16%tothepersistence ofunemployment,predictingthe50%oftherecoveryin8quarters. Here,theemployedworkersare learningslowlyaboutthetrueshockandtheyanticipatetheproductivitytobepersistentlyworse thanactual. Employedworkerssearcheffortremainsdampened,whichleadsfirmstopostfewer vacanciesandthusalsodampenstheunemployedworkerschancesoffindingajob. Thisleadsto dampenedjobfindingratesforlongerandhencehigherdurationofrecoveryofunemployment rate. Learninginteractswithstickywagesandgenerateshigherpersistenceevenwithouton-the-job search. Firmsanticipatetheproductivitytobepersistentlyworsethanactual,whichdecreases theirincentivestohirewhichisfurtheramplifiedbywageswhichareslowtoadjustdownwards. Therefore, hiring remains dampened as they slowly learn about the true shock. Workers also 38

decreasesearcheffortandcombinedwithlowerhiring,thisleadstodampenedjobfindingrates andhenceunemploymentratetakesmuchlongertorecover. Imperfectinformationwithsticky wagespredictthatitwouldtake11quarterstorecoverthe50%oftheriseinunemployment. This is65%oftherecoverydurationinthedataandis24%higherthanthefullinformationstickywage model. Finally,Ishowthatimperfectinformationwithstickywagesandon-the-jobsearchaddsa substantial5.5quarters(30%)totheaveragedurationtorecover50%oftheriseinunemployment, ascomparedtothefullinformationcounterpart. Inequilibrium,asfirmsandworkersanticipate theproductivitytobepersistentlylowerthanactualduetoimperfectinformation,stickywages declinetheincentivestohireevenfurther,whiledeclineinon-the-jobsearchmakesitharderfor unemployedworkerstofindjobs,thusendogenouslygenerating84%ofthepersistenceobserved inthedata. InAppendixSectionB.5.2,Idiscussthemodeldecomposition,comparingthefullinformation modelwiththeimperfectinformationmodelwithoutnoiseshocks. Thisanalysesestablishesthat learningendogenouslyleadstohigherpersistenceofunemployment,evenwithoutnoiseshocks. In the model with sticky wages and on-the-job search, imperfect information adds 18% to the persistenceofunemploymentrate(approximately3quarters). Thishighlightstheimportanceof incorporatingimperfectinformationinmodelsofsearchandmatchingtoaccuratelypredictthe durationofrecoveryofunemploymentfromrecessions. 5.2.2 UnemploymentForecastsintheModel. Themodelpresentsauniquefeaturewithrespecttotheunemploymentforecast. Whenfaced withapersistentproductivityshock,duetoimperfectinformation,agentsattributeapartofthe shocktobenoiseaswellastransitoryshockandhencetheirprojectionsunder-reacttotheactual unemploymentrate. However,thereversehappenswhentheyfaceanoiseshock. Agentsattribute somepartofanegativenoiseshocktobepersistentortransitoryproductivityandhenceinitially forecasttheunemploymentratetobehigherthanitactuallyis(sincetrueproductivityhasnot changed). Theyeventuallystartplacingmoreandmoreweightontheshockbeingnoiseandas theylearn,theirforecastsconvergestothetrueunemploymentrate. ThisisillustratedinFigure12 wherePanel(a)showsthe4-8quarteraheadunemploymentforecastsbyagentsinthemodelin responsetoanoiseshockalongwiththeimpulseresponseofunemploymentrate. Panel(b)shows theresponseofactualandforecastedunemploymentrateinresponsetoapersistentTFPshock. To illustrate that the noise shocks in this model can replicate the over-shooting of unemploymentprojectionsobservedinthedata,Isimulatethemodelandgeneratelongrunforecasts. Figure12bshowsthemodelgenerated,one,twoandthreeyearaheadunemploymentprojections inthemodelaftertheGreatRecession. Here,allthreeshocks,identifiedfromtheVAR,acttogether eachperiodwhilesimulationtheimperfectinformationmodelwithnoiseshocks. Sinceallthree shocksact,theprojectionsunder-reactifthecontributionofthepersistentshockdominatesthe contributionofthenoiseshocksaswellastransitoryshocks. Similarly,asthecontributionofthe noiseshocksdominates, theprojectionsover-estimatetheunemploymentrate. Asseeninthe historicaldecompositionoftheunemploymentrateinthedatainFigureA11,thecontributionof 39

Figure12: Long-RunUnemploymentProjectionsintheModel (a)ResponseofUnemploymenttoNoiseShock (b)UnemploymentProjections:GreatRecession Note:Panel(a)showsthemodelimplied4-8quartersaheadprojectionsinresponsetoanoiseshock.Thesolidthick blacklineistheactualresponseofunemploymentduetotheshock.Panel(b)showsthemodelimpliedforecastsfor unemploymentrate1,2and3yearsahead.Thedashedblacklineisthemodelsimulatedunemploymentrateforthe GreatRecession.Whilesimulatingthemodel,eachperiodallthreeshocksact. thenoiseshockstothemovementinunemploymentdominatesafter2012. Thus,inthemodel, initially,astheproductivityshockshavehigherweight,theunemploymentrateisunder-estimated bytheagentsinthemodel. However,from2012,thecontributionofthenoiseshocksincreasesbut theagentsareunabletodiscerntheshockfromatruepersistentproductivityshockandhence keepexpectinghigherunemploymentratesinthefuture. However,astheshockistrulynoise,the actualunemploymentrateislowerthanexpected. Thisissimilartothepatternseeninthedatain Figure2b. Itisimportanttonotethatthenoiseshocksareuniqueingeneratingover-estimationof longrununemploymentprojections. Forallstructuralshocks,thelongrunexpectationsunderestimatetheunemploymentrate. Thus,noiseshockscanbeapotentialsolutiontotheconsistent patternobservedinthedatawherethelong-rununemploymentforecastsareover-estimatedby professionalforecasters. 5.2.3 UnemploymentDynamicsduringCOVID-19Recession Finally,IconsidertherecoveryfromtheCovid-19recession,whichstandsoutasoneofthequickest recoveriesinpostwarhistory. Theunemploymentraterosefrom3.5%inFebruary2020toapeak of14.7%,anddeclinedto3.9%attheendof2021. Theriseinunemploymentwasprimarilydue to temporary layoffs. Typically, unemployment from temporary layoffs declines quickly once economicactivityimprovesasworkerscantoreturntoworkquicklywhenlabordemandimproves. Tounderstandtheroleofimperfectinformation,Ifirstlookattheshorterhorizonexpectations(1-4 quartersahead)oftheprofessionalforecastersinFigure13a,andthelongerhorizonexpectations (1-3 years ahead) in Figure A2b. Forecasters revised their shorter horizon expectations much fasterby2021q4andthelongerhorizonprojectionsby2021,suggestingthattheyexpectedthis recessiontobetransitoryrelativetootherrecessions. Inourframework,thissuggeststhatthe roleofnoiseshockswasnotveryhighduringtheCovid-19recession. Toformallyunderstandthe roleofimperfectinformation,IextendtheSVARbyincludingtheCovid19pandemicinmysample (1968-2022). Now,themax-shareidentificationassumesthatthepersistentTFPshockmaximizes 40

Figure13: COVID-19:ProjectionsandContributionofShocks (b) COVID-19: HistoricalContributionofShocksto (a)ShorterHorizonProjections UnemploymentRate Note:Panel(a)showsthemedian1-4quartersaheadprojectionsofunemploymentratefromtheSurveyofProfessional ForecastersduringCovid-19.Panel(b)showsthehistoricaldecompositionofunemploymentratefollowingequation16. TheblacklineisthecumulativecontributionoftheidentifiedpersistentandtransitoryTFPshockstothemovementsin demeanedunemploymentrate(redline).ThedashedbluelineisthecontributionoftheTFPshockandthenoiseshocks. theFEVofTFPatalongrun. However,giventhattheCovid-19pandemichappenedlessthan20 quarters ago, I assume that the persistent shock maximizes the forecast error variance ofTFP evenintheshortrun. However,sincetheidentificationofnoiseshocksdoesnotdependonthe max-shareidentification,thisexerciseisstillinformativeaboutthedynamicsofunemployment ratedrivenbynoiseshocks. InFigure13b,thenoiseshocksdidnotplayanimportantroleinthis recession,asmostofthefluctuationispickedupbytheTFPshocks. Thiscanbeexplaineddue totheexpectationsthatadjustedveryquicklytotherecoveryandintheshortrun, forecasters predictedatransitoryrecovery. Thissuggeststhatnoiseshocksdidnotplayanimportantroleand althoughtherewassomedegreeofmisperception,itwaslessthaninotherrecessionssuchas 2007-09. Inthiscase,thefundamentalshockscontributedtomostoftheriseandthequickrecovery. 6 Conclusion This paper assess the role of imperfect information in labor market fluctuations and recovery patterns. Usingatri-variatestructuralVARmodel,Iidentifynoiseshocksandtheirsignificant effectsonlabormarketdynamics. Idocumentthatnoiseshockscanbeanimportantdriverof theslowrecoveryofunemploymentduringrecessions. Ifindthatwithoutnoiseshocks,thelabor marketwouldhaverecoveredfasterbysixquartersonaverageinthedownturnsbetween1968-2019. Furthermore,noiseshocksaccountforone-thirdofthevarianceinunemployment,jobfinding rate,andvacancypostingsatthebusinesscyclefrequency. Theresponseoflabormarketoutcomes totheidentifiednoiseshocksissignificantatthebusinesscyclefrequencyandishump-shaped. Thequantitativelyandstatisticallysignificantresponsetonoiseshockssuggestthepresenceof informationfrictionsandthehump-shapeindicatesthatfirmsandworkersarelearningunder imperfectinformation. Theseresultsthenmotivatetheintroductionofimperfectinformationinto ageneralequilibriummodelofsearchandmatching. Theintroductionofimperfectinformationinageneralequilibriummodelprovidesamore 41

robust framework for explaining the phenomena observed in the labor market. The model is calibratedtomatchunconditionalmomentsinthedataaswellastheimpulseresponsestothe identifiedshocksfromtheSVAR.Theimperfectinformationmodelwithnoiseshockspredicts∼30 percenthigherpersistenceinrecoveryofunemploymentonaverageacrossrecessions,relative to the model with full information. During downturns, firms and workers receive a sequence ofalltheshocks,agentsoverestimatethepersistenceofthenegativeproductivityshockdueto presenceofnoiseshocksandperceivethenegativeproductivityshocktobepersistentlyworse than it actually is. Since they gradually learn whether a shock is persistent or transitory, they respond as if facing a more persistent negative productivity shock than the true shock. Firms thereforepostfewervacanciesforlongerandworkerssearchwithlowerintensity,whichleads topersistentlydampenedjobfindingrates. Thisleadstoaslowerdeclineinunemploymentrate, whichisfurtherexacerbatedbystickywageswhicharesluggishtoadjustinitiallyaswellasdueto on-the-jobsearch. Thisisparticularlytruefortheperiodafter1990,emphasizingtheincreased importanceofimperfectinformationinmorerecenteconomicconditions. Inconclusion,itis importantformodelsofbusinesscyclesinthelabormarkettoconsidertheroleofnoiseshocks. 42

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A Empirical Appendix A.1 DataSources Thissubsectiondescribesallthedatasourcesusedinthispaper. Thesampleperiodforallthe primaryanalysisis1968q4to2019q4. 1. Unemploymentrateandunemploymenttoemploymenttransitionratesareconstructedfrom theCurrentPopulationSurvey(CPS). 2. Since1994,theCPShasaskedindividualswhethertheystillworkatthesamejobasinthe previousmonth. However,itisnotpossibletoobservejob-to-jobtransitionspriorto1994 andmysamplechangesto1994-2019forthejob-to-jobtransitions. 3. Vacanciesaremeasuredasanindexconstructedbasedonthecompositehelp-wantedindex computedBarnichon,2010a,asitgoesbacktothebeginningofmysample(1968). 4. RealGDPandwage(averagehourlyearnings)seriesarefromBEA. 5. Aggregate productivity is measured as the Solow residual, for which I rely on the utility adjustedseriesfromFernald,2014,alsoupdatedbytheFederalReserveBankofSanFrancisco. 6. NowcastErrors: TheGDPnowcasterrorsareconstructedfromthemedianforecastfromthe SurveyofProfessionalForecasters,whichstartsin1968q4. 46

A.2 RecoveryPatternoftheLaborMarket The labor market recovery has been sluggish and typically lags behind the recovery of output. FigureA1illustratestherecoverypatternofunemploymentandvacancies. Inparticular,thelast three recessions before the pandemic have been slower. Specially the recessions occurring in 1990-91,2001,and2007-09,displaydistinctfeaturesfromthepostwarrecoveriesobservedbefore the1990s. Intheserecoveries,unemploymentlevelsremainedelevated,whilebothemployment growthandjobvacanciesweresluggishformultiplequartersfollowingthetroughinoutput. I computethedurationtorecovertheriseinunemploymentacrossvariousrecessions. Icalculate thefollowingurecovery durationtorecover25%,50%,75%and100%riseinunemploymentacross recessions. ThesearereportedinTable1. FigureA1: RecoveriesacrossRecessions (a)UnemploymentRate (b)Vacancies Note:Thisfigureplotstherecoveryofunemploymentrateandvacanciesfromrecessionstotheirpre-recessionlevels. ThevacancyseriesisfromBarnichon,2010a A.3 ProjectionsfromForecastersandPolicymakers Asnotedintheintroduction,thestandardassumptioninmostmacroeconomicmodelisthatagents immediatelyrecognizethenatureofsuchashockandadjusttheirexpectations(anddecisions). However,itcantakeagentsmuchlongertolearnaboutthetruenatureoftheshock. A.3.1 ProfessionalForecasters. SurveyofProfessionalForecasters. Toillustratethispoint,Ifirstpresentunemploymentprojections from the Survey of Professional Forecasters (SPF) during the recovery from the Great RecessioninFigureA2. Thisillustratesthata)theforecastersconsistentlypredictedunemploymenttobehigherthanitactuallywasduringtherecoveryfromtheGreatRecessionand,b)this misperceptionaboutthetruenatureoftheshocklikelycontributedtohigherpersistence,asthe historicaldecompositionofunemploymentrateinFigure??suggests. TheSPFdocumentslongrun projectionsstartingin2009andhencelongrunprojectionsarenotavailableforearlierrecessions. However,Iprovide1yearaheadforecastsinFigureA4a,whichdocumentsthatforecastersalmost alwayspredicttherecoverytobeslowerthanitis. 47

FigureA2: ProjectedUnemploymentRatefromSurveyofProfessionalForecasters: (a)Projections:GreatRecession (b)Projections:Covid-19 Note:Thevariouscoloredlinesrepresentsthemedian1year,2yearand3yearaheadprojectionofunemploymentrate fromtheSurveyofProfessionalForecasters.Thesolidredlineistheactualunemploymentrate LivingstonSurvey TheLivingstonsurveyisthelongestrunningsurveyofforecastersstarting in1946.20 Thesurveyisconductedtwiceayearandconsistsofforecastsof18differentvariables describingunemployment,output,prices,andothermacroeconomicdata. Theforecastsareby economistsfromindustry,government,banking,andacademia. Figure A3 depicts the 1 year and 2 year ahead median forecasts by the forecasters in the Livingstonsurveyforthe1973-74recessionintheleftpanelandthe2007-09recessionintheright panel. QualitativelytheresultsaresimilartowhattheSPFforecastersexpected. Acrossrecessions, forecasters over estimated the recovery of unemployment rate. This re-enforces the idea that agentscannotdistinguishbetweenapersistentandtransitoryshockimmediatelyandmaytake severalquarterstolearn. FigureA3: ProjectedUnemploymentRatefromtheLivingstonSurvey (a)1973-74Recession (b)2009-07Recession Note:Thevariouscoloredlinesrepresentsthemedian1and2yearaheadprojectionofunemploymentratefromthe LivingstonSurvey.Thesolidredlineistheactualunemploymentrate 20ItispubliclyavailableandisfieldedbytheFederalReserveBankofPhiladelphiawhotookoverfromJoseph Livingston. 48

Policymakers. Now,toillustratethatitcanbechallengingevenforpolicymakerstocorrectly assessthenatureoftherecession,IpresenttheprojectionsfromFederalReserve’sGreenbookin December2008inFigureA4b. Underallscenarios,theFederalReserveBoardpredictedamuch fasterrecoveryduringtheGreatRecession. FigureA4: ProjectedUnemploymentRatefromtheSPF (b) ProjectedUnemploymentRate: Fed. Greenbook,December2008 (a) 1yearaheadUnemploymentForecastfromSurveyof ProfessionalForecasters Note:Panel(a):Theblackdashedlinerepresentsthemedian1yearaheadprojectionofunemploymentratefromthe SurveyofProfessionalForecasters.Thesolidredlineistheactualunemploymentrate.Panel(b):Theblackdashedline representstheprojectionofunemploymentratefromtheDecember2008GreenbookreleasedbytheFederalReserve Board.ThedashedlinesrepresentprojectionsundervariousscenariosthattheFedsimulated.Thesolidredlineisthe actualunemploymentrate. 49

A.4 SVAR InthissectionIdiscusssomemoreresultsandrobustnessfromtheSVAR.First,Ipresenttheshock seriesidentifiedfromtheVARinFigureA5. FigureA5: ShockSeries Note:Thisfigureplotsthetimeseriesofnoiseshocks,persistentTFPshocks,andtransitoryTFPshocks,asidentifiedin theSVAR.ThepersistentTFPshockshavelowervolatilitypost1985whilethenoiseshockshavehighervolatility. 50

VARImpulseResponseFunctions. TheVAR’simpulseresponses,illustratedinFigureA6,align withboththemodel’sstipulatedassumptionsandeconomicreasoning. Specifically: 1. NoiseShocks: ThesedonotaffectTFPbutinstantlyreducenowcasterrorswhileboosting GDP.TheseareimposedbytheVARonimpact. However,weseeinthedynamicsthatthe nowcast errors remain negative for about 8 quarters which implies that the agents keep gettingsurprisedastheyexpectGDPtobehigherthanitis. Thissuggeststhattheydonot immediatelyrecognizetheshockasnoiseandattributeittoapersistentortransitorychange inproductivity. Thisisonceagainconsistentwithlearningunderimperfectinformation. 2. PersistentShocks: TheseraiseTFPinamannerthatalignswithalong-lastingshock. Concurrently,GDPandnowcasterrorsincreaseimmediately. Evenafterseveralquarters,nowcast errorsstayelevated,indicatingthatthesepersistentshockscontinuallysurpriseeconomic agentsandtheymisperceivetheshocks. 3. TransitoryShocks: ThesemomentarilyelevatebothTFPandoutput,buttheireffectisshortlived. Initially,nowcasterrorsriseduetotheseshocksbutsoonturnnegative. Thisindicates thatagentsmistakenlyviewtheshockaspersistentforadurationandcontinuetobesurprised sinceit’sactuallyatranbsitoryshockwithminimallong-termeffectsonproductivityand GDP. FigureA6: ImpulseResponsefromtheVAR Note: ThisfigureplotstheimpulseresponsefunctionsforthevariablesintheVARtoapersistentshock,atransitory shockandanoiseshock,identifiedusingtheSVARdescribedbytheoptimizationproblemin7.Thesampleperiodis 1968q4:2019q4. ThefigureA7plotstheforecasterrorvariancecontributionofeachshocktoTFP,GDPand thenowcasterrors(NCE).Theblueshadedareaisattributedtothepersistentshock,theyellowis 51

attributedtothetransitoryshockandthepurpleisattributedtothenoiseshocks. Asexpected,the maximumforecasterrorvarianceofTFPisexplainedbythepersistentshock. Wealsoseethat thenoiseshockdoesnotcontributesignificantlytotheforecasterrorvarianceofTFP,whichis consistentwiththeassumptionsoftheVAR.Thenowcasterrorsaremostlyexplainedbythenoise shockswhileGDPisintitallyexplainedtoalargeextentbythenoiseshocksbutasthehorizon increases,persistentshockbecomestheprimarydriverofforecasterrorvarianceofGDP.Thisis alsoinlinewitheconomicintutition–asnoiseshocksdiedown,GDPisexplainedbytheactual TFPshocksinthelongrun. FigureA7: FEVDafterMax-ShareIdentification TFP GDP 100 100 90 90 80 80 70 70 60 60 50 50 40 40 30 30 20 20 10 10 0 0 5 10 15 20 25 30 35 40 5 10 15 20 25 30 35 40 NCE 100 90 80 70 60 50 40 30 20 10 0 5 10 15 20 25 30 35 40 TFP GDP NCE Note: ThisfigureplotstheforecasterrorvariancedecompositionforTFP,GDPandtheNowcasterrors(NCE).The blueshadedareaisattributedtothepersistentshock,theyellowisattributedtothetransitoryshockandthepurpleis attributedtothenoiseshocks. A.5 SmoothLinearProjections Inthissection,IdescribethemethodusedtoestimatetheSLP,followingBarnichonandBrownlees (2019). They model the sequence of impulse response coefficients as a linear combination of B-splinesbasisfunctions. Theseareestimatedusingashrinkageestimatorthatshrinkstheimpulse responsetowardapolynomial. SLPcoincideswithLPwhenthedegreeofshrinkageislowand withpolynomialdistributedlagmodelwithahighdegreeofshrinkage. ConsideraLPoftheform: P j j j ∑ j j (39) y t+h = α h +β h u t + γ p ω t–p+µ h,t+h p=1 whereω j isthesetoflaggedvaluesofy anduj. t–p Following Barnichon and Brownlees, 2019, one can approximate β j ≈ ∑K b j B j (h) using h k=1 k k 52

alinearB-splinesbasisfunctionexpansionintheforecasthorizonh. Thus,thecorresponding smoothLinearProjectionscanbewrittenasEquation40. K K P K (40) y ≈ ∑ a j B (h) j + ∑ b j B j (h)u j + ∑ ∑ c j B j (h)ω j +µ j t+h k k k k t pk k t–p h,t+h k=1 k=1 p=1k=1 TheSLPisestimatedusinggeneralizedridgeestimation. Whentheshrinkageparameterissmall, it is close to the least square estimation and has zero bias but potentially large variance.When theshrinkageparameterislarge,theestimatorisbiasedbuthassmallervariancethantheleast squaresestimator. Theshrinkageparameterischosenusingk-foldcross-validation(Racine(1997)). Ipresenttheimpulseresponsesfromthelocalprojectionandtheirsmoothedcounterpartsforthe noiseshockinFigureA9andthepersistentshockinFigureA8. TheIRFsfromthesmoothedLPs arequalitativelyandquantitativelysimilartothelocalprojections. FigureA8: SmoothedImpulseResponsetoPersistentTFPShocks Note:Thisfigureplotstheimpulseresponsesfromthelocalprojectionandtheirsmoothedcounterpartsforthepersistent TFPshock.Theshadedarearepresentsa95%confidenceintervalforthelocalprojection. 53

FigureA9: SmoothedImpulseResponsetoNoiseShocks Note:Thisfigureplotstheimpulseresponsesfromthelocalprojectionandtheirsmoothedcounterpartsforthenoise shock.Theshadedarearepresentsa95%confidenceintervalforthelocalprojection. A.6 EmpiricalResults A.6.1 ImpulseResponsetoTransitoryTFPShock FigureA10illustratestheresponseoflabormarketvariablestoatransitoryTFPshock.Theresponse toatransitoryshockismutedandislessthanthatofnoiseshocksintermsofmagnitude. A.6.2 HistoricalDecomposition Tounderstandtheroleofimperfectinformationoverthebusinesscycle,itisusefultounderstand howmuchofthedeviationofthekeylabormarketoutcomesfromtheirpredictedpathcanbe explainedbytheproductivityshocks. Ifnoiseshocksarenotimportant,theproductivityshocks wouldexplainalmostallthefluctuationsinthesevariables. TheVARmodelinitsVectorMoving Averageformis (41) yt = et +Ψ 1 u t–1 +...+Ψ tu 1 +Ψ¯ t (42) = ψ 0 vt +ψ 1 v t–1 +...+ψ tv 1 +Ψ¯ t where,ψ 0 = Qandψ j = Ψ j QarefunctionsofA 1 ,...,Ap andQ. Ψ¯ t isthepuredeterministic 54

FigureA10: ImpulseResponsetoTransitoryTFPShocks Note:ThisfigureplotstheimpulseresponsefunctionsforkeylabormarketvariablestoatransitoryTFPshock.The shadedarearepresentsa95%confidenceinterval. component. Now,wecandecompose(yt –Ψ¯ t)asthesumofthecontributionofnshocks t t (43) yt –Ψ¯ t = ∑ ψ j v t 1 –j +...+ ∑ ψ j v t n –j j=0 j=0 InFigureA11,IpresentthecontributionofthetwoTFPshocks(inblue)andthenoiseshocks (in green) in the VAR to the movements in unemployment rate. The noise shocks contribute significantly to the unemployment rate during the recessions in 1990-91, 2001 and 2007-09. In FigureA12IplotthecumulativecontributionoftheTFPshocksandnoiseshockstotheoutflow fromunemploymentaswellasthevacancypostings. Thesethreeshocksexplainalmostallthe movementintheoutflowfromunemploymentandvacancypostingsacrossthebusinesscycle. WhencombinedwithFigure5bandFigure5c,thesegraphsestablishthatthenoiseshocksplayan importantroleindrivingthedynamicsofkeylabormarketoutcomesspeciallypost1985,asthe TFPshocksdonotfullyexplainthefluctuationswhilethenoiseshockscontributesubstantiallyto thesemovements. 55

FigureA11: HistoricalContributionofShockstoUnemploymentRate Note:Thisfigureplotsthehistoricaldecompositionofunemploymentratefollowingequation16.Thebluebarsare thecumulativecontributionoftheidentifiedpersistentandtransitoryTFPshockstothemovementsindemeaned unemploymentrate(blackline).Thegreenbarsarethecontributionofthenoiseshocks. FigureA12: HistoricalContributionofPersistent,TransitoryandNoiseShocks (a)JobFindingRate (b)VacancyPosting Note:Thisfigureplotsthehistoricaldecompositionforeachseriesfollowingequation16.Theblacklineisthecumulative contributionoftheidentifiedpersistent,transitoryandnoiseshockstothemovementsindemeanedunemployment rate(redline). 56

FigureA13: HistoricalContributionofPersistent,TransitoryandNoiseShocks (a)WageGrowth (b)PrivateInvestment Note:Thisfigureplotsthehistoricaldecompositionforeachseriesfollowingequation16.Forwagegrowth,heblackline isthecumulativecontributionoftheidentifiedpersistent,transitoryandnoiseshockstothemovementsindemeaned unemploymentrate(redline).Forprivateinvestment,thebluebarsarethecumulativecontributionoftheidentified persistentandtransitoryTFPshockstothemovementsindemeanedunemploymentrate(redline).Thegreenbarsare thecontributionofthenoiseshocks. 57

A.6.3 PersistenceofUnemployment Tounderstandthecontributionofthenoiseshockstothepersistenceofunemployment,Icompute foreachrecessionbetween1968-2019,theshareoftheriseinunemploymentduringtherecession thathasbeenreversedduringtheexpansion. Ithendefinepersistenceasthenumberofquarters torecover50%oftheriseinunemploymentduringarecession,thatisurecovery,t = 0.5. Now,from thehistoricaldecomposition,Icancalculatewhatfractionofthispersistencecanbeattributed toeachoftheshockbyfirstcomputingthepredictedunemploymentratefromeachshockand thencalculatingthepersistenceasdefinedabove. TheresultsaresummarizedinTableA1. Forthe greatrecession,noiseshocksaccountforabout35%ofthe50%oftheriseinunemploymentand onaveragenoiseshocksaccountfor32%ofthisrecoveryacrossrecessions. TableA1: ContributionofNoiseShockstoRecoveryofUnemploymentAcrossRecessions Noofquartersfor50%recovery Shareexplainedby Recession Data Noiseshocks 2007-09 20 35% 2001 15 33% 1990-91 18 28% 1981-82 18 33% 1973-75 17 29% Average 17.6 32% Note:Thistablereportsthenumberofquarterstorecover50%oftheriseinunemploymentduringarecession,thatis urecovery,t =0.5. A.6.4 ResponseofUnemploymentForecastErrors FigureA14: UnemploymentRate:ProjectionsandActual (a)UnemploymentProjections:NoiseShocks (b)UnemploymentProjections:PersistentShock Note:Panel(a)showstheresponseofactualunemploymentrate(solid,black)andexpectedunemploymentratefrom theSPF(dashed,blue)toaonestandarddeviationnoiseshock,fortheduration1968-2019.Panel(b)showstheresponseof actualunemploymentrate(solid,black)andexpectedunemploymentratefromtheSPF(dashed,blue)toaonestandard deviationpersistentTFPshock,fortheduration1968-2019. 58

FigureA15: ResponseofUnemploymentForecastErrortoShocks Note: ThisfigureplotstheresponseofunemploymentforecasterrorfromtheSPFfor1968-2019toNoiseShocksin Panel(a)andtoPersistentTFPShocksinPanel(b). A.6.5 Sub-SampleAnalysis In this subsection, I discuss a sub-sample analysis to address investigate whether there were structuralchangesinthebusinesscycleposttheGreatModerationin1985. Ifirstpresentsome simplestatisticsfromtheSVARidentifiedshocksaswellasthenowcasterrorsinTableA2. Ifind thatpost1990,thenoiseshockbecamemorevolatilewhilethepersistentshockshavebecomeless volatile. Interestingly,theunemploymentnowcasterrorsnotonlybecamemorevolatile,butalso theaverageflippedsignpost1990implyingthatforecastersonanaverage,predictunemployment tobehigherthanitisinthisperiod. Likewiseforoutputgrowth, forecasterspredictoutputto belowerthanitis. ThissuggestssomestructuralchangethatmighthavehappenedpostGreat Moderation,andIleaveitforfutureworktoinvestigateit’ssource. 59

TableA2: SummaryStatisticsPreandPost1990 1968-1989 1990-2019 Mean SD Mean SD UnemploymentRate 5.68 1.65 5.80 1.83 GDPNowcastError 0.06 1.72 0.25 2.59 UnemploymentNowcastError 0.07 0.695 -0.03 1.20 NoiseShock 0.04 0.745 -0.05 1.27 PersistentShock -0.21 1.34 0.03 0.68 TransitoryShock 0.18 0.83 0.27 0.89 Note:Thistablereportssummarystatisticsfrom1968-1989and1989-2019intheempiricalexercise. A.7 Robustness: ControllingforUncertaintyShocks. Fundamentalshocksdonotsatisfythesignrestrictionsusedtoidentifythenoiseshocks,asany fundamentalshockhasahigherimpactonactualoutputthanexpected,thusviolatingthesign restrictions. However,theremightbesomecaseswhereuncertaintyshocksmightbehavelike thenoiseshocks: generatealargerchangeinexpectedoutputthanactualoutput. Toaddressthis concern,IcontrolfortheuncertaintyshockseriesfromBloom,2009inthelinearprojections. This robustnessistotestwhetherthenoiseshocksareindependentofuncertaintyshocks. Ifthenoise shock was indeed capturing uncertainty shocks, controlling for uncertainty shock would then capturetheresponseotherwiseattributedtothenoiseshock. Ispecificallyestimatetheregression inEquation44. P (44) y t+h = α h +β˜ h u t noise +θ h u t uncertainty + ∑ γ˜ p ω˜ t–p+µ˜ j h,t+h p=1 whereω˜ j isthesetoflaggedvaluesofy,unoiseanduuncertainty. Ithenplottherespectivesmoothed t–p cumulativeimpulseresponsetothenoiseshock(β˜). ThebaselineisEquation39whereIcompute the smooth cumulative impulse response to a noise shock without controlling for uncertainty shock. TheresultsofthisexerciseareshowninFigureA16. Thisexerciseshowsthatcontrollingfor uncertaintyshocksdoesnotchangetheresponseofkeylabormarketoutcomestoastandardized negativenoiseshock. Theimpulseresponsesforalltheoutcomesinthelabormarketliewithinthe 90percentconfidenceintervalofthebaselineimpulseresponses. Furthermore,thehump-shape oftheresponsesareretained,whichareconsistentwithBayesianlearning. Thisexercisesuggests thatthenoiseshocksarenotcapturingtheuncertaintyshocks. 60

FigureA16: ImpulseResponsetoaNoiseShockWhenControllingforUncertainty Note: Thisfigureplotstheimpulseresponseofkeylabormarketoutcomestothenoiseshockswithandwithout controllingforuncertaintyshocks.ThesolidblacklineisthesmoothedcumulativecoefficientβfromEquation39.The bluedashedlineisthesmoothedcumulativecoefficientβ˜ fromEquation44.Theimpulseresponsesaresmoothedby followingEquation40respectively.Theerrorbandsplotthe90percentconfidenceinterval. 61

B Theoretical Appendix InthissectionIderivesometheoreticalresultsanddiscussvariousmechanismsindetail. Iconduct somesensitivityanalysiswithalternatecalibrationsthataredocumentedinthissection. B.1 InformationStructure Considerthefollowingstate-spacerepresentation: zt = xt +η t, η t ∼ iidN(0,σ2 η) xt = ρ xx t–1 +(cid:15) t, (cid:15) t ∼ iidN(0,σ2 (cid:15)) at = xt +nt, nt = ρ nn t–1 +ν t, ν t ∼ iidN(0,σ2 ν) Where: • zt istheobservedsum. • at istheobservedpublicsignal. • xt istheunderlyingstatevariable. KalmanGainDerivation TheKalmangainisderivedfromthefollowinggeneralequation: (cid:48) (cid:48) –1 (45) Kt = P t|t–1 H (HP t|t–1 H +R) Giventhesystem,thestatetransitionmatrixF: (cid:34) (cid:35) ρ x 0 (46) F = 0 ρ n ObservationmatrixH: (cid:34) (cid:35) 1 0 (47) H = 1 1 ProcessnoisecovariancematrixQ: (cid:34) (cid:35) σ2 (cid:15) 0 (48) Q = 0 σ2 ν MeasurementnoisecovariancematrixR: (cid:34) (cid:35) σ2 η 0 (49) R = 0 0 62

Usingthesematrices,theKalmangainis: (cid:34) (cid:35)(cid:32)(cid:34) (cid:35) (cid:34) (cid:35) (cid:34) (cid:35)(cid:33)–1 1 1 1 0 1 1 σ2 η 0 (50) Kt = P t|t–1 P t|t–1 + 0 1 1 1 0 1 0 0 Insteady-state,theerrorcovariancematrixdoesnotchangeovertime,i.e.,P = P = P¯. t+1|t t|t–1 Thesteady-stateRiccatiequationis: (51) P¯ = FP¯F T +Q–FP¯H T (HP¯H T +R) –1 HP¯F T Fromthisequation,thevarianceoftheestimationerrorforxt insteady-stateisgivenbyP¯ 11 . Giventhestructureoftheprocesses,theKalmangainmatrixisgivenby 1 1 (cid:34) (cid:35) σ2 σ2 z s (52) Kt = 1 1 1 ; 1 1 1 + + + + σ2 σ2 σ2 σ2 σ2 σ2 z x,t s z x,t s where,σ2 x,t istheconditionalforecastvarianceofx t+1 ≡ Vart(x t+1 ). Itisupdatedaccordingtothe standardRiccattiequation: (cid:16) 1 1 1 (cid:17)–1 (53) σ2 = ρ2 + + +σ2 x,t x σ2 σ2 σ2 x s z x,t–1 where, (54) σ2 z = Var(zt) = σ2 x +σ2 η σ2 (55) = (cid:15) +σ2 η 1–ρ2 z (56) σ2 s = Var(sˆ t) = σ2 x +σ2 a σ2 σ2 (cid:15) ν (57) = + 1–ρ2 1–ρ2 z a B.2 Estimation: FullInformation InthissectionIpresenttheresultsforre-calibrationofthefullinformationmodeltomatchthe impulseresponsesfromthepersistentTFPshocks. B.3 InternalValidityoftheSVAR Inthissection,IdescribetheinternalvalidityoftheSVAR.Todothis,Ifirstsimulatetheestimated modelfor10,000periods. Then,IusethismodelmodelgenerateddataintheSVARtoidentifythe 63

TableB1: EstimatedParametersfromIRFMatching:FullInformationModel Parameters Interpretation Value Target Ψ Matchefficiency 0.48 UnemploymentRate= 0.055 κ Costofhiring 8.23 U –E = 0.28 µ Scaleparameterofsearchcost 0.082 E–E = 0.025 1–σ Separationrate 0.010 E–U = 0.010 φ SSproductivityfrombadjob 0.68 ∆WageofE-E= 0.045 Parameters Interpretation Estimate Std. Error λ Renegotiationfrequency 0.88 0.13 ξ Probabilityoffindingagoodjob 0.18 0.05 η Hiringcostconvexity 0.34 0.09 h Note:Thistablereportstheestimatedparametersfromtheimpulseresponsematchingexerciseoutlinedinequation38 fortheFullinformationmodel.Thethirdcolumnreportstheestimatedvalueswhilethefourthcolumnreportsthe standarderrorsforthesevalues.TheimpulseresponsesarematchedbyGMMandthestandarderrorsarecalculated usingthedeltamethod. threeshocks. ThetestoftheSVARisthatiftheidentificationstrategyindeedrecoversthetrue shocks,thenforalargesample,themodelgeneratedimpulseresponsemustbeequivalenttothe impulseresponsesgeneratedbytheSVARimplementedonthesimulateddata. Theseimpulse responsefunctionsarepresentedinFigureB1,wherethesetwoIRFscoincide. Thisimpliesthat theidentificationstrategyindeedrecoversthetrueshocksinthemodel. FigureB1: InternalValidityoftheVAR Note:ThisfigureplotstheinternalvalidityfortheSVARintheestimatedmodelwithimperfectinformation.Thedashed blacklinesarethesimulateddataimpliedIRFsintheSVAR(p=8,T=10,000),whereasthesolidgreenlinesarethemodel impliedIRFs. 64

B.4 QuantitativeResults B.4.1 ImpulseResponsesFromtheModel HereIplotimpulseresponsefunctionsfromtheimperfectinformationstickywagewithon-the-job searchmodelforotherimportantoutcomeslikeoutput,investment. FigureB2: ModelImpliedImpulseResponseFunctionstoaNegativeNoiseshock Note:Thisfigureplotsthemodelimplied,impulseresponsefunctionstoanoiseshock. 65

FigureB3: ModelImpliedImpulseResponseFunctionstoaPositivePersistentProductivity 0 -10 -20 -30 -40 -50 0 5 10 15 20 25 30 35 40 Quarter after shock etats ydaets t.r.w noitaiveD % Unemployment 70 60 50 40 30 20 10 0 0 5 10 15 20 25 30 35 40 Quarter after shock etats ydaets t.r.w noitaiveD % Vacancies 2.5 2 1.5 1 0.5 0 0 5 10 15 20 25 30 35 40 Quarter after shock etats ydaets t.r.w noitaiveD % Employment 4.5 4 3.5 3 2.5 2 1.5 1 0 5 10 15 20 25 30 35 40 Quarter after shock etats ydaets t.r.w noitaiveD % Output 2.5 2 1.5 1 0.5 0 0 5 10 15 20 25 30 35 40 Quarter after shock etats ydaets t.r.w noitaiveD % Consumption 3.5 3 2.5 2 1.5 1 0.5 0 0 5 10 15 20 25 30 35 40 Quarter after shock etats ydaets t.r.w noitaiveD % Capital 100 80 60 40 20 0 0 5 10 15 20 25 30 35 40 Quarter after shock etats ydaets t.r.w noitaiveD % v/u 70 60 50 40 30 20 10 0 0 5 10 15 20 25 30 35 40 Quarter after shock etats ydaets t.r.w noitaiveD % 25 20 15 10 5 0 0 5 10 15 20 25 30 35 40 Quarter after shock etats ydaets t.r.w noitaiveD % Investment Note:Thisfigureplotsthemodelimplied,impulseresponsefunctionstoapersistentproductivityshock. B.5 ProjectionsfromtheModel Inthissection,Ipresentthe4-8quartersaheadprojectionsbytheagentsinthemodelinresponse toapersistentTFPshockandinresponsetoanoiseshock.WhenfacedwithapersistentTFPshock, duetoimperfectinformation,agentsattributeapartoftheshocktobenoiseaswellastransitory shockandhencetheirprojectionsunder-reacttotheactualunemploymentrate. However,the reversehappenswhentheyfaceanoiseshock. Theysimilarlyattributesomepartoftheshock tobepersistentortransitoryproductivityandhenceinitiallyexpectunemploymenttobehigher thanitactuallyis(sincetrueproductivityhasnotchanged). Theyeventuallystartplacingmoreand moreweightontheshockbeingnoiseandastheylearn,theirprojectionsareclosertotheactual. FigureB5ashowsthemodelgenerated,one,twoandthreeyearaheadunemploymentprojectionsinthemodelaftertheGreatRecession. Here,allthreeshocks,identifiedfromtheVAR, acttogethereachperiod whilesimulationtheimperfectinformationmodelwith noiseshocks. Sinceallthreeshocksact,theprojectionsunder-reactifthecontributionofthepersistentshock dominates the contribution of the noise shocks as well as transitory shocks. Similarly, as the contributionofthenoiseshocksdominates,theprojectionsover-estimatetheunemploymentrate. AsseeninthehistoricaldecompositionoftheunemploymentrateinthedatainFigureA11,the contributionofthenoiseshockstothemovementinunemploymentdominatesafter2012. Thus, inthemodel,initially,astheproductivityshockshavehigherweight,theunemploymentrateis under-estimatedbytheagentsinthemodel. However,from2012,thecontributionofthenoise shocksincreasesbuttheagentsareunabletodiscerntheshockfromatruepersistentproductivity shockandhencekeepexpectinghigherunemploymentratesinthefuture. However,astheshock 66

FigureB4: 4-8QuarterAheadProjectionsintheModel (a)NoiseShock (b)PersistentProductivityShock Note:Thisfigureshowsthe4-8quartersaheadprojectionsbytheagentsinthemodelinresponsetoaresponsetoa noiseshock(a)andapersistentTFPshock(b).Thesolidthickblacklineistheactualresponseofunemploymentdueto theseshocksrespectively. istrulynoise,theactualunemploymentrateislowerthanexpected. Thisissimilartothepattern seeninthedatainFigureB5b. Itisimportanttonotethatthenoiseshocksareuniqueingenerating over-estimationoflongrununemploymentprojections. Forallstructuralshocks,thelongrun expectationsunder-estimatetheunemploymentrate.Thus,noiseshockscanbeapotentialsolution totheconsistentpatternobservedinthedatawherethelong-rununemploymentforecastsare over-estimatedbyprofessionalforecasters. FigureB5: UnemploymentRate:ProjectionsandActual–ModelvsData (a)UnemploymentProjections:Model (b)UnemploymentProjections:Data Note: Panel(a)showsthemodelimpliedforecastsforunemploymentrate1,2and3yearsahead. Thedashedblack lineisthemodelsimulatedunemploymentratefortheGreatRecession.Whilesimulatingthemodel,eachperiodall threeshocksact.InPanel(b)thevariouscoloredlinesrepresentthemedianlong-run(1year,2yearand3yearahead) projectionsoftheunemploymentratefromtheSurveyofProfessionalForecastersduringtheGreatRecession.The dashedredlineistheactualunemploymentrate. 67

B.5.1 UnemploymentDynamicsacrossRecessions: DatavsModel Thecalibratedmodelissimulatedtogeneratecounterfactualunemploymentrateseriesfor5recessionsbetween1970-2019. Thisexerciseshowsthatimperfectinformationexplainstheslowrecovery ofunemploymentrateinthelastthreerecessions. Forthisexercise,themodelisnormalizedto matchthestartingunemploymentrateforeachoftherecessions. Whilesimulatingtheimperfect informationmodel,eachperiod,allthreeidentifiedshocksfromtheVARareincorporated. For thefullinformationmodel,Ionlyintroducethepersistentandthetransitoryshockseachperiod. Furthermore,thefullinformationmodelisre-estimatedasdescribedintheprevioussection,to match the empirical IRFs to the persistent TFP shocks. The estimated parameters for the full informationmodelispresentedintheAppendix. FigureB6: ModelImpliedRecoveryofUnemploymentforRecessions Note:Thisfigureplotsthemodelimplied,simulatedunemploymentrateforthere-calibratedfullinformationmodel (dashedblueline)andtheimperfectinformationmodel(solidgreenline)formajorrecessionsbetween1973-2019. B.5.2 ComparingMechanismsintheModel Inthissection,Icomparethepersistenceandvolatilityofunemploymentundervariousmechanismswithandwithoutimperfectinformation. Icomparethemodelunder4scenarios: a)flexible wageswithouton-the-jobsearch(OJS),b)flexiblewageswithOJS,c)stickywageswithoutOJS,and d)stickywageswithOJS. Persistence Tocapturethepersistenceofunemployment,Icomparetheaveragedurationto recoverthe50%oftheriseinunemploymentacrossrecessionsbetween1968-2019. FigureB7shows 68

thedecompositionforthefourdifferencemodelspecifications,andwithineachspecificationI furtherdecomposethere-calibratedfullinformationbenchmarktotheimperfectinformation modelwithoutnoise. FigureB7: AverageDurationtoRecover50%ofRiseinUnemploymentAcrossModels Note: Thisfigureplotsthemodelimplieddurationfromthebeginningoftherecessionstorecover50%oftherise inunemployment. Thisisaveragedacrosstherecessionsbetween1968-2019,forvariousmodelspecification. The percentagesarethepercentofthedata(18quarters)thattheparticularmodelspecificationexplains,whilethex-axisis theactualnumberofquartersexplainedbytheparticularspecification.Thegreenbarsareincrementalcontributions bylearning,whichimpliesthatthetotalcontributionoftheimperfectinformationmodelisthesumoftheblueandthe greenbar.Here,thefullinformationmodelisnotre-calibratedandthenoiseisshutdownintheimperfectinformation model.Further,Ishutdowneachmechanismonebyoneinbothmodels. Themaintakeawayofthisgraphisthatintroducinglearningendogenouslycontributesto persistence in unemployment rate in the model. This speaks to Wright, 1986, who finds that imperfectinformation(albeitaboutwages,andonworkerside),introduceslearningendogenously inpresenceofjobsearch. IalsopresentthefulldurationofrecoveryacrossrecessionsintheTableB2forthere-estimated fullinformationmodelandtheimperfectinformationmodel. Thismeasurecapturestheduration ofrecoverybycalculatingthenumberofquartersittooktheunemploymentratetoreturntoit’s pre-recessiontrough. 69

TableB2: DurationofRecoveryofUnemploymentRateAcrossRecessions Recession Data FullInformation ImperfectInformation 1973-75 22 14 17 1981-82 24 17 21 1990-91 28 16 24 2001 24 14 21 2007-09 37 22 32 Note:Thistablereportsthenumberofquartersittakesunemploymenttoreturntopre-recessiontroughacrossfive recessionsbetween1975-2019. Themodelisnormalizedtomatchthestartingunemploymentrateforeachofthe recessions.TheimperfectinformationmodelissimulatedeachperiodwithallthreeidentifiedshocksfromtheVAR activated.Forthefullinformationmodel,onlythepersistentandthetransitoryshocksareincorporatedeachperiod. Thefullinformationmodelisthenre-estimatedtomatchtheempiricalimpulseresponsestothepersistentTFPshocks. BusinessCycleStatisticsAcrossSpecifications TableB3comparesthebusinesscyclestatistics obtainedbysimulatingtheimperfectinformationmodelaswellthere-calibratedfullinformation model,tothestatisticsintheUSeconomyfrom1968-2019acrossmultiplelabormarketvariables suchasoutput(Y),unemploymentrate(U),jobvacancies(V),job-tp-jobtransitions(E–E),job transitionsfromunemploymenttoemployment(U–E),andhiringrate. Icomparefullinformation benchmarktoimperfectinformationmodelunder4scenarios: a)flexiblewageswithouton-the-job search(OJS),b)flexiblewageswithOJS,c)stickywageswithoutOJS,andd)stickywageswithOJS. Theimperfectinformationmodeloutperformsthefull-informationmodelacrossallspecifications,highlightingthatlearningisanimportantmechanismforvolatilityinthelabormarket. TableB3: BusinessCycleStatistics Data(SD) FlexWage,NoOJS FlexWage,OJS StickyWage,NoOJS StickyWage,OJS FullInfo ImperfectInfo FullInfo ImperfectInfo FullInfo ImperfectInfo FullInfo ImperfectInfo Y 0.019 0.009 0.014 0.011 0.017 0.013 0.021 0.018 0.027 U 0.162 0.029 0.068 0.052 0.098 0.087 0.128 0.121 0.153 V 0.182 0.032 0.091 0.072 0.136 0.101 0.176 0.131 0.193 U-E 0.069 0.019 0.031 0.027 0.042 0.032 0.061 0.048 0.077 E-E 0.102 0.017 0.039 0.042 0.063 0.036 0.055 0.069 0.086 Note:Thistablereportsstandarddeviationofkeylabormarketvariablesinthemodel.Thedataherehasbeensimulated fromthemodelandHP-filtered(100,00). 70