Temporal Aggregation Bias and Monetary Policy Transmission

Finance and Economics Discussion Series Federal Reserve Board, Washington, D.C. ISSN 1936-2854 (Print) ISSN 2767-3898 (Online) Temporal Aggregation Bias and Monetary Policy Transmission Margaret M. Jacobson, Christian Matthes, and Todd B. Walker 2022-054 Please cite this paper as: Jacobson, Margaret M., Christian Matthes, and Todd B. Walker (2023). “Temporal Aggregation Bias and Monetary Policy Transmission,” Finance and Economics Discussion Series 2022-054r1. Washington: Board of Governors of the Federal Reserve System, https://doi.org/10.17016/FEDS.2022.054r1. NOTE: Staff working papers in the Finance and Economics Discussion Series (FEDS) are preliminary materials circulated to stimulate discussion and critical comment. The analysis and conclusions set forth are those of the authors and do not indicate concurrence by other members of the research staff or the Board of Governors. References in publications to the Finance and Economics Discussion Series (other than acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers.

TEMPORAL AGGREGATION BIAS AND MONETARY POLICY TRANSMISSION * MargaretM.Jacobson† ChristianMatthes‡ ToddB.Walker§ March2023 Abstract Temporalaggregationbiasesestimatesofmonetarypolicyeffects. Wehypothesizethatinformationmismatchesbetweenprivateagentsandtheeconometrician—thesourceoftemporalaggregationbias—areasimportantasthemorestudiedmismatchbetweenprivateagentsandthecentral bank (the “Fed information effect”) in the study of monetary policy transmission. In impulse responses from both local projections and an unobserved components model, we find that the responseofdailyinflationtohigh-frequencymonetaryshocksconfirmstheoreticalpredictions.Ifthere isanadverse-signedresponsesuchthatinflationincreasesinresponsetoacontractionarymonetary shock,itismuchlessprominentthanpreviouslythoughtandexplainedbyfrequencymismatchesof shocksanddependentvariables. *ThismaterialreflectstheviewsoftheauthorsandnotthoseoftheFederalReserveBoardofGovernors.Theauthorsthank MiguelAcosta,JonasArias,andFranciscoRuge-Murciaforexcellentdiscussions.Additionally,theauthorsthankConnorBrennan,KairongChen,andJakeScottforexcellentresearchassistance,aswellasSeanD’Hoostelaereandseminarparticipantsat NotreDame,Osaka,theFederalReserveBoard,BankofCanadaWorkshoponMonetaryPolicyResearch,theFederalReserve SystemMacroMeeting,theNBERWorkshoponMethodsandApplicationsforDSGEModels,the25thAnnualDeNederlandscheBankResearchConference,SeoulNationalUniversity,theInauguralMacroeconometricCaribbeanConference,andthe FederalReserveBankofClevelandforhelpfulcomments.FinallytheauthorsthankMiguelAcostaforhelpextendingtheNakamuraandSteinsson(2018a)shockseriesandJohnRogersandWenbinWuforhelpreconstructingtheirshockseries. †FederalReserveBoard;Margaret.M.Jacobson@frb.gov ‡DepartmentofEconomics,IndianaUniversity,matthesc@iu.edu §DepartmentofEconomics,IndianaUniversity,walkertb@indiana.edu

1 INTRODUCTION Thispaperrevisitsafundamentalquestionofmonetaryeconomics: Whatisthetransmissionofmonetarypolicytotheeconomy? Empiricalworkoftenfindsthatresponsesofmacroeconomicvariablesto monetarypolicyshockshavetheoppositesignofwhatstandardtheorypredicts.Researcherstracethese adverseresponsestoinformationissues,withexistingsolutionsconsistingofeitheraddingmoreinformation[Sims(1992)]oremphasizinginformationmismatchesbetweencentralbanksandprivatesector agentsasa“Fedinformationeffect”.1 Weproposetemporalaggregationbiasasanewinformation-basedexplanationfortheadversetransmissionofmonetarypolicy.WhenusingthedailyCPIfromtheBillionPricesProject[CavalloandRigobon (2016)] as a temporally disaggregated macroeconomic indicator, we find that the adverse response of inflationisshort-lived, ifitispresentatall. Wearguethatexistingworkonmonetarypolicytransmissionfindsanadverseresponsebecauseofthefrequencymismatchbetweentheinformationsetsofthe econometricianandprivateagents.Atemporallydisaggregatedmeasureofinflationovercomesthismismatchbybetteraligningthefrequenciesofshocksanddependentvariables. Tounderstandhowonecanobtainasizableadverseresponsetomonetarypolicyshockswithmonthly orquarterlydatawhenonlyalimitedadverseresponseactuallyexists, wecombineasimplemodelof temporalaggregationbiaswithinformalandformalempiricalevidence.WebeginbyusingMonteCarlo evidence to show how there is no clean identification of monetary policy transmission when time aggregatingwithlocalprojections. Wethenuseawell-knownmodelfromthemonetarypolicyliterature consistingofanEulerequationandamonetarypolicyruletoshowhowtemporalaggregationcanexacerbateinitialimpulseresponsefunctions. Ourmainfinding—theresponseofinflationisconventionally-signedwithonlyashort-livedadverse responseifoneispresentatall—isobtainedfromthelocalprojectionspecificationadvocatedbyNakamura and Steinsson (2018b). The monetary policy shocks are identified via high-frequency variation inassetpricesaroundmonetarypolicyannouncements,asisstandardintheliterature[Kuttner(2001), Gürkaynaketal.(2005), Campbelletal.(2012), NakamuraandSteinsson(2018a), Buetal.(2021)]. We establish that temporally aggregated high-frequency measures of inflation correlate well with official lower-frequency measures (e.g. monthly CPI) over our sample period (July 2008 to August 2015). Our empiricaltestscorroboratetheclaimthatthehigh-frequencymeasureofinflationis“goodatanticipatingmajorchangesininflationtrends,”[emphasisadded,CavalloandRigobon(2016)].Wethusalignthe frequencyofourvariableofinterest(inflation)morecloselytothefrequencyofvariationusedtoidentify shocks. Impulseresponsefunctionsshowtheresponseofinflationtoacontractionarymonetarypolicy shockispositiveforafewweekswitha90%crediblesetcoveringzerooverthistimehorizonandnegative thereafter. Becausetheeffectoftemporalaggregationbiasinlocalprojectionsdependsonthetimingofhigh- 1BauerandSwanson(2023),Buetal.(2021),andCaldaraandHerbst(2019)alsoemphasizeaddingmoreinformation.Fora “Fedinformationeffect"seeRomerandRomer(2000),Campbelletal.(2012,2017),NakamuraandSteinsson(2018a),Jarocinski andKaradi(2020),Miranda-AgrippinoandRicco(2021),Lunsford(2020),Hoeschetal.(2021),CieslakandSchrimpf(2019), Acosta(2022),Sastry(2021),KarnaukhandVokata(2022),Lewis(2020),BundickandSmith(2020),AndradeandFerroni(2021), GolezandMatthies(2021).

JACOBSON,MATTHES&WALKER: TEMPORALAGGREGATIONBIAS frequencyshocks,webuildanunobservedcomponentsmodelthatexplicitlyincorporateswhenmonetarypolicyshocksoccurwithinamonth.ThisstatespacemodeladdsthedailyCPIanddailybreak-even inflationratesaswellaspossibleeffectsofmonetarypolicyshocksintoamodelofinflationdynamics alongthelinesofStockandWatson(2016)andNasonandSmith(2020). Theseimpulseresponsescorroborateourlocalprojectionresultsbyshowingconventionally-signedtransmissionofmonetarypolicy. Ourcontributionoftemporalaggregationbiasasanexplanationforthetransmissionofmonetary policyshocksprovidesfurthersupportfortheongoingclaim,datingbacktoatleastKuttner(2001),that monetarypolicyneedstobestudiedinahigh-frequencyenvironment.Eventhoughhigh-frequencyeconomicindicatorsandtemporalaggregationtheoryhavebeenavailablefordecades,wearethefirst—to ourknowledge—toapplythemtothestudyofmonetarypolicytransmission.2 Bypairinghigh-frequency shockswithhigh-frequencyresponsevariables,ourworkfollowsexistingspecificationsthatestimatethe transmissionofmonetarypolicyshockstofinancialindicators.3 Financialindicators,however,maynot beassusceptibletotemporalaggregationbiasasmacroeconomicindicatorsbecausetheformerareobservableathighfrequencies. Bycontrast,economicindicatorsareaccumulatedoverafixedtimeintervalandpublishedwithalag,resultinginaggregationbiasfrompotentiallymismatchedinformationsets betweenprivateagentsobservinghigh-frequencyindicatorsandaneconometricianrelyingonofficial releases.4 Unlikeotherstudies,wherecompetingmethodologiesorconditioningondifferentdataservestoobfuscateanalysis,adistinctadvantageofourapproachistheconsistencyininference. Weconditionon thesamedataandapplythesamemethodologywiththeonlydistinctionbeingthefrequencyofthedata. Anincreaseinthefrequencyofinflationobservationseliminatesadversemonetaryimpulseresponses. Becauseourtemporalaggregationresultsaregeneric,wearguethatthebenefitsofusinghighfrequency dataareneitherlimitedtothestudyofmonetarypolicytransmissionnorpricesandwillbeakeyfeature ofthenascentfieldofhigh-frequencymacro[Baumeisteretal.(2021),Lewisetal.(2021)]. Inamacroeconomicenvironmentcharacterizedbyfast-movingturningpoints,suchastheGreatFinancialCrisisor theCOVID-19recession,estimatesofpolicyeffectsmaybesensitivetothesamplingfrequencyofeconomic response variables. Although high-frequency observables may be susceptible to measurement noisebecausetheyareonlyproxiesoftheirlowerfrequencyofficialcounterparts, frameworkslikeour statespacemodelallowformeasurementerror.Wethusarguethatmeasurementnoiseisnotnecessarily moreimportantthanthebiasinducedbytemporalaggregation. 1.1 CONNECTION TO LITERATURE WhileCampbelletal.(2012)andNakamuraandSteinsson(2018a) findadverseresponseswhenestimatingthetransmissionofhigh-frequencymonetarypolicyshocksto lowerfrequencyforecastsofmacroeconomicaggregates,subsequentworkfindsthatproperlyaccount- 2Lewisetal.(2020a)discusshowtimeaggregationaffectstheirestimatesofmonetarypolicytransmissiontohousehold expectations.SeeShapiroetal.(2022),Aruobaetal.(2009),Lewisetal.(2020b)forotherhighfrequencyeconomicindicators. 3SeeGolezandMatthies(2021),AndradeandFerroni(2021),NakamuraandSteinsson(2018a),BauerandSwanson(2022), Gürkaynaketal.(2022),andGürkaynaketal.(2021). 4Forexample,StockandWatson(2007)notethattimeseriesestimatesoftheCPIaresusceptibletotemporalaggregation bias. 2

JACOBSON,MATTHES&WALKER: TEMPORALAGGREGATIONBIAS ingforinformationdeliversresultsthatareeitherambiguousorinlinewithstructuralpredictions.5 Closesttoourspecificationofhigh-frequencyinflationindicatorsrespondingtohigh-frequencymonetarypolicyshocksarespecificationsthatrelyonhigh-frequencyexpectedinflation(TIPS)[Nakamura and Steinsson (2018a)] or commodity prices [Velde (2009)]. Relative to these previously used proxies, wearguethattheBillionPricesProjectdailyCPIisarelativelymorecompletemeasureofinflationand hencebettersuitedtoassessthetransmissionofmonetarypolicyshocks. Expectedandrealizedinflationmayhavedifferentsensitivitiestomonetarypolicyshocksbecausetheformertendstobeanchored while the latter is more prone to fluctuations.6 Similarly, commodities are known to be more volatile thanmeasuresofinflationwhichmayresultindifferentsensitivitiestomonetarypolicyshocks. Ratherthanfollowingmuchoftheempiricalmonetarypolicytransmissionliteratureandfocusingon informationrefinementstopossibleexplanatoryvariables,weinsteadfollowBauerandSwanson(2023) andcontributerefinementstotheless-studiedmeasurementofresponsevariables.7 Manystudiesfind predictabilityandorbiasinstandardhigh-frequencymonetarypolicyshockssuchasthoseestimatedby NakamuraandSteinsson(2018a).ThesestudiesmainlyfocusontheresponseofGDPandarguethatthe adversesigndisappearsoncetheshocksareeitherorthogonalized[KarnaukhandVokata(2022),Bauer andSwanson(2022)]orconditionedonmissinginformation[CaldaraandHerbst(2019),Sastry(2021), Miranda-AgrippinoandRicco(2021),BauerandSwanson(2023)]. Many studies account for the adverse transmission of high-frequency monetary policy shocks by appealingtoRomerandRomer’s(2000)“Fedinformationeffect”whicharguesthatcentralbankshave aninformationadvantageoverprivateagents.8 Privateagentsthusreviseuptheirforecastsofinflation inresponsetotightermonetarypolicybecausetheyperceiveasignalthatthecentralbankhasrelatively optimisticnon-publicinformation.However,severalrecentpapersexplicitlytestforacentralbankinformationadvantageandfindnoevidence[Sastry(2021),BundickandSmith(2020)andBauerandSwanson(2023)]. Otherpaperstaketheinformationadvantageasgiven,controlforitdirectly,andfindthat iteitherchangesthetransmissionofmonetarypolicyshocks[Lunsford(2020),Buetal.(2021),Hoesch etal.(2021),CieslakandSchrimpf(2019),Acosta(2022)]oreliminatestheadversetransmissionentirely [Miranda-AgrippinoandRicco(2021),JarocinskiandKaradi(2020)].9 Incontrasttoexistingwork,wedo notexplicitlytestormodelhowtheinformationsetsofcentralbanksandprivateagentsaffectmonetary policytransmission.Weinsteadfocusontheless-studiedinformationmismatchbetweenprivateagents andtheeconometricianandhowthisbiasesestimates. Decadesofworksupportsourclaimthattemporalaggregationbiascanaffectboththedirectionand 5Uribe(2022)takesacontrastingstanceandarguesthatmonetarypolicyshocksmayactuallybeneo-Fisherian. 6CommonspecificationsthatrelyonthechangeinBlueChipforecastsmaythusbeunderstatingthetransmissionofmonetarypolicyshockstoinflationbecausetheycapturechangesinexpectedratherthancurrentinflation.Wepositthatthedifferent sensitivitiesofexpectationsandactualindicatorsislessofanissueforthetransmissionofmonetarypolicyshockstoGDP. 7BauerandSwanson’s(2023)surveyfindsthatBlueChipforecastersrarelychangetheirestimatesofeconomicindicators inresponsetomonetarypolicyannouncementswhichcallsforreexaminationofthesuitabilityoftheseforecastsasresponse variables. 8Faustetal.(2004)findthattheadverseresponseofinflationdisappearsoncetheVolckerdisinflationisexcludedfrom RomerandRomer’s(2000)study. 9Lewis(2020)andAcosta(2022)specificallyidentifyaFedinformationeffectshockandfindevidencethatiseithermixedor againstadversetransmissionofmonetarypolicyshocks. 3

JACOBSON,MATTHES&WALKER: TEMPORALAGGREGATIONBIAS magnitudeofmonetarypolicytransmission.WefollowMarcet(1991)indemonstratinghowthesystematiceffectoftimeaggregationistobiasthefirstfewcoefficientsofthemoving-averagerepresentation. CoupledwithresultsinAmemiyaandWu(1972),whoshowthattemporalaggregationofautoregressive processespreservesinvertibility,thesebiaseswouldinfiltratemodernapproachestoVARidentification. Thisputsourmainresult—thattheFedinformationeffectisanartifactoftemporalaggregation—onfirm theoreticalground.Whileapplicationsoftheseideasinappliedmacroeconomicsarestillrelativelyrare, ForoniandMarcellino(2016)highlighthowjointlyusingdatacollectedatdifferentfrequenciescanhelp with the identification of structural VARs with a focus on traditional recursive identification schemes, whereasForoniandMarcellino(2014)makeasimilarargumentfordynamicequilibriummodels. Arelated,butdistinct,literaturehasdevelopedtoolstoestimateregression-typemodelswhentheleft-hand sideissampledatadifferentfrequencythantherighthandside(Ghyselsetal.,2004). 2 A FEW PROPERTIES OF TEMPORAL AGGREGATION Weemploystylizedmodelsofmonetarypolicyinordertoestablishpropertiesoftemporalaggregation designedtoshedlightontheempiricalresultsofSection4. Usingsimulateddataandlocalprojections, weshowhowashort-livedadverseresponse(i.e.,positiveresponseofinflationtoacontractionarymonetarypolicyshock)canseepintolowerfrequenciesduetotemporalaggregationbias. Wethenprovide amoretheoreticalframeworktodemonstratehowtemporalaggregationleadstosubstantialbiasinimpulseresponsefunctions;specifically,intheinitialvaluesofmoving-averagerepresentations. Wekeep themodelssufficientlysimpleinordertoprovideclearintuition,acknowledgingthattheseareexamples asopposedtotheorems. However,weconjecturerobustnessofourresultsbyappealingtoanearlierliteraturethatoperatesincontinuoustime,andbydiscussingnecessaryconditionsofourresultsinaNew Keynesiansetup. 2.1 TEMPORALAGGREGATIONWITHLOCALPROJECTIONS Considerthedata-generatingprocessofinflation, 59 π = (cid:88) Θ εmp +u (1) t j t−j t j=0 u t =ρuu t−1 +εu t wheret isassumedtobedaily,andthemonetarypolicyshockεmp∼N(0,1)isuncorrelatedwiththepert sistentshocku ∼N(0,σ2). Weassumethemonetarypolicyshockoccursonlyoncepermonth,while t u u occurs every day. We examine three alternative specifications of the timing of the monetary policy t shock—ashockthatoccursatthebeginning(day1),middle(day15),andend(day30)ofthemonth. To approximatepopulationmoments,wesimulatethreemilliondailyobservations,taking30-dayaverages ofshocksandtheinflationprocess(1)toobtaincorrespondingmonthlydata.Localprojectionsareused to estimate monthly responses of inflation to the monetary policy shock, controlling for lagged inflationoutcomes. Wesetρ =0.99andσ =1tocapturetheideathatothershocksarejustasimportant u u asmonetarypolicyfortheevolutionofinflationatthedailyfrequency. Theparametersgoverningthe 4

JACOBSON,MATTHES&WALKER: TEMPORALAGGREGATIONBIAS reactionofinflationtomonetarypolicyaregivenbyΘ =1for j =0,...,9andΘ =−1for j =10,...,59. j j Ourparameterizationaccomplishestwotasks: first,itintroduceswhatwerefertoasaninitial“adverse"policyresponseofinflation;thatis,thefirsttendailyobservationsofinflationfollowingamonetarypolicyshockareinconsistentwithstandardtheoryinthatacontractionaryshockwouldleadtoan increaseininflation.Second,theaverageeffectoverthe30-dayperiodisconsistentwiththeory.Theremainingtwo-thirdsofthedailyobservationsoverthemonthenterwithanegativecoefficient,implying acontractionaryshockwouldleadtoafallininflation.Notealsothatthemagnitudesofthefirst10days andlast20daysaresimilar. Theimplicationofourcalibrationisthatonewouldnotexpecttheadverse inflationaryresponsetomaterializeintheaggregate(monthly)data. PanelA:Beginning PanelB:Middle PanelC:End εmp Π εmp εmp Π εmp εmp Π εmp t t−1 t−1 t t−1 t−1 t t−1 t−1 Π -0.40 0.82 0.29 0.82 0.03 0.82 t Π -0.50 -1.16 0.38 -0.85 -0.06 -0.26 t Π t+1 -1.09 0.61 -0.92 0.60 -0.19 0.60 Table 1: Local Projection Results. Three million observations of daily inflation simulated via (1) and aggregated to monthly (30 day) frequency were estimated using local projections. The panels denote whenthemonetarypolicyshockhitstheeconomy,atthebeginning(day1),middle(day15)orend(day 30)ofthemonth.Dependentvariablesareinthefirstcolumn,theothercolumnsdisplaythecoefficients oftheright-hand-sidevariablegivenatthetopofeachcolumnwithinapanel.Thefirstrowoftheresults istheresponseofinflationtothemonetarypolicyshocksfromthecurrentandpreviousmonths. The secondandthirdrowofresultsarethelocalprojectionsattimet=0andt=1,respectively. Table 1 shows results for three local projection specifications and various timing of the monetary policyshock.Intwoofthethreespecifications,theeconometricianwouldfindapositiveinitialresponse of inflation to a monthly monetary policy shock, despite the fact that the time-averaged response is negative. Onlywhenthemonetarypolicyshockhitstowardsthebeginningofthemonthdoesthesign of the response of inflation match the temporally aggregated negative value. The lagged shock, ϵ t−1 , doesenterwithanegativesign,sowhiletheinitialresponsecouldbeadverse,thesubsequentmovesare standard. Thetimingofthemonetarypolicyshockisimportant.Figure1(leftpanel)plotsthetime-aggregated monthlymovingaveragecoefficients(i.e. theaccumulatedresponsetoamonetaryshock)(left,y-axis) againstthetimingofthemonetarypolicyshock(x-axis).10 Thetime-aggregatedMAcoefficientsforany month can be written as Ψ ≡ (cid:80)2 t= 9− 0 j (1 t≤9 −1 t>9 ) where j = 0,...,29 is the day of the month when the monetary policy shock occurs. For example, when j =29 so that the shock occurs on the last day of themonth,Ψ=1. Theaggregatedresponseisthusinitiallyincreasingaswedecrease j =29,...,21(the shockoccursearlierinthemonth)withthelargestpositiveimpactΨ=10onday j =21. Thereafter,the negativeMAcoefficientsenterintothemonthlyaggregationandthelargestnegativeimpactΨ=−10is whentheshockoccursonday j =0attheverybeginningofthemonth.Thehistogramplottedontheleft 10Notethatthisaccumulatedresponseisnotdirectlycomparabletoourestimatesreportedintable1sinceweassumeinour simulationsthatthereisonlyonemonetaryshockpermonth. 5

JACOBSON,MATTHES&WALKER: TEMPORALAGGREGATIONBIAS Histogram of the days of the month for FOMC announcements, 8/2008 to 1/2016 10 10 mean/median FOMC day= 19 8 5 6 0 4 -5 2 -10 0 0 10 20 30 Time aggregated monthly MA coefficients (left axis) Distribution of the days of the month, frequency (right axis) Figure1: Robustness. ImpactIRFestimatedvialocalprojectionsasafunctionofAutocorrelation(ρu) andStandardDeviation(σu)(rightpanel)andaccumulatedresponsesasafunctionofshocktiming(left panel). panelofFigure1showsthatoveroursampleperiod,thetimingofFOMCannouncementsisconsistent withtheshockhittingduringthemiddleofthemonth. ThemeanandmedianFOMCannouncement occurredonthe19thdayofthemonth,andamajorityoftheannouncementsoccuredafterthe10thday ofthemonth.Thissimpleexampleshowshowresearchersusingaggregateddatacanestimateapositive responseofinflationtoacontractionarymonetarypolicyshockeventhoughmostofthedisaggregated responsecoefficientsarenegative. Finally,wenotethattheresultsarenotcontingentontheparameterizationofthedailyprocess,u . t Figure 1 (right panel) plots the initial response using the middle of the month timing as in Panel B of 1againsttheserialcorrelationcoefficientandstandarddeviation. Itshowsthatsizeofthepositivecoefficient in the LP regression is increasing in the correlation of the non-monetary policy shock and its standard deviation, but remains substantial (0.13) when these values are close to zero. These results confirmourempiricalfindings—ashort-livedadverseresponseatdailyfrequencycanbepersistentand significantatmonthlyfrequency. 2.2 TEMPORAL AGGREGATION IN A STRUCTURAL MODEL. Wenowprovideamoretheoreticalframeworktodemonstratehowtemporalaggregationleadstosubstantialbiasinimpulseresponsefunctions. Consideranominalbondthatcosts$1atdatet andpaysoff(1+i )atdatet+1.Theasset-pricingequat tionforthisbondcanbewritteninlog-linearizedformasaFisherequation, i t =r +E[π t+1 |I t ], where therealinterestrateisassumedtobeconstantandE[π t+1 |I t ]istheprivateagents’expectationofnext period’s (t+1) inflation. Monetary policy follows a Taylor rule, adjusting the nominal interest rate in responsetoinflation, i =r +φ[π |I ]+x , wherethemonetarypolicyshockfollowsanAR(1)process, t t t t x t =ρx t−1 +ε t ,withρ∈(0,1)andε t distributedasGaussianwithmeanzeroandvarianceσ2 ε.Weassume theinformationsetofthemonetaryauthorityisconsistentwithprivateagents’(I )sothatwecanisolate t theeffectsoftheinformationmismatchbetweenprivateagentsandtheeconometricianwithoutaconfounding “Fed information effect." The unique equilibrium rate of inflation is well known and follows 6

JACOBSON,MATTHES&WALKER: TEMPORALAGGREGATIONBIAS m=1 m=2 m=5 m=10 m=20 m=30 m=40 m=50 ρm 0.990 0.980 0.951 0.904 0.818 0.740 0.669 0.605 θ 0.000 0.171 0.250 0.264 0.265 0.266 0.266 0.267 σ2 0.028 0.041 0.085 0.160 0.288 0.391 0.476 0.542 u σ2 1.397 1.389 1.374 1.351 1.307 1.266 1.226 1.186 Π Table2: EstimatesoftheARMA(1,1)(4)usingtemporallyaggregatedobservationsof(2). Notethatfor m=1(notemporalaggregation),σ2 =σ2 . u w fromimplementingtheTaylorprinciple(φ>1), x π t =− φ− t ρ =ρπ t−1 +w t (2) wherew =−ε /(φ−ρ). t t We assume the econometrician observes realizations of the equilibrium processes at a frequency thatislowerthanprivateagents. Specifically,lett =mT anddefinethetemporallyaggregatedinflation processas Π T = (cid:181) 1 (cid:182) (cid:195) m (cid:88) −1 Lj (cid:33) π mT = (cid:181) 1 (cid:182) (π mT +π mT−1 +···+π mT−m−1 ) T =1,2,3,... (3) m m j=0 For example, if t is a month and m = 3, then T is a quarter. Inflation, π , could be interpreted as a t monthly year-over-year percentage change, and the three-month non-overlapping arithmetic mean is onepossiblewayofaggregating.Alternatively,wecouldassumetoobservemonth-over-monthinflation and the direct summation yields quarterly inflation. Our analysis below is robust to these alternative aggregationmethods. Appendix C shows that temporally aggregating the AR(1) inflation process given by (2) yields an ARMA(1,1)representation, (1−ρmL)Π T =u T +θu T−1 u T ∼N(0,σ2 u ) (4) where,forlagoperatorL,theautocorrelationcoefficientisraisedtothepowerofm(thenumberofaggregatecomponents),andtheestimatedshocks(u )willbefundamentalfortheΠ process(Amemiyaand t t Wu(1972)). Thelastfactensuresthatanautoregressive(orVAR)representationwillaccuratelyestimate theARMAprocess. AnanalyticalmappingbetweentheaggregatedinflationprocessandtheARMA(1,1) parametersisnotfeasiblebutTable2providesestimatesoftheparametersforvariousvaluesofmusing simulateddata.Wesetρ=0.99,φ=1.05,σ2 ε =0.01,anduseonemilliondisaggregatedobservations. TheestimatesofTable2revealimportantpropertiesofthemappingbetweenanAR(1)processandits temporallyaggregatedARMA(1,1)counterpart: [i.] theautocorrelationcoefficientdecaysexponentially 7

JACOBSON,MATTHES&WALKER: TEMPORALAGGREGATIONBIAS 0.15 − 0.2 − 0.25 − 0.3 − 0.35 − 0.4 − 0.45 − 0.5 − 0.55 − 0.6 − 0.65 − 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 NumberofAggregations FRIfotcapmIlaitinI 1 0.8 0.6 0.4 0.2 0 0 /2 (a)ImpactResponseforVariousm (b)SpectrumofMAFilter(m=2,....,30) Figure2: InitialImpulseResponseandMoving-AverageFilterforVariousm. Panelashowsthedecline intheinitialimpactcoefficientasm increasesfrom1to30. Panelbplotsthespectrumform=2(blue) throughm=30(green),demonstratingwhylowfrequencypropertiesarepreserved. atratem;[ii.]thevarianceoftheaggregateinflationprocess, σ2 σ2 = π (cid:161) m+2[(m−1)ρ+(m−2)ρ2+···+ρm−1] (cid:162) (5) Π m2 declinesmultiplicativelyinm(seeAppendixCforderivation). Takentogether,[i]and[ii]implythatthe variance of the innovation process σ2 and the moving average parameter θ must compensate for the u fasterdeclineintheautocorrelationcoefficient, ρm. Table2showsthatthevarianceoftheinnovation (σ2)increases46%form=2andbyafactoroftenform=20,andthemoving-averageparameteralso u increaseswithm. Theincreaseintheestimatedvariancewilltranslateintoamorepronouncedinitial impactoftheimpulseresponseofinflationtoamonetarypolicyshock.Figure2aplotstheinitialimpulse responsetoaone-standarddeviationshock(σ )forvariouslevelsofaggregation. Notethattheunitsof u thex-axiscorrespondtothedegreeofaggregationm.Thedisaggregatedimpulse(m=1)showsaninflationprocesswithanimpactresponsethatissubstantiallymitigatedrelativetothetemporallyaggregated responses.Evenaslightincreaseinthedegreeofaggregationleadstoasubstantialchangeintheimpact responsetoamonetarypolicyshock—temoporallyaggregatingoversixperiodsmorethandoublesthe initialimpact.ThisdynamicisconsistentwithourempiricalfindingsinSection4,seeFigures6and7.11 (cid:179) (cid:180) Figure2bplotsthemoving-averagefilter (cid:161)1(cid:162) (cid:80)m−1Lj inthefrequencydomainovertherangeof m j=0 0toπ. ThefigureshowsthataMAfilterisalow-passfilter,allowinglowerfrequenciestopassthrough whileattenuatingmediumandhigherfrequencies.Whatiscriticalforunderstandingthebiasassociated withtemporalaggregationishowthereallocationofthespectrumisdistributedacrossvariousparam- 11Onedistinctionbetweenthisexerciseandourempiricsisthenormalizationofthevariance. Ifoneweretonormalizethe varianceforthetemporallyaggregatedseriestomatchthedisaggregatedvalue,thecorrectionwouldcomethroughthemoving averagetermandFigure2continuestoberelevant. 8

JACOBSON,MATTHES&WALKER: TEMPORALAGGREGATIONBIAS eters of the estimated ARMA(1,1) process. Lower frequencies are preserved when aggregation occurs despitethedeclineintheautocorrelationcoefficient(fromρtoρm).AmemiyaandWu(1972)showthat, foranystationaryAR(p)representation,temporalaggregationpreservestheorderoftheautoregressive process (i.e., an AR(p) becomes an ARMA(p,q))12 with the autoregressive roots all raised to the power m.Theseseeminglyconflictingproperties—adeclineinthevalueofthe(positive)autocorrelationroots coupledwithnosubsequentchangeinthelowfrequencypropertiesofthetimeseriesprocess—leadsto asubstantialchangeintheinitialimpulseresponsecoefficientsthroughanincreaseinthevarianceof theinnovationprocessandappearanceofpositivemoving-averageparameters. 2.3 ROBUSTNESS The purpose of this section was to establish how temporal aggregation can substantially alter initial moving-average coeffcients. An econometrician, time-aggregating the data, will attributeastructuralinterpretationtothesignificantandpotentiallyadverseinitialreactionofinflation toamonetarypolicyshock, whenthelion’sshareoftheresponseisduetotemporalaggregationbias. Whilewebelievethissectionhasestablishedcompellingintuitionforourresults,themodelsarestylized andsowebrieflydiscussrobustness. First,appealingtoMarcet(1991),ourprimaryresultisnotanartifactofspecificassumptionsunderlyingourmodelbutisduetothemoregenericpropertiesoftemporal aggregation. Workinginacontinuous-timeframeworkandwithgenericWoldrepresentations, Marcet (1991)findsthe“systematiceffectoftimeaggregationistoincreasetheabsolutesizeofthefirstfewcoefficientsoftheMAR(moving-averagerepresentation)(emphasisadded).”Thisresult,coupledwiththefact thattemporalaggregationpreservesinvertibilityforautoregressiveprocesses(AmemiyaandWu(1972)), suggeststhatourresultsarerobusttoalternativespecifications. Second,howdowesquareourresultswiththeubiquitousprice-stickinessfrictionsnowstandardin theNewKeynesianliterature? Ourresultwillcontinuetogothroughundertheassumptionthatsome firms have the ability to adjust prices at a frequency higher than monthly. Building a New Keynesian modelwithmultipleCalvoadjustmentfrequencies,wecanshowthattemporalaggregationbiaswillbe substantialifonly1/5offirmschangepricesatfrequencieshigherthanmonthly.13 Observingdataata monthlyorquarterlyfrequency,theeconometricianwillbesusceptibletotemporalaggregationbias. 3 DATA OuranalysisusestheBillionPricesProjectDailyCPI(BPP).Severalpapershavealreadyestablishedthe abilityoftheBPPtoimproveforecastsoftheCPI[CavalloandRigobon(2016),AparicioandBertolotto (2020)andHarchaouiandJanssen(2018)].Wealsoreferreaderstothesepapersforadetaileddiscussion of BPP construction. Our analysis below confirms that the BPP contains additional information that helpsforecasttheCPIoveroursampleperiod. 12StramandWei(1986)showthisconditionholdsaslongastheARrootsaredistinctfromtheMAroots. 13Resultsavailableuponrequest. 9

JACOBSON,MATTHES&WALKER: TEMPORALAGGREGATIONBIAS 3.1 DAILY INFLATION DATA We define daily inflation as the 30-day percentage change in the BPP.14 TheBPPisconstructedfromoverfivemilliononlinepricesfrom300retailersin50countrieswebscraped daily.Whileweprovideabriefoverviewhere,ameticulousdescriptionofthedataisprovidedinCavallo andRigobon(2016).Ourdataconsistsof(publiclyavailable)observationsfrom2008to2015.Advantages ofthedataare[i.]thehigherfrequency(daily)vis-a-vistheCPI(monthlyorbi-monthly)orscannerdata (weekly);and[ii.] thenumberofpricescollectedfarexceedstheCPI(500kvs. 80k). Thedisadvantages are[i.] pricesareonlycollectedfromonlineretailersandthereforethesampleisnotrepresentativeof allconsumerprices; specifically, thesamplecontainsnopricingfromtheservicessector.15 According to Cavallo and Rigobon (2016), the data contain at least 70 percent of the weights in Consumer Price Index(CPI)basketsofroughly25countries;[ii.] Becausepricesarewebscraped,thedatadoesnotcontaininformationonquantitiessold. Thus,onlinepricesmustbecoupledwithweightsfromconsumer expendituresurveysorothersourcestoyieldexpenditure-weighteddata.16 Eventhoughpricesobtained byphysicallyvisitingstoresmaynotnecessarilycoincidewiththoseobservedonline,Cavallo(2017)finds a70percentmatchrate. 3.2 CONNECTIONTOCPIINFLATION ToalleviateconcernsthatBPPdatamaynotalignwellwiththe USCPI,wenowconductseveralteststoshowthattheBPPiseffectiveatanticipatingchangesininflation, afactthatwewillexploitinoureconometricanalysis. Statistic Releasedelay(days) Mean 16.97 Standarderror 2.73 Min 13 Max 30 Table3:SummarystatisticsonCPIreleasedelaysfromJuly2008toAugust2015. Panel3aplotsthepercentagechangeofthemonthlyCPIandtheBPPdailyindex;Panel3bplotsthe percentagechangeofthemonthlyCPIagainsttheaggregatedmonthlyBPP.Whilethecorrelationofthe twoseriesplottedinPanel3bisonly0.64,severalstudieshaveshownthattheBPPindexisparticularly adeptatpickingupturningpointsintheCPI,whichleadstoimprovedforecasts[CavalloandRigobon (2016),AparicioandBertolotto(2020)andHarchaouiandJanssen(2018)].Toshowthisresultholdsover oursampleperiod,weusethemonthlyaggregatedBPPseriestoconductaNowcastoftheCPIbyestimating,∆CPI =β +β ∆BPP +e .DespitebothindicesbeingdenotedwithsubscriptT,theCPIatdateT T 0 1 T T isannouncedwithaslightdelayasshownbyTable3,whichdocumentsthesummarystatisticsofrelease 14Incontrasttoday-over-daypercentage,30-daypercentagechangeallowsfortheunitsofdailyinflationtobecomparable tothoseofofficialinflationwhicharemeasuredatmonthlyfrequency. 15AlthoughcomparingtheBPPtoaversionoftheCPIwiththesamecoverageofcategorieswouldbeanidealexercise,we arelimitedbydataavailability. Wehaveinsteadrepeatedsomeofthecalculationsofthissectionusingsub-categoriesofthe CPIandtheresultsarebroadlysimilarasshowninAppendixA.Thesesub-categoriesincludethecommoditypriceindex,the commodityplusshelterindex,theofficialindexlessenergy,andtheofficialindexlessmedicalservices. 16TheBPPonlydisclosesweightspooledacrossallcountrieswheretheycollectdata. Theydonotdisclosecountryspecific weights.Seehttps://www.pricestats.com/approach/data-composition. 10

JACOBSON,MATTHES&WALKER: TEMPORALAGGREGATIONBIAS 1 1 0 0 1 − 1 − Daily Index (BPP) 2 Monthly Index (CPI) − 2 9 0 1 2 3 4 5 − 9 0 1 2 3 4 5 0 1 1 1 1 1 1 0 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 2 2 2 2 2 2 2 2 2 2 2 2 2 (a) DailyIndex(BPP)vsCPI (b)MonthlyAveragedDailyIndex(BPP) Figure 3: Official and daily inflation, monthly and 30-day percentage change. For month T, ∆CPI = T 100×(logCPI T −logCPI T−1 ) and for day t, ∆BPP t =100×(logBPP t −logBPP t−30 ) so that ∆BPP T = m 1 (cid:80)m t=1 100×(logBPP t −logBPP t−30 )fort=1,...,mdaysinmonthT. delaysindays(e.g., June2008CPIwasreleasedJuly16). Giventhatourinterestliesinhigh-frequency changesininflation,theslightdifferenceintimingisrelevantasonecanusethemonthlyaverageofthe BPPtopredictthatmonth’sCPInumber. Acoefficientequaltounity(β =1)suggeststheBPPperfectly 1 predictstheCPI.Theestimatedvalueis0.94withanR-squaredof0.58,implyingsubstantialpredictive power,seePanel4b.Panel4aplotsthein-samplepredictedvaluesagainsttherealizedvalues. Giventhepersistenceofinflation,weaddressthefollowingquestion:IsthereanyadditionalpredictivepoweroftheBPPbeyondthatcontainedinpastvaluesoftheCPI?Table4comparestheNowcastto anautoregressiverepresentationoftheCPI.ColumnonereportstheAR(1)specificationresults.Columns twoandthreeconditiononlyonpastvaluesoftheBPP,andshowasubstantialincreaseintheR-squared value when conditioning on the contemporaneous BPP, while the lagged BPP has less predictive contentthanlastmonth’sCPI.Columnsfourandfivedemonstrateanaffirmativeanswertothequestionof additionalpredictivepoweroftheBPP:ThecoefficientsonthecontemporaneousBPParepositiveand statisticallysignificant.TheR-squaredvalueistwiceashighastheautoregressivespecification.17 4 EMPIRICAL RESULTS 4.1 MEASURES OF HIGH-FREQUENCY MONETARY POLICY SHOCKS Beforeestimatingmonetarypolicy transmissionwithdisaggregatedinflationdata,webrieflydescribeourchoiceofmonetarypolicyshocks andtheirrespectivetimingandidentification. Wediscusstwosuchconstructionsindetail—Nakamura 17WeconductseveralrobustnesschecksinAppendixAwhichcorroborateourfindingsthattheBPPindexiseffectiveat predictingchangesininflation. Forexample,weconstructalternativemetricsforcomputinginflation(levels,end-of-month values)andexaminedifferenttypesofseasonality(day-of-the-week). 11

JACOBSON,MATTHES&WALKER: TEMPORALAGGREGATIONBIAS 1 1 0 0 1 1 − − ∆CPI = -0.09 + 0.94 ∆BPP Predicted CPI T T (0.37) (0.12) 2 Monthly Index (CPI) − 2 2009 2011 2013 2015 − 1.5 1 0.5 0 0.5 1 − − − (a)PredictedvsActual (b)ModelFit Figure4:NowcastofCPIusingmonthlyaggregatedBPP,monthlypercentagechange.Standarderrorsin parenthesesonPanel4b.FormonthT,∆CPI T =100×(logCPI T −logCPI T−1 )andfordayt andmonth T,∆BPP T = m 1 (cid:80)m t=1 100×(logBPP t −logBPP t−30 )fort=1,...,mdaysinmonthT. ∆CPI T (1) (2) (3) (4) (5) ∆CPI T−1 0.558 ∗∗∗ 0.178 (0.143) (0.107) ∆BPP 0.937 ∗∗∗ 0.878 ∗∗∗ 0.828 ∗∗∗ T (0.129) (0.097) (0.106) ∆BPP T−1 0.591 ∗∗ 0.109 −0.03 (0.248) (0.193) (0.222) R2 0.32 0.58 0.23 0.59 0.61 Adj.R2 0.31 0.58 0.22 0.58 0.6 Standarderrorsinparentheses. ∗ (p<.10), ∗∗ (p<.05), ∗∗∗ (p<.01) Table4:NowcastofBPPvs.autoregressiveCPI.FormonthT,∆CPI T =100×(logCPI T −logCPI T−1 )and fordayt andmonthT,∆BPP T = m 1 (cid:80)m t=1 100×(logBPP t −logBPP t−30 )fort=1,...,mdaysinmonthT. andSteinsson(2018a)(NS)andBuetal.(2021)(BRW).Wefocusontheseshocksbecausetheyarecharacterizedbyasinglefactorthatcanbeparsimoniouslyembeddedintomorecomplexframeworkslike ourstatespacemodel. EventhoughtheNSshockiswidelyused,thereareknownconcernsaboutpredictabilityandbias.Forthisreason,wealsoincludeestimatesusingtheBRWshockasitclaimstocontrol forsomeoftheseconcerns.NSfindasubstantialadversetransmissionofmonetarypolicyshocks,while BRWclaimtoovercomesuchdynamics.Ouraggregatedresultsreplicatethesefindings. NSdefinea“policynewsshock”asthefirstprincipalcomponentofthechangeinfiveinterestrates /futuresarounda30-minutewindowofFOMCannouncements: theexpectedfederalfundsrateatthe endofthemonthoftheFOMCannouncement,theexpectedfederalfundsrateattheendofthemonth 12

JACOBSON,MATTHES&WALKER: TEMPORALAGGREGATIONBIAS of the next scheduled FOMC announcement, and expected 3-month Eurodollar interest rates at horizonsoftwo,threeandfourquarters. Thelastthreefuturesaremeanttocapturetheeffectsofforward guidance as it impacts expectations beyond the federal funds rate. Our extension of this shock series isconstructedfromtheChicagoMercantileExchangefuturestickdatatoassureascloseofamatchas possibletotheoriginalseries.BRWusetheFamaandMacBeth(1973)two-stepproceduretoextractunobservedmonetary policy shocksfrom thecommoncomponentofzero-couponyields encompassing thefullyieldcurve. Thefirststepintheprocedureestimatesthesensitivityofyieldsofdifferentmaturitytomonetarypolicyviastandardtime-seriesregressions. Filteringoutnon-monetarypolicynewsis done through the heteroskedasticity-based estimator of Rigobon (2003) and Rigobon and Sack (2004), implementedbyemployinginstrumentalvariables(IV). 0.2 Nakamura-Steinsson Bu-Rogers-Wu 0.15 0.1 0.05 0 0.05 − 0.1 − 0.15 − 0.2 − 2009 2010 2011 2012 2013 2014 2015 Figure5: ExtractedshockseriesfromNakamuraandSteinsson(2018a)andBuetal.(2021). Theshocks arescaledsothattheireffectsequalunityonnominalTreasuryyieldsoftenuresequaltooneyear(NS) andtwoyears(BRW). Figure5plotstheextractedshockseriesforeachapproachoveroursampleperiod.AsnotedinBRW, their shock series has “moderately high correlation" with that of NS in addition to those of Swanson (2021)andJarocinskiandKaradi(2020).WhatisevidentfromthefigureisthattheBRWshockserieshas muchmoredispersionwhichislikelyattributedtothedifferentfrequencies,methods,tenures,andasset pricesusedintheconstruction.Despitethesedifferencesindispersion,ourempiricalanalysisconfirms thattemporalaggregationexacerbatesinitialimpulseresponsesforbothshockseries. 13

JACOBSON,MATTHES&WALKER: TEMPORALAGGREGATIONBIAS Daily Inflation Monthly Inflation 1.5 0.2 1 0.1 0.5 0 0 -0.1 -0.5 -0.2 -1 -0.3 -1.5 10 20 30 40 50 60 2 4 6 8 10 12 (a)Dailyinflation (b)Aggregated(dailytomonthly)inflation Figure 6: Impulse response of daily inflation (30-day percentage change) to a one standard deviation Nakamura and Steinsson (2018a) shock: aggregated vs disaggregated. For a given month, the aggregatedseriesarethesumofthemonetarypolicyshocksandtheaverageof30-dayannualizedpercentage changeofdailyinflation,BPP T = 12 m 00(cid:80)m t=1 (logBPP t −logBPP t−30 )fordayst=1,...,mofmonthT. 4.2 LOCAL PROJECTIONS WeemploylocalprojectionsusingthemethodologyofCanovaandFerroni (2022)toestimatetheimpulseresponsesofdisaggregatedandaggregatedinflation.Lety t+h bethevalue ofdailyinflationoverthepast30daysatdayt+h,x t−1 bethemonetarypolicyshock,andz t bethevector ofcontrolswhicharethe30lagsofdailyinflation.Giventhemodel, y t+h =α (h) +β (h) x t−1 +Γ (h) z t +e t (h), e t (h)∼N(0,σ (h) ) estimates are computed via instrumental variables with robust heteroskedasticity and autocorrelation consistent(HAC)standarderrors. Wereport90%confidencebandsandexaminetheresponseofinflationobservedatvariousfrequencies.Wenormalizetheshockseriestohaveunitvariance. Figure6plotstheimpulseresponsetoaone-timecontractionaryNSmonetarypolicyshockatboth thedailyandmonthlyfrequency. Panel6ashowsmediandisaggregateddailyinflationrespondspositivelyinitially;however,the90%confidenceintervalsubstantiallyoverlapszeroforperiodszerothrough 33.Afterroughly30periods(onemonth),theinflationresponseturnsnegativeandissignificantlysofor theremainingperiodsshown. Byitselfthisimpulseresponseismerelysuggestive. Atadailyfrequency, theNSshocksequencedoesnotproduceasubstantialandlong-lastingpositiveresponseofinflationtoa contractionarymonetarypolicyshock.Themagnitudeoftheinitialpositiveresponseisroughlyhalfthat ofthenegative(andmuchmorepersistent)response. SincetheNSmonetarypolicyshockisassociated withinitiallyadverseresponses,feedinginthissequencegivesusthebestchanceofrecoveringone. At dailyfrequency,sucharesponsematerializesonlytemporarily. 14

JACOBSON,MATTHES&WALKER: TEMPORALAGGREGATIONBIAS Daily Inflation, BRW Monthly Inflation, BRW 0.2 0.5 0.15 0.1 0 0.05 0 -0.5 -0.05 -0.1 -1 -0.15 -0.2 -1.5 -0.25 10 20 30 40 50 60 2 4 6 8 10 12 (a)Dailyinflation (b)Aggregated(dailytomonthly)inflation Figure7:Impulseresponseofdailyinflation(30-daypercentagechange)toaonestandarddeviationBu etal.(2021)shock:aggregatedvsdisaggregated.Foragivenmonth,theaggregatedseriesarethesumof themonetarypolicyshocksandtheaverageof30-dayannualizedpercentagechangeofdailyinflation, BPP T = 12 m 00(cid:80)m t=1 (logBPP t −logBPP t−30 )fordayst=1,...,mofmonthT. Panel6baggregatesthedailyindextoamonthlyfrequency.18 Theadverseresponseemerges. When aggregated,thedatasuggesttheinitialadverseresponseisquantitativelylargeandoneofthefewcomponentsoftheimpulseresponsefunctionforwhichtheconfidencebanddoesnotcoverzero. Incontrast, the disaggregated initial response of the daily frequency was dominated by the larger and more significantnegativeresponseofthelatertimeperiods.Figure6bbehoovesresearcherstoprovideanexplanationforthisadverseresponsewheninfactitisnottheprominentfeatureofthedataataslightly higherfrequency. Ourmodelingresultsinsection2canreconcilethesediscrepanciesintheestimated adverseresponseviasubstantiallyalteredmoving-averagecoefficientsduetotemporalaggregationbias. Oneexplanationisthatmonetarypolicyannouncementscontainnovelinformationabouteconomic fundamentalsandprivateagentsarereactingtothisnews.19 Whatwerefertoasan“adverse"shockor onethatrunscountertostandardtheorycouldbeexplainedbyintroducingadiscrepancyininformationbetweentheFederalReserveandprivateagents,NakamuraandSteinsson(2018a).However,testing forthiseffectrequireshighfrequencydata.Previousstudies[e.g.,JarocinskiandKaradi(2020),Lunsford (2020)] examined the reaction of asset prices, such as stocks and bonds, but we are the first to study themostintriguingeconomicfundamental—inflation—athighfrequency. Figure6definitivelydemonstratesthattheadverseresponsetoinflationcouldbeduetoaninformationdiscrepancybetweenthe econometricianandprivateagents,andnotjusttheFederalReserveandprivateagents. Figure 7 plots the impulse response of inflation to a contractionary BRW monetary policy shock. 18Weaveragethedailyshocksforeachmonthandthennormalizetheresultingshocktohaveunitvariance. 19RomerandRomer(2000), Campbelletal.(2012,2017), NakamuraandSteinsson(2018a), JarocinskiandKaradi(2020), Miranda-AgrippinoandRicco(2021),Lunsford(2020),Hoeschetal.(2021),CieslakandSchrimpf(2019),Acosta(2022),Lewis (2020),BundickandSmith(2020),AndradeandFerroni(2021),GolezandMatthies(2021). 15

JACOBSON,MATTHES&WALKER: TEMPORALAGGREGATIONBIAS Panel7ashowsthat,atdailyfrequency,themedianresponseofinflationisclosetozeroorslightlynegativeuntilaboutperiod60(twomonths)whenitbecomesmorenegativeandonthemarginoftheconfidencebands. Incontrast,whenthedailyindexisaggregatedtoamonthlyfrequency,thepointestimate oftheimpactresponseispositive,albeitzeroiswellcontainedinthe90%crediblesets.Theresultsusing BRW shocks are not surprising given that the “information effect" is not a feature of the shock series. Indeed, theprimarytakeawayofBuetal.(2021)isthatthelong-endoftheyieldcurveisnecessaryto eliminatetheinformationeffect: Whereasalternativemeasuresareconstructedfromonlyshortrates,weusetheentireyield curve. ThisisimportantbecausewefindthattheFedinformationeffectisessentiallynonexistentinmaturitiesoffiveyearsandlonger. While we certainly agree that there could be additional information in interest rates of duration longerthantwoyears, itisnotclearthatthisadditionaldataisthesolereasonforeliminatingtheFed information effect, especially when the methodologies generating the shock sequences are drastically different. Fromatheoreticalperspective, onewouldhavetoassumethatFOMCannouncementscontainsubstantialinformationabouteconomicfundamentalsathorizonslongerthantwoyearsandthat thishorizonismostrelevantforexplainingimpact impulseresponses,whichseemshighlyunlikely. We instead take the same shock sequence and temporally aggregate the same inflation data to construct alternative impulse response functions. The adverse response, often attributed to the “Fed informationeffect",isabsentathigherfrequenciesbecausethemismatchbetweeneconometricianandprivate agentsiseliminated.Temporalaggregationbiasexplainsthisresponseatlowerfrequencies,asopposed toaninformationaldiscrepancybetweenpolicymakersandprivateagents. 4.3 UNOBSERVEDCOMPONENTSMODEL Tostudytheresponseofhigh-frequencyinflationtoamonetarypolicyshockmoresystematically,wenowintroduceanunobservedcomponentsmodel.Weemploy thismethodologyforseveralreasons.First,thepermanent-transitorydecompositionscastinstatespace formhaveprovenveryusefulforinflationatlowerfrequencies[StockandWatson(2020)].Second,there is transparency in modeling assumptions. Relative to the local projections methodology, which relies onIVandHACerrors,themodelingassumptionsherearemorestraightforward. Thisallowsustotake amoredefinitivestanceonourfindingofaconventionally-signedtransmissionofmonetarypolicy,as opposedtodisentanglinghowtemporalaggregationmightinteractwith, say, ourIVestimation. Third and relatedly, the model specification is parsimonious. Finally and most importantly, the state space / estimation methodologies allow us to more easily handle data observed at different frequencies and withobservationsmissingatdifferentdates—weusedailyinflationdata, dataonbreak-eveninflation ratesthatisavailabledailyexceptforholidaysandweekends, infrequentmonetarypolicyshocks, and monthlyinflationrates.Furthermore,theunobservedcomponentsapproachallowsustoexplicitlytake intoaccounttheexacttimingofmonetarypolicyshocksandreleasesofofficialinflationreleaseswithin amonth. Ourmodelconsistsofthefollowingstateequations:[i.]UnobserveddailyCPIinflation,π =τ +g + t t t e π ,brokendownintoapermanentcomponentτ,atransitorycomponentg,andi.i.d.shocke π .Thepert 16

JACOBSON,MATTHES&WALKER: TEMPORALAGGREGATIONBIAS # Parameter Prior Notes 1 σπ Γ(1,0.5) standarddeviationofi.i.d.componentofunderlyinginflation 2 στ Γ(1,0.5) standarddeviationofinnovationtorandomwalkpermanentcomponent 3 ρ g β(4,4) persistenceofstationarypart 4 σ g Γ(1,0.5) standarddeviationofinnovationtostationarypart 5 αm N(0,0.00012) interceptofmeasurementequationofmonthlyCPIinflation 6 σmonthly Γ(1,0.5) standarddeviationofmeasurementerrorofmonthlyCPIinflation 7 αdaily N(0,52) interceptofmeasurementequationofdaily(30-day)inflation 8 σdaily Γ(1,0.5) standarddeviationofmeasurementerrorofdailyinflation 9 αBE N(0,52) interceptofmeasurementequationofdailyBEinflation 10 σBE Γ(1,0.5) standarddeviationofmeasurementerrorofdailyBEinflation 11 θg N(0,0.252) contemporaneousimpactofmonetaryshockong 0 12 θτ N(0,0.252) contemporaneousimpactofmonetaryshockonτ 0 13 σm,obs Γ(1,0.5) standarddeviationofmonetaryshock 14∼72 θg N(0,(0.25∗0.95i)2) vectorofeffectsof59dayslaggedmonetaryshocksong i 59×1 73∼131 θτ N(0,(0.25∗0.95i)2) vectorofeffectsof59dayslaggedmonetaryshocksonτ i 59×1 Table5:PriorSpecification manentandtransitorycomponentsfollow,τ t =τ t−1 +(cid:80)K k=0 θ k τ m t−k +e t τ andg t =ρg t−1 +(cid:80)J j=0 θ j m t−j + e g ,respectively.20 Thepermanentcomponentofinflationallowsforaunit-rootspecificationandaset quenceofmonetarypolicyshocksfor60periods(K =J =60). Thetransitorycomponentpermitsautocorrelation and the same number of monetary policy shocks. We assume monetary shock dynamics m =em with all shocks e being i.i.d. and Gaussian. The observation equations are the monthly obt t servationofCPI(real-timevintages): πm t =αm+π t−p +e t monthly ,wherep ispublicationlagmentioned in Section 3 (which can vary over time as shown in Table 3). At higher frequencies, we use the daily measureofmonthly(30-day)inflation: πdaily =αdaily+π +e daily , andthe10-yearbreak-evenrates: t t t πB t E,h=αBE+E t π t,t+h +e t BE.Weassumethatthemonetarypolicysurpriseisanoisymeasurementofthe truemonetarypolicyshock: mobs=m +e m,obs ,alongthelinesofCaldaraandHerbst(2019). Notethat t t t themodelimpliesE t π t+h =E t (τ t+h +g t+h )=τ t +ρhg t ≈τ t , wherethelastapproximationisimposed ontheestimationprocedure(ourpriorimposesthatthedailypersistenceofthetransitorycomponent |ρ|<1,andhrepresentsthe10yearhorizon). The estimation is Bayesian with the likelihood function evaluated using the Kalman filter. To effectivelyexploretheposteriordistribution,asequentialMonteCarloalgorithmisimplemented[Herbst andSchorfheide(2016)]. Weuse15,000particleswith200stepstogofromthepriortothefullposterior andfiveMetropolisHastingsstepsperiterationofthealgorithm.Table5reportsourpriordistributions, which are largely uninformative. The one are where we impose somewhat informative priors are the effectsofmonetarypolicyshocksonthetransitoryandpermanentcomponentsofinflation. Wecenter thosepriorsat0tonotbiasourresultsfororagainstfindingadverseeffects,butwedoimposeshrinkage 20Incontrasttopreviousworkusingstatespacemodelstodescribeinflationdynamics,weexplicitlyincorporatearolefor monetarypolicyshocks.Weallowtheseshocks(whicharemeasuredwitherror)toaffectbothtransitoryandpermanentcomponentsofinflation.Thisisimportantbecausemovementsininflationthatmightseempermanentatthedailyfrequencycan correspondtopersistent,butnon-permanentcomponentsatalowerfrequency. 17

JACOBSON,MATTHES&WALKER: TEMPORALAGGREGATIONBIAS 1 0.5 0 0.5 − 1 − 1.5 − 0 5 10 15 20 25 30 35 40 45 50 55 60 (a) OverallImpulseResponse 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0 0.2 0.2 − − 0.4 0.4 − − 0.6 0.6 − − 0.8 0.8 − 0 5 10 15 20 25 30 35 40 45 50 55 60 − 0 5 10 15 20 25 30 35 40 45 50 55 60 (b)ResponseofTransitoryComponent(gt) (c)ResponseofPermanentComponent(τ t) Figure8: ImpulseresponsestoaonestandarddeviationNakamuraandSteinsson(2018a)shock. Error bandsare68%and90%posteriorbandscenteredatthemedian. - the further a monetary policy shock is in the past, the more we shrink its effect towards zero. In the Appendixweshowthatourfindingsarerobusttoimposinglessshrinkage. Panel 8a plots the overall impulse response function of inflation to a contractionary NS monetary policy shock, while Panels 8b-8c plot the response of the transitory and permanent components, respectively.21 Darkershadederrorbandsare68thpercentiles, whilelightershadesare90th. Theinitial observation is that inflation—at a daily frequency—does not contain an adverse response. The initial reactionofinflationtoaonestandarddeviationmonetarypolicyshockisnegative,evenatthe90thpercentile,followedbyanincreaseandanerrorbandthatcontainszeroovertheremaininghorizon. The permanentcomponentresponseofPanel8cshowsthatthestandardandtheory-consistentresponseof inflationispresentinourdailydata. Theseresultsfurthercorroborateourfindingsfromthelocalprojections;namely,thatthepositivereactionofinflationtoamonetarypolicyshockisdifficulttodetectat thedailyfrequency. Bydecomposingintopermanentandtransitorycomponents,weareabletoparse theconventionallysignedimpulseresponseaspermanent. Mostimportantly,thetransitoryresponseis showntobequantitativelysmallrelativetotrend.22 21ResultsaresimilarfortheBRWshockseriesandareavailableuponrequest. 22AppendixBshowsthatourresultsarerobusttolessshrinkageoftheestimators.Infact,thepermanentcomponentshows amoresubstantialconventionallysignedresponseatlongerhorizonsinthatcase. 18

JACOBSON,MATTHES&WALKER: TEMPORALAGGREGATIONBIAS Thevariancedecomposition,plottedinFigure9,showsthatthelion’sshareofvolatilityisexplained bythepermanentcomponentofinflationasopposedtothetransitorycomponent.Takentogether,these figures suggest that methodologies that de-trend inflation prior to analysis could miss conventionally signed responses. More germane to our argument, the transitory component when evaluated at daily frequenciesdoesnotdisplayasubstantialadverseresponsedespitethefactthattheshocksfedintothe systemgeneratesubstantialadverseresponsesatmuchlower(monthly)frequencies. τ 0.15 0.1 0.05 0 5 10 15 20 25 30 35 40 45 50 55 60 g · 10− 4 1 0.5 0 5 10 15 20 25 30 35 40 45 50 55 60 TotalInflation 8· 10− 4 6 4 2 0 5 10 15 20 25 30 35 40 45 50 55 60 Figure9:VarianceDecompositionassociatedwithMonetaryPolicyShockasafractionoftotalvariance. 5 CONCLUDING THOUGHTS Thispaperrevisitsafundamentalquestionofmonetaryeconomics: Whatisthetransmissionofmonetarypolicytotheeconomy? Weintroducetemporalaggregationbiasasanewinformation-basedexplanationfortheadversetransmissionofmonetarypolicyshocks. WhenusingthedailyCPIfromthe BillionPricesProjectasatemporallydisaggregatedmacroeconomicindicator,wefindaconventionallysigned response with only a short-lived adverse sign when present at all. To understand how one can obtainasizableadverseresponsetomonetarypolicyshockswithmonthlyorquarterlydatawhenonly a limited adverse response actually exists, we combine a simple model of temporal aggregation bias 19

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JACOBSON,MATTHES&WALKER: TEMPORALAGGREGATIONBIAS A APPENDIX: BPP ROBUSTNESS CHECKS A.1 ALTERNATIVE CONSTRUCTIONS OF BPP INFLATION This section shows an alternative version of figure4. 1 1 CPI=-.02+.85BPP R^2=.42 0 0 -1 -1 -2 -2 -1 -.5 0 .5 1 2008 2010 2012 2014 2016 Official Index Prediction by Daily Data Fitted Values Figure10:NowcastofCPIusingendofmonthvaluesoftheBPP,monthlyand30-daypercentagechange. For month T, ∆CPI T = 100×(logCPI T −logCPI T−1 ) and for day m of of month T, ∆BPP T = 100× (logBPP m −logBPP m−30 ). 110 110 105 105 CPI=11.8 7 R + 2 . = 8 . 8 9 D 9 ailyIndex 100 100 95 95 2008 2010 2012 2014 2016 95 100 105 110 Official Index Prediction by Daily Data Fitted Values Figure 11: Nowcast of CPI using aggregated monthly values of the BPP, index. For month T, CPI = T logCPI andfordayt ofmonthT,∆BPP =(cid:80)m logBPP fort=1,...mdaysinmonthT. T T t=1 t 24

JACOBSON,MATTHES&WALKER: TEMPORALAGGREGATIONBIAS A.2 CPISUB-CATEGORIES Table6showshowtheBPPNowcastoftheheadlineCPIcomparestoother CPIsub-categories. ∆CPIi ,variousCPISub-categories,i T (1) (2) (3) (4) (5) Commodities Headline Headline Headline Commodities &Shelter exenergy exMedical ∆BPP 0.937 ∗∗∗ 1.618 ∗∗∗ 0.53 ∗∗∗ 0.18 ∗∗∗ 1.001 ∗∗∗ T (0.129) (0.283) (0.121) (0.052) (0.137) R2 0.58 0.48 0.36 0.21 0.59 Adj.R2 0.58 0.47 0.36 0.2 0.58 Standarderrorsinparentheses. ∗ (p<.10), ∗∗ (p<.05), ∗∗∗ (p<.01) Table6: NowcastofCPIsub-categoriesusingtheBPP.FormonthT andsub-categoryi,∆CPIi =100× T (logCPI T i −logCPI T i −1 )andfordayt andmonthT,∆BPP T = m 1 (cid:80)m t=1 100×(logBPP t −logBPP t−30 )for t=1,...,mdaysinmonthT. A.3 SEASONALITY BPP =trend + (cid:88) αday 1dayof week+ϵ t t j t j Sunday Monday Tuesday Wednesday Thursday Friday Saturday 96.7 96.8 96.9 97 Figure12:DayofweekeffectsoftheBillionPricesProjectdailyinflation. 25

JACOBSON,MATTHES&WALKER: TEMPORALAGGREGATIONBIAS B APPENDIX: IMPULSE RESPONSE FUNCTIONS WITH LESS SHRINKAGE This Appendix shows the impulse responses from the state space model under the assumption of less shrinkage-thepriorstandarddeviationoflaggedcoefficientsisnow0.25∗0.99i,wherei isthelag. 1.5 1 0.5 0 0.5 − 1 − 1.5 − 2 − 2.5 − 3 − 3.5 − 0 5 10 15 20 25 30 35 40 45 50 55 60 Days Figure 13: Impulse response of inflation (π ) to a one standard deviation Nakamura and Steinsson t (2018a) monetary policy shock . Error bands are 68 % and 90 % posterior bands centered at the median. 26

JACOBSON,MATTHES&WALKER: TEMPORALAGGREGATIONBIAS 2 1.5 1 0.5 0 0.5 − 1 − 0 5 10 15 20 25 30 35 40 45 50 55 60 Days Figure 14: Impulse response of the transitory component of inflation (g ) to a one standard deviation t NakamuraandSteinsson(2018a)monetarypolicyshock.Errorbandsare68%and90%posteriorbands centeredatthemedian. 0.5 0 0.5 − 1 − 1.5 − 2 − 2.5 − 3 − 0 5 10 15 20 25 30 35 40 45 50 55 60 Days Figure15: Impulseresponseofthepermanentcomponentofinflation(τ )toaonestandarddeviation t NakamuraandSteinsson(2018a)monetarypolicyshock.Errorbandsare68%and90%posteriorbands centeredatthemedian. 27

JACOBSON,MATTHES&WALKER: TEMPORALAGGREGATIONBIAS C APPENDIX: TEMPORAL AGGREGATION Theorem1. Thetemporallyaggregatedinflationprocessgivenby (3)and (2)satisfiesthefollowingtwo properties: 1. Thetemporallyaggregatedinflationseries,Π ,followsanARMA(1,1)process. T (1−ρmL)Π T =u T +θu T−1 (6) 2. TheinnovationoftheARMA(1,1)process(6)isfundamentalforthetemporallyaggregatedinflation sequence,Π . T ThistheoremiswellknownanddatesbacktoatleasttoAmemiyaandWu(1972); thus, wedonot offeracompleteproofbutprovideintuitionandreferences. Tounderstandpart(1),letπ t =ρπ t−1 +w t , wherew t isGaussianwithmeanzeroandvarianceσ2 w =σ2 ε/(φ−ρ)2,andnote σ2 γ(0)=Var(Π )= π (cid:161) m+2[(m−1)ρ+(m−2)ρ2+···+ρm−1] (cid:162) (7) T m2 σ2 γ(s)=Cov(Π t ,Π t−s )= m π 2 ρm(|s|−1)+1(1+ρ+ρ2+···+ρm−1)2 s̸=0 (8) γ(s)=ρmγ(s−1) |s|≥2 (9) whereσ2 π =σ2 w /(1−ρ2),seeWeiandAhsanullah(1984). Theintuitionof(7)–(8)comesfromthecorre- σ2 lationstructureofanautoregressiveprocess, whereallelementsaremultipliedby π. Thus, thereare m2 (m−1)“neighbors",(m−2)elementstwoperiodsremoved,etc.Giventhestrengthoftheautocorrelation ofmanymacroaggregates,thefollowinglimitsareuseful.Asρ→1,theterminbracketsin(7)converges to m(m−1)/2 and therefore, Var(Π T )→σ2 π and Var(Π T )∈(0,σ2 π). Further, the parenthetic term in (8) convergestomasρ→1,andCov(Π t ,Π t+s )→σ2 π. π π π ··· π t t−1 t−2 t−m π 1 ρ ρ2 ··· ρm−1 t π t−1 ρ 1 ρ ··· ρm−2 π t−2 ρ2 ρ 1 ··· ρm−3 . . . π t−m ρm−1 ρm−2 ρm−3 ··· 1 Thecovariancedifferenceequation(9)identifiestheautocorrelationcoefficientoftheΠ processas T ρm.Wecanthenmultiply[(1−ρmL)/(1−ρL)] (cid:80)m−1Lj tobothsidesofπ togive, j=0 t (cid:195) (1−ρL)(1−ρmL) (cid:80)m−1Lj(cid:33) (cid:195) (1−ρmL) (cid:80)m−1Lj(cid:33) j=0 π = j=0 w 1−ρL t 1−ρL t m−1 (1−ρmL)Π T = (cid:88) (ρL)jw t =u T +θu T−1 (10) j=0 28

JACOBSON,MATTHES&WALKER: TEMPORALAGGREGATIONBIAS whereu ∼N(0,σ2).Theerrorsdefinedbythethemmoving-averageterms (cid:80)m−1(ρL)jw arecorrelated T u j=0 t andthereforecannotbeusedtoobtaintheWoldinnovationsassociatedwithpredictingΠ linearlyfrom T itspast. Theorem1ofAmemiyaandWu(1972)provesthatwithm≥2,thenthemoving-averageterms areatmostoforderone,whichestablishesthefinalequality. TheproofofPart2alsoreliesonargumentsinAmemiyaandWu(1972).Inorderfortheprocesstobe fundamental,onemustshowthattherootsof1−θzlieoutsideoftheunitcircle(i.e.,|θ|<1).Giventhat theinitialAR(1)processispositivedefinite(ρ∈(0,1)),thenithasapositivespectraldensity. Asshown inAmemiyaandWu(1972),temporalaggregatemaintainsthepositivedefinitestructureandhencethe rootsofthemoving-averagerepresentationmustlieoutsidetheunitcircle. C.1 MOVING-AVERAGEFILTERS Supposewehaveastationarystochasticprocessx t thatisaggregated accordingto X T = (cid:181) 1 (cid:182) (cid:195) m (cid:88) −1 Lj (cid:33) x mT = (cid:181) 1 (cid:182) (x mT +x mT−1 +···+x mT−m−1 ) (11) m m j=0 Notethat1+L+L2+···+Lm−1=(1−Lm)/(1−L).Thus,thecovariancegeneratingfunctionofX isrelated T tox by t 1 (cid:181) 1−zm(cid:182)(cid:181) 1−z −m(cid:182) g (z)= g (z) (12) X m2 1−z 1−z −1 x Inthefrequencydomain(z=e −iω ), g (e −iω )= 1 (cid:181) 1−e −iωm(cid:182)(cid:181) 1−eiωm(cid:182) g (e −iω ) X m2 1−e −iω 1−eiω x = 1 (cid:181) 1−cos(ωm) (cid:182) g (e −iω ) (13) m2 1−cos(ω) x where (1−e −iωm)(1−e −iωm) = 2−(eiωm+e −iωm) = 2−2cos(ωm) = 2(1−cos(ωm)) because eiωm = cos(ωm)+isin(ωm) and e −iωm = cos(ωm)−isin(ωm). Plotting this function over the range of [0,π] givesFigure2b. 29

JACOBSON,MATTHES&WALKER: TEMPORALAGGREGATIONBIAS D APPENDIX: DATA Thissectionliststhesourceanddescriptionofeachseriesusedinthispaper. OFFICIAL CPI INDEX Analysis in section 3.2 use the BLS’ seasonally adjusted Consumer Price Index (FRED: CPIAUCSL) at a monthly frequency. Results in section (4) use the seasonally adjusted (PCPI) andnotseasonallyadjusted(CPIN)real-timeConsumerPriceIndexwhichisaccessedviatheReal-time DataResearchCenterattheFederalReserveBankofPhiladelphia.23 Ineachreal-timespreadsheet,the columnsarethedateofthevintageandtherowsarethetimeseriesforthatvintage.Wethenconstructa timeseriesbycalculatingthemonthlypercentagechangeforthelasttwoentriesforeachvintage. DAILYCPI TheBillionPricesProjectpubliclyavailabledailyinflationindexcanbeobtainedviaCavallo andRigobon(2016)forJuly2008throughAugust2015.24 Theindexisobtainedbywebscrapingprices frommultichannelretailersthatsellbothonlineandoffline. BREAK-EVEN INFLATION RATES 10-year spot breakeven inflation rates are the daily 10-year treasury yieldatconstantmaturity(FRED: BC_10YEAR)lessthedaily10-yearTIPSatconstantmaturity(FRED: TC_10YEAR).TheseratesareobtainedfromtheU.S.TreasuryDepartmentviaFRED. ZERO-COUPONTREASURYYIELDS Continuouslycompoundedzero-couponyields(mnemonic:SVENYXX) areobtainedviatheFederalReserveBoard.25 NAKAMURAANDSTEINSSON(2018A)MONETARYPOLICYSHOCK High-frequencymonetarypolicyshocks areoriginallyavailablefrom1995to2014.26 Weextendthisshockseriesfrom1994topresentusingfuturestickdataaccessedviaCMEGroupInc.DataMine(https://datamine.cmegroup.com/)attheFederal ReserveBoard.27 TheconstructionoftheshockseriesfollowsthatofGürkaynaketal.(2005)asdescribed inNakamuraandSteinsson(2018a)andreliesonthechangesinfiveshort-terminterestratefutures.Let 23WethankTomStarkforhelpobtainingtheseseries. https://www.philadelphiafed.org/surveys-and-data/real-time-dataresearch/real-time-data-set-full-time-series-history 24Series indexCPI for country==USA in spreadsheet pricestats_bpp_arg_usa.csv in folder all_files_in_csv_format.zipatwebsite https://dataverse.harvard.edu/dataset.xhtml?persistentId=doi%3A10.7910%2FDVN%2F6RQCRS. Alternatively, the data are also available from the pricestats_bpp_ar_usa.dta file in the RAWDATA folder on the website https://www.openicpsr.org/openicpsr/project/113968/version/V1/view. 25Seehttps://www.federalreserve.gov/data/yield-curve-tables/feds200628_1.htmlorasacsvfile. 26SeriesFFR_shockfromthespreadsheetPolicyNewsShocksWeb.xlsx https://eml.berkeley.edu/∼jsteinsson/papers/PolicyNewsShocksWeb.xlsx 27https://eml.berkeley.edu/∼jsteinsson/papers/realratesreplication.zip 30

JACOBSON,MATTHES&WALKER: TEMPORALAGGREGATIONBIAS t indexFOMCannouncementsandthechangesinthefiveinterestratefuturesbegivenas: mp1 : changeinfederalfundfuturesexpiringattheendofthemonthoftheFOMCannouncement t mp2 : changeinfederalfundsfuturesexpiringattheendofthemonthofthenextscheduledFOMC t announcement ed2 : changeineurodollarfuturesexpiringinthenextquarterfromtheFOMCannouncement t (called2ndcontract) ed3 : changeineurodollarfuturesexpiringtwoquartersaftertheFOMCannouncement t (called3rdcontract) ed4 : changeineurodollarfuturesexpiringthreequartersaftertheFOMCannouncement t (called4thcontract) The calculations underlying the above series are given below.28 Let s index the month of the current FOMC announcement and s ′ index the month of the next FOMC announcement. For example, s = March2014 and s ′ =April2014 for the March 19, 2014 FOMC announcement where s and s ′ need not beconsecutivemonths.Wedefinet morepreciselyas20minutesaftertheFOMCannouncementwhile t−∆t isdefinedas10minutesbefore theFOMCannouncement.29 FortheMarch19, 2014FOMCannouncementwhichoccurredat14:00,t =March19,201414:20andt−∆t =March19,201413:50. Letq indexthequarterofthecurrentFOMCannouncementandq+1indextheofthenextFOMCannouncement. For example, q =2014:Q1, q+1=2014:Q2, and q+2=2014:Q3 for the March 19, 2014 FOMC announcement. D1 mp1 t = D1−d1 (f f1 s,t −f f1 s,t−∆t ) (14) (cid:183) (cid:184) D2 d2 mp2 t = D2−d2 (f f2 s′,t −f f2 s′,t−∆t )− D2 mp1 t (15) ed2 t =ed2 q+1,t −ed2 q+1,t−∆t (16) ed3 t =ed3 q+2,t −ed3 q+2,t−∆t (17) ed4 t =ed4 q+3,t −ed4 q+3,t−∆t (18) FormonthsindexingthecurrentFOMCannouncement... 28NakamuraandSteinsson(2018a)explainthat,“Aeurodollarfuturescontractexpiringinaparticularquarter(say2ndquarter2004)isanagreementtoexchange,onthesecondLondonbusinessdaybeforethethirdWednesdayofthelastmonthofthe quarter(typicallyaMondaynearthe15thofthemonth),thepriceofthecontractpfor100minusthethencurrentthree-month USdollarBBALIBORinterestrate." 29Inpractice,thewindowsarenotalwaysthispreciseandwefollowtheonlineAppendixofNakamuraandSteinsson(2018a). Forthet−∆t contact,weusethecontractasclosetothe10minutesbeforethepolicyannouncementaspossibleandonly considertradesonthedayinquestion. Forthet contract,wesimilarlyusethecontractasclosetothe20minutesafterthe announcementaspossibleandconsidertradesaslateasnoononthefollowingday.Iftherearenoeligibletradestoconsider, thechangeissettozero(i.e.,weinterpretnotradingasnopricechange).Wesourcethetimeoftheannouncementsfromthe FederalReserveBoardandthenfromGürkaynaketal.(2005)andBloombergNewsWire.Ifthereisaconflictinannouncement times,wefollowthisorderofpriority. 31

JACOBSON,MATTHES&WALKER: TEMPORALAGGREGATIONBIAS mp1 : monetarypolicysurprise t D1: numberofdaysinmonth d1: dayofmonth f f1 s,t−∆t : federalfundsfuturescontractatmost10minutesbefore f f1 : federalfundsfuturescontractatleast20minutesafter s,t ′ Formonths indexingthenextscheduledFOMCannouncement... mp2 : monetarypolicysurprise t D2: numberofdaysinthemonth d2: dayofmonth f f2 s′,t−∆t : federalfundsfuturescontractatmost10min.before f f2 s′,t : federalfundsfuturescontractatleast20min.after Forquarterq indexingthecurrentFOMCannouncement... ed2 q+1,t−∆t : 2ndexpiringeurodollarfuturescontractatmost10minutesbefore ed2 q+1,t : 2ndexpiringeurodollarfuturescontractatleast20minutesafter ed3 q+2,t−∆t : 3rdexpiringeurodollarfuturescontractatmost10minutesbefore ed3 q+2,t : 3rdexpiringeurodollarfuturescontractatleast20minutesafter ed4 q+3,t−∆t : 4thexpiringeurodollarfuturescontractatmost10minutesbefore ed4 q+3,t : 4thexpiringeurodollarfuturescontractatleast20minutesafter IfthecurrentFOMCannouncementoccursinthelast7daysofthemonththenthescalingisnotusedas itmaybequitelargetowardstheendofthemonth.Instead,thefuture’scontractforthemonthfollowing thatofthecurrentFOMCannouncementisused, s+1. Forexample,thefuturescontractforFebruary 2015wouldbeusedinsteadofJanuary2015fortheJanuary28,2015announcement. mp1 t =(f f1 s+1,t −f f1 s+1,t−∆t ) (1.a) And similarly if the next scheduled FOMC announcement occurs in the last 7 days of the month, the followingmonth’sfuturescontractisisused, s ′+1. Forexample,theannouncementfollowingthaton March18,2015isonApril29,2015wouldusethefuturescontractforMay2015insteadofApril2015. mp2 t =(f f2 s′+1,t −f f2 s′+1,t−∆t ) (2.a) Themonetarypolicyshockisthenthefirstprincipalcomponentofexpressions(14)-(18)scaledso thatitseffectonone-yearnominalTreasuryyieldsisequaltoone. 32

JACOBSON,MATTHES&WALKER: TEMPORALAGGREGATIONBIAS BU ET AL. (2021) MONETARY POLICY SHOCK Dailymonetarypolicyshockareavailablefrom1994to 2020.30 ThisshockseriesisconstructedbyaFamaandMacBeth(1973)two-stepprocedurethatextracts unobserved monetary policy shocks ∆i from the common component of the change in zero-coupon t yields∆R . j,t 1. estimatesensitivityofyieldswithmaturity j =1,...,30tomonetarypolicyviatime-seriesregressions ∆R =α +β ∆i +ϵ j,t j j t j,t assume∆i isone-to-onewithtwo-yearyield∆R toallowfornormalization t 2,t ∆R =θ +β ∆R +ϵ −β ϵ j,t j j 2,t j,t j 2,t (cid:124) (cid:123)(cid:122) (cid:125) ξ j,t corr(∆R ,ξ )duetoβ ϵ reconciledwithIVortheheteroskedasticity-basedestimatorofRigobon j,t i,t j 2,t (2003). 2. recoveralignedmonetarypolicyshock∆i aligned formcross-sectionalregressionsof∆R onthe t j,t sensitivityindexβˆ foreachFOMCannouncementt j ∆R =α +∆i alignedβˆ +v , t=1,...,T j,t j t j j,t 3. re-scaletheshock. WefollowBuetal.(2021)anduse2-yearTreasuries,butourresultsarerobust toscalingby1-yearTreasuriestomatchthescalingoftheNSshocks. 30SeriesBRW_fomcofspreadsheetbrw-shock-series.csvhttps://www.federalreserve.gov/econres/feds/files/brw-shockseries.csv 33