Which Output Gap Estimates Are Stable in Real Time and Why?

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. Which Output Gap Estimates Are Stable in Real Time and Why? Alessandro Barbarino, Travis J. Berge, Han Chen, Andrea Stella 2020-102 Please cite this paper as: Barbarino, Alessandro, Travis J. Berge, Han Chen, and Andrea Stella (2020). “Which Output Gap Estimates Are Stable in Real Time and Why?,” Finance and Economics Discussion Series 2020-102. Washington: Board of Governors of the Federal Reserve System, https://doi.org/10.17016/FEDS.2020.102. 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.

Which Output Gap Estimates Are Stable in Real Time and Why? AlessandroBarbarino* TravisJ.Berge HanChen FederalReserveBoard FederalReserveBoard FederalReserveBoard AndreaStella FederalReserveBoard December11,2020 Abstract Output gaps that are estimated in real time can differ substantially from those estimated after the fact. We aim to understand the real-time instability of output gap estimates by comparing a suite of reduced-formmodels. WeproposeanewstatisticaldecompositionandfindthatincludingaOkun’slaw relationshipimprovesreal-timestabilitybyalleviatingtheend-pointproblem. Modelsthatincludethe unemploymentratealsoproduceoutputgapswithrelevanteconomiccontent. However,wefindthatno modeloftheoutputgapisclearlysuperiortotheothersalongeachmetricweconsider. JELclassifications: E24,E32,E52. Keywords: Outputgap;real-timedata;trend-cycledecomposition;unobservedcomponentsmodel. *We thank without implication Gianni Amisano for helpful comments and suggestions. Michael Kister and Niko Paulson providedexcellentresearchassistance. WealsothankseminarparticipantsatCBF2018, SETA2019, andIAAE2019fortheir comments.TheviewsexpressedhereareourownanddonotnecessarilyreflecttheviewsoftheBoardofGovernorsoftheFederal ReserveortheFederalReserveSystem. 1

1 Introduction Theoutputgap—thedeviationofobservedoutputfromitspotentiallevel—isaconceptfundamentaltothe conductofeconomicpolicy. Indeed,thecyclicalstateoftheeconomyisoftenattheforefrontofpolicymakers’ thinking, and they often refer to the output gap when discussing policy decisions.1 Yet the output gap is never observed and must be estimated. And in an influential paper, Orphanides and van Norden (2002) document that output gap estimates are unreliable in real time, throwing into question their adequacy for policymakers. In light of these findings, a large literature has taken on the challenge of improving the stability of real-timeoutputgapestimates. Garratt,Lee,MiseandShields(2008)amelioratetheend-pointproblemby detrending output data that has had forecasts appended to it, while Clements and Galva˜o (2012) explicitly model data revisions alongside the output gap. Another approach is to include highly cyclical variables in multivariate models. Trimbur (2009) finds manufacturing capacity utilization improves the real-time properties of output gap estimates. Fleischman and Roberts (2011) extract an output gap estimate from a large number of macroeconomic variables; their model also improves real-time stability. Finally, Edge and Rudd (2016) consider the real-time properties of the Federal Reserve staff’s judgmental output gap estimates. They show that, in contrast to model-based approaches, the Board staff produced stable output gapestimatesinrealtimebetweenthemid-1990sandthemid-2000s(whentheirsampleends). Inthispaper,weaimtounderstandwhatcontributestothereal-timestabilityofanoutputgapestimate. To do so, we produce output gap estimates from a suite of models, including the models considered in OrphanidesandvanNorden(2002),therelativelynewde-trendingmethodsfromMuellerandWatson(2017) and Hamilton (2018), the judgmental output gap from the Federal Reserve’s staff, as well as a number of univariate and multi-variate unobserved components models. We next propose a decomposition of the estimates from models that can be written as a linear Gaussian state-space system. The decomposition estimatesthecontributionofeachobservableseriestothereal-timeinstabilityoftheoutputgapfromthree distinct sources: revisions to the underlying data, parameter instability, and the end-point problem. Lastly, weconsidertheperformanceofthevariousoutputgapsintermsoftheireconomicvaluebycomparingthem toajudgmentalestimateproducedbythestaffoftheFederalReserve,theirabilitytoforecastinflation,and whethertheyareconsistentwithex-postmonetarypolicydecisions. 1See,forexample,theJanuary19,2017speechofJanetYellen“TheEconomicOutlookandtheConductofMonetaryPolicy,” madewhenshewasChairoftheFederalOpenMarketCommitteeoftheFederalReserve. 1

Theresultsconfirmthedifficultyofestimatingtheoutputgapinreal-time. First,wefindthattheFederal Reservestaff’soutputgapisasstableasanyofthemodel-basedestimatesweconsider,reconfirmingEdge andRudd(2016)withadatasetthatnowincludestheGreatRecession. WealsofindthattheTealbookoutput gap has economic content, in the sense that it forecasts inflation no worse than any of the other models we considerandimpliesaninterestratepolicythatisrelativelyclosetotheactualoutcome.2 Amongthemodels weconsider,thosethatincludeanOkun’slawrelationship,relatingtheunemploymentrate’sdeviationfrom its trend to the output gap, are overall quite similar to the Tealbook output gap estimate. They provide a stable and reliable estimate of the cyclical position of the economy, and are consistently among the best performers along a number of economic considerations, suggesting that they have the ability to inform policyinameaningfulway. Arecentrelatedliteraturehasfocusedonestimatingtheoutputgapusinglargemacroeconomicpanels. AastveitandTrovik(2014)andBarigozziandLuciani(2018)uselargedatasetstoestimatetheoutputgapvia adynamicfactormodel. MorleyandWong(2019)estimatetheoutputgapwithaBayesianVARcomprising a large macroeconomic dataset. Importantly, their methodology allows them to inspect the contribution of VARforecasterrorstothecycle. Theyfindthattheunemploymentrateisthesinglemostimportantvariable thatcontributestothecycle. However,thesepapersdonotfocusonthereal-timestabilityoftheoutputgap estimate. The next section introduces the real-time data and various models used to estimate the cyclical state of the economy. Section 3 documents the real-time stability of the suite of models we consider, while in Section 4 we decompose the model’s relative instability. In Section 5 we evaluate the economic relevance oftheoutputgapinstabilitieswedocument,andthefinalsectionprovidesaconcludingdiscussion. 2 Empirical design 2.1 Real-timedata We use real-time vintage data throughout the paper. Our primary data source is the Real-time Dataset for MacroeconomicoftheFederalReserveBankofPhiladelphia,whereweobtainreal-timedatavintagesofreal 2TheFederalReserveBoardstaffestimateiscontainedinadocumentknownastheTealbook,preparedpriortoeachFederal OpenMarketCommittee(FOMC)meeting.TheTealbookwaspreviouslyknownastheGreenbook.Throughout,however,wewill refertotheFederalReservestaffforecastastheTealbook.Tealbookforecastscanbefoundat:https://www.philadelphiafed. org/research-and-data/real-time-center/greenbook-data/. 2

GDP,theunemploymentrate,theGDPpricedeflator,andmanufacturingcapacityutilization.3 Thedatabase containsquarterlyreal-timevintagesforeachseries. Thatis,thereal-timevintagegivesthedataseriesasit would have been seen by a practitioner as of the middle of each quarter. Vintage T +1 contains data that typicallybeginsin1947:Q1andcontinuesthroughquarterT. ForrealGDP,inflation,andtheunemployment rate, our first vintage is 1965:Q4, while the first available vintage for manufacturing capacity utilization is 1985:Q3. Thefinalvintageinourdatasetis2019:Q1,for214vintagesintotal. The1996:Q1real-timeGDP vintageismissingbecauseofafederalgovernmentshutdown. 4 We also consider the real-time stability of the output gap estimate produced by the staff of the Federal ReserveBoard. ThestaffestimateiscontainedinadocumentknownastheTealbook,preparedpriortoeach Federal Open Market Committee (FOMC) meeting. Tealbook output gap estimates since March 1996 are availablefromtheFederalReserveBankofPhiladelphia. FollowingOrphanidesandvanNorden(2002)we use the first Tealbook estimate available each quarter. However, Edge and Rudd (2016) provide real-time outputgapnowcasts—thatis,estimatesoftheoutputgapinthepreviousquarter—since1950:Q4,whichwe use to extend our sample earlier into history. Because Tealbook forecasts are made public with a five year lag,thelastTealbookoutputgapestimateinourdatasetisfromthefourthquarterof2013.5 2.2 Modelsoftheoutputgap We first consider several univariate trend-cycle decompositions explored by Orphanides and van Norden (2002) and Edge and Rudd (2016). Let y denote 100 times the log of real GDP. We first decompose log real GDP into a trend (y∗) and a cycle (C) using a linear time trend, a quadratic trend, and the HP filter. For completeness, we add other more recently developed models. The first is the trend-cycle model from Garratt et al. (2008) (GLMS). The GLMS methodology aims to reduce the end-point problem by applying an HP filter to a data series that has been augmented with a forecast from an AR model. The Mueller and Watson(2017)(MW)extractsatrendfromaunivariateseriesbyprojectingtheseriesontoasmallnumber oftrigonometricfunctionsintendedtoisolatethelow-frequencyinformationintheseries. Finally,Hamilton (2018)providesanewregression-basedapproachtotrendextraction. EachoftheseconsideronlyrealGDP 3See table A1. Croushore and Stark (2001) and the Federal Reserve Bank of Philadelphia’s website (https://www. philadelphiafed.org/research-and-data/real-time-center/real-time-data)provideadditionaldetails. 4Wesetoutputgapstomissingin1995:Q4sincemanyofthemodelscannotproduceagapestimatewithoutadditionalassumptions.Therearethen213observationswhenweworkwithreal-timedata. 5SeeCleve,LukeVanandLaforte,Jean-Philippe,andStella,Andrea(2019)foradditionaldetailsaboutvintagesoftheFederal Reservestaff’soutputgapestimate. 3

andarestraightforwardtoimplement. Our primary focus, however, is on output gap estimates produced by unobserved components (UC) models,becausewecancasttheminstatespaceformwhichoffersawaytoformallyassesspropertiesofthe outputgapestimates. TheUCmodelsdecomposethelogofrealGDPintoatrend,cycle,andidiosyncratic error(e ): y =y∗+C +e .Trendoutputfollowsrandomwalkwithdrift,τ,whichitselffollowsarandom y t t t yt walk: y t ∗ =y t ∗ −1 +τ t−1 +ε y∗t τ =τ +ε . t t−1 τt Thedrifttermcapturesslow-movingchangestothegrowthrateoftrendoutputdueto,forexample,changes to productivity growth or demographics. The error terms, ε y∗t and ε τt , are distributed as standard normals. Theoutputgap,C,followsastationaryAR(2)process C =φ C +φ C +ε . t 1 t−1 2 t−2 Ct wheretheerrortermε isdistributedasastandardnormal. Weconsiderseveralnestedversionsofthis Ct framework. Thefirstistheunivariatemodelabove,firststudiedinClark(1987). Wethengraduallyenlarge theinformationsetofthemodeltoincludeotherimportantmacroeconomicindicators. Weaddtotheframework above the unemployment rate, capacity utilization, and inflation one at a time. Considering results frombivariateUCmodelsallowsustosingleouttheindividualcontributionsofthevariousmacroeconomic indicators. Ineachcase,theonlychangetothemodelistheinclusionofanadditionalobservationequation andstate-transitionequationtothemodel. Considerthemodelthataddsvariablex. Thisvariablerelatesto theoutputgap: x =x∗+α C +α C +e , t t 0 t 1 t−1 xt where the parameters α and α measure the observable’s response to changes in the output gapC. In the 0 1 bivariatemodelwiththeunemploymentrate,thesewouldbeOkun’slawcoefficients,whereasinthemodel withinflation,thesewouldbePhillipscurveparametersinstead(Kuttner1994). Theothertrendcomponent, x∗,againfollowsarandomwalk. t ThelargestmodelincludesrealGDP,unemploymentandinflation. Whilewewillnotexplorethevarious 4

trends that result from our models, in principle, they are of interest. For example, the unemployment rate trend, u∗, could proxy the NAIRU (Staiger, Stock and Watson 1997), and the trend of inflation, π∗, could be interpreted as a proxy for inflation expectations (see Stock and Watson (2007), Kim, Manopimoke and Nelson (2014), and Clark and Doh (2014)). The fundamental aspect of our small multivariate models, whether they include unemployment rate or inflation, is the presence of a common cycle that affects the deviationofeachmacroeconomicvariablefromitstrend. For each data vintage, all models are estimated on an estimation sample that begins in 1960:Q1 and whichexpandsastimegoesby. TheUCmodelsarecastaslinearGaussianstate-spacemodels. Weestimate these models in a Bayesian way using a Gibbs sampler. Since the estimation is standard, we leave the details to Appendix A2.1. However, in our experience, Bayesian unobserved components models can be sensitive to the priors placed on the variance of shocks to model trends.6 Because we produce real-time estimates, we require a method for setting model priors in a systematic, data-driven manner. We take the followingapproach. Foragivenobservableanddatavintage,weproduceanestimateofthetrendusingan HP filter on a pre-sample period, 1947:Q1–1959:Q4. We then center our Inverse Gamma prior about the variance implied by this pre-sample trend. We set the degrees of freedom parameter in the IG distribution to Tv/20, where T v denotes the number of observations in data vintage v. Priors for other model parameters areconjugateanddiffuse. MoredetailscanbefoundinAppendixA2.2. 3 Estimating the output gap in real time FollowingOrphanidesandvanNorden(2002)andEdgeandRudd(2016),foreachdatavintage,definethe real-timeoutputgap,CR,astheestimateoftheoutputgapinthepreviousquarter. So,forexample,therealt timeoutputgapassociatedwiththe2019:Q1vintageistheestimateoftheoutputgapin2018:Q4. Because thefirstestimateofquarterlyNIPAdataisreleasedwithroughlyaone-monthlag, thisassumptionensures that a model estimatedat vintage vhas the first NIPA estimates for output and inflation in quarterv−1, as wellasthemonthlyestimatesoftheunemploymentrateandcapacityutilization. Figure 1 shows the sequence of real-time output gap estimates. While each output gap estimate is broadly pro-cyclical, at any point in time the models deliver a wide range of estimates. Table 1 provides summary statistics, which reinforce the observation that output gap estimates differ dramatically across 6When the models are estimated via maximum likelihood, in our experience this sensitivity manifests itself as sensitivity to startingvalues. 5

models: the real-time estimates have notably different means and volatilities, and often have different first derivativesandsigns. Figure1: Sequenceofreal-timeoutputgapestimates,1965:Q3–2018Q4. 1970 1980 1990 2000 2010 2020 5 0 5− 01− 51− % l Linear l UC (GDP) l Quad l UC (GDP/p ) l HP l UC (GDP/CU) l GLMS l UC (GDP/un.) l Hamilton l UC (GDP/p ,un.) l MW l Tealbook Notes: Figure shows the sequence of real-time output gap estimates for the suite of models we consider. GLMS refers to Garratt et al. (2008), and MW to Mueller and Watson (2017). Grey shaded areas denote NBER-definedrecession. FinalTealbookestimatesfor2013:Q3. Seetextfordetails. Table1: Summarystatisticsofreal-timeoutputgapestimates,1965:Q3–2018Q4. N.obs Mean Stddev Min Max %>0 Linear 213 -4.55 4.06 -12.53 2.47 8 Quadratic 213 0.41 3.24 -6.71 6.52 57 HP 213 -0.11 1.62 -6.63 3.84 56 GLMS 213 -0.40 1.06 -4.22 1.48 39 Hamilton 213 0.42 2.49 -8.11 6.24 62 MW 213 1.01 1.36 -3.05 3.89 77 UC(GDP) 213 -0.93 1.40 -6.72 1.91 25 UC(GDP/π) 213 -0.55 1.50 -5.24 3.40 36 UC(GDP/CU) 134 -0.49 1.54 -6.31 1.97 43 UC(GDP/u.) 213 -0.89 2.73 -8.86 3.26 46 UC(GDP/u./π) 213 -0.73 2.66 -8.01 3.17 46 Tealbook 192 -2.99 3.84 -16.21 2.90 27 Notes: Sample period is 1965:Q3–2018:Q4 except for UC(GDP/CU) (begins in 1985:Q2) and Tealbook (endsin2013:Q3). 1995:Q4settomissingforallmodels. %>0denotesthepercentofquarterswhenthe outputgapispositive. Seetextfordetails. 6

Table2describestherevisionsofeachoutputgapmodelweconsider. Wedefinetheoutputgaprevision inquarterttobethedifferencebetweenthein-sampleoutputgapestimatefromourlastvintage,V=2019:Q1, and the real-time estimate. The revision proxies for uncertainty that surrounds any given real-time output gapestimate. Thecolumnslabeled“NSR”reportnoise-to-signalratios: Thefirstisdefinedastheratioofthe standard deviation of the model revision to thestandard deviation of the final gap estimate, and the second is theratio ofthe rootmean squareof the revisions to thestandard deviationof thefinal gapestimate. The column “% sign agree” measures the fraction of observations where the real-time and final estimate have thesamesign,i.e.,whethertheeconomyisaboveorbelowpotential. Table2: Summarystatisticsofoutputgaprevisions,1965:Q3–2018:Q4. NSR NSR %sign N Mean Stddev RMSE (SD) (RMSE) agree Linear 213 5.48 2.04 5.85 0.43 1.22 34 Quadratic 213 -0.21 3.73 3.73 1.11 1.11 62 HP 213 0.16 1.53 1.54 1.05 1.05 59 GLMS 213 0.44 0.99 1.08 0.67 0.74 73 Hamilton 213 -0.48 1.25 1.34 0.48 0.51 84 MW 213 -1.01 1.72 1.99 1.11 1.28 57 UC(GDP) 213 1.91 2.08 2.82 0.73 1.00 65 UC(GDP/π) 213 0.80 1.31 1.53 0.70 0.81 77 UC(GDP/CU) 134 0.54 1.23 1.34 0.72 0.78 76 UC(GDP/u.) 213 0.82 1.38 1.60 0.49 0.56 81 UC(GDP/u./π) 213 0.71 1.22 1.41 0.43 0.50 83 Tealbook – – – – – – – Notes: Revision defined as final output gap estimate less real-time estimate, CV −CR andCV denotes the t t t output gap estimate using theV=2019:Q1 data vintage. Bold face entries indicate best performing model accordingtothatmeasure. Allstatisticsarefor1965:Q3–2018:Q4exceptforUC(GDP/CU),whichbegins in1985:Q2. TealbookrevisionstatisticsnotavailablebecauseTealbook2019:Q1outputgapestimateisnot yetavailable. Seetextfordetails. Most univariatemodels fare quitepoorly in terms oftheir real-time stability. Withthe exception of the GLMSandHamiltonmodels,theunivariatemodelshavenoise-signalratiosthatalmostalwaysexceedone. The procedure of Garratt et al. (2008) does improve real-time stability. We are among the first to evaluate the real-time properties of the newer detrending methods—see also Quast and Wolters (2019) and Berge (2020)—and find that while Hamilton’s (2018) method is relatively stable, the MW decomposition is less stablethanmanyotherunivariatemodelsinrealtime. AmongtheUCmodels,UC(GDP)andUC(GDP/π) are less stable than the other UC models, but are about as stable as the univariate models. Comparing the UC (GDP/π) model to the UC (GDP), and comparing UC (GDP/u./π) to UC (GDP/u.), adding inflation 7

does not improve real-time stability, in contrast to Planas and Rossi (2004). Finally, we note that output gapestimatesofmodelsthatincludetheunemploymentratehavesmallerrevisions,lessvariablerevisions, smaller noise-signal ratios, and tend not to change signs. Gonzalez-Astudillo and Roberts (2016) show that the unemployment rate helps identify the cyclical component of GDP; these results indicate it can also improve the real-time stability of that estimated cycle, despite the additional parameters that must be estimated. Table3: Summarystatisticsofoutputgaprevisions,selectedsubsamples. 1975:Q1–1997:Q4 1998:Q1–2013:Q3 NSR NSR %sign NSR NSR %sign N (SD) (RMSE) agree N (SD) (RMSE) agree Linear 91 0.57 2.31 24 63 0.27 0.67 48 Quadratic 91 0.75 1.68 45 63 0.42 0.53 95 HP 91 1.11 1.10 55 63 0.88 0.88 65 GLMS 91 0.72 0.74 75 63 0.67 0.76 73 Hamilton 91 0.40 0.50 93 63 0.28 0.28 90 MW 91 1.17 1.33 51 63 0.71 1.13 68 UC(GDP) 91 0.75 0.88 58 63 0.53 1.07 62 UC(GDP/π) 91 0.72 0.74 76 63 0.55 1.12 63 UC(GDP/CU) 50 0.63 0.64 72 63 0.44 0.77 78 UC(GDP/u.) 91 0.53 0.53 79 63 0.29 0.64 81 UC(GDP/u./π) 91 0.45 0.45 85 63 0.29 0.61 78 Tealbook 91 1.51 1.85 77 63 0.43 0.49 92 Notes: Revision defined as 2013:Q4 output gap estimate less real-time estimate, C2013:Q4−CR. Bolded t t entriesindicatebestperformingmodelaccordingtothatmetric. Real-timevintagesforUC(GDP/CU)begin in 1985:Q2; statistics for UC(GDP/CU) are in italics for the first subperiod because of the shorter period over which they are calculated. First subsample begins in 1975:Q1 because the 2013:Q4 Tealbook output gapestimatebeginsinthatquarter. Seetextfordetails. InTables3and4wecomputethesesummarystatisticsforselectedsubperiods. Table3splitsthesample into two roughly equal subperiods, 1975:Q3–1997:Q4 and 1998:Q1–2013:Q3, following Edge and Rudd (2016).7 We end the sample in 2013 so that we can include the Tealbook output gap estimates. Edge and Rudd(2016)documentthattheFederalReserve’soutputgapestimatesbecomemuchmorestablebeginning inthe1990s,andwefindthatremainstrueevenafterincludingtheGreatRecessionperiod. Themodel-based estimates are also more stable in the second half of the sample, but the improvement in stability is not as pronouncedasinthecaseofTealbookestimates. AmongtheUCmodels,themodelsusingunemployment 7Edge and Rudd (2016) consider two subsamples, 1980Q1:1992Q4 vs 1994Q1:2006Q4 and 1966Q1:1997Q4 vs 1998Q1:2006Q4. Our main takeaways are robust to splitting our sample in 1993 instead of 1997 and to starting the sample in 1980insteadof1975. 8

areagainthemoststable. Table4measuresthestabilityoftheoutputgapestimatesatturningpoints,presumablythemostimportantperiodsoftimeforpolicymakers. Wecomputeoutputgaprevisionsforthe12quarterspriortoNBER peaksandtroughs. Table4isbroadlysimilartothepreviousonesinthattheHamilton-trendimpliedoutput gapandtheoutputgapestimatedwiththeUCmodelsusingunemploymentareverystableatturningpoints. Interestingly, the Tealbook has produced relatively less stable output gap estimates prior to turning points, especiallybusinesscyclepeaks. Table4: Real-timeoutputgaprevisionsatturningpoints. NBERpeaks NBERtroughs NSR NSR %sign NSR NSR %sign N (SD) (RMSE) agree N (SD) (RMSE) agree Linear 91 0.65 3.23 15 91 0.41 1.75 30 Quadratic 91 1.48 1.47 69 91 1.00 1.00 73 HP 91 1.13 1.39 55 91 0.75 1.08 57 GLMS 91 0.86 1.18 66 91 0.64 0.86 67 Hamilton 91 0.61 0.64 87 91 0.43 0.49 90 MW 91 1.16 1.15 64 91 0.89 0.90 74 UC(GDP) 91 0.86 2.24 49 91 0.73 1.31 54 UC(GDP/π) 91 0.82 1.20 73 91 0.76 0.84 73 UC(GDP/CU) 39 1.34 2.65 85 39 0.44 0.85 85 UC(GDP/u.) 91 0.59 1.08 78 91 0.41 0.66 75 UC(GDP/u./π) 91 0.54 0.99 85 91 0.38 0.58 79 Tealbook 65 1.99 2.47 63 66 0.97 1.19 80 Notes: Revision defined as 2013:Q4 output gap estimate less real-time estimate, C2013:Q4−CR. Bolded t t entries indicate best performing model according to that metric. Italicized entries indicate the model has fewer observations. Real-time vintages for UC(GDP/CU) begin in 1985:Q2. There are fewer Tealbook observations because the 2013:Q4 Tealbook output gap estimate begins only in 1975:Q1. See text for details. 4 Decomposing output gap revisions Wenowturntounderstandingwhysomereal-timeestimatesoftheoutputgaparemorestablethanothers. We propose a decomposition that measures the contribution of each observable series to the instability of the output gap from three distinct components: revisions to the underlying data, parameter instability, and theend-pointproblem. 9

4.1 Sourcesofinstability The model parameters from a model estimated on data vintage v and periodst =1,...,τ are denoted Θv . 1:τ AlsoassumethatT +1isthelastavailablevintage. Wecanthendefinethefollowingobjects. • The final output gap estimate in periodt,CF (ΘT+1), is the smoothed estimate of the output gap in t|T 1:T periodt whenthemodelisestimatedusingthecompletefinaldatavintage,T +1. • The quasi-final output gap estimate, CQF(ΘT+1), is the filtered estimate of the output gap, again t|t 1:T usingparametersestimatedfromthefullsampleofthefinaldatavintage. • The quasi-real time estimate,CQR(ΘT+1), is the filtered estimate of the output gap produced using t|t 1:t parameterestimatesfromthefinaldatavintage,butestimatedusingdataonlythroughperiodt. • The real-time output gap estimate,CR(Θt+1), is the filtered estimate of the gap using all available t|t 1:t datafromvintaget+1. TherevisionCF −CR thatwasusedinSection3tocomputereal-timestabilitystatisticscanbedecomt|T t|t posed: CF −CR =CF (ΘT+1)−CQF(ΘT+1) Effectofend-pointproblem t|T t|t t|T 1:T t|t 1:T +CQF(ΘT+1)−CQR(ΘT+1) Effectofparameterinstability (1) t|t 1:T t|t 1:t +CQR(ΘT+1)−CR(Θt+1) Effectofdatarevisions t|t 1:t t|t 1:t The differenceCF −CQF is due to the end-point problem, since we are comparing two sets of estimates t|T t|t obtainedwithexactlythesameparameterestimatesandthesamedatavintages,butdatasamplesofdifferent lengths. ThedifferenceCQF−CQR isduetoparameterinstabilitysincetheonlydifferencebetweenthetwo t|t t|t estimatesiswhetherweusedatathroughT ort toestimatetheparameters. Finally,thedifferenceCQR−CR t|t t|t is due to data revisions, since the quasi-real estimate is obtained with the final vintage and the real-time estimate with the real-time vintage, but both estimates are obtained filtering the data through period t and usingparametersestimatedwithdatathroughperiodt. We can further track the source of each element of the decomposition in (1) into the effects from each observable included in the model. The unobserved states from a model estimated in state-space form are weighted averages of the observables, where the weights are functions of the parameters (Koopman and Harvey 2003). For the purposes of illustration, consider the decomposition of the UC (GDP/u.) model (of 10

course,thealgebraisgeneralizable). Thevariousestimatesoftheoutputgapinperiodt are: T CF = ∑[wu(ΘT+1)u +w y(ΘT+1)y ] t τ 1:T τ τ 1:T τ τ=1 t CQF = ∑[wu(ΘT+1)u +w y(ΘT+1)y ] t τ 1:t τ τ 1:t τ τ=1 (2) t CQR = ∑[wu(ΘT+1)u +w y(ΘT−1)y ] t τ 1:t τ τ 1:t τ τ=1 t CR = ∑[wu(Θt+1)u +w y(Θt+1)y ]. t τ 1:t τ τ 1:t τ τ=1 The equations in (2) decompose the level of any given estimate of the output gap. We can now write output gap estimate instability in terms of the observables by implementing a variance decomposition of theoutputgaprevisions. Byplugging(2)into(1)andtakingthestandarddeviation,wecandecomposethe standard deviation ofCF −CR into the contributions of the end-point problem, parameter instability, and t|T t|t datarevisions,anddecomposeeachcomponentintothecontributionsoftheobservablesseries. 4.2 Decompositionresults Tomotivatethedecompositionoftherevisions,Figure2plotsthesequenceofreal-timeoutputgapestimates from the multivariate unobserved component models, and decomposes the level of those estimates into the contribution of the observables following equation (2). As shown in the upper-left panel, the UC (GDP/π) model relies heavily on inflation to make inference about the output gap in the 1960s and early 1970s, but almostexclusivelyreliesontheinformationcomingfromrealGDPtoestimatetheoutputgapsincethemid- 1990s. After1980,inflationcarriesverylittleinformationabouttheoutputgapestimateineithermodelthat includesit. TheUCmodelsthatconditionontheunemploymentrateorcapacityutilizationheavilyrelyon them to estimate the cyclical state of the economy. Real GDP also influences the output gap estimate, but itsinfluenceisrelativelysmall,especiallywhencomparedtothecyclicalinformationintheunemployment rate. Table 5 decomposes the change in the output gap estimate from the real-time estimate to the final. We decomposethenoise-to-signalratiothatusesstandarddeviations. Todoso, wetakethestandarddeviation ofbothsidesofequation(1),andthendividebythestandarddeviationofthefinalestimate. Theend-point problem explains the bulk of the revisions to the output gap for all models. Indeed, it often explains all or 11

Figure2: Decompositionofreal-timeoutputgapestimates,selectedUCmodels. l ll ll l llll lllllllllllll l l l ll l l lll ll l l l l ll l ll ll ll lllllll lll l llll lll lll ll l l l l lllllllllll llll llllllll ll ll l llllllllllllll lll llllll lll lllllllllll l l l llllllllllllllllllllll llll ll l l l ll llll l llllllll l l llllllll l l llllll lllll l 1970 1980 1990 2000 2010 2020 6 4 2 0 2− 4− 6− 8− 01− Real GDP p l Output gap llllllllll lllllllll lll l l lllllll llll lllll ll llllllllllllllll llll l l l llllllll ll ll lllllllllllll l l l l ll llll llllllllllllllllllllllllllll l ll l ll 1970 1980 1990 2000 2010 2020 (a)DecompositionofUC(GDP/π). 6 4 2 0 2− 4− 6− 8− 01− Real GDP Cap utilization l Output gap (b)DecompositionofUC(GDP/CU). l l llllllllllllllll l l l l lllll l l l l lll l l l l lll l ll l lllllllll l lllll l l l ll llllllllll lll lllllllllll l l l ll llll ll l llll llll lllllllll lllllllllll l l l ll llllll llllllllllllllllll l l l l l llll ll l l l l lll lllllllll lllll l l l lll l ll l l l l l llll l l l l l 1970 1980 1990 2000 2010 2020 6 4 2 0 2− 4− 6− 8− 01− Real GDP Unemployment rate l Output gap l l llllllllllllll lll l l l lll l lll l l lll l l l l l llll ll l llllll lll l lllll l l l l l lllllllllll llll lllllllll l l l l llllll lllllll llll lllllll ll lllllll llll l l l ll lllllll lllllllllllllllll l l l l l ll llll l l l lllll llllllll lllll l l ll l llll l l l l lllll l ll 1970 1980 1990 2000 2010 2020 (c)DecompositionofUC(GDP/u.). 6 4 2 0 2− 4− 6− 8− 01− Real GDP Unemployment rate p l Output gap (d)DecompositionofUC(GDP/u./π). Notes: Figuresshowdecompositionofreal-timeoutputestimatefromselectedUCmodels. Contributionof observable(inp.p.) shownasshadedbar,whichsumtooutputgapestimate. Seetextfordetails. more than all of the noise-to-signal ratio. Decomposing the variance of equation (1) involves computing thecovariancesamongthethreecomponents,whichwesummarizeintheresidualatthebottomofTable5; thesecovariancesturnouttobemostlynegativeandexplainforafewmodelslargeportionsofthenoise-tosignalratio. Forinstance,whenwecomparetheGLMStotheHPfilter,theimprovementinnoise-to-signal ratiodoesnotseemtocomefromareductionintheend-pointproblem,burratheralargernegativeresidual, whichisdrivenbyanegativecorrelationbetweentheend-pointproblemandthedatarevisioncomponents. ThecontributionofparameterinstabilityislargestfortheUC(GDP/π)model,andespeciallyfrominflation. Inprevioussectionsweshowedthatmodelsthatincludelabormarketdataproduceamorestableoutput gap in real time. Table 5 shows that the improvement in stability is obtained by reducing the influence of theend-pointproblem. Thecontributionofdatarevisionsdoesnotbecomesmallerwhenotherobservables are added to the univariate model, and the contribution of parameter instability may become larger. For 12

example,comparingthevariabilityoftheUC(GDP)outputgaptothatfromUC(GDP/π/u.),weseethatthe variabilityasmeasuredbythenoise-to-signalratioismorethanathirdlower. Mostofthatimprovementis duetoareductionintheend-pointproblem. Table5: Decomposingthenoise-to-signalratiointoobservables. UC HP GLMS UC UC UC UC (GDP) (GDP) (GDP) (GDP/π) (GDP/CU) (GDP/u.) (GDP/π/u.) NSR(SD) 0.7 1.0 0.7 0.7 0.7 0.5 0.4 Ofwhichisdueto: DataRevisions 0.2 0.4 0.7 0.4 0.2 0.1 0.2 GDP 0.2 0.4 0.1 0.3 0.3 0.1 0.1 π 0.4 0.1 CU 0.3 u. 0.1 0.2 Parameterinstability 0.1 0.0 0.0 0.4 0.1 0.1 0.3 GDP 0.1 0.0 0.0 0.3 0.5 0.1 0.1 π 0.5 0.2 CU 0.4 u. 0.1 0.2 End-pointproblem 0.7 1.0 1.0 0.7 0.7 0.5 0.5 GDP 0.7 1.0 1.0 0.7 0.6 0.5 0.4 π 0.1 0.0 CU 0.3 u. 0.3 0.2 Residual -0.3 -0.3 -1.0 -0.8 -0.3 -0.3 -0.5 Notes:Tabledecomposesthestandarddeviationoftherevisionsoftheoutputgapestimatesanddividesitbythe standarddeviationofthefinalestimate. Residualisduetocovariancesamongcomponents. Allstatisticsarefor 1965:Q3–2018:Q4exceptforUC(GDP/CU),whichbeginsin1985:Q2. Seetextfordetails. 5 Economic relevance In addition to being stable in real-time, output gap estimates should also be meaningful: a model that estimates the output gap to be a constant is stable but entirely useless. We examine three practical uses of output gap estimates. First, we compare model output gap estimates to the Federal Reserve Board’s judgmental one. We also evaluate whether output gaps are useful in the context of Phillips curve forecasts ofinflation,andiftheycandescribeabenchmarkinterestrateforpolicymakers. 13

5.1 HowdothemodelestimatescomparetoTealbookestimates? Since Edge and Rudd (2016) show that the Fed’s output gap measure is a reliable input to the FOMC’s decision-makingprocess,inTable6,wecomparethereal-timeoutputgapestimatestothereal-timeoutput gapestimatefromtheTealbook. Table7comparesthemtotheOctober2013Tealbookestimateinstead. With the exception of the linear trend in the more recent sub-sample, model-based output gaps are, on average,lowerthantheTealbookestimate,althoughthestandarddeviationofthatdifferenceisusuallylarge enoughthatonewouldnotrejectanullhypothesisthatitiszero. InSection3,wefoundthatsomeunivariate modelsproducedstablereal-timeoutputgapestimates,forexample,theoutputgapfromHamiltonwasvery stable in real-time across different subsamples. However, the Hamilton filter’s output gap is very different from the Tealbook’s estimate. Indeed, when the final Tealbook estimate is used as “truth,” Table 7, the noise-to-signal ratios are higher than one in the first half of the sample and more than twice the magnitude ofmanyoftheothermodelsinthesecondhalf. In terms of noise-to-signal ratios, the UC models that include the unemployment rate tend to perform best. Close inspection of Figure 1 shows that UC (GDP/u.) and UC (GDP/u./π) are both very close to the Tealbook output gap estimate starting from the late 1980s. Prior to that, the two model estimates are quite similar,butthemodelthatincludesinflationhasaslightlyhigheroutputgapestimatethroughoutthe1970s. Table6: Evaluatingmodel-basedestimatesrelativetoreal-timeTealbookestimates. 1975:Q1–1997:Q4 1998:Q1–2013:Q3 NSR NSR NSR NSR N Mean SD (SD) (RMSE) N Mean SD (SD) (RMSE) Linear 91 -0.03 3.15 0.72 0.72 63 2.75 2.57 0.83 1.22 Quadratic 91 -5.78 3.15 0.72 1.51 63 -1.87 1.66 0.54 0.81 HP 91 -3.84 4.20 0.96 1.30 63 -1.73 3.07 1.00 1.14 GLMS 91 -3.49 3.82 0.88 1.18 63 -1.31 2.39 0.77 0.88 Hamilton 91 -4.76 4.11 0.94 1.44 63 -2.04 2.33 0.76 1.00 MW 91 -4.82 4.36 1.00 1.49 63 -3.33 3.05 0.99 1.46 UC(GDP) 91 -3.00 3.38 0.78 1.03 63 -0.73 1.60 0.52 0.57 UC(GDP/π) 91 -3.07 3.78 0.87 1.11 63 -0.90 1.75 0.57 0.64 UC(GDP/CU) 50 -1.31 1.62 0.72 0.93 63 -0.56 1.93 0.63 0.65 UC(GDP/u.) 91 -2.38 2.08 0.48 0.72 63 -0.69 0.73 0.24 0.33 UC(GDP/u./π) 91 -2.47 2.31 0.53 0.77 63 -0.64 0.72 0.23 0.31 Notes: Table shows summary statistics for difference between real-time model-based output gap estimate relativetoreal-timeTealbookestimate,CR,TB−CR,i. Boldedentriesdenotebestperformingmodelaccording t t toeachmetric. Real-timeestimatesfromUC(GDP/CU)beginin1985:Q2;shortenedsampleindicatedwith italics. Seetextfordetails. 14

Table7: Evaluatingmodel-basedestimatesrelativetoOctober2013Tealbook. 1975:Q1–1997:Q4 1998:Q1–2013:Q3 NSR NSR NSR NSR N Mean SD (SD) (RMSE) N Mean SD (SD) (RMSE) Linear 91 2.41 1.33 0.59 1.22 63 3.34 2.65 1.09 1.76 Quadratic 91 -3.33 1.49 0.66 1.61 63 -1.28 1.95 0.81 0.96 HP 91 -1.39 2.33 1.03 1.20 63 -1.14 2.38 0.98 1.08 GLMS 91 -1.04 1.69 0.75 0.88 63 -0.72 1.72 0.71 0.77 Hamilton 91 -2.32 2.46 1.09 1.49 63 -1.45 1.77 0.73 0.94 MW 91 -2.38 2.64 1.17 1.57 63 -2.74 2.34 0.97 1.48 UC(GDP) 91 -0.56 1.57 0.70 0.73 63 -0.14 0.97 0.40 0.40 UC(GDP/π) 91 -0.62 1.66 0.73 0.78 63 -0.31 1.21 0.50 0.51 UC(GDP/CU) 50 -1.24 1.06 0.73 1.12 63 0.03 1.41 0.58 0.58 UC(GDP/u.) 91 0.07 1.74 0.77 0.77 63 -0.10 1.12 0.46 0.46 UC(GDP/u./π) 91 -0.03 1.50 0.66 0.66 63 -0.05 1.13 0.47 0.46 Tealbook 91 2.44 3.41 1.51 1.85 63 0.59 1.05 0.43 0.49 Notes: Table shows summary statistics for difference between real-time model-based output gap estimate relativetoreal-timeTealbookestimate,CF,TB−CR,i. Boldedentriesdenotebestperformingmodelaccording t t toeachmetric. Real-timeestimatesfromUC(GDP/CU)beginin1985:Q2;shortenedsampleindicatedwith italics. Seetextfordetails. 5.2 Forecastinginflation We now consider the importance of real-time instability to predictions of inflation. We fit Phillips curve modelstoeachoutputgapestimatethatweconsider: 6 πt+h =α+∑βπ +γCi +e (3) t i t−i t−1 t i=1 where π = 400×log(P/P ) is the annualized growth rate of the core PCE price index and πt+h is the t t t−1 t average inflation rate from quartert tot+h;Ci is the real-time output gap estimate from model i that is t−1 availableinquartert. Weconstrainthecoefficientsonlaggedinflationtosumtooneandestimateequation (3) via ordinary least squares. We forecast three horizons: two, four, and eight quarters ahead. For each horizon-model pair, we consider the sequence of real-time, quasi-real time, and the 2013:Q4 output gap estimate. Becauseweareinterestedinisolatingtheimpactoftheoutputgapwhenforecastinginflation,we use the current vintage of inflation data. To keep the estimation sample the same for all models, our first out-of-sampleforecastismadeusingmodelsestimatedfrom1985:Q2,thefirstavailablereal-timeestimate fromtheUC(GDP/CU)model,andthefirstout-of-sampleforecastisproducedforthefirstquarterof1990. Afterthat,weexpandthewindowusedtoestimate(3)untilourfinalforecastquarter,2013:Q3. 15

Table8showstheratiooftheRMSEforeachmodelrelativetotheRMSEfromthemodelthatusesthe October2013Tealbookoutputgap. TherawRMSEfortheOctober2013Tealbookisshowninboldedtext. For other models, values less than one indicate superior inflation forecasts relative to the null model. We test for superior forecast ability using a Diebold and Mariano (1995) test and using Newey-West standard errors. Table8: PhillipscurvebasedforecastsofcorePCEinflation,1990:Q1–2013:Q3. Twoquartersahead Fourquartersahead Eightquartersahead RT QRT 2013:Q4 RT QRT 2013:Q4 RT QRT 2013:Q4 Linear 1.02 1.03 1.02 1.02 1.02 1.03 1.01 1.02 1.02 Quadratic 1.05 1.04 1.01 1.04 0.97 1.00 1.03 0.99 1.00 HP 0.99 0.96 0.98 0.88 0.89 0.91 0.86 0.93 0.93 GLMS 1.05 1.04 0.98 0.96 0.95 0.91 0.92 0.92 0.93 Hamilton 1.01 0.97 0.95 0.91 0.92 0.87 0.89 0.97 0.88 MW 1.07 1.05 1.04 1.03 1.01 0.98 0.99 0.98 1.02 UC(GDP) 1.08 1.07 1.01 1.05 1.01 1.00 1.01 0.99 1.00 UC(GDP/π) 1.02 1.07 1.03 1.00 1.03 1.00 0.98 0.99 0.98 UC(GDP/CU) 1.05 1.07 1.00 1.05 1.06 1.01 1.06 1.06 1.02 UC(GDP/u.) 1.07 1.07 1.02 1.06 1.06 1.02 1.05 1.05 1.03 UC(GDP/u./π) 1.06 1.07 1.03 1.05 1.07 1.03 1.05 1.05 1.03 Tealbook 1.04 – 0.46 1.02 – 0.44 1.04 – 0.46 Notes: Absolute RMSE of inflation forecast from equation 3 using October 2013 Tealbook output gap estimate shown as bolded values. All other entires are the ratio of that RMSE to the bolded value. Outof-sampleperiodis1990:Q1–2013:Q3andmodelsareestimatedbeginningin1985:Q2usinganexpanding window. Asterisks indicate dominance of indicated model over the null Tealbook estimate is statistically significantat10(∗)or5(∗∗)percentlevelandwithNewey-Weststandarderrors. Seetextfordetails. As has been shown in other empirical applications—e.g., Stock and Watson (2009), Berge (2018), and Dotsey, Fujita and Stark (2018), among many others—no single output gap estimate produces clearly superiorPhillipscurvebasedinflationforecasts. Table8showsaperiodwhereininflationhasbeenquiescent (Coibion and Gorodnichenko 2015). In this period, although some output gap estimates may slightly outperform the null Tealbook output gap-based forecast, the improvement is never statistically meaningful. However, when we evaluate other out-of-sample periods, and if we substitute total PCE price inflation for core,wefindthatnooutputgapestimateclearlydominatestheothers. (SeeAppendixA3.) Unsurprisingly,focusingoneachmodel’sindividualresults,thebestinflationforecastisproducedwhen usingthe2013:Q4outputgapestimate. Forecastperformancetendstodeterioratewhenreal-timeorquasireal-timeestimatesareusedtoforecastinflationinstead. Thereductioninforecastability,however,israther small. Forexample,whenwefocusonthefour-quarteraheadforecasts,thedeteriorationinforecastability from using real-time output gap estimates to the 2013:Q4 output gap estimate is around 2-3 percentage 16

pointsrelativetotheTealbookestimate,andinfact,forsomeoftheunivariatefilters,thereal-timeestimate actuallyproducessuperiorinflationforecaststothe2013:Q4estimate. 5.3 Outputgapsasinputtopolicy Lastly, we evaluate the implications of the different output gap estimates for monetary policy. Because policy is not set in a rules-based manner, and because it is likely that policymakers implicitly consider an output gap that may be described as some combination of the output gaps presented above, comparing the Taylor-ruleimpliedinterestratestotheactualfederalfundsrateisatransparentandeasy-to-understandway tounderstandtherelativeperformanceoftheoutputgaps. Foreachoutputgap,wecomputetheshort-term interestrateprescribedbytheTaylor(1993)rule: r =r∗+π +w C +w (π−π∗), (4) t t t C t π t wherer∗ istheneutralinterestrate,π ispercentchangeininflationfromoneyearago,y istheoutputgap, t t t and π t ∗ is the trend rate of inflation. In the 1993 paper, Taylor sets w C = w π = 1/2, and assumes that r t ∗ and π∗ are time-invariant and equal to two percent. Results using a “balanced” Taylor rule with w =1; t C w π =1/2arepresentedinSectionA4oftheAppendix. Figure 3 plots the Taylor-rule implied policy rates using our various real-time output gap estimates. The Taylor rule uses time-invariant estimates of π∗ and r∗, so that the prescribed interest rates tend to be belowrealizedfederalfundsrateinthefirsthalfofthesampleandaboveitinthesecondhalf. However,the patternofTaylor-rule-impliedinterestratesbroadlyfollowthatoftherealizedeffectivefundsrate—giventhe patternofinflation,therulesprescribehighinterestratesinthe1970sfollowedbyasteadymovelower. The prescribedinterestratesalsotendtomovelowerduringorimmediatelyfollowingNBER-definedrecessions. EachoftheoutputgapsimplynotabledownwardmovementstotheFederalfundsrateinthe1990,2001and 2007-2008recessions. Notably,andincontrasttotheactualpolicyrate,someoutputgapsimplyanincrease in the Federal funds rate at the end of the Great Recession. For example, the Mueller-Watson output gap impliesaminimumvalueofthefederalfundsrateofroughly1percentin2009:Q3beforeincreasingto53⁄ 4 percent in 2011:Q3. The interest rate implied by the Tealbook’s output gap estimate increases much more graduallyaftertheGreatRecession,andisjust1.5percentinthefinalavailableperiod,2013:Q3. Table9calculatestheroot-mean-squareerrorforeachTaylor(1993)ruleimpliedinterestraterelativeto 17

Figure3: Taylor(1993)ruleimpliedfederalfundsrate,1965:Q3–2018:Q4. 1970 1980 1990 2000 2010 2020 02 51 01 5 0 5− % l Linear l UC (GDP) l Quad l UC (GDP/p ) l HP l UC (GDP/CU) l GLMS l UC (GDP/un.) l Hamilton l UC (GDP/p ,un.) l MW l Tealbook Notes: Figure shows prescribed interest rate from Taylor 1993 rule, calculated as in equation 4 with w = π w C =1/2, π equal to the four quarter change in the real-time GDP price deflator, y t as the real-time output gap estimate, and π∗ =r∗ =2 percent. Effective federal funds rate shown as thick black line. See text for details. theeffectivefederalfundsrate,againforthefullsampleandaroughlyequalsplitsample. Asbefore,wetest for statistically meaningful differences relative to the interest rate implied by the October 2013 Tealbook output gap estimate. The real-time Tealbook output gap produces an implied interest rate that is closest to policy in the most recent period; in fact, the real-time output gaps imply an interest rate path that is closer to actual policy than the output gap estimate from 2013:Q4. However, in the first half of the sample, the interestrateimpliedbyTealbookisamongtheleastsimilartoactualpolicy. Nomodelclearlyoutperformstheothersintermsofdeliveringaninterestratepaththatmatchesactual policy. Manyofthemodelsthatwererelativelystableinrealtimewouldhaveimpliedinterestratepolicies that were ultimately ignored by policymakers. Again consider the Hamilton model as an example, since it delivers stable output gaps in real time. Yet in late 2010, this model would have implied an interest rate policy more than 300 basis points higher than the actual federal funds rate, and nearly 350 basis points higher than the policy implied by the Tealbook’s output gap. Similarly, the linear detrending of real GDP 18

resultedinrelativelystablereal-timeoutputgapestimates,butwouldhaveprescribedsizableandpersistent negativepolicyratesthroughatleast2018. Table9: RMSEofprescribedinterestratefromTaylor(1993). 1965:Q3–2013:Q3 1975:Q1–1997:Q4 1998:Q1–2013:Q3 RT QRT 2013:Q4 RT QRT 2013:Q4 RT QRT 2013:Q4 Linear 2.84 2.66 2.86 3.56 3.34 3.11 2.41 2.17 2.22 Quadratic 2.28 2.32 3.14 2.79 2.88 3.44 1.68 1.59 2.51 HP 2.58 2.61 2.59 2.83 2.89 2.88 2.55 2.49 2.27 GLMS 2.49 2.47 2.57 2.83 2.83 2.88 2.23 2.22 2.23 Hamilton 2.56 2.59 2.53 2.86 2.94 2.92 2.44 2.36 2.24 MW 2.90 2.88 2.56 2.88 2.88 2.77 3.11 3.04 2.33 UC(GDP) 2.34 2.38 3.17 2.83∗ 2.87 3.20 1.76 1.74 2.66 UC(GDP/π) 2.51 2.48 2.75 2.94∗ 2.84∗ 3.00 1.90 1.89 2.35 UC(GDP/CU) – 2.46 2.74 – 2.90∗∗ 3.08 2.08 1.98 2.08 UC(GDP/u.) 2.47 2.51 2.90 2.92 2.98 3.24 2.02 2.00 2.24 UC(GDP/u./π) 2.51 2.55 2.89 2.92∗ 2.99 3.23 2.04 2.00 2.18 Tealbook 2.59 – 2.67 3.49 – 3.23 1.27 – 1.54 Notes: Table shows RMSE of implied interest rate from Taylor (1993) rule calculated using each output gap estimate relative to actual effective federal funds rate. UC (GDP/CU) is excluded from the real-time evaluationinthefirsttwosub-samplesduetoshortenedsampleavailability. AsterisksdenotethatDMWtest statistic for dominance of indicated model over the null October 2013 Tealbook output gap is statistically significantat10(∗)or5(∗∗)percentlevel. Seetextfordetails. 6 Conclusions We consider several statistical models of the U.S. economy and evaluate their estimates of the output gap along several dimensions. First, we document that output gap estimates can be wide ranging and that real-timeestimatesfromthesemodelscanbeheavilyrevisedafterthefact. Weproposeastatisticaldecomposition of output gap revisions, and find that the end-point problem is the primary reason that univariate modelsfailtoachievereal-timestability. Thefindingsuggeststhatmethodsaimedatreducingtheimpactof theend-pointproblemmaybeparticularlyusefultopolicymakers. WhileresearcherssuchasGarrattetal.(2008)orClementsandGalva˜o(2012)havetackledthisproblem explicitly,wefindanotherpossibleavenue: theuseoflabormarketinformationwhenproducingoutputgap estimates. The models we consider that contain an Okun’s law relationship are able to provide an estimate of the output gap that isrelatively stable in real time, in spite of the fact thatthey contain more parameters to be estimated. These output gap estimates also have useful economic properties. Models that include 19

theunemploymentrateareconsistentlyamongthebest-in-classofthemodelsweconsiderintermsoftheir usefulnessasaharbingeroffutureinflationandasagaugeforthestanceofmonetarypolicy. Andsincetheir output gap estimates are relatively stable, this performance does not substantially deteriorate when used in realtimerelativetowhenusedex-post. Finally, the real-time stability of the output gap—as was emphasized by Orphanides and van Norden (2002)—isnottheonlymetricbywhichtojudgeoutputgapestimates. Nooutputgapestimateweconsider is clearly superior to the others along each metric. While the Mueller-Watson gap produced relatively accurate forecasts between 1984 and 2013, when used in a Taylor rule, it also prescribed a Federal Funds rateofmorethan5percentonlyafewyearsaftertheendoftheGreatRecession. Similarly,whiletheGarratt et al. (2008) procedure does reduce the end-point problem inherent when filtering in real time, the output gapproducedbythatprocedureisoftenoutperformedintermsofeconomicsignificance. Ofcourse, atthe time of writing the U.S. economy has been subject to an unprecedented economic and social trauma: the outbreakoftheCOVID-19inlate2019and2020. Itwillbeinterestingtocontinuetomonitorthereal-time behaviorofthesevariousoutputgapestimatesandtheirusefulnesstopolicymakers. References Aastveit,KnutAreandTørresTrovik,“Estimatingtheoutputgapinrealtime: Afactormodelapproach,” TheQuarterlyReviewofEconomicsandFinance,2014,54(2),180–193. Barigozzi, Matteo and Matteo Luciani, “Measuring US Aggregate Output and Output Gap Using Large Datasets,”SSRN,2018. Berge, Travis J., “Understanding survey-based inflation expectations,” International Journal of Forecasting,2018,34(4),788–801. , “Time-varying uncertainty of the Federal Reserve’s output gap estimate,” Finance and Economics Discussion Series 2020-012, Board of Governors of the Federal Reserve System, Washington, D.C. February2020. 20

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Appendix A1 Data details TableA1: Descriptionofreal-timedatavintages Firstreal-time Finalreal-time Variable Source vintage vintage RealGDP 1965:Q4 2019:Q1 FRB-PhRT GDPdeflator 1965:Q4 2019:Q1 FRB-PhRT Unemploymentrate 1965:Q4 2019:Q1 FRB-PhRT Man. capacityutilization 1985:Q3 2019:Q1 FRB-PhRT Tealbookgap 1975:Q1 2013:Q4 FRB-PhGB Notes: FRB-Ph RT denotes real-time dataset from the Federal Reserve Bank of Philadelphia, available at https://www.philadelphiafed.org/research-and-data/real-time-center/ real-time-data. FRB-Ph GB denotes Greenbook/Tealbook dataset from Federal Reserve Bank of Philadelphia, available at https://www.philadelphiafed.org/research-and-data/ real-time-center/real-time-data/greenbook. ThePhiladelphiaFedGreenbookdatahasreal-time outputgapestimatessince1975:Q1. TothisdataweappendearlierFederalReservestaffnowcastsbeginning in1950:Q4,obtainedfromEdgeandRudd(2016). A2 Estimation details A2.1 OverviewofEstimation TheUCmodelscanbecastaslinearGaussianstate-spacemodels,theestimationofwhichcanbeachieved usingstandardmethods. Hereweoutlinetheestimationprocedure. Group the observables into the vectorY and the unobserved states into the vector s . For example, the t t UC(3) model would haveY = (y ,u ,π )(cid:48) and s = (y∗,τ∗,u∗,π∗,C,C )(cid:48) . The model is then compactly t t t t t t t t t t t−1 writteninstate-spaceform: Y =Fs +e ; e ∼N(0,Σ) (1) t t t t s =Gs +ε ; ε ∼N(0,Ω) (2) t t−1 t t Measurementerrorsareuncorrelatedwitheachotheraswellaswiththeshockstothestates. WeestimatethemodelusingBayesianmethodsandusingaGibbssampler. Theintuitionofthesampler isstraightforward: conditionalonthedataandstates,equations1and2areasetofindependentlinear,Gaus- 24

sianregressionsforwhichparameterscanbesampledfromtheirknownposteriordistribution. Conditional on the model’s parameters, the states can be drawn using the Kalman filter. Thus, we estimate the model usingaGibbssampleralternatesbetweendrawsofmodelparametersthatareconditionedonthestatesand drawsofthestatesthatareconditionedonmodelparameters. Eachstepofthesamplerisstandard;see,e.g.,KimandNelson(1999). Ourpriorsareconjugatesothat alltheposteriorsareknowninclosedform. Thealgorithmconsiststhefollowingfivesteps: 1. Sample F. Conditional on the data, states and Σ, draw the slope terms in equation 1. Because Σ is diagonal,wedothisequation-by-equation. 2. SampleΣ. Conditionalonthedata,states,andF,drawthevariancesofthemeasurementerrors. 3. SampleG. ConditionalonstatesandΩ,drawtheARparametersofthecycle. Weimposestationarity oftheautoregressiveprocessviarejectionsampling. 4. Sample Ω. Conditional on the data (yT), states (sT) and G, draw the variances of the shocks to the states. 5. Sample the states. Conditional on the data and parameters, we draw the states using the Durbin and Koopman(2002)andasimplementedinJarocin´ski(2015). We produce 12,000 draws from this sampler, ignoring the first 2,000 and using a thinning factor of 10 foratotalof1,000drawseachmodel’sposteriordistribution. A2.2 Priors Wesetourpriorsinthefollowingdata-drivenmanner. Asdescribedinthetext,ingeneralourpriorsfollow thetypicalNormal-InverseGaussianscheme. Ourpriorsaresetforeachvintageasfollows. • Forthepriorsofthevarianceforinnovationstotrends(e.g.,ε y∗ ,ε u∗ ,ε π∗ ),wesetthescaleparameter ofthedistribution,whichcontrolsthetightnessoftheprior,tobeTvintage/20,asomewhatinformative distribution. WethenHPfilterapre-sample(1947–1959)ofthevintagedata. Theshapeparameteris thensetsothatthepriordistributioniscenteredabouttheHP-filter-impliedtrendvariance. • Wefollowananalogousprocedureforsettingthepriorforthevarianceoftheinnovationstothedrift intrendoutput,ε . τ 25

• The prior for the variance of the innovations to the cycle is centered about the variance of the presample’s HP-filter implied cycle, and with scale parameter equal to one, a much less informative prior. • Measurementerrorinnovationsareverylooselyspecified,assumedtobeIG(1,1). • Ourpriorsfor“slopecoefficients”areGaussianandnottightlyspecified. – Reflecting our belief that the cycle is quite persistent, our priors for the AR coefficients of the cycle, φ φ are centered at (3/2, -2/3)’ with variance-covariance matrix equal to the identity 1 2 matrix. – Theα parametersaregiveneconomicallysensiblepriorswithvariance-covariancematrixequal totheidentity. Forinflation,themeanofα andα aresetto1⁄ and1⁄ . Whentheobservableis 0 1 10 20 unemployment,theyarecenteredabout-1⁄ and-1⁄ ,andforcapacityutilizationtheyarecentered 2 4 at0andmuchmorediffuse,withvariance-covariancematrixequaltotentimestheidentity. TableA2showsthepriorsandposteriorsforthe2019:Q1vintageoftheUC(GDP/u./π)model. TableA2: PriorsandposteriordistributionofUC(GDP/u./π)model. Priors Posteriors Parameter 5% Mean 95% 5% Mean 95% α -2.14 -.5 1.14 -.65 -.41 -.23 0 α -1.89 -.25 1.39 -.14 .01 .15 1 β -1.54 .10 1.74 -.32 -.03 .27 0 β -1.59 .05 1.69 -.07 .22 .49 1 φ -.14 1.50 3.14 1.54 1.68 1.81 1 φ -2.31 -.67 .98 -.84 -.72 -.58 2 σ y∗ .03 .24 .44 .20 .34 .57 σ .00 .02 .05 .02 .02 .03 τ σ u∗ .00 .07 .16 .13 .21 .30 σ π∗ .00 .08 .19 .43 .52 .65 σ .29 1.08 1.82 .35 .45 .56 C σ .23 1.00 1.73 .23 .29 .36 e,y σ .23 1.00 1.73 .13 .15 .17 e,u σ .23 1.00 1.73 .56 .65 .73 π,u Notes: This table shows the priors and posteriors of UC(GDP/u./π) model estimated on the for 2019:Q1 vintage. σ denotesstandarddeviationofshockprocess. 26

A3 Alternative inflation forecast experiments TableA3: PhillipscurvebasedforecastsoftotalPCEinflation,1990:Q1–2013:Q3. Twoquartersahead Fourquartersahead Eightquartersahead RT QRT 2013:Q4 RT QRT 2013:Q4 RT QRT 2013:Q4 Linear 1.01 0.99 0.98 1.00 0.98 0.98 1.00 0.96 0.98 Quadratic 1.01 0.96 1.04 1.00 0.94 1.04 0.98 0.95 1.04 HP 0.97 0.95 0.99 0.92 0.92 0.97 0.89 0.90 0.98 GLMS 0.97 0.97 0.99 0.93 0.93 0.97 0.92 0.93 0.99 Hamilton 0.98 0.94 0.94 0.94 0.94 0.92 0.90 0.93 0.89 MW 1.03 1.00 1.02 1.02 0.98 1.00 0.99 0.95 1.01 UC(GDP) 1.02 1.00 1.03 1.01 0.99 1.02 0.98 0.97 1.02 UC(GDP/π) 1.01 0.99 1.02 1.01 0.97 1.01 1.00 0.97 1.00 UC(GDP/CU) 1.03 1.01 1.01 1.04 1.02 1.02 1.04 1.02 1.03 UC(GDP/u.) 1.05 1.04 1.04 1.05 1.05 1.05 1.04 1.04 1.05 UC(GDP/u./π) 1.05 1.04 1.04 1.05 1.04 1.05 1.04 1.03 1.05 Tealbook 1.04 – 1.20 1.03 – 1.00 1.04 – 0.88 Notes: Absolute RMSE of inflation forecast from equation 3 using October 2013 Tealbook output gap estimate shown as bolded values. All other entires are the ratio of that RMSE to the bolded value. Outof-sampleperiodis1990:Q1–2013:Q3andmodelsareestimatedbeginningin1985:Q2usinganexpanding scheme. Asterisks denote that DM test statistic for dominance of indicated model over the null Tealbook estimateisstatisticallysignificantat10(∗)or5(∗∗)percentlevel. Seetextfordetails. TableA4: PhillipscurvebasedforecastsofcorePCEinflation,1984:Q1–2013:Q3. Twoquartersahead Fourquartersahead Eightquartersahead RT QRT 2013:Q4 RT QRT 2013:Q4 RT QRT 2013:Q4 Linear 1.07 1.06 1.09 1.05 1.08 1.11 1.02 1.08 1.12 Quadratic 1.06 1.05 1.03 1.05 1.06 1.05 1.03 1.06 1.10 HP 0.98 0.96 1.01 0.96 0.96 1.00 0.93 0.95 0.95 GLMS 1.00 1.01 1.01 0.98 1.00 1.00 0.99 1.00 0.95 Hamilton 0.98 0.96 0.96 0.95 0.94 0.93 0.92 0.95 0.94 MW 0.96 0.93 1.01 0.99 0.94 0.99 1.03 0.97 0.92 UC(GDP) 0.98 1.05 1.03 0.98 1.07 1.04 1.01 1.08 1.08 UC(GDP/π) 0.94 1.05 1.04 0.92 1.05 1.06 0.97 1.04 1.08 UC(GDP/CU) – 1.05 1.01 – 1.05 1.02 – 1.06 1.03 UC(GDP/u.) 1.07 1.09 1.06 1.08 1.11 1.09 1.05 1.10 1.12 UC(GDP/u./π) 1.08 1.10 1.06 1.08 1.12 1.09 1.05 1.10 1.11 Tealbook 1.03 – 0.63 – 0.99 – 0.66 – 0.92 – 0.75 Notes: Absolute RMSE of inflation forecast from equation 3 using October 2013 Tealbook output gap estimate shown as bolded values. All other entires are the ratio of that RMSE to the bolded value. Outof-sampleperiodis1984:Q1–2013:Q3andmodelsareestimatedbeginningin1975:Q1usinganexpanding scheme. Real-timeUC(GDP/CU)modelomittedduetoshortsample. AsterisksdenotethatDMteststatistic for dominance of indicated model over the null Tealbook estimate is statistically significant at 10 (∗) or 5 (∗∗)percentlevel. Seetextfordetails. 27

TableA5: PhillipscurvebasedforecastsoftotalPCEinflation,1984:Q1–2013:Q3. Twoquartersahead Fourquartersahead Eightquartersahead RT QRT 2013:Q4 RT QRT 2013:Q4 RT QRT 2013:Q4 Linear 1.04 1.04 1.04 1.02 1.05 1.06 1.00 1.06 1.07 Quadratic 1.04 1.03 1.03 1.02 1.03 1.05 1.01 1.03 1.10 HP 1.02 1.00 1.02 0.99 0.98 1.01 0.96 0.97 0.97 GLMS 1.02 1.03 1.01 1.00 1.01 1.01 0.98 0.99 0.97 Hamilton 1.01 0.99 1.01 0.98 0.98 0.99 0.94 0.96 0.97 MW 1.03 1.00 1.02 1.04 1.01 1.00 1.06 1.03 0.93 UC(GDP) 1.02 1.05 1.03 0.99 1.05 1.04 0.99 1.06 1.07 UC(GDP/π) 1.01 1.04 1.04 0.98 1.04 1.05 0.97 1.03 1.07 UC(GDP/CU) – 1.04 1.01 0.78 1.04 1.02 – 1.04 1.03 UC(GDP/u.) 1.06 1.07 1.05 1.06 1.08 1.07 1.04 1.08 1.11 UC(GDP/u./π) 1.06 1.07 1.05 1.07 1.08 1.08 1.04 1.08 1.10 Tealbook 1.02 – 1.36 0.98 – 1.26 0.89 – 1.27 Notes: Absolute RMSE of inflation forecast from equation 3 using October 2013 Tealbook output gap estimate shown as bolded values. All other entires are the ratio of that RMSE to the bolded value. Outof-sampleperiodis1984:Q1–2013:Q3andmodelsareestimatedbeginningin1975:Q1usinganexpanding scheme. Asterisks denote that DM test statistic for dominance of indicated model over the null Tealbook estimateisstatisticallysignificantat10(∗)or5(∗∗)percentlevel. Seetextfordetails. 28

A4 Alternative Taylor rule implied interest rates Here,werepeattheanalysisofsection5.3butuseinsteadarulewithw =1andw =1/2. C π FigureA1: “Balanced”Taylorruleimpliedfederalfundsrate,1965:Q3–2018:Q4. 1970 1980 1990 2000 2010 2020 02 51 01 5 0 5− % l Linear l UC (GDP) l Quad l UC (GDP/p ) l HP l UC (GDP/CU) l GLMS l UC (GDP/un.) l Hamilton l UC (GDP/p ,un.) l MW l Tealbook Notes: Figure shows prescribed interest rate from Taylor 1993 rule, calculated as in equation 4 with w = π 1/2,w C =1,π equaltothefourquarterchangeinthereal-timeGDPpricedeflator,y t asthereal-timeoutput gapestimate,andπ∗=r∗=2percent. Thickblacklinedenotesactualeffectivefederalfundsrate. Seetext fordetails. 29

TableA6: Performanceofprescribedinterestratefrom’‘balanced”Taylorrule. 1965:Q3–2013:Q3 1975:Q1–1997:Q4 1998:Q1–2013:Q3 RT QRT 2013:Q4 RT QRT 2013:Q4 RT QRT 2013:Q4 Linear 5.24 4.80 4.21 5.27 4.62 3.77 6.18 5.80 4.72 Quadratic 2.92 3.02 4.29 3.43 3.68 4.39 2.61 2.50 3.82 HP 2.74 2.77 2.73 3.04 3.11 3.08 2.73 2.66 2.32 GLMS 2.42 2.39 2.70 2.90 2.90 3.08 2.04 2.03 2.23 Hamilton 3.03 3.06 3.08 3.46 3.60 3.56 2.79 2.70 2.69 MW 3.33 3.31 2.74 3.05 3.09 2.87 3.93 3.82 2.57 UC(GDP) 2.28∗∗ 2.30 4.27 2.98∗∗ 2.99 3.78 1.37 1.29 3.87 UC(GDP/π) 2.60 2.54 3.13 3.14 2.93∗∗ 3.30 1.62 1.60 2.74 UC(GDP/CU) – 2.67 3.25 – 3.27∗ 3.63 2.06 1.94 2.25 UC(GDP/u.) 3.02 3.04 3.81 3.69 3.69 4.03 2.50 2.50 3.23 UC(GDP/u./π) 3.05 3.07 3.78 3.57 3.66 4.00 2.55 2.40 3.13 Tealbook 4.25 – 3.23 5.75 – 4.01 1.90 – 1.58 Notes: Table shows RMSE of implied interest rate from “balanced” Taylor rule (w C =1, w π =1/2) calculatedusingeachoutputgapestimaterelativetoactualeffectivefederalfundsrate. UC(GDP/CU)isexcluded fromthereal-timeevaluationinthefirsttwosub-samplesbecauseoflimitedsample. Asterisksdenotethat DMW test statistic for dominance of indicated model over the null October 2013 Tealbook output gap is statisticallysignificantat10(∗)or5(∗∗)percentlevel. Seetextfordetails. 30