Forecasting US inflation in real time

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. Forecasting US inflation in real time Chad Fulton and Kirstin Hubrich 2021-014 Please cite this paper as: Fulton, Chad, and Kirstin Hubrich (2021). “Forecasting US inflation in real time,” Finance and Economics Discussion Series 2021-014. Washington: Board of Governors of the Federal Reserve System, https://doi.org/10.17016/FEDS.2021.014. 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.

Forecasting US inflation in real time ChadFulton KirstinHubrich FederalReserveBoard FederalReserveBoard December9,2020, firstdraft: May2019 Abstract We perform a real-time forecasting exercise for US inflation, investigating whether andhowadditionalinformation–additionalmacroeconomicvariables,expertjudgment, orforecastcombination–canimproveforecastaccuracyandrobustness. Inouranalysis we consider the pre-pandemic period including the Global Financial Crisis and the following expansion – the longest on record – featuring unemployment that fell to a rate not seen for nearly sixty years. Distinguishing features of our study include the use of published Federal Reserve Board staff forecasts contained in Tealbooks and a focus on forecastingperformancebefore,during,andaftertheGlobalFinancialCrisis,withrelevance also for the current crisis and beyond. We find that while simple models remain hard to beat, the additional information that we consider can improve forecasts, especiallyinthepost-crisisperiod. Ourresultsshowthat(1)forecastcombinationapproaches improveforecastaccuracyoversimplermodelsandrobustifyagainstbadforecasts,aparticularlyrelevantfeatureinthecurrentenvironment;(2)aggregatingforecastsofinflation componentscanimproveperformancecomparedtoforecastingtheaggregatedirectly;(3) judgmental forecasts, which likely incorporate larger and more timely datasets, provide improvedforecastsatshorthorizons. Keywords: Inflation,Surveyforecasts,Forecastcombination ∗Chad Fulton and Kirstin Hubrich: Board of Governors of the Federal Reserve System, 20th and Constitution Ave NW, Washington, DC 20551 (e-mail: chad.t.fulton@frb.gov and kirstin.hubrich@frb.gov). †Theviewsexpressedinthispaperarethoseoftheauthorsanddonotnecessarilyreflectthoseof theFederalReserveBoardortheFederalReserveSystemoritsstaff. WethankNeilEricssonandtwo anonymousrefereesforusefulsuggestionsaswellasparticipantsoftheInternationalAssociationfor AppliedEconometrics2019conferenceandtheCFE2019forhelpfulcomments.

1 Introduction After a slower-than-usual recovery from the Great Recession, the unemployment rate fell to 3.5% in December 2019, its lowest reading since December 1969. At the same time, wage growth, while firming, remained only moderate, and consumer price inflation only briefly reachedthe2%targetoftheFederalOpenMarketCommittee(FOMC).Theserestrainedprice movements in the face of dramatic swings in labor market data, illustrated in Figure 1, have beenhistoricallypuzzling. Thecurrentdebateaboutpossibleinflationarypressuresdeveloping highlightstheincreaseduncertaintyaboutthefuturebehaviorofinflationandtheimportance of taking into account a broad information set. Our interest, therefore, is to consider what information,ifany,maybeusedtoguideinflationforecastsgoingforward. One popular framework for analyzing and forecasting inflation is based on the Phillips curve, the predicted negative relationship between economic slack and inflation. In addition totheextensiveliteratureexploringtheempiricalandtheoreticalpropertiesofthesemodels– includingthediscussionoftherecentflatteningofthePhillipsCurve–formerFederalReserve BoardChairJanetYellenandcurrentChairJeromePowellhaveinrecentspeechesreferenced anexpectations-augmentedeconometricPhillipscurvespecificationasaframeworkformodeling and forecasting consumer price inflation.1 At the same time, however, recent literature oninflationforecastinghasmostlyemphasizedsimpler,oftenunivariate,models. Inthispaper weinvestigateifandhowadditionalinformation–additionalmacroeconomicvariables,expert judgment,orforecastcombination–canimproveforecastaccuracy. Ourapproachisinformedbythreerecentstrandsoftheliteratureoninflationforecasting. First,AtkesonandOhanian(2001)andStockandWatson(2007)showthatwhileinflationhas becomeeasiertoforecastoverallinrecentdecades–inthesenseoflowerout-of-samplemean square errors across a variety of univariate and multivariate models mainly due to the overall lower variability of inflation – it has at the same time become more difficult to effectively incorporate information other than inflation itself in producing forecasts that improve over simplebenchmarkmodels. Inparticular,theynotethattheusefulnessofPhillipscurvemodels, inwhichslackcanbeusedtopredictfutureinflation,appearstohavedeclined. A second strand of the literature shows that survey forecasts have predictive power for 1See,forexample,Yellen(2015)andPowell(2018). 1

10 8 6 4 1989 1994 1999 2004 2009 2014 2019 tnecreP Unemployment rate 6 4 2 1989 1994 1999 2004 2009 2014 2019 tnecreP Core PCE price inflation Wage inflation (average hourly earnings) Figure1: HistoricalUSunemployment,wageinflation,andpriceinflation inflation,bothwhenincludedasanexpectationsterminPhillipscurvemodelsandwhenconsideredasdirectforecasts. FaustandWright(2013)distillfrompreviousresultsandtheirown real-timeforecastingexercisethefollowinglessons: (1)Subjectiveforecastsdobest;(2)Good forecasts must account for a slowly varying local mean; (3) Good forecasts begin with high quality nowcasts; (4) One of the best forecasting techniques is to simply produce a smooth path between the best available nowcast (as the forecast for the first horizon) and the best availablelocalmean(astheforecastforthelasthorizon). Weviewtheseresultsaspromisingsincealthoughallofthesepapersemphasizethesuperiorityofsimplemodels,eachactuallyincorporatesmoreinformationinitsforecaststhanthe last. Atkeson and Ohanian (2001) forecast inflation using only its own last four lags, while the unobserved components model with stochastic volatility model introduced by Stock and Watson(2007)allowsfortime-varyingparametersinordertoemploytheentirehistoryofinflation. In the language of Faust and Wright (2013), each of these papers presented methods forestimatinga“localmean”ofinflation. FaustandWright(2013)thenextendthelocalmean to make use of variables other than inflation itself, including subjective nowcasts and longterm forecasts from surveys that potentially incorporate a large – although poorly defined – additionaldataset. A third strand of the literature explores whether forecast combination can improve inflation forecasts. Forecast combination of different forecasts of the same variable have been 2

showntoimproveoverthebestsingleforecastincertainsituations(seeHendryandClements (2004)). Furthermore, combining forecasts from disaggregate component models to forecast an aggregate has been found to improve over forecasts from an aggregate model under certainconditions(seee.g. Lütkepohl(1984), Granger(1987), Hubrich(2005), andHendryand Hubrich(2011)). In this paper, we build on these literatures, exploring if and how additional information should inform inflation forecasts. First, we consider incorporating additional information in the form of multivariate inflation forecasting models. We begin by adding specific macroeconomic variables explicitly to econometric models, focusing on resource utilization and inflationexpectationsasincorporatedinanempiricalPhillipscurve. Theeconomicinformation containedinthesevariablesiswell-definedandcanbematcheduptotheoreticalPhillipscurve models. We next consider incorporating information from judgmental sources, in particular the Survey of Professional Forecasters (SPF) forecast and the Federal Reserve Board staff forecast presented in the Tealbook (prior to 2010 referred to as Greenbook). The economic information contained in these forecasts is less-well-defined, since it captures both subjective judgment and an unknown range of models and data from a potentially large number of unknownsources. Second,weinvestigateincorporatingadditionalinformationintheformofmultipleeconometricmodels,consideringboththecombinationofforecastsfrommultiplemodelsofoverall priceinflationandtheconstructionofoverallpriceinflationforecastsbyaggregatingforecasts of price subcomponents. Specifically, we investigate whether a Phillips Curve specification foroverallpriceinflationimprovesoverforecastingcore,energy,andfoodpriceinflationseparatelyandthenaggregatingthoseforecasts. Wealsocomparethiswithforecastcombination ofdifferentmodelsforoverallpriceinflationusingdifferentweightingschemes. Previous literature has mainly focused on aggregation of forecasts from the same model or model class (see, for example, Hubrich (2005), Hendry and Hubrich (2011), and Stock and Watson (2016)). In contrast, we investigate whether forecast performance for US price inflation can be improved by aggregating forecasts with different specifications for each underlying inflation component, allowing us to capture particular time series characteristics of eachseries. Inaddition,weinvestigatewhethercombiningdifferentforecastsoftotalUSprice 3

inflationimprovesforecastperformanceoverthesinglebestforecast. Thisisparticularlyrelevantintimesofeconomicuncertainty,sinceforecastcombinationcanpotentiallybeatoolto improveforecastperformanceinthepresenceoflargechangessuchastheglobalfinancialcrisis. HubrichandSkudelny(2017)findthatforEuroareainflation,forecastcombinationhelps to robustify the forecast, since forecast combination for euro area inflation helps improving overtheworstforecasts. Toaddressthesequestions,weperformareal-timeforecastingexercise,focusingonprice inflationasmeasuredbythepersonalconsumptionexpenditures(PCE)chain-typepriceindex employed by the Federal Reserve to evaluate the inflation objective. We extend the real-time forecastevaluationbyFaustandWright(2013)inanumberofrespects: weexplicitlycompare differentforecastcombinationandaggregationstrategiesandincludeinthisanalysisSPFand Tealbook forecasts. We also include more recent sample periods and we focus on PCE price inflation (as opposed to other inflation measures such as those based on the GDP deflator or the consumer price index) motivated by its importance for monetary policy in the US. We explore which additional pieces of information were most useful before, during, and after the global financial crisis, and so shed light on which methods are most promising now for constructing and robustifying inflation forecasts. This is particularly relevant in light of the surprisingbehaviorofinflationduringtherecentexpansionandtheadditionaluncertaintythat hasbeenintroducedbythecurrent,pandemic-induced,economiccrisis. 2 Data Our forecasting exercise focuses on U.S. inflation, measured by the quarter-over-quarter percentchangeinthepersonalconsumptionexpenditures(PCE)chain-typepriceindexproduced by the Bureau of Economic Analysis (BEA). PCE prices are particularly significant from the perspectiveofmonetarypolicy,becausethelonger-runinflationobjectiveoftheFederalOpen MarketsCommittee, firstadoptedinJanuary2012andlaterrevisedinAugust2020, isstated intermsofPCEinflation.2 Nonetheless,othermeasuresofinflationremainimportant,bothas economic indicators and for our exercise here. In Figure 2, we show the evolution of several ofthesemeasures. 2SeeYellen(2015)foradditionaldiscussionofthePCEpriceindexinthecontextofmonetarypolicy. 4

8 6 4 2 0 2 4 6 8 1989 1994 1999 2004 2009 2014 2019 tnecreP PCE price inflation CPI inflation Core PCE price inflation 10 8 6 4 2 0 2 4 1989 1994 1999 2004 2009 2014 2019 tnecreP 60 40 20 0 20 40 Food PCE price inflation 60 1989 1994 1999 2004 2009 2014 2019 tnecreP Energy PCE price inflation Figure 2: Annualized quarterly inflation rates for various price indexes using data from the 2020Q3vintage. TheprimaryalternativemeasureofU.S.consumerpriceinflationisbasedontheConsumer PriceIndex(CPI)publishedbytheBureauofLaborStatistics(BLS).WhilethismeasurediffersfromthePCEpriceindexinseveralimportantways,ithashistoricallybeenanimportant measure for monetary policymakers.3,4 Moreover, while attention has recently shifted to the PCE price index, its construction by the BEA largely relies on source data from disaggregate CPI series collected by the BLS. This fact has implications for our forecasting exercise, because it implies that monthly CPI releases provide information about quarterly PCE price 3WhiletheCPIandPCEpriceindexshareasimilarlow-frequencyevolution,differencesinformula,weight, andscope–asdiscussedin,forexample,McCullyetal.(2007)–canresultinpersistentdifferencesinmeasured inflation. One commonly noted implication of the formula effect is that the CPI – which employs a Laspeyres index concept – is slower to accommodate consumer substitution between goods, and so tends to increase at a fasterpacethanthePCEpriceindex. 4 Indeed,theCPIwastheonlymeasureofinflationexplicitlyincludedintheprojectionsofFederalReserve BanksandBoardmembersproducedaspartofthesemi-annualMonetaryPolicyReport(MPR)toCongressduring theperiod1992-1999.WethanktheguesteditorNeilEricssonforpointingthisouttous. 5

inflationthatcanbeexploitedinforecasting.5,6 sesaeler ataD stsaceroF GDP (Q3, advance) GDP (Q4, advance) PIO (Sep) PIO (Oct) PIO (Nov) PIO (Dec) PIO (Jan) PIO (Feb) CPI (Oct) CPI (Nov) CPI (Dec) CPI (Jan) CPI (Feb) November December January 2012 February March SPF (Q4) SPF (Q1) TB (Dec) TB (Jan) TB (Mar) h=1 corresponds to 2011Q4 h=1 corresponds to 2012Q1 Gross Domestic Product Personal Income & Outlays Consumer Price Index Survey of Professional Forecasters Tealbook Figure3: ExampleofdatareleaseandforecasttimingfortheperiodNovember2011through March2012 WhiletheoverallPCEpriceindexprovidesthebroadestmeasureofconsumerprices,there is also considerable interest in core inflation measures, which exclude the volatile food and energysubcategories.7 Onecommonlycitedbenefitofcoremeasuresofinflationisthat,since theyexcludevolatilecomponents,theyarebetterpredictorsoffutureinflation. Inourexercise, weincludeamodelthataimstotakeadvantageofthisbyfirstseparatelyproducingforecasts for core, food, and energy prices, and then aggregating to produce a forecast of overall PCE priceinflation. Whileseveralofourforecastingmodelsaredesignedtopredictfutureinflationusingonly consumerpricedata,manyoftheforecaststhatweconsidermakeuseofothermacroeconomic variables, including data on oil prices, prices of imported goods, inflation expectations, and realeconomicactivity. Thesevariablesaredescribedinmoredetailbelow,whenweintroduce our forecasting models. We also consider real-time judgmental inflation forecasts produced bytheSurveyofProfessionalForecastersandtheFederalReserveBoard,andthisintroduces 5While our econometric forecasting models are specified at the quarterly frequency and so do not take this higher-frequencyinformationintoaccount,weincludejudgmentalforecastsfromtheSurveyofProfessionalForecasters and Federal Reserve Board Tealbooks that do incorporate this information. Recent work on mixed frequencyeconometricmodelsthatcanbeusedforthepurposeof"nowcasting"inflationincludesModugno(2013) andKnotekandZaman(2017). 6It is worth noting that the raw price data that underlies both of these measures is known to be subject to measurementerror,asdocumentedinShoemaker(2011)andEichenbaumetal.(2014). Whilewearenotableto correctfortheseerrors,theyprimarilyaffectthemostdisaggregateinflationseries,andarelessofaconcernfor thehigh-levelaggregatesthatweuseforforecasting. 7Indeed,theMPR(seefootnote4)replacedoverallPCEpriceswithcorePCEpricesin2004,andtheFOMC SummaryofEconomicProjections,introducedin2007,includesbothmeasures. 6

severalissuesrelatedtoforecasttiminganddataavailability,whichwediscussnow. TheSurveyofProfessionalForecasters(SPF)isaquarterlysurveypublishedbytheFederal Reserve Bank of Philadelphia, with timing based around the release schedule for Gross Domestic Product (GDP) and quarterly PCE prices, which are both part of the National IncomeandProductAccounts(NIPA).8 Inparticular,surveysaretypicallysenttoandduetobe returned by respondents early in the second month of a given quarter. This is timed to occur shortly afterthe first – or"advance" – releaseof the NIPAdata for the previousquarter. This timing is illustrated in Figure 3, which shows the evolution of data releases and judgmental forecasts for the end of 2011 and beginning of 2012. For example, the advance release of 2011Q3 NIPA data, annotated as "GDP (Q3, advance)", occurred on October 27, 2011, and surveyresponsesforthefourth-quarterSPFweredueonNovember8. The Federal Reserve Board, meanwhile, produces inflation forecasts as part of the "Tealbook"forecasts(called"Greenbook"forecastspriorto2010)thatarepreparedbystaffeconomists in advance of each of eight annually scheduled Federal Open Markets Committee meetings. While this typically results in two Tealbook forecasts per quarter, they are not synchronized specificallytoNIPAdatareleases,andsotiminganddataavailabilitycanvarybetweenTealbooks. For example, the advance release of 2011Q4 GDP occurred after the publication of boththeDecember2011andJanuary2012Tealbooks,bothofwhichwerepublishedwellafterfourth-quarterSPF.ArchivedTealbookdataismadeavailablebytheFederalReserveBank ofPhiladelphiaReal-TimeDataResearchcenter. InadditiontothequarterlydatareleasedaspartoftheNIPAs,amonthlyPCEpriceindex is available as part of the BEA’s Personal Income and Outlays (PIO) release, and the CPI is similarly released monthly by the BLS. Depending on the timing of the SPF and Tealbook releases,thiscanintroduceadifferenceinthedatasetavailablewhenthesedifferentforecasts wereproduced. Forexample, Figure3showsthatbetweenthe2011Q4SPFduedateandthe December2011Tealbook,pricedataforOctober–thefirstmonthofthefourthquarter–was released for both the CPI and PCE measures. More generally, high frequency data that may be relevant for inflation forecasting – such as daily data on oil and gasoline prices – accrues over the course of each quarter. As a result, even though each quarterly PCE release corre- 8Croushoreetal.(2019)providearecentoverviewofthedetailsofthissurvey. 7

sponds to one SPF forecast and (typically) two Tealbook forecasts, there is a clear difference in the available information set at the time each forecast was produced. In order to alleviate this difference as much as possible, in our exercise we only consider the forecasts produced forthefirstTealbookfollowingeachadvanceGDPrelease. IntheexamplefromFigure3,we comparethe2011Q4SPFforecastsagainstthosefromtheDecember2011Tealbook,anddiscardthosefromtheJanuary2012Tealbook,sincethelatterincorporatesevenmoreadditional updatedinformationincomparisontotheSPF,whiletheformerTealbookhasaninformation setrelativelymorecomparabletotheSPF. Themodel-basedforecaststhatweconsideroperateonlyonaquarterlybasis,andassuch they do not incorporate monthly-frequency data on prices. To fix the timing, we assume that these models were run on the day of the included Tealbook forecast, although since they are estimated only using data through the previous quarter, the specific timing within the quarter mattersonlytoalittle.9 Specifically,intheexamplefromFigure3,themodel-basedforecasts that we compare against the 2011Q4 SPF and the December 2011 Tealbook only include data through 2011Q3, based on the vintage available at the time of the December Tealbook’s publication.10 3 Forecasting Methodology Ourfocusisprimarilyontherootmeansquarederror(RMSE)ofout-of-sampleforecastsfor quarterly inflation measured by the PCE price index. Our results are usually shown relative to a benchmark model, where a relative RMSE number less than one indicates improvement comparedtothebenchmark. Becauseoursourcedata–bothforPCEpricesandmanyofthe other variables we use, such as GDP – is subject to potentially large revisions, a real-time forecastingexerciseisnecessary.11 ThedataweuseisdrawnfromarchivedTealbookdatabasesunderlyingpubliclyavailable Tealbook forecasts and from Alfred (the real-time data repository maintained by the Federal 9Forexample,iftheestimateofthehistoryoftheunemploymentgapwasrevisedduringagivenquarter,then thespecifictimingoftheforecastcouldhaveasmalleffectonmodelsthatincludethatvariable. 10Becausethemodel-basedforecastsdonothaveanyabilityto"nowcast",wealwaysrefertothefirstforecast horizonash=1,ratherthanash=0(whichmightbemorenaturalinanowcastingcontext). 11Measurementerrorstoaparticularvariablemightbesystematic,andonelineofresearchhasdistinguished between“news”and“noise”intherevisionprocessofdata.Inpractice,datarevisionsaredifficulttomodel. 8

Reserve Bank of St. Louis). The timing of forecasts is as follows: once PCE prices are published through period t, we produce forecasts for each period t+h up to two years ahead (h=1,2,...,8 quarters).12 The two judgmental forecasts that we consider, however, may alreadyhaveadditionalinformationaboutthequartert+1,andsothoseforecastsatthehorizon h=1aremoreaccuratelydescribedasnowcasts. Toconducttheout-of-sampleforecastevaluation,weestimateallmodelsbasedonarecursivelyexpandingsample,startingin1988,withthefirstestimationsampleendingin1999Q3. Ourfinalforecastemploysdatathrough2019Q4, basedonTealbooksupto2020Q1.13 Since PCEpricedataarerevised,thereisnosinglesourceoftruedataagainstwhichtocompareour forecasts. WefollowTulip(2009)andFaustandWright(2013)inusingPCEpriceinflationas measured in the release two quarters after the reference quarter as the true value from which forecasterrorsareconstructed. 3.1 Model-basedforecasts Webeginourreal-timeexercisebyconstructingforecastsofinflation,denotedπ ,fromparat+h metriceconometricmodels. Theseprovideanexplicitspecificationofbothincludedvariables andinflationdynamics. Sincethereareanunlimitednumberofpotentialforecastingmodelsto consider,wefocusourattentionontheclassesofmodelsthat(a)havebeenshowntoproduce competitiveinflationforecastsinpreviousstudies,(b)areparsimonious,and(c)mostdirectly speak to the role of additional information in inflation forecasting.14 We present a unifying framework in equation (4) after introducing the first set of different models employed in this paper. 12Adiscussionofthetimingandinformationsetsassociatedwithjudgmentalforecasts,includingthedefinition oftheh=1forecast,isavailableinsection2. 13Note that we included a different number of forecasts for each horizon to use as much information about the forecasts as possible since otherwise our evaluation sample would be based on rather small samples for all horizons. 14Alternatively,wecouldhaveconsideredtostartfromageneralunrestrictedmodelusingageneral-to-specific modelselectionstrategyinvolvingmultiplepathsearches,encompassingtestsandasetofdiagnostictests,ashas beenadvocatedbyDavidHendryandisimplementedinAutometrics(seee.g. Doornik,2009). Moregenerally, modelselectioncanbeconsideredasastrategywheresmallermodelsaretestedagainstmoregeneralmodel.Our comparisonofforecastingmodelsandmethodsusingasmallerinformationsetwithmodelsandmethodsusing largerinformationsetsisinthatspirit. 9

AutoregressiveModel(AR) Thefirstmodelweconsiderhasaverysimplespecification,in whichtheinflationforecastsπ areproducedfromtheAR(p)model t+h p π =ρ +∑ρ π +ε (1) t 0 j t−j t i=1 We then iteratively apply a one-step-ahead forecast h times to construct the desired forecast π . Thelagorderthatwepresentresultsfor, p=1,wasselectedusingtheBayesInformation t+h Criteriaoverthelargestsampleperiod.15 Thismodelisunivariateininflationforecasting,and soincludestheleastadditionalinformationofallmodelsthatweconsider.16 Inflation gap model (AR-Gap) A useful way to incorporate some additional information whilemaintainingaparsimoniouseconometricmodelistomodelinflationasexhibitingshorttermfluctuationsaroundsomeunderlyingtrend,denotedτ . Thisrequiresspecificationofthe t inflation trend and an econometric model for modeling the dynamics of the “inflation gap”. Theinflationgap,denotedg =π −τ ,isthedifferencebetweeninflationanditstrend. Here, t t t weusetheSurveyofProfessionalForecastersforecastofaveragePCEinflationoverthenext 10 years as a proxy for trend inflation, while we model the inflation gap as an autoregressive process. Relativetothesimplerautoregressivemodelpresentedabove,thismodelincorporates additional information from survey forecasts to help pin down the “local mean” of inflation. Specifically,theforecastingmodelis p g =ρ +∑ρ g +v (2) t 0 j t−j t i=1 We then proceed as in Faust and Wright (2013) by taking the predictions of the gap – the forecastsg –andaddingbackthefinalobservationofthetrendtogettheimpliedprediction T+h ofinflation. Wepresentresultsforlagorder p=1,thesameasforthesimpleautoregression modelabove. 15WealsoexaminedboththeBayesInformationCriteriaandtheAkaikeInformationCriteriainreal-time. Our selectedlagorderp=1iscompetitiveacrossmostofthesampleperiodforbothcriteria,anditisthemodelmost preferredbytheBayescriteriasinceabout2009. 16Wealsoexaminedothercommonparsimoniousunivariatemodels,includingtherandomwalkforecastandthe modelofAtkesonandOhanian(2001).WereportresultsfortheAR(p)modelsinceitexhibitedbetterforecasting performanceinoursample. 10

Phillipscurvemodels Wenowexplicitlyincorporateintoourforecastsadditionalinformation from macroeconomic variables other than inflation, in the form of an empirical Phillips curve model. This class of models is appealing in that it uses macroeconomic variables to forecastinflationandhaslinkstotheoreticalmodelsofprice-setting. Thegeneralformofthe Phillipscurvemodelsthatweconsideris p q π =α+∑ρ π +βτ +γy +∑φ(cid:48)x +ω (3) t j t−j t t j t t j=1 j=1 whereτ isanestimateoftheinflationtrendattimet,y isameasureofeconomicslackattime t t t,andx isavectorofcontrols. Byvaryingthespecificationsoftheinflationtrend,economic t slack,andthevectorofcontrols,wecanaccommodateawiderangeofadditionalinformation. Asintheconstructionoftheinflationgapmodelabove,wemodeltheinflationtrendusing long-runinflationexpectationsfromtheSurveyofProfessionalForecasters. Economicslackis modeledasthedistancebetweentheunemploymentgapandanestimateofthenaturalrateof unemployment. ForallforecastsmadethroughDecember2014,weusetheTealbookestimate of the natural rate of unemployment, while for the period January 2015 to the present we use the estimate of the natural rate of unemployment produced by the Congressional Budget Office, since Tealbook estimates from this latter period have not yet been made public. The vector of controls that we include contains relative core import price inflation and relative energy price inflation.17 Note that in this model relative import price inflation captures the impactofinternationalinflationdevelopmentsonUSinflation.18 Forecastsofπ basedonthisequationrequireforecastsoftheright-hand-sidevariables. t+h For results that we report, we apply a random walk forecast for the inflation trend and the forecastfromanAR(1)modelfortheothervariables. As a unifying conceptual framework to think about how the different forecasting models use additional information to forecast inflation, one can consider the following extended version of the Phillips curve model that nests the AR, AR-Gap, and standard Phillips curve 17ThereareahugenumberofempiricalPhillipscurvespecificationsthathavebeenconsideredintheliterature, and although we report results for only one specification, we considered many alternatives. For example, we considered models with the inflation trend derived from different survey measures or from the Federal Reserve Boardstaffforecastsandforeconomicslackweconsideredvariousmeasuresofboththeunemploymentgapand outputgap. 18Intheaggregatedmodelpresentedbelowweenergyinflationasafunctionofoilpricestocaptureadifferent dimensionofinternationalinfluencesoninflation. 11

modelsdescribedintheparagraphsabove: q q p π =α+βτ +∑ρ π +γy +∑φ(cid:48)x +∑η τ +ξ (4) t t j t−j t j t j t−j t j=1 j=1 j=1 When β = γ = φ = 0, then the AR model is obtained, while when β = 1, γ = φ = 0 and j j η =−ρ we obtain the AR-Gap model that we will use as our benchmark forecast model j j in the forecast comparison. Finally, if β=η =0 then we obtain the Phillips curve model j discussedabove. VectorAutoregressiveModel(VAR) WealsoconsiderforecastsfromaVectorAutoregressive Model (VAR) that can be thought of as another extension of the unifying Phillips-Curve frameworkdiscussedwheretheright-handsidevariablesofthesingle-equationPhillipscurve modelareincludedasendogenousvariablesratherthanconditioningonthem. To facilitate the comparison, we use the same variables as we did in our Phillips curve model. As in the simple univariate autoregression, we estimate the parameters of the vector autoregression and then iteratively apply the one-step-ahead forecast h times. We present resultsforthelagorder1selectedusingtheBayesInformationCriteriaoverthelargestsample period. 3.2 Aggregatingforecastsofdisaggregateinflation As another extension of the unifying Phillips-Curve framework outlined above, we also produce forecasts of the primary disaggregate series that make up total PCE price inflation and then combine them as a weighted average to forecast the aggregate. Here, our primary focus is on including additional information in the model specification, and we are able to allow differentpricesubcomponentstodependondifferentmacroeconomicvariablesandtoexhibit different dynamics. In particular, we separately make forecasts for core PCE price inflation, food PCE price inflation, and energy PCE price inflation, and then combine them using their relativesharesinPCEasweights. The forecast of core PCE price inflation is based on a Phillips curve model similar to the empirical Phillips Curve model described above. The forecast for food PCE price inflation is also based on a similar Phillips curve model, except that in this case, no control variables 12

are included so that the term with x is dropped. Energy PCE price inflation is modeled as t πe = α+∑4 φ πoil+ζ , where πoil is oil price inflation. Forecasts are then produced by t j=1 j t t t assuming that oil price inflation follows a random walk. This aggregated forecast approach potentiallyimprovestheaccuracyofforecastingtheaggregatebyareductionintheestimation uncertaintyandmisspecification. 3.3 SubjectiveForecasts We include in our forecasting exercise two sets of forecasts that are not based on an explicit forecasting model. Relative to model-based forecasts, these subjective forecasts are likely basedonamuchlargerinformationset. Survey of Professional Forecasters (SPF) First, we include forecasts based on responses to the Survey of Professional Forecasters (SPF). Since 2007, the SPF has included forecasts of total PCE price inflation at quarterly horizons h=0,1,2,3,4 as well a forecast of average annual inflation over the next ten years, intended to capture expected long-run inflation. To constructforecastsforthehorizonsh=5,...,12,wefollowanapproachalongthelinesofthat suggested by Faust and Wright (2013) and linearly interpolate between a near-term forecast andalong-termforecast.19 Inparticular,wesettheh=12forecastequaltotheSPFforecast for long-run inflation and then linearly interpolate between the h=4 and h=12 values. In all cases, we use median SPF forecasts. Prior to 2007, the SPF only produced forecasts of consumerpriceindex(CPI)inflation,andsowemustuseimpliedforecastsforPCE.Asnoted earlier,differencesintheconstructionofthesetwopriceindicestendtolendanupwardsbias toCPIinflationcomparedtoPCEinflation.20 ToimputeforecastsforPCEpriceinflationprior to2007,ateachperiodwestartwiththeCPIforecastprovidedbytheSPFandthensubtractthe historicalwedge(aswouldhavebeencomputedatthattime)betweenpublishedCPIinflation andpublishedPCEpriceinflation. 19ThegoodperformanceofthisinterpolationapproachnotedbyFaustandWright(2013)suggeststhatitwould alsobeinterestingtoapplyitusingourmodel-basedforecastsinplaceoftheSPFforecasts,althoughweleavethis forfuturework. 20Thisfeatureisdiscussedinfootnote3andcanbeseeninFigure2. 13

Tealbookforecastsof FederalReserveBoardstaff Second, we include theforecastsprovidedineachTealbookfortotalPCEpriceinflation,whicharejudgmentalforecastsproduced bystaffoftheFederalReserveBoardofGovernors. Duetothe5-yearlagbetweenfinalizing the Tealbook and its public release, Federal Reserve Board staff forecasts are only available prior to 2015, and our primary exercise therefore only considers forecasts made through the endof2014. Asecondaryexerciseexpandsthesampletoincludeforecastsmadethroughthe end of 2019, although it excludes Tealbook forecasts. It should be noted that the Tealbook forecast takes into account that the US is an open economy given the staff discussions and takingonboardconditionalassumptions,forinstanceaboutoilpricesandtrade. 3.4 ForecastCombination As a final part of our analysis we include a forecast for PCE price inflation generated by takingaweightedaverageoftheforecastsfromthemodelsdescribedabove,exceptexcluding the Tealbook forecasts. We consider two methods for generating the forecast combination weights: in the first case (referred to below as "simple" combination), the weights are set equalforeachmodel,whileinthesecondcase(referredtobelowasthe"MSE"combination) the weight for a given model at the time t is set to be the inverse of the root mean squared error generated by the model over the preceding 8 quarters. Combining different forecast of the same variable can improve over the best forecast when forecasts are biased in opposite direction.21 Also, the forecast combination method with time-varying weights helps to shed lightonthetime-varyingrelativeforecastperformanceofthedifferentmodelsincludedinthe forecastcomparison. 4 Results: Forecasting US PCE inflation in real time We begin by presenting results for our comparison of forecasts of US PCE inflation in realtime for the portion of our sample period for which public Tealbook forecasts are available, 21NotethatcombiningforecastswithoverlappinginformationcontentcanalsoleadtoimprovedMSFEdueto differentiallymis-specifiedforecasts. Comparingin-sampletoout-of-sampleweightingintermsofequalweights versusMSEweightswouldbeaninterestingextension 14

1999Q3 - 2016Q4, and then discuss the pre- and post-crisis periods for that sample.22 Our focus will be on the root mean square error (RMSE) of our forecasts relative to the AR(1) model in the inflation gap. Our RMSE evaluation is based on quarterly inflation.23 This is also the model that Faust and Wright (2013) use as their benchmark, and we found that it outperforms other candidate benchmarks (such as the AR(1) model in inflation). We have also carried out Diebold-Mariano (DM) tests (see Diebold and Mariano, 1995, West, 1996, Diebold, 2015) to investigate whether the RMSE improvements over the benchmark model weresignificant.24 4.1 Real-timeanalysisincludingpublicTealbookforecast,Sampleuntil2016 We first consider a comparison of our selected forecast models and methods for the sample period where our second source of judgmental forecasts – those produced by the staff of the Federal Reserve Board (FRB) and recorded in Tealbooks – is publicly available. Due to the embargoonrecentTealbooks, fortheresultsinthissectionwerestrictoursamplesothatthe final forecasts were produced in 2014Q4. The relative RMSE results are shown in Figure 4, andtherelevantDMtestresultsarepresentedinTable1. AkeytakeawayoverthefullsampleforwhichtheTealbookisavailableisthattheautoregressive model in the inflation gap is generally difficult to improve upon except for horizon h=1. Indeed, themodel-basedforecaststhatincorporatespecificadditionalmacroeconomic variables – the Phillips curve and vector autoregression – do no better than this benchmark, except for the Phillips curve model during the post-crisis period. However, improvements from some of the forecasting methods we employ, in particular subjective forecasts, forecast aggregationandforecastcombination,dostandout. First, the aggregated forecast – which forecasts core, food, and energy prices separately beforeaggregatingthemtoproducethetotalPCEpriceinflationforecast–isabletoimprove onthegapmodelatthehorizonh=1. Thisisparticularlynoteworthysincetheotherforecast methodsthatimproveatthishorizonincorporatemixedfrequencydata(seethediscussionof 22Notethatthissampleperioddescribestheincludedforecastperiods. WhilethefinalpublicTealbookisfrom December2014,itsh=8forecastcorrespondsto2016Q4. 23Notethattherankingbetweenthedifferentforecastingmodelsandmethodsmightdifferbasedonadifferent transformation,suchasannualinflation. 24Statementsinthetextrefertothe10percentsignificancelevel. 15

Full sample with public Tealbooks (1999Q3 - 2016Q4) 1.0 0.8 0.6 0.4 0.2 0.0 Post-crisis with public Tealbooks (2010Q2 - 2016Q4) 1.2 1.0 0.8 0.6 0.4 0.2 0.0 Pre-crisis (1999Q3 - 2008Q2) 1.2 1.0 0.8 0.6 0.4 0.2 0.0 A R( 1), G a p A R( 1) P hilli ps C ur v e V A R( 1) A g gr e g ati o n S P F T e al b o o k C o m b (s i n i m at p i o l e n ) C o m bi ( n M a S ti E o ) n Horizon 1 2 4 8 Figure 4: Relative RMSE of Selected Inflation Forecasts with public Tealbook - through 2016Q4 16

Table1: Forecastcomparison,sampleincludingpublicTealbooks Horizon AR(1),Gap AR(1) PhillipsCurve VAR(1) Aggregation SPF Tealbook Combination Simple MSE FullsamplewithpublicTealbooks(1999Q3-2016Q4) 1 1.65 1.73 1.65 1.68 1.46 1.05 0.47 1.49 1.43 – -1.75 0.23 -0.71 1.81* 1.57 2.03* 1.68* 1.73* – [0.086] [0.388] [0.310] [0.078] [0.117] [0.051] [0.097] [0.090] 2 1.73 1.86 1.73 1.81 1.75 1.67 1.35 1.73 1.74 – -1.86 -0.08 -1.66 -0.63 0.90 1.49 -0.23 -0.39 – [0.070] [0.398] [0.100] [0.326] [0.265] [0.132] [0.388] [0.370] 4 1.67 1.77 1.69 1.73 1.68 1.70 1.77 1.68 1.68 – -1.72 -0.44 -1.68 -0.31 -1.01 -1.34 -0.73 -0.48 – [0.092] [0.362] [0.097] [0.380] [0.240] [0.163] [0.306] [0.356] 8 1.71 1.85 1.67 1.75 1.67 1.51 1.79 1.70 1.68 – -2.33 1.03 -0.51 1.40 0.96 -0.81 0.96 2.00* – [0.026] [0.235] [0.350] [0.150] [0.251] [0.287] [0.252] [0.054] Post-crisiswithpublicTealbooks(2010Q2-2016Q4) 1 1.11 1.17 1.07 1.17 1.05 0.91 0.32 1.05 1.02 – -1.23 0.46 -0.49 1.41 2.77** 4.51** 1.61 1.95* – [0.187] [0.359] [0.353] [0.147] [0.009] [0.000] [0.109] [0.059] 2 1.43 1.59 1.37 1.43 1.47 1.40 1.05 1.42 1.42 – -2.07 0.54 -0.00 -0.53 0.39 1.25 0.22 0.19 – [0.047] [0.345] [0.399] [0.347] [0.370] [0.183] [0.389] [0.391] 4 1.35 1.54 1.33 1.36 1.29 1.34 1.34 1.33 1.34 – -2.36 0.16 -0.10 0.94 0.13 0.05 0.33 0.13 – [0.025] [0.394] [0.397] [0.257] [0.396] [0.399] [0.378] [0.396] 8 1.40 1.66 1.25 1.45 1.26 1.38 1.31 1.37 1.34 – -4.15 2.01* -0.36 3.55** 0.57 0.62 1.43 1.86* – [0.000] [0.053] [0.374] [0.001] [0.339] [0.328] [0.143] [0.070] Pre-crisis(1999Q3-2008Q2) 1 1.30 1.39 1.36 1.31 1.13 1.00 0.56 1.20 1.16 – -1.50 -1.95 -0.09 2.54** 1.96* 3.28** 2.44** 2.54** – [0.129] [0.060] [0.397] [0.016] [0.058] [0.002] [0.021] [0.016] 2 1.31 1.37 1.39 1.37 1.35 1.30 1.26 1.32 1.31 – -0.58 -1.65 -1.72 -0.64 0.06 0.38 -0.56 0.05 – [0.338] [0.102] [0.091] [0.325] [0.398] [0.371] [0.342] [0.398] 4 1.31 1.31 1.39 1.38 1.35 1.34 1.51 1.32 1.30 – -0.02 -1.53 -1.72 -0.72 -0.53 -1.53 -0.49 0.44 – [0.399] [0.124] [0.091] [0.307] [0.346] [0.123] [0.353] [0.362] 8 1.45 1.50 1.50 1.52 1.47 1.51 1.74 1.46 1.43 – -0.33 -0.63 -0.85 -0.24 -2.42 -1.75 -0.33 1.81* – [0.378] [0.326] [0.279] [0.388] [0.022] [0.086] [0.378] [0.077] Note:thistablereportsforecastperformanceandcomparisoninformationfornineforecastingmodels,fourforecasthorizons,and threesubsamples. Foreachmodel/horizon/subsamplecombination,wereportthreevalues:therootmeansquareforecasting error,theteststatisticfromaDiebold-Marianotestagainstthebaseline"AR(1),gap"model,andtheassociatedp-value. Cases inwhichtherootmeansquareerrorissignificantlylowerthanthebaselinemodelatthe10and5percentsignificancelevelsare denotedwithoneortwoasterisks,respectively. 17

theSPF,Tealbook,andcombinationmodelsbelow),whilethismethoddoesnot. Theimprovementsfortheaggregatedforecastarestatisticallysignificantatthishorizon,accordingtoDM testresults,butnotforotherhorizons. Second, the SPF forecast shows a dramatic reduction in forecasting error at the horizon h=1. As described above, SPF respondents generally produce this forecast at the beginning of the second month of the quarter being forecasted, and so it can be labeled as a nowcast. Thus, this enhanced forecasting performance reflects both the judgmental expertise of the forecastersandthefactthatSPFforecastershaveaccesstoalargerinformationset,including some information about the h=1 quarter, when making their forecast. At a horizon of two yearsahead(h=8)theSPFforecastisalsosuperiortothebenchmark,whileforhorizonsh=2 andh=4thereislittleornoimprovementinSPFforecastperformanceoverthebenchmark. Theimprovementsinforecastperformanceforh=1areclearlystatisticallysignificantforthe pre- and post-crisis period, and borderline significant for the full sample period according to theDMtest. Third, the forecasts produced by forecast combination methods show improvements over thebenchmark. TheseincludetheSPFasoneoftheconstituentforecasts,andtheirimprovementrelativetothebenchmarkatthehorizonh=1showthattheyareabletotakeadvantage of SPF forecast improvements. At the same time, the combination forecasts provide a robust forecast, as they do not degrade as much as the SPF at longer horizons, and always improve over the worst models, including both the AR(1) and VAR(1). Finally, the combination incorporating time-varying weights performs slightly better than the equal-weight counterpart, suggesting that it can be useful to take into account variation over time in forecasting performance. TheRMSEimprovementsofthecombinationmethodwithtime-varyingweightsover thebenchmarkmodelarestatisticallysignificantforboth1quarterand2yearhorizons(h=1 andh=8). Finally, the public Tealbook forecasts by Federal Reserve staff provide substantial and statisticallysignificantforecastingimprovementsathorizonsh=1,butdonotoutperformthe benchmark model at the longer horizons. The most conspicuous result is the performance of the nowcast (h = 1) contained in the Tealbook, even compared to that from the Survey of Professional Forecasters. Although striking, this result likely largely reflects the fact that 18

FederalReservestaffnowcaststakeintoaccountamuchlargerinformationsetthanthemodels weconsider,whichincludeatmostahandfulofexplanatoryvariables. Althoughsomeofthis improvementisnodoubtduetotheuseofhigher-frequencyvariables,theTealbooknowcasts alsosubstantiallyoutperformthoseoftheSurveyofProfessionalForecasters,whowouldhave hadaccesstosimilardata,althoughasnotedinsection2,theTealbookforecastshaveaslightly updated information compared to the SPF forecasts. Altogether, this suggests that Tealbook nowcastsprovideanupperboundforforecastingimprovements,andshowsthateventhequitegoodSPFnowcastsstillhaveroomtoimprove. Asecondnotableresultfromourout-of-sampleforecastcomparisonisthestrongimprovement of the Tealbook forecast compared to the benchmark model at the horizon h=2 – the only forecast to outperform the benchmark at this horizon, although only statistically significantona15percentsignificancelevel. Oneexplanationforthisresultis,asnotedbyFaustand Wright(2013),thatagoodforecastforh=1canhelpimprovetheforecastforh=2. Thissuggests that there are gains still available in near-term inflation forecasting (even beyond those gainedbyusinghigher-frequencydatatoproducenowcasts),eitherfromadditionaldatawith predictivepowerorfromimprovedmodels. 4.2 Pre-vspost-crisisanalysisincludingpublicTealbooks The global financial crisis and subsequent Great Recession substantially disrupted the US economy; this raises several questions relevant for inflation forecasting. First, forecasting errorsmadeduringthecrisis–atimeinwhichinflationwasquitevolatile–mightbeinfluencing ourresults. Second,astructuralbreakmighthaveoccurredininflationdynamics,sothatforecasting methods or sources of information that improved forecast accuracy in comparison to the benchmark model prior to the crisis might not provide similarly accurate forecasts after the crisis. For instance, it might be argued that the slow labor market recovery following the global financial crisis (see Figure 1) was evidence of an altered economic climate compared to the pre-crisis period, with implications for inflation. To address these issues, we consider subsample analyses of the pre- and post-crisis periods, with results shown in the middle and lowerpanelofFigure4. Comparison of the forecasting performance in the pre- and post-crisis periods suggests 19

thatourprimaryqualitativeresultsdescribedforthefullsampleagreewiththosefromtheprecrisisperiod,butbegintobreakdownduringthepost-crisisperiodasmanymodelsoutperform thebenchmarkatbothshortandlonghorizonsinRMSEterms. These results suggest that there can be room for the use of additional information in improvinginflationforecasting,particularlyinthepost-crisisperiod,andespeciallyatveryshort and very long horizons.25 Moreover, we are able to find several different methods of incorporatingadditionalinformationintoeconometricmodelsthatproducetheseimprovements. Notably,andunlikeinpreviouswork,herewefindthatPhillipscurvemodelscanstillimprove onsimpleforecastingmodelswhenforecastinginflationinsomesituations. Tosummarizetheresultsofthesampleandsub-samplesthatincludethepublicTealbook forecasts: we find that the methods that include richer information sets (compared to simple forecastsbasedonjustoneunivariateormultivariateforecastmodel)allsignificantlyimprove over the benchmark inflation gap model for the full sample including public Tealbooks and the pre-crisis period at the shortest horizon. These models include the aggregation, the combination methods – including both the simple average and the time-varying MSE-weighted combination – as well as the judgmental forecasts – including both the SPF and Tealbook forecasts. In the post-crisis period including public Tealbooks, we continue to find that the timevarying,MSE-weightedcombinationandbothjudgmentalforecastsimprovesignificantlyover thebenchmarkmodel,whiletheimprovementsoftheaggregationmethodandsimplecombination are not significant at the short horizon. Meanwhile, the aggregation forecast and the time-varying,MSE-weightedcombinationforecastbothsignificantlyimproveoverthebenchmarkatthe2-yearhorizon. 4.3 Fullsamplethrough2019 Toinvestigatewhetherourresultsforthebaselinesampleperiod(theperiodthatincludespubliclyavailableTealbooks)alsoholdforanextendedsampleperiod, wecomparetheforecasts from the forecast models and methods other than the Tealbook for the full sample including recent history up to 2019 as well as the post-crisis period including these more recent years. 25Indeed,thesewerethehorizonsemphasizedasmostimportantbyFaustandWright(2013). 20

Full sample (1999Q3 - 2019Q4) 1.0 0.8 0.6 0.4 0.2 0.0 Post-crisis (2010Q2 - 2019Q4) 1.2 1.0 0.8 0.6 0.4 0.2 0.0 A R( 1), G a p A R( 1) P hilli ps C ur v e V A R( 1) A g gr e g ati o n S P F T e al b o o k C o m b (s i n i m at p i o l e n ) C o m bi ( n M a S ti E o ) n Horizon 1 2 4 8 Figure5: RelativeRMSEofSelectedInflationForecasts-Fullsample 21

Table2: Forecastcomparison,fullsample Horizon AR(1),Gap AR(1) PhillipsCurve VAR(1) Aggregation SPF Tealbook† Combination Simple MSE Fullsample(1999Q3-2019Q4) 1 1.53 1.61 1.52 1.56 1.34 0.95 – 1.38 1.31 – -2.40 0.39 -0.76 2.16** 1.81* – 1.91* 1.98* – [0.023] [0.369] [0.298] [0.039] [0.078] – [0.065] [0.056] 2 1.56 1.68 1.56 1.62 1.58 1.51 – 1.56 1.56 – -2.34 0.02 -1.57 -0.75 0.77 – -0.32 -0.49 – [0.026] [0.399] [0.117] [0.301] [0.297] – [0.379] [0.354] 4 1.52 1.63 1.53 1.57 1.54 1.56 – 1.53 1.53 – -2.13 -0.39 -1.51 -0.50 -1.18 – -0.91 -0.66 – [0.042] [0.369] [0.127] [0.352] [0.199] – [0.263] [0.322] 8 1.57 1.70 1.53 1.60 1.53 1.40 – 1.56 1.55 – -2.46 0.99 -0.52 1.29 0.93 – 0.81 1.67* – [0.019] [0.244] [0.349] [0.173] [0.258] – [0.288] [0.100] Post-crisis(2010Q2-2019Q4) 1 1.10 1.17 1.06 1.13 0.96 0.75 – 1.00 0.95 – -2.36 0.78 -0.60 2.26** 2.49** – 2.65** 3.06** – [0.025] [0.294] [0.334] [0.031] [0.018] – [0.012] [0.004] 2 1.16 1.31 1.11 1.15 1.19 1.15 – 1.16 1.16 – -3.10 0.71 0.12 -0.76 0.15 – 0.14 0.09 – [0.003] [0.310] [0.396] [0.299] [0.395] – [0.395] [0.397] 4 1.13 1.31 1.11 1.13 1.11 1.14 – 1.13 1.13 – -3.03 0.22 -0.04 0.52 -0.25 – 0.08 -0.10 – [0.004] [0.389] [0.399] [0.348] [0.386] – [0.398] [0.397] 8 1.22 1.44 1.11 1.26 1.11 1.21 – 1.19 1.17 – -3.80 1.79* -0.37 2.74** 0.08 – 1.13 1.42 – [0.000] [0.080] [0.373] [0.009] [0.398] – [0.210] [0.145] Note:thistablereportsforecastperformanceandcomparisoninformationfornineforecastingmodels,fourforecast horizons,andtwosubsamples.Foreachmodel/horizon/subsamplecombination,wereportthreevalues:theroot mean square forecasting error, the test statistic from a Diebold-Mariano test against the baseline "AR(1), gap" model, and the associated p-value. Cases in which the root mean square error is significantly lower than the baselinemodelatthe10and5percentsignificancelevelsaredenotedwithoneortwoasterisks,respectively. †ResultsforTealbookforecastsareunavailableforthefullsampleduetotheir5-yearembargoperiod. 22

The results for the full sample are shown in the upper panel of Figure 5, while the results for the post-crisis period up to 2019 are shown in the lower panel. Results for the DM tests are showninTable2. Itisnoteworthythatthereislittlechangeintherelativeforecastperformanceofthemodels when adding five additional years. For the full sample, the inflation gap is still difficult to improve upon, apart from the horizon h = 1 where the SPF, aggregation, and combination methodsprovideasignificantlybetterforecastthanthebenchmarkmodel,andthehorizonh= 8,wherethetime-varyingMSE-weightedcombinationmethodimprovesoverthebenchmark. For the post crisis period through 2019 we get the same results, except that the aggregated forecast and the Phillips Curve model significantly improve over the benchmark for h = 8 whilethetime-varyingMSE-weightedcombinationmethodimprovementisnotsignficant. Thetime-varyingrelativeperformanceintermsofMSEisnicelyillustratedinthefollowing two figures for horizons h=1 and h=8, respectively. It also illustrates the relevance of the largerinformationsetincorporatedintheSPFforh=1inepisodeswithhigheruncertaintyand volatility,forinstanceduringtheGlobalFinancialCrisis. Figure6: MSEbasedtime-varyingweightsincombinationforecast,h=1 23

Figure7: MSEbasedtime-varyingweightsincombinationforecast,h=8 4.4 Summaryofresults Overall, there are two main takeaways that we think are worth highlighting: First, the timevarying MSE-weighted combination method consistently and significantly improves on the benchmarkacrossdifferentsamplesforhorizonsof1quarterand8quarters(thelatterexcept forthepost-crisisfullsampleperiod,wheretheaggregationmethodisbetter). Second,forthe nowcast h=1 it should be noted that the SPF and Tealbook forecasts improve significantly overthebenchmark,sotheadditionalinformationintheinformationsetsusedinthoseforecast helpsimprovingtheforecast. 5 Remarks 1. Forecast encompassing, forecast combination and forecast accuracy tests: Having the smallest RMSE comparisons of a set of forecasts is a necessary but not sufficient condition for forecast encompassing (see Ericsson, 1992). The concept of forecast encompassinghasbeenproposedbyChongandHendry(1986)andcanbetestedbyinves- 24

tigatingwhethertheregressorofthealternativeforecastmodelcanexplaintheforecast error of a benchmark forecast model of interest. We explore forecast combination as onepossibleforecastmethod. Forecastcombinationiscloselyrelatedtotheconceptof forecast encompassing. Evidence that forecast combination of two forecasting models providessmallerRMSEthanthebenchmarkmodelimpliesthatthebenchmarkforecast doesnotencompassthealternativemodelforecast. Ourresultthatforecastcombination doesimproveoversimplebenchmarkmodelsandalsooverPhillipscurvemodelsdoes suggest that some of the alternative models contain additional predictive content. This is confirmed by our result of the Diebold and Mariano test that forecast combination significantly outperforms the simple benchmark model for horizons of one quarter and two years across all samples and most pre- and post-crisis subsamples. One extension forfurtherresearchwouldbetoapplythetestsuggestedbyHubrichandWest(2010)to comparesmallnestedmodelsetsviaadjustMSFEsrelevanttosomeofthecomparisons, thatcanbeviewedasaforecastencompassingtestforsmallnestedmodelsets. 2. OthermodelsWehaveincluded(butdonotpresent)inourforecastcomparisonarandom walk model that has often been used as a benchmark model in the literature. We alsoconsideredforecastsbasedonanAR(1)modelestimatedwitharollingestimation windowinsteadofarecursivelyexpandingestimationwindow. Wefindthattherolling windowARmodelperformsslightlybetterthanthebenchmarkandalltheothermodels for a one year horizon for the post-crisis period that includes the published Tealbook as well as the full post-crisis period, and performs better than the benchmark for most horizonsforthepre-crisisperiod. Otherthanthesefewinstances,neitherofthesemodels does outperform our benchmark inflation gap model in RMSE terms except at very fewhorizonsandinthosecasestheimprovementwasnegligibleandinanycaseclearly outperformedbyourbestforecastingmethods. 3. RMSEscomparisons: Wecomparethedifferentforecastmodelsandmethodsinterms of RMSE. As Clements and Hendry (1993) have pointed out, RMSE are not invariant to certain transformations. For example, different transformations (first differences or annual differences) might affect the RMSE ranking of the forecast models. We have 25

focused on the forecast performance for quarterly inflation, and note that the RMSE basedforecastcomparisonmightbedifferentforannualinflation. However,wechoose out-of-sampleRMSEcomparisonsbecauseparameterestimationuncertaintyandstructural breaks often imply that good in-sample fit does not translate into out-of-sample forecasting(seee.g. ClementsandHendry,1998,andGiacominiandRossi,2009). 4. SPF It should be noted that the SPF is itself an average (or a median) and so may already benefit from any aggregation effects due to differentially misspecified models ormethodsbyforecastersinthesample. 6 Conclusion Inthispaper,weperformareal-timeforecastingexercise,focusingonpriceinflationasmeasured by the personal consumption expenditures (PCE) chain-type price index that is most relevant for monetary policy decisions. We investigate whether and how additional information – additional macroeconomic variables, expert judgment, or forecast combination – can improveforecastaccuracy. Weanalyzepre-andpost-crisisperformanceofdifferentinflation forecasting models as well as judgmental forecasts from the SPF and Tealbook. We show whichforecastingmethodsaremostusefulbefore,during,andaftertheglobalfinancialcrisis, andsoaimtoshedlightonwhichmethodsaremostpromisingforconstructingandrobustifyinginflationforecasts. Ouranalysisisalsorelevantinlightofthecurrentcrisisthathasposed challengesforforecasting,giventheunprecedentednatureofthepandemic. Hence,strategies torobustifyforecasts, suchastheoneswehaveconsideredhere, arelikelytobeincreasingly important. Ourresultsprovideinterestingnewinsightsforinflationforecastingfromrecentepisodes, while some of our results confirm previous literature. Our key finding is that while simple models remain generally hard to beat, careful introduction of additional information can improve forecasts, particularly in the post-crisis period. Three types of additional information stand out as useful. First, forecast combination of different models for overall inflation arecompetitiveandrobustifyagainstbadforecasts. Second,aggregatingforecastsofinflation componentscanimproveperformancecomparedtoforecastingtheaggregatedirectly,suggest- 26

ingthattherearegainstobehadfromthecarefulspecificationofthedynamicsofdisaggregate inflationseries. Finally,thelargeinformationsetavailabletoprofessionalforecastersandthe Federal Reserve Board staff can substantially improve forecasting performance, especially at shorthorizons,suggestingthatmultivariatemodels,includingthosecapableofhandlinglarge datasets,canplayanimportantroleininflationforecasting. 27

References Atkeson, A. and Ohanian, L. E. (2001). Are phillips curves useful for forecasting inflation?, FederalReservebankofMinneapolisquarterlyreview25(1):2–11. Chong, Y. Y. and Hendry, D. F. (1986). Econometric evaluation of linear macro- economic models,ReviewofEconomicStudies53(4):671–690. Clements, M. P. and Hendry, D. F. (1993). On the limitations of comparing mean square forecasterrors,JournalofForecasting12(8):617–637. Clements, M. P. and Hendry, D. F. (1998). Forecasting Economic Time Series, Cambridge UniversityPress,Cambridge. Croushore, D., Stark, T. et al. (2019). Fifty years of the survey of professional forecasters, EconomicInsights4(4):1–11. Diebold, F. X. (2015). Comparing predictive accuracy, twenty years later: A per- sonal perspectiveontheuseandabuseofdiebold—marianotests,JournalofBusinessandEconomic Statistics33(1):1–9. Diebold,F.X.andMariano,R.S.(1995). Comparingpredictiveaccuracy,JournalofBusiness andEconomicStatistics13(3):253–263. Eichenbaum, M., Jaimovich, N., Rebelo, S. and Smith, J. (2014). How frequent are small pricechanges?,AmericanEconomicJournal: Macroeconomics6(2):137–55. Ericsson,N.R.(1992). Parameterconstancy,meansquareforecasterrors,andmeasuringforecast performance: An exposition, extensions, and illustration, Journal of Policy Modeling 14(4):465–495. Faust, J. and Wright, J. H. (2013). Forecasting inflation, Handbook of economic forecasting, Vol.2,Elsevier,pp.2–56. Giacomini,R.andRossi,B.(2009). Detectingandpredictingforecastbreakdowns,reviewof economicstudies76(2),2009. 28

Granger,C.W.(1987).Implicationsofaggregationwithcommonfactors,EconometricTheory 3(2):208–222. Hendry,D.F.(2006). Robustifyingforecastsfromequilibrium-correctionsystems,Journalof Econometrics135(1):399–426. Hendry, D. F. and Clements, M. P. (2004). Pooling of forecasts, The Econometrics Journal 7(1):1–31. Hendry, D. F. and Hubrich, K. (2011). Combining disaggregate forecasts or combining disaggregate information to forecast an aggregate, Journal of business & economic statistics 29(2):216–227. Hubrich,K.(2005). Forecastingeuroareainflation: Doesaggregatingforecastsbyhicpcomponentimproveforecastaccuracy?,InternationalJournalofForecasting21(1):119–136. Hubrich, K. and Skudelny, F. (2017). Forecast combination for euro area inflation: a cure in timesofcrisis?,JournalofForecasting36(5):515–540. Hubrich, K. and West, K. (2010). Forecast evaluation of small nested modelsets, Journal of AppliedEconometrics25:574–594. Knotek, E. S. and Zaman, S. (2017). Nowcasting US headline and core inflation, Journal of Money,CreditandBanking49(5):931–968. Lütkepohl, H. (1984). Forecasting contemporaneously aggregated vector arma processes, JournalofBusiness&EconomicStatistics2(3):201–214. McCully, C. P., Moyer, B. C. and Stewart, K. J. (2007). Comparing the consumer price index and the personal consumption expenditures price index, Survey of Current Business 87(11):26–33. Modugno, M.(2013). Now-castinginflationusinghighfrequencydata, InternationalJournal ofForecasting29(4):664–675. Powell, J. (2018). Monetary policy and risk management at a time of low inflation and low unemployment. At the "Revolution or Evolution? Reexamining Economic Paradigms" 29

60th Annual Meeting of the National Association for Business Economics, Boston, Massachusetts. Shoemaker, O. J. (2011). Variance estimates for price changes in the consumer price index, BureauofLaborStatisticsReport. Stock, J. H. and Watson, M. W. (2007). Why has us inflation become harder to forecast?, JournalofMoney,Creditandbanking39:3–33. Stock, J. H. and Watson, M. W. (2016). Core inflation and trend inflation, Review of EconomicsandStatistics98(4):770–784. Tulip, P.(2009). Hastheeconomybecomemorepredictable? changesingreenbookforecast accuracy,JournalofMoney,CreditandBanking41(6):1217–1231. West, K. D. (1996). Asymptotic inference about predictive ability, Econometrica 64: 1067– 1084. Yellen, J. L. (2015). Inflation dynamics and monetary policy, Philip Gamble Memorial Lecture,UniversityofMassachusetts,Amherst,Massachusetts. 30