Measuring the Level and Uncertainty of Trend Inflation

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. Measuring the Level and Uncertainty of Trend Inflation Elmar Mertens 2011-42 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.

Measuring the Level and Uncertainty (cid:3) of Trend Inflation y Elmar Mertens Federal Reserve Board September 28, 2011 (cid:3)I would like to thank Michiel DePooter, David Lo´pez-Salido, Edward Nelson, Seth Pruitt, Jeremy Rudd, Mark Watson,MinWeiandJonathanWrightforhelpfulconversationsaboutthisproject. Excellentresearchassistancehas beenprovidedbyBenjaminBrookinsandChrstineGarnier. yForcorrespondence:ElmarMertens,BoardofGovernorsoftheFederalReserveSystem,WashingtonD.C.20551. emailelmar.mertens@frb.gov. Tel.:+(202)4522916. Theviewsinthispaperdonotnecessarilyrepresentthe viewsoftheFederalReserveBoard,oranyotherpersonintheFederalReserveSystemortheFederalOpenMarket Committee. Anyerrorsoromissionsshouldberegardedassolelythoseoftheauthor. 1

Measuring the Level and Uncertainty of Trend Inflation Abstract Firmly-anchored inflation expectations are widely viewed as playing a central role in the successfulconductofmonetarypolicy. Thispaperpresentsestimatesoftrendinflation,based oninformationcontainedinsurveyexpectations, thetermstructureofinterestrates, andrealizedinflationrates. Myapplicationcombinesavarietyofdatasourcesatthemonthlyfrequency and it can flexibly handle missing data arising from infrequent observations and limited data availability. In order to assess whether inflation expectations are anchored, uncertainty surrounding future changes in trend inflation — measured by a time-varying volatility of trend shocks—isestimatedaswell. Not surprisingly, the estimates suggest that trend inflation in the U.S. rose and fell again over the 1970s and 1980s, accompanied by increases in uncertainty. Considering the recent crisis, full-sample estimates of trend inflation fell quite a bit, but not too dramatically. In contrast, real-time estimates recorded sizeable increases of trend uncertainty during the crisis of2007/2008,whichhaveabatedsincethen. JELClassification: C53,E37,E47,E58 Keywords: TrendCycleModel,InflationTarget,StochasticVolatility,Surveys,BayesianEconometrics 2

1. INTRODUCTION Firmly-anchored inflation expectations are widely viewed as playing a central role in the successful conduct of monetary policy.1 This paper presents estimates of trend inflation, based on a fairly broad information set, spanned by survey expectations, the term structure of interest rates, and realized inflation rates. In order to assess whether inflation expectations are anchored, I estimate not only the level of trend inflation, but also the uncertainty surrounding future changes in trendinflation. While the task of monitoring inflation expectations is important, it is typically hampered by the lack of a comprehensive summary measure of inflation expectations. For example, a variety of survey measures exist, which often differ from one another in several aspects. Some indicators measureshorttermexpectations,andsomelonger-termexpectationsoffutureinflationrates;some refer to changes in the CPI, others to the GDP or PCE deflators. In some cases, the forecast horizon may only vaguely be specified, in others the relevant price index may be left open. The series are often reported at different frequencies, and typically have different starting points. As analternativedatasource,inflationexpectationsmayalsobeextractedfromfinancialmarketdata, likethelevelsofnominalinterestrates. Whileallthesemeasuresarepotentiallyvaluable,theyare alsolikelysubjecttonoiseandmeasurementerrors,andmayattimesconveyconflictingsignals. This paper uses a time series model to condense information from a variety of data series — inflationrates,surveyresponsesaboutfutureinflation,andnominalinterestrates—intoacommon trend measure. Kalman filtering techniques and the Gibbs sampler allow to account for missing observationsarisingfromtheinfrequentpublicationorlimitedavailabilityofsomeseries. 1For example, in testimony presenting the Monetary Policy Report of March 2011, Federal Reserve Chairman Bernanke emphasized that “[s]ustained rises in the prices of oil or other commodities would represent a threat both toeconomicgrowthandtooverallpricestability, particularlyiftheyweretocauseinflationexpectationstobecome lesswellanchored. Wewillcontinuetomonitorthesedevelopmentscloselyandarepreparedtorespondasnecessary to best support the ongoing recovery in a context of price stability.” Similar views have been expressed by Federal ReserveViceChairYellen(2011),FederalReserveBankPresidentKocherlakota(2011),ECBPresidentJean-Claude Trichet (Trichet and Constancio, 2011) and Bank of England Governor Mervyn King (Feldstein et al., 2004; King, 1997) — to name but a few. Further, Woodford (2005) provides a discussion of the interplay between inflation expectations and monetary policy in the context of theoretical models, and Mishkin (2007) reviews the channels throughwhichchangingbehavioroftrendinflationmayhaveaffectedinflationdynamics. 3

Adopting the trend concept of Beveridge and Nelson (1981), trend inflation is defined as the model’s long-run forecast of PCE headline inflation. What makes this a common trend is the assumptionthattrendinflationmovesinlock-stepwiththelong-runforecastsforallothervariables — like levels of nominal yields, core inflation as well as survey responses. Equivalently, this assumption requires survey errors to be stationary, but allows them to have non-zero mean, and alsorequiresthatinflationdifferentials,say,betweenheadlineandcoreinflationarestationary,and thattermpremiaandrealyieldsarestationary. Inaddition,Iwillconsiderresultsfromanextended versionofthemodel,whichallowsfordriftinrealyields. Uncertainty in the trend of inflation expectations is measured by the volatility of trend shocks, which is allowed to vary over time as in Stock and Watson (2007) and similar to Cogley et al. (2010). When the volatility of trend shocks is low, the trend behaves like a constant and we can speakofwell-anchoredinflationexpectations. Whenthevolatilityoftrendshocksishigh,inflation expectations will likely become unmoored, and trend movements will start to become a major source of variations in actual inflation. By tracking time-variation in the uncertainty measure, the modelcandocumentwhetherandtowhatextentinflationexpectationshavebecomeunanchoredat timesinthepast,aswellasprovidinganestimateofthecurrentriskofchangesintrendinflation. The remainder of this paper is structured as follows. The next section offers a brief discussion of the Beveridge-Nelson trend with stochastic volatilitybefore turning over to a description of my empirical model in Section 3. Section 4 presents trend estimates extracted from surveys and inflation, while Section 5 adds nominal yields to the conditioning set used for constructing the trend estimates. Section 6 discusses in more detail trend estimates for the most recent years and compares real-time estimates of trend inflation with the kind of in-sample estimates shown elsewhere inthispaper. Section7presentsestimatesoftrendinflationderivedfrommodelswithsmallerconditioning sets, which permits incorporation of richer time-varying dynamics in the persistence of thedataandadditionalsourcesofstochasticvolatility. Adetailedreviewoftherelatedliteratureis given in Section 8. Section 9 concludes the paper with a brief summary and an outlook on further research. 4

2. TRENDCONCEPT Following Beveridge and Nelson (1981), this paper identifies trend inflation from long-term forecasts of inflation. This section provides a brief discussion of the trend concept, the role of stochastic volatility and the use of multivariate information in identifying the trend made in this paper. A description of the time-series model from which forecasts are generated in this paper will be deferred until Section 3. A more detailed discussion of the related literature is given in Section8. 2.1. TheBeveridge-NelsonTrendwithStochasticVolatility An important motivation for monitoring inflation expectations is to detect shifts in people’s beliefaboutaneconomy’snominalanchor(orlackthereof). TheBeveridge-Nelsontrendisparticularly suited for this task, since it is an expectation of future inflation conditional on some current information set. Formally, the Beveridge-Nelson trend ((cid:28) ) of inflation ((cid:25) ) is identified as the t t forecastofinflationattheinfinitehorizon: E (cid:25) = (cid:28) (1) t t+1 t andactualinflationisassumedtobethesumofthetrendandastationarycomponent,(cid:25)~ . Adopting t theterminologyofCogleyetal.(2010),(cid:25)~ willbecalledthe“inflationgap”. t (cid:25) = (cid:28) +(cid:25)~ (cid:25)~ (cid:24) I(0) E((cid:25)~ ) = 0 (2) t t t t t While the inflation gap has an unconditional mean equal to zero, it may have arbitrary serial correlation(withingthelimitsofstationarity).2 In this context it is important to notice that trend inflation is not a forecast of average inflation between now and some long-dated maturity, but rather the forecast of inflation at a long-dated 2Setting the unconditional mean of the gap to zero is a normalization, since the mean gap cannot be identified independentlyfromtheinitialtrendlevel. 5

point in time (namely the infinite horizon). As will be seen further below, focusing the trend on the infinite horizon turns out to be very convenient in order to derive a common trend restriction, whichholdsinalargeclassofmodelenvironments.3 As a simple example, consider first an economy with a well established and credible inflation target, where “credible” is understood such that policymakers will stabilize inflation around the target forever and that the public knows about this. In this economy the Beveridge-Nelson trend will be constant and identical to the target.4 As another example, suppose that an economy has just credibly adopted an inflation target, but that the target is different from average past inflation. In this case, the trend measure will crucially depend on the information set used to generate longterm inflation forecasts. When the information set contains knowledge about the new inflation target, the Beveridge-Nelson trend will instantaneously adjust to the new target rate — even when the transition to the new target could be expected to take a while. If forecasts were however generated by extrapolating past inflation behavior, say with an autoregressive time-series model, trend estimates should converge only over time to the new target, where the rate of convergence would depend on the weight given by the forecast to more recent inflation behavior and on the length of the adjustment period to the new target regime.5 By using an estimated time-series model, this paper will invariably resort to generating forecasts by extrapolating from the past. But by using forward-looking information variables — like surveys and financial market data — the procedure should also be capable of detecting shifts in the inflation outlook not yet captured in realizedinflationdata. Defining the trend measure as an expectation has immediate consequences for the implied dynamics of inflation itself. Differencing the trend definition in equation (1) yields a unit root 3Inaddition,Equation(3)belowshowshowtheinfinitehorizonofallowstoabstractfromroll-overissuesastime evolves,sincechangesintrendinflationmerelyreflectchangesininformation,butnotchangesintheforecast’starget date—whichalwaysremainsequaltotheinfinitehorizon. 4Such a setting would be consistent with DSGE models with credible monetary policies and a constant rate of inflation in steady state, as for example in Rotemberg and Woodford (1997), Christiano et al. (2005) or Smets and Wouters(2007). 5The relationship between the Beveridge-Nelson concept of trend inflation and theoretical models of monetary policyisfurtherdiscussedinSection8. 6

processforthetrend: (cid:28) = (cid:28) +(E (cid:0)E )(cid:25) (3) t t(cid:0)1 t t(cid:0)1 t+1 = (cid:28) +e(cid:22) t(cid:0)1 t where the trend shocks, e , form a martingale-differencesequence under the conditioning set used t togenerateexpectations;E e(cid:22) = 0. Unlesstrendshockswerealwayszero,thetrendthusfollows t(cid:0)1 t a random walk, which via (2) will be inherited by the process for actual inflation. As will be discussednext,avitalingredientinmymodelistoassumeatime-varyingvolatilityoftrendshocks, e(cid:22) (cid:24) N(0;(cid:27)(cid:22) ) (4) t t to allow for periods, when trend shocks are essentially zero and inflation is close to a stationary process,aswellastoallowforsituationswheninflationexpectationsmaybecomeunanchoredand trendshocksaresizable. Taken at face value, the notion of a random walk component in the inflation process could seemtroubling. Apartfromstatisticalconcerns,assuminganon-stationaryinflationprocesswould imply that monetary policy has failed in its task of keeping inflation rates stable. Ideally, the analysis should neither preclude the possibility of well anchored inflation expectations nor should suchrisksberuledout.6 An inflation model with a Beveridge-Nelson decomposition as in (2), will always assign some weight to a non-stationary component in inflation. But as long as the weight is very small, the inflation process could arbitrarily well be represented by a stationary process; see for example Cochrane (1991). By estimating a time-varying volatility of trend shocks — and thus a timevarying importance of the unit root component in inflation — this paper will be able to capture episodes of stable as well as unanchored inflation expectations. As will be seen below, the model 6Forinstance,KozickiandTinsley(2001),Gu¨rkaynaketal.(2010)andBeecheyetal.(2011)provideevidence— based on the term structure of interest rates — suggesting that long-term inflation expectations in the U.S. are time varyingandfarfromconstant. 7

usesthesametrendconceptforinflationasStockandWatson(2007)—aBeveridge-Nelsontrend withstochasticvolatility—whichisalsoapproximatedbythemodelCogleyetal.(2010). Butthe modeldiffersinitsuseofamultivariatedataset,combininginformationfromsurveysandtheterm structureofinterestratesinestimatingthetrend. 2.2. CointegrationandtheMultivariateBeveridge-NelsonTrend Akeyaspectofthispaperistheuseofmultivariateinformation,containedinvariousmeasures ofrealizedinflation,surveyresponsesaswellasnominalinterestrates,toidentifychangesintrend inflation. For concreteness, the analysis will always be concerned with identifying the trend in headline PCE inflation.7 As will be discussed next, a tight link will be imposed amongst the trend levels of different inflation measures, survey responses and nominal interest rates, requiring that allvariablesaresubjecttothesametrendshocks. Thedatasetusedinthispapercanbroadlybeclassifiedinthreegroupsofvariables: 1. Realizedinflationrates,likePCEheadlineandcoreinflationandCPIinflation. 2. Surveyexpectationsoffutureinflation,liketheLivingstonSurvey’sCPIforecastforthenext yeartheMichiganSurvey’sexpectedpricechangeoverthenext5-to-10years.8 3. Nominalinterestrates,liketheyieldsonnominalTreasurysecuritiesatdifferentmaturities. AllvariablesarealsolistedinTable1.9 7Amongst others, the Monetary Policy Report to the Congress from the Board of Governors of the Federal ReserveSystemdescribestheBoard’soutlookforinflationintermsofthePCE,sinceitsconstructionbetterreflectsthe changing composition of spending than other measures, like the CPI. McCully et al. (2007) also provide a detailed comparisonofthePCEandCPIpriceindices. 8TheMichigansurveydoesaskrespondentstorefertoaspecificpricebasketliketheCPIorthePCEdeflator. 9Inprinciple,itisalsostraightforwardtoincludeTIPS-basedmeasuresofinflationcompensation,alsoknownas thebreak-eveninflationrateofaTIPSsecurity,inthisframework. However,sincedataoninflationcompensationis available only for roughly ten years, and at least initially, but also during the recent crisis, the underlying securities priceswereheavilyaffectedbyliquiditypremia,IhavechosennottoincludeTIPS.Resultsnotshownheresuggestthat theseissuesseemtohaveinducednear-permanenteffectsonTIPSdata,whichdistortthecommontrendextractionin significantways. AllowingforaseparateyieldtrendasdiscussedinSection5.2seemstoprunetheseeffectshowever quitewell,andtheinclusionofTIPSyieldsresultssimilartowhatisshownthere. 8

Table1: DataDescriptionandAvailability Variable Since Frequency InflationRates PCEDeflator 02/1959 Monthly CorePCEDeflator 02/1959 Monthly ConsumerPriceIndex 02/1947 Monthly GDPDeflator Q1/1947 Quarterly SurveyExpectationsofInflation BlueChip,CPI4-quarterahead 06/1980 Monthly BlueChip,CPIFive-to-tenyear 03/1987 MarchandOctober Livingstonsurvey,CPInext12month 12/1946 JuneandDecember MichiganSurvey(cid:3) 1year 01/1978 Monthly MichiganSurvey(cid:3) 5-to-10year 02/1975 Monthly(since1990)(cid:3)(cid:3) SPF,CPI4-quartersahead 08/1981 February,May,August andNovember SPF,CPInext10years 11/1991 February,May,August andNovember NominalInterestRates 10-yearTreasuryYield 04/1953 Monthly (average) 30-yearTreasuryYield 02/1977 Monthly(cid:3)(cid:3)(cid:3) (average) Nine-to-tenyearforwardrate 08/1971 Monthly (firstdayofmonth) Note: ThemodelusesmonthlyobservationsfromJanuary1960throughAugust2011ofalldatareceivedbytheend ofAugust2011. SPFdenotestheSurveyofProfessionalForecasters. (cid:3) TheMichigansurveydoesnotspecificallyrefertoanyspecificpricestatisticorconsumptionbasket. (cid:3)(cid:3) FromFebruary1975andApril1990,theMichigan5-to-10-yearsurveywasconductedonlysporadically. (cid:3)(cid:3)(cid:3) FromMarch2001toJanuary2006,dataisunavailableforthe30-yearTreasuryYield. 9

Figure1: InflationandotherIndicatorSeries (a)PCEInflation: HeadlineandCore (b)10-yearTreasuryYield 15 12 16 Headline (y−o−y) 10−year Treasury yield (right) Headline (monthly) Headline PCE (left) Core (y−o−y) 10 14 10 8 12 5 6 10 0 4 8 −5 2 6 −10 0 4 −1159 60 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010 −11299 6600 1965 11997700 1975 11998800 1985 11999900 1995 22000000 2005 220011002 (c)LivingstonSurvey (d)MichiganSurvey(1year) 12 12 Livingston Survey Michigan Survey (1−year) Headline PCE Headline PCE 10 10 8 8 6 6 4 4 2 2 0 0 −129 60 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010 −129 60 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010 Note: Monthly data, where available, since 1960, expressed in annualized percentage points. Unless stated otherwise,inflationratesareshownastwelve-monthtrailingmovingaverages. (Themodel’sinformationsetusesmonthly inflationrates—asshowninPanel(a)forheadlinePCEinflation—whichspanthesemovingaverages.) Someindicatorsmeasureshorttermexpectations,andsomelongertermexpectationsoffuture inflation rates. Most surveys refer explicitly to changes in the CPI, while the price index relevant to bond investors cannot be observed. A basic tenet of this paper is the assumption that differencesbetweenthe variousindicatorsshould notbe expectedtolast forever. Avisual inspectionof Figure 1, which shows time series data for a few variables from this paper’s data set in comparison with realized PCE inflation, suggests indeed some commonality in low frequency movements of inflation, survey and interest rates. However, it should also be noted that a closer inspection reveals that deviations between interest rates and inflation can be very persistent, and possibly non-stationary. Giventhelowpowerofformaltestsforstationarityandcointegration,andastrong 10

prior in favor of the cointegrating assumptions imposed by a vast body of theoretical models, I have chosen to impose the common trend assumption outlined below, trading off ease of interpretationagainststatisticalflexibility. Afurtherimpedimenttotheapplicationofformalcointegration tests is the regular occurrence of missing data values throughout the data set used here. There is however no problem of missing observations for PCE inflation and the 10-year Treasury yield – showninPanel(b)ofFigure1—andaugmentedDickey-FullerTestssoundlyrejectaunitrootin thedifferencebetweenthe10-yearTreasuryyieldandheadlinePCEinflation.10 Formally, I assume that a) differences between different inflation measures are stationary, b) survey errors are stationary, and c) real interest rates and risk premiums are stationary. (All variables are expressed in logs.) As a consequence of the Beveridge-Nelson decomposition (2), forecasterrorsofinflationatanyhorizonarestationary. Togetherwithassumptionsa)andb)itfollows that differences between current headline PCE inflation, (cid:25) , and survey responses are stationary t — irregardless of the survey’s inflation measure and forecast horizon. Based on the Fisher equation, nominal interest rates are the sum of a real rate, expected inflation and a residual, which I will refer to as a risk premium. Assumptions a) and c) ensure then that the difference between (cid:25) t and the current nominal yield is stationary. Closer inspection of Panel (b) in Figure 1’s suggests that deviations between nominal yields and inflation may at times be near permanent, contrary to assumption c), and Section 5 investigates the consequences of allowing for drift in the real rate of interestforestimatesoftrendinflation.11 3. EMPIRICALMODEL This section lays out my basic time-series model. The model assumes that there is a common trend in inflation rates, surveys and nominal yields. Extensions will be considered in Section 5 10UsingdatafromJanuary1960untilJune2011,augmentedDickey-Fullertests(withandwithoutintercept)reject thenullhypothesisofaunitrootinthedifferencebetweenheadlinePCEinflationandthe10-yearTreasuryyieldwith p-valuesbelow1%. 11Inparticularoverthelasttenyears,deviationsbetweenthenominalyieldsofshorterdatedTreasurysecurities— like 5-year Treasuries — have deviated quite substantially from inflation data and yields on longer-dated securities, whichiswhyonlylongerdatedTreasuryyieldsandthenine-to-tenyearforwardrateisincludedinthedataset. 11

(separate trend shocks affecting real rates) and Section 7 (time-varying gap dynamics). Initially, the presentation will assume that the entire data set can be observed without missing values. The handling of missing data will be described at the end of this section. Let Y denote a vector t containing inflation rates, nominal yields and survey expectations.12 Each variable is supposed to beintegratedoforderoneandamultivariateBeveridge-Nelsondecompositionholds: Y = (cid:28) +Y~ lim E Y = (cid:28) (5) t t t t t+k t k!1 withstationary“gaps”,whichareunconditionallymeanzero;formallyY~ (cid:24) I(0)andE(Y~ ) = 0. t t ReflectingthediscussionofSection2,itwillbeassumedthatacommontrendshockisdriving eachvariables’trendlevel: (cid:28) = (cid:28) +1e(cid:22) e(cid:22) = (cid:27)(cid:22) "(cid:22) "(cid:22) (cid:24) N(0;1) (6) t t(cid:0)1 t t t t t The trend levels of individual variables may differ only in their initial values (cid:28) . Changes in the 0 trendsofeachvariableareidentical.13 Theinitialtrendlevelscandiffer,forexampleinreflectionof averagerealyields,averagetermpremiumsorbiasesinsurveyexpectations,whichareallassumed tobestationary,butnotnecessarilymeanzero. As discussed in Section 2, the model accounts for time-variation in the importance of trend movements with stochastic volatility in the trend process. As in Stock and Watson (2007) it is assumedthatthelog-varianceoftrendshocksfollowsadriftlessrandomwalk. log(cid:27)(cid:22)2 (cid:17) h h = h +(cid:27) (cid:24) (cid:24) (cid:24) N(0;1) (7) t t t t(cid:0)1 h t t 12Throughoutthispaper,vectorvariableswillbedenotedwithboldfaceletters,whilescalarsareprintedinstandard font. 13An alternative representation of the common trend assumption would be to write the multivariate Beveridge- Nelsondecompositionintermsofascalartrend,Y = 1(cid:28) +Y~ andtoallowfornon-zeromeansinthegapsinstead t t t ofvariable-specificinitialtrendlevels. WhenitcomestoestimatingthemodelwithaGibbssamplerasdiscussedin AppendixA,therepresentationaboveturnsouttobemoreefficient,sinceitallowstorecovertheinitialtrendlevels jointlywiththemodel’slatenttrendprocess. 12

Thegapsarestationary,butnotnecessarilyiid. Inordertohandlealargecross-sectionofdata withmissingdata,thedynamicsofthegapsarerequiredtofollowa time-invariant VAR. A(L)Y~ = e~ (8) t t wheretherootsofthelag-polynomialA(L)arerestrictedtobeoutsidetheunitcircle. Inthespirit ofmodernbusinesscycletheory,themodelallowsforcorrelationbetweentrendandgaps:14 e~ = (cid:12)"(cid:22) +"~ "~ (cid:24) N(0;(cid:6)~) (9) t t t t Pleasenotethat,thiscorrelationisconstantandpertainstothestandardizedtrendshocks,"(cid:22) instead t of e(cid:22). Allowing the gap shocks to load onto e(cid:22) would induce stochastic volatility in the gaps, but t t onlyviathetrendshockswhichislikelytoorestrictive.15 Choosing to model gap dynamics as being time-invariant enables the model to handle a larger set of variables with partly missing data, than what would be possible when gap dynamics were time-varyingasinCogleyetal.(2010). Forcomparison,Section7willconsidermodelsestimated from smaller data sets, with drifting coefficients in the gap VAR and stochastic volatility in gap innovations. Before describing the handling of missing data, it will be useful to summarize the model in its state space form. For a given realization of the stochastic volatility process, the state vector of the model consists of the trend vector (cid:28) and the gaps Y~ as well as any lagged gaps needed for t t the companion form of (8). For the sake of exposition, it will be assumed below that the VAR in equation (8) has one lag, such that A(L) = I (cid:0) AL. Denoting the state vector by X the state t 14Thesecorrelationsallowfortransitoryresponsesinthegapstotrendshocks. Forexample,iftrendshockswere interpreted as exogenous shocks to the inflation target as in Ireland (2007), these gap responses would reflect the adjustmentsinstickypricesandwagestoanewtrendlevel. 15Estimatesbasedonsuchaspecificationwoulddisplayapoorabilitytodistinguishshockswhicharepermanent fromshockswhichareheteroscedasticbutshort-lived. Forexample,afewinflationmeasurescontainlargebuttransitoryspikesaround9/11/11duetodistortionsinnon-marketbasedpricecomponents,whichcouldthenerroneouslybe attributedtothetrend. 13

spacesystemcanthenbewrittenas 2 3 6(cid:28) 7 t X = 4 5 (10) t Y~ t 2 3 2 3 6I 07 61(cid:27)(cid:22) 0 7 t = 4 5X +4 5w (11) t(cid:0)1 t 0 A (cid:12) (cid:6)~1=2 Y = CX (12) t t where C = [I I], w (cid:24) N(0;I), and (cid:6)~1=2 denotes an arbitrary factorization of the variance t covariancematrixofthegapresiduals. So far, it has been assumed that observations of Y are regularly available. In the case of t missing data, the state space system (10) and (12) can be modified as follows. Denote the actual dataasZ ,withtypicalelementZ —similarly,individualelementsofY willbedenotedY — t t;i t t;i andencodemissingobservationsas 8 > > < Y ifavailable t;i Z = (13) t;i > > : 0 otherwise andreplacetheobserverequation(12)with Z = C X (14) t t t where C is a deterministically varying measurement matrix. If data on Y are available, the ith t t;i row of C is identical to the ith row of C and zero otherwise. The only variable with missing obt servations,whichistreateddifferentlythandescribedaboveisthequarterlyinflationseriesderived from the GDP deflator. As explained in Appendix B, each quarterly observation of GDP inflation ismodeledasthethree-monthmovingaverageofanunobservablemonthlyinflationrate.16 16Intermsifinformationalcontent,missingdataforGDPinflationcouldalsobemodeledasin(14),whichwould 14

The model is estimated with a Gibbs sampling algorithm described in Appendix A. The algorithmyieldsnotonlyestimatesofthelatentfactors(cid:28) and(cid:27)(cid:22) butalsofortheparametersofthegap t t VAR(8). Inaddition,theGibbssamplerrecoverstheposteriordistributionofmissingdataentries, conditionalonthemodelandallobserveddatavalues. Examplesofestimatesfortheposteriordistribution of missing data values are shown Appendix B. The only fixed parameter is the volatility p ofshockstothelog-variancesin(7),(cid:27) ,whichhasbeensetequalto0:2= 3,correspondingtothe h valueof0:2usedbyStockandWatson(2007)intheirquarterlymodel.17 4. THECOMMONTRENDININFLATIONANDSURVEYS Thissectionpresentsestimatesaboutthelevelanduncertaintyoftrendinflation,extractedfrom the survey expectations and realized inflation rates listed in Table 1.18. The model described in Section 3 is estimated from monthly data since 1960, covering several complete cyclical episodes anddifferentregimesfortheconductofmonetarypolicy. Data on inflation and nominal yields are taken from the FRED database, maintained by the FederalReserveBankof St. Louis, forwardrate data fromthe Federal ReserveBoard’swebsite,19 and survey responses were obtained from the various survey providers. All variables have been transformed into annualized percentage rates using continuous compounding.20 If available, all monthly observations since January 1960 are used and a detailed list of all variables as well as theiravailabilityisgiveninTable1. As shown in Figure 2, trend estimates based on surveys and inflation broadly track the “Great Inflation”ofthe1970sandthesubsequentdisinflationunderFederalReserveChairmanPalVolcker during the first half of the 1980s. Starting at about 1% in the early 1960s, the trend measure rises however induce a slightly different pattern for the persistence of the latent monthly gap series implied by such a representation. 17Estimatingthevaluefromthedatayieldssimilartrendestimates,butconsiderablymorevolatileestimatesofthe log-variancesh . t 18Table2below,referstothissetofvariablesalsoas“SURV.” 19http://www.federalreserve.gov/econresdata/researchdata/feds200628 1.html 20For example, using monthly observations of the PCE deflator p , PCE inflation is computed as (cid:25) = 1200 (cid:1) t t (logp t (cid:0)logp t(cid:0)1 ),andtheannualizedpercentageyieldonanominalTreasurysecurity,I t ,istransformedintoi t = 100(cid:1)log(1+I =100). t 15

Figure2: InflationTrendbasedonSurveysandInflationRates(“SURV”) LEVEL 10 8 6 4 2 0 1960 1970 1980 1990 2000 2010 UNCERTAINTY 0.15 0.1 0.05 0 1960 1970 1980 1990 2000 2010 Note: Thetoppanelshowsthesmoothedestimatesofthetrendandthebottompanelshowsuncertaintyabouttrend shocks. The estimates combine information from the surveys and inflation rate variables listed in Table 1. Reddashedlinesshow90%confidenceintervalsbasedonthemodel’sposteriordistributionconditionalonalldata. NBER recessiondatesareshaded. to a peak of about 8% percent in late 1980 from which it gradually descends until it reaches about 21/ 4 %percentby2000. Very strikingly — though not surprisingly — large changes in trend inflation tend to be accompanied by increases in trend uncertainty, notably around 1974 and the late 1970s and early 1980s. Thosewerenotonlytimeswhentrendinflationwasunacceptablyhigh,butalsowhentrend inflation has become unanchored. Interestingly, between 1974 and 1977, the measure of trend uncertainty decreased quite a bit, while the level estimate remained fairly stable at around 51/ 2 %. Taken at face value, this result suggests that the unmooring of inflation expectations that had in- 16

tensified around 1974 had temporarily abated — albeit at a quite elevated level of trend inflation. However, it should be emphasized that the estimated volatilities will by design mostly mirror the sizeofchangesintheestimated levels. Hence,theestimationisnotlikelytopickuppureinflation scares, i.e. periods in which an increase in trend uncertainty did not lead to an eventual change in the trend. While such events are possible under the model’s data generating process, the kind of levels data used here cannot be very informative about their occurrence, at least not when using the model to construct retrospective estimates of uncertainty — why estimate a high shock volatility for a date in the past when nothing seems to have happened after all? This consideration notwithstanding, stochastic volatility matters when estimating the model, since it helps to account for actual changes in inflation persistence, rather than its ability to detect mere changes in uncertaintywhichdidnotaffectthelevel. AswillbeseeninSection6,inflationscares,likein1994and someoftheexamplesdiscussedbyGoodfriend(1993),doaffecttheestimatesinreal-time. The volatility of trend shocks — labeled “uncertainty” in the lower panel of Figure 2 — is measuredbythestandarddeviationofamonthlytrendshock,wherethetrenditselfisexpressedin unitsofannualizedinflationrates. During“normal”times,likethe1960sortheGreatModeration period(1980sto2007),theestimatesofuncertaintytypicallystandatvaluescloseto4basispoints. Amonthlystandarddeviationofthissizecumulatestoastandarddeviationofabout15and40basis pointsoverperiodsofoneandtenyearsrespectively-providedthattheuncertaintyoftrendshocks remainsat4basispointspermonth.21 Turningtothemorerecentyears,thetrendestimateshavehoveredjustabove2percentbetween 2000 and the onset of the recent crisis, accompanied by historically low values of uncertainty. (Figure6belowdepictsthetrendestimatesforthelastdecadeinanenlargedpicture.) Thefinancial crisishasleftaclearimprintonestimatesoftrendanduncertaintywhichwillbediscussedfurther inSection6. Figure 3 displays the SURV estimates of trend inflation alongside some select indicator variables,whicharepartoftheunderlyingconditioningset. Asexpected,theestimatedinflationtrend 21As the trend is modeled as a random walk, the standard deviation of cumulated changes grows with the square rootoftime. 17

Figure3: DataandtheSURVTrend (a)Livingston (b)Michigan1-year-ahead 12 12 10 10 8 8 6 6 4 4 2 2 1 0 960 1970 1980 1990 2000 2010 1 0 960 1970 1980 1990 2000 2010 (c)SPF4-quarterahead (d)PCEInflation 12 12 10 10 8 8 6 6 4 4 2 2 1 0 960 1970 1980 1990 2000 2010 1 0 960 1970 1980 1990 2000 2010 Note:GreyshadedbandsandthesolidbluelinedisplaytheposteriordistributionofSURVtrendestimates,alsoshown inFigure2. broadlytrackslow-frequencymovementsofinflationratesandsurveyresponses. Interestingly,the trend measure follows more closely long-term survey expectations — like the Livingston survey shown in Panel (a) of the figure — than some short-term expectations. In particular, the one-year MichigansurveyshowninPanel(b)isknowntobeaverynoisymeasureofinflationinthemedium term,seevanderKlaauwetal.(2008)andArmantieretal.(2011),andthemodel’strendestimate do not take much signal from these survey responses. In contrast, the year-ahead expectations from the SPF appear to be tracked quite closely by the trend estimates. Interestingly, the SURV estimatesinterprettheburstsinrealizedinflationratesaround1974onlyaspartofagradualrisen 18

trendinflation,withtheSURVtrendpeakingin1980butnot1974/75. AsreportedinAppendixC, a model conditioned on inflation rates attribute the inflation burst of 1974 much more strongly to a rise in the trend. The more gradual increase in the SURV trend is shaped by the responses to the Livingston survey — the only survey for which observations are available for this period — which rose only briefly above 6% during this period while annual inflation rates reached double digitlevels. 5. TRENDINFLATIONANDNOMINALYIELDS 5.1. CommonTrend The previous section presented estimates of trend inflation which were conditioned on survey responses and realized inflation rates. This section compares these results with model estimates conditioned on nominal yields and inflation rates. These conditioning sets will be referred to as “SURV” and “YLD” respectively. Table 2 in the appendix, describes the subsets of the data used throughout this paper to study the sensitivity of the trend measure to different conditioning sets. Ineachcase,realizedinflationratesareincludedintheinformationsetaswell,suchthatthetrend level can be aligned with the model’s long-term forecast of headline inflation in the PCE deflator. A detailed list of the variables used in each information set is given in Table 2. (Estimates based onallvariables,whichareclosetotheYLDestimates,arereportedinAppendix C.) Considering the rise and fall of inflation in the 1970s and 1980s, the extent to which interest rates and inflation diverged over this period is notable, as can be seen from Panel (b) of Figure 1. TheFisherrelationwouldsuggestthatbothmeasuresshouldstronglyco-move,atleastatintermediate horizons. In practice, inflation rates peaked twice, around 1975 and 1980, and came down fairly swiftly during the early 1980s, whereas interest rates rose only belatedly during the 1970s and stayed elevated at persistently higher levels well into the early 1990s, with the behavior of surveyexpectationsroughlyfallinginbetweenthesetwopatterns. Figure4comparestrendestimatesbasedontheYLDdatasetwiththeSURVtrendandthe10yearTreasuryyield. AscanbeseeninPanel(a)ofthefigure,trendestimatesYLDandSURVpeak 19

Table2: ModelInformationSets InformationSet Variables ALL allvariableslistedinTable1 SURV allsurveysandinflationrateslistedinTable1 YLD allnominalyieldsandinflationrateslistedinTable1 INF allinflationrateslistedinTable1 SMALL PCEDeflator(vintagedata) ConsumerPriceIndex(non-seasonallyadjustedvintages) Nine-to-ten-yearforwardrate Livingstonsurvey TVP1 PCEDeflator CorePCEDeflator ConsumerPriceIndex(seasonallyadjusted) Livingstonsurvey TVP2 PCEDeflator Livingstonsurvey 10-yearTreasuryYield Note: IndividualvariablesandtheavailabilityofeachseriesisdescribedinTable1. in late 1980, at around 8%, and gradually decline until they reach about 21/ 4 % by 2000. However, the persistently elevated level of long-term interest rates during the early 1980s is reflected in a somewhat slower decline in the YLD trend during the 1980s. These differences between the two trend estimates are consistent with the prolonged skepticisms of financial markets concerning the durabilityofthedisinflationeffortsoftheFederalReserveintheearly1980s,whichledGoodfriend andKing(2005)torefertothisepisodeasthe“incredibleVolckerdisinflation.” Notably,theYLD trend records also a marked peak in 1974/75, coinciding with the burst in inflation rates discussed at the end of the previous section. Estimates of trend and uncertainty based on the combined data of SURV and YLD, that is all variables listed in Table 1, are very similar to the YLD estimates (showninAppendix C). 20

Figure4: Trendestimates“YLD”basedonNominalYieldsandInflationRates (a)YLDvsSURV (b)YLDand10-yearTreasuryyield 12 16 14 10 12 8 10 6 8 4 6 2 4 0 2 1960 1970 1980 1990 2000 2010 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010 Note: Thetoppanelshowsthesmoothedestimatesofthetrendandthebottompanelshowsuncertaintyabouttrend shocks. The estimates are derived from the nominal yields and inflation rates, denoted “YLD” in Table 1. Reddashedlinesshow90%confidenceintervalsbasedonthemodel’sposteriordistributionconditionalonalldata. NBER recessiondatesareshaded. 5.2. ASeparateYieldTrend The common-trend model for YLD depends on the assumed stationarity of real rates and risk premia; an assumption typically embodied in modern macroeconomic models, for example Smets and Wouters (2007) or Edge et al. (2008). However, it is open to question whether deviations between nominal yields, like the 10-year yield shown in Panel (b) of Figure 4, and the YLD trend aremerelypersistent(butstationary),orwhetheritmightnotbestatisticallymoreaccuratetoallow fordeviationsbetweentrendinflationandyieldtrends. This section considers an extended model, where long-term forecasts of yields are driven not only by changes in trend inflation, but also by a second shock, common only to yields. Nominal yields can be decomposed into the sum of expected inflation, real yield and a risk premium.22 The additional trend component could thus be related to shifts in real rates.23 In many macro- 22Thisdecompositioniswithoutlossofgeneralityaslongasthelabel“riskpremium”isunderstoodlooselyasthe residualbetweenthenominalyieldandtheyieldimpliedbytheFisherequation. 23Inprinciple,shiftsintermpremiacouldalsoaccountforthisadditionaltrendcomponent. 21

economic models, long-term forecasts of the real yield depends on expected productivity growth. It is beyond the scope of this paper to assess the extent to which the estimates of (cid:28)r are related to t changes in perceived productivity growth. For the present purpose, it shall merely be stressed that (cid:28)r isameasureofdeviationsbetweenlong-termforecastsofinflationandnominalyields,andthus t ameasureofthegoodnessoffitofthehithertoassumedcointegrationbetweeninflationandyields. Based on the Fisher equation and the maintained assumption of stationary risk premia, the modelassumesthefollowingdecompositionofthetrendinnominalinterestrates,denotedi t E i = (cid:28)(cid:25) +(cid:28)r (15) t t+1 t t where(cid:28)(cid:25) istrendinflation —asdefined above— and(cid:28)r isthe long-termforecast forthereal rate t t ofinterest,whichfollowsarandomwalk,independentofshockstotrendinflation. Asdiscussedin Section 2, the non-stationarity implied by the random walk is mitigated by specifying a stochastic volatilityprocessforthetrendshocks. (cid:28)r = (cid:28)r +e(cid:22)r e(cid:22)r (cid:24) N(0;((cid:27)(cid:22)r)2) E(e(cid:22)r;(cid:28)(cid:25) ) = 0 8k (16) t t(cid:0)1 t t t t t+k The model allows for stochastic volatility in both trends, while maintaining the assumption of time-invariantVARdynamicsforthegapsasin(8),allowingforarbitrarycorrelationbetweengap shocksand(standardized)trendshocks. Panels (a) and (b) of Figure 5 depict the estimated trend components, (cid:28)(cid:25) and (cid:28)r, as well as t t estimatesoftheirunderlyingvolatilitiesfromthisextendedmodel. Theestimatesarederivedfrom nominal yields and inflation rates; the variable set denoted “YLD” in Table 2. The initial level of theseparateyieldtrendhasbeennormalizedtozero, (cid:28)r = 0. 0 Byandlarge,theestimatedinflationtrendshowninPanel(a)ofthefigureisverysimilartothe estimatesfromthecommon-trendmodel,discussedinSection5.1. Notably,thevolatilityestimates shown in the lower half of Panel (a) display the protracted hump shape, extending over most of the 1970s and the early 1980s, which has also been found by Stock and Watson (2007), while 22

the single-trend model discussed above documented more two distinct peaks during this period. The trend estimates from the two-trend model are slightly more volatile, with the “twin peaks” of 1974-75and1980beingabitmorepronouncedthaninFigure4. Thetrendestimatesfortherecent crisisarediscussedinSection 6. Not surprisingly, given the choice of fitting separate trend shocks onto the yield process, the modelchoosestodoso. Accordingto(15),deviationsbetweentheinflationtrendandthecommon trend in yields are measured by (cid:28)r. As can be seen in Panel (b), these deviations are significantly t negativeformostofthe1970s,andsignificantlypositiveduringtheVolckerdisinflationintheearly 1980s. Loosely speaking, these estimates reflect the well known pattern of nominal interest rates having been “too low” relative to inflation, during the 1970s and having been “very high” during the 1980s; see, for example, Taylor (1999). According to the model estimates, these deviations from trend inflation have been so persistent, that the model prefers to interpret them as permanent effectsdriving(cid:28)r. (AswillbeseeninSection7,similartrendestimatesareobtainedwhenallowing t fortime-varyinggapdynamics.) AsdiscussedinSection2,mydatasetdisregardsTIPS-basedmeasuresofinflationcompensation, since historical data is only available for about ten years, and at least during the initial years, the TIPS market was fraught with illiquid trading. For the model discussed in this section, which allows for an additional, common trend component in yields, estimates including TIPS data (not reportedhere)arehoweververyclosetowhatisshowninFigure5. 23

sdleiYnitnenopmoCdnerTlanoitiddAnadnanoitaflnIdnerT :5erugiF r(cid:28)skcohSdleiYtnenamreP)b( (cid:25)(cid:28)dnerTnoitaflnI)a( t t LEVEL LEVEL 6 01 4 8 2 0 6 2− 4 4− 2 6− 8− 0 0102 0002 0991 0891 0791 0691 0102 0002 0991 0891 0791 0691 YTNIATRECNU YTNIATRECNU 5.1 1 9.0 52.1 8.0 1 7.0 6.0 57.0 5.0 4.0 5.0 3.0 52.0 2.0 1.0 0102 0002 0991 0891 0791 069 0 1 0102 0002 0991 0891 0791 069 0 1 dna sdleiy lanimon no desab ,2.5 noitceS ni debircsed ledom dnert-owt eht ni ,(cid:25)(cid:28) ,dnert noitaflni eht fo ytniatrecnu dna level detamitse swohs )a( lenaP :etoN t nileveldnertlaitiniehT.r(cid:28),sdleiyfotnenopmocdnertlanoitiddaehtniytniatrecnudnaleveldetamitseswohs)b(lenaP.)2elbaTni”DLY“detoned(setarnoitaflni t REBN .atadllanolanoitidnocnoitubirtsidroiretsops’ledomehtnodesabslavretniecnedfinoc%09wohssenildehsad-deR .orezotdezilamronneebsah)b(lenaP .dedahserasetadnoissecer 24

6. TRENDESTIMATESINREAL-TIMEANDTHERECENTCRISIS ThissectionreviewstheestimatedtrendsderivedfromSURVdata(Section4)aswellasYLD (Section5)fortherecentyears. Afterdiscussingthefullsampleestimates—alreadyshowninthe previoussections—thesectionpresentsreal-timesimulations. 6.1. Full-sampleEstimatesfortheLastDecade Panel (a) of Figure 6 compares estimates SURV, YLD and ALL for the last decade. The estimates are based on available data since 1960 and thus identical to what has been shown in the previous two sections. Estimates based on SURV and ALL are fairly similar, and as before, the discussion will mostly focus on SURV and YLD. The financial crisis has left a clear imprint on the various estimates of trend and uncertainty. Coincident with concerns about rising commodity prices during the first half of 2008, the trend estimates from SURV and YLD initially have edged higherbefore dropping by about 1/ 4 and3/ 4 percentagepoints during the second half of 2008, when thecrisisbecamemoresevere. Consideringthehistoricalestimatessince1960,showninPanel(a) ofFigure4,suchsteepchangeshaveoccurredbefore,buttheyarenotthenormeither. Since2009, thetrendestimatesfromSURVandYLDhavestayedatvaluescloseto2%,abitlowerthanbefore thecrisis. Incidentally,FederalReserveChairmanBernankehasrecentlycharacterisedtheFederal Reserve’s “mandate consistent” inflation rate to be “2 percent or a bit less” (Board of Governors oftheFederalReserveSystem,2011).24 Over the course of 2008, there is also a noticeable, though not dramatic, uptick in the uncertainty measure, in particular for YLD; shown in Panel (c) of the figure. Naturally, the model estimates see the highest volatility of trend shocks during the second half of 2008, when — accordingtothelevelestimates—thelargestshockstothetrendoccurred. Quantitatively,theuptick in uncertainty is modest. As can be seen from Figure 2 for the SURV estimates, it registers on a scale below the increase seen after the first oil shock in 1973/74, and with much less persistence. 24Chairman Bernanke made this remark in the context of the release of the summary of economic projections by theFOMC,inwhichtheprojectionsforPCEinflationshowedacentraltendencybetween1.7and2.0percent. 25

Figure6: RecentInflationTrends (a)Level(SingleInflationTrend) (b)Level(Modelw/SeparateYieldTrend) 3.5 3.5 SURV YLD w/Separate Yield Trend YLD ALL w/Separate Yield Trend ALL 3 3 2.5 2.5 2 2 1.5 1.5 1 1 0.5 0.5 0 0 −0.2250 00 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 −0.2250 00 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 (c)Uncertainty(SingleInflationTrend) (d)Uncertainty(w/SeparateYieldTrend) 0.1 0.6 0.3 SURV YLD w/Separate Yield Trend (left scale) YLD ALL w/Separate Yield Trend (right scale) ALL 0.29 0.28 0.27 0.5 0.26 0.05 0.25 0.24 0.4 0.23 0.22 0.21 2 0 0 00 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 022.300 0000 22000011 22000022 22000033 22000044 22000055 22000066 22000077 22000088 22000099 22001100 22001111 0.2 Note: Panels (a) and (b) depict estimates of the inflation trend (cid:28)(cid:25) and Panels (c) and (d) show the corresponding t measuresoftrendshockvolatility. Allestimatesarederivedfromavailabledatasince1960. Theconditioningsetsare describedinTable2. The uncertainty in the YLD measure rose quite a bit more, but remained below its estimates for the1970sand1980s(showninAppendixC). More importantly, any such risks of unanchored inflation expectations seem to have abated in the period since, as the volatility measures has come down over the course of 2009 and stands againatthehistoricallylowlevelofabout4basispoints. Panel (b) of Figure 6 displays estimates of level and uncertainty of trend inflation estimated from YLD and ALL, while allowing for a separate yield trend, as described in Section 5.2. Both measures are more volatile than those derived from the model with a single, common trend in in- 26

flation and yields; and both dropped more vigorously in late 2008 than the single-trend estimates shown in Panel (a). However there is considerable uncertainty attending these estimates, in particular in the case of the YLD two-trend model whose estimates appear excessively volatile. Still, theposteriordistributionsofbothestimates(notshowninthefigure)placemorethan90%oftheir mass on trend values below 2%, but not significantly below 11/ 2 %, in late 2008. Both measures haverecoveredsincethen,atabout2%,slightlybelowtheirpre-crisisaverage. 6.2. Real-timeEstimates So far, this paper has presented full-sample estimates of level and uncertainty in the inflation trend based on data available through August 2011. Borrowing terminology from Kalman filtering, these results will be referred to as “smoothed” estimates. Smoothed trend measure for, say, December 2008 reflect all observations received through the end of the data sample in August 2011. Evenwhenabstractingfromdatarevisions,thehindsightknowledgecontainedinsmoothed estimatesmaybesubstantial. Toassesstheissue,thissectionshowsestimatesbasedonareal-time simulation,wherethemodelhasbeenre-estimatedforeachmonthoverthelastdecade. Sincethecomputationalcostsinvolvedinre-estimatingthemodelforeachmontharenotnegligible, the real-time simulations have been limited to a small subset of variables, which captures the salient features of the trend estimates seen so far. This “SMALL” conditioning set comprises twoinflationrates(PCEandCPI),theLivingstonsurveyandthenine-to-tenyearforwardrate(see also Table 2). In order to avoid hindsight bias from data revisions, the model uses vintage data for the PCE deflator and the non-seasonally adjusted CPI. All estimates use available data since 1960 andthefirstreal-timeestimationissimulatedforJanuary1970.25 As shown in Figure 7, there are some marked differences between real-time and smoothed estimatesoftrendlevelanduncertainty. Bothestimatesseetrendinflationhoveringwellanchored 25Between1970andDecember1979,thevintagedataprovidesonlyquarterlyreadingsforthePCEdeflator. Estimatessimulatedforthisperiodarederivedfromameasurementequation,whichinterpretsthequarterlyPCEreadings as the moving average of an unobserved monthly series. This is analogous to the treatment of the GDP deflator as describedinAppendixB. 27

Figure7: InflationTrendEstimatesinReal-Time(“SMALL”) LEVEL 10 Full Sample Information Real Time 8 6 4 2 0 1970 1975 1980 1985 1990 1995 2000 2005 2010 UNCERTAINTY 0.4 0.3 0.2 0.1 0 1970 1975 1980 1985 1990 1995 2000 2005 2010 Note: Estimates are based on available real-time data for the for the SMALL data set described in Table 2, using vintagedatasince1960anduntiltheendofanymonth.Greyshadedbandsandthesolidbluelinedisplaytheposterior distribution of “smoothed” trend estimates conditional on all data received through August 2011. Red-solid lines denoteposteriormeanand90%confidenceintervalsofreal-timeestimates. around 2 percent between 2001 and 2007, and both measures register a drop in trend inflation accompaniedbyanincreaseinuncertaintyduringlate2008. However,real-timeestimatesoflevel and uncertainty changed much stronger in late 2008 than the smoothed estimates, with the level ofthereal-timeestimatesdroppingbyalmost1percentagepoint,whereasthesmoothedestimated decreased only by about half as much. This difference reflects the mostly short-lived nature in the drop of real-time estimates of the trend level, leading the smoothed estimates to attribute a larger part of the decreases in the data around the crisis to the gaps instead of the trend. Also, the realtime estimate of trend uncertainty rises much stronger and more sharply in late 2008, than what is 28

suggestedbythesmoothedestimates. Another critical period occurred during the late 1970s, before the onset of the Volcker disinflation. In real-time, the trend estimates reacted more dramatically to incoming data— this time overshooting the smoothed estimates by almost 2 percentage points over the course of 1979-82.26 Both measures pick up marked increases in trend inflation between 1973 and 1975. However, for most of the 1970s, the real-time estimates are about 50 basis points lower than the smoothed trend measure, rendering the increase of the real-time trend in early 1980 even more pronounced. Strikingly, during the recent crisis the real-time measure of uncertainty reached about the same heightsasduringtheearlyVolckeryearsandwellinto1983/84,aperiodwhichGoodfriend(1993) also characterized as an inflation scare. Interestingly, the inflation scare of 1994 registers clearly in real-time estimates of trend and uncertainty. Since the bond market turmoil of 1994 proved short-lived,itbarelyshowsupinthesmoothedestimates. 7. SMALLERMODELSWITHMORETIME-VARYINGPARAMETERS TheempiricalmodeldescribedinSection3assumestime-invariantdynamicsforgapvariables. While this seems to be a practical choice given the size of the data set and the amount of missing values, which are modeled as latent factors, it stands in contrast to the lessons from Cogley and Sargent (2005b) and Cogley et al. (2010) who documented important variations, for example, in the persistence of the inflation gap. As a robustness check, this section extends the basic model to incorporatedriftingcoefficientsinthegapVAR,aswellasstochasticvolatilityingapinnovations. Thisextendedmodelisthenestimatedwithtwosmallersetsofvariables. InthetheextendedmodelthegapVAR(8)isreplacedby A (L)Y~ = e~ (17) t t t 26The increase in the real-time measure of uncertainty also dwarfs movements in the smoothed estimates. Please noticeaswell,thatforthisdatasetthesmoothedestimatedofuncertaintypeaksmorenoticeablyin1974insteadofthe late1970s,asithasbeenthecaseforthelargermodeldiscussedintheprevioussections. 29

Figure8: Time-varyingGapPersistence (a)TVP1 (b)TVP2 1 0.95 0.9 0.95 0.85 0.9 0.8 0.85 0.75 0.8 0.7 0.75 0.65 0.7 0.6 0.65 0.55 1960 1970 1980 1990 2000 2010 1960 1970 1980 1990 2000 2010 Note: Posteriormeansand90%confidenceintervalsforthemaximuminverserootofthegapVAR’slagpolynomial, denoted A (L) in (17). Panel (a) shows estimates for the common trend model TVP1. Panel (b) depicts estimates t derivedfromtheconditioningsetTVP2,whichincludesthenominalyieldon10-yearTreasuries.TheTVP2estimates allowforaseparateyieldtrendasdescribedinSection5.2. TheconditioningsetsTVP1andTVP2aredescribedin Table2, where the coefficients of A (L) follow independent random walks, subject to a reflecting barrier, t toensurethateachdrawofA (L)hasallrootsoutsidetheunitcircle. Themodelisestimatedwith t three lags and the volatility of changes in the VAR coefficients is estimated as well. To facilitate stability, only the volatility of changes in the VAR coefficients along the main diagonal of the first lagmatrixisestimatedwithavagueprior. The gap innovations have stochastic volatility, but constant correlations. As before, they are alsocorrelatedwiththetrendshocks. 2 3 6(cid:27)~ t;1 0 :::7 6 7 6 7 e~ = (cid:12)(cid:27)(cid:22) "(cid:22) +D6 0 (cid:27)~ 7"~ (18) t t t 4 t;2 5 t . . . ... whereD isalowertriangularmatrixwithonesonitsmaindiagonal. Inamanneranalogoustothe trendshockvolatility(7),thelogsof(cid:27)~ followindependentrandomwalks. t;i 30

Two sets of conditioning variables are considered. TVP1 comprises inflation rates (PCE headlineandcoreaswellCPI)andtheLivingstonsurvey,whileTVP2combinesthe10-yearyieldwith headline PCE inflation and the Livingston survey. In the case of TVP2, draws of the VAR coefficients frequently violated the stability requirement and a separate yield trend was allowed for, as discussedinSection 5.2. Themainresultsfromthemodelswithtime-varyinggapdynamicsaretwofold: First,thetrend estimates — shown in the Appendix — are broadly similar to what has been reported above for the larger models with time-invariant gap dynamics. However, since the time-varying parameter models have additional flexibility in accounting for time-variation in the spectra for each variable, the estimated trends are more volatile than in the models with time-invariant gap dynamics, and surroundedbyadditionaluncertainty. Second, time-variation in the persistence of the gaps appears to have been limited. Figure 8 plotstheevolutionofthemaximuminverserootofA (L)forTVP1andTVP2,whichvariedquite t abit,butnotbyverymucheither. 8. RELATEDLITERATURE While my paper builds very closely on the studies by Stock and Watson (2007), Cogley and Sargent (2005b) and Cogley et al. (2010), it differs from these studies in using a much broader set of multivariate information variables to generate inflation forecasts. In order to handle such a larger data set, I have chosen to neglect some dimensions of the time-varying dynamics, which were embedded in these previous studies, while focusing my model on time-variation in the size of trend shocks. Compared with the quarterly models used by Stock and Watson (2007), Cogley and Sargent (2005b) and Cogley et al. (2010), the model used in this paper also handles missing observations,and combinesmonthly dataseries withless frequently sampledvariables—notably surveys,butalsotheinflationmeasurederivedfromthequarterlyGDPdeflator. Stock and Watson (2007) use a univariate inflation model with time-varying inflation persistence to elucidate changing patterns in the forecastability of inflation in postwar data for the U.S., 31

and document how time-varying volatility in trend shocks can be a very useful representation of the low frequency movements in U.S. inflation, which can already be discerned from Figure 1. To make their point, Stock and Watson (2007) use a simple representation of the inflation gap as a white noise process with time-varying volatility. In a similar spirit, Kiley (2008) estimates a bivariatetrend-cyclemodel,withcoreinflationratesforPCEandCPI,whererollingestimatesare usedtouncovertime-variationintherelativeimportanceofshockstotrendandcycles. Cogley and Sargent (2005b) and Cogley et al. (2010) extend this framework on two notable dimensions. First they use a trivariate system (with inflation, a nominal short rate and unemployment),imposingthatinflationandthenominalshortratearecointegratedasdescribedinSection2 above. In addition, Cogley and Sargent (2005b) and Cogley et al. (2010) allow for persistent, but stationary, gap processes which are modeled as VARs with drifting transition coefficients and stochastic volatility in the VAR innovations. Both papers compute measures of trend inflation from time-varying parameter VARs, whose coefficients follow driftless random walks, and their trend estimates are derived from local “time t” approximations of long-term inflation forecasts, in the spirit of the Beveridge-Nelson concept. While Cogley and Sargent (2005b) assume a constant varianceofshockstotheVARparameters,Cogleyetal.(2010)estimatedriftingvolatilitiesforthe parameter processes as well. Hence, innovations to the trend in Cogley and Sargent (2005b) have constantvariance,whilethemodelofCogleyetal.(2010)explicitlyallowsforatime-varyingsize of trend changes. In both models, the time-varying importance of trend movements is influenced bystochasticvolatilityintheVARresiduals. Incontrast,mymodeltracksthetime-varyingimportanceoftrendmovementsdirectlybyestimatingastochasticvolatilityprocessfortrendshocks,as inequations(3)and(4). Cogleyetal.(2010)emphasizenon-negligibletime-variationinthepersistenceoftheinflation gap. Their approach is however very expensive to compute since the time-varying coefficients of the gaps’ VAR process are latent variables themselves, thus adding N2 (cid:2)p latent variables to any model with N variables and p lags, which is feasible in the case of N = 3 but less so when trying touseamorediversedataset,asitisdoneherewithuptoN = 14andinfrequentlysampleddata. 32

Computationalissuesaside,themissingdatainsomeoftheseriesusedheremakesthedataalso lessinformativeaboutthekindoftime-varyingpersistencestudiedbyCogleyandSargent(2005b) and Cogley et al. (2010). To the extent that my paper is more concerned with characterizing movements in the common inflation trend — which is identified from common low-frequency movementsofmydatapanel—andlesswithforecastinghigherfrequencydynamicsofindividual gap variables, the approach chosen in this paper offers a tractable and potentially useful approach forextractingtrendinformationfromdiversedatasources. Section7comparesmybaselineresults with a model using much less variables, but allowing for time-varying gap dynamics in the spirit ofCogleyandSargent(2005b)andCogleyetal.(2010). The importance of low-frequency variations in U.S. inflation has been widely documented by previous studies, using at times very different methods. This evidence has motivated my choice to focus the model on time-variations in the low-frequency component of inflation. For example, Levin and Piger (2003) argue that changes in mean inflation — which are closely related to the Beveridge-Nelson concept — seem to account for a large part of the time-variation in inflation dynamicsintheU.S.andothercountries. Similarly,FaustandWright(2011)documenttheimportanceofaccountingformeandriftinforecastinginflation. Theirsimulatedout-of-sampleforecasts also suggest that the simple Stock and Watson (2007) model is a very competitive forecasting model,whichdespiteitssimplicityseemstocaptureasalientfeatureoftheinflationprocess. Faust andWright(2011)findthattrackingdriftinaverageinflationimprovestheperformanceofvarious other models as well. Kozicki and Tinsley (2001) document drifting means in nominal yields — which they call “shifting endpoints” in the term structure of interest rates — arguing that these shiftingendpointsreflectthepublic’slearningaboutlong-termgoalsofmonetarypolicy.27 Another difference between between my work and the studies of Stock and Watson (2007), Cogley and Sargent (2005b) and Cogley et al. (2010) is to condition inflation forecasts on survey data. A variety of studies has found survey expectations of future inflation useful for constructing 27KozickiandTinsley(2001)alsoestimateaunitrootmodelfornominalrates,called“movingaverageendpoint” specification,butwithaconstantvarianceoftheunitrootshocks,suchthatthemodelcannotmatchthetime-varying importanceoflow-frequencymovementsinnominalyields,seeforexampleFigure1above. 33

inflation forecasts, see for example Ang et al. (2007) and Gil-Alana et al. (2011). In evaluating the accuracy of survey expectations of inflation, Grant and Thomas (1999) argue that cointegration between survey responses and realized inflation is a weak requirement of rationality and find support for this hypothesis. While individual surveys may be biased and inefficient in that their forecast errors might be non-zero on average and predictable based on ex-ante information — for examplebecauseofimperfectinformationprocessingorlimitedinformationofsurveyrespondents — survey responses should not permanently deviate from trend inflation. Clark and Davig (2011) have also investigated the relationship between realized inflation and survey expectations, emphasizingtheroleoftime-varyingdynamicsinthespiritofCogleyetal.(2010). Theirstudyidentifies “long-term expectations” directly with a time series spliced together from the Survey of Professional Forecasters (SPF) 10-year forecast of inflation (since 1991) and a similar series from the Blue Chip surveys, based on which they document a marked decline in the volatility of long-term expectations. Kozicki and Tinsley (2006) use survey data to model a “term structure of inflation expectations” with particular emphasis on long-term expectations. Their model imposes cross-equation restrictions on survey processes — in the spirit of approximating survey expectations with autoregressive time-series forecasts — which may help to the extent that such restrictions provide a goodcharacterizationofactualsurveyresponses. Similarly,Haubrichetal.(2011)combinesurvey expectationswithtermstructuredatainaformalassetpricingmodelforTreasurysecurities—imposing no-arbitrage restrictions on the estimated dynamics of the data — from which they extract long-term inflation expectations. As will be seen in Section 3, my paper allows for arbitrary serial dependenceandcross-correlationsamongsttrenddeviationsofindividualinputvariables. Theimplicationsofcointegrationamongstnominalyieldsfortestsoftheexpectationshypothesis of the term structure of interest rates has been studied by Campbell and Shiller (1987). More recently, their work has been updated by King and Kurmann (2002). The common yield trend consideredbythesestudiesarisespresumably—butnotnecessarily—fromtrendinflation. Cogley (2005) has conducted a similar analysis using a VAR with time-varying parameters. While all 34

of these studies have documented significant deviations from the expectations theory of the term structure,theyhavealsofoundtheassumedcointegrationtobeusefulformodelingnominalyields in U.S. data. In adopting similar cointegrating assumptions, my paper will allow for deviations from the expectations hypothesis — in the form of time-varying but stationary risk premia — while interpreting the common trend in yields as arising from trend inflation. (Section 5 presents alsoresultsfromanextendedmodel,allowingforanadditionaltrendcomponentinyields.) Finally, it might be worthwhile to relate the Beveridge-Nelson measure of trend inflation to structural shocks in theoretical business cycle models. Many popular workhorse models of monetarypolicy—likeRotembergandWoodford(1997),Christianoetal.(2005)orSmetsandWouters (2007)—assumetheexistenceofaconstantinflationrateinsteady-state,whichcorrespondstothe specialcaseofaconstanttrendwithzeroshocks. Toaccountfortheriseandfallofinflationinthe U.S. during the 1970s and 1980s, Ireland (2007) augments a New-Keynesian DSGE model with a time-varying inflation target, driven by exogenous shocks with permanent effects.28 The inflation target process in Ireland’s model is identical to a Beveridge-Nelson trend with constant-variance shocks. Similarly, Cogley and Sbordone (2008) estimate a New-Keynesian Phillips Curve model withtrendinflationapproximatelyequaltoahomoscedasticBeveridge-Nelsontrend.29 Other studies, like Erceg and Levin (2003), Cogley and Sargent (2005a), Primiceri (2005) and Goodfriend and King (2005) have analyzed the consequences of opaque policy targets and the potential lack of credibility of monetary policy, and found these potentially useful in explaining U.S.inflationbehaviorofthe1970sand1980s. Eventhoughnoneofthesemodelsliterallyimplies the existence of a (non-degenerate) unit root process for trend inflation as in Ireland (2007), they give rise to low-frequency comovements between inflation and nominal rates which — as argued by Cogley and Sargent (2005b) — are close to the kind of trend model discussed above. Viewed from this perspective, it should be noted that the trend shocks in (3) may not be structural shocks, 28Indifferentversionsofhismodel,Ireland(2007)considersthecaseinwhichtargetshocksareorthogonaltoother fundamentalshocksandthecaseofcorrelationsbetweentargetshocksandotherexogenousdrivingvariablesofthe model. Ineachcase,theinflationevolvesasarandomwalk,drivenbyhomoscedasticshocks. 29Cogley and Sbordone (2008) identify trend inflation from a VAR with drifting coefficients as in Cogley and Sargent(2005b),howeverwithoutallowingforstochasticvolatility. 35

but rather the outcome of evolving policy communications and learning dynamics in response to morefundamentaleconomicdisturbances. 9. CONCLUSIONS Thispaperhaspresentedestimatesofthelevelanduncertaintyoftrendinflation,extractedfrom surveyexpectations,thetermstructureofinterestratesandrealizedinflationratessince1960. The application combines a variety of data sources at the monthly frequency and it can flexibly handle missingdataarisingfrominfrequentobservationsandlimiteddataavailability. Estimates of trend uncertainty typically rise with estimates of trend inflation itself. This result confirmsthatepisodesofhightrendinflationtendtobeperiodsinwhichinflationexpectationshave becomeunanchored,raisingtheriskoffurtherdriftinthetrend’slevel. Theresultalsounderscores theneedfordetectingchangesinlevelanduncertainty. Inthedecadepriortotherecentcrisis,inflationexpectationsappeartohavebeenwellanchored at around 2 percent. But in late 2008, my estimates record a noticeable increase in trend uncertainty accompanied by a marked drop in the trend level. By historical standards, this increase in uncertainty was close to, but below, levels seen shortly after the oil crisis of 1973/74, and well belowthepeakintrenduncertaintywitnessedduringtheearlystagesoftheVolckerdisinflationin 1980. These results are based on “smoothed” estimates, which enjoy the benefit of hindsight knowledge about the full data sample (up to August 2011). In contrast, when derived from real-time forecasts, trend estimates have dropped quite vigorously during the recent crisis, accompanied by considerable increases in uncertainty. The smoothed estimates condition on the knowledge that a persistent deflation has eventually been averted — presumably due to active monetary and fiscal policies — explaining the smaller, though still noticeable, reaction in smoothed estimates of level anduncertaintytothecrisis. 36

APPENDIX A. THEGIBBSSAMPLER The model is estimated with a Gibbs sampler and Bayesian MCMC methods, using multiple chains as in Gelman et al. (2003). Convergence is assessed by the scale reduction test of Gelman et al. (2003). The only fixed parameter is the volatility of shocks to the log-variances, (cid:27) in (7), h which is set at a value consistent with the quarterly model of Stock and Watson (2007), (cid:27) = h p 0:2= 3. The VAR of the gaps — see equation (8) — is estimated with rejection sampling to ensure stability. The baseline results use three lags, similar results are also obtained when using oneorsixlags. Thefollowingpriorswereusedintheestimation: (cid:15) Avaguepriorfortheinitialvaluesofthetrendvector(cid:28) (cid:24) N(0;1000(cid:1)I). 0 (cid:15) A vague prior for the initial value of the trend’s stochastic variance, which is log-normally distributedwithE(eh0) = 1=12,V(eh0) = 100. (cid:15) The correlation between standardized trend shocks and innovations to the gaps have a fairly vagueprior,centeredonzero,(cid:12) (cid:24) N(0;10(cid:1)I). (cid:15) Eachcoefficientofthelagpolynomialin(8)hasanormalprior,centeredaroundzero,which is restricted such that the posterior draws are more likely to generate a stable transition matrix. The prior assumes zero correlation amongst the coefficients, and postulates that the diagonal elements of A have a standard deviation of 0:1 and all other elements have a prior standard deviation of 0:01. As can be seen from, Figure 9, the posterior distribution of the maximum root in the companion matrix of the gap VAR, is not severely restricted by this prior—placingsubstantialmassonmaximumrootsof0:9andhigher. Ifanything,thisprior avoidstoomanydrawsofthemaximumroottopileupatorneartheunitroot. 37

Figure9: PersistenceofGapVARs (a)SURV (b)YLD λ λ 700 600 600 500 500 400 400 300 300 200 200 100 100 0 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0 0 .93 0.94 0.95 0.96 0.97 0.98 0.99 1 (c)ALL (d)ALL(w/separateYieldTrend) λ λ 500 450 450 400 400 350 350 300 300 250 250 200 200 150 150 100 100 50 50 0 0 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1 0.8 0.82 0.84 0.86 0.88 0.9 0.92 0.94 0.96 0.98 1 Note: Frequency of posterior draws of the maximum inverse root of A(L), the lag polynomial of the gap VAR, equation(8),ConditioningsetsaredescribedinTable2. (cid:15) The variance-covariance matrix of the gaps has a completely vague prior, assuming an inverse Wishart distribution with N +2 degrees of freedom. Given N gap variables, this is y y the minimum amount of degrees of freedom to ensure a well defined mean, but not a finite variance. TheGibbssampleriteratesoverthefollowingsteps: (cid:12) 1. Draw X (cid:12) h ;A(L);(cid:12);(cid:6)~;ZT. This is simply a draw from the posterior distribution of a t t Kalmansmoother,implementedasinDurbinandKoopman(2002). 38

(cid:12) 2. Draw A(L) (cid:12) "(cid:22);Y~ ;(cid:12);(cid:6)~ . This is a draw from the posterior distribution of a Bayesian ret t gression with normal priors and known residual variances. The draw also implies a set of gap shocks e~ . As in Cogley and Sargent (2005b), rejection sampling is used to ensure t stationarityoftheVAR. (cid:12) 3. Draw (cid:12) (cid:12) "(cid:22);e~ ;(cid:6)~ . This is a draw from the posterior distribution of a Bayesian regression t t withnormalpriorsandknownresidualvariances. Thedrawimpliesasetofresiduals"~. t (cid:12) 4. Draw (cid:6)~(cid:12) "~. This is a draw from the posterior distribution of a Bayesian regression with t normalpriorsandknownresidualvariances. 5. Drawthelog-variancesh (cid:24) f(h j"(cid:22))usingthealgorithmdescribedbyKimetal.(1998). t t t For the models with time-varying gap dynamics, described in Section 7, the second step is replaced by a Kalman Filter, drawing the latent coefficient dynamics, conditional on the gaps, followed by a step drawing the innovation variances of each coefficient’s random walk (which are assumedtobeindependent). Likewise,thestochasticvolatilitystepisaugmentedbyincludingthe orthogonalized innovation variances of each gap. The innovation variances are orthogonalized by recursiveapplicationofdrawsfromaBayesregressionasinStep3above. The Gibbs sampler was run 8 times, with the number of draws depending on the size of each model. Each of the 8 independent runs was initialized at different starting values, which were drawn from the prior distribution of model parameters. In the case of the basic model with all variables, each run had 4;000 draws, of which the first 2;000 were discarded. Convergence of the draws was assessed using the scale reduction test of Gelman et al. (2003), and for each model parameter convergence was achieved at statistics below 1:1 (values close to 1 indicate good convergence). B. MISSINGDATAFORTHEGDPDEFLATOR ThisappendixdescribeshowthemodelofSection3isaugmentedtohandlemissingdataforthe GDP deflator. The GDP deflator is available only at the quarterly frequency, whereas the model 39

Figure10: ThePosteriorDistributionofMonthlyGDPInflationandtheLivingstonSurvey (a)GDPInflation (b)LivingstonSurvey 14 12 Actual Data Actual Data Implied Data = Trend + Gap Implied Data = Trend + Gap 12 10 10 8 8 6 6 4 4 2 2 0 0 −2 −4 −2 1960 1970 1980 1990 2000 2010 1960 1970 1980 1990 2000 2010 Note:PosteriordistributionofmonthlyvaluesderivedfromthebasicmodeldescribedinSection3(usingSURVdata). Actual observations for GDP inflation and the Livingston survey — which are respectively available on a quarterly andbi-annualbasis—aredotted. Byconstruction,theposteriordistributioncollapsesforthemissingdatavaluesto theactualvalues,whenavailable. is monthly. What makes the GDP deflator different from, say, survey responses with missing observations, is that the GDP deflator measures price changes which accrue over the span of a quarter. SupposethattheGDPdeflatorisobservedinmonthst(cid:0)3;t;t+3etc. anddenotethelogarithm of the GDP deflator at the end of a quarter by pGDP and the implied quarterly inflation data by t zGDP = 400 (cid:1) (pGDP (cid:0) pGDP), and it becomes clear that it represents the trailing three-month t t t(cid:0)3 movingaverageofalatentmonthlyseries, (cid:25)GDP +(cid:25)GDP +(cid:25)GDP zGDP = t t(cid:0)1 t(cid:0)2 ; (19) t 3 wherethelatentvariable(cid:25)GDP tracksannualizedGDPinflationatthemonthlyfrequency. t 40

Formally,thissimplyrequiresaddingtwolagsof (cid:25)GDP tothestatevector, t 2 3 (cid:28) 6 t 7 6 7 6 7 6 Y~ 7 t Xt = 6 7 ; 6 7 6(cid:25)GDP7 4 t(cid:0)1 5 (cid:25)GDP t(cid:0)2 and to use (19) as measurement equation for GDP inflation in the months of March, July, October and December and zGDP = 0 otherwise. Based on the SURV model of Section 4, Figure 10 t depictstheposteriordistributionofthemodelestimatesfor(cid:25)GDP. Byconstruction,thedistribution t collapsestotheactualobservationforGDPinflationattheendofeachquarter. 41

C. ADDITIONALRESULTS Theresultspresentedinthisappendixareintendedonlyforwebpublication. C.1. Alternativeconditioningsets Panel(a)ofFigure11showsestimatesforYLD;thelevelestimatesarealsoshowninFigure4 of the main text. Panel (b) of Figure 11 shows the corresponding measures for INF, and Panel (c) displays estimates of trend level and uncertainty extracted from the joint data set of all variables listed in Table 1. These estimates are fairly close to the YLD estimates discussed in Section 5 of themaintext. C.2. TrendEstimatesfromModelswithTime-varyingGapDynamics Figures12and13reportestimatesoflevelanduncertaintyintrendinflationinthemodelswith time-varyinggapdynamicsdescribedinSection 7ofthemainpaper. C.3. StochasticVolatilityofGapsinTVPModels Figures 14 and Figures 15 report estimates of the stochastic volatility series for the gap variablesinthetime-varyingparametermodels(Section 7)formodelsTVP1andTVP2respectively. 42

steSgninoitidnoCevitanretlAnodesabdnerTnoitaflnI :11erugiF LLA)c( FNI)b( DLY)a( LEVEL LEVEL LEVEL 21 21 01 01 01 8 8 8 6 6 6 4 4 4 2 2 2 0 0 0 0102 0002 0991 0891 0791 0691 0102 0002 0991 0891 0791 0691 0102 0002 0991 0891 0791 0691 YTNIATRECNU YTNIATRECNU YTNIATRECNU 52.0 52.0 2.0 2.0 2.0 51.0 51.0 51.0 1.0 1.0 1.0 50.0 50.0 50.0 0 0 0 0102 0002 0991 0891 0791 0691 0102 0002 0991 0891 0791 0691 0102 0002 0991 0891 0791 0691 eht morf devired era setamitse ehT .skcohs dnert tuoba ytniatrecnu swohs lenap mottob eht dna dnert eht fo setamitse dehtooms eht swohs lenap pot ehT :etoN llanolanoitidnocnoitubirtsidroiretsops’ledomehtnodesabslavretniecnedfinoc%09wohssenildehsad-deR .1elbaTnidetsilsetarnoitaflnidnasdleiylanimon .dedahserasetadnoissecerREBN .atad 43

Figure12: InflationTrendfromTVP1model LEVEL 14 12 10 8 6 4 2 0 −2 1960 1970 1980 1990 2000 2010 UNCERTAINTY 3 2.5 2 1.5 1 0.5 0 1960 1970 1980 1990 2000 2010 Note: Thetoppanelshowsthesmoothedestimatesofthetrendandthebottompanelshowsuncertaintyabouttrend shocks. TheseestimatesarederivedfromtheextendedmodeldescribedinSection7, allowingfortime-varyinggap dynamics, stochastic volatility in gap innovations, using data on headline PCE, core PCE, the seasonally adjusted CPIaswellastheLivingstonsurvey—theconditioningsetcalled“TVP1”inTable2. Red-dashedlinesshow90% confidence intervals based on the model’s posterior distribution conditional on all data. NBER recession dates are shaded. 44

dnerTdleiYetarapeShtiwledoM-2PVTnisdnerT :31erugiF skcohSdleiYtnenamreP)b( dnerTnoitaflnI)a( LEVEL LEVEL 7 41 6 5 21 4 3 01 2 8 1 0 6 1− 2− 4 3− 4− 2 5− 0 6− 7− 2− 0102 0002 0991 0891 0791 0691 0102 0002 0991 0891 0791 0691 YTNIATRECNU YTNIATRECNU 57.1 5.1 5.1 52.1 1 1 57.0 5.0 5.0 52.0 0102 0002 0991 0891 0791 069 0 1 0102 0002 0991 0891 0791 069 0 1 ehtmorfdevirederasetamitseesehT .skcohsdnerttuobaytniatrecnuswohslenapmottobehtdnadnertehtfosetamitsedehtoomsehtswohslenappotehT :etoN tnenopmocdnertlanoitiddanasallewsa,snoitavonnipagniytilitalovcitsahcots,scimanydpaggniyrav-emitrofgniwolla,7noitceSnidebircsedledomdednetxe eht—seirusaerTraey-01nodleiylanimonehtsallewsayevrusnotsgniviLeht,ECPenildaehnosnoitavresboelbaliavallamorfdetamitsesiledomehT .sdleiyni ytniatrecnudnalevelfosetamitseswohs)b(lenaPsaerehw,(cid:25)(cid:28)dnertnoitaflniehtfoytniatrecnudnalevelstciped)a(lenaP .2elbaTni”2PVT“detonedteselbairav t nolanoitidnocnoitubirtsidroiretsops’ledomehtnodesabslavretniecnedfinoc%09wohssenildehsad-deR .)51(ni r(cid:28)detoned,tnenopmocdnertlanoitiddaehtrof t .dedahserasetadnoissecerREBN .atadlla 45

Figure14: StochasticVolatilityinGapsoftheTVP1Model (a)PCE (b)CorePCE 5 2.2 4.5 2 4 1.8 3.5 1.6 3 1.4 2.5 1.2 2 1 1.5 0.8 1 0.6 0.5 0.4 0 0.2 1960 1970 1980 1990 2000 2010 1960 1970 1980 1990 2000 2010 (c)CPI(SA) (d)LivingstonSurvey 4 6 3.5 5 3 4 2.5 2 3 1.5 2 1 1 0.5 1 0 960 1970 1980 1990 2000 2010 1 0 960 1970 1980 1990 2000 2010 Note: Posterior means and 90% confidence intervals for stochastic volatilities in the gap innovations, denoted (cid:27) t;i in (18). The estimates are derived from the time-varying parameter version of the common trend model described inSection7,usingavailabledatasince1960forheadlinePCE,corePCE,theseasonallyadjustedCPIaswellasthe Livingstonsurvey. Thisistheconditioningsetcalled“TVP1”inTable2. 46

Figure15: StochasticVolatilityinGapsoftheTVP2Model (a)PCE (b)LivingstonSurvey 6 4.5 4 5 3.5 4 3 2.5 3 2 2 1.5 1 1 0.5 1 0 960 1970 1980 1990 2000 2010 1 0 960 1970 1980 1990 2000 2010 (c)10-yearTreasuryYield 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 1960 1970 1980 1990 2000 2010 Note: Posterior means and 90% confidence intervals for stochastic volatilities in the gap innovations, denoted (cid:27)~ t;i in(18). Theestimatesarederivedfromthetime-varyingparameterversionofthecommontrendmodeldescribedin Section7,whenallowingforaseparateyieldtrendasdescribedinSection5. Themodelisestimatedfromdatasince 1960onheadlinePCE,theLivingstonsurveyandnominalyieldson10-yearTreasuries—theconditioningsetcalled “TVP2”inTable2. 47

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