Time Series Model of Interest Rates With the Effective Lower Bound

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. A Time Series Model of Interest Rates With the Effective Lower Bound Benjamin K. Johannsen and Elmar Mertens 2016-033 Please cite this paper as: Johannsen, Benjamin K., and Elmar Mertens (2016). “A Time Series Model of Interest Rates With the Effective Lower Bound,” Finance and Economics Discussion Series 2016-033. Washington: Board of Governors of the Federal Reserve System, http://dx.doi.org/10.17016/FEDS.2016.033. 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.

A Time Series Model of Interest Rates With the Effective Lower Bound Benjamin K. Johannsen∗ Elmar Mertens Federal Reserve Board Federal Reserve Board April 4, 2016 Abstract ModelinginterestratesoversamplesthatincludetheGreatRecessionrequirestakingstock of the effective lower bound (ELB) on nominal interest rates. We propose a flexible time– series approach which includes a “shadow rate”—a notional rate that is less than the ELB duringtheperiodinwhichtheboundisbinding—withoutimposingno–arbitrageassumptions. The approach allows us to estimate the behavior of trend real rates as well as expected future interestratesinrecentyears. ∗For correspondence: Benjamin K. Johannsen, Board of Governors of the Federal Reserve System, Washington D.C.20551. emailbenjamin.k.johannsen@frb.gov. Tel.: +(202)5306221. Wewouldliketothankseminar andconferenceparticipantsattheFederalReserveBanksofClevelandandRichmond,theFederalReserveBoard,the University of Texas at Austin, the World Congress of the Econometric Society (2015), and the Dynare Conference (2015),fortheirusefulcommentsandsuggestions. Theviewsinthispaperdonotnecessarilyrepresenttheviewsof theFederalReserveBoard,oranyotherpersonintheFederalReserveSystemortheFederalOpenMarketCommittee. Anyerrorsoromissionsshouldberegardedassolelythoseoftheauthors. 1

1 Introduction Thispapermodelsnominalinterestrates,alongwithothermacroeconomicdata,usingaflexibletime-seriesmodelthatexplicitlyincorporatestheeffectivelowerbound(ELB)onnominal interest rates. We employ a modeling device that we refer to as a “shadow rate”—the nominal interest rate that would prevail in the absence of the ELB—which is conceptually similar to the shadow rates studied in the dynamic term–structure literature, as in Kim and Singleton(2011),Krippner(2013),Priebsch(2013),IchiueandUeno(2013),BauerandRudebusch (2014), Krippner (2015), and Wu and Xia (2016). Our time–series approach allows us to estimate the relationship between interest rates and macroeconomic data in a flexible way and, similar to the approach taken in Diebold and Li (2006), does not impose rigid no–arbitrage restrictionsacrosstheterm–structureofinterestrates. We use our approach to estimate a trend–cycle model of U.S. data on interest rates, unemployment, and inflation over a sample that includes the recent spell at the ELB. Since the global financial crisis of 2008, real interest rates have been historically low, prompting some — for example, Summers (2014) and Rachel and Smith (2015) — to argue that the long–run normallevelofrealinteresthasfallen. Usingourestimatedmodel,wefindthatthetrendcomponentofthenominalinterestratehasdeclinedalmostcontinuouslysincetheearly1980s. The declineisduetolong–standingdownwardtrajectoriesofthetrendcomponentofbothinflation andtherealshort–terminterestrate. Whileuncertaintybandsaroundourestimateofthetrend real rate are wide, we find that any decline since the global financial crisis of 2008 is best characterizedasacontinuationofalongertermdownwardtrajectory. SimilartoLaubachand Williams (2003, 2015), Clark and Kozicki (2005), Hamilton et al. (2015), Kiley (2015), and Lubik and Matthes (2015) we find large uncertainty bands notwithstanding some differences inpointestimates.1 However,noneofthesepapersexplicitlymodelstheELB. WhenweignoretheconstraintsimposedbytheELBonthedataandproceedbyapplying a standard linear–state–space version of our model, we estimate a larger decline in the trend interest rates in recent years than in our model with the ELB. The reason is that the gap be- 1Notably,ourestimatedtrendrealratedisplayslessmovementthanthetrendestimatesreportedbyLaubachand Williams(2003,2015),anddoesnotnotdipaslowduringtherecentrecession. 2

tween the observed interest rates and the estimated trend is not as large as the gap between our estimated shadow rate and the trend. Without a large cyclical movement associated with the Great Recession, the model has an easier time explaining the prolonged period with the short–terminterestrateattheELBasachangeintrend. Additionally,explicitlymodelingthe ELBhaslargeeffectsoninferenceaboutout–of–sampleexpectedshort–terminterestratesand termpremiumsoverthepastseveralyears. OurestimatedshadowratesarelessthantheELB byconstruction,andourshadow–ratemodeldeliverspredictedpathsforfutureshort–terminterest rates that include extended periods at the ELB. By contrast, when we ignore the ELB, themodelpredictsrelativelyprecipitousincreasesinshort–termrates. Wefindthatincludingamedium–terminterestrateinourmodel—whichisnotconstrained by the ELB over our data sample—disciplines the behavior of the shadow rate, reflecting the model’s estimated co–movement between yields of different maturity. Without the medium– term yield, our shadow rate estimates would merely reflect the dynamic relationship between short–terminterestratesandthemacroeconomicvariablesinourdataset(unemploymentand inflation). However, inasmuch as medium-term rates contain useful information about the path of expected future short–term rates, the medium-term interest rate helps us identify and forecasttheshadowrateaboveandbeyondtheinformationgleanedfromthemacroeconomic datainoursample. ThewayweincorporatetheELBandestimatethemodelcanbeextendedtoabroadclassof timeseriesmodels. Withshort–termnominalinterestratesatorneartheirELBsinmanyparts oftheworld,timeseriesmodelsthatincludeinterestratesbutignoretheELB—likeastandard vector autoregression—have been unable to adequately address the data. Moreover, reducedformexplorationsoftheempiricalrelationshipbetweenshort–andlonger–terminterestrates— such as Campbell and Shiller (1991)—have often ignored the truncation in the distribution of future short–term interest rates. Our modeling approach overcomes these shortcomings in a wide class of otherwise conditionally–linear Gaussian state–space models. Examples include the vector autoregressions studied in Sims (1980) and the models with time–varying parametersstudiedinPrimiceri(2005)andCogleyandSargent(2005b). Following work by Black (1995), the no–arbitrage dynamic term–structure models stud- 3

ied in Kim and Singleton (2011), Krippner (2013), Priebsch (2013), Ichiue and Ueno (2013), BauerandRudebusch(2014),Krippner(2015),andWuandXia(2016)identifyshadowrates byimposingno–arbitragecross–equationrestrictions. Thesestudiesofferinterestinginsights, yettheno–arbitrageassumptionsthattheseauthorsmaintainmayprecludecertainmodelfeatures, like stochastic model parameters. Our time series approach naturally incorporates time variation in parameters, and thus, for some purposes—like including time–varying trends in inflation and interest rate data or modeling stochastic volatility—offers a flexible alternative. Inaddition,ourshadow–rateestimatesdonotonlyreflectinformationembeddedinalonger– term yield but also condition on direct readings about business cycle conditions embedded in macrovariablessuchastheunemployment–rategapandinflation.2 IwataandWu(2006),Nakajima(2011),ChanandStrachan(2014)aretheclosestpapersin theliteraturetoours. ThesepapersalsoestimatetimeseriesmodelsthatincorporatetheELB. In all of these studies, lagged observed interest rates (rather than shadow rates) are explanatoryvariablesinthedynamicsystem. Weinsteadallowlaggedshadowratestobeexplanatory variables. In doing so we are able to more closely align our approach with the no–arbitrage term–structure literature, and, in additional, connect the concept of the shadow rate with the leveloftheshort–termratethatwouldprevailintheabsenceoftheELBbecauseweallowitto have the same persistence and co–variance properties as short–term interest rates. Nevertheless,ourapproachisflexibleenoughtoincludebothshadowratesandobservedratesaslagged explanatoryvariables. 2 A Model of Interest Rates and the ELB In this section we describe our time–series model, which explicitly includes the ELB. The model includes inflation, a short– and a medium–term nominal interest rate, and the 2Inanalternativetime-seriesapproachLombardiandZhu (2014)useadynamicfactormodeltoderiveestimates ofthestanceofmonetarypolicy—labeled“shadowrate”—frominterestrates,monetaryaggregatesandvariables characterizingtheFederalReserve’sbalancesheet. However,assuch,theirunderlyingshadow–rateconceptisquite differentfromwhatisusedhereorinthedynamicterm-structureliteratureinthattheirmeasureneedsnotbeidentical toobservedinterestrates,evenwhentheELBisnotbinding,noristheirshadowrateconstrainedtoliebelowtheELB whentheboundisbinding. 4

unemployment–rategapasmeasuredbytheCBO. 2.1 The Shadow Rate Approach Ourdatasetincludesashort–terminterestrate,whichhasbeenattheELBduring28quarters in our sample. We model the interest rate (i ) as the observation of a censored variable. In t particular,weassumethatthenominalinterestrateisthemaximumoftheELBandashadow rate(s )sothat t i = max(s ,ELB) . (1) t t The ELB might arise because of an arbitrage between bonds and cash, though the world has seen negative short–term nominal interest rates in a number of countries. It also might be thought of as a level below which monetary authorities are unwilling to push short–term interest rates. For our purposes, it is taken as an exogenous known constant (which could be madetime–varying). Weproceedbymodelingtheshadowrate, inconjunctionwiththeother variables in the model, using standard time–series methods, and account for the ELB when conditioningtheposteriordistributionofourmodelonobservedinterestratedata. 2.2 A Time Series Model with Shadow Rates We assume that inflation (π ), the medium–term yield (y ), and the shadow rate (s ) can be t t t decomposedintotrendandcyclicalcomponents. Thatis,foreverydataseries,x ,weassume t that x = x¯ +x˜ where x¯ = lim E (x ) and E(x˜ ) = 0 (2) t t t t t t+h t h→∞ The defining feature of the cyclical (gap) component, x˜ , is that it has a zero ergodic mean. t For the unemployment rate (u ) we assume that the gap measure derived from subtracting t the CBO’s measure of the long-run natural rate from the actual unemployment rate reflects a detrendedseriesakintothex˜ componentin(2). t 5

Thetrendcomponentsx¯ aresimilarinspirittothetrendconceptofBeveridgeandNelson t (1981);however,bytreatingthetrendsasunobservedcomponentsweallowfortheconditional expectations,E (·)in(2),toreflectapossiblywiderinformationsetthanwhatisknowntoan t econometrician at time t.3 Defining the trend components as infinite-horizon expectations implies that changes in x¯ follow martingale-difference processes; and, as a result, the trend t components have unit root dynamics. As documented, for example, by Stock and Watson (2007) and Cogley and Sargent (Forthcoming), U.S. inflation dynamics are well captured by such a trend-cycle decomposition when trend shocks have time-varying volatility. So, for the trendcomponentofinflation,wewrite: π¯ = π¯ +σ (cid:15) . (3) t t−1 π¯,t π¯,t where(cid:15) ∼ N(0,1). (Throughoutthetext(cid:15) andη indicateuncorrelatedstandard–normal π¯,t ·,t ·,t randomvariables.) Away from the ELB, our shadow rate is identical to the short–term nominal interest rate. Weassumethatthetrendshadowratehastwocomponents s¯ = π¯ +r¯ (4) t t t where r¯ will be our measure of the trend in real interest rates, discussed below. We assume t thatr¯ evolvessothat t r¯ = r¯ +σ (cid:15) . (5) t t−1 r¯ r¯,t To capture a connection between short– and medium–term interest rates, we assume that the medium–term rate in our model (y ) shares a common trend with the shadow rate, adjusted t foranaverageterm–premium.4 Byassumingthatyieldspreadsarestationary, weimposethe 3SeealsothediscussioninMertens(forthcoming). 4Specifically,thetrendinthenominalmedium-termyieldiswrittenasy¯ =s¯ +p¯ wheretheconstantp¯ represents t t 0 0 the average term–premium, and its estimated value reflects the average spread between medium- and short–term nominalinterestratesinoursample(awayfromtheELB).Asforallgapvariables,themeanofthemedium-termyield gap, E(y˜), has been normalized to zero. In principle, the distinction between shadow rates and observed nominal t 6

same cointegrating relationship on nominal yields that has also been used by Campbell and Shiller (1987) and King and Kurmann (2002). In sum, there are two stochastic trends in our model: π¯ andr¯. t t Prior evidence suggests that shocks to trend inflation have been most likely highly heteroscedastic (Stock and Watson, 2007) in U.S. postwar data, possibly reflecting changes in the anchoring of public perceptions of long–term inflation expectations and the credibility of policymaker’s inflation goals. By contrast, the variability of the trend real rate is more likely to reflect changes in long–term growth expectations, demographic trends and other secular drivers(RachelandSmith,2015)andaccountsprobablyonlyforasmallshareofthevariabilityinrealratesasdiscussed,forexample,byHamiltonetal.(2015),whichcautionsusagainst fitting a stochastic volatility process for changes in this trend. Accordingly, shocks to trend inflation are assumed to be affected by stochastic volatility in our model, whereas we have chosen to specify a constant variance for the shocks to r¯; in addition, both trend shocks are t supposedtobemutuallyuncorrelated. We assume that the gap components of the four series in our model follow a joint autoregressiveprocess. Thatis,   π˜ t     u˜ t A(L)  = BΣ (cid:15) (6)   t t s˜   t   y˜ t whereA(L)isapolynomialinthelagoperator,whichhasrootsoutsidetheunitcircle,B isa unit–lower–triangularmatrix,andΣ isadiagonalmatrix. t Weincludetime–varyingvolatilityintheshockstothetrendcomponentofinflationandin thegap–vectorautoregression. Weassumethatσ2 andthediagonalelementsofΣ followa π¯,t t rates could also be extended here to the nominal medium-term yield and its trend; however, in our application the distinctionwouldbemootsincetheELBneverbindsforthemedium-termyieldinourdata. 7

processgivenby log (cid:0) σ2 (cid:1) −µ = ρ (cid:0) log (cid:0) σ2 (cid:1) −µ (cid:1) +φ η (7) j,t j j j,t−1 j j j,t whereσ2 denotesaparticulardiagonalelementofΣ orσ2 ,µ isthemeanofthelogofthe j,t t π¯,t j variablej,ρ isthepersistenceoftheprocess,andσ isthevolatilityofinnovations. Stochastic j j volatilityhelpsuscapturetherunupinaverageinflationinthelate1970sandearly1980sand gives the model flexibility to capture large changes in the gap components over the business cycle. Finally,weassumethatobservedinflationisthesumofπ¯ ,π˜ ,andmeasurementerrorwith t t stochasticvolatility: π = π¯ +π˜ +eπ, eπ = σ (cid:15) (8) t t t t t e,t e,t Theinclusionofmeasurementerrorhelpsuscapturehighlytransitory, one–offmovementsin headlineinflationthatdonotfeedbackintorealactivity. 2.3 Relationship Between Shadow and Interest Rates Weconceptualizetheshadowrateasthenominalinterestratethatwouldprevailintheabsence of the ELB. On a period–by–period basis, the interest rate is either equal to the shadow rate or equal to the ELB. The key distinction between shadow rates and interest rates is thus that shadowrateshaveunboundsupport. In the model we presented in the previous section, we modeled the shadow–rate gap, as well as its lags, as part of a joint dynamic system, which allows the shadow rate to have the same persistence properties when the ELB is binding and when it is not. By contrast, IwataandWu(2006)andNakajima(2011)modelthevariablesintheirmodelsasfunctionsof lagged observed interest rates. This means that, in those papers, at the ELB the value of the shadow–rate in the previous period has no direct effect on its value today. This approach is instarkcontrasttotheshadow–ratedynamicsfromdynamicterm–structureliterature;see,for 8

example,WuandXia(2016). Similar to the term-structure literature, we embed the shadow rate into a state vector with auto–regressive dynamics, such that the persistence of the shadow rate does not depend on whethertheELBbinds. WhentheELBisbindingonobservedinterestrates,theshadowrateis intendedtocapturethehypotheticallevelofthenominalratethatwouldprevailintheabsence oftheELBconstraint;accordingly,wedeemitbeneficialthattheestimatedpersistenceofthe shadowrateinourspecification, reflectstoalargedegreethepersistenceofobservedinterest rateswhenthoseareawayfromtheELB. 2.4 Interpretation of r¯ t Because interest rates are truncated shadow rates, the expected interest rate is necessarily weaklylargerthantheexpectedshadowrate. Inturn,itisalsothecasethat, lim E (i ) ≥ lim E (s ) = s¯ = π¯ +r¯. (9) t t+h t t+h t t t h→∞ h→∞ Inourmodel,s¯ isthemedianforecastoflim i ,offeringadirectconnectionbetween t h→∞ t+h far-aheadshadowratesandinterestrates.5 Further,assumingthattheFisherhypothesisholds, this connection gives r¯ the interpretation of the median forecast of the real interest rate in t the long run.6 Notably, the same relationship holds for the medium–term yield in our model, up to a constant offset, because of the co-integrating relationship we have assumed. For the remainderofthepaper,werefertor¯ asthetrendrealinterestrate. t Importantly, the co-integrating relationship between the short–term interest rate and the medium–termyieldinourmodelallowsthemedium–termyieldtoofferdirectevidenceonr¯. t Thisdisciplinesmovementsinr¯ becausethemedium–termyieldisabovetheELBthroughout t oursample. 5Inourmodelinsection(2.2)lim E (i )>lim E (s ).However,onecouldconceptualizemodels h→∞ t t+h h→∞ t t+h inwhichitneednotbestrict. 6 Solongass¯ ≥0,whichitisinourestimates,thenourinterpretationofr¯ applies. However,ifs¯ <0,thenthe t t t medianforecastofthelim i −π isELB−π¯ . Thenr¯ wouldhelpdeterminetheprobabilitythatinterest h→∞ t+h t+h t t rateswouldbeabovetheELBoverthebusinesscycleandwouldbelessthanthelong–runmedianrealinterestrate. 9

2.5 Estimation Procedure To estimate the parameters and unobserved states of the model, we use Bayesian methods.7 Thenovelmodelingcontributionofthispaperliesinthesamplingoftheunobservedtrendand gap components of the data when the interest rate data are at the ELB, so we focus the text on this step of the estimation procedure. Conditional on parameter values and a sequence of volatilities,ourmodelcanbeputintotheform ξ =Aξ +B ε (10) t t−1 t t X =Cξ (11) t t i =max(s ,ELB) (12) t t where X ≡ [s ,y ,π ,u˜ ](cid:48), ε is a vector of standard normal random variables, ξ contains t t t t t t t thestochastictrendandcyclicalcomponentsofourmodel, aswellastheappropriatenumber oflags,thematricesA,{B }T ,andC areconstructedaccordinglyfromtheparametersand t t=1 volatilitiesinourmodel,8 andthemaxoperatorencodestheELBintheobservationequation fortheinterestrate. WesetthevalueoftheELBtozero,andassumethatthefederalfundsrate was at the ELB for every quarter in which the target range for the federal funds rate was 0 to 25basispointsthroughoutthequarter. Our approach for drawing from the posterior of ξ ≡ [ξ ,ξ ,...,ξ ](cid:48) is to first treat the 1 2 T interest rate data at the ELB as missing and take draws from the posterior distribution of ξ, whichisstraightforwardusingstandardfilteringandsmoothingtechniques. Knowingthatthe interestrateisattheELB(notsimplymissing)inperiodtamountstoknowingthatthevalues of ξ that are consistent with the data imply values of s that are less than the effective lower t t bound during that period. Thus, we can draw from the posterior of ξ by first treating interest rate data at the ELB as missing, and then rejecting draws until we find a ξ that is consistent withtheELB. OurestimationprocedureisageneralizationofParketal.(2007)thatappliesthemethod- 7Ourdataarequarterly,andweincludetwolagsinA(L). 8InAppendixAweshowhowAandCcaneasilybemadetime–varying. 10

ology of Hopke et al. (2001). Appendix A explains in further detail how to construct a draw fromtheposteriordistributionofξ,conditionalonthedata,inaconditionally–linearGaussian state–spacemodellikeours. Withadrawofξinhand,theposteriordistributionoftheparameterscanbesampledusingstandardmethodsintheliteratureonconditionally–lineartimeseries modelswithtime–varyingparametersandstochasticvolatility,suchasthoseusedinPrimiceri (2005) or Cogley and Sargent (2005b). We jointly estimate the parameters and unobserved states of the model using Bayesian MCMC techniques; our priors and details of the MCMC stepsaredescribedinAppendixB. 3 Shadow Rate and Trend Estimates In this section we describe the posterior distribution of our model with regard to estimated shadow rates and trends. Our model is estimated using quarterly data from 1960:Q1 to 2015:Q4, which includes the recent period at the ELB. All data are publicly available from the FRED database maintained by the Federal Reserve Bank of St. Louis. Inflation is measuredbythequarterlyrateofchangeinthePCEheadlinedeflator(expressedasanannualized percentage rate). Readings for the federal funds rate and the 5-year nominal bond yields are constructed as quarterly averages of the effective federal funds rate and the Treasury’s 5-year constant maturity rate, respectively. The unemployment gap is computed as the difference between the quarterly average rate of unemployment and the CBO’s measure of the natural long–term rate of unemployment for a given quarter. All computations are based on the vintage of FRED data available that has been available at the end of January 2016.9 Figure 1 displaysthedataseriesweuseforestimation,alongwithourestimatedtrendsandcorrespondinguncertaintybands. [Figure1abouthere.] 9See https://research.stlouisfed.org/fred2/. The PCE headline deflator is available at https://research.stlouisfed.org/fred2/data/PCECTPI.txt. The daily federal funds rate and theconstant–maturity5–yearTreasuryyieldareavailableathttps://research.stlouisfed.org/fred2/ data/DFF.txtandhttps://research.stlouisfed.org/fred2/data/GS5.tx. Theunemployment rate and the CBO’s estimate of the natural rate of unemployment in the long run are available at https:// research.stlouisfed.org/fred2/series/UNRATEandhttps://research.stlouisfed.org/ fred2/series/NROU. 11

3.1 The Shadow Rate Ourmodeldeliversestimatesoftheshadowrateofinterestthatare,byconstruction,lessthan the ELB during the period in which the bound is binding. Panel (a) of Figure 2 shows our posterior estimates of the shadow rate, along with uncertainty bands. Our model estimates indicatethattheshadowratefellsharplyinearly2009,andbeganmovinguptowardtheELB in early 2013. The last observation in our sample, 2015:Q4, coincides with the last quarter beforeshort–terminterestratedataintheU.S.hasbeguntoriseagainabovetheELBinrecent years. Indeed, our end-of-sample estimate puts the shadow rate for 2015:Q4 just below the ELB. [Figure2abouthere.] Panel (b) of Figure 2 shows the posterior estimate of the shadow rate when we treat the interest rate data as missing over the period in which the ELB binds. Interestingly, the ELB does not pose much of a binding constraint for our sampling technique from 2009-2013, becausethemodelwouldhavepredictedshadowrates below theELB,evenwithoutknowledge thattheELBwasbindinginthedata. Startingin2013,thereisnon-negligiblemassabovethe ELB when we treat the data as missing, meaning that the information content in the fact that theELBwasbindingisgreatestinthispartoftheELBsample. Notably,evenattheendofour sample,theELBiswellwithintheuncertaintybandsourmodelproducesfortheshadowrate whenwetreattheinterestratedataasmissing. Forcomparison,Figure3showstheposteriormeanofourshadowrate,alongwithuncertainty bands, on the same plot as estimates from Wu and Xia (2016), and Krippner (2013).10 Twofeaturesareworthnoting. First,ourestimatedshadowrate,whichalsoconditionsonthe unemployment gap and inflation as business cycle indicators, is lowest during 2009, near the troughoftheGreatRecession,accordingtotheNBER.Bycontrast,theotherestimatesreach low points much later. Second, all three estimates are remarkably similar at the end of 2015, just before the Federal Reserve’s departure from the ELB, even though our model has access 10Measures of the shadow rate from Priebsch (2013) and Ichiue and Ueno (2013) are not shown because their sampleendsin2013. Theirmeasuresarequalitativelysimilarinthattheyreachlowpointswellafterthetroughofthe recentrecession. 12

to many fewer yields, we do not impose the rigid cross–equation restrictions associated with no–arbitrage assumptions in the dynamic term–structure literature, and our data sample does notincludetheperiodofdeparturefromtheELB. [Figure3abouthere.] 3.2 The Real Rate in the Long Run Figure 4 displays the posterior median of our estimates of the trend real rate, r¯, along with t uncertaintybands. Panel(a)ofFigure4showsquasireal–timeestimatesofr¯,whicharecont ditionedsolelyondatathroughperiodt. Panel(b)ofFigure4showsthesmoothedestimates, inthattheentiredatasampleisusedtoestimatetheparametersandr¯. Notably,theuncertainty t bandssurroundingourestimatesofr¯ arewide. Thisresultisconsistentwithresultsreported t byHamiltonetal.(2015),Kiley(2015),andLubikandMatthes(2015). [Figure4abouthere.] Forboththepseudo–real–timeandthesmoothedestimates,anyestimateddownwardtrend intherealratestartedwellbeforetheonsetoftheGreatRecession. StudieslikeLaubachand Williams (2015) and Lubik and Matthes (2015) also document downward trajectories in the trendrealrate;however,ourestimatesdonotdipnearlyasmuchastheirestimates. Onereason forthedifferencesisourinclusionofstochasticvolatility,whichallowsforlargemovementsin thegapcomponentsofourmodelin2008and2009. Anotheristhatourestimationprocedure explicitly models the ELB. If we instead ignore the ELB and treat the interest rate data as observations, we estimate a lower trend nominal interest rate, shown in Figure 5. When the ELB is ignored, the model is unable to generate a large cyclical movement in interest rates because it sees the interest rate stop falling at the ELB. As a result the model has difficulty explaining the prolonged period at the ELB without changes in the trend nominal rate. One margin by which the trend nominal rate can change is the trend real rate. By contrast, our modelisabletoproducealargeshadow–rategapin2009,whichhelpsexplainwhyobserved nominalinterestratesremainedattheELB. [Figure5abouthere.] 13

3.3 The Effect of Including the Medium–Term Yield Theposteriordistributionoftheshadow–rateattheELBisparticularlyinformedbytheinclusionofthemedium–termyieldinourmodel. Whenweestimatethemodelwithoutamedium– term yield, our posterior distribution of the shadow rate, shown in Panel (a) of Figure 6, is markedly more dispersed and lower on the eve of the Federal Reserve’s departure from the ELB in 2015:Q4. The difference from our baseline model, which includes the medium–term yield, is particularly striking when we consider estimates that treat short–term rates as missing data during the ELB period, shown in Panel (b) of Figure 6, which is analogous to our discussionofFigure2above. When shadow–rate estimates for the ELB period are solely informed by data on inflation and the unemployment–rate gap (and interest rate data up to 2008), the model would have expected the ELB to bind only for a couple of years, starting to place substantial odds on positive shadow rates around 2011. In contrast, when the medium–term yield is included in the estimation, the model predicts negative shadow rate at least until 2014, when short– term rate data is treated as missing as of 2009. Inclusion of the medium–term yield in our baseline model thus makes the ELB much less binding for our sampling routine than in the modelwithout. Insofarasmedium-termyieldsembedinformationaboutexpectedfutureshort rates, it makes sense that including a medium–term yield will greatly inform our shadow rate estimates. In the next section we show that our model with the medium–term yield forecasts short–terminterestrateslargelybetterthanthemodelwithout. [Figure6abouthere.] As shown in Panel (c) of Figure 6, the estimated trend real rate is fairly similar to our baseline estimates when the medium–term yield is excluded from the data. However, the estimated level of trend inflation is quite different, as shown in panel (d) of the figure. At the end of our sample, trend inflation derived from the model without the medium-term yield is markedly lower; which together with the mostly similar trend real rate implies a lower nominal short–term rate trend estimate in line with the model’s lower estimate of the shadow ratetrajectoryshowninPanel(a). Overall,inthemodelwithoutthemedium–termyield,trend 14

inflationismorevariableandmorecloselyalignedwith4-quarterchangesinthePCEheadline deflatorthaninthebaselinemodelasthepresenceofthemedium-termyieldhelpsthemodel torationalizeamorepersistentinflationgapprocess. 4 Forecasting Interest Rates Our shadow–rate approach has significant implications for forecasting interest rates. In this section we offer a number of ways to evaluate our shadow–rate approach by analyzing the model’sout–of–samplepredictions. 4.1 Predictive Density at Selected Dates Tocreateout–of–sampleforecastsfromourmodel,ateachdateweusethepredictivedensities fromtheposteriordistribution(usingdataonlyuptothatdate)toforecastfutureinterestrates. Our forecasting procedure thus captures uncertainty about both the parameter values and the unobservedstatesinourmodel; detailsaredescribedinAppendixC.Wecomparetheseforecasts to the forecasts from the model when we ignore the ELB and feed the interest rate data intothemodelwithoutaccountingfortheELB. [Figure7abouthere.] The left panels of Figure 7 display statistics from the posterior predictive density of the short–termnominalinterestratefromourbaselinemodelatdifferentdates. Theforecasthorizon extends for five years, and, in addition to mean and median predictions, shaded areas indicate 50 and 90 percent uncertainty bands. The dashed lines that extend below the ELB indicateposteriorquantilesoftheshadowratedistribution(asopposedtotheinterestratedistribution). The predictive density of the interest rate is a truncated version of the predictive densityoftheshadowratedistribution,sothequantilesoftheshadowratedistributionbecome exactly the quantiles of the interest rate distribution if the value is larger than the ELB. The truncation of the shadow–rate distribution causes substantial asymmetry in the interest rate 15

distributionleadingtomarkeddifferencesinthepredictivemeansandmediansofourbaseline model. The right panels of Figure 7 display statistics from the posterior predictive density of the short–term nominal interest rate when we ignore the ELB. Here, the predicted interest rate distribution can have negative support because the ELB is ignored. Because the posterior distribution of the shadow rate is roughly symmetric in our model, the posterior predictive meanandmediandonotdifferwhenweignoretheELB. In2008:Q4,thefirstperiodbeforetheELB(Panels(a)and(b)ofFigure7),themodelthat ignorestheELBdeliversthesameforecastsfortheshadowratesasourbaselinemodelbecause therehavebeennoELBperiodsintheestimationandallinterestratedatainthesamplehave beenpositive. WhentheELBisignored,shadow–rateforecastsareinterestrateforecasts,and the model predicts future negative rates due to the substantial decline in real activity. Our baseline model takes the ELB into account and truncates the distribution of expected future shadow rates to produce interest rate forecasts. In doing so, the mean interest–rate forecast risesappreciablyabovethemedianforseveralperiods. In2009:Q1,theperiodtheELBbeginstobind(Panels(c)and(d)ofFigure7),themodel thatignorestheELBproducesadifferentpredictivedensityfortheshadowratethanourbaselinemodel. Thereasonisthatweallowlaggedshadowratestodynamicallyaffectthecurrent shadow rate. When we ignore the ELB, the lagged shadow rate is the lagged interest rate, whichishigherinthisperiodthanourestimatedshadowratebecauseoftheELB.Thisshifts upthedistributionofexpectedshadowratesrelativetoourbaselinemodel,producingnotably more probability of positive interest rates in the subsequent few quarters even as the model predictsnegativeinterestrates. AsshowninPanel(c),accountingfortheELBproducesinterestrateforecaststhatplacesubstantialprobabilityonremainingexactlyattheELBforseveral quarters. As in Panel (a), the truncation of the shadow rate distribution in order to produce interest rate forecasts creates a divergence of mean and median estimates of interest rates for severalyears. [Figure8abouthere.] 16

In 2010:Q4, after the ELB had been binding for some time (panel (a) of figure 8), our baselinemodelstillpredictssubstantialprobabilityofinterestratesattheELBbecauseofthe estimated negative shadow rate. Moreover, the median interest rate forecast remains at the ELB for a number of quarters. By contrast, when the ELB is ignored (panel (b) of figure 8), the model predicts imminent departure from the effective lower bound because it does not estimate negative shadow rates and instead includes lagged observed interest rates in the dynamicsystem,whichbyconstructionarehigherthanourbaselinemodel’sestimatedshadow rates. This difference persists throughout the ELB episode, leading to an extended period in which ignoring the ELB in the estimation of our model would lead one to predict immediate departurefromtheELB.Towardtheendofoursample(2015:Q4,showninpanels(c)and(d) ofFigure8),theforecastsfromthetwomodelsaresimilar,inlargepartbecauseourestimated shadowrateisonlyslightlylessthantheELBatthatpoint. 4.2 Forecasting Performance We utilize the posterior predictive densities to calculate summary statistics in order to assess theforecastingperformanceofourbaselinemodelatitsvariants. Wefocusonthreestatistics: the root–mean–squared error, evaluated at the posterior predictive density’s mean; the mean absolutiondeviation,evaluatedattheposteriorpredictivedensity’smedian;andthepredictive scoreoftheposteriordistribution(GewekeandAmisano,2010). Weusethemeanforecastfor the root–mean–squared error statistic because the mean forecast minimizes expected square loss, and we use the median forecast for the mean absolute deviation statistic because the median minimizes expected absolute loss. In addition to our baseline model, we consider the version of the model where the ELB is ignored and a version of the model that includes the ELBintheestimationbutdoesnotincludethemedium–termyield.11 Tables 1 and 2 display forecast evaluation statistics for each model variant during our sample period to 2007 (prior to the ELB) and post 2007 for forecasts of future short–term interestrates, aswellasfuturemedium–termyields. Thestatisticsforourbaselinemodelare 11IntheversionofthemodelthatignorestheELB,laggedinterestratedataaretreatedaslaggedshadowratesand interestrateforecastsdonotaccountfortheELB,asintheright–handpanelsofFigures7and8. 17

shown as calculated. The statistics for the alternative model specifications are shown on a relativebasistothebaselinemodel. [Table1abouthere.] A key message from Table 1 is that the inclusion of the medium–term yield appears to help for forecasting short–term interest rates before 2007. At the one–quarter horizon, we findstatisticallysignificantdeclinesinforecastperformanceforallthreeofourstatisticswhen exclude the medium–term yield from our model. Because there are no periods at the ELB priorto2007,theroot–mean–squarederrorandmeanabsolutedeviationstatisticsareidentical for the baseline model and the model where the ELB is ignored for predictions about both short–term rates and medium–term yields. However, the predictive score statistic indicates that ignoring the ELB hurts forecasting performance. The reason is that the predictive score statistic incorporates information about the entire posterior density. During periods in which the nominal interest rate was low (for example, in 2003), accounting for the ELB has non– negligibleeffectsontheshapeofthepredictivedensity. Thepredictivescorestatisticaccounts for this affect, and the results illustrate the benefits of accounting for the ELB for density forecastsevenwhentheELBdoesnotyetbindfortheobserveddata. [Table2abouthere.] During the sample that includes the ELB, our baseline model almost uniformly out performs the model that ignores the ELB when predicting future short–term interest rates, as indicatedinPanelAofTable2.12 Thebaselinemodeldoesparticularlywellwhenevaluating forecastsbasedontheposteriormedianbecausethisisoftenexactlyattheELBoverthesample. At the 8–quarter horizon, ignoring the ELB appears to perform somewhat better for the root–mean–squarederrorstatistic, thoughnotintermsofaverageabsolutedeviationfromthe predictivemedian. Thisfindinghighlightstheimplicationsofusingdifferentforecaststatistics in samples affected by the ELB. In particular, the use of mean or median forecasts should be 12PointforecastsfromthealternativemodelthatignorestheELBturnouttobenegativeduringtheearlystagesof theELBperiodin2009. EvenwhentheseforecastsaresetequaltotheELB,theyareoutperformedbyourbaseline modelatsimilarsignificancelevelsbothintermsofRMSEandMAD. 18

carefully chosen for the relevant application in a model like ours. For forecasting short–term interestrates,duringthesamplethatstartsafter2007,ourbaselinemodelperformssimilarlyto themodelwithoutthemedium–termyield. Thereasonisthatbothmodelspredictsubstantial probabilityofremainingattheELBbecauseeachmodelproducesestimatesoftheshadowrate thatarelessthanzero. PanelBofTable2displaysourforecastevaluationstatisticsforforecastsofthemedium– termyieldduringthesamplethatstartsin2008. Foreachstatistic,ourbaselinemodelperforms statisticallybetterthantheversionthatignorestheELBforone–quarteraheadforecasts. The predictivescorestatisticalsoshowsthataccountingfortheELBimprovesforecastingperformanceovereachofthenextfourquarters. Thus,ourestimatesindicatethatinformationabout short–term rates and medium–term yields jointly improve the forecasts of the other, meaningthatthestatisticalconnectionisnon–negligibleandthataccountingfortheELBimproves densityforecasts. [Figure9abouthere.] 4.3 Forecast Uncertainty Naturally, the ELB has important effects on the predictive density for nominal interest rates when the predictive density for shadow rates has non-negligible coverage below the ELB. To illustratetherelevanceoftheseeffects,Figure9comparesforecastuncertaintyinourbaseline model against the alternative when the ELB is ignored. For the purpose of this figure, we measure forecast uncertainty by the conditional standard deviation of the predictive densities describedaboveforthenominalshort–termrateoneandeight-quartersahead. Overall, near– andmedium–termuncertaintyaboutfutureshort–termrateshasmostlydeclinedsincethemid- 1980s. Nevertheless, asthe levelof nominalrates hasbeen trendingdownover thisperiod as well, the probability of reaching the ELB has become more and more non-negligible; this is particularly true for longer–horizons forecasts made since 2000, causing forecast uncertainty inthebaselinemodeltodifferfromcomputationswhentheELBisignored. Not surprisingly, the onset of the last NBER recession in 2007 is reflected in higher esti- 19

mated levels of stochastic volatility to all shocks in our model, leading to increased shadow– rate uncertainty. When the ELB is ignored, this directly translates into larger near–term uncertaintyaboutthenominalshort–terminterestrate. ByaccountingfortheELB,ourbaseline model recognizes that during the last recession, increased shadow–rate uncertainty is accompanied by a marked downward shift of the shadow–rate distribution to values below the ELB such that the truncated distribution of actual nominal rates almost collapses at values at or slightlyabovetheELB.Consequently,near–termuncertaintyforshort–termnominalratesdeclines during the last recession when properly accounting for the ELB, as shown in Panel (a) of Figure 9. In contrast, as shown in Panel (b) of the figure, medium–term uncertainty about nominal interest rates increases with the increasing shadow–rate uncertainty, though not by quite as much, as nominal rates are projected to return to their estimated non–negative trend level. 5 Expected Interest Rate Paths Our model’s predicted paths for future short–term interest rates imply estimates of expected interest rate paths, and thus term–premiums for yields along the term–structure of interest rates. We define the expectations component of a yield on a bond with maturity h periods in thefutureas   h−1 1 (cid:88) e t,t+h ≡ E t i t+j. h j=0 Theobservedlonger-terminterestrateisgivenby e +p t,t+h t,t+h where p represents premiums. Note that for any j, E (i ) ≥ ELB and the posterior t,t+h t t+j expectationcanbeconstructedthroughsimulationofthepredictivedensityfornominalshort– terminterestratesasdescribedinAppendixC. 20

[Figure10abouthere.] We focus on the expectations component of the 5–year Treasury yield (y ), which is in t ourdataset. Weconstructtheexpectationscomponentofy usingtheout–of–sampleposterior t predictivedensitiesattimet. Figure10displaysourestimated5–yearexpectationscomponent, along with uncertainty bands. Also shown are the expectations component from Kim and Wright(2005)aswellasthe5–yearTreasuryyieldinourdatasample. Despitenotimposingtheno–arbitragecross–equationrestrictionsusedinKimandWright (2005), our model produces similar estimates of the expectations component of the 5–year yield over the sample. The largest deviations are in the early 1990s and during the period in whichtheELBbinds. Oneadvantageofourapproachtomodelinginterestratesisthatwecan easily incorporate non–stationarity into the inflation process. Given that our estimated trend inflationratehasfallensince1990,thismayhelpexplainwhyourestimatesoftheexpectation component of the 5–year yield are higher than the estimates of Kim and Wright (2005), who assumethatinflationisastationaryprocess. Interestingly,ourmodelproduceslargerpremiums than the model of Kim and Wright (2005) during the period in which the ELB binds. While the truncation of future shadow rates could have produced a higher expectations component for our model, it appears that the lagged shadow rates keep the interest rate low enough for longenoughtoproducerelativelylowpathsforinterestrates. Notably, the uncertainty bands in Figure 10 are not of constant width. The stochastic volatilityinourmodelleadstochangesinthewidthofuncertaintybands,whichisespecially pronounced at the beginning of the Great Recession, a period in which our model estimates thatvolatilityroseappreciably. 6 Conclusion In this paper, we develop a methodology to account for the ELB in otherwise conditionally–linear Gaussian time series models. Further, we demonstrate how to estimate theparametersandlatentstatesofsuchamodelwithanotherwisestandardBayesianMCMC sampler. 21

We document that including the ELB can have drastic effects for interest rate forecasts, as well as the expectations component of longer–term yields and thus also the computation ofterm–premiums. Further,accountingfortheELBusingourshadow–rateapproachappears to improve forecast performance. We also estimate changes in the trend real rate, defined as a long–term forecast of the real interest rate, and find that any decline in the trend real rate since the onset of the Great Recession is best characterized as a continuation of a downward trajectorythatbeganwellbefore. APPENDIX A Sampling States with Censored Data OurGibbssamplingprocedureisageneralizationofParketal.(2007)thatappliesthemethodology of Hopke et al. (2001). Assume that the vector ξ is a random variable that evolves so t that ξ = A ξ +B ε (13) t t t−1 t t where ε is a vector of standard normal random variables of appropriate length and the set quenceofmatrices{A }T and{B }T aregiven.13 t t=1 t t=1 Definethevector X = C ξ (14) t t t where the sequence of matrices {C }T are known.14 We assume that X has a partition t t=1 t made up of a vector shadow rates, (S ), and a partition of variables that are unconstrained by t 13Inourapplication,describedinthemainbodyofthepapernotethatwehaveaconstantA =A. t 14Inourapplication,describedinthemainbodyofthepapernotethatwehaveaconstantC =C. t 22

theELB,(M ).15 Thatis, t   S t X =   (15) t   M t Theobserveddataare   max(S ,ELB) t Z =   (16) t   M t wherethemaxoperatorisappliedelementbyelement. TheELBactsasacensoringfunction in the model through the max operator, though more general censoring functions could be used. DefineX ≡ [X(cid:48),X(cid:48),...,X(cid:48) ](cid:48),andZ ≡ [Z(cid:48),Z(cid:48),...,Z(cid:48) ](cid:48). WesplitZ intotwoparts, 1 2 T 1 2 T onepartcontainingallnon-interestratedataandallobservationsforinterestratesthatarenot constrained by the ELB, ZNC, and another part with the interest rate data constrained at the ELB,ZC.16 ThecorrespondingelementsofX areXNC andXC. Notethat,theelementsof XC areallshadowratesthatarelessthantheELB. Givenanormaldistributionforξ ,itfollowsthatthevectorsXNC andξ = [ξ(cid:48),ξ(cid:48),...,ξ(cid:48) ](cid:48) 0 1 2 T haveamultivariatenormal(prior)distribution       XNC µ V V   ∼ N  X, XX X,ξ  (17)       ξ µ V V ξ ξ,X ξ,ξ andwecanderivetheposteriordistributionforξ conditionalonobservedinterestrates,when theELBisnotbinding,aswellasallobservationsformacroeconomic,non-interest-ratevariables(M ): t ξ (cid:12) (cid:12) (cid:0) XNC = ZNC(cid:1) ∼ N (cid:16) µˆ ξ ,Vˆ ξ,ξ (cid:17) . (18) 15In the application described above, there are in principle two shadow rates, one associated with the short–term interest rate and one associated with the medium–term yield described; in practice, the ELB constraint has been bindingonlyfortheformer,however. 16Accordingly,ZC consistssolelyofobservationsforinterestratesthatareequaltoELB. 23

Ingeneral,theposteriormomentsin(18)aregivenby µˆ = µ +K (cid:0) ZNC −µ (cid:1) with K = V V−1 (19) ξ ξ X ξ,X XX and Vˆ = V −V V−1 V . (20) ξ,ξ ξ,ξ ξ,X XX X,ξ Typically, these posterior moment matrices will by quite large; µ is, for example, a vector ξ of length T ×N = 224×12 = 2,688 in our application, However, the Kalman smoother, ξ adapted for handling missing observations for interest rates when the ELB binds, provides a convenient way to recursively compute the moments in (19) and (20) while recovering the distribution of ξ conditional on observations for ZNC. To this point, our procedure amounts totreatingtheobservationsforZC asmissingdata.17 We then note that the information contained in the interest rate data at the ELB is that XC ≤ ELB (foreveryelementofXC). Theposteriordistributionofξ,conditionalonZ,is then (cid:16) (cid:17) ξ|(X = Z) ∼ TN µˆ ,Vˆ ;XC ≤ ELB (21) ξ ξ,ξ whereTN standsforatruncatednormalsuchthatitsdensityfunctionis (cid:16) (cid:17) Pr(ξ|Z) ∝ φ µˆ ,Vˆ 1(XC ≤ ELB) (22) ξ ξ,ξ whereφisthemultivariatenormaldensityfunctionandXC isashadowratedrawwhereevery element is below the ELB. To sample from the posterior distribution of ξ conditional on all observations in Z, we first draw ξ from Pr (cid:0) ξ (cid:12) (cid:12)XNC = ZNC(cid:1) . We then reject draws until we find a draw that satisfies the requirement that XC ≤ ELB for every element. Rejection samplingisthusdoneonanentiredrawofξ,whichcorrespondstoanentiredrawofthetime seriesforξ . t Inourbaselineframework,laggedvaluesofξ appearasexplanatoryvariablesandarenot t 17Alternatively,ChanandJeliazkov(2009)describewaystoefficientlycomputethemomentsin(19)and(20)based onsparsematricesthatexploitthestatespacestructureinherentin(17). 24

censored. AstraightforwardextensionistoincorporateagivennumberofplagsofZ ,which t includesinterestratedatathatcanbeconstrainedtheELB.Inthiscase,wechange(13)tobe ξ = A ξ +F ζ +B ε (23) t t t−1 t t−1 t t ζ ≡ [Z(cid:48) ,Z(cid:48) ,...,Z(cid:48) ](cid:48) (24) t−1 t−1 t−2 t−p where{F }T areconformablematricesthatareknown. Theposteriorofξcanbeconstructed t t=1 exactlyasinourbaselinemodel,treating{ζ }T asexogenousineveryperiodbecausethe t−1 t=1 rejection step will ensure that the sampled values of ξ are consistent with ζ for all t. For t−1 comparison,themodelsofIwataandWu(2006)andNakajima(2011)canbecast,conditional on parameter values, as special cases of this setup in which the matrix A = 0. A notable t difference in the posterior simulation of the model is that the truncated distributions in Iwata and Wu (2006) and Nakajima (2011) can be cast as period–by–period truncated normals. By contrast,ourposteriorestimatesrequirerejectionsamplingonanentiretimeseriesdrawofξ. B Priors and Posterior Sampling MCMCestimatesofthemodelareobtainedfromaGibbssampler. Thesamplerisrunmultiple times with different starting values and convergence is assessed with the scale reduction test of Gelman et al. (2003).18 For each run, 20,000 draws are stored after a burn-in period of 100,000draws;thepost-burnindrawsfromeachrunarethenmerged. Ourmodelsconsistoftwovectorsoflatentstates: (cid:20) (cid:21)(cid:48) ξ = π¯ r¯ s¯ y¯ π˜ u˜ s˜ y˜ e t t t t t t t t t π,t (cid:20) (cid:21)(cid:48) and h t = log(σ π 2 ¯,t ) log(σ π 2 ˜,t ) log(σ u 2 ˜,t ) log(σ s˜ 2 ,t ) log(σ y 2 ˜,t ) log(σ e 2 ,t ) 18Specifically, for every model, 10 independent runs for the Gibbs sampler were evaluated; each run initialized with different starting values drawn from the model’s prior distribution. Convergence is deemed satisfactory when the scale reduction statistics for every parameter and latent variable are below 1.2; (values close to 1 indicate good convergence). 25

aswellastheparametersgoverningthemeansandvolatilityofshockstoh ,see(7),stackedin t thevectorsµandφ2,19 thevarianceofshockstothetrendrealrate,denotedσ2,thetransition r¯ coefficients of the gap VAR (6), stacked in a vector a, and lower diagonal elements of gap shock loadings B in (6) that can be stacked in a vector denoted b. For ease of reference, (cid:20) (cid:21)(cid:48) all parameters are collected in the vector θ = a(cid:48) b(cid:48) σ2 φ2(cid:48) , µ(cid:48) . Furthermore, since r¯ MCMCestimatesofhT willbeobtainedfromthemulti–movefilterofKimetal.(1998), the useofasetofdiscreteindicatorvariables,sT,isrequiredtotoapproximatelogη2 in(7)with j,t a mixture of normals. Conditional on draws for the parameters, (θ), and log-volatilities hT, we can construct matrices A, {B }T , and C and to obtain the linear, Gaussian state space t t=1 systemdescribedbyequations(13)and(14)inAppendixA. Fortheinitialvaluesofthelatentstates,thefollowingpriorswereused:     ξ¯ Ω¯ 0 ξ ∼ N (E(ξ ),Ω) with E(ξ ) =   and Ω =   (25) 0 0 0     0 0 Ω˜ An uninformative prior for the initial gap levels is obtained by setting Ω˜ equal to the ergodic variance-covariance matrix of the gaps implied by the VAR (6), evaluated at the time zero drawsforthestochasticvolatilities,encodedinΣ ,foreveryMCMCdraw.20 Thepriorforthe 0 initialtrendlevelsaresettobeconsistentwith       π¯ 2 16 −16 −16 0             s¯  ∼ N 4,−16 116 16  (26)  0           y¯ 5 −16 16 116 0 which implies a fairly vague prior level of the real-rate trend, r¯ = s¯ − π¯ while taking t t t correlated signals about the initial level of trend inflation from readings on all three nominal variables. 19Theauto-regressivecoefficientsρ ineachofthestochasticvolatilityequationsin(7)arefixedat0.99. j 20InthecaseofaVAR(1),theergodicvariance-covariancematrixsolvesΩ˜ =AΩ˜A(cid:48)+BΣ B(cid:48) forgivenvalues 0 ofA,B,andΣ . 0 26

Thepriorfortheaveragelevelofthelog-variancesis: (cid:18) (cid:19) 1 25 1 µ ∼ N 2·log − ,25 ⇒ E(σ¯ ) = (27) j t 10 2 10 The prior distribution for each of the the parameters for the parameters φ2 is a univariate inverse–Wishartdistributionwith30degreesoffreedom(correspondingtoaninversegamma with a shape parameter set equal to a value of 15) and a mean equal to 0.22 which coincides with the fixed coefficient-value of 0.2 used by Stock and Watson (2007) in their univariate modelforinflation. The parameter governing the variability of real-rate trend shocks, σ2, has a univariate r¯ inverse–Wishart distribution with 3 degrees of freedom (corresponding to an inverse–gamma distribution with a shape parameter equal to 1.5 degrees of freedom) and is centered around a prior mean of 0.22. While this vaguely informative prior embeds the belief that the trend shocks explain only a small share of variations in real rates, it also helps to avoid the pile–up problem—known,forexample,fromStockandWatson(1998)andconsideredinthecontext ofestimatingσ2 alsobyLaubachandWilliams(2003)aswellasClarkandKozicki(2005)— r¯ bykeepingposteriordrawsfortheparameterawayfromzero. While the priors for a and b are taken to be completely diffuse, starting values for the Gibbssampleraredrawnfromthefollowingnormaldistributions: a ∼ N (cid:0) 0,0.32·I (cid:1) b ∼ N (0,I) (28) The Gibbs sampler is initialized with values drawn from the prior for hT and θ and then generatesdrawsfromthejointposteriordistribution p (cid:0) ξT,hT,a,b,σ2,µ,φ2,sT(cid:12) (cid:12)YT(cid:1) r¯ byiteratingoverdrawsfromthefollowingconditionaldistributions: 1. Draw from p (cid:0) ξT(cid:12) (cid:12)hT,a,b,σ2,φ2,µ,YT(cid:1) with the disturbance smoothing sampler of r¯ Durbin and Koopman (2002) and rejection sampling for the shadow rate when the ob- 27

servednominalshort–termrateisattheELBasdescribedinAppendixA. 2. Drawfromp (cid:0) a (cid:12) (cid:12)ξT,hT,b,σ2,φ2,µ,YT(cid:1) = p (cid:0) a (cid:12) (cid:12)ξT,hT,b (cid:1) ,anormalconjugateposr¯ terior for a VAR with known heteroscedasticity, with rejection sampling to ensure a stationaryVAR(CogleyandSargent,2005a;Clark,2011) 3. Draw from p (cid:0) b (cid:12) (cid:12)ξT,hT,a,σ2,φ2,µ,YT(cid:1) via recursive Bayesian regressions with r¯ knownheteroscedasticitytoorthogonalizethegapshocksoftheVARin(6). 4. Drawfromtheinverse-gammaconjugateposteriorsforσ2: r¯ p (cid:0) σ2(cid:12) (cid:12)ξT,hT,a,b,φ2,µ,YT(cid:1) = p (cid:0) σ2(cid:12) (cid:12)ξT(cid:1) r¯ r¯ 5. Drawfromtheinverse-gammaconjugateposteriorsforφ2: p (cid:0) φ2(cid:12) (cid:12)ξT,hT,a,b,σ2,µ,YT(cid:1) = p (cid:0) φ¯2(cid:12) (cid:12)hT(cid:1) r¯ 6. DrawthemixtureindicatorssT from: p (cid:0) sT(cid:12) (cid:12)ξT,hT,a,b,σ2,φ2,µ,YT(cid:1) r¯ 7. Drawfromp (cid:0) hT,µ (cid:12) (cid:12)sT,ξT,a,b,σ2,φ2,YT(cid:1) = p (cid:0) hT(cid:12) (cid:12)sT,ξT,φ¯2(cid:1) embeddingthedisr¯ turbance smoothing sampler of Durbin and Koopman (2002) in a linear state space for (cid:0) hT,µ (cid:1) asinKimetal.(1998).21 Strictly speaking, this is not a simple Gibbs sampler consisting of steps 1 – 7, but rather a Gibbs-within-GibbssamplerwiththeouterGibbssampleriteratingbetween p (cid:0) sT,ξT,θ (cid:12) (cid:12)hT,YT(cid:1) (thus,ablockconsistingofsteps1through6) and p (cid:0) hT,µ (cid:12) (cid:12)sT,ξT,a,b,σ2,φ2,YT(cid:1) (step7), r¯ similartothediscussionbyDelNegroandPrimiceri(forthcoming). 21Theconstantµisembeddedinthestatespaceasaunitrootwithoutshocks,whichimprovestheefficiencyofthe GibbssamplerbyjointlysamplinghT andµ. 28

C Computation of Predictive Densities Inordertoderivepredictivedensitiesforinterestrates(andotherdatainZ ),wefirstproceed t by characterizing the predictive density for the shadow rate, included in the non-censored vector of variables X described in Appendix A. Apart from handling the truncation issues t related to the ELB, our approach is fairly standard, building, for example, on the work by Geweke and Amisano (2010), Christoffel et al. (2010), and Warne et al. (2015). Given the truncation issues for interest rates and the fat tails introduced into the predictive density by thestochasticvolatilityspecification,wehavechosentocomputethepredictivedensitybased on the mixture of normals that is implied by the draws from our MCMC sampler, instead of approximating the predictive density solely based on its first two moments, treating the predictive density as a normal distribution, as has been done, for example, by Adolfson et al. (2007)inthecaseoflinearized,constant-parameterDSGEmodels. In order to compute the predictive density for Z jumping off data at time t, we first t+h employtheMCMCsamplerdescribedinAppendixBtore-estimateallmodelparametersand latentvariables(θ,ξ andh )conditionalondataavailablethroughtimet. Drawsfromthis 1:t 1:t MCMCsamplerwillhenceforthbeindexedbyk. Conditionalondraws(ξk,hk,θk),itisstraightforwardtocomputethepredictivemeanfor t t uncensoredvariables: E (cid:16) X t+h (cid:12) (cid:12)ξk t ,hk t ,θk (cid:17) = Ck (cid:16) Ak (cid:17)h ξk t (29) and the predictive mean, conditional solely on data through t, can then be approximated by averagingoverthemeansderivedfromeachMCMCdraw: E(X t+h |Z 1:t ) ≈ (cid:88) E (cid:16) X t+h (cid:12) (cid:12)ξk t ,hk t ,θk (cid:17) (30) k However, inordertocharacterizetheentirepredictivedensityforuncensoredvariablesor even the predictive density for interest rates, which are subject to censoring due to the ELB constraint, we need to account for non-linearities in the distribution for future ξ arising t:t+h 29

from the stochastic volatility shocks in our model. To do so, we simulate J = 100 trajectories, each indexed by j, of hk,j for each draw k from the MCMC sampler. Conditional on t:t+h (ξk,hk,θk) as well as hk,j , ξ and X are normally distributed. The means of the t t t:t+h t:t+h t:t+h conditionalnormalsarestillgivenby(29),andthevariancesaregivenby Var (cid:16) ξ t+h (cid:12) (cid:12)ξk t ,hk t ,hk t+ ,j 1:t+h ,θk (cid:17) = (cid:88) h (cid:16) Ak (cid:17)i Σ t+i k,j (cid:18) (cid:16) Ak (cid:17)i (cid:19)(cid:48) (31) i=0 Var (cid:16) X t+h (cid:12) (cid:12)ξk t ,hk t ,hk t+ ,j 1:t+h ,θk (cid:17) = CkVar (cid:16) ξ t+h (cid:12) (cid:12)ξk t ,hk t ,hk t+ ,j 1:t+h ,θk (cid:17)(cid:16) Ck (cid:17)(cid:48) (32) Recall that X contains shadow rates S and the predictive density for nominal t:t+h t:t+h interest rates can be constructed by computing the moments and density function of the truncated normal distribution conditional on every pair of draws (k,j). Specifically, consider the short–termshadowrates anddenoteitsconditionalpredictivemeanandvariancefordraw t+h (k,j), computed according to (29) and (32), by mk,j = mk and vk,j. The predictive density fori = max(s ,ELB),conditionalondraw(k,j),isthencharacterizedby: t t Pk,j ≡ Pr(s < ELB|·) (33) t+h pk,j ≡ φ(s < ELB|·) (34) t+h (cid:18) √ pk,j (cid:19) E(i |·) = Pk,j ·ELB+(1−Pk,j)· mk,j + vk,j (35) t+h Pk,j (cid:32) (ELB−mk,j)pk,j (cid:18) pk,j (cid:19)2(cid:33) Var(i |·) = (1−Pk,j)·vk,j 1+ √ + (36) t+h vk,jPk,j Pk,j f(i |·) = 1(i = ELB)Pk,j...+ (37) t+h t+h (cid:16) √ (cid:17) φ (i −mk,j)/ vk,j t+h +1(i > ELB)(1−Pk,j)· √ (38) t+h vk,jPk,j (cid:16) (cid:17) where “·” in “(i |·)” is a placeholder for the conditioning set ξk,hk,hk,j ,θk , and t+h t t t+1:t+h φ(·)isthestandardnormalpdf. Basedontheseconditionaldistributionforeverydraw(k,j)weconstructpredictivemeans andvariancesbyaggregatingacrossdrawsusingthelawofiteratedexpectationsandthelawof 30

totalvariance,respectively. Predictivemediansareobtainedfromsimulateddrawsofi for t+h every draw (k,j) and computing the 50% quantile across all draws. The time t contribution tothepredictivescoresarecomputedfromaveragingacrossdrawstheconditionallikelihoods f(i |·); as in Geweke and Amisano (2010) the predictive score for an evaluation window t+h rangingfromt = T throught = T isthengivenas 1 2 1 (cid:88) T2 PS = logf(i |Z ) . (39) t+h 0:t T −T +1 2 1 t=T1 References MalinAdolfson,JesperLinde,andMattiasVillani.ForecastingPerformanceofanOpenEconomyDSGEModel. EconometricReviews,26(2-4):289–328,2007. Michael D. Bauer and Glenn D. Rudebusch. Monetary policy expectations at the zero lower bound. FederalReserveBankofSanFranciscoWorkingPaperSeries,2014. StephenBeveridgeandCharlesR.Nelson.Anewapproachtodecompositionofeconomictime series into permanent and transitory components with particular attention to measurement ofthe’businesscycle’. JournalofMonetaryEconomics,7(2):151–174,1981. FischerBlack. Interestratesasoptions. TheJournalofFinance,50(5):1371–1376,1995. JohnY.CampbellandRobertJ.Shiller. Cointegrationandtestsofpresentvaluemodels. The JournalofPoliticalEconomy,95(5):1062–1088,October1987. John Y. Campbell and Robert J. Shiller. Yield spreads and interest rate movements: A bird’s eyeview. TheReviewofEconomicStudies,58(3):495–514,1991. JoshuaC.C.ChanandIvanJeliazkov. Efficientsimulationandintegratedlikelihoodestimation in state space models. International Journal of Mathematical Modelling and Numerical Optimization,1(1/2):101–120,2009. Joshua C.C. Chan and Rodney Strachan. The zero lower bound: Implications for modelling theinterestrate. WorkingPaper,2014. KaiChristoffel,AndersWarne,andGnterCoenen. ForecastingwithDSGEmodels. Working PaperSeries1185,EuropeanCentralBank,May2010. ToddE.Clark. Real-timedensityforecastsfrombayesianvectorautoregressionswithstochasticvolatility. JournalofBusiness&EconomicStatistics,29(3):327–341,2011. TimotheyCogleyandThomasJ.Sargent. Measuringpriceleveluncertaintyandinstabilityin theU.S.,1850–2012. ReviewofEconomicsandStatistics,Forthcoming. 31

TimothyCogleyandThomasJ.Sargent.Driftandvolatilities: Monetarypoliciesandoutcomes inthepostWWIIU.S. ReviewofEconomicDynamics,8(2):262–302,April2005a. Timothy Cogley and Thomas J. Sargent. Drifts and volatilities: monetary policies and outcomesinthepostWWIIUS. ReviewofEconomicDynamics,8(2):262–302,2005b. MarcoDelNegroandGiorgioE.Primiceri. Time-varyingstructuralvectorautoregressionand monetarypolicy: Corrigendum. ReviewofEconomicStudies,forthcoming. Fancis X. Diebold and Canlin Li. Forecasting the term sturcutre of government bond yields. JournalofEconometrics,(130):337–364,2006. Francis X Diebold and Roberto S Mariano. Comparing predictive accuracy. Journal of Business&EconomicStatistics,13(3):253–63,July1995. J.DurbinandS.J.Koopman. Asimpleandefficientsimulationsmootherforstatespacetime seriesanalysis. Biometrika,89(3):603–615,September2002. AndrewGelman,JohnB.Carlin,HalS.Stern,andDonaldB.Rubin. BayesianDataAnalysis. ChapmanandHall,2ndedition,2003. John Geweke and Gianni Amisano. Comparing and evaluating bayesian predictive distributions of asset returns. International Journal of Forecasting, 26(2):216–230, 2010. Special Issue: BayesianForecastinginEconomics. JamesD.Hamilton,EthanS.Harris,JanHatzius,andKennethD.West. Theequilibriumreal fundsrate: Past,presentandfuture. WorkingPaper,March2015. Philip K. Hopke, Chuanhai Liu, and Donald B. Rubin. Multiple imputation for multivariate datawithmissingandbelow-thresholdmeasurements: Time-seriesconcentrationsofpollutantsinthearctic. Biometrics,57(1):22–33,2001. ISSN0006341X. Hibiki Ichiue and Yoichi Ueno. Estimating Term Premia at the Zero Bound: An Analysis of Jpanese,US,andUKYields. BankofJapanWorkingPaperSeries,2013–E–8,2013. Shigeru Iwata and Shu Wu. Estimating monetary policy effects when interest rates are close tozero. JournalofMonetaryEconomics,53(7):1395–1408,2006. Michael T. Kiley. What can the data tell us about the equilibrium real interest rate? Finance andEconomicsDisucssionSeries,(2015-77),2015. DonKimandKennethJ.Singleton. Termstructuremodelsandthezerobound: Anemprirical investigationofJapaneseyields. Mimeo,2011. DonH.KimandJonathanH.Wright. Anarbitrage–freethree–factortermstrucutremodeland the recent behavior of long–term yields and distant–horizon forward rates. FEDS Working PaperSeries,2005. Sangjoon Kim, Neil Shephard, and Siddhartha Chib. Stochastic volatility: Likelihood inference and comparison with arch models. The Review of Economic Studies, 65(3):361–393, July1998. 32

Robert G. King and Andre´ Kurmann. Expectations and the term structure of interest rates: Evidenceandimplications. FederalReserveBankofRichmondEconomicQuarterly,(Fall): 49–95,August2002. ToddE.ClarkandSharonKozicki. Estimatingequilibriumrealinterestratesinrealtime. The NorthAmericanJournalofEconomicsandFinance,16(3):395–413,December2005. Leo Krippner. Measuring the stance of monetary policy in zero lower bound environments. EconomicsLetters,118(1):135–138,2013. LeoKrippner. ZeroLowerBoundTermStructureModeling: Apractitioner’sGuide. Palgrave Macmillan,2015. ThomasLaubachandJohnC.Williams. Measuringthenaturalrateofinterest. TheReviewof EconomicsandStatistics,85(4):1063–1070,November2003. Thomas A. Laubach and John C. Williams. Measuring the natural rate of interest redux. HutchinsCenteronFiscal&MonetaryPolicyatBrookings,(15),2015. MarcoJ.LombardiandFenZhu. AshadowpolicyratetocalibrateUSmonetarypolicyatthe zerolowerbound. BISWorkingPaper,425,2014. ThomasA.LubikandChristianMatthes.Calculatingthenaturalrateofinterest: Acomparison oftwoalternativeapproaches. EconomicBrief,(15-10),2015. ElmarMertens. Measuringthelevelanduncertaintyoftrendinflation. ReviewofEconomics andStatistics,forthcoming. Jouchi Nakajima. Monetary policy transmission under zero interest rates: An extended timevarying parameter vector autoregression approach. Bank of Japan Working Paper Series, 2011. Jung Wook Park, Marc G. Genton, and Sujit K. Ghosh. Censored time series analysis with autoregressive moving average models. The Canadian Journal of Statistics / La Revue CanadiennedeStatistique,35(1):151–168,2007. MarcelA.Priebsch.Computingarbitrage-freeyieldsinmulti-factorgaussianshadow-rateterm structuremodels. FinanceandEconomicsDisucssionSeries,(2013-63),2013. GiorgioE.Primiceri. Timevaryingstructuralvectorautoregressionsandmonetarypolicy. The ReviewofEconomicStudies,72(3):821–852,2005. LukaszRachelandThomasD.Smith. Seculardriversoftheglobalrealinterestrate. Bankof EnglandStaffWorkingPaper,(571),2015. ChristopherA.Sims. Macroeconomicsandreality. Econometrica,48(1):1–48,1980. James H. Stock and Mark W. Watson. Median unbiased estimation of coefficient variance in atime-varyingparametermodel. JournaloftheAmericanStatisticalAssociation,93(441): 349–358,1998. James H. Stock and Mark W. Watson. Why has U.S. inflation become harder to forecast? JournalofMoney,CreditandBanking,39(S1):3–33,February2007. 33

LawrenceH.Summers. U.S.economicprospects: Secularstagnation, hysteresisandthezero lowerbound. BuinessEconomics,49(2):65–73,2014. Anders Warne, Gnter Coenen, and Kai Christoffel. Marginalized predictive likelihood comparisonsoflineargaussianstate-spacemodelswithapplicationstoDSGE,DSGE-VAR,and VARmodels. mimeoEuropeanCentralBank,December2015. Cynthia Wu and Fan Dora Xia. Measuring the macroeconomic impact of monetary policy at thezerolowerbound. JournalofMoney,Credit,andBanking,48(2–3):253–291,2016. 34

Figure1: DataandEstimatedTrends 18 18 16 16 14 14 12 12 10 10 8 8 6 6 4 4 2 2 0 0 1960 1970 1980 1990 2000 2010 1960 1970 1980 1990 2000 2010 (a)FederalFundsRate (b)5–YearTreasuryYield 12 5 10 4 8 3 6 2 4 1 2 0 0 −1 −2 −4 −2 −6 −3 1960 1970 1980 1990 2000 2010 1960 1970 1980 1990 2000 2010 (c)PCEInflationRate (d)Unemployment–RateGap Note: Red solid lines are data. Shaded areas indicate 50 and 90 percent uncertainty bands, dashed lines are posterior means. The posterior distribution is derived from conditioning the model on the entire data sample 1960:Q1 – 2015:Q4. Uncertainty bands reflect the joint uncertainty about modelparametersandstates. DottedverticallinesindicateNBERrecessionpeaksandtroughs. 35

Figure2: ShadowRateEstimates 6 4 2 0 −2 −4 2007 2008 2009 2010 2011 2012 2013 2014 2015 (a)BaselineModel 6 4 2 0 −2 −4 2007 2008 2009 2010 2011 2012 2013 2014 2015 (b)TreatingELBDataasMissing Note: Shaded areas indicate 50 and 90 percent uncertainty bands, dashed lines are posterior means. Theposteriordistributionisderivedfromconditioningthemodelontheentiredatasample 1960:Q1 – 2015:Q4. Uncertainty bands reflect the joint uncertainty about model parameters and states. Results in Panel (b) are conditioned on a data set where observations on short–term rates aretreatedasmissingdataduringtheELBperiod. 36

Figure3: ShadowRateEstimates,AComparison 2 JM (baseline) Krippner (2013) 1 Wu−Xia (2016) 0 −1 −2 −3 −4 −5 −6 2009 2010 2011 2012 2013 2014 2015 Note: Shadedareasindicate50and90percentcrediblesets,dashedlinesareposteriormeansfrom our baseline model (“JM baseline”). The posterior distribution is derived from conditioning the modelontheentiredatasample1960:Q1–2015:Q4. Uncertaintybandsreflectthejointuncertainty aboutmodelparametersandstates. 37

Figure4: TheRealRateintheLongRun 5 4 3 2 1 0 −1 1985 1990 1995 2000 2005 2010 2015 (a)QuasiReal–TimeEstimate 5 4 3 2 1 0 −1 1985 1990 1995 2000 2005 2010 2015 (b)SmoothedEstimates Note: Shadedareasindicate50and90percentuncertaintybands,dashedlinesareposteriormeans. ResultsshowninPanel(a)reflecttheendpointsofsequentiallyre-estimatingtheentiremodelover growingsamplesofquarterlyobservationsstartingin1960:Q1,thusreflecting“filtered”estimates of the model’s latent variables. Results shown in Panel (b) reflect “smoothed” estimates using all available observations from 1960:Q1 through 2015:Q4. Uncertainty bands reflect the joint uncertaintyaboutmodelparametersandstates. 38

Figure5: TheTrendNominalRatewhentheELBisIgnored 5 Baseline ELB ignored 4 3 2 1 0 2009 2010 2011 2012 2013 2014 2015 Note: Shadedareaindicates50percentuncertaintybandsandthedashedlineisthemeanestimate for our baseline model. This solid red lines indicate 50 percent uncertainty bands and the thick solid red line is the mean estimate when we ignore the ELB. The posterior distribution is derived from conditioning the model on the entire data sample 1960:Q1 – 2015:Q4. Uncertainty bands reflectthejointuncertaintyaboutmodelparametersandstates. 39

Figure6: EstimatesWithandWithoutMedium–TermYield 8 8 Without medium−term yield Without medium−term yield Baseline Baseline 6 6 4 4 2 2 0 0 −2 −2 −4 −4 −6 −6 −8 −8 2007 2008 2009 2010 2011 2012 2013 2014 2015 2007 2008 2009 2010 2011 2012 2013 2014 2015 (a)ShadowRate (b)ShadowRate(missingELBData) 5 6 Without medium−term yield Baseline 4 4 3 2 2 1 0 0 Without medium−term yield Baseline PCE inflation (4 quarter) −1 −2 1985 1990 1995 2000 2005 2010 2015 1985 1990 1995 2000 2005 2010 2015 (c)RealRateTrend (d)InflationTrend Note: Shaded areas indicate 50 and 90 percent credible sets, dashed and solid lines are posterior means derived from the smoothed, full-sample posterior of the respective models. The posterior distributionisderivedfromconditioningthemodelontheentiredatasample1960:Q1–2015:Q4. Uncertaintybandsreflectthejointuncertaintyaboutmodelparametersandstates. ResultsinPanel (b)areconditionedondatasetswhereobservationsonshort–termratesaretreatedasmissingdata duringtheELBperiod. 40

Figure7: Short–TermInterestRateForecasts,StartofELB 8 8 6 6 4 4 2 2 0 0 −2 −2 −4 −4 −6 −6 −8 −8 Q1−2009 Q1−2010 Q1−2011 Q1−2012 Q1−2013 Q1−2009 Q1−2010 Q1−2011 Q1−2012 Q1−2013 (a)Forecastsfrom2008:Q4,Baseline (b)Forecastsfrom2008:Q4,IgnoringELB 8 8 6 6 4 4 2 2 0 0 −2 −2 −4 −4 −6 −6 −8 −8 Q1−2010 Q1−2011 Q1−2012 Q1−2013 Q1−2014 Q1−2010 Q1−2011 Q1−2012 Q1−2013 Q1−2014 (c)Forecastsfrom2009:Q1,Baseline (d)Forecastsfrom2009:Q1,IgnoringELB Note: Shaded areas indicate 50 and 90 percent credible sets, solid lines are posterior medians, widedashedlinesareposteriormeansoftheprojectedinterestrate. Intheleftpanels,dashedlines lessthantheELBaretheposteriormedianand50and90percentcrediblesetsoftheshadowrate. 41

Figure8: Short–TermInterestRateForecasts,DuringELB 8 8 6 6 4 4 2 2 0 0 −2 −2 −4 −4 −6 −6 −8 −8 Q1−2011 Q1−2012 Q1−2013 Q1−2014 Q1−2015 Q1−2011 Q1−2012 Q1−2013 Q1−2014 Q1−2015 (a)Forecastsfrom2010:Q4,Baseline (b)Forecastsfrom2010:Q4,IgnoringELB 8 8 6 6 4 4 2 2 0 0 −2 −2 −4 −4 −6 −6 −8 −8 Q1−2016 Q1−2017 Q1−2018 Q1−2019 Q1−2020 Q1−2016 Q1−2017 Q1−2018 Q1−2019 Q1−2020 (c)Forecastsfrom2015:Q4,Baseline (d)Forecastsfrom2015:Q4,IgnoringELB Note: Shaded areas indicate 50 and 90 percent credible sets, solid lines are posterior medians, widedashedlinesareposteriormeansoftheprojectedinterestrate. Intheleftpanels,dashedlines lessthantheELBaretheposteriormedianand50and90percentcrediblesetsoftheshadowrate. 42

Figure9: Time-varyingUncertaintyaboutFutureShort–TermInterestRates 1.4 baseline ZLB ignored 1.2 1 0.8 0.6 0.4 0.2 0 1985 1987 1990 1992 1995 1997 2000 2002 2005 2007 2010 2012 2015 (a)One-quarteraheaduncertainty 3.5 baseline ZLB ignored 3 2.5 2 1.5 1 0.5 0 1985 1987 1990 1992 1995 1997 2000 2002 2005 2007 2010 2012 2015 (b)Eight-quarteraheaduncertainty Note: Forecast uncertainty about future short–term interest rates as measured by the standard deviationofeachmodel’spredictivedensitiesfori . Thepredictivedensitiesarere-estimatedover t growing samples that all start in 1960:Q1. In the baseline model (black solid lines), the predictive density is truncated at the ELB, whereas no constraint is imposed on the predictive density in the alternativemodelthatignorestheELB(reddashedline). 43

Figure10: ExpectationsComponentofthe5-yearRate 8 Expectations Component (JM) Expectations Component (KW) 7 5−year Yield (FRED) 6 5 4 3 2 1 0 1990 1995 2000 2005 2010 2015 Note: Shaded areas indicate 50 and 90 percent credible sets and wide dashed lines are posterior means of the estimated 5–year term–premium in our model. The red solid line is the estimated 5–yearterm–premiumwhenweignoretheELB. 44

Table1: ForecastEvaluation(Through2007) Forecasthorizonh 1 2 3 4 8 PanelA:Short-terminterestratei t+h baseline RMSE 0.33 0.69 1.02 1.34 2.20 MAD 0.25 0.53 0.81 1.08 1.78 PS 0.58 −0.69 −1.37 −1.86 −2.81 ELBignored rel. RMSE 1.00 1.00 1.00 1.00 1.00 rel. MAD 1.00 1.00 1.00 1.00 1.00 rel. PS −0.99∗∗∗ −0.44∗∗∗ −0.15∗∗ 0.02 0.33∗∗∗ withoutmedium-termyield rel. RMSE 1.12∗∗ 1.06 1.02 1.00 0.98 rel. MAD 1.11∗ 1.04 1.00 0.97 0.98 rel. PS −0.15∗∗∗ 0.01 0.05 0.07 0.06 PanelB:Long-terminterestratey t+h baseline RMSE 0.45 0.73 0.91 1.09 1.40 MAD 0.37 0.61 0.74 0.89 1.10 PS 0.22 −0.65 −1.06 −1.37 −1.92 ELBignored rel. RMSE 1.00 1.00 1.00 1.00 1.00 rel. MAD 1.00 1.00 1.00 1.00 1.00 rel. PS −0.89∗∗∗ −0.49∗∗∗ −0.31∗∗∗ −0.20∗∗∗ 0.02 Note: RMSE are root-mean-squared errors computed from using each the means of each model’s predictive densities as forecasts; MAD are mean absolute deviations obtained from using the medians of each model’s predictive densities. PS are average predictive scores. Relative RMSE and MAD are expressed as ratios relative to the corresponding statistics from the baseline model (values below unity denoting better performance than baseline); predictive scores are expressed as differences relative to the baseline (positive values denoting better performance than baseline). Predictive densities are re-estimated over growing samples that all start in 1960:Q1. For the forecast evaluation, the first forecast jumps off in 1984:Q4 and the last in 2007:Q4. Stars indicate significant differences, relative to baseline, in squared losses, absolute losses and density scores, respectively, as assessed by the test of Diebold and Mariano (1995); ∗∗∗, ∗∗ and ∗ denote significanceatthe1%,5%respectively10%level. 45

Table2: ForecastEvaluation(Post2007) Forecasthorizonh 1 2 3 4 8 PanelA:Short-terminterestratei t+h baseline RMSE 0.21 0.36 0.61 0.84 1.82 MAD 0.05 0.11 0.23 0.36 1.27 PS −0.13 −0.41 −0.68 −0.88 −1.43 ELBignored rel. RMSE 1.64 1.63∗ 1.35∗ 1.18∗∗ 0.88∗∗ rel. MAD 4.41∗∗∗ 3.99∗∗∗ 2.78∗∗∗ 2.37∗∗∗ 1.16 rel. PS −0.13 −0.47∗∗∗ −0.58∗∗∗ −0.62∗∗∗ −0.55∗∗∗ withoutmedium-termyield rel. RMSE 1.21 1.08 1.09 1.07 1.05 rel. MAD 1.33 0.98 0.79 0.78 1.03 rel. PS −0.06 0.01 0.04 0.07 0.09 PanelB:Long-terminterestratey t+h baseline RMSE 0.36 0.63 0.78 0.90 1.52 MAD 0.28 0.46 0.62 0.76 1.24 PS 0.61 −0.33 −0.76 −1.03 −1.90 ELBignored rel. RMSE 1.05∗∗ 1.05 1.05 1.05 0.97 rel. MAD 1.06∗ 1.03 1.04 1.02 0.96 rel. PS −1.00∗∗∗ −0.59∗∗∗ −0.40∗∗∗ −0.28∗∗∗ 0.10 Note: RMSE are root-mean-squared errors computed from using each the means of each model’s predictive densities as forecasts; MAD are mean absolute deviations obtained from using the medians of each model’s predictive densities. PS are average predictive scores. Relative RMSE and MAD are expressed as ratios relative to the corresponding statistics from the baseline model (values below unity denoting better performance than baseline); predictive scores are expressed as differences relative to the baseline (positive values denoting better performance than baseline). Predictive densities are re-estimated over growing samples that all start in 1960:Q1. For the forecast evaluation, the first forecast jumps off in 2008:Q1 and the last in 2015:Q4. Stars indicate significant differences, relative to baseline, in squared losses, absolute losses and density scores, respectively, as assessed by the test of Diebold and Mariano (1995); ∗∗∗, ∗∗ and ∗ denote significanceatthe1%,5%respectively10%level. 46