Forecasting with Small Macroeconomic VARs in the Presence of Instabilities

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. Forecasting with Small Macroeconomic VARs in the Presence of Instabilities Todd E. Clark and Michael W. McCracken 2007-41 NOTE: Staff working papers in the Finance and Economics Discussion Series (FEDS) are preliminary materials circulated to stimulate discussion and critical comment. The analysis and conclusions set forth are those of the authors and do not indicate concurrence by other members of the research staff or the Board of Governors. References in publications to the Finance and Economics Discussion Series (other than acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers.

Forecasting with Small Macroeconomic VARs in the Presence of Instabilities ∗ Todd E. Clark Federal Reserve Bank of Kansas City Michael W. McCracken Board of Governors of the Federal Reserve System November 2006 Abstract Small–scaleVARsarewidelyusedinmacroeconomicsforforecastingU.S.output, prices,andinterestrates. However,recentworksuggeststhesemodelsmayexhibitinstabilities. As such, a variety of estimation or forecasting methods might be used to improve their forecast accuracy. These include using different observation windows for estimation, intercept correction, time–varying parameters, break dating, Bayesian shrinkage,modelaveraging,etc. Thispapercomparestheeffectivenessofsuchmethods in real time forecasting. We use forecasts from univariate time series models, theSurveyofProfessionalForecastersandtheFederalReserveBoard’sGreenbookas benchmarks. JELNos.: C53,E17,E37 Keywords: Real-timedata,prediction,structuralchange Clark (corresponding author): Economic Research Dept.; Federal Reserve Bank of Kansas City; 925 ∗ Grand; Kansas City, MO 64198; todd.e.clark@kc.frb.org. McCracken: Board of Governors of the Federal Reserve System; 20th and Constitution N.W.; Mail Stop #61; Washington, D.C. 20551; michael.w.mccracken@frb.gov.

1 Introduction In this paper we provide empirical evidence on the ability of several different methods to improve the real–time forecast accuracy of small-scale macroeconomic VARs in the presence of potential model instabilities. The 18 distinct trivariate VARs that we consider are each comprised of one of three measures of output, one of three measures of inflation, and one of two measures of short-term interest rates. For each of these models we construct real time forecasts of each variable (with particular emphasis on the output and inflation measures) using real–time data. For each of the 18 variable combinations, we consider 86 different forecasting methods or models, incorporating different choices of lag selection, observation windows used for estimation, levels or differences, intercept corrections, stochastically time–varying parameters, break dating, discounted least squares, Bayesian shrinkage, detrending of inflation and interest rates, and model averaging. We compare our results to those from simple baseline univariate models as well as forecasts from the SurveyofProfessionalForecastersandtheFederalReserveBoard’sGreenbook. We consider this problem to be important for two reasons. The first is simply that small-scale VARs are widely used in macroeconomics. Examples of VARs used to forecast output, prices, and interest rates are numerous, including Sims (1980), Doan, et al. (1984), Litterman (1986), Brayton et al. (1997), Jacobson et al. (2001), Robertson and Tallman (2001), Del Negro and Schorfheide (2004), and Favero and Marcellino (2005). More recently these VARs have been used to model expectations formation in theoretical models. Examples are increasingly common and include Evans and Honkapohja (2005) andOrphanidesandWilliams(2005). The second reason is that there is an increasing body of evidence suggesting that these VARsmaybepronetoinstabilities.1 ExamplesincludeWebb(1995),Boivin(1999,2006), KozickiandTinsley(2001b,2002),andCogleyandSargent(2001,2005). Stillmorestudieshaveexaminedinstabilitiesinsmallermodels,suchasARmodelsofinflationorPhillips curve models of inflation. Examples include Stock and Watson (1996, 1999, 2003, 2006), 1Admittedly,whiletheevidenceofinstabilitiesintherelationshipsincorporatedinsmallmacroeconomic VARsseemstobegrowing,theevidenceisnotnecessarilyconclusive.RudebuschandSvensson(1999)apply stabilityteststothefullsetofcoefficientsofaninflation–outputgapmodelandareunabletorejectstability. Rudebusch(2005)findsthathistoricalshiftsinthebehaviorofmonetarypolicyhaven’tbeenenoughtomake reducedformmacroVARsunstable. EstrellaandFuhrer(2003)findlittleevidenceofinstabilityinjointtests of a Phillips curve relating inflation to the output gap and an IS model of output. Similarly, detailed test resultsreportedinStockandWatson(2003)showinflation–outputgapmodelstobelargelystable. 1

Levin and Piger (2003), Roberts (2006), and Clark and McCracken (2006b). Although many different structural forces could lead to instabilities in macroeconomic VARs (e.g., Rogoff (2003) and others have suggested that globalization has altered inflation dynamics), much of the aforementioned literature has focused on shifts potentially attributable to changesinthebehaviorofmonetarypolicy. Given the widespread use of small-scale macro VARs and the evidence of instability, it seems important to consider whether any statistical methods for managing structural changemightbegainfullyusedtoimprovetheforecastaccuracyofthemodels. Ofcourse, whilestructuralchangesmightoccurduringtheforecasthorizon,inthispaperwefocuson the potential for breaks occurring in the estimation sample. Our results indicate that some of the methods do consistently improve forecast accuracy in terms of root mean square errors (RMSE). Not surprisingly, the best method often varies with the variable being forecast,butseveralpatternsdoemerge. Afteraggregatingacrossallmodels,horizonsand variablesbeingforecasted,itisclearthatmodelaveragingandBayesianshrinkagemethods consistently perform among the best methods. At the other extreme, the approaches of using a fixed rolling window of observations to estimate model parameters and discounted leastsquaresestimationconsistentlyrankamongtheworst. The remainder of the paper proceeds as follows. Section 2 provides a synopsis of the methods used to forecast in the presence of potential structural changes. Section 3 describes the real-time data used as well as specifics on model estimation and evaluation. Section4presentsourresultsonforecastaccuracy,includingrankingsofthemethodsused. Giventhelargenumberofmodelsandmethodsusedweprovideonlyasubsetoftheresults in tables and use the text to provide further information. Section 5 concludes. Additional tablescanbefoundinalongerworkingpaperversion,ClarkandMcCracken(2006a). 2 Methods Used This section describes the various methods we use to construct forecasts from trivariate VARsinthefaceofpotentialstructuralchange. Table1providesacomprehensivelist,with some detail, and the method acronyms we use in presenting results in section 4. For each model — defined as being a baseline VAR in one measure of output (y), one measure of inflation(π),andoneshort–terminterestrate(i)—weapplyeachofthemethodsdescribed below. Output is defined as either a growth rate of GDP (or GNP) or an output gap (we 2

deferexplanationofthemeasurementofoutputandpricestosection3). Unlessotherwise noted, once the specifics of the model have been chosen, the parameters of the VAR are estimatedusingOLS. We begin with the perhaps na¨ıve method of ignoring structural change. That is, we constructiteratedmulti-stepforecastsfromrecursivelyestimated—thatis,estimatedwith all of the data available up to the time of the forecast construction — VARs with fixed lag lengths of 2 and 4. While this approach may seem na¨ıve, it may have benefits. As shown in Clark and McCracken (2005b), depending on the type and magnitude of the structural change, ignoring evidence of structural change can lead to more accurate forecasts. This possibility arises from a simple bias-variance trade-off. While a fixed parameter model is obviously misspecified if breaks have occurred, by using all of the data to estimate the modelonemightbeabletoreducethevarianceoftheparameterestimatesenoughtomore thanoffsettheerrorsassociatedwithignoringthecoefficientshifts. A second approach constructs forecasts in much the same way but permits updating of the lag structure as forecasting moves forward. This method, also used in such studies as Stock and Watson (2003), Giacomini and White (2005), and Orphanides and van Norden (2005),permitstimevariationinthenumberoflagsinthemodel. Wedothisfourseparate ways. The first two consist of using either the AIC or BIC to select the number of model lags in the entire system. In two additional specifications, we allow the lag orders of each variable in each equation to differ (as is done in some of the above studies, as well as Keating(2000)),andusetheAICandBICtodeterminetheoptimallagcombinations. For each of the above methods, we repeat the process but with at least some of the variables in differences rather than in levels. One reason for taking this approach is based upon the observation that inflation and interest rates are sometimes characterized as being I(1), while each of the output-type variables is generally considered I(0) and hence in the absence of cointegration the predictive equations are likely to be unbalanced. A second is that, as noted in Clements and Hendry (1996), forecasting in differences rather than in levels can provide some protection against mean shifts in the dependent variable. As such, for each model considered above, we construct forecasts based upon two separate collections of the variables: one that keeps the output variable in levels but takes the first difference of the inflation and interest variables (we refer to these models as DVARs) and a second that takes the first difference of all variables (denoted as DVARs with output differenced). See Allen and Fildes (2006) for a recent discussion of forecasting in levels 3

vs.differences. WealsoconsiderselectBayesianforecastingmethods. Specifically,weconstructforecasts using Bayesian estimates of fixed lag VARs, based on Minnesota–style priors as described in Litterman (1986).2 We consider both BVARs in “levels” (in y, π, i) and BVARs inpartial–differences(iny,∆π,∆i),referringtothelatterasBDVARs. Forourparticularapplications,wegenerallyusepriormeansofzeroforallcoefficients, with prior variances that are tighter for longer lags than shorter lags and looser for lags of thedependentvariablethanforlagsofothervariablesineachequation. However,insetting prior means, in select cases we use values other than zero: in BVARs, the prior means for own first lags of π and i are set at 1; in BVARs with an output gap, the prior mean for the own first lag of y is set at 0.8; and in BVARs with output growth that incorporate an informative prior variance on the intercept, the prior mean for the intercept of the output equation is set to the historical average growth rate.3 Using the notation of Robertson and Tallman (1999), the prior variances are determined by hyperparameters λ (general tight- 1 ness), λ (tightness of lags of other variables compared to lags of the dependent variable), 2 λ (tightness of longer lags compared to shorter lags), and λ (tightness of intercept). The 3 4 prior standard deviation of the coefficient on lag k of variable j in equation j is set to λ1 . kλ3 Thepriorstandarddeviationofthecoefficientonlagkofvariableminequation jis λ1λ2 σj, kλ3 σm where σ and σ denote the residual standard deviations of univariate autoregressions esj m timated for variables j and m. The prior standard deviation of the intercept in equation j issettoλ σ . InourBVARsandBDVARs,weusegenerallyconventionalhyperparameter 4 j settingsofλ =.2,λ =.5,λ =1,andλ =1000(makingtheinterceptpriorflat). 1 2 3 4 Another common approach to estimating predictive models in the presence of structural change consists of using a rolling window of the most recent N (N <t) observations to estimate the model parameters. The logic behind this approach is that for models exhibiting structural change, older observations are less likely to be relevant for the present incarnationoftheDGP. Inparticular,usingolderobservationsimpliesatypeofmodelmisspecification (and perhaps bias in the forecasts) that can be alleviated by simply dropping thoseobservations. Weimplementthismethodology,recentlyadvocatedinGiacominiand 2We estimate the models with the common mixed approach applied on an equation–by–equation basis. AsindicatedinGewekeandWhiteman(2006),estimatingthesystemofequationswiththesameMinnesota priorswouldrequireMonteCarlosimulation. 3Inmodelestimatesforvintaget, usedforforecastinginperiodt andbeyond, theaverageiscalculated using data from the beginning of the available sample through period t 1 — data that would have been − availabletotheforecasteratthattime. 4

White (2005), for each of the above methods using a constant window of the past N =60 quartersofobservationstoestimatethemodelparameters. Ofcourse,itispossiblethatusing a sample window based on break test estimates could yield better model estimates and forecasts. In practice, however, difficulties in identifying breaks and their timing may rule outsuchimprovements(see,forexample,theresultsinClarkandMcCracken(2005b)). Whilethelogicbehindtherollingwindowsapproachhasitsappeal,itmightbeconsidered a bit extreme in its dropping of older observations. That is, while older observations might be less relevant for the present incarnation of the DGP, they may not be completely irrelevant. A less extreme approach would be to use discounted least squares (DLS) to estimate the model parameters. This method uses all of the data to estimate the model pat j rametersbutweightstheobservationsbyafactorλ− ,0<λ<1,thatplacesfullweighton the most recent observation (j =t) but gradually shrinks the weights to zero for older observations(j<t). Whilethismethodologyislesscommonineconomicforecastingthanis therollingscheme,recentworkbyStockandWatson(2004)andBranchandEvans(2006) suggestsitmightworkwellformacroeconomicforecasting. Withthisinmindweconsider four separate models estimated by DLS. The first two are the baseline VARs in y, π, i and DVARs in y, ∆π, ∆i with a fixed number of lags. The second two are VARs and DVARs with the number of model lags estimated using the AIC for the system. Our setting of the discount factor roughly matches the suggestions of Branch and Evans (2006): .99 for the outputequationand.95fortheinflationandinterestrateequations. Despite the appeal of both the rolling and DLS methods, one drawback they share is that they reduce the (effective) number of observations used to estimate each of the model parameters regardless of whether they have exhibited any significant structural change. There are any number of ways to avoid this problem. One would be to attempt to identify structuralchangeineveryvariableineachequation. Todosoonecoulduseanynumberof approaches,includingthoseproposedinAndrews(1993),BaiandPerron(1998,2003),and manyothers. However,inthecontextofVARs(forwhichtherearenumerousparameters), these tests can be poorly sized and exhibit low power, particularly in samples of the size often observed when working with quarterly macroeconomic data. This is precisely the conclusion reached by Boivin (1999). Instead, in light of the importance of mean shifts highlightedinsuchstudiesasClementsandHendry(1996),KozickiandTinsley(2001a,b), and Levin and Piger (2003), we focus attention on identifying structural change in the interceptsofthemodel. 5

To capture potential structural change in the intercepts, we consider several different methodsofwhatmightlooselybecalled‘interceptcorrections’. Themoststraightforward is to use pretesting procedures to identify shifts in the intercepts, introduce dummy variablestocapturethoseshifts,estimatetheaugmentedmodelandproceedtoforecasting. In particular, we follow Yao (1988) and Bai and Perron (1998, 2003) in using information criteria to identify break dates associated with the model intercepts. Specifically, at each forecastoriginwefirstchoosethenumberoflagsinthesystemusingtheAICandthenuse an information criterion to select up to two structural breaks in the set of model intercepts. For computational tractability, we use a simple sequential approach — a partial version of Bai’s(1997)sequentialmethod—toidentifyingmultiplebreaks. Wefirstusetheinformation criterion to determine if one break has occurred. If the criterion identifies one break, wethen searchfor asecondbreak thatoccurred betweenthe time ofthe firstbreak andthe end of the sample.4 The model with up to two intercept breaks is then estimated by OLS andusedtoforecast. Weusetwosuchmodels,onewithbreaksidentifiedbytheAICanda secondwithbreaksidentifiedusingtheBIC. Whilethisapproachmightproveusefulforidentifyingstructuralchangeintheinterior of the sample, it is likely to be less well behaved when the structural change occurs at the very end of the sample.5 Motivated by this observation, Clements and Hendry (1996) discuss several approaches to ‘correcting’ intercepts for structural change occurring at the very end of the sample. The approach we implement is directly related to one of theirs. Specifically, the intercept correction consists of adding the average of the past 4 residuals tothemodel(foreachequation)ateachstepacrosstheforecasthorizon. Equivalently,the forecast is constructed by adding a weighted average of the past 4 residuals (with weights thatdependupontheparametersoftheVARandtheforecasthorizon)tothebaselineforecast that ignores any structural change.6 We apply intercept correction to four different VAR systems. Two of the systems use a fixed lag order, and the other two use a lag order determined by applying AIC to the system. For each of these two baseline lag orders, we then construct intercept corrections once for the entire system of three equations and once makingadjustmentstoonlytheinflationandinterestrateequations. Our final variant of intercept correction draws on the approach developed by Kozicki 4Inthebreakidentification,weimposeaminimumsegmentlengthof16quarters. 5Weleaveasatopicforfutureresearchthepossibilitythatmethodsdesignedtoidentifybreaksattheend ofasample,suchasthoseofHendry,etal. (2004)andAndrews(2006),couldyieldbetterresults. 6Seeequation(40)ofClementsandHendry(1996)fordetails. 6

andTinsley(2001a,b). Intheir‘movingendpoints’structure,thebaselineVARismodeled as having time varying intercepts that allow continuous variation in the long run expectationsofthecorrespondingvariables. Ourprecisemethod,though,isperhapsmoreclosely related to Kozicki and Tinsley (2002).7 In the context of a small-scale macro VAR, the variables in their model are modeled as deviations from latent time varying steady states (trends). However,whereastheyusetheKalmanfiltertoextractestimatesofthisunknown trend,fortractabilityweusesimpleexponentialsmoothingmethodstogetestimates. Cogley (2002) develops a model in which exponential smoothing provides an estimate of a time–varying inflation target of the central bank, a target that the public doesn’t observe but does learn about over time. With exponential smoothing, the trend estimate can be easily constructed in real time and updated over the multi–step forecast horizon to reflect forecasts of inflation. As indicated in Figure 1, exponential smoothing yields a trend estimate quite similar to an estimate of long–run inflation expectations based on 1981-2005 data from the Hoey survey of financial market participants and the Survey of Professional Forecasters (for a 10–year ahead forecast of CPI inflation) and 1960-1981 estimates of long–run inflation expectations developed by Kozicki and Tinsley (2001a). We construct twodifferentsetsofforecastsusingtheexponentialsmoothingapproach.8 FollowingKozicki and Tinsley (2001b, 2002), in the first we use our exponentially smoothed inflation series to detrend both inflation and the interest rate measure. In the second we detrend the inflation and interest rate series separately. In either case we do not detrend the output variable. Anotherapproachtomanagingstructuralchangeinmodelparametersistointegratethe structural change directly into the VAR.9 Following Doan, et al. (1984) and more recent 7Insomesupplementalanalysis,wehaveconsideredmodelsoftheerrorcorrectionformusedin,among others,Brayton,etal.(1997)andKozickiandTinsley(2001b). Thesemodelsrelatey,∆π,and∆i tolags t t t anderrorcorrectiontermsπ π andi π ,whereπ denotestrendinflation(long–runexpected inflation). Weestimatedthe t m − 1 o − dels t∗ −w 1 ithfix t e − d 1 l − ags t∗ −o 1 f2and4a ∗ ndwithBayesianmethodsusingafixedlag of4(andflatpriorsontheerrorcorrectioncoefficients). WealsoconsideredBayesianestimatesofourVAR withinflationdetrending. Noneofthesemethodsprovedtoconsistentlybeattheforecastaccuracyofthebest performingmethodswedescribebelow. FortheapplicationscoveredinTables2-5,allofthesesupplemental methodsdeliveredaverageRMSEratios(correspondingtotheaveragesinTable7)above1.000. 8Weuseasmoothingparameterof.07fortheinterestrateandcorePCEinflationseriesandasmoothing parameterof.05fortheGDPandCPIinflationseries. Eachtrendwasinitializedusingthesamplemeanof thefirst20observationsavailable(since1947)fromthepresentvintage. 9AsnotedinDoan,etal.(1984),propermulti-stepforecastingwithVARswithTVPwouldinvolvetaking intoaccountthejointdistributionoftheresidualsintheVARequationsandthecoefficientequations. Inlight ofthedifficultyofdoingso,wefollowconventionalpracticeandtreatthecoefficientsasfixedattheirperiod t 1valuesforforecastinginperiodst andbeyond. − 7

work by Brainard and Perry (2000) and Cogley and Sargent (2001, 2005), we model the structural change in the parameters of a VAR in y, π, i with random walks.10 However, in light of the potentially adverse effects of parameter estimation noise on forecast accuracy and the potentially unique importance of time variation in intercepts (see above), we consider two different scopes of parameter change. In the first we allow time variation in all coefficients — both the model intercepts and slope coefficients. In the second, we allow forstochasticvariationinonlytheintercepts.11 We estimate each of these TVP specifications using Bayesian methods with a range of prior variances on the standard deviation of the intercepts and a range of allowed time variation in the parameters. In some cases we use informative priors on the intercepts (λ 4 = .5 or .1); in others we use flat priors (λ = 1000). The variance–covariance matrix of 4 the innovations in the random walk processes followed by the coefficients is set to λ times the prior variance of the matrix of coefficients, which is governed by the hyperparameters describedabove. DrawingonthesettingsusedinsuchstudiesasStockandWatson(1996) and Cogley and Sargent (2001), we consider λ values ranging from .0001 to .005. Note, however, that in those instances in which the intercept prior is flat, we follow Doan, et al. (1984)insettingthevarianceoftheinnovationintheinterceptatλtimesthepriorvariance of the coefficient on the own first lag instead of the prior variance of the constant. In the baselineTVPmodel,weuseλ =.1andλ=.0005. 4 The final group of methods we consider all consist of some form of model averaging. While model averaging as a means of managing structural change has its historical precedents—notablyMinandZellner(1993)—theapproachhasbecomeevenmoreprevalent in the past several years. Recent examples of studies incorporating model averaging include Koop and Potter (2003), Stock and Watson (2003), Clements and Hendry (2004), Maheu and Gordon (2004), and Pesaran, et al. (2006). We consider six distinct, simple forms of model averaging, in each case using equal weights.12 The first takes an average of all the VAR forecasts described above and the univariate forecast described below, for a giventripletofvariables. Morespecifically,foragivencombinationofmeasuresofoutput, 10Someotherstudies,suchasCanova(2002),imposestationarityonthecoefficienttimevariation. 11Allowing both the inflation and interest rate equations to have intercepts with TVP implies a non– stationary real interest rate. While some readers might prefer specifications that impose stationarity in the real interest rate, our specifications are consistent with evidence in such studies as Laubach and Williams (2003)andClarkandKozicki(2005)onnon–stationaritiesinrealinterestrates. 12Indoingso,weleaveasatopicforfutureresearchwhethermoresophisticatedapproachestoaveraging, suchasapproachesbasedonhistoricalaccuracy,wouldyieldimprovements. 8

inflation, and an interest rate (for example, for the combination GDP growth, GDP inflation, and the T-bill rate), we construct a total of 75 different forecasts from the alternative VARmodelsdescribedabove. Wethenaveragetheseforecastswithaunivariateforecast. We include a second average forecast approach motivated by the results of Clark and McCracken(2005b),whoshowthatthebias-variancetrade-offcanbemanagedtoproduce a lower MSE by combining forecasts from a recursively estimated VAR and a VAR estimated with a rolling sample. In the results we present here, for a given baseline fixed lag VAR we take an equally weighted average of the model forecast constructed using parameters estimated recursively (with all of the available data) with those estimated using a rollingwindowofthepast60observations. TwootheraveragesaremotivatedbytheClark andMcCracken(2005a)findingthatcombiningforecastsfromnestedmodelscanimprove forecastaccuracy. Inthispaper,weconsideranaverageoftheunivariateforecastdescribed below with the fixed lag VAR forecast, and an average of the univariate forecast with the fixed lag DVAR forecast. Finally, motivated in part by general evidence of the benefits of averaging,weconsidertwootheraveragesoftheunivariateforecastswithsomeoftheother forecaststhatprovetoberelativelygood. Oneisasimpleaverageoftheunivariateforecast withtheforecastoftheVARwithinflationdetrending. Theotherisasimpleaverageofthe univariateandfixedlagVAR,DVAR,andbaselineBVARwithtimevaryingparameters. To evaluate the practical value of all these methods, we compare the accuracy of the above VAR–based forecasts against various benchmarks. In light of common practice in forecasting research, we use forecasts from univariate time series models as one set of benchmarks.13 For output, widely modeled as following low-order AR processes, the univariatemodelisanAR(2). Inthecaseofinflation,weuseabenchmarksuggestedbyStock and Watson (2006): an MA(1) process for the change in inflation (∆π), estimated with a rolling window of 40 observations. Stock and Watson find that the IMA(1) generally outperforms a random walk or AR model forecasts of inflation. For simplicity, in light of some general similarities in the time series properties of inflation and short–term interest rates and the IMA(1) rationale for inflation described by Stock and Watson, the univariate benchmark for the short-term interest rate is also specified as an MA(1) in the first differ- 13Ofcourse,thechoiceofbenchmarkstodayisinfluencedbytheresultsofpreviousstudiesofforecasting methods. AlthoughaforecastertodaymightbeexpectedtoknowthatanIMA(1)isagoodunivariatemodel forinflation,thesamemaynotbesaidofaforecasteroperatingin1970. Forexample,Nelson(1972)usedas benchmarksAR(1)processesinthechangeinGNPandthechangeintheGNPdeflator(bothinlevelsrather thanlogs). NelsonandSchwert(1977)firstproposedanIMA(1)forinflation. 9

ence of the series (∆i). As described in section 4, we use the bootstrap methods of White (2000) and Hansen (2005) to determine the statistical significance of any improvements in VAR forecast accuracy relative to the univariate benchmark models. In light of our real time forecasting focus, we also include as benchmarks forecasts of growth, inflation, and interest rates from the Survey of Professional Forecasters (SPF) and forecasts of growth andinflationfromtheFederalReserveBoard’sGreenbook. 3 Data and Model details Asnotedabove,weconsiderthereal–timeforecastperformanceofVARswiththreedifferent measures of output, three measures of inflation, and two short–term interest rates. The outputmeasuresareGDPorGNP(dependingondatavintage)growth,anoutputgapcomputed with the method described in Hallman, et al. (1991), and an output gap estimated with the Hodrick and Prescott (1997) filter. The first output gap measure (hereafter, the HPS gap), based on a method the Federal Reserve Board once used to estimate potential outputforthenonfarmbusinesssector,isentirelyone–sidedbutturnsouttobeveryhighly correlated with an output gap based on the Congressional Budget Office’s (CBO’s) estimate of potential output. The HP filter of course has the advantage of being widely used and easy to implement. We follow Orphanides and van Norden (2005) in our real time application of the filter: for forecasting starting in period t, the gap is computed using the conventional filter and data available through periodt 1. The inflation measures include − the GDP or GNP deflator or price index (depending on data vintage), CPI, and PCE price index excluding food and energy (hereafter, core PCE price index).14 The short–term interest rate is measured as either a 3–month Treasury bill rate or the effective federal funds rate. Note, finally, that growth and inflation rates are measured as annualized log changes (from t 1 to t). Output gaps are measured in percentages (100 times the log of output − relativetotrend). Interestratesareexpressedinannualizedpercentagepoints. The raw quarterly data on output, prices, and interest rates are taken from a range of sources: the Federal Reserve Bank of Philadelphia’s Real–Time Data Set for Macroeconomists (RTDSM), the Board of Governor’s FAME database, the website of the Bureau of Labor Statistics (BLS), the Federal Reserve Bank of St. Louis’ ALFRED database, and 14Astheunivariateforecastresultssuggest, thesecompetingpriceindiceshavesomewhatdifferentcharacteristics. Differences appear to persist over long periods of time: there is little evidence of cointegration amongtheseandrelatedpriceindexes(see,forexample,Lebow,Roberts,andStockton(1992)). 10

various issues of the Survey of Current Business. Real–time data on GDP or GNP and the GDP or GNP price series are from the RTDSM. For simplicity, hereafter we simply use the notation “GDP” and “GDP price index” to refer to the output and price series, even thoughthemeasuresarebasedonGNPandafixedweightdeflatorformuchofthesample. For the core PCE price index, we compile a real time data set starting with the 1996:Q1 vintage by combining information from the Federal Reserve Bank of St. Louis’ ALFRED database (which provides vintages from 1999:Q3 through the present) with prior vintage data obtained from issues of the Survey of Current Business, following the RTDSM dating conventions.15 Because the BEA only begin publishing the core PCE series with the 1996:Q1 vintage, it is not possible to extend the real time data set further back in history withjustinformationfromtheSurveyofCurrentBusiness. In the case of the CPI and the interest rates, for which real time revisions are small to essentially non–existent (see, for example, Kozicki (2004)), we simply abstract from real timeaspectsofthedata. FortheCPI,wefollowtheadviceofKozickiandHoffman(2004) for avoiding choppiness in inflation rates for the 1960s and 1970s due to changes in index bases, and use a 1967 base year series taken from the BLS website in late 2005.16 For the T-billrate,weuseaseriesobtainedfromFAME. The full forecast evaluation period runs from 1970:Q1 through 2005; we use real time datavintagesfrom1970:Q1through2005:Q4. AsdescribedinCroushoreandStark(2001), thevintagesoftheRTDSMaredatedtoreflecttheinformationavailablearoundthemiddle ofeachquarter. Normally,inagiven vintaget,theavailableNIPA datarunthroughperiod t 1.17 The start dates of the raw data available in each vintage vary over time, ranging − from 1947:Q1 to 1959:Q3, reflecting changes in the samples of the historical data made available by the BEA. For each forecast origin t in 1970:Q1 through 2005:Q3, we use the real time data vintage t to estimate output gaps, estimate the forecast models, and then construct forecasts for periods t and beyond. The starting point of the model estimation sampleisthemaximumof1955:Q1andtheearliestquarterinwhichallofthedataincluded in a given model are available, plus the number of lags included in the model (plus one 15Inputtingtogethervintagesfor1996:Q1through1999:Q2,wealsoreliedonacoupleoffulltimeseries wehadonfilefrompriorresearch,seriesthatcorrespondtothevintagesfor1996:Q4and1999:Q2,obtained fromFAMEatthetimeoftheresearchprojects. 16TheBLSonlyprovidesthe1967baseyearCPIonanotseasonallyadjustedbasis.Weseasonallyadjusted theserieswiththeX-11filter. 17Inthe caseof the1996:Q1vintage, withwhichthe BEApublisheda benchmarkrevision, the datarun through1995:Q3insteadof1995:Q4. 11

quarterforDVARsorVARswithinflationdetrending). We present forecast accuracy results for forecast horizons of the current quarter (h = 0Q), the next quarter (h=1Q), and four quarters ahead (h=1Y). In light of the timet 1 − informationactuallyincorporatedintheVARsusedforforecastingatt,thecurrentquarter (t) forecast is really a 1–quarter ahead forecast, while the next quarter (t+1) forecast is really a 2–step ahead forecast. What is referred to as a 1–year ahead forecast is really a 5–step ahead forecast. In keeping with conventional practices and the interests of policymakers, the 1–year ahead forecasts for GDP/GNP growth and inflation are four–quarter rates of change (the percent change from period t+1 through t+4). The 1–year ahead forecastsforoutputgapsandinterestratesarequarterlylevelsinperiodt+4. As the forecast horizon increases beyond a year, forecasts are increasingly determined by the unconditional means implied by a model. As highlighted by Kozicki and Tinsley (1998, 2001a,b), these unconditional means — or, in the Kozicki and Tinsley terminology, endpoints — may vary over time. The accuracy of long horizon forecasts (two or three years ahead, for example) depend importantly on the accuracy of the model’s endpoints. Asaresult,weexaminesimplemeasuresoftheendpointsimpliedbyrealtime,1970-2005 estimates of a select subset of the forecasting models described above. For simplicity, we use10–yearaheadforecasts(forecastsforperiodt+39madewithvintaget dataendingin periodt 1)asproxiesfortheendpoints. − We obtained benchmark SPF forecasts of growth, inflation, and interest rates from the website of the Federal Reserve Bank of Philadelphia.18 The available forecasts of GDP/GNP growth and inflation span our full 1970 to 2005 sample. The SPF forecasts of CPI inflation and the 3-month Treasury bill rate begin in 1981:Q3. Our benchmark Greenbook forecasts of GDP/GNP growth and inflation and CPI inflation are taken from data on the Federal Reserve Bank of Philadelphia’s website and data compiled by Peter Tulip (someofthedataareusedinTulip(2005)). Wetake1970-99vintageGreenbookforecasts ofGDP/GNPgrowthandGDP/GNPinflationfromthePhiladelphiaFed’sdataset.19 Forecasts of GDP growth and inflation for 2000 are calculated from Tulip’s data set. Finally, wetake1979:Q4–2000:Q4vintageGreenbookforecastsofCPIinflationfromTulip’sdata 18TheSPFdataprovideGDP/GNPandtheGDP/GNPpriceindexinlevels,fromwhichwecomputedlog growth rates. We derived 1–year ahead forecasts of CPI inflation by compounding the reported quarterly inflationforecasts. 19Wederived1–yearaheadforecastsofgrowthandinflationbycompoundingthereportedquarterlypercent changes. 12

set.20 AsdiscussedinsuchsourcesasRomerandRomer(2000),Sims(2002),andCroushore (2006), evaluating the accuracy of real time forecasts requires a difficult decision on what to take as the actual data in calculating forecast errors. The GDP data available today for, say, 1970, represent the best available estimates of output in 1970. However, output as defined today is quite different from the definition of output in 1970. For example, today we have available chain weighted GDP; in the 1970s, output was measured with fixed weight GNP. Forecasters in 1970 could not have foreseen such changes and the potential impact on measured output. Accordingly, in our baseline results, we use the first available estimates of GDP/GNP and the GDP/GNP deflator in evaluating forecast accuracy. In particular, we define the actual value to be the first estimate available in subsequent vintages. In the case of h–step ahead (for h = 0, 1, and 4) forecasts made for period t+h with vintage t data ending in period t 1, the first available estimate is − normally taken from the vintage t+h+1 data set. In light of our abstraction from real time revisions in CPI inflation and interest rates, the real time data correspond to the final vintage data. In Clark and McCracken (2006a) we provide supplementary results using final vintage (2005:Q4 vintage) data as actuals. Our qualitative results remain broadly unchangedwiththeuseoffinalvintagedataasactuals. 4 Results Inevaluatingtheperformanceoftheforecastingmethodsdescribedabove,wefollowStock and Watson (1996, 2003, 2006), among others, in using squared error to evaluate accuracy and considering forecast performance over multiple samples. Specifically, we measure accuracywithrootmeansquareerror(RMSE).Theforecastsamplesaregenerallyspecified as 1970-84 and 1985-2005 (the latter sample is shortened to 1985-2000 in comparisons to Greenbook forecasts, for which publicly available data end in 2000).21 We split the full sample in this way to ensure our general findings are robust across sample periods, in light of the evidence in Stock and Watson (2003) and others of instabilities in forecast performance over time. However, because real time data on core PCE inflation only begin 20Year–aheadCPIforecastswereobtainedbycompoundingtheGreenbook’squarterlypercentchanges. 21Withforecastsdatedbytheendperiodoftheforecasthorizonh=0,1,4,theVARforecastsamplesare, respectively,1970:Q1+hto1984:Q4and1985:Q1to2005:Q3. 13

in1996,wealsopresentselectresultsforaforecastsampleof1996-2005.22 To be able to provide broad, robust results, in total we consider a very large number of modelsandmethods—fartoomanytobeabletopresentalldetailsoftheresults. Instead we use the full set of models and methods in providing only a high–level summary of the results, primarily in the form of rankings described below. In addition, we limit the presentationofdetailedresultstothosemodelsandvariablecombinationsofperhapsbroadest interest and note in the discussion those instances in which results differ for other specifications. Specifically, in presenting detailed results, we draw the following limitations. (1) Forthemostpart,accuracyresultsarepresentedforjustoutputandinflation. (2)Outputis measured with either GDP/GNP growth or the HPS gap. (3) The interest rate is measured with the 3-month Treasury bill rate. We provide results for models using the federal funds rate — results qualitatively similar to those we report in the paper — in supplemental tablesinClarkandMcCracken(2006a). (4)Thesetofforecastmodelsormethodsislimited to a subset we consider to be of the broadest interest or representative of the others. For example,whileweconsidermodelsestimatedwithafixednumberofeither2or4lags,we reportRMSEsassociatedonlywiththosethathave4lags. We proceed below by first presenting forecast accuracy results based on univariate and VAR models. We then compare results for some of the better–performing methods to the accuracyofSPFandGreenbookforecasts. Weconcludebyexaminingthereal–time,long– run forecasts generated by a subset of the forecast methods that yield relatively accurate short–runforecasts. 4.1 Forecast accuracy Tables2through5reportforecastaccuracy(RMSE)resultsforfourcombinationsofoutput (GDP growth and HPS gap) and inflation (GDP price index and CPI) and 27 models. In each case we use the 3-month Treasury bill as the interest rate. In every case, the first row of the table provides the RMSE associated with the baseline univariate model, while the others report ratios of the corresponding RMSE to that for the benchmark univariate model. Hence numbers less than one denote an improvement over the univariate baseline whilenumbersgreaterthanonedenoteotherwise. To determine the statistical significance of differences in forecast accuracy, we use a 22Specifically, the forecast sample is 1996:Q1+h to to 2005:Q3 (for forecasts dated by the end of the forecasthorizon). 14

non–parametric bootstrap patterned after White’s (2000) to calculate p–values for each RMSE ratio in Tables 2-5. The individual p–values represent a pairwise comparison of eachVARoraverageforecasttotheunivariateforecast. RMSEratiosthataresignificantly less than 1 at a 10 percent confidence level are indicated with a slanted font. To determine whether a best forecast in each column of the tables is significantly better than the benchmark once the data snooping or search involved in selecting a best forecast is taken into account, we apply Hansen’s (2005) (bootstrap) SPA test to differences in MSEs (for c each model relative to the benchmark). Hansen shows that, if the variance of the forecast loss differential of interest differs widely across models, his SPA test will typically have c much greater power than White’s (2000) reality check test. For each column, if the SPA c test yields a p–value of 10 percent or less, we report the associated RMSE ratio in bold font. Because the SPA test is based on t–statistics for equal MSE instead of just differc ences in MSE (that is, takes MSE variability into account), the forecast identified as being significantlybestbySPA maynotbetheforecastwiththelowestRMSEratio.23 c We implement the bootstrap procedures by sampling from the time series of forecast errorsunderlyingtheentriesinTables2-5. Forsimplicity,weusethemovingblockmethod of Kunsch (1989) and Liu and Singh (1992) rather than the stationary bootstrap actually usedbyWhite(2000)andHansen(2005);Whitenotesthatthemovingblockisalsoasymptotically valid. The bootstrap is applied separately for each forecast horizon, using a block size of 1 for the h=0Q forecasts, 2 for h=1Q, and 5 for h=1Y.24 In addition, in light of the potential for changes over time in forecast error variances, the bootstrap is applied separately for each subperiod. Note, however, that the bootstrap sampling preserves the correlationsofforecasterrorsacrossforecastmethods. While there are many nuances in the detailed results, some clear patterns emerge. The RMSEs clearly show the reduced volatility of the economy since the early 1980s, particularly for output. For each horizon, the benchmark univariate RMSE of GDP growth forecastsdeclinedbyroughly2/3acrossthe1970-84and1985-05samples;thebenchmark RMSEforHPSgapforecastsdeclinedbyabout1/2. Thereducedvolatilityislessextreme for the inflation measures but still evident. For each horizon, the benchmark RMSEs fell by roughly 1/2 across the two periods, with the exception that at the h = 1Y horizon the 23Formulti–stepforecasts,wecomputethevarianceenteringthet–testusingtheNeweyandWest(1987) estimatorwithalaglengthof1.5 h,wherehdenotesthenumberofforecastperiods. ∗ 24Foraforecasthorizonofτperiods,forecasterrorsfromaproperlyspecifiedmodelwillfollowanMA(τ − 1)process. Accordingly,weuseamovingblocksizeofτforaforecasthorizonofτ. 15

variabilityinGDPinflationdeclinednearly2/3. Consistent with the results in Campbell (2006), D’Agostino, et al. (2005), Stock and Watson (2006), and Tulip (2005), there is also a clear decline in the predictability of both outputandinflation: ithasbecomehardertobeattheaccuracyofaunivariateforecast. For example, for each forecast horizon, a number of methods or models beat the accuracy of the univariate forecast of GDP growth during the 1970-84 period (Tables 2 and 4). In fact, manyofthesedosoatalevelthatisstatisticallysignificant. Butoverthe1985-2005period, only the BVAR(4)-TVP models are more accurate, at only the 1–year ahead horizon. The reduction in predictability is almost, but not quite, as extreme for the HPS output gap (Tables 3 and 5). While several models perform significantly better than the benchmark in the 1970-84 period, only two classes of methods, the BDVARs and the BVAR-TVPs, significantlyoutperformthebenchmarkinthe1985-05period. The predictability of inflation has also declined, although less dramatically than for output. For example, in models with GDP growth and GDP inflation (Table 2), the best 1–yearaheadforecastsofinflationimproveupontheunivariatebenchmarkRMSEbymore than 10 percent in the 1970-84 period but only 5 percent in 1985-05. The evidence of a decline in inflation predictability is perhaps most striking for CPI forecasts at the h=0Q horizon. InbothTables4and5,mostofthemodelsconvincinglyoutperformtheunivariate benchmarkduringthe1970-84period,withstatisticallysignificantmaximalgainsof18%. But in the following period, many fewer methods outperform the benchmark, with gains typicallyabout4%. Reflecting the decline in predictability, many of the methods that perform well over 1970-84faremuchmorepoorlyover1985-05. Theinstabilitiesinperformanceareclearly evident in both output and inflation forecasts, but more dramatic for output forecasts. For example, a VAR with AIC determined lags and intercept breaks (denoted VAR(AIC), intercept breaks) forecasts both GDP growth and the HPS gap well in the 1970-84 period, with gains as large as 25% for 1–year ahead forecasts of the HPS gap. However, in the 1985-05period,theVARwithinterceptbreaksranksamongtheworstperformers,yielding 1–year ahead output forecasts with RMSEs 60 percent higher than the univariate forecast RMSEs. In the case of inflation forecasts, a DVAR(4) estimated with Bayesian methods and a rolling sample of data (denoted BDVAR(4)) beats the benchmark, by as much as 13 percent, at every horizon during the 1970-84 period. But in the 1985-05 period, the BDVAR(4)isalwaysbeatenbytheunivariatebenchmarkmodel,byasmuchas21%. 16

Thechangeinpredictabilitymakesitdifficulttoidentifymethodsthatconsistentlyimprove upon the forecast accuracy of univariate benchmarks. As noted above, none of the methodsconsistentlyimproveupontheGDPgrowthbenchmarkacrossthesubperiods. For forecasts of the HPS gap, the BVAR(4)-TVP models generally outperform the benchmark over both periods. However, the 1970-84 gains are not statistically significant. In the case of inflation forecasts, though, a number of the forecasts significantly outperform the univariatebenchmarkinbothsamples. Ofparticularnotearetheforecaststhataveragethe benchmark univariate projection with a VAR projection — either the VAR(4), DVAR(4), or VAR(4) with inflation detrending — and the average of the univariate forecast with (together) the VAR(4), DVAR(4), and TVP BVAR(4) projections. In the 1970-84 period, these averages nearly always outperform the benchmark, although without necessarily being the best performer. In the 1985-05 period, the averages continue to outperform the benchmarkandarefrequentlyamongthebestperformers. In Tables 6 and 7 we take another approach to determining which methods tend to performbetterthanthebenchmark. Acrosseachvariable,modelandhorizon,wecompute the average rank and RMSE ratio of the methods included in Tables 2-5, as well as the correspondingsamplestandarddeviations. Forexample,thefiguresinTable6areobtained by: (1)ranking,foreachofthe48columnsofTables2-5,the27forecastmethodsormodels considered; and (2) calculating the average and standard deviation of each method’s (48) ranks. Table 7 does the same, but using RMSEs instead of RMSE ranks. The averages in Tables 6 and 7 show that, from a broad perspective, the best forecasts are those obtained as averages across models. The best forecast, an average of the univariate benchmark with the VAR(4) with inflation detrending, has an average RMSE ratio of .943 in Tables 2-5, and an average rank of 5.1. Not surprisingly, orderings based on average RMSE ratios are closelycorrelatedwithorderingsbasedontheaveragerankings. Forinstance,thetopeight forecasts based on average rankings are the same as the top eight based on average RMSE ratios,withslightdifferencesinorderings. Tables6and7alsoshowthatsomeVARmethodsconsistentlyperformworse—much worse, in some cases — than the univariate benchmark. The univariate forecasts have the 9thbestaverageRMSEratioand11thbestaverageranking. Thus,onaverage,roughly2/3 oftheVARmethodsfailtobeattheunivariatebenchmark. Moreover,someofthemethods designedtoovercomethedifficultyofforecastinginthepresenceofstructuralchangeconsistently rank among the worst forecasts. Most notably, VAR forecasts based on intercept 17

corrections and DLS estimates are generally among the worst forecasts considered, yielding RMSE ratios that, on average, exceed the univariate benchmark by roughly 15 percent (weacknowledge,however,thatunderdifferentimplementations,theperformanceofthese methods could improve — we leave such analysis for future research).25 VARs estimated withrollingsamplesofdataalsoperformrelativelypoorly: ineverycase,aVARestimated with a rolling sample is, on average, less accurate than when estimated (recursively) with the full sample. In contrast, on average, standard Bayesian estimation of VARs generally dominates OLS estimation of the corresponding model. For example, the average RMSE ratio of the BVAR(4) forecast is 1.012, compared to the average VAR(4) RMSE ratio of 1.030. Tables 8-11 report RMSE results for models including core PCE inflation. As noted above, reflecting the real time core PCE data availability, the forecast sample is limited to 1996-05. As in Tables 2-5, we report results for models with two different measures of output, GDP growth and the HPS output gap, but a single interest rate measure, the Treasury bill rate. For comparison, we also report 1996-05 results for models using GDP inflation instead of core PCE inflation. As in the case of the results for 1970-84 and 1985-05, we use White (2000) and Hansen (2005) bootstraps to determine whether any of the RMSE ratios are significantly less than one, on both a pairwise (given model against univariate) and best–in–column basis. Individual RMSE ratios that are significantly less than 1 (10% confidence level) are indicated with a slanted font. Note, though, that once the search involved in selecting a best forecast is taken into account, the univariate model is never beaten in the 1996-05 results (that is, none of the data snooping–robust p–values arelessthan.10). Consistentwiththe1985-05resultsinTables2-5,theforecastresultsfor1996-05inTables 8-11 show that univariate benchmarks are difficult to beat. Of the inflation measures, thebenchmarkishardertobeatwithcorePCEinflationthanwithGDPinflation. For1996- 05, only a few forecasts (e.g., rolling VAR(4) or DVAR(4) forecasts for h=0Q) beat the univariate benchmark, and none statistically significantly. A few more forecasts are able to improve (some statistically significantly) on the accuracy of the univariate benchmark for GDP inflation. Importantly, for models with GDP inflation, those methods that per- 25Inourresults, interceptcorrectionsdon’tseemtoworkwitheitherGDPgrowthoroutputgaps. Inthe caseofgaps,however,thepersistenceandmeasurementerrorinherentinthemmaywarrantotherapproaches tointerceptcorrection. 18

formed relatively well over the samples covered in Tables 2-5 — such as the averages of the benchmarks with the VAR(4) or DVAR(4) models — also perform relatively well over the1996-05sample. Tables 12 and 13 provide aggregate or summary information on the forecast performanceofallthemethodsandnearlyallofthedatacombinationsconsidered. Thesummary information covers all of the variable combinations and models included in Tables 2-5, as well as variable combinations that include the HP measure of the output gap or the federal funds rate as the interest rate, models based on a fixed lag of two instead of four, and the full set of forecasting methods described in section 2 and listed in Table 1. Our summary approachfollowstherankingmethodologyofTables6and7. Thatis,inTables12and13 we present average rankings for every method we consider across every forecast horizon, various subclasses of models, and the 1970-84 and 1985-05 samples. Note, however, that we exclude the 1996-05 sample (and, as a result, results from models including core PCE inflation),inpartbecauseofitsoverlapwiththelonger1985-05period. While expanding coverage to all possible models and methods generates some additional nuances in results, the broad findings described above continue to hold. As shown in Table 12’s first column of ranks, across all combinations of variables the most robust forecasting methods are those that average the univariate model with one or a few VAR forecasts. Forexample,theaverageoftheunivariateforecastwithaforecastfromaVAR(2) withinflationdetrendinghasthebestaverageranking,of12.9(andthebestaverageRMSE ratio, not reported, of 0.937). Coming in behind these averaging methods, in the broad ranking perspective, are the fixed lag BVAR, BDVAR and BVAR-TVP models. Note that thefirstcolumnincludesinterestrateforecastresults—whichwereomittedfromprevious tables for brevity. The same classes of models that on average performed best for the outputandinflationseriescontinuetoperformamongthebestforinterestrateforecasts(andis anotherreasonwhywefeltcomfortableomittingthoseresults). Somewhatmoreformally, the Spearman rank correlation across the results in the first and second columns of Table 12—thesecondofwhichcontainstheranksofjusttheoutputandinflationforecasts—is arobust0.97. Columns 3and 4 of Table12 delineate theaverage impact ofthe choice of interestrate on forecast accuracy for the output and inflation measures. The rankings are extremely similar. The five best methods for forecasting output and inflation in models with the Tbill rate are also the five best methods in models with the federal funds rate. Moreover, 19

the Spearman rank correlation of the results conditioned on the T-bill rate and the results conditioned on the federal funds rate is 0.98. We should emphasize that this does not implythatthereweren’tdifferencesinthe nominaloutcomesacrossthesetwointerestrate measures. Rather, in light of our goal to identify those methods that are most robust in forecasting,thechoicebetweentheT-billandfederalfundsratesmakeslittledifference. Columns1-3inTable13delineatetheaverageimpactofthechoiceofoutputmeasurein forecastsofoutputandinflation. Theserankingsarequitesimilaracrossoutputmeasures, although not quite as similar as those comparing the impact of the interest rate measures. In each case the best methods generally continue to be averages of univariate benchmarks with VAR forecasts and BVARs with TVP. For example, in models with GDP growth, on average the best forecasts of output and inflation are obtained with an average of the univariate,VAR(4),DVAR(4),andTVPBVAR(4)forecasts. Perhapsthelargestdistinction among the three sets of rankings is that moving from GDP growth to HPS gap to HP gap, the concentration of best methods shifts from the averaging group to the BVAR-TVP with tight intercept priors group to the BVAR-TVP with loose intercept priors group. Even so, therankcorrelationsamongthethreecolumnsareveryhigh,between0.85and0.93. Similarly,columns4and5ofTable13provideaveragerankingsofforecastsforoutput andinflationthatconditionontheinflationmeasure,GDPinflationorCPIinflation. Again, the top performing methods remain the averages of univariate forecasts with select VAR forecasts and BVAR TVP forecasts. And, the results are very similar across inflation measures. In the average rankings, the top seven methods for models including GDP inflation are the same as the top seven for models including CPI inflation, with slight differences in orderings. Therankcorrelationacrossallmethodsis0.94. ThelasttwocolumnsofTable12comparetheperformanceofmethodsacrossthe1970- 84and1985-05periods. Asintheabovedetailedcomparisonsofasubsetofresults,across the two subperiods there are some sharp differences in the performance of many of even the better performing methods.26 Only four methods have an average ranking of less than 20 over the 1970-84 period (in order from smallest to largest): the average of all forecasts, the average of the univariate and VAR(4) with inflation detrending forecasts, the VAR(2) with full exponential smoothing detrending, and the average of the univariate, VAR(4), 26Inaddition,theaverageRMSEratios(notreported)associatedwitheachofthetop–performingmethods reflectthesharpreductioninpredictabilityin1985-05comparedto1970-84. ThebestaverageRMSEratio for 1970-84 is 0.873, from a VAR(2) with full exponential smoothing. The best average RMSE ratio for 1985-05is0.998,forthebaselineTVPBVAR(4). 20

DVAR(4), and TVP BVAR(4) forecasts. For the 1985-05 sample, a total of 11 methods have average rankings below 20, but only two of them — the average of the univariate and VAR(4) with inflation detrending forecasts and the average of the univariate, VAR(4), DVAR(4), and TVP BVAR(4) forecasts — correspond to the four methods that produce average rankings of less than 20 in the 1970-84 sample. Some of the models that perform relatively well in 1970-84 fare much more poorly in the second sample. For example, the averagerankingoftheVAR(2)withfullexponentialsmoothingdetrendingplummetsfrom 17.7 in 1970-84 to 63.9 in 1985-05. Not surprisingly, the rank correlation between these twocolumnsisrelativelylow,atjust0.58. In Clark and McCracken (2006a) we provide still more detailed information on which methods work the best individually for forecasting each output measure and the GDP and CPI measures of inflation. Perhaps not surprisingly, this further disaggregation of the results leads to modestly more heterogeneity in rankings of the best methods. This is particularly true for output forecast rankings compared to inflation rankings. For example, a DVARwithAIC–determinedlagshasanaveragerankingof15.4inforecastsofGDPinflationandanaveragerankingof48.5inforecastsofGDPgrowth. TheSpearmancorrelations ofoutputrankingswithinflationrankingsrangefrom0.46(forGDPgrowthandCPIinflation)to0.57(fortheHPSgapandCPIinflation). Bycomparison,thecorrelationsofoutput forecastrankingsacrossmeasuresofoutputaverage0.7,whilethecorrelationforGDPand CPIinflationrankingsis0.86. Despitethegreaterheterogeneityofthesemoredisaggregate rankings,therearesimilaritiesamongthebestperformers. Amongtheoutputvariables,on average,thebestforecastsaretypicallytheaveragesofunivariateforecastswithVARforecastsandtheBVAR-TVPforecasts. Forthetwoinflationmeasures,theaveragingmethods continuetoperformthebest,followedbyBVAR-TVPandDVARforecasts. Just as Tables 12 and 13 provide aggregate evidence on the best methods, they also show what methods consistently perform the worse in the full set of models, methods, and horizons. Perhaps most simply, not a single method on the second pages of the tables has an average rank less than 20! Consistent with the subset of results summarized in Tables 6 and 7, the lowest–ranked methods include: DLS estimation of VARs or DVARs, DVARs withoutput,inadditiontoinflationandtheinterestrate,differenced;andVARswithintercept correction. The consistency of the rankings for these worst–performing methods may be considered impressive. In addition, the average rankings of forecasts based on rolling estimation of VARs (and DVARs, BVARs, etc.) are generally considerably lower than the 21

average rankings of the corresponding VARs estimated with the full sample of data. For example, the average ranking of rolling DVAR(2) forecasts is 41.2, compared to the recursively estimated DVAR(2)’s average ranking of 32.8. While those methods generally falling in the middle ranks (between an average rank of, say, 20 and 50) may not be considered robust approaches to forecasting with the VARs of interest, in particular instances someofthesemethodsmayperformrelativelywell. Forexample,theDVARwithAIClags determinedforeachequationhasanaveragerankingof39.4,butyieldsrelativelyaccurate forecastsofGDPinflationin1985-05(Tables2and4). Table 14 compares the accuracy of some of the better time series forecasting methods with the accuracy of SPF projections. The variables we report are those for which SPF forecastsexist: GDPgrowth,GDPinflation,andCPIinflation(inthecaseofCPIinflation, the SPF forecasts don’t begin until 1981, so we only report CPI results for the 1985-05 period). WealsoreportresultsforforecastsoftheT-billratefromtheSPFandtheselected timeseriesmodels. Inparticular,Table14provides,forthe1970-84and1985-05samples, RMSEs for forecasts from the SPF and a select set of the better–performing time series forecasts: the best forecast RMSE for each variable in each period from those methods includedinTable2(Table4forCPIinflationforecasts),theunivariatebenchmarkforecast, several of the average forecasts, and the baseline TVP BVAR(4). To be sure, comparing forecasts from a source such as SPF against the best forecast from Table 2 or 4 gives the time series models an unrealistic advantage, in that, in real time, a forecaster wouldn’t knowwhichmethodismostaccurate. However,astheresultspresentedbelowmakeclear, ourgeneralfindingsapplytoalloftheindividualforecastsincludedinthecomparison. Perhaps not surprisingly, the SPF forecasts generally dominate the time series model forecasts. For example, in h = 0Q forecasts of GDP growth for 1970-84, the RMSE for the SPF is 2.571, compared to the best time series RMSE of 3.735 (in which case the best forecastistheallforecastaveragereportedinTable2). Inh=0QforecastsofGDPinflation for 1970-84, the RMSE for the SPF is 1.364, compared to the best time series RMSE of 1.727 (again, from the all–forecast average in Table 2). At such short horizons, of course, the SPF has a considerable information advantage over simple time series methods. As described in Croushore (1993), the SPF forecast is based on a survey taken in the second month of each quarter. Survey respondents then have considerably more information, on variables such as interest rates and stock prices, than is reflected in time series forecasts that don’t include the same information (as reflected in the bottom panel of Table 14, that 22

information gives the SPF its biggest advantage in near-term interest rates). However, the SPF’s advantage over time series methods generally declines as the forecast horizon rises. For instance, in h = 1Y forecasts of GDP growth for 1970-84, the SPF and best time series RMSEs are, respectively, 2.891 and 2.775; for forecasts of GDP inflation, the correspondingRMSEsare2.192and2.141. Moreover, the SPF’s advantage is much greater in the 1970-84 sample than the 1985- 05 sample. Campbell (2006) notes the same for SPF growth forecasts compared to AR(1) forecasts of GDP growth, attributing the pattern to declining predictability (other recent studies documenting reduced predictability include D’Agostino, et al. (2005), Stock and Watson (2006), and Tulip (2005)). In this later period, the RMSEs of h=0Q forecasts of GDPgrowthfromtheSPFandbesttimeseriesapproachare1.384and1.609,respectively. The RMSEs of h = 0Q forecasts of GDP inflation from the SPF and best time series approach are 0.831 and 0.926, respectively. Reflecting the declining predictability of output and inflation and the reduced advantage of the SPF at longer horizons, for 1–year ahead forecastsinthe1985-05period,theadvantageoftheSPFovertimeseriesmethodsisquite small. For instance, in 1–year ahead forecasts of GDP growth, the TVP BVAR(4) using GDP growth, GDP inflation, and the T-bill rate beats the SPF (RMSE of 1.218 vs. 1.274); in forecasts of GDP inflation, the TVP BVAR again beats the SPF (RMSE of 0.764 vs. 0.804). In light of the more limited availability of Greenbook (GB) forecasts (the public sample ends in 2000), in lieu of comparing VAR forecasts directly to GB forecasts, we simply compare the GB forecasts to SPF forecasts. As long as the GB and SPF forecasts are broadly comparable in RMSE accuracy, our findings for VARs compared to SPF will also apply to VARs compared to GB forecasts. Table 15 reports RMSEs of forecasts of GDP growth,GDPinflation,andCPIinflation,forsamplesof1970-84and1985-2000(weomit an interest rate comparison because, for much of the sample, GB did not include an unconditional interest rate forecast). Consistent with evidence in such studies as Romer and Romer (2000) and Sims (2002), GB forecasts tend to be more accurate, especially for inflation. For instance, the 1970-84 RMSEs of 1–year ahead forecasts of GDP inflation are 2.192 for SPF and 1.653 for GB. However, perhaps reflecting declining predictability, any advantage of GB over SPF is generally smaller in the second sample than the first. Regardless, the accuracy differences between SPF and GB forecasts are modest enough that comparing VAR forecasts against GB instead of SPF wouldn’t alter the findings described 23

above. 4.2 Long–run forecasts Asnotedinsection3,astheforecasthorizonincreasesbeyondtheoneyearperiodconsidered above, the so-called endpoints come to play an increasingly important role in determining the forecast. Kozicki and Tinsley (1998, 2001a,b), among others, have shown that these endpoints can vary significantly over time. In this section we examine which, if any of the forecast methods considered above, imply reasonable endpoints. For simplicity, we use a 10–year ahead forecast (the forecast in periodt+39, from a forecast origin oft using data through t 1) as a proxy for the endpoint estimate. Kozicki and Tinsley (2001b) use − asimilarmetric(KozickiandTinsleycompare10year–aheadforecaststosurveymeasures oflong-terminflationexpectations). Of course, an immediate question is, what is reasonable? Trend GDP growth is generallythoughttohaveevolvedslowlyovertime,(atleast)declininginthe1970sandrisingin the 1990s. The available real–time estimates of potential GDP from the CBO, taken from Kozicki (2004), show some variation in trend growth. CBO estimates of potential output growth rose from about 2.1 percent in 1991 vintage data to 3.2 percent in 2001 and 2.75 percent in 2004 vintage data.27 At the same time, the implicit inflation goal of monetary policymakers is thought to have trended up from the 1970s through the mid-1980s, and then trended down (see Figure 1 and the associated discussion in section 2). The trend in inflation implies a comparable trend in short-term interest rates. Accuracy in longer-term forecastingislikelytorequireforecastendpointsthatroughlymatchuptovariationinsuch trendsingrowthandinflation. For simplicity, in assessing the ability of VAR forecast methods to generate reasonable endpoints, we compare the estimated endpoint proxies to trends in growth, inflation, and interest rates estimated in real time with exponential smoothing. As noted above, exponential smoothing applied to inflation yields a trend quite similar to the shifting endpoint (or implicit target) estimate of Kozicki and Tinsley (2001a,b). Exponential smoothing appliedtoGDPgrowth(withasmoothingparameterof0.015)yieldsatrendmeasurethat,in line with many economists’ beliefs, shows trend growth gradually slowing over the 1970s 27Foreacheachvintaget,wecalculatetrendgrowthastheprojectedpercentchangeinpotentialGDPin yeart+5. Weuseafive–yearhorizonbecause,forsomeyears,theCBOdataonpotentialoutputextendonly five,ratherthan10,yearsintothefuture. 24

and 1980s before rising in the 1990s. Reflecting real time data availability, trends in each vintaget areestimatedusingdatathroughperiodt 1. − In light of space limitations, we present endpoint proxy results for just GDP growth andGDPinflation,foralimitedsetofforecastingmethodslikelytobeofthemostinterest. The reported forecasts are obtained from models in GDP growth, GDP inflation, and the T-bill rate. Qualitatively, results are similar across other measures of output, inflation, and the interest rate. We omit endpoint results for the T-bill rate because they are qualitatively very similar to those for inflation. The forecast methods or models include the univariate benchmarks, VAR(4), DVAR(4),VAR(4) with inflationdetrending, BVAR(4), BDVAR(4), rolling BDVAR(4), BVAR(4) with TVP, BVAR(4) with intercept TVP, the average of univariate and VAR(4) forecasts, and the average of the univariate and VAR(4) with inflation detrending. In light of the general value of shrinkage in forecasting and the potential success of inflation detrending in pinning down reasonable endpoints, we also include an approach not considered above: a VAR(4) with inflation detrending estimated with BVAR methods(BVAR(4)withinflationdetrending).28 Thissetofmethodsisintendedtoinclude those that work relatively well in shorter-term forecasting and particular approaches, such as differencing and rolling estimation, that are sometimes used in practice to try to capture non–stationaritiessuchasmovingendpoints. The results provided in Figures 2 (GDP growth) and 3 (GDP inflation) show that some forecast approaches fare very poorly, yielding endpoint proxies that are far too volatile to be considered reasonable (note that, in these charts, the scales differ between those methods that work reasonably well and those that don’t). These exceedingly volatile methods include the VAR, BVAR, BVAR with TVP, BVAR with intercept TVP, and the average of theunivariateandVAR(4). Forexample,inthecaseoftheVAR(4),the10–yearaheadforecast of GDP growth plummets to -15.2 percent in (vintage) 1975:Q1 and -12.8 percent in 1981:Q3; the forecast of inflation soars to 34.2 percent in 1981:Q3. In (vintage) 1980:Q2, the BVAR(4) forecasts of GDP growth and inflation reach the extremes of -9.4 and 25.8 percent, respectively. In the case of the BVAR(4) with TVP, the long–term projections of growth and inflation are -20.9 percent and 64.5 percent in 1980:Q2. Such extremes in forecastsofcoursesuggestexplosiverootsintheautoregressivesystems,whichareindeed evident in the system estimates. For example, the VAR(4) system has a largest root of 28WeobtaintheseestimatesusingtheBVARpriorvariancesdescribedinsection2andpriormeansof0 forallcoefficients. 25

1.005inthe1975:Q1estimates,1.002inthe1980:Q2estimates,and1.031inthe1981:Q3 estimates. The BVAR(4) system has a largest root of 1.011 in the 1981:Q3 estimates. As a result, for a practitioner interested in using these methods for forecasting in real time, somecareinadjustingestimatestoavoidexplosiverootswouldberequiredtoimprovethe endpointandlong–termforecastaccuracyofthemethods. Theotherforecastmethods—univariate,DVAR,VARwithinflationdetrending,BVAR with inflation detrending, BDVAR, rolling BDVAR, and the average of the univariate and VARwithinflationdetrending—producemuchlessvolatileandthereforemorereasonable endpoint estimates. For example, the univariate and BDVAR(4) 10–year ahead forecasts of GDP growth correspond pretty closely (at least in relative terms) to the exponentially smoothed trend. Of course, the exponentially smoothed measure may not be the best estimateoftrend. However,anybetterestimateoftrendgrowthisnotlikelytobesignificantly more volatile over time. As a result, even among this relatively better set of forecast methods, a smooth long–term forecast like that from the univariate model may be preferred to a modestly more volatile one, like the forecast from the VAR(4) with inflation detrending. Among inflation forecasts, the endpoint proxies from the univariate and BVAR with inflation detrending models provide the closest match to trend inflation. The endpoint proxy from the BVAR with inflation detrending includes less high frequency variation than does theestimatefromtheunivariatemodel,butisfartherfromtrendinflationinthe1970s. Two other results are worth noting. First, for both growth and inflation, rolling estimation of the BDVAR implies endpoints that are more volatile than the endpoints implied by the recursively estimated BDVAR. Second, compared to OLS estimation, Bayesian estimation of the VAR with inflation detrending helps to dampen volatility in the endpoint proxies(althoughnotincludedintheRMSEresultsabove,Bayesianestimationalsohelped tomodestlyimprovetheforecastaccuracyofVARswithinflationdetrending). 5 Conclusion In this paper we provide empirical evidence on the ability of several different methods to improve the real–time forecast accuracy of small-scale macroeconomic VARs in the presence of model instability. The 18 distinct trivariate VARs that we consider are each comprised of one of three measures of output, one of three measures of inflation, and one of two measures of short-term interest rates. For each of these models we construct real 26

time forecasts of each variable (with particular emphasis on the output and inflation measures). Foreachofthe18variablecombinations,weconsider86differentforecastmodels or methods, incorporating different choices of lag selection, observation windows used for estimation,levelsordifferences,interceptcorrections,stochasticallytime–varyingparameters,breakdating,discountedleastsquares,Bayesianshrinkage,detrendingofinflationand interestrates,andmodelaveraging. Wecompareourresultstothosefromsimplebaseline univariate models as well as forecasts from the Survey of Professional Forecasters and the FederalReserveBoard’sGreenbook. Ourresultsindicatethatsomeofthemethodsdoconsistentlyimproveforecastaccuracy intermsofrootmeansquareerrors(RMSE). Notsurprisingly,thebestmethodoftenvaries withthevariablebeingforecasted,butseveralpatternsdoemerge. Afteraggregatingacross all models, horizons and variables being forecasted, it is clear that model averaging and Bayesian shrinkage methods consistently perform among the best methods. At the other extreme,theapproachesofusingafixedrollingwindowofobservationstoestimatemodel parameters and discounted least squares estimation consistently rank among the worst. Of course, estimation methods that are unsuccessful in forecasting may nonetheless prove useful for other purposes. Perhaps not surprisingly, out–of–sample forecast accuracy does notseemtobestronglyrelatedtoin–samplefit. FormodelsinGDPgrowth,GDPinflation, and the T-bill rate, Figure 4 compares real time forecast RMSEs to in–sample fit estimates (for each forecasting model, in–sample fit is measured as the standard error of estimate, averaged over the forecasting sample). Except for some outlier observations, in–sample fit has little relationship (and sometimes a negative relationship) with forecast accuracy, at leastintheVARmodelsandmethodsweconsider. 27

Acknowledgments We gratefullyacknowledge helpful comments fromTaisuke Nakata, participants in the conference associated with the book, seminar participants at the Federal Reserve Bank of Kansas City and Board of Governors, and an anonymous reviewer. The views expressed hereinaresolelythoseoftheauthorsanddonotnecessarilyreflecttheviewsoftheFederal Reserve Bank of Kansas City, Board of Governors, Federal Reserve System, or any of its staff. 28

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Figure 1: Alternative estimates of CPI inflation trends 9 8 7 6 5 4 3 2 1 1961 1968 1975 1982 1989 1996 2003 exponential smoothing Kozicki-Tinsley/SPF

Figure 2: 10-year ahead forecasts of GDP growth (VAR in GDP growth, GDP inflation, and T-bill rate) univariate BVAR(4) BVAR(4) with TVP 5.0 5.0 5 4.5 2.5 0 4.0 0.0 -5 3.5 3.0 -2.5 -10 2.5 -5.0 -15 2.0 -7.5 -20 1.5 1.0 -10.0 -25 1971 1977 1983 1989 1995 2001 1971 1977 1983 1989 1995 2001 1971 1977 1983 1989 1995 2001 VAR(4) BVAR(4), inflation detrending BVAR(4) with intercept TVP 5.0 5.0 5 2.5 4.5 4 0.0 4.0 3 -2.5 3.5 2 -5.0 3.0 1 -7.5 2.5 0 -10.0 -12.5 2.0 -1 -15.0 1.5 -2 -17.5 1.0 -3 1971 1977 1983 1989 1995 2001 1971 1977 1983 1989 1995 2001 1971 1977 1983 1989 1995 2001 DVAR(4) BDVAR(4) avg. of univariate and VAR(4) 5.0 5.0 5.0 4.5 4.5 2.5 4.0 4.0 3.5 3.5 0.0 3.0 3.0 2.5 2.5 -2.5 2.0 2.0 -5.0 1.5 1.5 1.0 1.0 -7.5 1971 1977 1983 1989 1995 2001 1971 1977 1983 1989 1995 2001 1971 1977 1983 1989 1995 2001 VAR(4), inflation detrending BDVAR(4), rolling avg. of univariate and VAR(4) with infl. detrending 5.0 5.0 5.0 4.5 4.5 4.5 4.0 4.0 4.0 3.5 3.5 3.5 3.0 3.0 3.0 2.5 2.5 2.5 2.0 2.0 2.0 1.5 1.5 1.5 1.0 1.0 1.0 1971 1977 1983 1989 1995 2001 1971 1977 1983 1989 1995 2001 1971 1977 1983 1989 1995 2001 solid lines: forecasts dotted lines: exponentially smoothed trends

Figure 3: 10-year ahead forecasts of GDP inflation (VAR in GDP growth, GDP inflation, and T-bill rate) univariate BVAR(4) BVAR(4) with TVP 15.0 28 70 12.5 24 60 20 50 10.0 16 40 7.5 12 30 5.0 8 20 2.5 4 10 0.0 0 0 1971 1977 1983 1989 1995 2001 1971 1977 1983 1989 1995 2001 1971 1977 1983 1989 1995 2001 VAR(4) BVAR(4), inflation detrending BVAR(4) with intercept TVP 35 15.0 20.0 30 12.5 17.5 15.0 25 10.0 12.5 20 7.5 10.0 15 7.5 5.0 10 5.0 5 2.5 2.5 0 0.0 0.0 1971 1977 1983 1989 1995 2001 1971 1977 1983 1989 1995 2001 1971 1977 1983 1989 1995 2001 DVAR(4) BDVAR(4) avg. of univariate and VAR(4) 14 17.5 22.5 12 15.0 20.0 10 12.5 17.5 15.0 8 10.0 12.5 6 7.5 10.0 4 5.0 7.5 2 2.5 5.0 0 0.0 2.5 -2 -2.5 0.0 1971 1977 1983 1989 1995 2001 1971 1977 1983 1989 1995 2001 1971 1977 1983 1989 1995 2001 VAR(4), inflation detrending BDVAR(4), rolling avg. of univariate and VAR(4) with infl. detrending 15.0 20 15.0 12.5 12.5 15 10.0 10.0 10 7.5 7.5 5.0 5 5.0 2.5 0 0.0 2.5 -2.5 -5 0.0 1971 1977 1983 1989 1995 2001 1971 1977 1983 1989 1995 2001 1971 1977 1983 1989 1995 2001 solid lines: forecasts dotted lines: exponentially smoothed trends

Figure 4. In-sample fit vs. forecast RMSE (VAR in GDP growth, GDP inflation, and T-bill rate) GDP growth results for 1970-84 GDP growth results for 1985-05 5.0 2.6 4.8 2.4 4.6 2.2 4.4 4.2 2.0 4.0 1.8 3.8 1.6 3.6 3.4 1.4 3.50 3.75 4.00 4.25 4.50 3.4 3.6 3.8 4.0 4.2 GDP inflation results for 1970-84 GDP inflation results for 1985-05 2.4 1.4 2.3 1.3 2.2 2.1 1.2 2.0 1.1 1.9 1.0 1.8 1.7 0.9 1.0 1.2 1.4 1.6 1.8 2.0 2.2 1.25 1.50 1.75 2.00 T-bill results for 1970-84 T-bill results for 1985-05 1.84 0.600 1.76 0.575 1.68 0.550 1.60 0.525 1.52 0.500 1.44 0.475 1.36 0.450 1.28 0.425 1.20 0.400 1.12 0.375 0.5 0.6 0.7 0.8 0.9 1.0 0.7 0.8 0.9 1.0 1.1 1.2 1.3 average in-sample st. error of estimate Notes: 1. The figures compare forecast RMSEs for the indicated variable and sample to corresponding measures of in-sample fit. 2. All results are based on models in GDP growth, GDP inflation, and the T-bill rate. The forecast methods are listed in Table 1. The figures exclude results for the intercept correction methods of Clements and Hendry (1996), because it is not clear how best to measure in-sample fit for the associated forecasts. 3. The forecast RMSEs are based on the h = 0Q horizon. Starting with t = 1970:Q1, the in-sample fit of each model used to forecast is estimated as the conventional standard error of estimate (with the conventional degrees of freedom adjustment). For each model, the time series of in-sample fit estimates is averaged over the 1970-84 and 1985- 05 forecast samples. The charts use these average estimates of in-sample fit. 4. In the case of forecasts based on rolling sample model estimates, we fit the same model to the sample of data preceding the rolling sample (assuming, in effect, a break in the model’s coefficients at the time of the rolling sample start). We then estimate insample fit as the (square root of the) sum of squared residuals over the whole period divided by the total sample size less the total number of parameters.

Table1: Forecastingmethods method details VAR(4) VARiny,π,iwithfixedlagorderof4 VAR(2) sameasabovewithfixedlagorderof2 VAR(AIC) VARwithsystemlagdeterminedbyAIC VAR(BIC) VARwithsystemlagdeterminedbyBIC VAR(AIC,byeq.&var.) VARiny,π,iallowingdifferent,AIC-det.lagsfor eachvar.ineacheq. VAR(BIC,byeq.&var.) sameasabove,withBIC-determinedlags DVAR(4) VARiny,∆π,∆iwithfixedlagorderof4 DVAR(2) sameasabovewithfixedlagorderof2 DVAR(AIC) VARiny,∆π,∆iwithsystemlagsetbyAIC DVAR(BIC) VARiny,∆π,∆iwithsystemlagsetbyBIC DVAR(AIC,byeq.&var.) VARiny,∆π,∆iallowingdifferent,AIC-det.lags foreachvar.ineacheq. DVAR(BIC,byeq.&var.) sameasabove,withBIC-determinedlags DVAR(4),outputdiff. VARin∆y,∆π,∆iwithfixedlagorderof4 DVAR(2),outputdiff. sameasabovewithfixedlagorderof2 DVAR(AIC),outputdiff. VARin∆y,∆π,∆iwithsystemlagsetbyAIC DVAR(BIC),outputdiff. VARin∆y,∆π,∆iwithsystemlagsetbyBIC BVAR(4) VAR(4)iny,π,iest.withMinnesotapriors, usingλ =.2,λ =.5,λ =1,λ =1000 1 2 3 4 BVAR(2) sameasabovewithfixedlagorderof2 BDVAR(4) VAR(4)iny,∆π,∆iest.withMinnesotapriors, usingλ =.2,λ =.5,λ =1,λ =1000 1 2 3 4 BDVAR(2) sameasabovewithfixedlagorderof2 VAR(4),rolling VARiny,π,iwithfixedlagorderof4,est. witharollingwindowof60observations VAR(2),rolling sameasabovewithfixedlagorderof2 VAR(AIC),rolling sameasabovewithAIC–determinedlag VAR(BIC),rolling sameasabovewithBIC–determinedlag VAR(AIC,byeq.&var.),rolling VARiny,π,iallowingdifferent,AIC-det.lagsfor eachvar.ineacheq.,est.witharollingsample of60obs. VAR(BIC,byeq.&var.),rolling sameasabovewithBIC-determinedlags DVAR(4),rolling VARiny,∆π,∆iwithfixedlagorderof4,est. witharollingsampleof60observations DVAR(2),rolling sameasabovewithfixedlagorderof2 DVAR(AIC),rolling sameasabovewithAIC–determinedlag DVAR(BIC),rolling sameasabovewithBIC–determinedlag 39

Table1,continued: Forecastingmethods method details DVAR(AIC,byeq.&var.),rolling VARiny,∆π,∆iallowingdifferent,AIC-det.lagsfor eachvar.ineacheq.,est.witharollingsample of60obs. DVAR(BIC,byeq.&var.),rolling sameasabovewithBIC-determinedlags DVAR(4),outputdiff.,rolling VARin∆y,∆π,∆iwithfixedlagorderof4, est. witharollingsampleof60observations DVAR(2),outputdiff.,rolling sameasabovewithfixedlagorderof2 DVAR(AIC),outputdiff.,rolling sameasabovewithAIC–determinedlag DVAR(BIC),outputdiff.,rolling sameasabovewithBIC–determinedlag BVAR(4),rolling BVAR(4)iny,π,iwithλ =.2,λ =.5,λ =1, 1 2 3 λ =1000,est.witharollingsampleof60obs. 4 BVAR(2),rolling sameasabovewithfixedlagorderof2 BDVAR(4),rolling BVAR(4)iny,∆π,∆iwithλ =.2,λ =.5,λ =1, 1 2 3 λ =1000,est.witharollingsampleof60obs. 4 BDVAR(2),rolling sameasabovewithfixedlagorderof2 DLS,VAR(4) VAR(4)iny,π,i,est.withdiscountedleastsquares (DLS),usingdis.ratesof.99foryeq. .95forπandieq. DLS,VAR(2) sameasabovewithfixedlagof2 DLS,VAR(AIC) sameasabovewithlagorderdet. fromAICappliedto OLSestimatesofsystem DLS,DVAR(4) VAR(4)iny,∆π,∆i,est.withDLS, usingdis.ratesof.99foryeq.,.95for∆πand∆ieq. DLS,DVAR(2) sameasabovewithfixedlagof2 DLS,DVAR(AIC) sameasabovewithlagordersetbyAICappliedto OLSestimatesofsystem VAR(AIC),AICinterceptbreaks VARiny,π,iwithAIC-det.lags,allowinguptotwo breaksinthesetofintercepts,withthenumberand datesthatminimizetheAIC VAR(AIC),BICinterceptbreaks sameasabove,usingtheBICtodeterminethebreaks VAR(4),interceptcorrection VAR(4)forecastsadjustedbytheaverageofthe last4residuals(ClementsandHendry(1996),eq.40) VAR(2),interceptcorrection sameasabovewithfixedlagorderof2 VAR(AIC),interceptcorrection VAR(AIClag)forecastsadjustedbytheaverage ofthelast4residuals(Clements andHendry(1996),eq.40) VAR(4),partialint.corr. VAR(4)forecastsofπandiadjustedbytheaverage ofthelast4residuals(yresidualstreatedas0) VAR(2),partialint.corr. sameasabovewithfixedlagorderof2 VAR(AIC),partialint.corr. VAR(AIClag)forecastsofπandiadjustedbythe averageofthelast4residuals (yresidualstreatedas0) 40

Table1,continued: Forecastingmethods method details VAR(4),inflationdetrending VAR(4)iny,π π ,andi π ,where − ∗1 − ∗1 π =π +α(π− π ),α=.−05forGDPand ∗ ∗1 − ∗1 CPIinfl−ation,.07fo−rcorePCEinflation VAR(2),inflationdetrending sameasabovewithfixedlagof2 VAR(AIC),inflationdetrending sameasabovewithAIC–det.lagforthey, π π ,andi π system − ∗1 − ∗1 VAR(BIC),inflationdetrending sameas−abovewithB−IC–det.lagforthey, π π ,andi π system − ∗1 − ∗1 VAR(4),fullESdetrending VAR(4)−iny,π π −,andi i ,where − ∗1 − ∗1 π =π +α(π− π )(α=−.05or.07, ∗ ∗1 − ∗1 depend−ingonπmea−sure), i =i +.07(i i ) ∗ ∗1 − ∗1 VAR(2),fullESdetrending sameas−abovewithfi−xedlagof2 VAR(AIC),fullESdetrending sameasabovewithAIC–det. lagforthey, π π ,andi i system − ∗1 − ∗1 VAR(BIC),fullESdetrending sameas−abovewith−BIC–det. lagforthey, π π ,andi i system − ∗1 − ∗1 TVPBVAR(4) TVPBV−AR(4)iny,−π,iwithλ =.2,λ =.5, 1 2 λ =1,λ =.1,λ=.0005 3 4 TVPBVAR(2) sameasabovewithfixedlagof2 TVPBVAR(4),λ =.5,λ=.0025 TVPBVAR(4)iny,π,iwithλ =.2,λ =.5, 4 1 2 λ =1,λ =.5,λ=.0025 3 4 TVPBVAR(2),λ =.5,λ=.0025 sameasabovewithfixedlagof2 4 TVPBVAR(4),λ =1000,λ=.005 TVPBVAR(4)iny,π,iwithλ =.2,λ =.5, 4 1 2 λ =1,λ =1000,λ=.005 3 4 TVPBVAR(2),λ =1000,λ=.005 sameasabovewithfixedlagof2 4 TVPBVAR(4),λ =1000,λ=.0001 TVPBVAR(4)iny,π,iwithλ =.2,λ =.5, 4 1 2 λ =1,λ =1000,λ=.0001 3 4 TVPBVAR(2),λ =1000,λ=.0001 sameasabovewithfixedlagof2 4 InterceptTVPBVAR(4) BVAR(4)iny,π,i,TVPinonlyintercepts, λ =.2,λ =.5,λ =1,λ =.1, 1 2 3 4 λ=.0005 InterceptTVPBVAR(2) sameasabovewithfixedlagof2 InterceptTVPBVAR(4),λ =.5,λ=.0025 BVAR(4)iny,π,i,TVPinonlyintercepts, 4 λ =.2,λ =.5,λ =1,λ =.5, 1 2 3 4 λ=.0025 InterceptTVPBVAR(2),λ =.5,λ=.0025 sameasabovewithfixedlagof2 4 41

Table1,continued: Forecastingmethods method details averageofallforecasts simpleaverageofalloftheaboveforecasts avg.ofVAR(4),rollingVAR(4) averageofforecastsfromrecursiveandrolling estimatesofVAR(4)iny,π,andi avg.ofVAR(2),rollingVAR(2) sameasaboveusingVARswithfixedlagof2 avg.ofunivariate,VAR(4) averageofforecastsfromunivariatemodeland VAR(4)iny,π,andi avg.ofunivariate,VAR(2) sameasaboveusingVARwithfixedlagof2 avg.ofunivariate,DVAR(4) averageofforecastsfromunivariatemodeland VAR(4)in∆y,∆π,andi avg.ofunivariate,DVAR(2) sameasaboveusingVARwithfixedlagof2 avg.ofuniv.,IDTRVAR(4) averageofforecastsfromunivariatemodel andVAR(4)withinflationdetrending avg.ofuniv.,IDTRVAR(2) sameasaboveusingVARwithfixedlagof2 avg.ofuniv.,VAR(4),DVAR(4), simpleaverageofunivariate,VAR(4),DVAR(4), TVPBVAR(4) andTVPBVAR(4)(λ =.1,λ=.0005) 4 forecasts avg.ofuniv.,VAR(2),DVAR(2), sameasaboveusingVARswithfixedlagof2 TVPBVAR(2) univariate AR(2)fory,rollingMA(1)for∆π, rollingMA(1)for∆i Notes: 1. The variables y, π, and i refer to, respectively, output (GDP growth, the HPS gap, or the HP gap), inflation (GDP inflation, CPI inflation, or core PCE inflation), and the interest rate (T-bill or federalfunds). 2. Unless otherwise noted, all models are estimated recursively, using all data (starting in 1955 or later)availableuptotheforecastingdate. 3. Therollingestimatesoftheunivariatemodelsfor∆πand∆iuse40observations. 4. TheAICandBIClagordersrangefrom0(theminimumallowed)to4(themaximumallowed). 5. Section 2 details the hyperparameterization (and λ notation above) used in BVAR estimation. In BVAR estimation, prior means for all coefficients are generally set at 0, with the following exceptions: (a) prior means for own first lags of π and i are set at 1 in models with levels of inflation and interest rates; (b) prior means for own first lags of y are set at 0.8 in models with an output gap; and (c) prior means for the intercept of GDP growth equations are set to the historical averageofgrowthinBVARestimatesthatimposeinformativepriors(λ =.1or.5)ontheconstant 4 term. 6. The time variation in the coefficients of the TVP BVARs takes a random walk form. In time– varyingBVARswithflatpriorsontheintercepts(λ =1000),thevariationoftheinnovationinthe 4 intercept is set at λ times the prior variance of the coefficient on the own first lag instead of the priorvarianceoftheconstant. 7. The exponential smoothing used in the models with detrending is initialized with the average valueofinflationoverthefirstfiveyearsofeachsample. 42

Table2: Real-timeRMSEresultsforGDPgrowthandGDPinflation (RMSEsinfirstrow,RMSEratiosinallothers) GDPgrowthforecasts 1970-84 1985-2005 forecastmethod h=0Q h=1Q h=1Y h=0Q h=1Q h=1Y univariate 4.183 4.761 3.652 1.609 1.668 1.293 VAR(4) 1.022 .912 .936 1.184 1.200 1.110 VAR(4),interceptcorrection 1.038 .944 1.047 1.177 1.209 1.325 VAR(AIC) 1.024 .921 .969 1.169 1.188 1.105 DVAR(4) 1.039 .932 .760 1.260 1.298 1.152 DVAR(AIC) .974 .847 .798 1.208 1.240 1.108 VAR(AIC,byeq.&var.) .948 .902 .989 1.113 1.122 .998 DVAR(AIC,byeq.&var.) 1.019 .943 .783 1.204 1.260 1.155 BVAR(4) .919 .875 .949 1.077 1.090 1.005 BDVAR(4) .988 .956 .956 1.045 1.045 1.013 VAR(4),inflationdetrending .956 .837 .797 1.247 1.283 1.162 VAR(AIC),interceptbreaks .994 .894 .891 1.378 1.478 1.562 VAR(4),rolling 1.175 1.062 1.091 1.222 1.306 1.385 DVAR(4),rolling 1.077 1.003 .773 1.115 1.221 1.143 VAR(AIC,byeq.&var.),rolling 1.014 .943 1.019 1.296 1.301 1.321 BVAR(4),rolling .945 .880 1.004 1.196 1.220 1.193 BDVAR(4),rolling 1.008 .993 1.003 1.024 1.040 1.066 TVPBVAR(4) .927 .896 .955 1.025 1.024 .941 InterceptTVPBVAR(4) .922 .891 .940 1.019 1.013 .914 DLS,VAR(4) 1.081 1.005 1.068 1.154 1.183 1.143 DLS,DVAR(4) 1.078 1.028 .949 1.167 1.208 1.159 averageofallforecasts .893 .815 .816 1.078 1.093 1.015 avg.ofVAR(4),rollingVAR(4) 1.070 .957 .953 1.158 1.212 1.210 avg.ofunivariate,VAR(4) .958 .901 .900 1.057 1.056 .988 avg.ofunivariate,DVAR(4) .945 .882 .796 1.086 1.096 1.027 avg.ofuniv.,IDTRVAR(4) .931 .871 .849 1.060 1.061 .952 avg.ofuniv.,VAR(4), .922 .850 .804 1.078 1.084 .995 DVAR(4),TVPBVAR(4) 43

Table2,continued: RMSEresultsforGDPgrowthandGDPinflation (RMSEsinfirstrow,RMSEratiosinallothers) GDPinflationforecasts 1970-84 1985-2005 forecastmethod h=0Q h=1Q h=1Y h=0Q h=1Q h=1Y univariate 1.825 2.153 2.389 .951 1.016 .760 VAR(4) 1.022 1.033 1.061 1.001 .948 .959 VAR(4),interceptcorrection 1.020 1.054 1.142 1.133 1.134 1.439 VAR(AIC) 1.037 1.066 1.057 1.024 .977 .982 DVAR(4) 1.007 .946 .896 .989 .946 1.006 DVAR(AIC) .964 .955 .912 .994 .950 .985 VAR(AIC,byeq.&var.) 1.028 1.085 1.120 1.014 .965 .992 DVAR(AIC,byeq.&var.) 1.027 1.033 .998 1.003 .965 1.031 BVAR(4) .971 1.047 1.093 1.023 1.039 1.161 BDVAR(4) .969 .985 .936 1.030 1.034 1.069 VAR(4),inflationdetrending 1.024 1.013 1.006 1.011 .979 1.081 VAR(AIC),interceptbreaks 1.032 1.013 .996 1.085 1.098 1.438 VAR(4),rolling 1.016 1.083 1.080 1.156 1.128 1.407 DVAR(4),rolling 1.026 1.000 .900 1.066 .990 1.151 VAR(AIC,byeq.&var.),rolling 1.016 1.165 1.212 1.159 1.152 1.504 BVAR(4),rolling .950 1.022 1.050 1.090 1.174 1.482 BDVAR(4),rolling .965 .991 .939 1.075 1.101 1.191 TVPBVAR(4) .975 1.053 1.108 .992 .977 1.006 InterceptTVPBVAR(4) .975 1.047 1.081 1.007 1.004 1.079 DLS,VAR(4) 1.129 1.334 1.290 1.173 1.132 1.243 DLS,DVAR(4) 1.300 1.251 1.070 1.170 1.109 1.161 averageofallforecasts .946 .989 .970 1.025 1.015 1.057 avg.ofVAR(4),rollingVAR(4) 1.009 1.052 1.063 1.055 1.014 1.131 avg.ofunivariate,VAR(4) .967 .985 .996 .980 .958 .942 avg.ofunivariate,DVAR(4) .967 .952 .931 .974 .954 .967 avg.ofuniv.,IDTRVAR(4) .971 .979 .974 .985 .969 .980 avg.ofuniv.,VAR(4), .959 .978 .980 .977 .951 .953 DVAR(4),TVPBVAR(4) Notes: 1. ThevariablesineachmultivariatemodelareGDPgrowth,GDPinflation,andtheT-billrate. 2. The entries in the first row are RMSEs, for variables defined in annualized percentage points. All other entries are RMSE ratios, for the indicated specification relative to the corresponding univariatespecification. 3. Individual RMSE ratios that are significantly below 1 according to bootstrap p–values are indicated by a slanted font. In each column, if a forecast is significantly better (in MSE) than the benchmark according to data snooping–robust p–values (bootstrapped as in Hansen (2005)), the associatedRMSEratioappearsinaboldfont. 4. The forecast errors are calculated using the first–available (real–time) estimates of output and inflationastheactualdataonoutputandinflation. 5. In each quarter t from 1970:Q1 through 2005:Q4, vintage t data are used to form forecasts for periodst (h=0Q),t+1 (h=1Q), andt+4 (h=1Y). The forecasts of GDP growth and inflation fortheh=1Y horizoncorrespondtoannualpercentchanges: averagegrowthandaverageinflation fromt+1throught+4. 6. SeeTable1fordetailoneachforecastmethod. 44

Table3: Real-timeRMSEresultsfortheHPSoutputgapandGDPinflation (RMSEsinfirstrow,RMSEratiosinallothers) HPSoutputgapforecasts 1970-84 1985-2005 forecastmethod h=0Q h=1Q h=1Y h=0Q h=1Q h=1Y univariate 1.039 1.988 3.891 .702 1.028 2.044 VAR(4) 1.051 .960 .944 1.110 1.159 1.204 VAR(4),interceptcorrection 1.079 1.010 1.110 1.066 1.048 1.060 VAR(AIC) 1.016 .966 .991 1.108 1.155 1.207 DVAR(4) 1.068 .942 .743 1.102 1.127 1.084 DVAR(AIC) 1.039 .947 .866 1.099 1.105 1.059 VAR(AIC,byeq.&var.) .985 .946 .995 1.077 1.133 1.176 DVAR(AIC,byeq.&var.) 1.088 1.005 .880 1.071 1.110 1.085 BVAR(4) 1.012 .931 .922 1.077 1.151 1.176 BDVAR(4) 1.064 1.002 .994 1.002 .997 .991 VAR(4),inflationdetrending 1.030 .920 .892 1.060 1.077 1.012 VAR(AIC),interceptbreaks 1.008 .929 .754 1.189 1.320 1.267 VAR(4),rolling 1.190 1.110 1.032 1.116 1.237 1.305 DVAR(4),rolling 1.103 .993 .802 1.029 1.074 1.008 VAR(AIC,byeq.&var.),rolling 1.170 1.129 1.064 1.099 1.181 1.211 BVAR(4),rolling 1.060 .968 .986 1.087 1.172 1.186 BDVAR(4),rolling 1.093 1.047 1.059 .993 1.005 .995 TVPBVAR(4) 1.020 .957 .947 .982 .970 .921 InterceptTVPBVAR(4) 1.015 .944 .923 .977 .957 .908 DLS,VAR(4) 1.100 1.041 .935 1.053 1.067 1.108 DLS,DVAR(4) 1.106 1.020 .919 1.061 1.066 1.056 averageofallforecasts .948 .872 .824 1.025 1.036 1.000 avg.ofVAR(4),rollingVAR(4) 1.091 1.005 .931 1.089 1.162 1.218 avg.ofunivariate,VAR(4) .974 .912 .876 1.034 1.041 1.028 avg.ofunivariate,DVAR(4) .973 .904 .804 1.038 1.045 1.024 avg.ofuniv.,IDTRVAR(4) .954 .878 .841 1.011 1.003 .950 avg.ofuniv.,VAR(4), .966 .888 .809 1.028 1.027 .992 DVAR(4),TVPBVAR(4) 45

Table3,continued: RMSEresultsfortheHPSoutputgapandGDPinflation (RMSEsinfirstrow,RMSEratiosinallothers) GDPinflationforecasts 1970-84 1985-2005 forecastmethod h=0Q h=1Q h=1Y h=0Q h=1Q h=1Y univariate 1.825 2.153 2.389 .951 1.016 .760 VAR(4) 1.020 1.037 1.075 .973 .923 .933 VAR(4),interceptcorrection 1.017 1.043 1.116 1.109 1.108 1.436 VAR(AIC) 1.020 1.046 1.050 .992 .968 .975 DVAR(4) 1.003 .942 .904 .990 .960 1.132 DVAR(AIC) .941 .931 .879 .992 .967 1.130 VAR(AIC,byeq.&var.) 1.054 1.112 1.130 .989 .934 1.007 DVAR(AIC,byeq.&var.) 1.008 .993 .906 .992 .972 1.202 BVAR(4) .967 1.026 1.048 .993 .986 1.042 BDVAR(4) .960 .954 .879 1.031 1.047 1.209 VAR(4),inflationdetrending .982 .978 .942 .970 .910 .897 VAR(AIC),interceptbreaks .975 .973 .930 1.022 1.014 1.101 VAR(4),rolling 1.024 1.108 1.139 1.136 1.134 1.437 DVAR(4),rolling 1.013 1.017 .942 1.059 .971 1.123 VAR(AIC,byeq.&var.),rolling 1.017 1.166 1.167 1.145 1.152 1.579 BVAR(4),rolling .958 1.010 1.022 1.088 1.190 1.525 BDVAR(4),rolling .966 .978 .917 1.076 1.107 1.261 TVPBVAR(4) .959 1.010 1.043 .996 1.001 1.169 InterceptTVPBVAR(4) .958 1.004 1.018 .998 1.000 1.153 DLS,VAR(4) 1.139 1.311 1.322 1.208 1.176 1.368 DLS,DVAR(4) 1.350 1.257 1.236 1.166 1.100 1.251 averageofallforecasts .935 .957 .907 1.005 .991 1.035 avg.ofVAR(4),rollingVAR(4) 1.014 1.065 1.098 1.023 .986 1.081 avg.ofunivariate,VAR(4) .968 .982 .990 .967 .944 .930 avg.ofunivariate,DVAR(4) .963 .947 .926 .966 .946 .982 avg.ofuniv.,IDTRVAR(4) .954 .957 .924 .963 .934 .894 avg.ofuniv.,VAR(4), .951 .960 .954 .966 .942 .983 DVAR(4),TVPBVAR(4) Notes: 1. The variables in each multivariate model are the HPS output gap, GDP inflation, and the T-bill rate. 2. SeethenotestoTable2. 46

Table4: Real-timeRMSEresultsforGDPgrowthandCPIinflation (RMSEsinfirstrow,RMSEratiosinallothers) GDPgrowthforecasts 1970-84 1985-2005 forecastmethod h=0Q h=1Q h=1Y h=0Q h=1Q h=1Y univariate 4.183 4.761 3.652 1.609 1.668 1.293 VAR(4) 1.039 .945 .926 1.155 1.172 1.103 VAR(4),interceptcorrection 1.054 .952 1.004 1.156 1.197 1.367 VAR(AIC) .981 .948 1.031 1.142 1.157 1.084 DVAR(4) 1.093 .959 .767 1.236 1.264 1.159 DVAR(AIC) 1.058 .983 .947 1.236 1.264 1.159 VAR(AIC,byeq.&var.) .937 .873 .926 1.113 1.121 .974 DVAR(AIC,byeq.&var.) 1.043 .944 .773 1.200 1.254 1.151 BVAR(4) .919 .871 .917 1.061 1.071 .982 BDVAR(4) .987 .958 .958 1.035 1.041 1.014 VAR(4),inflationdetrending .977 .863 .793 1.324 1.380 1.341 VAR(AIC),interceptbreaks .935 .925 .963 1.413 1.504 1.498 VAR(4),rolling 1.135 1.061 1.049 1.363 1.348 1.333 DVAR(4),rolling 1.114 1.019 .813 1.179 1.190 1.178 VAR(AIC,byeq.&var.),rolling 1.011 .976 1.078 1.343 1.297 1.311 BVAR(4),rolling .935 .872 .971 1.224 1.236 1.211 BDVAR(4),rolling 1.009 .991 1.004 1.036 1.045 1.066 TVPBVAR(4) .925 .893 .929 1.009 1.015 .952 InterceptTVPBVAR(4) .921 .888 .916 1.007 1.007 .919 DLS,VAR(4) 1.071 1.077 1.017 1.170 1.170 1.129 DLS,DVAR(4) 1.104 1.041 .909 1.191 1.182 1.186 averageofallforecasts .904 .843 .826 1.090 1.100 1.037 avg.ofVAR(4),rollingVAR(4) 1.067 .982 .952 1.210 1.225 1.192 avg.ofunivariate,VAR(4) .969 .914 .879 1.044 1.042 .985 avg.ofunivariate,DVAR(4) .976 .909 .807 1.075 1.080 1.031 avg.ofuniv.,IDTRVAR(4) .937 .873 .807 1.083 1.091 1.019 avg.ofuniv.,VAR(4), .944 .872 .800 1.063 1.070 1.003 DVAR(4),TVPBVAR(4) 47

Table4,continued: RMSEresultsforGDPgrowthandCPIinflation (RMSEsinfirstrow,RMSEratiosinallothers) CPIinflationforecasts 1970-84 1985-2005 forecastmethod h=0Q h=1Q h=1Y h=0Q h=1Q h=1Y univariate 2.117 2.733 2.970 1.347 1.475 1.247 VAR(4) .866 .957 1.016 .975 1.028 1.078 VAR(4),interceptcorrection .885 1.046 1.152 1.188 1.393 1.963 VAR(AIC) .895 1.001 1.045 .975 1.022 1.064 DVAR(4) .847 .888 .854 .952 1.006 1.095 DVAR(AIC) .868 .917 .889 .952 1.006 1.095 VAR(AIC,byeq.&var.) .907 .993 1.045 .970 1.022 1.095 DVAR(AIC,byeq.&var.) .851 .894 .869 .952 .982 1.066 BVAR(4) .926 1.037 1.120 .986 .985 .999 BDVAR(4) .848 .912 .933 .977 1.009 1.065 VAR(4),inflationdetrending .824 .889 .822 .985 1.054 1.191 VAR(AIC),interceptbreaks .895 1.024 1.063 1.025 1.081 1.208 VAR(4),rolling .880 1.020 1.094 1.127 1.242 1.430 DVAR(4),rolling .847 .939 .916 1.025 1.093 1.255 VAR(AIC,byeq.&var.),rolling .950 1.099 1.181 1.113 1.173 1.383 BVAR(4),rolling .928 1.026 1.066 1.028 1.056 1.170 BDVAR(4),rolling .869 .933 .955 1.005 1.042 1.114 TVPBVAR(4) .914 1.014 1.090 .979 .970 .936 InterceptTVPBVAR(4) .914 1.001 1.043 .986 .981 .979 DLS,VAR(4) 1.007 1.357 1.603 1.262 1.264 1.407 DLS,DVAR(4) 1.031 1.153 1.082 1.194 1.216 1.451 averageofallforecasts .831 .931 .962 .989 1.025 1.099 avg.ofVAR(4),rollingVAR(4) .863 .983 1.047 1.011 1.075 1.138 avg.ofunivariate,VAR(4) .868 .920 .935 .959 .989 .997 avg.ofunivariate,DVAR(4) .862 .898 .894 .944 .980 1.013 avg.ofuniv.,IDTRVAR(4) .857 .895 .863 .962 .993 1.021 avg.ofuniv.,VAR(4), .851 .915 .933 .950 .978 .990 DVAR(4),TVPBVAR(4) Notes: 1. ThevariablesineachmultivariatemodelareGDPgrowth,CPIinflation,andtheT-billrate. 2. SeethenotestoTable2. 48

Table5: Real-timeRMSEresultsfortheHPSoutputgapandCPIinflation (RMSEsinfirstrow,RMSEratiosinallothers) HPSoutputgapforecasts 1970-84 1985-2005 forecastmethod h=0Q h=1Q h=1Y h=0Q h=1Q h=1Y univariate 1.039 1.988 3.891 .702 1.028 2.044 VAR(4) 1.066 .980 .943 1.097 1.142 1.162 VAR(4),interceptcorrection 1.096 1.030 1.092 1.054 1.038 1.063 VAR(AIC) .991 .979 1.017 1.086 1.135 1.155 DVAR(4) 1.146 1.014 .790 1.088 1.123 1.091 DVAR(AIC) 1.036 .992 .998 1.077 1.097 1.043 VAR(AIC,byeq.&var.) 1.011 1.007 1.003 1.074 1.124 1.136 DVAR(AIC,byeq.&var.) 1.064 .945 .760 1.069 1.109 1.085 BVAR(4) 1.022 .941 .906 1.060 1.123 1.127 BDVAR(4) 1.065 1.005 .997 .995 .996 .992 VAR(4),inflationdetrending 1.037 .916 .839 1.070 1.111 1.089 VAR(AIC),interceptbreaks .990 .961 .778 1.132 1.233 1.186 VAR(4),rolling 1.206 1.170 1.163 1.143 1.228 1.274 DVAR(4),rolling 1.163 1.062 .909 1.076 1.098 1.056 VAR(AIC,byeq.&var.),rolling 1.170 1.097 1.133 1.119 1.190 1.190 BVAR(4),rolling 1.068 .983 .998 1.093 1.173 1.189 BDVAR(4),rolling 1.093 1.049 1.063 .999 1.008 .995 TVPBVAR(4) 1.031 .971 .953 .972 .961 .916 InterceptTVPBVAR(4) 1.025 .957 .926 .972 .959 .913 DLS,VAR(4) 1.089 1.085 .973 1.055 1.059 1.084 DLS,DVAR(4) 1.156 1.094 .961 1.076 1.084 1.055 averageofallforecasts .967 .909 .864 1.026 1.036 1.003 avg.ofVAR(4),rollingVAR(4) 1.108 1.051 1.022 1.100 1.157 1.190 avg.ofunivariate,VAR(4) .992 .937 .892 1.029 1.035 1.015 avg.ofunivariate,DVAR(4) 1.012 .944 .828 1.032 1.043 1.028 avg.ofuniv.,IDTRVAR(4) .964 .885 .817 1.012 1.013 .985 avg.ofuniv.,VAR(4), .998 .927 .838 1.020 1.021 .988 DVAR(4),TVPBVAR(4) 49

Table5,continued: RMSEresultsfortheHPSoutputgapandCPIinflation (RMSEsinfirstrow,RMSEratiosinallothers) CPIinflationforecasts 1970-84 1985-2005 forecastmethod h=0Q h=1Q h=1Y h=0Q h=1Q h=1Y univariate 2.117 2.733 2.970 1.347 1.475 1.247 VAR(4) .906 1.012 1.108 .967 1.012 1.016 VAR(4),interceptcorrection .910 1.077 1.187 1.180 1.372 1.856 VAR(AIC) .874 .937 1.027 .960 .987 .960 DVAR(4) .902 .959 .946 .964 1.005 1.055 DVAR(AIC) .821 .896 .934 .974 1.007 1.063 VAR(AIC,byeq.&var.) .943 1.002 1.087 .962 1.021 1.052 DVAR(AIC,byeq.&var.) .880 .921 .938 .955 1.006 1.075 BVAR(4) .925 1.021 1.089 .981 .976 .949 BDVAR(4) .847 .901 .928 .983 1.030 1.123 VAR(4),inflationdetrending .860 .932 .900 .944 .964 .865 VAR(AIC),interceptbreaks .889 .943 .963 1.021 1.101 1.176 VAR(4),rolling .912 1.105 1.250 1.141 1.237 1.331 DVAR(4),rolling .912 1.046 1.072 1.018 1.052 1.116 VAR(AIC,byeq.&var.),rolling .946 1.064 1.135 1.067 1.113 1.200 BVAR(4),rolling .940 1.029 1.062 1.019 1.041 1.127 BDVAR(4),rolling .883 .946 .992 1.007 1.050 1.143 TVPBVAR(4) .916 1.010 1.102 .996 1.019 1.069 InterceptTVPBVAR(4) .915 .998 1.060 .997 1.016 1.058 DLS,VAR(4) 1.062 1.375 1.623 1.287 1.331 1.523 DLS,DVAR(4) 1.132 1.216 1.258 1.178 1.202 1.399 averageofallforecasts .834 .920 .939 .984 1.011 1.038 avg.ofVAR(4),rollingVAR(4) .897 1.052 1.167 1.017 1.065 1.050 avg.ofunivariate,VAR(4) .882 .945 .968 .957 .985 .981 avg.ofunivariate,DVAR(4) .886 .932 .931 .944 .972 .976 avg.ofuniv.,IDTRVAR(4) .861 .894 .828 .941 .950 .877 avg.ofuniv.,VAR(4), .873 .948 .981 .951 .977 .972 DVAR(4),TVPBVAR(4) Notes: 1. The variables in each multivariate model are the HPS output gap, CPI inflation, and the T-bill rate. 2. SeethenotestoTable2. 50

Table6: Averageforecastaccuracyrankings, acrossapplicationsandmethodsinTables2-5 (sortedlowtohigh) method average st.dev. avg. ofuniv.,IDTRVAR(4) 5.1 2.8 avg. ofuniv.,VAR(4),DVAR(4),TVPBVAR(4) 5.7 2.6 avg. ofunivariateandDVAR(4) 6.8 3.1 avg. ofunivariateandVAR(4) 7.7 2.9 averageofallforecasts 8.0 4.9 InterceptTVPBVAR(4) 9.8 6.4 BDVAR(4) 10.7 6.4 TVPBVAR(4) 10.8 6.9 VAR(4),inflationdetrending 10.8 7.5 DVAR(AIC) 11.2 6.6 univariate 12.1 6.7 DVAR(4) 12.2 7.9 DVAR(AIC,byeq.&var.) 12.5 6.2 BVAR(4) 12.6 6.3 BDVAR(4),rolling 14.2 7.1 VAR(AIC,byeq.&var.) 14.4 6.2 VAR(4) 14.8 5.6 VAR(AIC) 15.0 5.8 DVAR(4),rolling 15.9 6.3 VAR(AIC),AICinterceptbreaks 17.3 7.9 BVAR(4),rolling 18.5 5.9 avg. ofVAR(4)androllingVAR(4) 19.1 3.7 VAR(4),interceptcorrection 21.0 4.6 DLS,DVAR(4) 21.4 5.2 DLS,VAR(4) 22.3 5.4 VAR(AIC,byeq.&var.),rolling 23.9 2.6 VAR(4),rolling 24.4 2.8 Notes: 1. The figures in the table are obtained by: (1) ranking, for each of the 48 columns of Tables 2-5, the 27 forecast methods or models considered; and (2) calculating the average and standard deviationofeachmethod’s(48)ranks. 51

Table7: AverageRMSEs,acrossapplications andmethodsinTables2-5 (sortedlowtohigh) method average st.dev. avg. ofuniv.,IDTRVAR(4) .943 .070 avg. ofuniv.,VAR(4),DVAR(4),TVPBVAR(4) .955 .068 avg. ofunivariateandDVAR(4) .960 .072 averageofallforecasts .967 .082 avg. ofunivariateandVAR(4) .968 .050 InterceptTVPBVAR(4) .981 .056 TVPBVAR(4) .987 .058 BDVAR(4) .995 .064 univariate 1.000 .000 VAR(4),inflationdetrending 1.001 .143 DVAR(AIC) 1.004 .109 DVAR(4) 1.009 .130 DVAR(AIC,byeq.&var.) 1.011 .117 BVAR(4) 1.012 .076 BDVAR(4),rolling 1.025 .072 VAR(AIC,byeq.&var.) 1.025 .074 VAR(4) 1.030 .087 VAR(AIC) 1.031 .078 DVAR(4),rolling 1.036 .107 avg. ofVAR(4)androllingVAR(4) 1.068 .088 BVAR(4),rolling 1.081 .132 VAR(AIC),AICinterceptbreaks 1.088 .196 DLS,DVAR(4) 1.141 .113 VAR(4),interceptcorrection 1.149 .204 VAR(AIC,byeq.&var.),rolling 1.157 .132 VAR(4),rolling 1.173 .128 DLS,VAR(4) 1.184 .156 Notes: 1. The figures in the table are simple averages and standard deviations, across the 48 columns of Tables 2-5, of each forecast method’s RMSE ratios. Note that the RMSE ratio of the univariate forecastisalways1. 52

Table8: Real-time1996-2005RMSEresultsforGDPgrowthandGDPinflation (RMSEsinfirstrow,RMSEratiosinallothers) GDPgrowthforecasts GDPinflationforecasts forecastmethod h=0Q h=1Q h=1Y h=0Q h=1Q h=1Y univariate 1.624 1.691 1.283 .762 .841 .717 VAR(4) 1.228 1.223 1.104 .964 .965 .973 VAR(4),interceptcorrection 1.249 1.260 1.295 1.105 1.156 1.476 VAR(AIC) 1.228 1.223 1.104 .964 .965 .973 DVAR(4) 1.254 1.231 1.065 1.013 1.032 1.084 DVAR(AIC) 1.245 1.230 1.065 1.011 1.028 1.085 VAR(AIC,byeq.&var.) 1.176 1.193 1.052 .970 .976 .970 DVAR(AIC,byeq.&var.) 1.184 1.193 1.049 1.015 1.023 1.077 BVAR(4) 1.102 1.132 1.065 1.015 1.038 1.110 BDVAR(4) 1.053 1.032 .981 1.017 1.040 1.097 VAR(4),inflationdetrending 1.222 1.228 1.057 .974 .968 .953 VAR(AIC),interceptbreaks 1.263 1.314 1.204 .977 .980 .995 VAR(4),rolling 1.000 1.044 1.051 1.117 1.115 1.184 DVAR(4),rolling 1.058 1.099 1.176 1.069 1.022 1.047 VAR(AIC,byeq.&var.),rolling 1.125 1.131 1.039 1.105 1.081 1.201 BVAR(4),rolling 1.033 1.052 .986 1.042 1.094 1.255 BDVAR(4),rolling 1.036 1.037 1.080 1.030 1.064 1.156 TVPBVAR(4) 1.065 1.083 1.012 .998 1.001 1.016 InterceptTVPBVAR(4) 1.058 1.072 .987 1.008 1.019 1.051 DLS,VAR(4) 1.178 1.183 1.091 1.200 1.129 1.173 DLS,DVAR(4) 1.180 1.179 1.089 1.215 1.176 1.160 averageofallforecasts 1.082 1.084 .991 1.000 1.004 1.046 avg.ofVAR(4),rollingVAR(4) 1.073 1.101 1.049 1.018 1.021 1.058 avg.ofunivariate,VAR(4) 1.084 1.069 .975 .949 .949 .933 avg.ofunivariate,DVAR(4) 1.100 1.080 .989 .967 .975 .979 avg.ofuniv.,IDTRVAR(4) 1.070 1.057 .925 .952 .951 .923 avg.ofuniv.,VAR(4), 1.108 1.098 .991 .963 .968 .970 DVAR(4),TVPBVAR(4) Notes: 1. ThevariablesineachmultivariatemodelareGDPgrowth,GDPinflation,andtheT-billrate. 2. SeethenotestoTable2. 53

Table9: Real-time1996-2005RMSEresultsforGDPgrowthand corePCEinflation (RMSEsinfirstrow,RMSEratiosinallothers) GDPgrowthforecasts corePCEforecasts forecastmethod h=0Q h=1Q h=1Y h=0Q h=1Q h=1Y univariate 1.624 1.691 1.283 .646 .602 .460 VAR(4) 1.223 1.174 1.077 1.233 1.339 1.630 VAR(4),interceptcorrection 1.238 1.237 1.180 1.316 1.599 2.301 VAR(AIC) 1.223 1.174 1.077 1.233 1.339 1.630 DVAR(4) 1.171 1.134 .976 1.200 1.297 1.322 DVAR(AIC) 1.171 1.134 .976 1.200 1.297 1.322 VAR(AIC,byeq.&var.) 1.251 1.239 1.151 1.253 1.455 1.949 DVAR(AIC,byeq.&var.) 1.204 1.173 1.019 1.186 1.252 1.264 BVAR(4) 1.175 1.165 1.130 1.224 1.376 1.819 BDVAR(4) 1.049 1.007 .958 1.167 1.234 1.243 VAR(4),inflationdetrending 1.231 1.195 1.061 1.212 1.284 1.394 VAR(AIC),interceptbreaks 1.425 1.536 1.604 1.222 1.384 1.578 VAR(4),rolling 1.014 1.034 1.076 .981 1.166 1.580 DVAR(4),rolling .982 1.002 1.137 .938 1.077 1.060 VAR(AIC,byeq.&var.),rolling 1.157 1.115 1.174 1.024 1.261 1.670 BVAR(4),rolling 1.067 1.071 1.053 1.176 1.314 1.764 BDVAR(4),rolling 1.024 1.034 1.079 1.105 1.159 1.162 TVPBVAR(4) 1.090 1.081 1.028 1.161 1.257 1.459 InterceptTVPBVAR(4) 1.089 1.073 1.001 1.198 1.319 1.624 DLS,VAR(4) 1.168 1.146 1.051 1.122 1.458 1.551 DLS,DVAR(4) 1.150 1.108 1.072 1.123 1.505 1.387 averageofallforecasts 1.093 1.068 .988 1.117 1.199 1.326 avg.ofVAR(4),rollingVAR(4) 1.081 1.072 1.052 1.052 1.172 1.489 avg.ofunivariate,VAR(4) 1.074 1.042 .947 1.089 1.137 1.260 avg.ofunivariate,DVAR(4) 1.064 1.038 .955 1.076 1.120 1.108 avg.ofuniv.,IDTRVAR(4) 1.069 1.038 .921 1.081 1.117 1.156 avg.ofuniv.,VAR(4), 1.091 1.061 .960 1.123 1.187 1.275 DVAR(4),TVPBVAR(4) Notes: 1. The variables in each multivariate model are GDP growth, core PCE inflation, and the T-bill rate. 2. SeethenotestoTable2. 54

Table10: Real-time1996-2005RMSEresultsfortheHPSoutputgapand GDPinflation (RMSEsinfirstrow,RMSEratiosinallothers) HPSgapforecasts GDPinflationforecasts forecastmethod h=0Q h=1Q h=1Y h=0Q h=1Q h=1Y univariate .714 1.036 2.075 .762 .841 .717 VAR(4) 1.121 1.131 1.155 .981 .990 1.073 VAR(4),interceptcorrection 1.091 1.052 1.114 1.098 1.163 1.527 VAR(AIC) 1.118 1.130 1.158 .976 .985 1.067 DVAR(4) 1.119 1.086 1.084 1.029 1.056 1.226 DVAR(AIC) 1.130 1.088 1.089 1.025 1.062 1.246 VAR(AIC,byeq.&var.) 1.075 1.117 1.137 .980 .995 1.085 DVAR(AIC,byeq.&var.) 1.077 1.075 1.098 1.003 1.053 1.287 BVAR(4) 1.057 1.116 1.147 1.013 1.040 1.166 BDVAR(4) 1.007 .985 .983 1.031 1.075 1.274 VAR(4),inflationdetrending 1.088 1.097 1.073 .976 .959 1.005 VAR(AIC),interceptbreaks 1.066 1.041 .959 1.047 1.073 1.217 VAR(4),rolling 1.040 1.147 1.234 1.116 1.179 1.335 DVAR(4),rolling 1.010 1.024 1.060 1.081 1.046 1.161 VAR(AIC,byeq.&var.),rolling 1.037 1.071 1.160 1.116 1.171 1.322 BVAR(4),rolling 1.043 1.128 1.246 1.052 1.124 1.324 BDVAR(4),rolling .991 1.000 1.063 1.043 1.091 1.259 TVPBVAR(4) .994 1.004 .997 1.032 1.078 1.276 InterceptTVPBVAR(4) .989 .991 .971 1.030 1.071 1.266 DLS,VAR(4) 1.062 1.047 1.060 1.270 1.309 1.483 DLS,DVAR(4) 1.067 1.022 1.050 1.282 1.245 1.402 averageofallforecasts 1.028 1.029 1.049 1.004 1.022 1.123 avg.ofVAR(4),rollingVAR(4) 1.052 1.091 1.134 1.025 1.059 1.178 avg.ofunivariate,VAR(4) 1.047 1.038 1.026 .956 .961 .982 avg.ofunivariate,DVAR(4) 1.051 1.032 1.031 .971 .979 1.028 avg.ofuniv.,IDTRVAR(4) 1.033 1.028 1.003 .948 .937 .926 avg.ofuniv.,VAR(4), 1.044 1.030 1.019 .978 .995 1.082 DVAR(4),TVPBVAR(4) Notes: 1. The variables in each multivariate model are the HPS output gap, GDP inflation, and the T-bill rate. 2. SeethenotestoTable2. 55

Table11: Real-time1996-2005RMSEresultsfortheHPSoutputgapand corePCEinflation (RMSEsinfirstrow,RMSEratiosinallothers) HPSgapforecasts corePCEforecasts forecastmethod h=0Q h=1Q h=1Y h=0Q h=1Q h=1Y univariate .714 1.036 2.075 .646 .602 .460 VAR(4) 1.053 1.078 1.165 1.162 1.216 1.384 VAR(4),interceptcorrection 1.020 1.006 1.088 1.267 1.472 2.071 VAR(AIC) 1.071 1.110 1.190 1.129 1.200 1.429 DVAR(4) 1.033 1.021 1.041 1.161 1.289 1.409 DVAR(AIC) 1.050 1.025 1.027 1.153 1.242 1.362 VAR(AIC,byeq.&var.) 1.083 1.128 1.206 1.198 1.315 1.703 DVAR(AIC,byeq.&var.) 1.071 1.055 1.078 1.147 1.230 1.232 BVAR(4) 1.071 1.145 1.219 1.172 1.275 1.553 BDVAR(4) .985 .974 .987 1.153 1.248 1.358 VAR(4),inflationdetrending 1.061 1.084 1.102 1.117 1.161 1.093 VAR(AIC),interceptbreaks 1.055 1.112 1.161 1.252 1.423 1.905 VAR(4),rolling .999 1.104 1.312 1.006 1.155 1.622 DVAR(4),rolling .938 .954 1.023 .925 1.081 1.087 VAR(AIC,byeq.&var.),rolling 1.075 1.138 1.312 1.018 1.165 1.529 BVAR(4),rolling 1.054 1.138 1.307 1.196 1.355 1.894 BDVAR(4),rolling .975 .996 1.061 1.110 1.175 1.210 TVPBVAR(4) .985 .997 1.009 1.151 1.260 1.515 InterceptTVPBVAR(4) .981 .986 .989 1.160 1.265 1.499 DLS,VAR(4) .997 1.005 1.046 1.165 1.504 1.689 DLS,DVAR(4) .994 .987 1.040 1.115 1.572 1.505 averageofallforecasts 1.007 1.019 1.070 1.093 1.174 1.290 avg.ofVAR(4),rollingVAR(4) .999 1.048 1.187 1.044 1.141 1.452 avg.ofunivariate,VAR(4) 1.010 1.011 1.028 1.057 1.074 1.114 avg.ofunivariate,DVAR(4) 1.008 .999 1.012 1.048 1.086 1.032 avg.ofuniv.,IDTRVAR(4) 1.014 1.015 1.015 1.032 1.042 .937 avg.ofuniv.,VAR(4), 1.003 .998 1.011 1.089 1.139 1.182 DVAR(4),TVPBVAR(4) Notes: 1. The variables in each multivariate model are the HPS output gap, core PCE inflation, and the T-billrate. 2. SeethenotestoTable2. 56

Table12: Averagerankingsofallmethodsin1970-84and1985-2005forecasts, acrossallmodelsanddata ally,p ally,p ally,p ally,p all all using using 70-84 85-05 y,p Tbill FFR avg.ofuniv.,IDTRVAR(2) 12.9 16.7 15.5 18.0 21.1 12.4 avg.ofuniv.,IDTRVAR(4) 13.2 13.4 12.7 14.1 15.1 11.6 avg. ofuniv.,VAR(2),DVAR(2),TVPBVAR(2) 15.7 19.0 18.8 19.1 22.2 15.7 avg. ofuniv.,VAR(4),DVAR(4),TVPBVAR(4) 17.6 16.2 16.6 15.7 17.9 14.4 avg. ofunivariate,VAR(2) 18.8 23.7 22.3 25.1 31.3 16.0 averageofallforecasts 19.7 18.8 19.1 18.5 11.5 26.1 avg. ofunivariate,VAR(4) 20.3 20.6 20.9 20.3 26.2 15.0 avg. ofunivariate,DVAR(4) 21.3 19.9 19.8 20.1 21.3 18.6 avg. ofunivariate,DVAR(2) 22.9 24.1 23.9 24.2 27.1 21.0 InterceptTVPBVAR(4) 25.1 28.1 27.4 28.9 38.4 17.9 VAR(2),inflationdetrending 25.2 29.0 27.0 31.0 21.8 36.1 InterceptTVPBVAR(4),λ =.5,λ=.0025 26.4 27.4 27.4 27.3 27.1 27.7 4 BDVAR(4) 27.0 28.7 27.2 30.1 30.8 26.5 TVPBVAR(4),λ =.5,λ=.0025 28.2 23.8 23.5 24.0 30.4 17.1 4 TVPBVAR(4),λ =1000,λ=.005 29.1 23.4 22.9 23.9 30.0 16.8 4 TVPBVAR(4),λ =1000,λ=.0001 29.4 31.2 31.0 31.3 36.2 26.1 4 TVPBVAR(4) 29.7 29.3 28.7 29.9 42.8 15.8 BVAR(4) 30.1 32.9 33.0 32.8 37.3 28.4 InterceptTVPBVAR(2),λ =.5,λ=.0025 30.6 36.4 35.3 37.5 35.1 37.7 4 InterceptTVPBVAR(2) 31.1 38.5 36.9 40.1 50.3 26.7 TVPBVAR(2),λ =.5,λ=.0025 31.8 31.6 31.0 32.2 36.4 26.9 4 BDVAR(2) 32.0 34.4 33.2 35.6 36.9 31.9 TVPBVAR(2),λ =1000,λ=.005 32.2 30.2 29.2 31.1 36.3 24.0 4 VAR(4),inflationdetrending 32.6 31.6 31.2 32.0 25.8 37.4 DVAR(2) 32.8 31.3 31.1 31.5 25.0 37.6 avg. ofVAR(2),rollingVAR(2) 33.3 40.0 38.9 41.0 38.3 41.6 TVPBVAR(2),λ =1000,λ=.0001 33.3 40.0 38.9 41.0 45.7 34.2 4 univariate 33.6 36.5 34.0 38.9 52.2 20.8 BVAR(2) 34.2 41.6 40.5 42.8 46.9 36.3 TVPBVAR(2) 34.7 38.6 37.1 40.2 53.5 23.7 DVAR(AIC) 34.8 33.1 32.5 33.7 29.1 37.1 VAR(AIC),inflationdetrending 34.9 32.8 32.9 32.7 25.7 39.9 VAR(BIC),inflationdetrending 35.0 40.0 38.7 41.2 36.3 43.6 BDVAR(4),rolling 35.2 38.4 36.9 39.9 40.3 36.5 VAR(2) 35.5 41.0 37.7 44.3 48.7 33.3 DVAR(BIC,byeq.&var.) 37.6 33.5 33.6 33.4 36.6 30.4 VAR(AIC,byeq.&var.) 38.8 37.2 38.6 35.8 44.8 29.6 DVAR(BIC) 38.9 37.2 37.6 36.8 34.7 39.7 DVAR(AIC,byeq.&var.) 39.4 34.4 35.3 33.5 33.1 35.7 DVAR(4) 39.4 35.2 35.9 34.6 31.0 39.5 BDVAR(2),rolling 39.6 44.3 42.9 45.7 45.5 43.1 VAR(4) 40.8 41.0 41.9 40.1 45.6 36.4 DVAR(2),outputdiff. 41.0 42.8 44.2 41.3 39.9 45.6 DVAR(2),rolling 41.2 39.2 39.8 38.7 32.9 45.6 57

Table12,continued: Averagerankingsacrossallresults ally,p ally,p ally,p ally,p all all using using 70-84 85-05 y,p Tbill FFR VAR(AIC) 41.3 40.0 40.8 39.3 45.1 35.0 VAR(2),fullESdetrending 41.3 40.8 41.4 40.2 17.7 63.9 VAR(BIC,byeq.&var.) 43.8 44.8 43.7 45.9 55.7 33.8 DVAR(AIC),outputdiff. 44.2 45.1 45.6 44.6 43.4 46.9 DVAR(4),outputdiff. 45.9 45.1 45.5 44.6 43.5 46.6 VAR(BIC),fullESdetrending 46.1 46.4 46.9 45.9 29.0 63.8 VAR(BIC) 46.2 52.2 50.1 54.3 61.5 42.9 DVAR(AIC),rolling 46.5 40.2 39.1 41.3 37.2 43.2 VAR(AIC),fullESdetrending 47.6 42.3 45.2 39.4 24.6 60.1 VAR(4),fullESdetrending 48.8 45.6 49.6 41.7 27.9 63.4 BVAR(4),rolling 49.3 51.9 52.8 51.0 41.5 62.4 BVAR(2),rolling 49.6 54.8 54.9 54.7 43.5 66.1 DVAR(BIC),rolling 49.6 49.1 50.0 48.2 45.0 53.2 DVAR(BIC),outputdiff. 49.9 52.2 54.6 49.9 55.2 49.3 DVAR(AIC,byeq.&var.),rolling 50.3 41.1 44.2 38.1 39.6 42.7 DVAR(2),outputdiff.,rolling 51.3 54.4 56.9 51.9 51.5 57.3 DVAR(BIC,byeq.&var.),rolling 51.8 46.9 48.7 45.0 48.3 45.5 VAR(AIC),BICinterceptbreaks 52.7 47.5 47.7 47.3 30.7 64.4 avg. ofVAR(4),rollingVAR(4) 53.6 52.0 52.7 51.3 53.8 50.1 VAR(AIC),AICinterceptbreaks 55.4 49.0 47.7 50.3 32.9 65.1 DVAR(4),rolling 55.9 47.9 47.3 48.6 44.7 51.2 DLS,VAR(2) 56.1 56.8 54.8 58.7 65.2 48.3 VAR(2),interceptcorrection 56.9 60.0 59.5 60.6 61.2 58.8 DVAR(BIC),outputdiff.,rolling 57.8 64.9 66.9 63.0 63.8 66.1 DVAR(AIC),outputdiff.,rolling 59.0 57.3 57.7 56.8 55.2 59.3 DLS,DVAR(2) 59.4 55.7 56.7 54.8 57.2 54.3 VAR(2),rolling 62.5 65.3 65.3 65.3 56.4 74.2 DVAR(4),outputdiff.,rolling 63.5 60.5 59.4 61.6 56.3 64.7 DLS,DVAR(AIC) 63.9 59.3 59.3 59.3 63.7 54.9 VAR(AIC),interceptcorrection 64.0 63.6 62.7 64.5 61.1 66.2 VAR(4),interceptcorrection 64.9 64.4 64.3 64.5 63.8 65.1 DLS,VAR(AIC) 65.5 63.4 64.0 62.7 73.2 53.6 VAR(BIC),rolling 65.7 69.9 71.8 68.0 66.0 73.8 DLS,DVAR(4) 68.7 63.6 64.5 62.6 67.9 59.3 VAR(BIC,byeq.&var.),rolling 68.8 69.3 69.7 68.9 65.5 73.1 VAR(AIC,byeq.&var.),rolling 69.2 69.2 71.4 67.1 64.7 73.8 VAR(AIC),rolling 69.7 69.7 70.2 69.2 66.5 72.8 DLS,VAR(4) 69.8 66.7 68.5 64.9 75.8 57.5 VAR(2),partialint.corr. 72.1 67.4 67.1 67.7 72.2 62.6 VAR(4),rolling 72.4 71.8 71.8 71.9 68.1 75.5 VAR(AIC),partialint.corr. 76.4 72.9 72.8 73.0 74.8 71.0 VAR(4),partialint.corr. 76.5 73.1 73.9 72.2 74.9 71.3 #ofrankingobservations 216 144 72 72 72 72 58

Notes: 1. The table reports average rankings of the full set of forecast methods or models listed in Table 1. The average rankings in the first column of figures are calculated, for each forecast method, across a total of 216 (= 3 2 2 3 2 3) forecasts of output (3: GDP growth, HPS gap, HP × × × × × gap), inflation (2: GDP inflation, CPI inflation), and interest rates (2: T-bill rate, federal funds rate) at horizons (3) of h=0Q, h=1Q, and h=1Y and sample periods (2) of 1970-84 and 1985- 05. The average rankings in remaining columns are based on forecasts with models that include particular variables or forecasts of a particular variable, etc. For example, the average rankings in thesecondcolumnarebasedon144forecastsofjustoutputandinflation,withforecastsofinterest ratesomittedfromtheaveragerankingcalculation. 2. SeethenotestoTable2. 59

Table13: Averagerankingsin1970-84and1985-2005forecasts, conditionedonoutputandinflationmeasures using using using using using ∆ln HPS HP GDP CPI GDP gap gap π π avg.ofuniv.,IDTRVAR(2) 21.1 12.7 16.3 16.7 16.7 avg.ofuniv.,IDTRVAR(4) 16.8 8.9 14.4 13.0 13.8 avg. ofuniv.,VAR(2),DVAR(2),TVPBVAR(2) 16.9 19.0 20.9 18.8 19.2 avg. ofuniv.,VAR(4),DVAR(4),TVPBVAR(4) 13.3 15.6 19.6 14.1 18.2 avg. ofunivariate,VAR(2) 23.6 24.6 22.9 24.1 23.2 averageofallforecasts 19.9 17.7 18.8 16.0 21.6 avg. ofunivariate,VAR(4) 18.5 20.6 22.7 19.7 21.5 avg. ofunivariate,DVAR(4) 17.9 18.4 23.6 18.0 21.9 avg. ofunivariate,DVAR(2) 23.4 22.6 26.2 23.7 24.5 InterceptTVPBVAR(4) 24.7 30.3 29.4 27.1 29.1 VAR(2),inflationdetrending 37.4 20.2 29.3 33.5 24.5 InterceptTVPBVAR(4),λ =.5,λ=.0025 30.1 29.6 22.5 27.1 27.6 4 BDVAR(4) 28.2 32.3 25.6 29.8 27.6 TVPBVAR(4),λ =.5,λ=.0025 28.2 22.8 20.3 22.0 25.5 4 TVPBVAR(4),λ =1000,λ=.005 28.3 20.7 21.3 21.2 25.7 4 TVPBVAR(4),λ =1000,λ=.0001 30.5 39.1 23.9 31.5 30.8 4 TVPBVAR(4) 25.4 34.1 28.5 27.8 30.8 BVAR(4) 31.5 41.7 25.4 33.3 32.4 InterceptTVPBVAR(2),λ =.5,λ=.0025 35.2 40.6 33.4 36.7 36.1 4 InterceptTVPBVAR(2) 32.9 41.7 40.9 38.3 38.7 TVPBVAR(2),λ =.5,λ=.0025 31.5 33.7 29.7 29.8 33.4 4 BDVAR(2) 29.2 40.8 33.2 34.7 34.1 TVPBVAR(2),λ =1000,λ=.005 30.6 30.1 29.8 27.0 33.4 4 VAR(4),inflationdetrending 36.0 25.8 33.1 31.9 31.3 DVAR(2) 28.8 31.9 33.1 33.0 29.6 avg. ofVAR(2),rollingVAR(2) 36.2 46.6 37.1 43.8 36.1 TVPBVAR(2),λ =1000,λ=.0001 36.0 49.5 34.4 41.5 38.4 4 univariate 35.8 37.0 36.6 34.3 38.6 BVAR(2) 36.9 52.0 36.0 43.2 40.0 TVPBVAR(2) 32.2 43.9 39.8 37.9 39.4 DVAR(AIC) 31.9 32.4 34.9 30.6 35.5 VAR(AIC),inflationdetrending 40.5 25.7 32.3 36.6 29.1 VAR(BIC),inflationdetrending 47.7 33.5 38.7 43.9 36.0 BDVAR(4),rolling 38.2 42.9 34.1 38.6 38.2 VAR(2) 37.2 47.2 38.5 47.3 34.7 DVAR(BIC,byeq.&var.) 40.8 37.0 22.8 34.8 32.2 VAR(AIC,byeq.&var.) 31.6 44.2 35.9 39.0 35.5 DVAR(BIC) 41.0 34.9 35.7 38.6 35.8 DVAR(AIC,byeq.&var.) 35.9 35.1 32.2 37.4 31.4 DVAR(4) 34.1 35.3 36.3 33.4 37.1 BDVAR(2),rolling 40.3 51.7 41.0 43.7 45.0 VAR(4) 37.5 45.4 40.0 40.0 41.9 DVAR(2),outputdiff. 43.6 37.2 47.4 44.5 41.0 DVAR(2),rolling 39.2 40.4 38.1 40.0 38.5 60

Table13,continued: averagerankings, conditionedonoutputandinflationmeasures using using using using using ∆ln HPS HP GDP CPI GDP gap gap π π VAR(AIC) 40.1 44.1 36.0 45.4 34.7 VAR(2),fullESdetrending 43.2 32.6 46.6 41.6 40.0 VAR(BIC,byeq.&var.) 45.5 53.2 35.7 47.1 42.5 DVAR(AIC),outputdiff. 47.9 38.4 49.1 38.5 51.7 DVAR(4),outputdiff. 56.4 35.9 42.9 42.5 47.7 VAR(BIC),fullESdetrending 47.7 42.3 49.2 44.5 48.2 VAR(BIC) 47.1 63.8 45.7 57.3 47.0 DVAR(AIC),rolling 41.2 38.4 41.2 33.9 46.6 VAR(AIC),fullESdetrending 42.0 35.8 49.2 43.3 41.4 VAR(4),fullESdetrending 41.4 42.0 53.5 44.5 46.8 BVAR(4),rolling 48.8 57.0 49.9 50.8 53.0 BVAR(2),rolling 47.5 60.1 56.7 53.5 56.1 DVAR(BIC),rolling 50.4 49.8 47.0 49.7 48.4 DVAR(BIC),outputdiff. 50.3 47.7 58.7 55.1 49.4 DVAR(AIC,byeq.&var.),rolling 42.9 39.8 40.7 42.0 40.2 DVAR(2),outputdiff.,rolling 55.1 47.2 60.9 54.3 54.4 DVAR(BIC,byeq.&var.),rolling 50.0 49.2 41.5 49.2 44.5 VAR(AIC),BICinterceptbreaks 51.8 44.4 46.4 53.1 41.9 avg. ofVAR(4),rollingVAR(4) 50.4 57.2 48.3 49.6 54.3 VAR(AIC),AICinterceptbreaks 56.2 45.7 45.1 51.5 46.5 DVAR(4),rolling 45.3 46.9 51.7 45.9 50.0 DLS,VAR(2) 56.5 57.5 56.3 56.5 57.0 VAR(2),interceptcorrection 59.1 51.0 70.0 59.0 61.0 DVAR(BIC),outputdiff.,rolling 64.9 62.2 67.8 65.5 64.4 DVAR(AIC),outputdiff.,rolling 61.5 47.8 62.5 48.8 65.7 DLS,DVAR(2) 55.4 57.7 54.1 54.4 57.0 VAR(2),rolling 61.6 67.7 66.6 65.3 65.3 DVAR(4),outputdiff.,rolling 67.6 49.7 64.2 58.5 62.5 DLS,DVAR(AIC) 62.3 60.3 55.3 56.2 62.4 VAR(AIC),interceptcorrection 63.6 59.0 68.3 64.7 62.5 VAR(4),interceptcorrection 64.3 60.0 68.9 62.9 65.9 DLS,VAR(AIC) 66.8 62.5 60.9 62.6 64.2 VAR(BIC),rolling 64.4 72.8 72.6 72.0 67.8 DLS,DVAR(4) 64.8 62.9 63.0 61.7 65.4 VAR(BIC,byeq.&var.),rolling 64.3 74.8 68.8 69.8 68.8 VAR(AIC,byeq.&var.),rolling 66.6 72.6 68.5 71.0 67.5 VAR(AIC),rolling 70.5 72.9 65.5 69.0 70.3 DLS,VAR(4) 68.8 64.5 66.8 65.5 67.8 VAR(2),partialint.corr. 66.8 57.8 77.5 68.0 66.8 VAR(4),rolling 70.7 75.0 69.7 70.5 73.1 VAR(AIC),partialint.corr. 70.7 67.4 80.5 74.2 71.6 VAR(4),partialint.corr. 72.1 66.1 81.2 73.7 72.5 #ofrankingobservations 48 48 48 72 72 61

Notes: 1. The results in this table are based on just forecasts of output and inflation (excluding forecast resultsforinterestrates). 2. SeethenotestoTables2and12. 62

Table14: AccuracyofselectVARforecastscomparedtoSPFforecasts (RMSEsinallcases) GDPgrowthforecasts 1970-84 1985-2005 h=0Q h=1Q h=1Y h=0Q h=1Q h=1Y SPF 2.571 3.699 2.891 1.384 1.635 1.274 bestforecastfromTable2 3.735 3.878 2.775 1.609 1.668 1.182 univariateforecast 4.183 4.761 3.652 1.609 1.668 1.293 TVPBVAR(4) 3.876 4.267 3.487 1.650 1.708 1.218 avg.ofallTable2forecasts 3.735 3.878 2.978 1.734 1.824 1.312 avg.ofuniv.,DVAR(4) 3.953 4.199 2.906 1.747 1.828 1.328 avg.ofuniv.,IDTRVAR(4) 3.893 4.145 3.101 1.705 1.770 1.232 GDPinflationforecasts 1970-84 1985-2005 h=0Q h=1Q h=1Y h=0Q h=1Q h=1Y SPF 1.364 1.917 2.192 .831 .922 .804 bestforecastfromTable2 1.727 2.036 2.141 .926 .961 .716 univariateforecast 1.825 2.153 2.389 .951 1.016 .760 TVPBVAR(4) 1.779 2.267 2.646 .944 .993 .764 avg.ofallTable2forecasts 1.727 2.129 2.318 .974 1.032 .803 avg.ofuniv.,DVAR(4) 1.764 2.051 2.224 .926 .970 .735 avg.ofuniv.,IDTRVAR(4) 1.772 2.108 2.328 .937 .985 .744 CPIinflationforecasts 1970-84 1985-2005 h=0Q h=1Q h=1Y h=0Q h=1Q h=1Y SPF .823 1.278 .969 bestforecastfromTable4 1.744 2.427 2.441 1.272 1.431 1.167 univariateforecast 2.117 2.733 2.970 1.347 1.475 1.247 TVPBVAR(4) 1.935 2.772 3.238 1.319 1.431 1.167 avg.ofallTable4forecasts 1.758 2.544 2.856 1.333 1.511 1.370 avg.ofuniv.,DVAR(4) 1.825 2.456 2.656 1.272 1.446 1.262 avg.ofuniv.,IDTRVAR(4) 1.815 2.447 2.564 1.296 1.465 1.273 T-billrateforecasts 1970-84 1985-2005 h=0Q h=1Q h=1Y h=0Q h=1Q h=1Y SPF .310 1.436 2.589 .104 .460 1.543 bestforecastfromTable2 1.173 1.879 2.669 .371 .742 1.418 univariateforecast 1.305 2.098 2.821 .379 .777 1.633 TVPBVAR(4) 1.239 1.959 2.981 .407 .781 1.529 avg.ofallTable2forecasts 1.182 1.920 2.834 .386 .764 1.555 avg.ofuniv.,DVAR(4) 1.215 1.908 2.725 .389 .805 1.680 avg.ofuniv.,IDTRVAR(4) 1.206 1.910 2.719 .371 .742 1.473 Notes: 1. The forecast errors are calculated using the first–available (real–time) estimates of output and inflationastheactualdataonoutputandinflation. 2. RMSEs for SPF forecasts of CPI inflation are not reported for the 1970-84 sample because the SPFdatadon’tbeginuntil1981. 3. SeethenotestoTable2. 63

Table15: AccuracyofSPFforecastscomparedtoGreenbookforecasts, inrealtimedata (RMSEsinallcases) GDPgrowthforecasts 1970-84 1985-2000 h=0Q h=1Q h=1Y h=0Q h=1Q h=1Y SPF 2.571 3.699 2.891 1.334 1.543 1.352 Greenbook 2.434 3.783 2.832 1.309 1.650 1.485 GDPinflationforecasts 1970-84 1985-2000 h=0Q h=1Q h=1Y h=0Q h=1Q h=1Y SPF 1.364 1.917 2.192 .849 .932 .834 Greenbook 1.330 1.626 1.653 .691 .852 .670 CPIinflationforecasts 1970-84 1985-2000 h=0Q h=1Q h=1Y h=0Q h=1Q h=1Y SPF .700 1.206 .984 Greenbook .603 1.160 .949 Notes: 1. The forecast errors are calculated using the first–available (real–time) estimates of output and inflationastheactualdataonoutputandinflation. 2. RMSEs for forecasts of CPI inflation are not reported for the 1970-84 sample because the SPF andGreenbookdatadon’tbeginuntilcirca1980. 3. SeethenotestoTable2. 64