The Conditional Predictive Ability of Economic Variables

Transcription

1 The Conditional Predictive Ability of Economic Variables Eleonora Granziera y and Tatevik Sekhposyan zx December 13, 2014 Abstract The relative performance of forecasting models is known to be rather unstable over time. However, it is not very well understood why the forecasting performance of economic models changes. We propose to address this question by evaluating the predictive ability of a wide range of economic variables for key U.S. macroeconomic aggregates: output growth and in ation. We take a conditional view on this issue, attempting to identify situations where particular kind of models perform better than simple benchmarks. We therefore test whether the relative forecasting performance of models depend on economic activity, nancial conditions, uncertainty or past relative performance. Furthermore, we investigate whether using the conditioning information as a model selection criteria for model averaging could improve the accuracy of the predictions. Keywords: Conditional Predictive Ability, Macroeconomic Forecasting J.E.L. Codes: C22, C52, C53 preliminary and incomplete, do not cite y Bank of Canada, 234 Wellington Street, Ottawa, ON, K1A 0G9, Canada; egranziera@bankofcanada.ca z Texas A&M University, 3059 Allen Building, 4228 TAMU College Station, TX USA; tsekhposyan@tamu.edu. x The views expressed here are those of the authors only. No responsibility for them should be attributed to the Bank of Canada. 1

2 1 Introduction Instability in the accuracy of many forecasting models is well documented in the literature (see Stock and Watson, 2003 and Rossi, 2013 for overviews). Changes in relative performance have been found in the early to mid 80s (Rossi and Sekhposyan 2010) and more recently during the Great Recession (Del Negro et al 2014). Despite this evidence, it is still little understood why the forecasting performances change or why they change at di erent rates across the models. In this paper we try to answer this question by identifying patterns in the data that can help us to predict the relative performance of the models. To this end we employ the test of conditional predictive ability proposed by Giacomini and White (2006). Tests of unconditional predictive ability ask whether the forecasting models performed equally well on average over the out of sample period. Examples of such tests are Diebold and Mariano (1995), West (1996), Clark and McCracken (2001) and Clark and West (2007). The null then states that the loss di erences have zero mean. However researchers might be interested in knowing whether relative performance depends on some observables for example whether a model is particularly good in recessions vs expansions. Then, it might be more appropriate to use a test for conditional predictive ability which asks whether there is any information available at the time the forecasts are made, above and beyond past average performance, that can explain the relative performance of the models. Accordingly, the null states that loss di erences have mean zero conditional on some information set, for example, conditional on the state of the economy or conditional on past relative performance. This paper considers the conditional predictive ability of a wide range of economic variables that represent asset prices, measures of real economic activity, wages and prices, as well money supply. The set of models we consider are bivariate autoregressive distributed lag (ADL) models, a small-scale vector autoregression (VAR), as well as various model averages. The ADL models are analyzed to insure parsimony and a tractable framework where the marginal impact of each economic variable can be traced separately. The model averages are useful since they are known to help produce relatively more accurate forecasts. In addition, we investigate the forecasting performance of a simple multi-variate model. We do not consider an elaborate set of multivariate models, as the literature, represented by Clark and McCracken (2008, 2010) and Stock and Watson (2003, 2007), among others, has shown that multivariate models do not improve the point forecasts in a statistically signi cant way over univariate ones at least beyond mid-1980s. 2

3 We analyze the unconditional as well as conditional predictive ability of the models relative to a simple benchmark. The rst conditioning variable that we consider is the state of the business cycle: we assess the relative performance of the models in recessions versus expansions. Motivated by the recent nancial crises we also use a nancial stress index as a conditioning variable. We also consider a measure of macroeconomic uncertainty. Approaching from a more time series perspective, we include the past performance of the models in the conditioning set. In addition, we take a step-forward and consider the usefulness of the conditional tests. In particular, we experiment whether using the conditioning information as a criteria for model averaging in fact improves the forecasting performance of the models. To this end we consider two simple rules: a model selection rule introduced in Gacomini and White (2006) and a novel model averaging rule. In line with previous literature we nd that rejections when using the unconditional test are rare, suggesting that the benchmark and the alternative models are equally good on avegare over the sample. When applying the conditional test, our general nding is that the relative performance of the models can be predicted by a measure of nancial stress at short horizons, while at longer horizons the past relative perfomance is a good predictor of future performance. Moreover using the conditioning information in a simple decision rule performs at least as well as the single models or a simple average of them. The literature has approached the understanding of the reversals of the predictive abilities of the models largely from two perspective. For instance, Rossi and Sekhposyan (2010), Clark and Doh, (2011), Manzan and Zerom (2013) document the reversals in the relative predictive abilities of the models. The papers further seek to identify the economic events that might have occurred at the times of reversals. Some papers have approached the problem reversely by identifying certain recurring periods of economic signi cance and evaluating the forecasting performance of the models in those periods. For instance, Dotsey, Fujita and Stark (2011) consider the relative forecasting performance of models of in ation throughout various states of the economy. We add to this literature by identifying for a large set of models the variables that can best explain the relative performance of the models. The rest of the paper is organized as follows: section 2 presents the econometric framework, section 3 describes the data and conditional variables, section 4 describes the models, section 5 reports the results and section 6 concludes. 3

4 2 Econometric Framework Suppose that fy s+ ; x s g t s=1 are time series variables at each forecast origin t = R + 1; :::; R + T and that one is interested in forecasting a scalar y t+, 1; using two alternative models. 1 Denote by f t;r ^0;t = f y t; x t ; x t 1 ; :::; ^ 0;t and g t;r ^1;t = g y t; x t ; x t 1 ; :::; ^ the period ahead forecasts obtained from the estimated models through either the xed or rolling window scheme with window size R. The sequence of out-of-sample forecasts are evaluated by a loss function L t. The testing framework proposed by Giacomini and White (2006) can accommodate point as well as density forecasts, and generic loss functions. In this paper we focus on evaluating point forecasts and we use the squared error loss as our measure of accuracy. The null hypothesis is expressed as: H 0 : E hl t+ y t+ ; f t ^0;t L t+ y t+ ; g t ^1;t j G t i = 0 (1) Note that because the null is formulated in terms of the parameter estimates rather than their population values, the null is a statement on the forecasting methods rather than the forecasting models: models, size of the estimation window and estimation procedure are all subjected to evaluation. Also, by conditioning on the information set G t the null implies that the forecasting methods are equally accurate given the information available in G t. An unconditional predictive ability test would instead test whether the models are equally accurate on average. The unconditional predictive ability test can be considered as a special case of (1) where the conditioning set G t = f;; g ; the trivial - eld, rather than G t = ff t g is the information set at time t. Giacomini and White (2006) show that testing the null hypothesis (1) when G t = ff t g is equivalent to test for cov(h t L R;t+ ; h t j L R;t+ j ) = 0 for all j and for all test functions h t : They propose the following test statistic:! T 0 X TR;P; h = P P 1 1 h t L R;t+ ^V P 1 T t=r t=r! X h t L R;t+ X 1 ^V = ^0 + 1 j=1 j ^j + ^0 j ( 1) This framework allows data to be non-stationary. However, the type of non-stationarity considered rules out unit roots, but allows for changes that could be induced by distributions changing over time. 4

5 P T and ^i = 1 P +1 t=r (h tl R;t+ ) (h t L R;t+ ) 0 : The test statistic is compared against critical values from a chi-squared distribution with q degrees of freedom; the test rejects when T h R;P; > 2 q;1 : In the case of unconditional predictive ability the null of interest is where L R;t+ test statistic is: = L t+ y t+ ; f t ^0;t H 0 : E [L R;t+ ] = 0 L t+ y t+ ; g t ^1;t is the loss di erence. The t R;P; = p P L R;P ^ P where L R;P = P P 1 T t=r L R;t+; P = T h R is the size of the out-of-sample, ^ P 2 is an pp i estimator of the asymptotic variance P 2 =var LR;P for example in this paper we use the Newey West (1987) estimator. The test is two-sided and for a given level of signi cance it rejects the null when jt R;P; j > z =2 where z =2 is the (1 normal distribution. =2) quantile of a standard The test statistic in this case is analogous to the one proposed by Diebold and Mariano (1995), however note that the null is di erent, as it is expressed in terms of the parameter estimates rather than their population values. 3 Data and Models The data we consider are at quarterly frequency, for majority of the variables from 1959:I to 2013:I, although some series have a later starting date due to data availability constraints. 2 The data are transformed to eliminate stochastic or deterministic trends, as well as seasonality. The variables are in percentage points, and the growth rates have been annualized. We use a xed rolling window estimation scheme with a window size of 60 observations. This rolling window estimation implies a time-varying nature for the parameters, though in the subsequent analysis we omit the time-subscript for simplicity. We consider four alternative conditioning variables: a measure of economic activity, one of nancial conditions, one of macroeconomic uncertainty and past performance. Chauvet and Potter (2012) and Stock and Watson (2010), among others, nd that relative forecasting performance di ers across phases of the business cycle for output growth and in ation 2 Data at monthly frequency are aggregated to quarterly frequency as follows: for nancial variables we consider the end of period, for the remaining variables we take the simple average. 5

6 respectively so we consider as conditioning variable a dummy that takes the value one in periods of economic recessions and zero during expansions, as indicated by NBER Business Cycle dating committee. Our measure of nancial conditions is the Chicago Fed National Financial Condition Index (NFCI) which takes positive (negative) values when conditions are tighter (looser) than average. We also consider an indicator of macroeconomic uncertainty which we construct based on the methodology described in Bloom (2009). Finally, the last conditioning variable is the lagged loss di erence in the squared forecast errors between the benchmark and the alternative model. A comprehensive description of the data is provided in Table 1. We consider a variety of parsimonious bivariate forecasting models in a form of autoregressive distributed lag models. In these models we consider the marginal predictive content of various economic fundamentals that could be largely classi ed as either asset prices, measures of real economic activity, wages and prices, and money. The data used is similar to that considered in Stock and Watson (2003), albeit updated to 2014:I. We also include one small multivariate model that allows for interaction between output, in ation and interest rate. In addition, we consider various model averages, as well as a bivariate model, where we augment the autoregression with a factor structure extracted from the whole dataset at hand. Model averages and factor models are particularly important given the fairly robust result in the forecasting literature that variety of parsimonious ways of pooling information could result in an improvement in forecast accuracy. In what follows, we describe the autoregressive distributed lag models, the small multivariate model as well as the variety of models that pool information either at the estimation stage (factor models) or when constructing the forecast (model averages). The set of models are very similar to the ones considered in Rossi and Sekhposyan (2012). 3.1 Autoregressive Distributed Lag (ADL) Models We consider forecasting quarterly output growth and in ation h-periods into the future using lags of one predictor at a time in addition to the lagged dependent variable. The forecasting model is: Y h t+h = k;0 + k;1 (L) X t;k + k;2 (L) Y t + u t+h ; t = 1; :::; T (2) where the dependent variable is either Yt+h h = (400=h) ln(rgdp t+h=rgdp t ) or Yt+h h = 400=h ln(p GDP t+h =P GDP t ) 400 ln(p GDP t =P GDP t 1 ); RGDP t+h and P GDP t+h are the 6

7 real GDP and GDP de ator, respectively. X t;k denotes the k-th explanatory variable, for k = 1; :::; K described in Table 1, and u t+h is the error term. The total number of individual economic variables considered in our application is K = Y t is either the period t output growth, that is Y t = 400 ln(rgdp t =RGDP t 1 ), or the period t change in in ation, that is Y t = 400 ln(p GDP t =P GDP t 1 ) 400 ln(p GDP t 1 =P GDP t 2 ). 4 We consider h = 1; 4 corresponding to one-quarter ahead and one-year ahead forecast horizons. 1 (L) = P p j=0 1jL j and 2 (L) = P q j=0 2jL j, L being the lag operator. We estimate the number of lags (p and q) recursively by BIC, rst selecting the lag length for the autoregressive component, then augmenting with an optimal lag length for the additional predictor. As a particular case, we consider the autoregressive model, where we use only the lagged dependent variable to forecast output growth and in ation. In what follows next, the autoregressive model is considered to be the benchmark model. 3.2 Model Averages We consider several averaging schemes for the ADL models discussed previously. 5 (i) Simple Average Model. The rst pooling strategy we consider is the simple model average which has been shown to perform well for point forecasts by Stock and Watson (2003, 2004). More speci cally, we estimate the ADL models in eq. (2) for all the regressors (oneat-a-time), i.e. for k = 1; :::; K, and consider linear combinations of their forecasts, where each forecast is weighted with an equal weight (1=K). The equal-weight pooled prediction is: ^Y c t+h = 1 K KX k=1 ^Y h t+h;k; (3) where the k subscripts indicates that the forecast corresponds to the k-th ADL regression. (ii) Bayesian Model Averaging (BMA). The second averaging method we consider is the Bayesian Model Average, which also pools from the set of simple models, yet assigns weights that are proportional to the models posterior probabilities. BMA puts more weight on more likely models as opposed to putting equal weight on all the models. We consider two variants of BMA models following Wright (2009). - BMA-OLS. The rst version is very similar to the simple model average (eq. 3) as it uses 3 The dataset for output growth includes historical data for in ation, but not output growth (and vice versa), as the lagged dependent variable is automatically included in eq. (2). 4 Note that, like Stock and Watson s (2003) approach, this relies on the assumption that in ation is I(2). 5 We omit the autoregressive model from the set of models being averaged. 7

8 the OLS estimates of the respective model s parameters. It is di erent however from the simple model average since it has time-varying weights P t (M k jd t ); which represent the posterior probability of model k denoted by M k, given the data D t = fy t ; X t ; Y t 1 ; X t 1 ; :::; Y t R ; X t R g. The average predition in this case is: ^Y BMA OLS t+h = KX k=1 P t (M k jd t ) ^Y h t+h;k (4) -BMA. The second version of BMA we consider is the full Bayesian version, where the estimated parameters are not the OLS counterparts (in the Bayesian framework this would be equivalent to obtaining coe cients under a at prior), but rather they are posterior estimates and, thus, are in uenced by the choice of the prior distribution. Let ~: indicate estimates associated with the fully Bayesian estimation, then the model average would be de ned as ^Y BMA t+h = KX k=1 P t (M k jd t ) ~ Y h t+h;k; (5) where M k denotes the k-th model and P t (M k jd t ) is the posterior probability of the k-th model given the data D t. Rossi and Sekhposyan (2012) provide the explicit expressions for the posterior model probability, as well as posterior mean of the regression parameters used to construct the point forecast. The priors considered for the Bayesian estimation are similar to that in Wright (2009), i.e. the prior is parameterized such that it puts equal weight on the prior and the data in the posterior density of the regression coe cients, while the prior on the precision parameter is fairly at. We should note that when considering the ADL models or the simple model average, p and q (the lag length) are selected recursively via BIC. We keep p and q xed at their recursively selected levels for both the BMA-OLS as well as the BMA speci cations. Furthermore, as noted in Wright (2009), the analytical results for the Bayesian calculations presented in this section work under the assumption of strict exogeneity of the regressors and do not allow for serial correlation in the error terms, which is important for multi-step forecasts. 3.3 Models with Principal Components We further consider a variant of the ADL model, eq. (2), where instead of considering each individual regressor one-by-one, we consider one model augmented with factors extracted 8

9 from the set of all regressors. More in detail, we estimate a static factor model: 6 Y h t+h = 0 + ^F t + 2 (L) Y t + u h t+h; t = 1; :::; T; (6) where ^F t is the (1 m) vector of estimated rst m principal components of the K variables we consider in this paper. We recursively select the number of factors m over each rolling window R such that the total number of factors explain at least 60% of the variation contained in the K macroeconomic data series. This results in 2-3 factors for output growth and in ation at di erent estimation periods. 3.4 Multivariate Models As multivariate model we consider a trivariate VAR(p) of the form: Z h t+h = C + A (L) Z t + u t+h ; t = 1; :::; T where u t+h is the error term, A(L) is a lag polynomial, Z t consists of in ation, real GDP growth and the Federal Funds rate and in ation and the real GDP growth are de ned as above. The lag lenght p is determined with the BIC criterion and the model is estimated via OLS. 4 Results We evaluate the unconditional and conditional predictive ability of the models and methods described in the previous section using the testing framework introduced in section 2. This framework allows for comparison of nested as well as non-nested models and of Bayesian as well as classical estimation procedures. Table 2 shows the results for the unconditional predictive ability test. Results for the conditional ability test are provided in Table 3 for GDP growth and in Table 4 for in ation. For the unconditional test the tables report the test statistic, while for the conditional tests the tables shows an indicator that computes the proportion of times the alternative model outperforms the benchmark. For each model m and foreach conditional variable, the indicator is constructed as follows: rst regress the loss di erences L m R;t+ on the conditioning variable h t over the out-of-sample and denote the 6 The static factor model could, in principle, be extended to a dynamic factor model, although, as Bai and Ng (2007) note, there is little gain to be expected from moving from static to dynamic factor models from a forecasting standpoint. 9

10 regression coe cients as ^ P m : The indicator is computed as I m = (1=P ) TX t=r+1 o 1 n^p h t > 0 where 1 fg takes the value 1 if ^ P h t > 0 and 0 otherwise. As ^ P h t E [L R;t+ j F t ] the indicator measures the relative success of the alternative model. In line with previous literature we nd that unconditional equal predictive ability tests reject only in a handful of cases. For both variables and forecasting horizons, the models which use a single indicator perform signi cantly worse than the benchmark as indicated by a negative t-stat. Pooling together the forecasts from other models seems a goeod strategy, as suggested by the relative success of the simple average and BMA procedures. In general the conditional test rejects more frequently than the unconditional test at both horizons, especially for output growth at one step ahead. At this forecasting horizon conditioning on current nancial conditions provides with further information regarding the future relative predictive ability of the models. rejections are more frequent when conditioning on past performance. At four steps ahead, for both variables We interpret rejection of the null of conditional equal predictive ability as indication of misspeci cation of the models, as the conditioning variable represents information available at the time the forecasts are made that is able to explain the relative performance of the models. Following a rejection then, a researcher aiming at improving the accuracy of the forecasts can adopt two strategies: (i) modify the original models to incorporate the information provided by the conditioning variable or (ii) adopt the simple model averaging rule described in the next subsession. The rst strategy requires to formulate a speci cation of a new forecasting model as well as to estimate the new model, while the second strategy is based on the forecasts of the benchmark and alternative models which are already available. 4.1 Model Averaging Giacomini and White (2006) propose the following two steps model selection rule: rst regress the loss di erences ^L m;t+ on the conditioning variables h t over the out-of-sample and denote the regression coe cient i as ^ P m; second predict y T + using the forecast of the m0 alternative model if E h^l t+ j G t ^ P h T > 0 or the forecast of the benchmark model if i m0 E h^l t+ j G t ^ P h T 0: The above strategy selects only one model. We suggest a novel rule for model averaging 10

11 where the weight assigned to the alternative model is: w m = (1=P ) TX t=r+1 n^m0 o 1 P h t > 0 and the weight of the benchmark model is w 0 = 1 w m : We evaluate the performance of the two simple rules against the performance of the single forecasting models or a simple average of them. We estimate the parameter over rolling samples of 60 observations starting from 1974Q1 and we evaluate the decision rules over the sample 1989Q1-2014Q1. We report the relative RMSFE in Tables 5 through 10: the two decision rules generally perform as good as the single models or their simple average, especially when using our novel model averaging rule, however gains are modest. Then, we further compare the decision rules to the single model forecasts using a measure of dispersion of the forecast errors, the interquartile range shown in Table 11 through 16. For both horizons, the relative interquartile range is well below one for output growth while is generally above one for in ation. This result points in favor of our decision rule for the prediction of gdp growth: the rules provide slightly higher forecast accuracy and smaller interquartile range. Intuitively this means that using our decision rule we make frequent, small forecast errors but we are insulated from large errors. ^ m0 P 5 Conclusions In this paper we conducted a systematic evaluation of the conditional predictive ability of various economic variables that represented asset prices, measures of real economic activity, wages and prices, as well money. The models considered ranges from bivariatautoregressive distributed lag (ADL) models,to a small-scale vector autoregression (VAR), as well as various model averages.we ask whether the relative performance of the models depends on the state of the economy, nancial conditions, macroeconomic uncertainty or whether it can be predictive based on past out-of sample relative accuracy. We nd that for both variables at longer horizons the past relative performance is a good indicator of future relative performance, while for the short run nancial conditions have predictive content for next period relative performance. Our results suggest using conditional test in an informative way. In particular, we document that using the conditioning information as a criteria for model selection in fact performs better than the single forecasting models. 11

12 6 Figures and Tables Table 1. Description of Data Series Label Trans Period Name Description Source Asset Prices rovnght level 59:M1-14:M5 FEDFUNDS Int. Rate: Fed Funds (E ective) F rtbill level 59:M1-14:M5 TB3MS Int. Rate: 3-Mn Tr. Bill, Sec Mkt Rate F rbnds level 59:M1-14:M5 GS1 Int. Rate: US Tr. Const Mat., 1-Yr F rbndm level 59:M1-14:M5 GS5 Int. Rate: US Tr. Const Mat., 5-Yr F rbndl level 59:M1-14:M5 GS10 Int. Rate: US Tr. Const Mat., 10-Yr F stockp ln 59:Q1-14:Q1 SP500 US Share Prices: S&P 500 F exrate ln 73:M1-14:M5 EXRUS Trade Weighted US Dollar Index F crespr level 59:M1-14:M5 AAA10YM Aaa Corporate Bond vs 10-Yr Treasury F Real Activity rgdp ln 59:Q1-14:Q1 GDPC96 Real GDP, sa F ip ln 59:M1-14:M5 INDPRO Industrial Production Index, sa F capu level 59:M1-14:M5 CAPUB04 Capacity Utilization Rate: Man., sa F emp ln 59:M1-14:M5 CE16OV Civilian Employment: thsnds, sa F unemp level 59:M1-14:M5 UNRATE Civilian Unemployment Rate, sa F hours ln 59M1-14M5 AWHMAN Manufacturing Average Weekly, sa F Wages and prices pgdp ln 59:Q1-13:Q1 GDPDEF GDP De ator, sa F cpi ln 59:M1-13:M6 CPIAUCSL CPI: Urban, All items, sa F ppi ln 59:M1-13:M6 PPIACO Producer Price Index, nsa F earn ln 59:M1-13:M6 AHEMAN Hourly Earnings: Man., nsa F Money mon0 ln 59:M1-13:M6 AMBSL Monetary Base, sa F mon1 ln 59:M1-13:M6 M1SL Money: M1, sa F mon2 ln 59:M1-13:M6 M2SL Money: M2, sa F Notes: Sources are abbreviated as follows: F - Federal Reserve Economic Data (FRED). W hen the names in the table are preceded with a pre x r, it indicates real variable adjusted either by the G D P de ator pgdp (sto ck variables) or pgdp in ation ( ow variables). Interest rate spread is calculated as the di erence b etween rbndl and rovnght. 12

13 Table 2. Unconditional Tests of Equal Predictive Ability, h=1,4 Output Growth In ation Model t-stat h=1 t-stat h=4 Model t-stat h=1 t-stat h=4 rovnght rovnght -1.91* rtbill rtbill -1.87* rbnds rbnds -1.98* rbndm rbndm rbndl rbndl rspread rspread stockp stockp exrate -1.70* -1.63* exrate crespr crespr rrovnght rrovnght rrtbill rrtbill rrbnds rrbnds rrbndm rrbndm rrbndl rrbndl rstockp rstockp rexrate * rexrate ip rgdps capu ip emp capu unemp emp pgdp unemp hours hours cpi cpi ppi ppi earn earn mon mon mon mon mon mon rmon rmon rmon rmon rmon rmon simple ave 2.12* 2.72* simple ave * BMA-OLS BMA-OLS BMA BMA 1.70* 1.51 factor model factor model VAR VAR * 13

14 Table 3. Conditional Tests of Equal Predictive Ability, Output Growth h=1 h=4 Model I-rec I- n I-unc I-lag I-rec I- n I-unc I-lag rovnght 0.17* 0.32** rtbill 0.17* 0.31** * rbnds 0.17** 0.32** * ** rbndm 0.17** 0.32** * ** rbndl 0.17** 0.30** ** rspread stockp * exrate * ** crespr 0.17* 0.36** * rrovnght rrtbill 0.17* 0.23** * rrbnds 0.17** 0.23** ** rrbndm * * rrbndl rstockp * rexrate * ip capu ** emp unemp ** ** pgdp * ** hours cpi ppi * earn * mon * mon * mon rmon * rmon rmon simple ave 1.00* 0.75** 1.00* ** 0.69** 1.00** 0.94** BMA-OLS ** * BMA 0.17* 0.80** * * factor model 0.17** VAR 0.17* 0.27**

15 Table 4. Conditional Tests of Equal Predictive Ability, In ation h=1 h=4 Model I-rec I- n I-unc I-lag I-rec I- n I-unc I-lag rovnght * rtbill * rbnds * rbndm ** ** rbndl ** ** rspread stockp ** ** 0.12 exrate crespr rrovnght * rrtbill ** rrbnds ** rrbndm ** rrbndl ** rstockp ** * 0.34 rexrate rgdp * * ip ** ** capu emp 0.16* * 0.78 unemp ** 0.87 hours * ** 0.12 cpi 0.84* ppi earn * * mon * 0.11** mon ** mon rmon rmon * * rmon ** * 0.37** simple ave * * 0.97* BMA-OLS * BMA factor model * VAR *

16 Notes: The benchmark model is the autoregressive model. Positive (Negative) t-stat indicates that the alternative model is better (worse) than the benchmark. p-value is that of Giacomini and W hite (2006) asymptotic critical values. References [1] Clark T.E. and T. Doh (2011), A Bayesian Evaluation of Alternative Models of Trend In ation, Cleveland FED wp [2] Clark T.E. and M.W. McCracken (2001), Tests of Equal Forecast Accuracy and Encompassing for Nested Models, Journal of Econometrics 105, [3] Clark T.E. and K.D. West (2007), Approximately Normal Tests for Equal Predictive Accuracy in Nested Models, Journal of Econometrics 138, [4] Diebold, F. X. and R. S. Mariano (1995), Comparing Predictive Accuracy, Journal of Business and Economic Statistics 13(3), [5] Dotsey, M., Fujita, S. and T. Stark (2011), Do Phillips Curves Conditionally Help to Forecast In ation, Philadelphia FED wp [6] Giacomini, R. and H. White (2006), Tests of Conditional Predictive Ability, Econometrica 74(6), [7] Rossi, B. (2013), Advances in Forecasting Under Instabilities, in G. Elliott and A. Timmermann (eds.), Handbook of Economic Forecasting, Volume 2, Elsevier-North Holland Publications. [8] Rossi, B. and T. Sekhposyan (2010), Have Economic Models Forecasting Performance for US Output Growth and In ation Changed over Time, and When, International Journal of Forecasting, 26(4), [9] Rossi, B. and T. Sekhposyan (2014), Evaluating Predictive Densities of U.S. Output Growth and In ation in a Large Macroeconomic Data Set, International Journal of Forecasting, 30(3), [10] Stock, J.H. and M.W. Watson (2003), Forecasting Output and In ation: The Role of Asset Prices, Journal of Economic Literature 41(3),

17 [11] Stock, J.H. and M.W. Watson (2004), Combination Forecasts of Output Growth in a Seven Country Data Set, Journal of Forecasting 23(6), [12] West, K.D. (1996), Asymptotic Inference about Predictive Ability, Econometrica 64, [13] Wright, J.H. (2009), Forecasting US In ation by Bayesian Model Averaging, Journal of Forecasting 28(2),