Determinants of Bond Risk Premia

Transcription

1 Determinants of Bond Risk Premia Jing-zhi Huang and Zhan Shi Penn State University First draft: November 2009 This version: April 24, 2012 Abstract In this paper, we provide new and robust evidence on the power of macro variables for forecasting bond risk premia by using a recently developed model selection method the supervised adaptive group least absolute shrinkage and selection operator (lasso) approach. We identify a single macro factor that can not only subsume the macro factors documented in the existing literature but also can substantially raise their forecasting power for future bond excess returns. Specifically, we find that the new macro factor, a linear combination of four group factors (including employment, housing, and price indices), can explain the variation in excess returns on bonds with maturities ranging from 2 to 5 years up to 43%. The new factor is countercyclical and furthermore picks up unspanned predictability in bond excess returns. Namely, the new macro factor contains substantial information on expected excess returns (as well as expected future short rates) but has negligible impact on the cross section of bond yields. We thank Yakov Amihud, Charles Cao, Long Chen, Laura Field, Raymond Kan, Anh Le, Hong Liu, Marco Rossi, Joel Vanden, Wei Zhong, Hao Zhou, and seminar participants at Fordham, National University of Singapore, Penn State, Penn State Math, Singapore Management University, Temple, Wisconsin-Milwaukee, the 20th Annual Derivatives Securities and Risk Management Conference at FDIC, the 2010 CICF (Beijing), the 2010 FMA (New York), the 2010 SAIF and CKGSB Summer Research Conference, the 2011 AFA (Denver), and the 2012 Darla Moore Fixed Income Conference for helpful comments and suggestions. We especially want to thank our discussants: Peter Feldhütter (AFA), Matt Pritsker (FIC), Ilona Shiller (FMA), Hong Yan (SAIF-CKGSB), and Weina Zhang (CICF), for their valuable and detailed comments and suggestions that helped to improve the paper substantially. We acknowledge a grant from the Penn State Institute for Real Estate Studies for partial financial support. Both authors are at the Smeal College of Business, Penn State University, University Park, PA 16802, USA. addresses: jxh56@psu.edu (Huang) and zus116@psu.edu (Shi).

2 1 Introduction Recent empirical evidence has documented that some financial and macroeconomic variables can be used to predict the excess returns of the U.S. Treasury bonds. For instance, financial variables found to have such predictive power include forward rates or spreads (Fama and Bliss, 1987; Stambaugh, 1988; Cochrane and Piazzesi, 2005) and yield spreads (Campbell and Shiller, 1991). In particular, Cochrane and Piazzesi show that a tent-shaped linear combination of five forward rates can explain between 30 and 35 percent of the variation in one-year excess returns on bonds with 2-5 years to maturity. On the other hand, Ludvigson and Ng (2009) obtain a macro factor (extracted from a monthly panel of 131 macroeconomic variables using dynamic factor analysis) that has forecasting power for bond excess returns, above and beyond the power contained in the aforementioned financial variables. Specifically, Ludvigson and Ng find that their factor alone can explain percent of the one-year excess returns and can raise it to percent when augmented with the Cochrane-Piazzesi factor. Interestingly, Cochrane and Piazzesi, Ludvigson and Ng, and Duffee (2011) all document empirically the presence of a so-called hidden factor, namely, a factor that contains substantial information about expected excess returns but has negligible impact on the cross section of bond yields. These findings have generated important insights and implications for term structure modeling, and spawned a fast growing literature on the determinants of bond risk premia. Nonetheless, some recent studies have raised concerns about the robustness of the documented power of those financial and macro variables for predicting bond risk premia (see, e.g., Duffee, 2007). In this paper, we reexamine the potential power of macro variables for forecasting bond risk premia using a recently developed model selection method, namely, the supervised adaptive group least absolute shrinkage and selection operator (lasso) approach (referred to as the SAGLasso approach, hereafter). We first extract a new macro factor from a standard monthly panel of macro variables the same set of macro variables used in Ludvigson and Ng (2009) using the SAGLasso approach. We then examine the intuition of the new macro factor and, in particular, investigate whether the new factor has any forecasting power for bond risk premia above and beyond the predictive power contained in those financial and macro factors identified in the literature. Finally, as a robustness check, we address two issues raised recently by Duffee (2007, 2011) about the empirical literature on the prediction of bond excess returns. The new macro factor that we obtain is a linear combination of four non-overlapping group factors, each of which itself is a linear combination of a small number of closely related macro variables (a subset of the original 131 macro variables) and thus has a clear interpretation. More specifically, the four group factors represent employment, housing, price indices, and financial, respectively. As such, our new macro factor is easy to interpret. 1

3 We find that the new macro factor can predict excess returns on 2- to 5-year maturity bonds with (in sample) R 2 up to 43 percent. This is significantly higher than that found by either Cochrane and Piazzesi (2005, CP hereafter) or Ludvigson and Ng (2009, LN hereafter). Furthermore, our new macro factor is found to subsume the LN factor. However, like the LN factor, our factor does not subsume the CP factor and contains information about bond risk premia that is not contained in the CP factor. Augmenting our factor with the CP factor can increase the R 2 of the forecasting regression to 47 percent. Like the CP and LN factors, the new macro factor is found to be countercyclical. We also find that our new factor has strong out-of-sample forecasting power as well and moreover has significantly incremental predictive power beyond that in the LN factor. Overall, results from both in-sample and out-of-sample analysis indicate that our new macro factor contains information about future bond excess returns beyond what captured by the CP and LN factors. We also find that our employment group factor can subsume the output gap factor found by Cooper and Priestley (2009) that can predict excess returns on 2- to 5-year maturity bonds with R 2 equal to 2 percent. As such, our macro factor goes beyond output gap and inflation (two main macro variables considered in existing studies) and, in particular, includes a component of macro risk tied to economic measures in the housing sector, that is consistent with the implication of the Piazzesi, Schneider, and Tuzel (2007) model. 1 To explore further the information content in the new macro factor, we include the realized jump-mean factor of Wright and Zhou (2009) in predictive regressions of the bond risk premium on the macro factor and the Duffee (2011) hidden factor, both jointly and separately. Regression results indicate that these three factors are all significant and jointly can predict excess returns on 2- to 5-year maturity bonds with R 2 up to 66 percent (where the sample period used is , a sub sample period over which the jump factor can be constructed). Finally, we conduct a robustness test of the empirically documented predictability of our new macro factor by addressing two issues raised by Duffee (2007, 2011). He argues in the former study that all existing predictive regression studies actually test a restrictive null hypothesis that excess bond returns are unforecastable, whereas the more relevant null hypothesis should be that expected excess returns are stochastic, persistent, and independent of the macroeconomy. We attempt to distinguish between these two nulls by documenting the strong forecasting power of the lagged value of excess returns themselves. We construct tests for both restrictive and general null hypotheses and find that the general null, which Duffee cannot reject in the finite sample analysis, is rejected regardless whether asymptotic or simulated critical values are used. The other issue is whether the evidence for predictability of excess returns is simply a symptom of small-sample biases in estimated t-statistics or R 2. 1 Piazzesi et al. (2007) focus on excess stock returns. But the same mechanism applies to excess bond returns because risk premia on bonds and stocks are largely driven by the same business-cycle factors (Fama and French, 1989). 2

4 We find that all the evidence of return predictability (based on regressions with asymptotic theories) in our new macro factor persists even after we adjust the estimated test statistics for their finite-sample properties. To sum, we provide new and robust evidence on the link between expected excess bond returns and macroeconomic variables. The new macro factor identified in our analysis is intuitive, includes a housing component, subsumes both the Ludvigson-Ng macro factor and the output gap identified in Cooper and Priestley, and contains the information about future bond excess returns that is not contained in the Cochrane-Piazzesi forward rate factor, the Duffee hidden factor (referred to as an expectation factor by some researchers), and the Wright-Zhou realized jump-mean factor. Furthermore, our analysis indicates that sources of bond risk premium predictability include macro variables, jumps, and an expectation factor. The SAGLasso approach, used to extract our bond return forecasting factor from a large set of 131 macro variables using, has some advantages over the standard principal component analysis (PCA) or factor analysis. First, the SAGLasso approach selects predictors based on their association with the dependent variable (the bond risk premium in our case), whereas principal components may contain most information with respect to the data matrix of independent variables, but this information may not be most correlated with the dependent variable to be forecasted. 2 Second, the SAGLasso approach picks only a few most important ones (out of those 131 macro variables) as explanatory variables by shrinkage, whereas principal components or factors estimated using the PCA method are linear combinations of all 131 macro variables. In particular, due to cluster structure of macroeconomic data, we can divide 131 macro variables into groups and then apply the SAGLasso approach at the group level to help us select group factors (which are informative and easy to interpret) and thus identify underlying economic determinants of bond risk premia. Finally, predictive regressions of excess bond returns tend to exhibit autocorrelation (due to both high serial and cross-sectional correlations of bond prices) and the SAGLasso approach provides a robust way to correct for autocorrelated disturbances with an unspecified structure in such regressions. Our study builds directly on the insightful studies by Cochrane and Piazzesi (2005) and, in particular, Ludvigson and Ng (2009), respectively, the latter of which documents among other things that macro factors have important forecasting power for future bond excess returns, above and beyond the predictive power contained in yield curve factors (identified by the former study). We extend LN in several directions. First, we extract macro factors using the SAGLasso approach instead of dynamic factor analysis, and identify a new factor the housing factor. Secondly, we identify more sources of bond risk premium predictability. Finally, we address the concerns raised by Duffee (2007, 2011) on the robustness of such a 2 However, the information on the dependent variable can be used to help select a particular combination of those principal components (and/or their higher order terms) as a predictor in predictive regressions. See Ludvigson and Ng (2009). 3

5 predictability. Overall, we provide new and robust evidence to support LN s findings. Several recent studies of dynamic term structure models (DTSM) document that factors unspanned by bond yields have predictive content for bond excess returns. For instance, Cochrane and Piazzesi (2008) and Duffee (2011) focus on unspanned yield-curve risks by allowing yield factors, other than the traditional level, slope, and curvature factors to drive variation in expected excess returns. Joslin, Priebsch, and Singleton (2009) develop a model that incorporates macro factors but allows for components of macroeconomic risks orthogonal to the yield curve. Our empirical analysis sheds more light on the nature of unspanned predictability documented in the aforementioned studies. Specifically, we identify macroeconomic risk over and beyond that associated with variations in output gap and inflation, the focus of current literature. Also, our regression results highlight the importance of incorporating both yield-curve evolution and macroeconomic fundamentals in the extension of the conventional three-factor DTSMs. Indeed, based on both information sets our estimate of term premium precisely tracks its true value, as shown in our decomposition of the yield curve into expectation and risk premium components (Bernanke, 2006). The organization of the paper is as follows: The next section lays out the econometric framework and introduces the Supervised Adaptive Group Lasso (SAGLasso) method. Section 3 reports empirical results. In particular, we first extract those macro factors with significant predictive power for excess bond returns and then conduct both in-sample and out-of-sample forecasting regression analysis. We also present some applications of the constructed macro factors. Section 4 summarizes the results of our investigation. The appendix provides a list of macroeconomic variables used in the empirical analysis and also describes the dynamic term structure model used in the bootstrap analysis. 2 The Empirical Method This section introduces the supervised adaptive group lasso (SAGLasso) method and illustrates how to use it to select macroeconomic factors that can help forecast excess bond returns. Below we first describe the lasso method and its two variations - the adaptive lasso and group lasso. We then propose our SAGLasso procedure based on these two variations. 2.1 Basic Setup Following Fama and Bliss (1987), we use continuously compounded annual log returns on an n-year zero-coupon Treasury bond in excess of the annualized yield on a 1-year zerocoupon Treasury bond. For t = 1,, T, excess returns are defined rx (n) t+1 = r (n) t+1 y (1) t = ny (n) t (n 1)y (n 1) t+1 y (1) t, where r (n) t+1 is the one-year log holding-period return on an n-year bond purchased at time t and sold at time t + 1, and y (n) t is the log yield on the n-year bond. We assume that there are N macroeconomic measures observed for T time periods. Let 4

6 X be the T N panel of macroeconomic data with elements x it, i = 1,, N, t = 1,, T. As in dynamic factor analysis, the cross-sectional dimension here, N, is large and possibly greater than the number of observations, T. Following Ludvigson and Ng (2009), we consider the following predictive regression: rx (n) t+1 = β F t + γ Z t + e t+1, (1) where F and Z represent macroeconomic and other factors, respectively. Here, F can be either predetermined macroeconomic measures, such as the GDP growth, NAPM price index and personal consumption expenditure, or factors extracted from a set of macroeconomic series. 2.2 Supervised Adaptive Group Lasso For a T 1 response vector y, consider the following penalized least squares function: N f PLS (β) = y Xβ 2 + λ β i, (2) where λ 0 is a tuning parameter used to penalize the complexity of the model, and is the l 2 -norm, namely, η := (η η) 1/2, η R T. The l 1 -norm penalty β i used here induces sparsity in the solution and defines the least absolute shrinkage and selection operator (lasso) method (Tibshirani, 1996). 3 The lasso estimate is given by: ˆβ lasso = arg min β f PLS (β). If λ is zero, then ˆβ lasso equals the ordinary least squares (OLS) estimate, ˆβ ols, provided that the OLS would be feasible. Recall that none of ˆβ ols s components are zero. However, as λ increases, some components of ˆβ lasso will shrink to zero and, as a result, the corresponding useless explanatory variables will be dropped and the resulting regression model becomes more parsimonious. The lasso method has several advantages over the OLS. First, by construction, lasso reduces the variance of the predicted value and thus improves the overall (out-of-sample) forecasting performance. Second, the OLS is known to have poor finite sample property when the dimension of parameters to be estimated is comparable with the number of observations. 4 The lasso approach is developed to handle such problems. Lastly, the lasso leads to a much more parsimonious and easier to interpret model than the OLS. In fact, the parsimonious or sparse feature of lasso also distinguishes it from ridge regression, another shrinkage method. However, despite lasso s popularity, one limitation of the method is that lasso estimates can be biased. Zou (2006) shows that this problem can be fixed by using adaptive lasso, which minimizes the following objective function N y Xβ 2 + λ i β i (3) 3 The word lasso rather than LASSO is usually used to refer to this method in the statistics literature. 4 For our empirical study, there are 131 macroeconomic series with only 528 observations. i=1 i=1 5

7 where different tuning parameters {λ i } are introduced to penalize different regression coefficients separately. For a large-scale macroeconomic data set, economic series are usually organized in a hierarchical manner. Given such cluster structure of macroeconomic variables, we can divide these variables into different groups and then apply the lasso at the group level to select those important ones. We can implement this idea using the Group Lasso developed by Yuan and Lin (2006) to deal with situations where covariates are assumed to be clustered in groups. See the appendix for more details of the group lasso. As such, in order to construct a macro-based predictor of bond excess returns, we select a small subset of the original large set of macroeconomic variables in two steps. In the first step, we divide these macroeconomic variables into groups and consider variable selection separately within each group/cluster. Specifically, we screen out less important or irrelevant individual economic series and identify informative ones within each cluster using the Adaptive Lasso method. The analysis at the within-cluster level is desirable because even variables within the same group may represent certain quantitative measurements of different economic sectors. For instance, in our context, it is natural to conjecture that the Industrial Production (IP) Index for Consumer Goods and the IP Index of Materials might be connected to bond risk premia in a different manner. The analysis in this step allows to identify macroeconomic measures that are jointly significantly associated with risk premia in bond returns. Importantly, this analysis selects only a small number of variables within each cluster and makes it possible to construct parsimonious models using those selected variables. In the second step, we consider all the groups together, each of which now consists of only those macroeconomic variables selected in step one, and then conduct variable selection at the group level. Specifically, we select important clusters using the group lasso. This helps identify influential economic sectors in addition to influential factors selected in the first step. For ease of reference, we refer to this two-step procedure as the supervised adaptive group lasso (SAGLasso) approach. See the appendix for more details of this procedure and its implementation. To our knowledge, the SAGLasso is the first to consider penalized time series selection at both the within-cluster level and the cluster level. 3 Empirical Analysis In this section we extract macro factors from a monthly panel of 131 measures of economic activity over the period and then examine their power for forecasting excess bond returns. Specifically, Section 3.1 describes the data used in our empirical analysis. Section 3.2 discusses two different null hypotheses on the relationship between term premia and the 6

8 macroeconomy that will be tested. Section 3.3 constructs our SAGLasso predictor and presents results from in-sample analysis. Finally, Section 3.4 examines the forecast power of the SAGLasso predictor out of sample. 3.1 Data We use monthly data on bond returns and macroeconomic variables over January 1964 to December This sample period is the same as the one used in Duffee (2011) and Ludvigson and Ng (2011) but four-year longer than the period used in Cochrane and Piazzesi (2005) and Ludvigson and Ng (2009). Monthly prices for 1-year through 5-year zero coupon U.S. Treasury bonds from the CRSP are used to construct annual excess returns, as specified in Section 2.1. Following the literature, we construct annual returns by continuously compounding monthly return observations, rather than constructing monthly excess returns. In spite of the well-known statistical problem associated with regressions involving overlapping observations, there may truly be more information on predictability of excess returns using the annual excess returns because they subtract the one-year yield instead of the one-month yield. The macroeconomic data set used in this study consists of a balanced panel of 131 monthly macroeconomic times series, each spanning the entire sample period. This data set is the same one used in Ludvigson and Ng (2011) and is an updated version of the macro data set used in Stock and Watson (2002, 2005) and Ludvigson and Ng (2009) that include one more economic series no longer available. These 131 macroeconomic series are initially transformed to induce stationarity. They represent 15 broad categories: real output and income; employment and hours; real retail, manufacturing and trade sales; consumption; housing starts and sales; real inventories; orders; commercial credit; stock indexes; exchange rates; interest rates and spreads; money and credit quantity aggregates; inflation indexes; average hourly earnings; and miscellaneous. The appendix provides the complete list of series and their transformation. 3.2 Null Hypotheses Duffee (2007) argues that the existing literature does not test the relevant null hypothesis that expected excess bond returns are persistent and uncorrelated with macroeconomic measures. Instead, previous studies examine the null that excess returns are unforecastable. In this study, we follow Duffee and refer to the former null hypothesis as the general null and the latter null hypothesis as the restrictive null. Statistically, the well-known spurious regression problem is exemplification of their difference and the general null can be typically incorporated into the linear regressions by adjusting the covariance matrix of parameter estimates. However, in predictive regression the critical issue associated with different null hypotheses is whether the small-sample properties of test statistics are close to standard asymptotic 7

9 properties. Therefore, when analyzing the finite-sample properties of their techniques, existing studies usually makes bootstrap inference based on the restrictive null hypothesis. Put differently, the model used to generate simulation data offers only a choice between term premia that co-vary with macroeconomic variables and term premia that are serially uncorrelated. The difference between the two null hypotheses also has an important implication as the following. If we focus on predictors from the financial sector, using the restrictive null may not yield any conclusive result on whether expected excess returns are correlated with the macroeconomy. As noted by Duffee (2007), least squares regression detects partial correlations instead of unconditional correlations. Therefore, if macroeconomic series are correlated with the noise in financial variables derived from prices of risky securities, these macro series would exhibit forecasting power in regressions even if they are independent of excess returns. To assess the independent predictive power of macroeconomic inputs, we test both restrictive and general null hypothesis in this study. 3.3 In-Sample Analysis We first replicate the main analysis of CP and LN using our longer sample, for comparison purposes. We then construct our single marco factor the SAGLasso factor and present empirical evidence on forecast power of this factor based on an in-sample analysis The CP and LN Single Factors We construct both CP s and LN s single return-forecasting factors by running their main predictive regressions. Consider the annualized log forward rates f (n) t = p (n 1) t p (n) t, n = 1,..., 5 or in their vector form fwd t = (f (1) t,..., f (5) t ), and the average excess return arx t+1 = n=2 rx(n) t+1. The CP forward-rate factor is formed as the fitted value from the following regression: arx t+1 = γ 0 + γ fwd t + ɛ t+1, (4) We denote the CP factor by ĈP t = γ fwd t. We now follow Ludvigson and Ng (2009, 2011) to construct their single macro factor. First, we extract the first eight principal components (static factors), ( ˆf it ) 1 i 8, from the set of 131 macro series using the asymptotic PCA. We can then perform best-subset selection among different subsets of { ˆf 3 1t, { ˆf it, ˆf 2 it; i = 1,..., 8}} using the BIC criterion (Schwarz, 1978). Ludvigson and Ng (2011) find the solution to be the nine-factor subsect: F 9t = 8

10 ( ˆf1t,... ˆf 8t, ˆf ) 1t 3. 5 As such, we run the following regression: arx t+1 = δ 0 + δ F 9t + ɛ t+1. (5) It follows that the LN single macro factor LN t = ˆδ F 9t. Given the constructed CP and LN factors, we can then run in-sample regressions of the form (1) for a given maturity bond. For each regression, we compute standard errors using two separate corrections. The first one is the Hansen-Hodrick (1980) GMM correction for overlapping observations; and the other is the Newey-West (1987) (with 18 lags) correcting for serial correlation. Note that these two correction methods correspond to different null hypotheses of the Wald test. The HH method is used to estimate the covariance matrix of parameter estimates for the restrictive null which states that forecast errors are serially uncorrelated, whereas the NW method is used for the general null that forecast errors contain persistent components independent of the macroeconomy. Under both null hypotheses the test is asymptotically distributed as a χ 2 (1). Results reported in Table 1 show that regressions against ĈP t have an adjusted-r 2 of percent, slightly lower than reported by CP, who use data through The adjusted-r 2 of the macro factor LN t ranges from 0.26 to 0.28, very similar to what is obtained by LN. Augmenting with ĈP t raises R 2 to about 0.4. We also conduct a Ljung-Box refined Q test to see if the first-order autocorrelation in the error term of this augmented regression is zero. Test results shown in the last two columns indicate a strong rejection of the null. The results shown in Table 1 serve as the benchmark and starting point of our empirical analysis Construction of the SAGLasso Factor In this subsection we construct parsimonious macro-based models for predicting bond excess returns using the SAGLasso approach proposed in Section 2. Following Ludvigson and Ng (2011), we divide the set of 131 macro series into eight groups: (1) output, (2) labor market, (3) housing sector, (4) orders and inventories, (5) money and credit, (6) bond and FX, (7) prices, and (8) stock market. Column 2 of Table A.1 shows the group id of each of the 131 macro series. We then implement the SAGLasso method in two steps. We first apply the adaptive lasso to each of the eight groups separately. For a given group, we perform model selection using only macro variables within the same group along with their lagged values (up to six lags). We also include lagged bond excess returns in the set of candidate predictors but we do not penalize the coefficients associated with these 5 We use this solution because our sample period is the same as the one used in Ludvigson and Ng (2011) and four-year longer than what used in Ludvigson and Ng (2009). The solution identified in the latter study is F 6 t = ( ˆf 1t, ˆf 3 1t, ˆf 2t, ˆf 3t, ˆf 4t, ˆf 8t). 9

11 lagged dependent variables. In this first step of SAGLasso, we can screen out a large portion of exogenous explanatory variables (candidate predictors) within each group and thus significantly reduce the number of candidate predictors. For instance, consider the largest group, the labor market group, that originally contains 32 series (as shown in column 2 of Table A.1) and thus 32 7 candidate explanatory variables (once six lagged variables of each series included). The last column of Table A.1 indicates only four series including #42, #44, #45, and #49 are selected. More specifically, only 11 of these 224 variables are selected, including #42 itself; #44 along with its lag-1, lag-2, lag-3, lag-4, and lag-6; lag-4 of #45; and #49 along with its lag-3 and lag-6. In total, only 38 macroeconomic series (out of 131) remain and have non-zero coefficients on their contemporaneous and/or lagged values after the adaptive lasso is applied; and the number of the selected macro variables is 58 (out of 131 7). In the second step, we select relevant clusters/groups using the Group Lasso. The results from this analysis show that coefficients of groups (1), (4), (5) and (8) are shrunk to exactly zero. The selected groups are labor market (#2), housing (#3), bond and FX (#6), and prices (#7). Note that each of the four selected groups has a clear economic interpretation by construction. As indicated in the last column, labeled Ĝt, in Table A.1, the housing group includes 5 macro series and 11 macro variables, the bond and FX group 8 series and 9 variables, and the prices group 4 series and 7 variables. In total, the number of selected macro series is 21 (= ) and the number of selected macro variables is 38 (= ). Namely, out of the original 131 macroeconomic series, we identify 21 series associated with labor market, housing, interest rates and prices that show strongest connection with bond risk premia. We also find evidence that many series have lagged effect on bond risk premia. In particular, shocks to consumer prices or labor market seem to require a long lag to manifest their impact on the bond market. As such, using the SAGLasso approach we obtain a parsimonious and intuitive macrobased model whose factors are also easy to interpret. Below we first provide a more clear picture of the relationship between excess bond returns and each of the four selected macroeconomic groups. We then construct a single factor using these group factors and investigate its predictive power Evidence from Macro Group Factors Given the group lasso solution obtained from (16), we can construct four group factors as follows: g ht = X h βh, h = 2, 3, 6, 7, (6) where h denotes the index of the group whose beta coefficient is not zero. For easy reference, we relabel these four group factors as ĝ ht, h = 1,..., 4, which represent the employment, housing, interest rates and inflation factors, respectively. 10

12 Table 2 presents results from in-sample predictive regressions of bond excess returns on the group factors as well as the CP and LN factors, for 2-, 3-, 4-, and 5-year bonds. Panel A reports coefficient estimates, t-statistics, and R-squared of univariate regressions on individual group factors. The results indicate that all group factors exhibit significant unconditional predictive power. The R 2 ranges from 20 to 30 percent for the first three group factors and from 11 to 13 percent for the last group factor (the inflation factor). Results from multivariate regressions reported in Panel B indicate that the four group factors together have an adjusted R 2 ranging from 38 percent for the 5-year bond to 43.6 percent for the 2-year bond, substantially higher than those with the LN factor (see Table 1). This provides evidence that the factors extracted using the SAGLasso approach can significantly improve the explanatory power of macro factors. We also run multivariate regressions augmented with the ĈP factor in order to see whether the group factors have any marginal predictive power. Results shown in Panel B indicate that the four group factors have statistically significant and economically important predictive power beyond that contained in the forward-rate factor ĈP. For instance, the R 2 statistic increases from 0.3 with ĈP (Table 1) to when the group factors are included. Moreover, unlike the interest rate factor ˆf 2t constructed in LN, the ĝ 4t factor (our interest rate group factor) does not lose its marginal predictive power when ĈP t is included as a predictor. On the other hand, ĈP t remains significant in the augmented regressions and thus is not subsumed by the macroeconomic factors, a finding consistent with LN s Evidence from the Single SAGLasso Macro Factor We now follow CP and LN to construct a single macro factor as follows: Ĝ 4 h=1 X h βh. (7) We refer this predictor as to the SAGLasso factor. Note that Ĝt is basically a linear combination of macroeconomic series as well as their lagged values. The SAGLasso factor differs from the LN macro factor in that (1) the former involves only 21 macroeconomic series and consists of four easier-to-interpret group factors; and (2) the SAGLasso factor takes into account the dynamic response of bond risk premia to macroeconomic innovations. Untabulated results indicate that the SAGLasso factor has statistically and economically significant predictive power conditional on lagged bond excess returns, regardless of the bond maturity. This implies that the SAGLasso factor contains information about future returns that is not contained in its own historical path. This test directly corresponds to the general null hypothesis, as it demonstrates that the lagged returns per se, especially the first lag, have considerable forecasting power for the future ones. The Ljung-Box Q test does not indicate the presence of AR(1) serial correlation in our model. The Q test is known to be 11

13 less powerful when the null hypothesis is false. As a robustness check, we also perform the Lagrangian multiplier test (Breusch, 1978; Godfrey, 1978) and the results are the same. Next, we examine whether the SAGLasso factor has unconditional predictive power for excess returns. Table 3 presents the estimation results of univariate predictive regressions on Ĝt. The results show that Ĝt is significant regardless of the bond maturity. In addition, the regression ˆR 2 ranges from for the 5-year bond to for the 2-year bond. This provides evidence that the single SAGLasso factor has substantially higher explanatory power than either LN s or CP s single factor (Table 1). The estimation results also show that augmenting ĈP t with Ĝt can substantially improve the forecasting power. For instance, the regression R 2 increases from against ĈP t alone to against ĈP t and Ĝt together, for the 5-year bond. In addition, both ĈP t and Ĝt are significant regardless of the bond maturity. In another word, like LN t, our Ĝt provides additional forecasting power over the forward rate factor ĈP t but does not subsume it. We also observe from Table 3 that augmenting Ĝt with LN t improves forecasting power only marginally under the ˆR 2 statistic, and that LN t becomes insignificant under either the HH or NW t-statistic. This indicates that the SAGLasso factor can capture the macroeconomic information about term premia contained in the LN macro factor and thus can subsume LN t. Following LN, we explore the economic interpretation of our macro factor on other fronts as well. For example, like LN t, the SAGLasso factor can capture the countercyclic component in risk premia, as illustrated by Figure 1 which plots the six-month moving average of Ĝt and the Growth Rate of Industrial Production (GIP) vs. time over our sample period. Shaded areas indicate the periods designated by the National Bureau of Economic Research (NBER) as recession periods, and associated with low GIPs and high values of Ĝt. Indeed, Ĝ t is strongly negatively correlated with GIP, with a correlation coefficient of The SAGLasso factor falls to troughs in the mid-to-late stage of economic expansions and reaches its peaks at the end of recessions. Cooper and Priestley (2009) document that the output gap can predict excess returns on 2- to 5-year maturity bonds with R 2 equal to 2 percent, where the output gap is measured as the deviations of the log of the Industrial Production Index from a quadratic and linear trend. Notice that the output gap is not among the 131 series used in our analysis (and LN s). To investigate potential connection between this new macro variable and our SAGLasso factor, we include the output gap in the analysis. The top two panels in Table 4 replicate the results on the predictive power of this factor (denoted gap) reported in Cooper and Priestley, where ĈP t is the Cochrane-Piazzesi (2005) factor orthogonalized relative to gap. Results reported in the third panel of the table show that if the part of our factor Ĝt orthogonal to gap is used together with gap in regressions, the forecasting power of gap also remains. However, as shown in the bottom panel of Table 4, our employment group factor (when not orthogonalized 12

14 to gap) can subsume the output gap factor Evidence on SAGLasso Being A New Hidden Factor As shown earlier, like LN t, the SAGLasso factor Ĝt contains information about future bond excess returns that is not contained in ĈP t that summarizes the time-t yield curve. In this subsection we first provide further evidence that Ĝt is a hidden factor; we then provide evidence that verifies Duffee s (2011) conjecture about roles of a hidden factor in the bond market; and finally we show that Ĝt is different from the hidden factor identified in Duffee (2011). We firstly examine the cross sectional relation between the SAGLasso factor and bond yields based on the following regression: y (n) t = θ 0 + θ 1 Ĝ t + ε t, n = 1,..., 5. (8) We find that the SAGLasso factor explains only 0.09% of the variance of yield changes, a level indistinguishable from noise in yields. Recall that the percentage of variance of yield changes explained by Ĝt is defined to be 100 trace(cov(θ 1 Ĝ t ))/trace(cov( y t )). As such, although the SAGLasso factor contains substantial information about future excess bond returns, its contribution to the overall variance of the cross section of bond yields is imperceptible. This provides additional evidence that Ĝt is a macro factor that is hidden from the yield curve at time t. A natural question is what mechanisms through which a macro factor can have strong forecasting power for future yields and excess returns but at the same time have little impact on current bond yields. Duffee (2011) points out one such potential mechanism based on the following decomposition of bond yields: ) y (n) t = 1 n E t ( n 1 i=0 y (1) t+i + T P (n) t, (9) where the two terms on the right-hand side of equation represent the expectations of future short rates and the term premium (or the yield risk premium), respectively. Duffee observes that if there exists one factor whose impacts on these two terms cancel each other, then the factor has no impact on current bond yields. Consider one such scenario. If aggregate risk aversion is time varying in response to news about aggregate consumption growth, as indicated by Campbell and Cochrane (1999), shocks to consumption growth temporarily raises risk aversion and cause investors to demand higher risk premia on risky long-term bonds. On the other hand, investors believe that the Fed will attempt to offset these short-lived macroeconomic shocks with monetary policy actions. This in turn drives down the expectations of future short rate. Thus the net effect of the 13

15 macro shocks on current yields becomes insignificant because the expected change in short rates and the change in risk premia have opposite effects. To see if our SAGLasso factor can capture market expectations and affect the bond market as described above, we consider a factor-augmented vector autoregressive (FAVAR) model and examine impulse response functions (IRF) of certain key macro series. Following Bernanke, Boivin, and Eliasz (2005), we assume that Z t = Φ(L)Z t 1 + u t ; (10) X t = ΛZ t + ɛ t. (11) where Φ(L) is a lag polynomial of order d, Z t = [F t, Y t ], F t denotes a K 1 vector of unobservable factors, Y t an M 1 vector of observable economic variables, and X t the set of the 131 macroeconomic time series as before. Suppose also K+M 131. Vector F t can capture the information set of policy-makers and the private sector, which is not in the econometrician s information set. The econometrician, however, can identify the effects of monetary policy by utilizing X t. One attractive feature of the FAVAR framework is that we can construct IRF for any observable economic variables included in X t. In our implementation, we set K+M to 8 as the Bai and Ng (2002) information criteria indicates that eight factors are needed to capture the majority of information contained in X t ; and we use d = 7 following Bernanke et al. (2005). We consider the specification of the above FAVAR model where Y t = Ĝt in order to illustrate the risk channel through which the SAGLasso factor affects expectations of future rates and the term premium, and yet is orthogonal to the yield curve. We estimate the model using likelihood-based Gibbs sampling, as outlined in Bernanke et al. Figure 2 plots the IRFs (along with 90 percent confidence intervals) of selected economic indicators to a one-standard-deviation shock in the SAGLasso factor Ĝt. Consider first the response of the 3-month Treasury bill yield. The SAGLasso factor has little impact on the short rate in Month 0, but leads to a nearly 40 basis-point drop in the short rate after 18 months. Moreover, the response of the short rate is quite persistent, remaining to be 25 basis points after four years. In contrast, the response of the 5-year bond yield seems insignificant regardless of the time horizon, especially for the first 18 months in which the IRF is flat. As a result, the responses of the short rate and long-maturity bond yield jointly imply an increase in the risk premium. The IRF of Consumer Expectation Index (of the University of Michigan) plotted in the last panel of the figure provides further evidence on the expectation nature of Ĝt. The illustrated immediate effect of Ĝt on the index confirms that the unspanned predictability of Ĝt is associated with market expectations. The responses of the other economic indicators shown in the figure are generally of the expected sign and magnitude. In particular, variables that typically exhibit stickiness, such as CPI, investment and Unemployment, do have slow-moving responses to Ĝt, while the housing start index 14

16 and dividend, which are sensitive to future changes in the short rate, initially jump and eventually wear off. As such, IRFs show that the SAGLasso factor indeed has no contemporaneous effect on the term structure but contains substantial information about expected bond excess returns. The evidence here also verifies the mechanism pointed out by Duffee (2011) through which a macro factor can become a hidden one. Finally, we replicate the analysis of Duffee (2011) and examine whether the SAGLasso factor contains any information of future excess returns beyond that contained in Duffee s hidden factor. Consistent with his findings, results reported in the top panel of Table 5 show that Duffee s five latent factors extracted from the yield data contain information about future excess returns. The R 2 ranges from 0.31 for the 2-year bond to 0.34 for the 4-year bond (slightly lower than those documented in Duffee (2011), who use data through 2006), is markedly higher than ĈP t s R 2 (Table 1), and is notably lower than Ĝt s (Table 3). Untabulated results show that Duffee s five factors subsume ĈP t. This is likely because those five factors are extracted from yield curves. Results reported in the bottom panel of Table 5 show that both the SAGLasso factor and Duffee (2011) s hidden factor, Ĥ 5t, remain highly significant and do not subsume each other. This finding indicates that the two hidden factors capture different aspects of bond risk premia. One implication of this result is that macroeconomic risk underlies, but does not perfectly capture, the variation in excess returns that is not contained in the first three principal components of yields. As such, both unspanned yield-curve risk and unspanned macro risk are priced and bond risk premia are not fully identified in absence of either information set. This also implies that the number of risk factors driving expected excess returns is not necessarily the same as the number of state variables that determine bond yields, an idea first proposed by Duffee (2011) The Cycle Factor In a parallel study, Cieslak and Povala (2010) identify a cycle factor that contains almost all term premium information that the level factor embeds. They find that unspanned factors documented in CP and LN hold little incremental predictive power after controlling for their cycle factor (which they refer to as ĉf). Moreover, this factor is found to be reflected in cross-sectional yields, whose impact increases in the bond maturity. In this subsection we follow Cieslak and Povala s procedure and construct ĉf, and then investigate whether ĉf encompasses the predictors considered earlier in our analysis. Consistent with Cieslak and Povala (2010), results from univariate regressions reported in the top panel of Table 6 confirm that the cycle factor is both economically and statistically significant regardless of the bond maturity. The R 2 ranges from for the 2-year bond 15

17 to for the 5-year bond. 6 Results from regressions against both ĉf and Ĝ show that both factors are significant across all maturities. Augmenting ĉf with Ĝ can increase the R 2 from to In addition, regression coefficients of Ĝ are in line with those reported in Tables 3 and 5. These results indicate that the SAGLasso factor contains information about future excess returns beyond what is contained in the cycle factor. This finding is not inconsistent with the result of Cieslak and Povala that ĉf comprises most predictability associated with LN s first eight principal components (whose nonlinear functions are not included in their analysis). Recall that a key difference between Ĝ and LN is the housing component of Ĝ. As such, it is likely the housing factor ĝ 2 that preserves Ĝ s significance and contributes to its explanatory power in the presence of ĉf. Results from regressions against our group factors and ĉf (shown in the top panel of Table 6) supports this explanation. Notice that when ĉf is controlled for, only the housing factor remains highly significant while the other three group factors becomes either insignificant or marginally significant. We now focus on the information content of the cycle factor about yield curves. Panel B reports the results from regressions against both the cycle factor and Duffee s five filtered factors. Indeed, the level factor Ĥ1 is found to largely lose its predictive power in the presence of ĉf. This implies that the latter summarizes the information contained in the first principal component of the yield curve, consistent with Cieslak and Povala. Notice from the panel that Duffee s hidden factor Ĥ5, however, remains highly significant and accounts for 18-24% of the total variance (untabulated). This indicates that the Duffee hidden factor contains information of future excess returns beyond what is contained in the cycle factor Realized Jump Factors Wright and Zhou (2009) construct a jump factor that can nearly double the R 2 when included in the benchmark predictive regression on forward rates. In this subsection we examine if this jump factor can capture any information beyond what contained in macro-based predictors, given that realized jumps in bond markets are often associated with macroeconomic announcements, and evidence on connections between term structure model-implied jumps and macroeconomic shocks (Das, 2002; Johannes, 2004). Following Wright and Zhou (hereafter WZ), we construct monthly realized volatility, jump intensity, jump mean, and jump volatility using data on 30-year Treasury bond futures at the five-minute frequency from CBOT. Untabulated results from regressions of bond excess returns on these four factors and forward rates over August 1984-December 2007 are almost identical to those reported in WZ, whose sample period ends 16-month earlier. 7 In 6 These R 2 values are lower than the levels of 41% 46% reported in Cieslak and Povala, likely due to bond data used in the analysis. They use constant maturity Treasury yields from the Fed and we use the CRSP Fama-Bliss data in this study. 7 The sample starts from August 1984 as high-frequency Treasury bond future data are not available until July 16

18 particular, augmenting CP s regression with the jump mean raises the adjusted R 2 up to 59 percent and the CP factor and jump mean do not subsume each other. We identify the timings of the 20 largest realized jumps (untabulated) during our sample period (high-frequency data allow more timely jump filtering) and examine if each of these jumps coincides with unexpected macroeconomic news arrivals. We find that most jumps occur when there are regularly scheduled announcements, consistent with the findings of Fleming and Remolona (1997) and Balduzzi, Elton, and Green (2001). Based on the absolute jump size, largest moves occur when unemployment announcements are made, implying that surprises about the current state of the real economy significant affect long-term bond prices. As such, realized jumps may capture the impact of macroeconomic shocks on the Treasury bond market. We now examine regressions on the SAGLasso and other predictors augmented with the jump mean factor over August 1984-December Notice first from Panel A of Table 7 that the univariate R 2 of the SAGLasso factor drops to around 30 percent from more than 40 percent estimated over the full sample. Notice also that augmenting the SAGLasso regressions with the jump mean factor (denoted ĴM) raises the R2 to about percent. Moreover, the coefficients on Ĝ and ĴM are in line with their counterparts under separate univariate regressions. This implies that unlike our macroeconomic factor, the realized jump mean measure captures a high-frequency relation between macroeconomic variables and bond yields Unspanned Predictability of SAGLasso: Discussion Intuitively, we can classify predictors of bond excess returns discussed so far into three groups: those based on yield curves (including ĈP, {Ĥi}, and ĉf), those macro based (including LN and Ĝ), and those based on interest rate derivatives including ĴM. Given the evidence that {Ĥi} and Ĝ can subsume ĈP and LN, respectively, we now focus on the forecasting power of {Ĥi}, ĉf, Ĝ, and ĴM. Results reported in Panel B of Table 7 indicate that WZ s jump mean factor, Duffee s five yield factors, and our SAGlasso factor jointly can explain about 61-66% of the variation in bond excess returns. Moreover, ĴM, Ĥ 5 (a hidden factor), and Ĝ all remain statistically significant, regardless of the bond maturity. Next, we regress bond excess returns on Ĥ5, ĝ 2, ĉf, and ĴM. Results reported in Panel C of the table show that these four predictors are still statistically significant across bonds with different maturities, except in the case of 5-year bonds where Ĥ5 becomes marginally 1982 and the three jump measures are 24-month rolling averages due to the presumption that jumps are rare and large. 8 Interestingly, this subsample, over which high-frequency data on Treasury bond futures are available, covers the post monetary experiment period. This considerable evidence of a regime switch during the late 1970s and early 1980s. 17

19 insignificant. In addition, the four predictors together can explain about 54-58% of the variation in bond excess returns. These findings indicate that there are at least two primary conduits through which the information about macro economy can enter the term structure of interest rates. One is captured by macro factors such as the LN and our SAGLasso factors, which describe relation between macroeconomic variables and yields a low-frequency level. Realized jump measures capture the other channel through which bond yields directly respond to unexpected shocks from real economy and monetary policy. As pointed out by LN, empirical evidence of macro factors predictive power has some implications for affine term structure modeling, in which cross-section of yields is often assumed to summarize completely the predicability of bond returns. Macro-finance models incorporate the first conduit by allowing macroeconomic factors to enter state variables (Ang and Piazzesi, 2003). A further extension may be required in order to deal with the jumpinduced misspecification. However, even a jump-diffusion macro-finance model may not be able to account for all of these stylized empirical facts. As discussed by WZ, the fact that jump risk factors are not spanned by the current yield curve may have something to do with unspanned stochastic volatility documented in Collin-Dufresne and Goldstein (2002). 3.4 Out of Sample Analysis The results reported so far are obtained from in sample analysis. It is known that significant in-sample evidence of predictability does not guarantee significant out-of-sample predictability (see, e.g., Goyal and Welch, 2008). In this subsection we examine the out-ofsample forecasting performance of several predictors considered in the previous subsection. To avoid forward looking bias, we estimate all the factors and parameters using only information available at the time of the forecast. And we do this recursively and re-estimate both factors and parameters when the new information becomes available. The predictors to be implemented in the out-of-sample analysis include the Cochrane- Piazzesi forward-rate factor, Ludvigson-Ng macro factor, Cieslak-Povala cycle factor, and our single SAGLasso macro factor. The Wright-Zhou jump mean factor is not considered because, as mentioned earlier, the series starts in August The Duffee filtered factors are not included due to computational constraints; it would be impractical to recursively estimate his five-factor term structure model to extract his factors at every step when they are used to predict bond excess returns. We denote recursively constructed factors by a tilde, in order to differentiate them from their in-sample counterparts (denoted by a hat) estimated using the full sample. We now describe the procedure of the out-of-sample analysis. We divide the full sample into in-sample (training/estimating) and out-of-sample (testing) portions. The former consists of R > 1 observations. Namely, at time t = R we have available the following set 18

20 of monthly observations of macro series and bond excess returns over the training period: F R = {X t, {rx (n) t, n = 2,..., 5}, t = 1,..., R}. 9 We construct a predictor, say, the SAGLasso predictor ( G t=r ) based on the information set F R and use it to help forecast one-step ahead yearly bond excess returns. Next, we calculate the realized forecast error of this model. We then repeat this exercise at t = R + 1 using added observations. If P denotes the number of 1-step ahead predictions, then T=R+P+12, where T is the total number of observations of macroeconomic series. We conduct model comparisons to assess the incremental predictive power of the SA- GLasso factor ( G) above and beyond the predictive power in factors including CP, LN, and the cycle factors. Consider the CP factor ( CP ) for illustration. In this case, we compare the out-of-sample forecasting performance of an unrestricted specification including both the SAGLasso and CP factors to the performance of the restricted benchmark model (the null) including only the CP factor. Another example of the restricted benchmark model is a constant term as a benchmark, apart from an MA(12) error term, which has been strongly rejected even out-of-sample (see, e.g., LN). As such, in order to investigate whether the SA- GLasso factor has any additional predictive power conditional on information contained in past returns, we also use a simple AR(6) specification as a restricted benchmark model. Given a pair of the restricted and unrestricted specifications, we can obtain their time series of realized forecast errors over the entire initial out-of-sample period and then conduct a model comparison. One simple and intuitive metric used for model comparisons is the ratio, MSE u /MSE r, where the numerator and denominator denote the mean-squared forecasting errors of the unrestricted and restricted models, respectively. We also implement two encompassing tests for nested models, namely, the Ericsson (1992) test and the ENC- NEW test of Clark and McCracken (2001). Both tests examine the null hypothesis that the restricted benchmark model encompasses the unrestricted model with additional predictors. However, the precise asymptotic distribution of the test statistics in these two tests depends on parameter π (the asymptotic ratio of P/R), where π = lim P,R inf P/R. The Ericsson test critical values from a standard normal distribution are conservative if π > 0. As the value of π is unknown in our case, we use the Ericsson test as a robust check. For the full sample 1964:1 through 2007:12, the initial estimation period spans 1964:1 to 1984:12 (dependent variables from 1965:1 to 1984:12 and independent variables from 1964:1 to 1983:12). This choice of in-sample portion follows LN; alternative choices lead to qualitatively similar results. Table 8 reports results from our out-of-sample analysis. Column labeled Ericsson in the table reports the ENC-REG test statistic, whose 95 percent critical value is Similarly, column labeled Clark-McCracken presents the ENC-NEW test statistic, whose 95 9 In the conventional definition of rx (n) t used in the literature and followed in Section 2.1, the time index t refers to year t although it also implicitly refers to month t when used to denote time-t observations. 19

21 percent critical value is Notice from panel A that the forecasting model including the SAGLasso factor improves remarkably over the constant expected return model. For instance, the unrestricted model s forecast error variance is only percent of the restricted model. We observe from panel B that the general null hypothesis that the expect excess return follows an AR(6) is strongly rejected by both encompassing tests. 10 Augmenting the original AR(6) model with the SAGLasso factor can reduce the mean-squared error by about percent. Results reported in panel C (D) indicate that including the SAGLasso factor can improve substantially over the restricted benchmark that includes a constant and LN ( CP ), consistent with the results from the in-sample analysis shown in Table 3. In particular, the increase in the forecast power is strongly significant, as indicated by both test statistics. These results based on out of sample provide further evidence that the SAGLasso factor contains information about future excess returns that is not captured by either the CP or LN factor. Notice from panels C and D that G s improvement over LN is weaker than over CP, which is not surprising as G and LN are essentially constructed using the same data set. Nonetheless, the evidence presented here demonstrates that the shrinkage based SAGLasso approach can help significantly improve the forecast power of macro variables, not only in sample but more importantly also out of sample. To illustrate further the role of shrinkage here, consider a simple OLS predictor which is constructed as follows: Run a forecasting regression of form (4) using the 38 macroeconomic variables included in Ĝ (an in-sample predictor) as explanatory variables and then use the fitted value from this regression as a single naive predictor. Untabulated results indicate that although the OLS forecasting regression can largely retain the in-sample performance documented earlier, the naive factor is found to have little incremental forecasting power over LN in the out-of-sample analysis. 11 To some extent, this is not surprising given the property of the OLS estimates reviewed in Section 2.2. We now take CP t + G t as the benchmark (that captures the information from both bond yields and macro variables), and test if adding the cycle factor, cf t, can improve the (outof-sample) forecasting power. Notice from panel E that the unrestricted model in this case actually increases the mean-squared forecast error by around percent relative to the restricted benchmark (the null). ENC-REG tests cannot reject the null (the test statistics below the 20% critical value). ENC-NEW statistics shown in the panel appear significant based on the asymptotic results. However, as demonstrated in Section 3.5, the ENC-NEW test suffers from the small sample size problem in our setting and the true critical value is greater than 24. We can replace cf t by ĉf t in the unrestricted model and then redo outof-sample tests. This ought to improve the performance as the latter predictor is estimated 10 The Ericsson test may not apply to the general null because the forecasts are not truly out of sample. 11 We thank Peter Feldhütter for the idea of the naive predictor and for suggesting this exercise. 20

22 using the full sample. Results shown in panel F show that indeed the ratio, MSE u /MSE r, is markedly lower and the two test statistics are much higher than those shown in panel E. However, MSE u /MSE r is still greater than one and the two tests fail to reject the null. To sum, we find strong out-of-sample evidence that augmenting either the CP factor or the LN factor with the SAGLasso factor can improve forecasting power of the model. And we fail to reject the null that the Cieslak-Povala cycle factor does not contain any information of future excess returns beyond what is contained in the CP forward rate factor and the SAGLasso macro factor. 3.5 Finite-Sample Property In this subsection we discuss small-sample inference. This is largely motivated by Hodrick (1992), Stambaugh (1999), and Ferson, Sarkissian, and Simin (2003), who identify two cases where asymptotic inference in return forecasting regressions could be problematic. In the first case where the standard instruments employed as predictors are highly persistent and/or contemporaneously correlated with the idiosyncratic noise in returns, serious over-rejection could result. The other case is when overlapping observations are used in constructing dependent variables, estimates of standard errors for regression coefficients show strong bias. In our case, some factors estimated with SAGLasso display high persistence, with first-order autoregressive coefficient up to ; and annual excess returns used in regressions involve overlapping yields data. We proceed with a finite-sample analysis by generating bootstrap samples of the yields as well as of the exogenous predictors {ĝ ht }, to answer the question raised by Duffee (2007). We generate simulation data for finite-sample inference using no-arbitrage dynamic term structure models (DTSMs) as well as traditional time-series models. The term structure model we use is an eight-factor model in the spirit of Duffee (2007). The time-series model we consider is a vector moving average model of order 12, namely, a VMA(12) model. Following Duffee (2007), we conduct small-sample inference under either the restrictive or the general null hypothesis. See Appendix C for technical details of data simulation and the DTSM used for simulation. To test the restrictive null hypothesis we drawing random samples from the empirical distribution of the residuals from a VMA(12) model, so that annual returns are forecastable up to an MA(12) error structure. For the restrictive null we pre-estimate factors by {ĝ ht } re-sampling the T N panel of data. This procedure creates bootstrapped samples of the predictors themselves. We conduct both in-sample and out-of-sample analysis. Tests considered in the analysis include the Wald test, the Ericsson test, and the Clark-McCracken test. The first test is used for in-sample analysis whereas the latter two tests are used in an out-of-sample analysis. For each of these three test statistics, we obtain its rejection rate and the five-percent critical 21

23 value under both restrictive and general nulls; and we do this for 2-, 3-, 4-, and 5-year bonds. Table 9 reports finite-sample properties of test statistics. Column labeled Rejection Rates presents finite-sample rejection rates of tests of the null hypothesis when using the asymptotic five percent critical value, which is 9.49 for the Wald test, for the Ericsson test, and for the Clark-McCracken test, respectively. Column labeled 5% CV reports true finite-sample critical values at a five percent rejection rate. The main conclusion is that the results based on bootstrap inference are broadly consistent with those based on asymptotic inference in Tables 3 and 7. Our results support Duffee s finding that small-sample distributions of test statistics associated with the general null markedly diverge from their counterparts under the restrictive null, as well as the asymptotic distributions of these test statistics. However, even after adjusting the estimated test statistics for their finite-sample properties, all regression evidence of return predictability presented earlier in Section 3 remains robust. For example, the actual values of Wald, Ericsson and Clark-McCracken test statistics are 67.5, 3.36, and 129, respectively, all of which are much higher than the 95% small-sample critical values even under the general null, namely, 19.2, 2.23 and Indeed, both null hypotheses are rejected regardless of the type of regression or the test used. Table 10, the finite-sample distributions based counterpart of Table 2, reports the evidence from the in-sample analysis. The magnitude of predictability found in historical data, measured in R 2 s and χ 2 tests, is too large to be accounted for by sampling error in samples of the size we currently have. Moreover, P-values computed with the empirical distributions of 50,000 bootstrapped samples are almost zero as well. 3.6 Yield Curve Decomposition The analysis so far focuses on the excess return-forecasting power of certain predictors, such as unspanned yield factors (e.g., the Duffee hidden factor) and macro factors (i.e., the LN and SAGLasso factors). In this subsection we relate our results on return predictability to the recent literature concerning yield curve decomposition, more specifically on how to decompose current yields or forward rates into long-term expectations of future interest rates and term premia (Kim and Wright, 2005; Cochrane and Piazzesi, 2008; Joslin, Priebsch, and Singleton, 2009). We show that conditioning on the SAGLasso factor can improve the realtime estimation of term premia and thus our understanding of the time series behavior of term premia. One commonly used approach to extracting the risk premium component from the yield curve is based on affine DTSMs, which among other things provide many insights about the risk-neutral dynamics of bond yields. However, the model performance depends on the specification and estimation of the risk-neutral dynamics, as well as restrictions imposed on market prices of risk. In addition, forward looking bias involved in some cases may 22

24 be a concern. Indeed, Piazzesi and Schneider (2010) find that real-time estimates of bond risk premia show very different time-series properties from those obtained from in-sample predictions. Below we consider an alternative approach that allows us to estimate term premia using the SAGLasso factor. Consider the decomposition in (9). If the SAGLasso factor truly captures market expectation of excess returns beyond what is contained in yield curves, then including G should reduce the following estimation error of yield risk premia: 1 n [ ( n 1 Ẽ t i=1 rx (n i+1) t+i ) ( n 1 E t i=1 rx (n i+1) t+i )] = η (n) t η (n) 1,t where E t [ ] and Ẽt[ ] represent the true term premium and its estimate, respectively; η (n) t is the overall decomposition error; and η (n) 1,t is the estimation error of the yield-expectation [Ẽt ( n 1 i=0 y(1) t+i ) )]. Interpretation of empirical results is nev- ( n 1 component 1 E n t i=0 y(1) t+i ertheless complicated by the fact that only η (n) t is observable. However, given a statistic model used to generate out-of-sample forecasts of 1-year rates, the difference between measured and true expectations of future rates η (n) 1,t, is fixed. 12 (n i+1) Therefore, if to estimate Ẽt(rx t+i ) by including G t leads to a uniformly better cross-sectional fitting than to do it excluding (1) G t, regardless of Ẽt(y t+i ) used, then the better performance of a G t -based identification is unlikely due to sampling error. We focus on the five-year bond in the balance of this subsection. To identify the first expectation component in the decomposition (9), we need 1- through 4-year ahead forecasts of one-year T-bill yields. We consider two widely used yield-forecasting models: the Nelson and Siegel (1987) model (NS hereafter) with autoregressive factor dynamics (Diebold and Li, 2006) and a VAR(1) model of bond yield levels, inflation, and personal saving rates. Panel A of Table 11 reports the root of mean squared error (RMSE) of these two models for their forecast of future one-year yields. Consistent with Diebold and Li (2006), we find that the NS model performs better for longer forecast horizons, while the VAR model outperforms in 1- and 2-year ahead forecasts. Moreover, to validate our use of these model-generated forecasts as proxies of historical predictions, we also report the RMSE relative to survey forecasts as well as to in-sample forecasts. Results for 1-year ahead forecasts indicate that these out-of-sample forecasts are rather close to investors true expectations ( than today s n 1 ) statistical predictions. As such, in the subsequent analysis we consider 1 nẽt i=0 y(1) t+i estimated using both the NS and VAR(1) models. 12 Theoretically, historical survey forecast of interest rates should constitute the most effective measure of Ẽt(y(1) t+i ) (Kim and Orphanides, 2005; Chun, 2011). A lack of availability of survey data for long forecast horizons, however, largely limits the applicability of this approach to our decomposition. As to two popular sources of survey data, Survey of Professional Forecasters only reports quarterly projections for 10-year Treasury bond yields; Blue Chip Financial Forecasts do provide forecasts of 12-month T-bill rate, but with expectation horizons only up to six quarters. 23

25 Our estimates of the risk premium component the second component in the decomposition (9) are constructed from linear forecasts of excess returns, as outlined in LN. Specifically, we estimate a monthly VAR(2) model to generate multi-step ahead forecasts, where the order of the VAR is determined by information criteria (Hannan and Quinn, 1979; Schwarz, 1978). Our VAR system involves observations on excess returns and return-forecasting factors [rx (5) t, rx (4) t, rx (3) t, rx (2) t, Z t ], where implies that predictors are constructed month-bymonth, with information up to time t. We begin with a restricted model where Z t includes only CP t, and then extend the vector to include G t and other predictors. We then examine fitting errors for the cross-section of bond yields resulting from different specifications of Z t. These errors provide a measure of how well these factors span the information sets (on term premia) of financial market participants. We estimate models and make return forecasting recursively, using only the information available at the time of the forecast. The forecast period is The first column in Panel B of Table 11 shows that the CP t -based term premium identification gives a reasonably good approximation to observed 5-year yields. If we use NS model-implied forecast as the measure of the expectation component, the corresponding mean squared decomposition errors are about ) 42 basis points. Adoption of the alternative, VAR-based estimation of generates slightly higher fitting errors. 1 nẽt ( n 1 i=0 y(1) t+i Results shown in the second column address the key question of our decomposition analysis whether estimation of the premium component is more accurate conditional on the information in our SAGLasso factor. We observe significantly improved model ( fits when n 1 ) G t are included in the VAR, this result holds up for different choices of 1 nẽt i=0 y(1) t+i. Particularly, if the NS model is used to measure expected one-year rates, the mean squared decomposition errors is reduced to only 33 basis points by incorporating macroeconomic factors in premium estimation. Interestingly, the improvements in fitting cross-sectional yields are of the same magnitudes for both measures of expected short rates. It is also important to realize that the errors analyzed in Table 11 are in some respects quite small, almost comparable to those resulting from DTSMs. 14 Therefore, our results indicate that we are able to exploit the cross-sectional composition of bond yields with remarkable accuracy using an ease-to-implement procedure. Note that as different risk-neutral dynamics can generate equally good cross-sectional fits, we may use alternative measures to the root of mean square error (RMSE) to assess the performance of our decomposition exercise. To check the time-series properties of estimated historical term premium, we plot in Figure 3 the six-month moving average of estimates 13 This is longer than what is used in the out-of-sample analysis discussed in Section 3.4. The reason is to include the early-1980s recession in the forecast period so that we can obtain a more clear picture of the cyclical behavior of estimated risk premia in long-term yields. 14 For the 5-year bonds, RMSE computed from typical three-factor models usually ranges from tens to more than thirty basis points, depending on the model structure and the sample period used. 24

26 based on Z t = [ CP t, G t ]. To compare it with the statistical risk premium, Figure 3 also shows in-sample estimates of the premium component in the 5-year bond. Consistent with LN s finding, the statistical term premium is highly countercyclical, in the sense that the yield risk premium tends to rise over the course of a recession and peak just after the recession period. In contrast, the month-by-month estimated premium is less volatile and responds to business cycles to a less extent. Also, our historical premium appears very similar to that estimated by Piazzesi and Schneider (2010) with survey forecasts. For example, it moves at a frequency lower than the business cycle. It exhibits rather a trend than cycles: it spiked in early 1980s and became lower towards the end of our sample. More importantly, the largest discrepancies between historical and statistical term premia occur during and after the early 1990s and early 2000s recessions. These discrepancies are attributed by Piazzesi and Schneider (2010) to adaptive learning of investors who react slowly to shocks to the level of interest rates. To summarize, our out-of-sample estimates of term premium are much closer to investors historically perceived values than to in-sample predictions obtained real time. 4 Conclusion Although it is now believed that expected excess bond returns are time-varying and forecastable, the empirical evidence on the correlation between term premia and their macroeconomic underpinnings is mixed so far. In this paper, we reassess the predictive power of macroeconomic indicators using the supervised adaptive group lasso (SAGLasso) approach, a new model selection approach that allows us to explore the underlying structure of macroeconomic variables with respect to risk premium in bond markets. Our empirical analysis provides new and robust evidence on the explanatory power of macroeconomic fundamentals for variations in term premia, which is even stronger than previously documented in the literature. Furthermore, we find evidence of an unspanned predictor extracted from macroeconomic variables, and discuss its practical applications in finite sample analysis and in identification of the term premium component in long-term yields. Overall, our study provides further support for the implication from Ludvigson and Ng (2009) that we should look beyond observable bond yields when building term structure models, as well as predicting future returns. 25

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29 Ludvigson, S.C. and S. Ng (2009), Macro factors in bond risk premia, Review of Financial Studies, vol. 22, Ludvigson, S.C. and S. Ng (2011), A factor analysis of bond risk premia, in Handbook of Empirical Economics and Finance, (edited by Ullah, Aman and David E. A. Giles), pages , CRC Press. Nelson, C. R. and A. F. Siegel (1987), Parsimonious modeling of the yield curves, Journal of Business, vol. 60, Newey, Whitney K. and Kenneth D. West (1987), A simple, positive semi-definite, heteroskedasticity and autocorrelation consistent covariance matrix, Econometrica, vol. 55, Piazzesi, M. and M. Schneider (2010), Trend and Cycle in Bond Premia, Working Paper, Stanford University. Piazzesi, M., M. Schneider, and S. Tuzel (2007), Housing, consumption and asset pricing, Journal of Financial Economics, vol. 83, Schwarz, G. (1978), Estimating the dimension of a model, Annals of Statistics, vol. 6, Stambaugh, R.F. (1988), The information in forward rates: Implications for models of the term structure, Journal of Financial Economics, vol. 21, Stambaugh, R.F. (1999), Predictive regressions, Journal of Financial Economics, vol. 54, Stock, J.H. and M.W. Watson (2002), Forecasting using principal components from a large number of predictors, Journal of the American Statistical Association, vol. 97, Stock, J.H. and M.W. Watson (2005), Implications of dynamic factor models for VAR analysis, NBER working paper. Tibshirani, R. (1996), Regression shrinkage and selection via the lasso, Journal of the Royal Statistical Society. Series B (Methodological), pages Wright, J.H. and H. Zhou (2009), Bond risk premia and realized jump risk, Journal of Banking and Finance, vol. 33, Yuan, M. and Y. Lin (2006), Model selection and estimation in regression with grouped variables, Journal of Royal Statistical Society Series B (Statistical Methodology), vol. 68, Zou, H. (2006), The adaptive lasso and its oracle properties, Journal of the American Statistical Association, vol. 101,

30 A Macroeconomic Series Used in the Analysis Table A.1 lists macroeconomic series used in our empirical analysis. Following Ludvigson and Ng (2009), we provide the short name of each series, its mnemonic (the series label used in the source database), its transformation code, and a brief data description. The transformation codes are defined as follows. Code 1: no transformation applied to the series; 2: the first difference applied; 3: the second difference; 4: the logarithm; 5: the first difference of logarithm; and 6: the second difference of logarithm. The Ĝt column specifies whether the macroeconomic variable has a non-zero coefficient for contemporaneous and/or lagged value in the SAGLasso regression. The value of 0 under Ĝt corresponds to the contemporaneous variable and the value of 1 through 6 denotes corresponding lagged values. B Supervised Adaptive Group Lasso Method We first briefly review the group lasso (Yuan and Lin, 2006). We begin with the following liner model Y = Xβ 0 + e (12) where e is assumed to be a T -dimensional vector of iid errors (we will relax this assumption later). The main assumption of the Group Lasso is that some subvectors of the true coefficients β 0 are zero. we denote by h H 1 = {h : βh 0 0} the unknown index set of non-zero subvectors of β 0. Hence, the Group Lasso involves identifying H 1 and estimating β 0. The method is usually implemented by estimating the following restrictive form { } min β R N y Xβ 2 + λ h β h. (13) Note that expression (13) reduces to the Lasso when H = N and each h corresponds to the 1-dimensional subspace of R T spanned by the corresponding column of the design matrix X. In our implementation, we consider the general Group Lasso and more specifically, the adaptive group lasso as follows: { } min β R N y Xβ 2 + λ h w h β h Next, we describe the Supervised Adaptive Group Lasso (SAGLasso) method for predicting excess returns proposed earlier in Section The method consists of two steps. Step I: For cluster h H, compute ˆβ h the cluster-wise Adaptive Lasso estimate of β h. Namely, { } ˆβ h = argmin β h arx X h β h 2 + j 29 λ h ŵ hj β h j (14), (15)

31 where arx is a vector of average excess bond returns across maturity and ŵ hj the j-th OLS component of ŵ h, the vector of the (adaptive) weights. Zou (2006) recommends using ˆβ RID to construct ŵ h. As collinearity is a concern in our case, we set ŵ h = 1/ ˆβ h γ h, where ˆβ h RID is the best ridge regression fit of arx on X h. That is, for cluster h we only use macroeconomic variables within that cluster to construct predictive models. The optimal pairs of (γ h, λ h ) are determined using two-dimensional cross-validations. It is worth noting that tuning parameters λ h are selected for each cluster separately in order to have different degrees of regularization for different clusters. This flexibility allows us to uncover subtle structures that otherwise will be missed when applying the (adaptive) lasso method to all the series/clusters at the same time. Notice that for each cluster h H, the adaptive lasso ˆβ h has only a small number of nonzero components. Let β h = ˆβ h \ 0, the vector of nonzero estimated components of ˆβ h given by the cluster-wise model (15), and denote the corresponding part of X h by X h. In our case, a typical cluster size (dim(x h )) of 80 variables may reduce to a dim( X h ) of Namely, the number of macroeconomic measures selected in Step I is significantly smaller than the original number to begin with. Step II: Construct the joint predictive model under the Group Lasso constraint as the following: { } ˆβ = argmin β arx Xβ 2 + λ h H w h β h, (16) where X is formed by concatenating the design matrices X h. λ is also chosen by cross validation. With λ, estimates of some components of β h s can be exactly zero. Following Yuan and Lin (2006), we obtain the solution to the Group Lasso problem (16) efficiently by using a modified least angle regression selection (LARS) algorithm of Efron, Hastie, Johnstone, and Tibshirani (2004) (and using 2-fold cross validation to determine tuning parameters). Note that the SAGLasso approach differs from the supervised principal component analysis (SPCA) another two-step supervised learning approach proposed by Bair, Hastie, Paul, and Tibshirani (2006) in a biological setting, which has been applied to inflation forecasts in Bai and Ng (2008). 15 For instance, the former takes into account the underlying cluster structure of candidate variables whereas SPCA does not and considers all the candidates simultaneously. Also, variables selected in SPCA are the principal components whose economic interpretations may not be obvious even though they may have satisfactory prediction performance. As shown in Section 3.3, factors identified under SAGLasso are easier to interpret. 15 Bai and Ng (2008) also implement the LARS although the purpose of their paper is to forecast inflation. Gibson and Pritsker (2000) use partial least squares to choose risk factors of fixed-income portfolios. Goto and Xu (2010) apply the graphical lasso to portfolio selection. 30

32 C Data-Generating Processes In the literature, there are two major methods used to generate simulation data for finitesample inference of macroeconomic variables forecasting power. One approach is to construct a time-series model for the log yields and rerun the predictive regressions. 16 Residuals are bootstrapped to form the empirical distribution. Note that for this method the independent variables (macro factors) are not simulated and enter the predictive regression as their actual values. The other one is based on a DTSM where state variables are also simulated from the model with parameters estimated from the actual sample data. Notably, Duffee (2007) constructs a class of discrete-time DTSMs to make finite-sample inference for both restrictive and general null hypotheses. For conservative reasons, we follow both types of simulation procedures and compare the empirical properties of their generated regression statistics. Only regressions on four group factors ĝ ht are examined here because the number of latent term premia factors must be equal to the number of macro factors in the DTSM. 17 For the general null hypothesis, along the spirit of the first approach we run a VAR(12) for the yield process that imposes a single unit root (one common trend). As to the second approach we construct a DTSM resembling Duffee s in that macro factors determine the dynamics of short rate but not the price of risk. In particular, we specify the short rate as an affine function of a vector of both macroeconomic and latent factors as follows: f t = ( m t, x t), where mt is a 4 1 vector of macroeconomic factors (to be estimated by supervised principle components), and x t a 4 1 vector of latent factors. The structure imposed by Duffee restricts the latent factors from driving the dynamics of short rate r, r t = δ 0 + δ 1f t, where δ 1 = (δ m, ) (17) In other words, their only role here is to drive the risk compensation for corresponding macro factors, [ ] [ ] λ0m λ1m I λ 0 = and λ 0 1 = 4 4, (18) where λ 0 and λ 1 are parameters that characterize the market prices of risk Σλ t = λ 0 + λ 1 f t. To distinguishing between macro and non-macro influences on term premia, independence between the macro and term premia factors is also crucial. We assume that the state vector 16 For example, CP (2005) run an unconstrained VAR(12) model of all 5 yields, and LN (2008) use an MA(12) process to test the restrictive null. 17 If the single factor Ĝ is used, the resulting model with only one macro factor and one latent factor may not fully capture the joint variation in the macro variables and yield data. 31

33 follows a Gaussian VAR(1) process f t = µ + Φf t 1 + Σɛ t [ ] [ ] [ ] [ ] [ ] µm Φm 0 = mt 1 Σm ɛ m t Φ x x t Σ x ɛ x t (19) where shocks ɛ t N(0, 1). With the restriction imposed on Φ and Σ, the evolution of the latent term premia factors depends only on the latent factors. Moreover, innovations in m t, and thus innovations in the short rate, are by construction orthogonal to the latent state vector. As such, the compensation required by investors depends on latent factors that evolve independently of the macro factors but affect the risk compensation for macro factors through 1-to-1 mappings. 18 As a result, the functional form of λ t can be written as [ ] λ0m + λ Σλ t = 1m m t + x t. (20) The risk compensation depends on the macro factors only through λ 1m. Therefore, with the restriction λ 1m = 0, shocks to the macroeconomic factors have no impact on expected excess returns at all leads and lags, and thus the model corresponds to the general null that excess bond returns are stochastic and persistent, but independent of the macroeconomy. Otherwise, we cannot recover latent factors solely from the market price of risk. It follows that zero-coupon bond prices are exponential affine functions of the state variables (Duffie and Kan, 1996). When taking yield data to the model, we assume that all bonds are priced with error, and thus use Kalman filters to obtain fitted states in maximum likelihood estimation (Duffee and Stanton, 2004). The parametrization of expected excess returns for the model under general null requires 61 free parameters, including five measurement error parameters. Some of the initial values (those for Φ m and Σ m ) are set to those from OLS estimation of the VAR(1), and analytic derivatives are used in the derivative-based optimization routine. Table A.2 shows that the two simulation procedures yield almost indistinguishable empirical distributions for our test statistics. A close scrutiny of our simulation data suggests the reason from this consistent result. First of all, as the group factors estimated from actual data are highly persistent variables (the most persistent factor has a monthly AR(1) coefficient of ), the Gaussian VAR(1) process, specified in the term structure model, may be a good approximation for their joint dynamics. More important, as both data-generating processes satisfy the general null of independence, what is relevant for the test statistics is the time series properties of the excess returns, instead of how the returns are generated (VAR or term structure model), because that is how the distribution of our test statistics are determined. Indeed, in the two different settings that we use to generate excess bond Latent factor i affects only the risk compensation for macro factor i. 32

34 returns, they give similar time series properties (when we plot the average ACFs for the simulated excess returns generated by two simulation model, they look very close). Though not reported here, the finite-sample distributions generating by these two approach under the restrictive null are also qualitatively identical. Therefore, for our smallsample inference we employ a vector moving average (VMA) model of order 12, namely, a VMA(12) model, to form the bootstrap samples under the restrictive null hypothesis, and use the eight-factor DTSM specified earlier for the general null. The use of VMA model is due to the MA(12) error structure in the series of excess returns, which induced in the construction of continuously compounded annual returns. For the DTSM-based simulations, an initial draw of the state variables is taken from their unconditional multivariate normal distribution. Subsequent draws use their conditional multivariate normal distribution. The finite-sample distributions are constructed based on 50,000 Monte Carlo simulations, and the length of each simulation is 528 months, the same as the length of the full sample used in our empirical analysis. 33

35 Table A.1 Data Description Series No. Group Mnemonic Description Short Name tran Ĝt 1 1 a0m052 Personal income (AR, bil. chain 2000 $) PI A0M051 Personal income less transfer payments (AR, bil. chain 2000 $) PI less transfers A0M224R Real Consumption (AC) A0m224/gmdc Consumption A0M057 Manufacturing and trade sales (mil. Chain 1996 $) M & T sales A0M059 Sales of retail stores (mil. Chain 2000 $) Retail sales IPS10 INDUSTRIAL PRODUCTION INDEX - TOTAL INDEX IP: total IPS11 INDUSTRIAL PRODUCTION INDEX - PRODUCTS, TOTAL IP: products IPS299 INDUSTRIAL PRODUCTION INDEX - FINAL PRODUCTS IP: final prod IPS12 INDUSTRIAL PRODUCTION INDEX - CONSUMER GOODS IP: cons gds IPS13 INDUSTRIAL PRODUCTION INDEX - DURABLE CONSUMER GOODS IP: cons dble IPS18 INDUSTRIAL PRODUCTION INDEX - NONDURABLE CONSUMER GOODS iip:cons nondble IPS25 INDUSTRIAL PRODUCTION INDEX - BUSINESS EQUIPMENT IP:bus eqpt IPS32 INDUSTRIAL PRODUCTION INDEX - MATERIALS IP: matls IPS34 INDUSTRIAL PRODUCTION INDEX - DURABLE GOODS MATERIALS IP: dble mats IPS38 INDUSTRIAL PRODUCTION INDEX - NONDURABLE GOODS MATERIALS IP:nondble mats IPS43 INDUSTRIAL PRODUCTION INDEX - MANUFACTURING (SIC) IP: mfg IPS307 INDUSTRIAL PRODUCTION INDEX - RESIDENTIAL UTILITIES IP: res util IPS306 INDUSTRIAL PRODUCTION INDEX - FUELS IP: fuels PMP NAPM PRODUCTION INDEX (PERCENT) NAPM prodn A0m082 Capacity Utilization (Mfg) Cap util LHEL INDEX OF HELP-WANTED ADVERTISING IN NEWSPAPERS (1967=100;SA) Help wanted indx LHELX EMPLOYMENT: RATIO; HELP-WANTED ADS:NO. UNEMPLOYED CLF Help wanted/emp LHEM CIVILIAN LABOR FORCE: EMPLOYED, TOTAL (THOUS.,SA) Emp CPS total LHNAG CIVILIAN LABOR FORCE: EMPLOYED, NONAGRIC.INDUSTRIES (THOUS.,SA) Emp CPS nonag LHUR UNEMPLOYMENT RATE: ALL WORKERS, 16 YEARS & OVER (%,SA) U: all LHU680 UNEMPLOY.BY DURATION: AVERAGE(MEAN)DURATION IN WEEKS (SA) U: mean duration LHU5 UNEMPLOY.BY DURATION: PERSONS UNEMPL.LESS THAN 5 WKS (THOUS.,SA) U 5 wks LHU14 UNEMPLOY.BY DURATION: PERSONS UNEMPL.5 TO 14 WKS (THOUS.,SA) U 5-14 wks LHU15 UNEMPLOY.BY DURATION: PERSONS UNEMPL.15 WKS + (THOUS.,SA) U 15+ wks LHU26 UNEMPLOY.BY DURATION: PERSONS UNEMPL.15 TO 26 WKS (THOUS.,SA) U wks LHU27 UNEMPLOY.BY DURATION: PERSONS UNEMPL.27 WKS + (THOUS,SA) U 27+ wks A0M005 Average weekly initial claims, unemploy. insurance (thous.) UI claims CES002 EMPLOYEES ON NONFARM PAYROLLS - TOTAL PRIVATE Emp: total CES003 EMPLOYEES ON NONFARM PAYROLLS - GOODS-PRODUCING Emp: gds prod CES006 EMPLOYEES ON NONFARM PAYROLLS - MINING Emp: mining CES011 EMPLOYEES ON NONFARM PAYROLLS - CONSTRUCTION Emp: const CES015 EMPLOYEES ON NONFARM PAYROLLS - MANUFACTURING Emp: mfg CES017 EMPLOYEES ON NONFARM PAYROLLS - DURABLE GOODS Emp: dble gds CES033 EMPLOYEES ON NONFARM PAYROLLS - NONDURABLE GOODS Emp: nondbles CES046 EMPLOYEES ON NONFARM PAYROLLS - SERVICE-PROVIDING Emp: services CES048 EMPLOYEES ON NONFARM PAYROLLS - TRADE, TRANSPORTATION, AND UTILITIES Emp: TTU CES049 EMPLOYEES ON NONFARM PAYROLLS - WHOLESALE TRADE Emp: wholesale CES053 EMPLOYEES ON NONFARM PAYROLLS - RETAIL TRADE Emp: retail CES088 EMPLOYEES ON NONFARM PAYROLLS - FINANCIAL ACTIVITIES Emp: FIRE 5 0,1,2,3,4, CES140 EMPLOYEES ON NONFARM PAYROLLS - GOVERNMENT Emp: Govt CES151 AVERAGE WEEKLY HOURS OF PRODUCTION OR NONSUPERVISORY WORKERS ON PRIVATE NONFAR Avg hrs CES155 AVERAGE WEEKLY HOURS OF PRODUCTION OR NONSUPERVISORY WORKERS ON PRIVATE NONFAR Overtime: mfg aom001 Average weekly hours, mfg. (hours) Avg hrs: mfg PMEMP NAPM EMPLOYMENT INDEX (PERCENT) NAPM empl 1 0,3, HSFR HOUSING STARTS:NONFARM( );TOTAL FARM & NONFARM(1959-)(THOUS.,SA HStarts: Total HSNE HOUSING STARTS:NORTHEAST (THOUS.U.)S.A. HStarts: NE 4 5, HSMW HOUSING STARTS:MIDWEST(THOUS.U.)S.A. HStarts: MW 4 2,3,4,5, HSSOU HOUSING STARTS:SOUTH (THOUS.U.)S.A. HStarts: South 4 0, HSWST HOUSING STARTS:WEST (THOUS.U.)S.A. HStarts: West HSBR HOUSING AUTHORIZED: TOTAL NEW PRIV HOUSING UNITS (THOUS.,SAAR) BP: total HSBNE HOUSES AUTHORIZED BY BUILD. PERMITS:NORTHEAST(THOU.U.)S.A BP: NE HSBMW HOUSES AUTHORIZED BY BUILD. PERMITS:MIDWEST(THOU.U.)S.A. BP: MW HSBSOU HOUSES AUTHORIZED BY BUILD. PERMITS:SOUTH(THOU.U.)S.A. BP: South HSBWST HOUSES AUTHORIZED BY BUILD. PERMITS:WEST(THOU.U.)S.A. BP: West PMI PURCHASING MANAGERS INDEX (SA) PMI PMNO NAPM NEW ORDERS INDEX (PERCENT) NAPM new ordrs PMDEL NAPM VENDOR DELIVERIES INDEX (PERCENT) NAPM vendor del PMNV NAPM INVENTORIES INDEX (PERCENT) NAPM Invent A0M008 Mfrs new orders, consumer goods and materials (bil. chain 1982 $) Orders: cons gds A0M007 Mfrs new orders, durable goods industries (bil. chain 2000 $) Orders: dble gds A0M027 Mfrs new orders, nondefense capital goods (mil. chain 1982 $) Orders: cap gds 5 34

36 67 4 A1M092 Mfrs unfilled orders, durable goods indus. (bil. chain 2000 $) Unf orders: dble A0M070 Manufacturing and trade inventories (bil. chain 2000 $) M & T invent A0M077 Ratio, mfg. and trade inventories to sales (based on chain 2000 $) M & T invent/sales FM1 MONEY STOCK: M1(CURR,TRAV.CKS,DEM DEP,OTHER CK ABLE DEP)(BIL$,SA) M FM2 MONEY STOCK:M2(M1+O NITE RPS,EURO$,G/P&B/D MMMFS&SAV&SM TIME DEP(BIL$, M FM3 MONEY STOCK: M3(M2+LG TIME DEP,TERM RP S&INST ONLY MMMFS)(BIL$,SA) M FM2DQ MONEY SUPPLY - M2 IN 1996 DOLLARS (BCI) M2 (real) FMFBA MONETARY BASE, ADJ FOR RESERVE REQUIREMENT CHANGES(MIL$,SA) MB FMRRA DEPOSITORY INST RESERVES:TOTAL,ADJ FOR RESERVE REQ CHGS(MIL$,SA) Reserves tot FMRNBA DEPOSITORY INST RESERVES:NONBORROWED,ADJ RES REQ CHGS(MIL$,SA) Reserves nonbor FCLNQ COMMERCIAL & INDUSTRIAL LOANS OUSTANDING IN 1996 DOLLARS (BCI) C&I loans FCLBMC WKLY RP LG COM L BANKS:NET CHANGE COM L & INDUS LOANS(BIL$,SAAR) C&I loans CCINRV CONSUMER CREDIT OUTSTANDING - NONREVOLVING(G19) Cons credit-nonrevolving A0M095 Ratio, consumer installment credit to personal income (pct.) Inst cred/pi FSPCOM S&P S COMMON STOCK PRICE INDEX: COMPOSITE ( =10) S&P FSPIN S&P S COMMON STOCK PRICE INDEX: INDUSTRIALS ( =10) S&P: indust FSDXP S&P S COMPOSITE COMMON STOCK: DIVIDEND YIELD (% PER ANNUM) S&P div yield FSPXE S&P S COMPOSITE COMMON STOCK: PRICE-EARNINGS RATIO (%,NSA) S&P PE ratio FYFF INTEREST RATE: FEDERAL FUNDS (EFFECTIVE) (% PER ANNUM,NSA) FedFunds CP90 Cmmercial Paper Rate (AC) Commpaper FYGM3 INTEREST RATE: U.S.TREASURY BILLS,SEC MKT,3-MO.(% PER ANN,NSA) 3 mo T-bill FYGM6 INTEREST RATE: U.S.TREASURY BILLS,SEC MKT,6-MO.(% PER ANN,NSA) 6 mo T-bill FYGT1 INTEREST RATE: U.S.TREASURY CONST MATURITIES,1-YR.(% PER ANN,NSA) 1 yr T-bond FYGT5 INTEREST RATE: U.S.TREASURY CONST MATURITIES,5-YR.(% PER ANN,NSA) 5 yr T-bond FYGT10 INTEREST RATE: U.S.TREASURY CONST MATURITIES,10-YR.(% PER ANN,NSA) 10 yr T-bond FYAAAC BOND YIELD: MOODY S AAA CORPORATE (% PER ANNUM) Aaabond FYBAAC BOND YIELD: MOODY S BAA CORPORATE (% PER ANNUM) Baa bond scp90 cp90-fyff CP-FF spread sfygm3 fygm3-fyff 3 mo-ff spread sfygm6 fygm6-fyff 6 mo-ff spread sfygt1 fygt1-fyff 1 yr-ff spread sfygt5 fygt5-fyff 5 yr-ffspread sfygt10 fygt10-fyff 10yr-FF spread sfyaaac fyaaac-fyff Aaa-FF spread sfybaac fybaac-fyff Baa-FF spread EXRUS UNITED STATES;EFFECTIVE EXCHANGE RATE(MERM)(INDEX NO.) Ex rate: avg 5 5, EXRSW FOREIGN EXCHANGE RATE: SWITZERLAND (SWISS FRANC PER U.S.$) Ex rate: Switz EXRJAN FOREIGN EXCHANGE RATE: JAPAN (YEN PER U.S.$) Ex rate: Japan EXRUK FOREIGN EXCHANGE RATE: UNITED KINGDOM (CENTS PER POUND) Ex rate: UK EXRCAN FOREIGN EXCHANGE RATE: CANADA (CANADIAN P ERU.S.) EX rate: Canada PWFSA PRODUCER PRICE INDEX: FINISHED GOODS (82=100,SA) PPI: fin gds PWFCSA PRODUCER PRICE INDEX:FINISHED CONSUMER GOODS (82=100,SA) PPI: cons gds PWIMSA PRODUCER PRICE INDEX:INTERMED MAT.SUPPLIES & COMPONENTS(82=100,SA) PPI: int matls PWCMSA PRODUCER PRICE INDEX:CRUDE MATERIALS (82=100,SA) PPI: crude matls PSCCOM SPOT MARKET PRICE INDEX:BLS & CRB: ALL COMMODITIES(1967=100) Commod: spot price PSM99Q INDEX OF SENSITIVE MATERIALS PRICES (1990=100)(BCI-99A) Sens matls price PMCP NAPM COMMODITY PRICES INDEX (PERCENT) NAPM com price 1 0,4,5, PUNEW CPI-U: ALL ITEMS (82-84=100,SA) CPI-U: all PU83 CPI-U: APPAREL & UPKEEP (82-84=100,SA) CPI-U: apparel PU84 CPI-U: TRANSPORTATION (82-84=100,SA) CPI-U: transp PU85 CPI-U: MEDICAL CARE (82-84=100,SA) CPI-U: medical PUC CPI-U: COMMODITIES (82-84=100,SA) CPI-U: comm PUCD CPI-U: DURABLES (82-84=100,SA) CPI-U: dbles PUS CPI-U: SERVICES (82-84=100,SA) CPI-U: services PUXF CPI-U: ALL ITEMS LESS FOOD (82-84=100,SA) CPI-U: ex food PUXHS CPI-U: ALL ITEMS LESS SHELTER (82-84=100,SA) CPI-U: ex shelter PUXM CPI-U: ALL ITEMS LESS MEDICAL CARE (82-84=100,SA) CPI-U: ex med GMDC PCE,IMPL PR DEFL:PCE (1987=100) PCE defl GMDCD PCE,IMPL PR DEFL:PCE; DURABLES (1987=100) PCE defl: dlbes GMDCN PCE,IMPL PR DEFL:PCE; NONDURABLES (1996=100) PCE defl: nondble GMDCS PCE,IMPL PR DEFL:PCE; SERVICES (1987=100) PCE defl: services CES275 AVERAGE HOURLY EARNINGS OF PRODUCTION OR NONSUPERVISORY WORKERS ON PRIVATE NO AHE: goods CES277 AVERAGE HOURLY EARNINGS OF PRODUCTION OR NONSUPERVISORY WORKERS ON PRIVATE NO AHE: const CES278 AVERAGE HOURLY EARNINGS OF PRODUCTION OR NONSUPERVISORY WORKERS ON PRIVATE NO AHE: mfg HHSNTN U. OF MICH. INDEX OF CONSUMER EXPECTATIONS(BCD-83) Consumer expect 2 35

37 Table A.2: Finite-Sample Distributions Based on Different Data Generating Processes This table summarizes results from 50,000 Monte Carlo simulations based on a VAR (12) process or a macro-finance term structure model. Both data-generating processes satisfy the general null hypothesis that expected excess bond returns are time-varying but independent of the macroeconomy. And 528 months of data are generated for each simulation. Excess returns are regressed on the month-t values of four group macroeconomic factors. The in-sample test statistic is a Wald test (with the New-West correction with 18 lags) of the hypothesis that the coefficients are jointly zero. The table reports the empirical rejection rate using the five percent critical value for a χ 2 (4) distribution, as well as the finite sample five-percent critical value. Similar statistics are reported for the out-of-sample ENC-REG test of Ericsson (1992) and ENC-NEW test of Clark and McCracken (2001). The ENC-REG test has an asymptotic N(0,1) distribution and the 95% asymptotic critical value is for the ENC-NEW test statistic. VAR (12) Term Structure Model Type of regression Test Statistic Maturity (yr) Rejection rate True 5% CV Rejection rate True 5% CV Wald In-sample Out-of-Sample Ericsson Clark-McCracken

38 Table 1: Predictability of the CP and LN Factors The return to an n-year zero-coupon Treasury bond from month t to month t+12 less the month-t yield on a one-year Treasury bond is regressed on ĈP t, the Cochrane-Piazzesi (2005) predictor (a linear combination of forward rates), and/or LN t, the Ludvigson and Ng (2009) factor (a linear combination of static factors estimated as the fitted values from OLS). Rows labeled HH report test statistics computed using standard errors with the Hansen-Hodrick GMM correction for overlap. Rows labeled NW report test statistics computed using standard errors with 18 Newey-West lags to correct serial correlation. Column labeled Joint Test reports Wald tests of the hypothesis that all coefficients equal zero. Asymptotic p-values, based on a χ 2 (1) distribution, are in brackets. Ljung-Box Q statistic is used to test autocorrelation in the error term in the multivariate regressions. The sample spans the period January 1964 to December maturity ĈP t R 2 LN t R 2 ĈP t LN t R2 Joint P-val Q- P-val (yr) Test Test [0.00] HH (5.851) (6.833) (4.277) (3.080) [0.00] NW (6.402) (7.349) (4.697) (3.474) [0.00] [0.00] HH (5.439) (6.805) (4.393) (3.137) [0.00] NW (6.011) (7.295) (4.830) (3.548) [0.00] [0.00] HH (5.419) (6.949) (4.335) (3.226) [0.00] NW (6.043) (7.295) (4.776) (3.669) [0.00] [0.00] HH (5.065) (7.015) (4.367) (2.959) [0.00] NW (5.638) (7.426) (4.792) (3.361) [0.00] 37

39 Table 2: Predictability of SAGLasso Group Factors The return to an n-year zero-coupon Treasury bond from month t to month t + 12 less the month-t yield on a one-year Treasury bond is regressed on ĝ it s, the four macroeconomic group factors estimated by the Group Lasso method. ĈP t denotes the Cochrane-Piazzesi (2005) forward-rate factor. Rows labeled HH report test statistics computed using standard errors with the Hansen-Hodrick GMM correction for overlap. Rows labeled NW report test statistics computed using standard errors with 18 Newey-West lags to correct serial correlation. Column labeled Joint Test reports Wald tests of the hypothesis that all coefficients equal zero. Asymptotic p-values, based on a χ 2 (4) distribution, are in brackets. The sample spans the period January 1964 to December Panel A: Univariate predictive regressions on individual group factors ĝ it maturity (yr) ĝ 1t ĝ 2t ĝ 3t ĝ 4t HH (4.496) (6.115) (5.537) (2.736) NW (5.040) (6.873) (5.916) (3.025) R HH (4.482) (5.326) (5.313) (2.563) NW (4.980) (5.985) (5.735) (2.843) R HH (4.364) (5.099) (5.324) (2.494) NW (4.824) (5.718) (5.751) (2.768) R HH (4.331) (4.856) (5.482) (2.365) NW (4.767) (5.428) (5.904) (2.622) R Panel B: Predictive regressions of excess returns on group factors {ĝ it } and ĈP t. maturity (yr) ĝ 1t ĝ 2t ĝ 3t ĝ 4t ĈP t R2 Joint Test P-val HH (1.415) (2.509) (3.938) (0.051) [0.00] NW (1.572) (2.765) (4.308) (0.057) [0.00] HH (1.490) (1.858) (4.057) (0.159) [0.00] NW (1.656) (2.062) (4.371) (0.176) [0.00] HH (1.346) (1.613) (4.189) (0.168) [0.00] NW (1.500) (1.799) (4.483) (0.186) [0.00] HH (1.333) (1.472) (4.426) (0.081) [0.00] NW (1.483) (1.641) (4.702) (0.089) [0.00] HH (1.992) (2.639) (2.716) (-0.263) (3.531) [0.00] NW (2.171) (2.832) (2.928) (-0.287) (3.896) [0.00] HH (2.129) (1.808) (2.639) (-0.170) (3.494) [0.00] NW (2.318) (1.959) (2.752) (-0.185) (3.839) [0.00] HH (2.016) (1.469) (2.840) (-0.203) (3.686) [0.00] NW (2.202) (1.603) (2.862) (-0.221) (4.082) [0.00] HH (1.917) (1.330) (3.449) (-0.255) (3.093) [0.00] NW (2.093) (1.455) (3.429) (-0.278) (3.418) [0.00] 38

40 Table 3: Predictability of the SAGLasso Single Factor The return to an n-year zero-coupon Treasury bond from month t to month t + 12 less the month-t yield on a one-year Treasury bond is regressed on Ĝt (the single predictor estimated by SAGLasso), ĈP t (the Cochrane-Piazzesi (2005) return-forecasting factor), and/or LN t (the Ludvigson and Ng (2009) factor, a linear combination of static factors estimated as the fitted values from OLS). Rows labeled HH report test statistics computed using standard errors with the Hansen-Hodrick GMM correction for overlap. Rows labeled NW report test statistics computed using standard errors with 18 Newey-West lags to correct serial correlation. Column labeled Joint Test reports Wald tests of the hypothesis that all coefficients equal zero. Asymptotic p-values, based on a χ 2 (1) distribution, are in brackets. The sample spans the period January 1964 to December maturity Ĝ t ĈP t LN t R2 Joint Test P-val HH (10.797) [0.000] NW (11.519) [0.000] HH (9.566) [0.000] NW (10.210) [0.000] HH (9.070) [0.000] NW (9.665) [0.000] HH (8.909) [0.000] NW (9.441) [0.000] HH (7.317) (2.170) [0.000] NW (8.110) (2.445) [0.000] HH (6.491) (2.300) [0.000] NW (7.196) (2.599) [0.000] HH (6.118) (2.526) [0.000] NW (6.722) (2.870) [0.000] HH (6.523) (2.230) [0.000] NW (7.081) (2.527) [0.000] HH (4.544) (1.230) [0.000] NW (5.111) (1.340) [0.000] HH (4.392) (1.405) [0.000] NW (4.918) (1.535) [0.000] HH (4.057) (1.356) [0.000] NW (4.535) (1.491) [0.000] HH (4.089) (1.291) [0.000] NW (4.564) (1.422) [0.000] 39

41 Table 4: Predictability of the Output Gap and SAGLasso Factors. The dependent variable is the return to an n-year zero-coupon Treasury bond from month t to month t + 12 less the month-t yield on a one-year Treasury bond. The independent variables include the output gap gap measured as the deviations of the log of industrial production index from a quadratic and linear trend as in Cooper and Priestley (2009), ĈP t the Cochrane-Piazzesi (2005) factor orthogonalized relative to gap, Ĝ t the SAGLasso orthogonalized relative to gap, and ĝ 1t the first macroeconomic group factor identified using the Group Lasso. Rows labeled NW report test statistics computed using standard errors with 18 Newey-West lags to correct serial correlation. Results based on the Hansen-Hodrick GMM correction for overlap are similar and not reported here for brevity. The sample spans the period January 1964 to December maturity gap t 1 ĈP t Ĝ t ĝ 1t R NW (-2.480) NW (-2.142) NW (-2.211) NW (-2.224) NW (-2.812) (5.241) NW (-2.482) (5.630) NW (-2.594) (5.832) NW (-2.606) (5.409) NW (-3.681) (11.356) NW (-3.159) (10.539) NW (-3.127) (10.309) NW (-3.094) (10.193) NW (-1.309) (3.741) NW (-0.959) (3.695) NW (-1.048) (3.519) NW (-1.062) (3.442) 40

42 Table 5: Comparison between the SAGLasso factor and Duffee s Yield Factors The return to an n-year zero-coupon Treasury bond from month t to month t + 12 less the month-t yield on a one-year Treasury bond is regressed on Ĥit, i = 1,..., 5, Duffee (2011) s five latent yield factors (the 5th factor being the hidden factor) estimated using Kalman filtering, alone with Ĝt, the single predictor estimated by SAGLasso. Rows labeled HH report test statistics computed using standard errors with the Hansen- Hodrick GMM correction for overlap. Rows labeled NW report test statistics computed using standard errors with 18 Newey-West lags to correct serial correlation. Column labeled Joint Test reports Wald tests of the hypothesis that all coefficients equal zero. Asymptotic p-values, based on a χ 2 (1) distribution, are in brackets. The sample spans the period January 1964 to December maturity Ĝ t Ĥ 1t Ĥ 2t Ĥ 3t Ĥ 4t Ĥ 5t R2 Joint Test P-val HH (2.228) (3.764) (-1.309) (2.570) (3.689) [0.00] NW (2.488) (4.099) (-1.461) (2.627) (3.652) [0.00] HH (1.608) (3.655) (-0.965) (2.329) (4.699) [0.00] NW (1.802) (4.052) (-1.075) (2.368) (4.543) [0.00] HH (1.346) (3.842) (-0.961) (2.151) (5.017) [0.00] NW (1.513) (4.297) (-1.059) (2.193) (4.797) [0.00] HH (1.145) (3.961) (-0.750) (1.745) (4.582) [0.00] NW (1.287) (4.446) (-0.821) (1.794) (4.354) [0.00] HH (6.999) (1.606) (0.677) (-0.755) (1.494) (2.480) [0.00] NW (7.633) (1.785) (0.747) (-0.717) (1.567) (2.608) [0.00] HH (6.421) (0.973) (0.881) (-0.019) (1.274) (3.457) [0.00] NW (6.997) (1.080) ( 0.981) (-0.018) (1.312) (3.560) [0.00] HH (5.771) (0.739) (1.291) (0.035) (1.126) (3.611) [0.00] NW (6.234) (0.823) ( 1.444) (0.031) (1.139) (3.717) [0.00] HH (5.730) (0.555) (1.441) (0.420) (0.757) (3.197) [0.00] NW (6.152) (0.617) ( 1.615) (0.378) (0.766) (3.282) [0.00] 41

43 Table 6: Results on the Predictability of the Cieslak and Povala (2010) Cycle Factor The return to an n-year zero-coupon Treasury bond from month t to month t+12 less the month-t yield on a one-year Treasury bond is regressed on the Cieslak and Povala (2010) cycle factor (ĉf t), the SAGLasso single factor (Ĝt), group factors, ĝ it, i = 1,..., 4, and Duffee (2011) s five latent yield factors (the 5th factor being the hidden factor), Ĥit, i = 1,..., 5. Rows labeled HH report test statistics computed using standard errors with the Hansen-Hodrick GMM correction for overlap. Rows labeled NW report test statistics computed using standard errors with 18 Newey-West lags to correct serial correlation. Column labeled Joint Test reports Wald tests of the hypothesis that all coefficients equal zero. Asymptotic p-values, based on a χ 2 (1) distribution, are in brackets. The sample spans the period January 1964 to December mat. (yr) ĉf t R 2 Ĝ t ĉf t R2 Panel A: The Cycle Factor vs. Macro Factors ĝ 1t ĝ 2t ĝ 3t ĝ 4t ĉf t R HH (5.013) (7.686) (3.241) (1.187) (3.279) (1.035) (0.157) (4.353) NW (5.634) (8.209) (3.497) (1.309) (3.541) (1.184) (0.168) (4.776) HH (5.170) (7.093) (3.680) (1.259) (2.469) (1.075) (0.288) (4.442) NW (5.815) (7.567) (3.971) (1.399) (2.704) (1.209) (0.311) (4.890) HH (5.253) (6.564) (3.905) (1.095) (2.123) (1.320) (0.296) (4.782) NW (5.920) (7.010) (4.227) (1.223) (2.347) (1.454) (0.321) (5.210) HH (5.451) (6.203) (4.047) (1.069) (1.963) (1.504) (0.185) (4.742) NW (6.124) (6.614) (4.368) (1.194) (2.171) (1.638) (0.201) (5.129) Panel B: The Cycle Factor vs. Yield Factors maturity Ĥ 1t Ĥ 2t Ĥ 3t Ĥ 4t Ĥ 5t ĉf t R HH (1.725) (0.902) (1.040) (0.971) (3.203) (3.327) NW (1.928) (0.966) (1.167) (1.018) (3.083) (3.565) HH (1.062) (0.848) (0.685) (0.626) (4.169) (3.540) NW (1.192) (0.921) (0.770) (0.661) (3.913) (3.776) HH (0.824) (1.182) (0.660) (0.483) (4.342) (3.396) NW (0.927) (1.291) (0.739) (0.512) (4.050) (3.595) HH (0.610) (1.195) (0.438) (0.149) (3.797) (3.473) NW (0.687) (1.313) (0.489) (0.159) (3.536) (3.671) 42

44 Table 7: Regressions of Annual Excess Bond Returns on Various Predictors over The return to an n-year zero-coupon Treasury bond from month t to month t + 12 less the month-t yield on a one-year Treasury bond is regressed on the jump mean (JMt), the SAGLasso factor (Ĝt), Duffee (2011) s five latent yield factors (Ĥit, i = 1,..., 5), and the cycle factor (ĉf t ). Rows labeled HH report test statistics computed using standard errors with the Hansen-Hodrick GMM correction for overlap. Rows labeled NW report test statistics computed using standard errors with 18 Newey-West lags to correct serial correlation. The sample spans the period August 1984 to December Panel A: Predictive regressions on the jump mean, SAGLasso, and yield factors. maturity Ĝt R2 JMt Ĝt R2 JMt Ĥ1t Ĥ2t Ĥ3t Ĥ4t Ĥ5t R HH (5.438) (-2.784) (4.353) (-4.526) (-1.130) (2.218) (1.539) (3.135) (4.057) NW (5.791) (-2.945) (4.737) (-4.709) (-1.148) (1.929) (1.639) (3.398) (4.605) HH (5.688) (-4.007) (5.009) (-5.196) (-1.405) (1.750) (0.982) (3.830) (3.874) NW (5.785) (-4.011) (5.198) (-5.253) (-1.448) (1.538) (1.045) (4.123) (4.401) HH (6.107) (-5.044) (5.831) (-5.312) (-1.324) (1.636) (0.789) (4.795) (3.879) NW (5.963) (-4.628) (5.733) (-5.337) (-1.372) (1.480) (0.839) (5.122) (4.408) HH (5.652) (-5.712) (5.779) (-5.537) (-0.878) (1.513) (0.489) (5.241) (3.794) NW (5.481) (-4.953) (5.525) (-5.553) (-0.916) (1.381) (0.517) ( 5.552) (4.302) Panel B: Predictive regressions the jump mean, SAGLasso, and yield factors together. maturity JMt Ĝt Ĥ1t Ĥ2t Ĥ3t Ĥ4t Ĥ5t R HH (-3.949) (2.024) (-1.095) (1.510) (1.379) (1.659) (3.601) NW (-4.203) (2.194) (-1.078) (1.355) (1.382) (1.779) (4.015) HH (-4.781) (2.123) (-1.404) (1.184) (0.680) (2.294) (3.360) NW (-4.950) (2.238) (-1.418) (1.051) (0.685) (2.451) (3.758) HH (-5.166) (2.223) (-1.313) (1.114) (0.454) (3.011) (3.299) NW (-5.291) (2.248) (-1.337) (1.011) (0.453) (3.207) (3.687) HH (-5.530) (2.017) (-0.837) (1.071) (0.134) (3.487) (3.201) NW (-5.658) (1.985) (-0.866) (0.971) (0.134) (3.686) (3.574) Panel C: Predictive regressions on the Duffee hidden, housing, cycle, and jump mean factors together maturity (yr) Ĥ5t ĝ2t ĉf t JMt R HH (2.143) (4.824) (3.520) (-2.153) NW (2.255) (5.163) (3.626) (-2.277) HH (2.117) (4.603) (3.919) (-2.895) NW (2.265) (5.041) (4.089) (-2.958) HH (1.942) (4.672) (4.270) (-2.958) NW (2.082) (5.171) (4.526) (-2.961) HH (1.464) (4.481) (4.544) (-3.144) NW (1.580) (4.998) (4.823) (-3.122) 43

45 Table 8: Out-of-Sample Predictive Power of Supervised Macro Factors This table reports results from one-year-ahead out-of-sample forecast comparisons of n-period log excess bond returns, rx (n) t+1. The sample spans the period January 1964 to December 2007, and a burn-in period of 21 years is used. indicates that the predictor is constructed month-by-month, using information up to that point, while predictors with are estimated once with the whole sample. Column labeled MSE u/mse r reports the ratio of the mean-squared forecasting error of the unrestricted model to the mean-squared forecasting error of the restricted benchmark model that excludes additional forecasting variables. Column labeled Ericsson reports the ENC-REG test statistics of Ericsson (1992), whose 95th percentile of the asymptotic distribution is Φ 1 = Column labeled Clark-McCracken reports the ENC-NEW test statistics of Clark and McCracken (2001), whose 95% critic value is for testing one additional predictor. Both tests share the same null hypothesis that the benchmark model encompasses the unrestricted model with excess parameter. maturity (yr) Ericsson MSE u /MSE r Clark-McCracken Ericsson MSE u /MSE r Clark-McCracken 44 Panel A: SAGLasso Factor G t v.s. constant Panel B: G t + AR(6) v.s. AR(6) Panel C: G t + LN t v.s. LN t + constant Panel D: CP t + G t v.s. CP t + constant Panel E: CP t + G t + cf t v.s. CP t + G t Panel F: CP t + G t + ĉf t v.s. CP t + G t

46 Table 9: Finite-Sample Properties of Forecasting Regressions This table summarizes results from Monte Carlo simulations based on a VMA(12) process or a macro-finance term structure model. Yield data generated from the VMA process satisfies the restrictive null hypothesis of no predictability. The term structure model satisfies the general null hypothesis that expected excess bond returns are time-varying but independent of the macroeconomy. And 528 monthly data are generated for each simulation. Excess returns are regressed on the month-t values of four group macroeconomic factors. The in-sample test statistic is a Wald test of the hypothesis that the coefficients are jointly zero. To mimic the choices used in Table 1, the covariance matrix of the parameter estimates is computed using the robust Hansen-Hodrick approach for the restrictive null, and the Newey-West procedure is used with 18 lags for the general null. The table reports the empirical rejection rate using the five percent critical value for a χ 2 (4) distribution, as well as the finite sample five percent critical value. Similar statistics are reported for the out-of-sample ENC-REG test of Ericsson (1992) and ENC-NEW test of Clark and McCracken (2001). The ENC-REG test has an asymptotic N(0,1) distribution and the 95% asymptotic critical value is for the ENC-NEW test statistic. Restrictive Null General Null Type of regression Type of Test Maturity Rejection rate True 5% CV Rejection rate True 5% CV Wald In-Sample Wald Wald Wald Out-of-Sample Ericsson Ericsson Ericsson Ericsson Clark-McCracken Clark-McCracken Clark-McCracken Clark-McCracken

47 Table 10: Small Sample Inference for the Predictability of Excess Bond Returns This table is the counterpart of Table 2 and based on finite-sample distributions for test statistics. 50,000 Monte Carlo simulations are run based on a VMA(12) process or a macro-finance term structure model. Yield data generated from the VMA process satisfy the restrictive null hypothesis of no predictability. The term structure model satisfies the general null hypothesis that expected excess bond returns are time-varying but independent of the macroeconomy. And 528 monthly data are generated for each simulation. 95-percent confidence intervals for R 2, under each null hypothesis, are reported in square brackets. Column labeled Joint Test reports the Wald test statistics computed with the actual data, but the P-values are based on the empirical distributions of 50,000 bootstrapped samples. Both the restrictive and general hypotheses are defined as in Duffee (2011). maturity (yr) Null Hypothesis R 2 Joint Test P-val Restrictive [0.0025,0.0559] [0.0000] General [0.0072,0.1272] [0.0002] Restrictive [0.0021,0.0441] [0.0000] General [0.0074,0.1277] [0.0003] Restrictive [0.0028,0.0547] [0.0000] General [0.0071,0.1220] [0.0010] Restrictive [0.0037,0.0543] [0.0000] General [0.0071,0.1190] [0.0004] 46

48 Table 11: Decomposition of the 5-Year Bond Yield This table summarizes results from our decomposition of 5-year Treasury bond yields into expected future interest rates and term premia as follows: [ y (n) t = 1 n 1 ] [ n Ẽt + 1 n 1 ] n Ẽt + η (n) t. i=0 y (1) t+i i=1 rx (n i+1) t+i Panel A reports two model-implied measures of short-rate expectations Ẽty(1) t+i and forecast errors of these two measures. The models used include the Nelson and Siegel (1987) model with AR(1) factor dynamics (denoted NS ), and a 5-dimensional VAR(1) model of yield levels and macroeconomic indicators. We estimate both models recursively from 1964 to the time of the forecast, beginning in 1981 and extending through RMSE denotes the root of the mean squared error. Panel B presents root mean squared decomposition error 2 1 T T t=1 η2 t with different VAR estimations of yield risk premium. All three VAR models contain observed excess returns but differ in choices of term premium factors Z t included. Panel A: Results for out-of-sample forecasts of 1-year T-bill rates RMSE (%) relative to Forecast Observed Yields Survey Forecasts In-Sample Forecasts Horizon (yr) VAR NS VAR NS VAR NS Panel B: Fitting errors for the 5-year yield (in percentage) Interest-Rate Zt Expectations { CP t } { CP t, G t } VAR NS

49 Figure 1: Time Variations of the SAGLasso Factor and the IP Growth This figure plots the normalized SAGLasso factor and the IP Growth over time. Shaded bars denote months designated as recessions by the National Bureau of Economic Research. 48