Real Time Macro Factors in Bond Risk Premium Dashan Huang Singapore Management University Fuwei Jiang Central University of Finance and Economics Guoshi Tong Renmin University of China September 20, 2018 Tsinghua PBCSF Workshop
This Paper This paper contributes to two broad literature Empirical macro finance Could we forecast time-varying bond risk premium with real time macro variables? Answer: Yes, but need a better econometric method Econometric methodology Could we incorporate forecasting target info. into the PCA factors? Answer: Yes, we develop a Scaled PCA (spca) method 2 / 25
Bond Return Predictability Fama and Bliss (1987): Expectation hypothesis Bond returns are predicted by yield curve: level, slope, and curvature Joslin, Priebsch, and Singleton (2014): Spanning hypothesis 1 Bond returns are predicted by yield curve and macro variables (e.g., inflation). 2 Expected bond returns are countercyclical. Ludvigson and Ng (LN, 2009) extract PCA factors from 130 macro variables and find forecasting power for the first 8 components. 1 The first factor is the most powerful. 2 Bond risk premium forecasted by macro factors is countercyclical. Google citations: Joslin, et al. (2014), 328; LN (2009), 693 It seems that macro data do have incremental forecasting power. 3 / 25
But... I. Macro data are released with lags: 1 Macro data of month t 1 are released on the first Friday of month t. 2 On June 30, to predict returns in July, we cannot use the data in June, but in May, which is released on June 1. Time stamp is important for return predictability with macro data. 4 / 25
But... II. Macro data are revised after release: April s nonfarm payrolls is 164k on May 31, but should be used as 159k on June 30. When using macro data for predictability, we should consider revisions over time. 5 / 25
Snapshot of Vintages of Total Nonfarm Payrolls Data PAYEMS PAYEMS PAYEMS PAYEMS... PAYEMS Date 20170203 20170310 20170407 20170505... 20171208 2016-07 144,457 144,457 144,457 144,457... 144,457 2016-08 144,633 144,633 144,633 144,633... 144,633 2016-09 144,882 144,882 144,882 144,882... 144,882 2016-10 145,006 145,006 145,006 145,006... 145,006 2016-11 145,170 145,170 145,170 145,170... 145,170 2016-12 145,327 145,325 145,325 145,325... 145,325 2017-01 145,554 145,563 145,541 145,541... 145,541 2017-02 145,798 145,760 145,773... 145,773 2017-03 145,858 145,852... 145,823 2017-04 146,063... 146,030 Joslin et al. (2014) and LN (2009) use the last column data (i.e., final revised). Ghysels et al. (2017) use real time data. 6 / 25
Ghysels, Horan, and Moench (2017) Forecasting bond returns without considering release lags and revisions would lead to a look-ahead bias. Using real time macro data, Ghysels et al. (2017) show that 1 the forecasting power of macro variables is substantially weakened and even insignificant. 2 bond risk premium forecasted by macro factors is less countercyclical. 3 The forecasting power of macro data, found in Joslin et al. and LN, is mainly from data revision. Real time macro data s forecasting power is limited, if there is any. 7 / 25
This Paper: Can Real Time Macro Data Predict Bond Returns? We agree with Ghysels et al. (2017) that real time data are more noisy than the final macro data. And argue that more efficient econometric approaches can separate the wheat from the chaff: 1 Bond returns can be predicted by real time macro variables. 2 Predicted bond returns are countercyclical. 3 Our real time macro factors capture the current and future macro conditions better. To some extent, we reconcile Joslin et al. (2014) and LN (2009) and Ghysels et al. (2017). 8 / 25
Why Do Econometric Approaches Matter? Because of measurement errors and structural breaks, jointly using multiple variables is preferred. Since macro data are highly dimensional, dimension-reduction is important to avoid over-fitting. However, the conventional PCA used in LN (2009) and Ghysels et al. (2017) is not necessarily efficient for predictability. 1 It does not incorporate the target forecasting bond returns when extracting factors. 2 The PCA treats all macro variables equally important for predictability. 9 / 25
The Forecasting Power Varies across Macro Variables 10 / 25
The PCA Factor Suppose (real time or final) macro data X = x 1 x 2... x N x 11 x 12... x 1N x 21 x 22... x 2N...... x T 1 x T 2... x TN The first PCA factor is a linear combination of (x 1, x 2,, x N ): F = w 1 x 1 + w 2 x 2 + + w N x N, where w = (w 1,, w N ) is the solution to max w Var(Xw), s.t. w w = 1. F does not use any information about bond returns. 11 / 25
Our Approach: the Scaled PCA (spca) Factor 1 Let β i be the slope of rx t+1 = α + β i x ti + ε t+1 for i = 1,, N. 2 Scale each x i by β i, i.e., β X = β 1 x 1 β 2 x 2... β N x N β 1 x 11 β 2 x 12... β N x 1N β 1 x 21 β 2 x 22... β N x 2N...... β 1 x T 1 β 2 x T 2... β N x TN 3 The spca factor is a linear combination of (β 1 x 1, β 2 x 2,, β N x N ): f = w 1 β 1 x 1 + w 2 β 2 x 2 + + w N β N x N = w 1 x 1 + w 2 x 2 + + w N x N where w is the solution to max w Var(β X w), s.t. w w = 1. 12 / 25
spca vs. PCA 1 The spca extracts factors from the space spanned by expected bond returns, since β i x ti is the expected bond return predicted by x i. (β 1 x 1, β 2 x 2,, β N x N ) In contrast, the PCA extracts factors from the space spanned by expected bond returns and the space spanned by others. (x 1, x 2,, x N ) 2 The spca puts more weights on those variables with higher predictive power. The PCA treats all macro variables equally. The spca factor should outperform the PCA factor. 13 / 25
Two Alternative Machine Learning Approaches 1 1 Target PCA (tpca) (Bai and Ng, 2008) Use LASSO to choose a subset of variables, say (x 1, x 2, x 3 ), from (x 1, x 2,, x N ), and extract PCA factors from (x 1, x 2, x 3 ). 2 Partial least squares (PLS, Kelly and Pruitt, 2013, 2015) where w is the solution to f = w 1 x 1 + w 2 x 2 + + w N x N, max w Cov 2 (R, Xw), s.t. w w = 1. Our spca is complementary to both tpca and PLS. 1 Gu, Kelly, and Xiu (2018) Empirical Asset Pricing via Machine Learning. 14 / 25
Data 1 Dependent variable: 2- to 5-year Treasury bond excess returns, rx (n) t+1, (Fama-Bliss data). 2 Independent variables: Real time data from the Archival Federal Reserve Economic Data (ALFRED) 60 macro variables used in Ghysels et al. (2017) Vintages over 1982:03 2016:12 Out-of-sample is over 1990:01 2016:12 3 Recursively estimate real time macro factors for prediction. F 1,t T : the first PCA factor extracted from final data F 1,t 1 t : the first PCA factor extracted from real time data f 1,t 1 t : the first spca factor extracted from real time data 15 / 25
Performance Evaluation Statistically 1 In-sample R 2 and regression slope 2 Out-of-sample ROS 2 (Campbell and Thompson, 2008) R 2 OS = 1 T 12 (n) t=m (rx m+12 T 12 (n) t=m (rx m+12 rx (n) m+12 )2 rx (n) m+12 )2. Economically (out-of-sample): mean-variance portfolio allocation Utility gain = utility with predictability utility without predictability 16 / 25
Macro Factors (i.e., risk premiums) 4 3 f 1,t 1 t F 1,t 1 t F 1,t T 2 1 0 1 2 3 4 5 1985 1990 1995 2000 2005 2010 2015 f 1,t 1 t : real time spca factor; F 1,t 1 t : real time PCA factor; F 1,t T : final revised PCA factor). 17 / 25
F 1,t T 0.91 [ 1.78] In-Sample Forecasting Results Dependent variable: 5-year bond return F 1,t 1 t 0.58 1.34 [ 1.16] [1.00] f 1,t 1 t 0.82 2.07 f + 1,t 1 t [ 2.05] [ 1.95] 0.76 [ 1.98] Adj-R 2 0.03 0.01 0.03 0.03 0.02 f 1,t 1 t : real time spca factor; F 1,t 1 t : real time PCA factor; F 1,t T : final revised PCA factor; f 1,t 1 t : residual of regressing f 1,t 1 t on F 1,t 1 t. The real time spca factor has comparable forecast power as the final macro PCA factor, and outperforms the real time PCA factor, which is not significant. 18 / 25
Out-of-Sample Forecasting Results ROS 2 (%) MSPE p-value R2 OS (%) MSPE p-value 2-year bond return: rx (2) t+12 3-year bond return: rx (3) t+12 F 1,t T 13.54 2.60 0.00 14.68 2.68 0.00 F 1,t 1 t 9.80 2.27 0.01 11.49 2.42 0.01 f 1,t 1 t 11.16 1.97 0.02 17.04 2.31 0.01 4-year bond return: rx (4) t+12 5-year bond return: rx (5) t+12 F 1,t T 13.95 2.66 0.00 12.78 2.62 0.00 F 1,t 1 t 11.42 2.44 0.01 10.85 2.42 0.01 f 1,t 1 t 19.03 2.50 0.01 19.54 2.61 0.00 f 1,t 1 t : real time spca factor; F 1,t 1 t : real time PCA factor; F 1,t T : final revised PCA factor. 19 / 25
Out-of-Sample Forecasting with tpca and PLS Factors R 2 OS (%) MSPE p-value R2 OS 2-year bond return (%) MSPE p-value 5-year bond return F 1,t 1 t 9.80 2.27 0.01 10.85 2.42 0.01 tpca H1 14.50 2.06 0.02 16.02 2.53 0.01 tpca H2 12.21 1.76 0.04 21.13 2.36 0.01 tpca H3 38.87 1.61 0.05 2.13 2.12 0.02 tpca S1 10.09 2.25 0.01 12.38 2.58 0.00 tpca S2 3.77 2.03 0.02 11.93 2.56 0.01 tpca S3 7.73 1.93 0.03 12.19 2.56 0.01 PLS 5.81 1.98 0.02 15.32 2.53 0.01 Finding I: Bond returns can be significantly predicted by real time macro factors, in- and out-of-sample. 20 / 25
Term Premium is Countercyclical tp (n) t = 1 n [Et (rx(n) t+1 ) + Et (rx(n 1) t+2 )+,..., +E t (rx (2) t+n 1 )], A: Term premium implied by f 1,t 1 t 2 Term premium CFNAI 0 2 4 corr= 0.22** 1985 1990 1995 2000 2005 2010 2015 B: Term premium implied by F 1,t 1 t 2 Term premium CFNAI 0 2 4 corr= 0.10** 1985 1990 1995 2000 2005 2010 2015 C: Term premium difference between f 1,t 1 tand F 1,t 1 t 2 Term premium difference CFNAI 0 2 corr= 0.37*** 4 1985 1990 1995 2000 2005 2010 2015 Term premium is implied by a VAR model that includes a macro factor as the predictor. IP growth: monthly growth in industrial production. 21 / 25
Correlation Between Expected Bond Return and Macro Condition corr(er t, Y t ) ER t Y t Output gap Cay Inflation uncertainty Consumption uncertainty Recession probability E[rx (5) t+12 F 1,t 1 t] 0.39 0.22 0.61 0.60 0.78 E[rx (5) t+12 f 1,t 1 t] 0.55 0.33 0.70 0.67 0.82 ER t 0.57 0.36 0.55 0.49 0.50 ER = E[rx (5) t+12 f 1,t 1 t] E[rx (5) t+12 F 1,t 1 t] Finding II: The predicted bond returns (i.e., term premium) are countercyclical. 22 / 25
Forecasting Macro Data Revision Dependent variable: revision t T = F 1,t T F 1,t 1 t f + 1,t 1 t 0.49 0.49 [3.98] [3.60] F 1,t 1 t 0.12 0.12 [ 1.05] [ 1.20] Adj-R 2 0.06 0.03 0.09 f + 1,t 1 t is the residual of regressing the real time spca factor f 1,t 1 t on the real time PCA factor F 1,t 1 t. Finding III a : The spca factor outperforms the PCA factor because it better captures current macro condition. 23 / 25
Forecasting Next Year Macro Conditions G t+12 = α + βf + 1,t 1 t + ɛ t+12 CFNAI Consumption Employment Macro Yield uncertainty spread β 0.20 0.14 0.23 0.23 0.10 t-stat 1.87 2.40 2.05 2.31 2.49 adj-r 2 0.02 0.03 0.03 0.06 0.04 f + 1,t 1 t is the residual of regressing the real time spca factor f 1,t 1 t on the real time PCA factor F 1,t 1 t. Finding III b : The spca factor predicts future bond returns because it significantly predicts future macro condition. 24 / 25
Conclusion Econometric approaches matter for extracting real time macro factors. 1 Bond returns can be significantly predicted by real time macro factors. 2 The predicted bond risk premium is countercyclical. 3 Our spca factor outperforms the PCA factor because it better captures current and future macro conditions. 25 / 25