Online Appendix for Short-Run and Long-Run Consumption Risks, Dividend Processes, and Asset Returns 1 More on Fama-MacBeth regressions This section compares the performance of Fama-MacBeth regressions between the simulated data and empirical data using 40 value-weighted portfolios 10 momentum (Mom), 10 long-term contrarian portfolios (Cont.), 10 valuation ratio portfolios (BM in the data and DP in the simulations), and 10 size portfolios as testing assets. We evaluate the performance of CAPM, consumption CAP- M (C-CAPM), and a two-factor model that uses the market and consumption growth as the risk factors. The results are reported in Table OA1. Panel A shows the estimated risk premia, mean absolute errors (MAE) of the 40 testing portfolios, the p-value of the overidentification test, and the OLS-R 2. Overall, the pattern of the results using simulated data are broadly consistent with that using the actual data: 1. The estimated risk premium for the market factor and consumption growth are positive and similar in magnitude across all three models; 2. CAPM outperforms C-CAPM in capturing the cross-sectional stock returns as given by a smaller MAE and a higher R 2. Even though C-CAPM fails to reject the overidentification test, this is likely due to the large pricing errors from C-CAPM; 3. The two-factor model improves the performance over both CAPM and C-CAPM. In addition, we also take a closer look at the pricing errors of individual strategies, that is, the momentum winner-minus-loser portfolio, the contrarian loser-minus-winner portfolio, the high- 1
minus-low valuation ratio portfolio, and the small-minus-big size portfolio. The absolute pricing errors are reported in Panel B of Table OA1. Both in the actual data and in the simulated data, CAPM performs better than C-CAPM for the long-term contrarian profit, the value premium, and the size premium, due to their exposures to the long-run consumption risk. However, we also note that the pricing error for the momentum strategy is also lower in CAPM than in C-CAPM, which seems to be inconsistent with our finding that momentum profits are mainly driven by shortrun consumption risk exposure. The reason for this result is because in our second-stage Fama- MacBeth regressions, we do not include an intercept, so that the estimated risk premium for the consumption growth in C-CAPM is mainly driven by the tension in matching the average portfolio returns, instead of portfolio returns in the cross section. In another word, the large estimated risk premium for the consumption growth in C-CAPM echoes the equity premium puzzle from Mehra and Prescott (1985) due to the misspecification of the empirical asset pricing model, and a back-of-the-envelope calculation for the estimated risk aversion is 3.09%/(2.84%) 2 = 38.3, much larger than the value of 10 in our calibration. The large estimated risk premium for the aggregate consumption growth overshoots the model predicted momentum profits compared with the average realized momentum profits, giving rise to a large pricing error in the cross section. 1 Compared with CAPM and C-CAPM, the two-factor model in general produces much smaller pricing errors for these investment strategies. Their magnitudes are similar for the simulated data and actual data. Part of the pricing errors from simulated data is due to the fact that, strictly speaking, our model is a three-factor model with aggregate dividend growth as the third risk factor. We find that including this factor into our Fama-MacBeth cross-sectional tests improves the performance of the model and further lowers the pricing errors. Overall, we find that the cross-sectional test results using the simulated data are very similar to those using the empirical data, providing further support to our model for the cross-sectional stock returns. 1 In an untabulated analysis, we recalculate the pricing errors by including an intercept in the cross-sectional regression, and consistent with our intuition, this procedure generates a much smaller pricing error in the C-CAPM than in the CAPM for the momentum profit. 2
2 More sensitivity analysis In Table OA2, we report the result from additional sensitivity analysis for our long-run consumption risk model. Moments from the benchmark are reported in Specification (0). We consider alternative values for the correlation between shocks to the exposure of long-run consumption growth and firm dividend growth shocks in Specifications (1), (2), and (3), the correlation between shocks to the exposure of short-run consumption growth and firm dividend growth shocks in Specifications (4), (5), and (6), the persistence of exposure to long-run consumption growth in Specifications (7), (8), and (9), the persistence in exposure to short-run consumption growth in Specifications (10), (11), and (12), the standard deviation of aggregate dividend growth shocks in Specifications (13) and (14), the average exposure to long-run risk in (15) and (16), the conditional standard deviation of exposure to long-run risk in (17) and (18), the average exposure to short-run risk in (19) and (20), the conditional standard deviation of exposure to short-run risk in (21) and (22), the persistence in firm-specific dividend growth in (23) and (24), and the conditional standard deviation of firmspecific dividend growth in (25) and (26). We report the mean, standard deviation, and the firstorder autocorrelation coefficient of market excess returns (both value-weighted and equal-weighted), risk-free rate, log price-dividend ratio, the profitability to momentum, contrarian, value, and size strategies. We simulate 100 samples with each sample representing 972 months and 1,000 firms. The cross-simulation median annualized moments are reported. 3 Earnings momentum In this section, we show that our model can reproduce positive earnings momentum, and the mechanism is the same as the mechanism in generating price momentum. In the literature of earnings momentum (e.g., Ball and Brown (1968), Chordia and Shivakumar (2006)), firms with positive innovations on earnings tend to outperform firms with negative innovations on earnings, and this phenomenon is short-lived. Panel A (Panel B) of Table OA3 reports the value-weighted (equalweighted) stock returns and CAPM performance for the decile portfolios sorted by the dividend shocks in the past three months from our simulated data. The result shows that firms with positive past earnings (dividend) surprises have a higher average return than firms with negative past earnings (dividend) surprises, and both the value-weighted and equal-weighted earnings momentum 3
is more than 13 % per year. CAPM fails to explain the return spread between earnings momentum winners and losers, giving rise to an abnormal return spread of 10.4% per year for the value-weighted strategy and 15.1% per year for the equal weighted strategy. Panel C of Table OA3 reports the characteristics of earnings momentum portfolios. Earnings momentum winners have better past short-term stock performance than earnings momentum losers. The result of portfolio risk exposures to the consumption risks indicates that the majority of the risk premium comes from the short-run consumption risk exposure (-10.4 for earnings momentum losers to 10.4 for earnings momentum winners). Therefore, both price and earnings momentum are primarily due to exposures to the short-run consumption risks within our model. 4
References Ball, Ray, and Philip Brown, 1968, An empirical evaluation of accounting income numbers, Journal of Accounting Research 6, 159 178. Chordia, Tarun, and Lakshmanan Shivakumar, 2006, Earnings and price momentum, Journal of Financial Economics 80, 627 656. Mehra, Rajnish, and Edward C. Prescott, 1985, The equity premium: A puzzle, Journal of Monetary Economics 15, 145 161. 5
Table OA1: Additional results on Fama-MacBeth regressions This table reports the Fama-MacBeth cross-sectional tests on the value-weighted returns of 40 portfolios 10 momentum (Mom), 10 long-term contrarian portfolios (Cont.), 10 valuation ratio portfolios (BM in the data and DP in the simulations), and 10 size portfolios using empirical data and simulated data. The empirical data is annual from 1931 to 2011. For the simulated data, we simulate 100 samples with each sample representing 972 months and 1,000 firms. The crosssimulation mean are reported. The tested models include CAPM, consumption CAPM, and a two-factor model with market return (MKT) and consumption growth (ConG) as the risk factors. In Panel A, the estimated risk premia, mean absolute error (MAE) of the 40 portfolios, p-value for the J T tests, and the OLS-R 2 are reported. In Panel B, the absolute values of pricing errors for the momentum winner-minus-loser portfolio, the contrarian loser-minus-winner portfolio, the high-minus-low valuation ratio portfolio, and the small-minus-big size portfolio are reported using both empirical and simulated data for each tested model. Panel A: Cross-sectional test results Data Simulation CAPM C-CAPM MKT+ConG CAPM C-CAPM MKT+ConG MKT 8.89 9.32 10.48 10.31 ConG 3.09 0.78 2.35 0.17 MAE 1.12 8.47 1.10 1.84 6.14 1.77 p(j T ) 0.00 0.61 0.00 0.06 0.45 0.09 R 2 42.36% -221.77% 55.47% 29.28% -189.95% 35.83% Panel B: Absolute pricing errors Data Simulation CAPM C-CAPM MKT+ConG CAPM C-CAPM MKT+ConG Mom 11.05 16.26 5.87 4.68 5.07 4.69 Cont. 4.13 15.65 6.18 4.10 5.42 3.78 BM(DP) 3.09 8.94 3.70 6.96 13.04 6.63 Size 3.23 10.39 3.63 5.64 7.08 5.33 6
Table OA2: Additional sensitivity analysis Specification (0) (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) Parameter ρfy ρfy ρfy ρhy ρhy ρhy ρf ρf ρf ρh ρh ρh Benchmark -0.97-0.97-0.97 0.875 0.875 0.875 0.989 0.989 0.989 0.781 0.781 0.781 Alternative -0.9-0.848-0.5 0.5 0.8 0.95 0.99 0.9 0 0 0.726 0.836 E(MKTVW) 8.20 8.77 8.94 10.92 7.23 8.03 8.26 7.47 13.41 13.75 6.81 8.05 8.71 Std(MKTVW) 25.44 25.91 26.28 28.80 25.36 25.32 25.45 25.14 31.77 32.41 23.30 24.76 26.50 AC1(MKTVW) 0.00 0.00 0.01 0.01 0.00 0.01 0.01 0.01-0.01-0.01 0.02 0.01-0.01 E(MKTEW) 11.66 11.72 11.76 12.02 11.73 11.68 11.65 11.64 12.42 12.65 11.28 11.55 11.85 Std(MKTEW) 28.70 28.82 28.91 29.47 28.84 28.73 28.67 28.65 30.07 30.51 28.01 28.50 29.08 AC1(MKTEW) 0.00 0.00 0.00 0.00 0.00-0.01 0.00-0.01 0.00 0.00 0.00 0.00-0.01 E(rf) 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 Std(rf) 1.24 1.24 1.24 1.24 1.24 1.24 1.24 1.24 1.24 1.24 1.24 1.24 1.24 AC1(rf) 0.83 0.83 0.83 0.83 0.83 0.83 0.83 0.83 0.83 0.83 0.83 0.83 0.83 E(p-d) 3.12 3.06 3.01 2.79 3.26 3.14 3.10 3.28 2.51 2.49 3.04 3.08 3.20 Std(p-d) 0.25 0.26 0.27 0.30 0.25 0.25 0.25 0.24 0.36 0.37 0.23 0.25 0.27 AC1(p-d) 0.64 0.64 0.65 0.68 0.64 0.64 0.64 0.63 0.73 0.73 0.63 0.64 0.65 Mom 7.54 7.44 7.37 7.77 2.43 6.41 8.62 7.63 8.08 8.59-2.57 6.07 8.54 Std(Mom) 37.17 36.43 35.99 33.35 25.22 34.28 40.29 37.82 33.00 32.94 8.90 31.36 45.04 Contrarian 5.13 4.64 4.31 2.04 5.49 5.27 5.06 5.59-1.05-1.25 6.08 5.44 4.48 Std(Contrarian) 12.50 11.72 11.20 8.55 12.76 12.49 12.48 13.34 3.82 3.91 12.87 12.47 12.91 Value 9.21 8.48 7.86 4.21 10.98 9.45 8.83 10.29-3.99-4.41 11.76 9.86 7.84 Std(Value) 26.39 25.13 24.26 19.56 24.91 25.94 26.89 28.06 14.07 14.04 22.80 24.75 29.90 Size 5.72 5.37 4.89 2.50 6.49 5.89 5.68 6.73-1.11-1.16 7.46 6.37 5.27 Std(Size) 14.27 13.29 12.50 8.91 14.47 14.26 14.25 15.83 4.90 4.92 14.80 14.18 14.69 7
Table OA2: Additional sensitivity analysis (Continued) Specification (0) (13) (14) (15) (16) (17) (18) (19) (20) (21) (22) (23) (24) (25) (26) Parameter σd σd f f σ f σf h h σ h σh ρy ρy σy σy Benchmark 0.0467 0.0467 5.857 5.857 0.351 0.351 0.008 0.008 4.935 4.935 0.979 0.979 0.0015 0.0015 Alternative 0.0447 0.0487 5.51 6.204 0.317 0.385-1.417 1.433 4.408 5.562 0.976 0.982 0.0014 0.0016 E(MKTVW) 8.20 8.22 8.18 7.06 9.33 9.22 6.85 7.43 8.94 8.04 8.41 8.58 7.60 8.50 7.84 Std(MKTVW) 25.44 24.91 26.09 24.50 26.84 26.33 24.89 25.10 26.28 24.89 26.22 25.74 26.01 25.63 25.61 AC1(MKTVW) 0.00 0.00 0.00 0.01 0.00 0.00 0.02 0.01 0.01 0.01 0.00 0.01 0.02 0.01 0.01 E(MKTEW) 11.66 11.59 11.68 10.79 12.45 11.82 11.50 10.63 11.76 11.52 11.84 11.66 11.79 11.65 11.68 Std(MKTEW) 28.70 28.15 29.24 27.20 30.29 28.99 28.32 29.58 28.91 28.44 29.08 28.69 28.81 28.68 28.77 AC1(MKTEW) 0.00 0.00-0.01 0.00 0.00 0.00 0.00 0.00 0.00 0.00-0.01 0.00-0.01 0.00-0.01 E(rf) 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 Std(rf) 1.24 1.24 1.24 1.24 1.24 1.24 1.24 1.24 1.24 1.24 1.24 1.24 1.24 1.24 1.24 AC1(rf) 0.83 0.83 0.83 0.83 0.83 0.83 0.83 0.83 0.83 0.83 0.83 0.83 0.83 0.83 0.83 E(p-d) 3.12 3.10 3.15 3.33 2.97 2.94 3.43 3.34 3.01 3.05 3.23 2.89 3.56 2.99 3.29 Std(p-d) 0.25 0.25 0.26 0.23 0.28 0.27 0.24 0.26 0.27 0.25 0.26 0.26 0.26 0.26 0.25 AC1(p-d) 0.64 0.65 0.63 0.62 0.65 0.66 0.63 0.65 0.65 0.63 0.63 0.64 0.65 0.64 0.64 Mom 7.54 7.48 7.57 7.56 7.39 7.37 7.32 7.51 7.37 6.82 8.09 6.86 7.83 7.08 7.80 Std(Mom) 37.17 37.10 37.25 37.90 36.55 36.25 38.55 37.42 35.99 34.15 40.71 35.70 39.13 36.25 38.15 Contrarian 5.13 5.09 5.19 5.55 4.82 4.27 6.31 5.46 4.31 5.29 5.11 4.62 5.90 4.78 5.57 Std(Contrarian) 12.50 12.41 12.59 13.23 11.85 10.73 14.74 13.20 11.20 12.48 12.58 11.64 13.91 11.87 13.19 Value 9.21 9.15 9.25 9.85 8.64 7.31 11.49 9.71 7.86 9.37 9.08 8.55 9.66 9.06 9.49 Std(Value) 26.39 26.28 26.51 27.51 25.43 23.46 30.12 27.23 24.26 25.47 27.63 25.43 27.87 25.79 27.12 Size 5.72 5.72 5.81 6.28 5.44 4.83 7.24 6.09 4.89 6.01 5.47 5.29 6.66 5.46 6.30 Std(Size) 14.27 14.19 14.36 15.18 13.56 12.24 17.01 14.94 12.50 14.25 14.34 13.23 15.74 13.60 15.06 8
Table OA3: Earnings momentum This table reports the earnings momentum from the simulated data. Earnings surprise is measured as the dividend innovations in the past three months. Panel A (Panel B) reports the mean and standard deviation of returns of value-weighted (equal-weighted) decile portfolios sorted by earnings momentum, CAPM α, CAPM β, and CAPM R 2. Newey-West t-stats given in parentheses control for heteroscedasticity and autocorrelation. Panel C reports the characteristics of the earnings momentum portfolios. DP is the dividend-price ratio. R t 6 t 2 is the short-run cumulative returns from t 6 to t 2. R t 60 t 13 is the long-run cumulative returns from t 60 to t 13. y is the firm-specific dividend growth rate, which is scaled by 100 times for convenience of exposition. y(l) is the cumulative change in y between t 60 and t 13. y(s) is the cumulative change in y between t 6 and t 2. f is the exposure to long-run consumption shocks. h is the exposure to short-run consumption shocks. We simulate 100 samples with each sample representing 972 months and 1,000 firms. The cross-simulation mean are reported. Panel A: Value-weighted portfolio returns and CAPM test VW Los 2 3 4 5 6 7 8 9 Win Mean 1.26 4.39 5.35 6.18 7.54 8.45 9.78 10.40 11.86 14.33 Std 38.17 30.72 27.96 26.58 25.92 26.15 26.94 28.67 31.87 40.21 α CAP M -4.64-1.97-1.28-0.69 0.53 1.21 2.36 2.77 3.91 5.80 (-1.22) (-0.72) (-0.56) (-0.33) (0.32) (0.72) (1.30) (1.37) (1.60) (1.66) β CAP M 0.72 0.78 0.82 0.85 0.87 0.89 0.92 0.95 0.99 1.06 (13.98) (20.93) (26.51) (32.51) (37.53) (40.59) (39.60) (34.54) (28.73) (20.97) R 2 (%) 20.27 36.50 48.04 57.16 62.85 66.01 65.47 61.08 53.61 38.82 Panel B: Equal-weighted Portfolio returns and CAPM test EW Los 2 3 4 5 6 7 8 9 Win Mean 4.03 6.84 8.19 9.33 10.35 11.37 12.43 13.42 14.87 17.59 Std 40.26 32.28 29.09 27.27 26.30 26.01 26.45 27.73 30.43 37.92 α CAP M -7.60-4.46-2.95-1.68-0.56 0.59 1.76 2.88 4.49 7.54 (-2.41) (-2.33) (-2.32) (-2.17) (-1.40) (1.50) (2.29) (2.27) (2.33) (2.36) β CAP M 1.08 1.04 1.03 1.02 1.01 0.99 0.98 0.97 0.96 0.92 (25.56) (40.64) (60.15) (97.63) (198.98) (208.01) (98.47) (57.53) (37.01) (21.26) R 2 (%) 47.80 70.19 83.84 93.20 98.11 98.08 92.81 82.25 66.18 39.91 Panel C: Portfolio characteristics Los 2 3 4 5 6 7 8 9 Win DP 0.075 0.075 0.075 0.075 0.075 0.075 0.075 0.075 0.075 0.075 R t 6 t 2-0.079-0.031-0.004 0.018 0.040 0.062 0.085 0.112 0.148 0.223 R t 60 t 13 0.581 0.583 0.579 0.582 0.582 0.582 0.581 0.582 0.580 0.581 y 100-0.444-0.263-0.173-0.098-0.032 0.032 0.098 0.172 0.265 0.445 y(l) -0.003 0.014 0.000 0.007-0.002 0.001-0.005-0.005-0.005 0.007 y(s) -2.018-1.201-0.782-0.453-0.144 0.145 0.446 0.782 1.209 2.017 f 6.876 6.459 6.251 6.082 5.931 5.783 5.632 5.461 5.250 4.834 h -10.417-6.216-4.031-2.306-0.751 0.759 2.315 4.051 6.246 10.433 9