Utilizing (4) and (10), we have βˆII= ˆβLS−σˆuvII the true Γand xt is covariance-stationary, xht will meet the stationary condition as well when
T →∞.8 By the law of large number (LLN), central limit theory (CLT; Corollary 5.25 of White, 1984) leads to
T−1/2
Combining (14) and (15) gives that√
T f ˆΓII, T → 0 as T →∞. And together with (13), it is
and making use of the similar arguments of (14) and (15) yields the desired results:
T−1
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Table 1: Bias of AR(1)-Based Estimator When Pre-dictor is AR(2) obtained from an AR(1) regression of the predictor.
3 10, 000 replications are conducted.
Table 2: Simulation Results: Autoregression of Predictor Predictor T p Largest-Root |Bias| Variance RMSE
b/m 88 1 0.886 LS 0.044 0.004 0.078
II 0.001 0.005 0.069
dfy 90 1 0.828 LS 0.041 0.005 0.081
II 0.000 0.006 0.074
d/y 137 1 0.980 LS 0.034 0.001 0.049
II 0.008 0.001 0.032
e/p 137 1 0.785 LS 0.026 0.003 0.062
II 0.000 0.003 0.058
i/k 62 1 0.773 LS 0.058 0.009 0.112
II 0.001 0.010 0.102
lty 90 1 0.977 LS 0.052 0.003 0.073
II 0.016 0.002 0.046
svar 124 1 0.711 LS 0.025 0.005 0.072
II 0.001 0.005 0.069
tms 89 2 0.407 LS 0.036 0.023 0.153
II 0.002 0.024 0.155
d/p 137 3 0.966 LS 0.049 0.026 0.163
II 0.006 0.027 0.164
infl 90 4 0.920 LS 0.098 0.054 0.241
II 0.006 0.059 0.243
tbl 89 4 0.997 LS 0.084 0.075 0.280
II 0.041 0.081 0.286
ntis 82 5 0.794 LS 0.078 0.082 0.289
II 0.006 0.091 0.302
d/e 137 6 0.927 LS 0.094 0.060 0.250
II 0.004 0.064 0.254
eqis 82 7 0.831 LS 0.153 0.110 0.338
II 0.011 0.125 0.354
1 Estimated model: xt=ρ0+ρ1xt−1+ ··· +ρpxt−p+ vtwhere xt is the predictor in the predictive regression model.
2 |Bias|, Variance, and RMSE are defined as∑pi=1|E( ˆρi−ρi)|,∑i=1p Var( ˆρi) and
∑i=1p E( ˆρi−ρi)21/2
, respectively.
Table 3: Simulation Results: Predictive Regression Predictor T β Bias( ˆβ) Var( ˆβ) RMSE( ˆβ) |Bias( ˆβ)|
std( ˆβ) SizeL5% SizeR5% SizeL10% SizeR10%
b/m 88 0.131 LS 0.049 0.007 0.097 0.575 1.9% 12.5% 3.3% 21.3%
II -0.002 0.008 0.088 0.022 10.5% 5.2% 16.8% 10.0%
dfy 90 -0.642 LS 1.005 6.454 2.732 0.396 2.6% 9.6% 5.75% 17.4%
II 0.015 6.781 2.604 0.006 6.1% 4.8% 12.5% 10.0%
d/y 137 0.077 LS -0.002 0.001 0.037 0.062 6.0% 4.7% 11.2% 9.2%
II 0.000 0.001 0.037 0.008 5.2% 5.5% 10.2% 10.7%
e/p 137 0.074 LS 0.005 0.001 0.039 0.139 4.2% 6.7% 8.1% 12.4%
II 0.000 0.001 0.039 0.006 5.5% 5.2% 10.6% 10.0%
i/k 62 -13.2 LS -0.149 37.710 6.143 0.024 5.5% 4.9% 11.0% 9.9%
II 0.038 37.974 6.162 0.006 5.3% 5.4% 10.4% 10.6%
lty 90 -0.584 LS 0.138 1.124 1.069 0.130 4.1% 6.5% 8.2% 11.9%
II 0.018 1.140 1.068 0.017 5.5% 5.3% 11.0% 10.0%
svar 124 0.085 LS 0.062 0.162 0.407 0.155 3.5% 6.4% 7.5% 11.8%
II -0.005 0.164 0.405 0.011 5.3% 4.8% 10.7% 9.4%
tms 89 1.497 LS 0.072 2.308 1.521 0.047 4.7% 5.6% 9.3% 11.1%
II 0.002 2.315 1.522 0.001 5.2% 5.0% 10.3% 10.3%
d/p 137 0.031 LS 0.030 0.002 0.050 0.761 0.7% 15.3% 2.1% 26.0%
II 0.001 0.002 0.042 0.014 10.9% 5.5% 18.0% 10.3%
infl 90 -0.218 LS 0.016 0.236 0.487 0.032 4.8% 5.4% 9.6% 10.4%
II 0.001 0.238 0.488 0.002 5.3% 5.1% 10.2% 10.1%
tbl 89 -0.592 LS -0.126 0.395 0.641 0.200 7.3% 3.4% 13.8% 7.0%
II -0.036 0.403 0.636 0.057 5.4% 5.1% 10.6% 9.9%
ntis 82 -1.450 LS -0.024 1.485 1.219 0.020 5.3% 5.0% 10.4% 9.8%
II -0.011 1.491 1.221 0.009 5.2% 5.1% 10.3% 10.1%
d/e 137 -0.001 LS 0.007 0.002 0.047 0.157 3.9% 6.8% 7.8% 12.7%
II 0.000 0.002 0.048 0.001 5.7% 5.3% 11.1% 10.0%
eqis 82 -0.470 LS 0.006 0.062 0.249 0.025 5.2% 5.5% 10.1% 10.5%
II 0.001 0.062 0.249 0.003 5.4% 5.3% 10.5% 10.1%
1 Boldface number denotes the value of |Bias( ˆβ)|
std( ˆβ) is larger than 0.2.
2 SizeLαand SizeRαare respectively the realized sizes of the left-tailed test and right-tailed test with a nominal size ofα.
Table 4: Predictors for S&P500 Equity Premium and the AR-Order Selections Predictor Definition Time Span AR-Order χ[1]2 χ[2]2 χ[3]2 χ[4]2
b/m Book to Market 1921-2008 1 1.511 3.447 3.920 5.890
dfy Default Yield Spread 1919-2008 1 2.488 2.472 2.837 3.612
d/y Dividend Yield 1872-2008 1 0.016 2.983 3.219 3.613
e/p Earning Price Ratio 1872-2008 1 0.694 1.688 1.723 3.431 i/k Investment Capital Ratio 1947-2008 1 1.592 2.108 3.596 5.545
lty Long Term Yield 1919-2008 1 1.466 1.416 2.434 3.575
svar Stock Variance 1885-2008 1 2.569 4.469 4.729 5.945
tms Term Spread 1920-2008 2 0.607 0.782 5.727 5.219
d/p Dividend Price Ratio 1872-2008 3 0.609 1.081 2.723 2.756
infl Inflation 1919-2008 4 0.150 1.413 2.856 3.317
tbl T-Bill Rate 1920-2008 4 2.538 2.580 2.906 3.617
ntis Net Equity Expansion 1927-2008 5 0.055 1.525 2.363 3.027 d/e Dividend Payout Ratio 1872-2008 6 0.536 2.185 5.484 7.250 eqis Pct Equity Issuing 1927-2008 7 2.193 2.412 2.343 4.192
1 See Goyal and Welch (2008) and Amit Goyal’s website for detailed variable description.
2 χ[q]2 is the Breusch-Godfrey LM test statistic, with a null hypothesis that there is no serial correlation up to order q.
3 Boldface number denotes significance at 10% level.
4 The lag-order is selected by continuously increasing the lag-order of the AR model from a initialization “0”, until all the fourχ[q]2 statistics are not significant.
Table 5: Estimation of the Autoregression
Predictor Method ρˆ0 ρˆ1 ρˆ2 ρˆ3 ρˆ4 ρˆ5 ρˆ6 ρˆ7 Largest-Root
b/m LS 0.089 0.842 − − − − − − 0.842
II 0.063 0.886 − − − − − − 0.886
dfy LS 0.003 0.787 − − − − − − 0.787
II 0.002 0.828 − − − − − − 0.828
d/y LS −0.179 0.946 − − − − − − 0.946
II −0.071 0.980 − − − − − − 0.980
e/p LS −0.647 0.760 − − − − − − 0.760
II −0.581 0.785 − − − − − − 0.785
i/k LS 0.010 0.717 − − − − − − 0.717
II 0.008 0.773 − − − − − − 0.773
lty LS 0.002 0.962 − − − − − − 0.962
II 0.001 0.977 − − − − − − 0.977
svar LS 0.010 0.685 − − − − − − 0.685
II 0.009 0.711 − − − − − − 0.711
tms LS 0.007 0.691 −0.181 − − − − − 0.426
II 0.007 0.709 −0.166 − − − − − 0.407
d/p LS −0.314 0.807 −0.164 0.261 − − − − 0.932
II −0.163 0.829 −0.158 0.281 − − − − 0.966
infl LS 0.007 0.671 −0.097 −0.120 0.303 − − − 0.866
II 0.005 0.700 −0.076 −0.125 0.351 − − − 0.920
tbl LS 0.003 1.062 −0.351 0.047 0.158 − − − 0.933
II −0.000 1.111 −0.349 0.045 0.190 − − − 0.997
ntis LS 0.005 0.626 −0.006 0.019 0.041 −0.025 − − 0.671
II 0.004 0.650 0.012 0.020 0.063 −0.012 − − 0.794
d/e LS −0.126 0.799 −0.250 0.115 −0.053 −0.215 0.362 − 0.892 II −0.088 0.818 −0.238 0.114 −0.036 −0.229 0.402 − 0.927 eqis LS 0.075 0.504 0.142 0.023 0.268 −0.280 −0.103 0.044 0.819 II 0.060 0.529 0.165 0.014 0.302 −0.309 −0.090 0.065 0.831
Table 6: Estimation of the Predictive Regression
Predictor T αˆ βˆ tβˆ χ[1]2 χ[2]2 χ[3]2 χ[4]2 b/m 88 LS −0.046 0.180 2.259 2.459 2.607 3.432 3.658
II −0.018 0.131 1.645 2.691 2.846 3.683 4.030
dfy 90 LS 0.052 0.327 0.122 0.751 2.314 2.085 4.328
II 0.064 −0.642 −0.240 1.067 2.531 2.283 4.379 d/y 137 LS 0.280 0.075 1.877 0.088 4.683 5.968 6.914 II 0.286 0.077 1.930 0.081 4.680 5.952 6.882 e/p 137 LS 0.255 0.079 1.843 0.533 7.102 8.830 10.148 II 0.241 0.074 1.725 0.569 7.149 8.863 10.171 i/k 62 LS 0.534 −13.343 −2.156 0.119 1.943 2.207 3.344 II 0.529 −13.225 −2.137 0.120 1.936 2.195 3.339 lty 90 LS 0.080 −0.460 −0.591 0.415 2.428 2.158 4.482 II 0.086 −0.584 −0.750 0.442 2.450 2.19 4.489 svar 124 LS 0.043 0.150 0.337 0.096 5.011 6.062 7.177 II 0.045 0.084 0.190 0.118 4.967 6.039 7.141
tms 89 LS 0.038 1.559 1.025 0.498 1.739 1.810 3.580
II 0.038 1.497 0.984 0.506 1.744 1.816 3.588 d/p 137 LS 0.239 0.061 1.655 0.939 5.317 6.773 8.181 II 0.143 0.031 0.837 1.819 6.166 7.745 9.180 infl 90 LS 0.062 −0.210 −0.458 0.671 2.637 2.729 4.701 II 0.062 −0.218 −0.475 0.686 2.651 2.751 4.718 tbl 89 LS 0.087 −0.714 −1.025 0.427 1.975 2.016 4.045 II 0.082 −0.592 −0.852 0.458 1.998 2.041 4.084 ntis 82 LS 0.080 −1.464 −1.764 0.151 1.110 0.963 7.879 II 0.080 −1.450 −1.748 0.164 1.123 0.961 7.842 d/e 137 LS 0.048 0.006 0.115 0.247 5.955 6.974 8.266 II 0.044 −0.001 −0.009 0.281 5.946 7.010 8.298 eqis 82 LS 0.141 −0.463 −2.408 0.121 0.692 0.738 2.406 II 0.143 −0.470 −2.441 0.114 0.687 0.752 2.435
1Boldface number denotes the significance at 10% level when a two-tailed test is conducted.
2The same as note 2 in Table 4.
Table 7: Estimation of the Residual Covariance Matrix
Predictor σˆu2 σˆv2 σˆuv σˆuv ˆσv2
b/m LS 3.574 × 10−2 1.992 × 10−2 −2.196 × 10−2 −1.103 II 3.590 × 10−2 2.005 × 10−2 −2.210 × 10−2 −1.103 dfy LS 3.811 × 10−2 2.784 × 10−5 −6.717 × 10−4 −24.131 II 3.812 × 10−2 2.794 × 10−5 −6.741 × 10−4 −24.131 d/y LS 3.257 × 10−2 1.984 × 10−2 1.453 × 10−3 0.073 II 3.257 × 10−2 2.002 × 10−2 1.464 × 10−3 0.073 e/p LS 3.260 × 10−2 7.081 × 10−2 −1.439 × 10−2 −0.203 II 3.260 × 10−2 7.090 × 10−2 −1.441 × 10−2 −0.203 i/k LS 2.697 × 10−2 5.917 × 10−6 2.033 × 10−5 3.436 II 2.697 × 10−2 5.956 × 10−6 2.041 × 10−5 3.427 lty LS 3.780 × 10−2 5.945 × 10−5 −1.425 × 10−4 −2.396 II 3.780 × 10−2 5.962 × 10−5 −1.438 × 10−4 −2.412 svar LS 3.447 × 10−2 9.273 × 10−4 −2.401 × 10−3 −2.590 II 3.447 × 10−2 9.282 × 10−4 −2.404 × 10−3 −2.590 tms LS 3.700 × 10−2 1.208 × 10−4 −3.096 × 10−4 −2.562 II 3.700 × 10−2 1.208 × 10−4 −3.096 × 10−4 −2.582 d/p LS 3.275 × 10−2 4.139 × 10−2 −3.015 × 10−2 −0.728 II 3.291 × 10−2 4.175 × 10−2 −3.042 × 10−2 −0.729 infl LS 3.802 × 10−2 9.331 × 10−4 −2.603 × 10−4 −0.279 II 3.802 × 10−2 9.430 × 10−4 −2.755 × 10−4 −0.292 tbl LS 3.700 × 10−2 1.904 × 10−4 3.344 × 10−4 1.756 II 3.701 × 10−2 1.957 × 10−4 3.439 × 10−4 1.757 ntis LS 3.767 × 10−2 1.919 × 10−4 9.756 × 10−5 0.509 II 3.767 × 10−2 1.930 × 10−4 5.340 × 10−5 0.277 d/e LS 3.342 × 10−2 5.095 × 10−2 −1.352 × 10−2 −0.265 II 3.342 × 10−2 5.120 × 10−2 −1.369 × 10−2 −0.267 eqis LS 3.647 × 10−2 4.165 × 10−3 −5.725 × 10−4 −0.137 II 3.647 × 10−2 4.195 × 10−3 −6.826 × 10−4 −0.163