This section focuses on the finite sample performance of the shrinkage estimator β as com-b pared to the OLS and FD counterparts for the regression model with stationary regressor and errors. Without loss of generality, only one regressor is considered in the experiment, i.e., we assume K = 1 throughout this section. Moreover, γ = 0 is assumed throughout this section.
We focus on the cases where εt and Zt are both generated as AR(1) processes:
(1− φεL)εt= vt, (1− φZL)Zt= wt, (14) such that vt and wt both are zero-mean normally i.i.d. white noise processes with:
E(vt2) = σv2, E(wt2) = σw2. (15) The value of σ2v and σw2 in (41) are chosen to ensure the variance of εtand Zt are both equal to 1. The values of φε and φZ ranges from 0.1 to 0.9.
In the context of stochastic regressor framework, we generate 5,000 replciation of Zt and εt based on the following model:
Ctl= β1Ztl+ εlt, t = 1, 2, . . . , n, l = 1, 2, . . . , 5000, (16) where l denotes the l-th replication of the data. β1 can be 1 or 0.9 for investigating the empirical powers of the shrinkage estimator given that the null hypothesis for β1 is always tested as:
H0 : β1 = β0 = 1. (17)
We adopt the the long-run variance estimator of Robinson (1998) to implement the shrink-age estimator and conduct inference for the OLS, FD, and shrinkshrink-age estimators, because it does not involve the difficult choices of kernel function, bandwidth parameter, or lag length of AR model typically used in the literature. Particularly, V11 in (10) can be estimated with
where and et are the residuals from the OLS estimation:
Ct− C2:n = (Zt− Z2:n)>βb2:n,OLS+ et, t = 2, 3, . . . , n. (20) Similarly, V22 can be estimated with Vb22:
Vb22=
and et,FD are the residuals from the FD estimation:
4Ct− 4C2:n=(4Zt− 4Z2:n
Table 1 contain the RMSE of the OLS, FD, and shrinkage estimators in estimating the regression coefficient β. The results shows that, for a given value of φZ, the performance of the OLS estimator deteriorates with the increasing value of φε. This is what we expect because we note that the OLS estimator achieve the Gauss-Markov bound when the error term is a Gaussian white noise. On the other hand, the efficiency of the FD estimator improves with the increasing value of φε. This corresponds to the findings in Chipman (1979) and Kr¨amer (1982) that the FD estimator is an approximation to the generalized least squares (GLS) estimator when estimating the coefficient of the linear trend.
For ease of comparison, we define RMSEξ as the RMSE of the estimator ξ in estimating β of the model in (6), and compare the finite sample relative efficiency of OLS estimator to its shrinkage counterpart as:
relative efficiency of OLS to shrinkage estimator in estimating β = RMSEOLS
RMSEbβ . (26) The shrinkage estimator is more efficient than the OLS counterpart in estimating β if we find the ratio in (16) is greater than 1.
Table 2 shows that the shrinkage estimator performs much better than the OLS estimator for the 81 cases considered in Table 2, especially when φεis larger. Indeed, we only find 10 out of 81 cases where the OLS can beat the shrinkage estimators when T = 100 . Even within these 10 cases, the relative efficiency of the OLS estimator as compared to the shrinkage estimator are very much close to each other, because the ratio are very close to 1. Moreover, we also find the relative performance of shrinkage estimator as compared to the OLS ones improves when the sample increases. For example, we now observe 9 out of 81 cases that the OLS estimator can beat the shrinkage estimator when n = 200. Moreover, there are only 3 cases that the shrinkage estimator is inferior to the OLS estimator as the sample size increases to be 400, and the ratios from these 3 cases are very close to 1.
Table 3 shows that the shrinkage estimator also performs much better than the FD esti-mator when φε is not close to the boundary of 0.9. Indeed, we only find 18 out of 81 cases where the FD can beat the shrinkage estimators when T = 100. Even within these 18 cases, the relative efficiency of the OLS estimator as compared to the shrinkage estimator are very much close to each other. Again, we find the relative performance of shrinkage estimator as compared to the OLS counterpart improves with an increasing sample size. For example, we observe only 10 out of 81 cases that the FD estimator outperform the shrinkage one when T = 200, the ratio are more close to 1 as compared to the case T = 100.
4 Conclusion
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Table 1. RMSE from Estimating the Regression Coefficient β: n = 100
φZ
φε Estimator 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.1 OLS 0.104 0.104 0.106 0.105 0.108 0.111 0.114 0.118 0.129 FD 0.122 0.128 0.132 0.139 0.152 0.166 0.191 0.229 0.312 βb 0.105 0.105 0.107 0.105 0.109 0.113 0.116 0.120 0.131 0.2 OLS 0.103 0.108 0.108 0.112 0.112 0.115 0.120 0.129 0.144 FD 0.112 0.119 0.124 0.130 0.140 0.158 0.178 0.213 0.299 βb 0.101 0.105 0.106 0.109 0.111 0.115 0.119 0.128 0.146 0.3 OLS 0.105 0.107 0.111 0.114 0.119 0.123 0.130 0.137 0.154 FD 0.102 0.109 0.113 0.121 0.131 0.145 0.164 0.197 0.274 βb 0.097 0.099 0.102 0.107 0.112 0.117 0.125 0.134 0.151 0.4 OLS 0.107 0.109 0.115 0.120 0.123 0.130 0.138 0.148 0.167 FD 0.095 0.098 0.104 0.114 0.120 0.135 0.154 0.183 0.253 βb 0.092 0.094 0.099 0.105 0.109 0.116 0.126 0.138 0.160 0.5 OLS 0.107 0.111 0.116 0.121 0.128 0.139 0.146 0.160 0.179 FD 0.084 0.089 0.094 0.099 0.109 0.120 0.137 0.166 0.234 βb 0.083 0.087 0.091 0.095 0.103 0.111 0.121 0.138 0.165 0.6 OLS 0.105 0.112 0.119 0.127 0.135 0.146 0.160 0.177 0.202 FD 0.074 0.078 0.083 0.090 0.096 0.108 0.123 0.147 0.208 βb 0.073 0.077 0.081 0.088 0.094 0.104 0.117 0.134 0.169 0.7 OLS 0.106 0.112 0.120 0.133 0.141 0.154 0.169 0.189 0.230 FD 0.062 0.066 0.071 0.077 0.083 0.091 0.104 0.128 0.178 βb 0.062 0.065 0.071 0.077 0.082 0.090 0.103 0.123 0.163 0.8 OLS 0.105 0.114 0.125 0.132 0.152 0.164 0.184 0.213 0.261 FD 0.050 0.053 0.058 0.061 0.067 0.073 0.086 0.104 0.147 βb 0.050 0.053 0.058 0.061 0.067 0.074 0.086 0.105 0.148 0.9 OLS 0.103 0.112 0.121 0.135 0.150 0.166 0.191 0.228 0.292 FD 0.035 0.037 0.039 0.042 0.045 0.052 0.061 0.072 0.103 βb 0.035 0.037 0.039 0.042 0.046 0.053 0.061 0.074 0.109
Notes: All the results are based on 5,000 replications of the simulated data defined in (14), (15), (16), and β1 = 1. β is the shrinkage estimator defined in (4).b
Table 2. Relative Efficiency of OLS Estimator to the Shrinkage Counterpart from Estimating the Regression Coefficient β
φZ
φε 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
n = 100
0.1 0.9927 0.9883 0.9927 0.9951 0.9875 0.9850 0.9883 0.9806 0.9853 0.2 1.0225 1.0225 1.0183 1.0201 1.0115 1.0031 1.0011 1.0034 0.9896 0.3 1.0851 1.0784 1.0812 1.0651 1.0592 1.0530 1.0428 1.0258 1.0200 0.4 1.1600 1.1549 1.1638 1.1399 1.1323 1.1271 1.0951 1.0754 1.0445 0.5 1.2903 1.2760 1.2732 1.2810 1.2463 1.2536 1.2088 1.1616 1.0894 0.6 1.4337 1.4530 1.4653 1.4416 1.4432 1.4012 1.3669 1.3200 1.1929 0.7 1.7015 1.7163 1.7066 1.7347 1.7184 1.7189 1.6359 1.5386 1.4060 0.8 2.0998 2.1511 2.1567 2.1750 2.2528 2.2238 2.1449 2.0226 1.7683 0.9 2.9363 3.0215 3.1114 3.1987 3.2812 3.1448 3.1218 3.0856 2.6802
n = 200
0.1 0.9988 0.9964 0.9983 1.0019 0.9965 0.9927 0.9993 0.9944 0.9950 0.2 1.0437 1.0298 1.0283 1.0202 1.0231 1.0200 1.0202 1.0051 0.9990 0.3 1.0900 1.0861 1.0933 1.0746 1.0658 1.0646 1.0515 1.0339 1.0095 0.4 1.1756 1.1680 1.1728 1.1604 1.1461 1.1402 1.1080 1.0811 1.0449 0.5 1.2875 1.2978 1.2954 1.2829 1.2725 1.2300 1.2179 1.1665 1.0916 0.6 1.4653 1.4743 1.4977 1.4716 1.4655 1.4293 1.3958 1.3167 1.2072 0.7 1.6822 1.7379 1.7563 1.7862 1.7523 1.7258 1.6646 1.5688 1.3979 0.8 2.1147 2.2099 2.2650 2.2325 2.2573 2.2542 2.2389 2.1143 1.8576 0.9 3.0284 3.1638 3.2378 3.3515 3.3323 3.3569 3.3152 3.3012 2.9232
n = 400
0.1 1.0040 1.0039 1.0051 1.0031 1.0016 0.9999 0.9998 1.0009 0.9984 0.2 1.0371 1.0379 1.0329 1.0283 1.0270 1.0247 1.0140 1.0145 1.0015 0.3 1.0905 1.0954 1.0891 1.0824 1.0757 1.0794 1.0557 1.0331 1.0241 0.4 1.1620 1.1803 1.1767 1.1758 1.1650 1.1569 1.1240 1.1032 1.0520 0.5 1.3104 1.3126 1.3058 1.2833 1.2833 1.2558 1.2285 1.1557 1.1012 0.6 1.4536 1.4726 1.4863 1.5035 1.4795 1.4403 1.4123 1.3148 1.2021 0.7 1.7533 1.7560 1.7767 1.7674 1.7785 1.7382 1.6775 1.5760 1.3966 0.8 2.1574 2.2205 2.3092 2.2825 2.2976 2.2803 2.2484 2.0852 1.8654
Table 3. Relative Efficiency of FD Estimator to the Shrinkage Counterpart from Estimating the Regression Coefficient β
φZ
φε 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
n = 100
0.1 1.1604 1.2109 1.2357 1.3193 1.3872 1.4704 1.6506 1.9038 2.3840 0.2 1.1093 1.1298 1.1718 1.1904 1.2684 1.3722 1.4948 1.6626 2.0467 0.3 1.0562 1.0946 1.1015 1.1307 1.1749 1.2377 1.3102 1.4724 1.8097 0.4 1.0334 1.0439 1.0538 1.0864 1.1056 1.1694 1.2283 1.3312 1.5780 0.5 1.0119 1.0221 1.0310 1.0470 1.0629 1.0815 1.1299 1.2060 1.4210 0.6 1.0071 1.0067 1.0165 1.0243 1.0267 1.0362 1.0532 1.1012 1.2281 0.7 0.9990 1.0031 1.0063 1.0002 1.0097 1.0127 1.0102 1.0450 1.0899 0.8 0.9984 0.9977 0.9967 0.9993 0.9953 0.9958 1.0014 0.9887 0.9916 0.9 0.9984 0.9962 0.9976 0.9966 0.9920 0.9930 0.9936 0.9791 0.9476
n = 200
0.1 1.1657 1.2067 1.2718 1.3342 1.4258 1.5191 1.7158 1.9365 2.5396 0.2 1.1080 1.1590 1.1801 1.2145 1.2657 1.3631 1.4989 1.6791 2.2373 0.3 1.0684 1.0911 1.1115 1.1383 1.1982 1.2629 1.3544 1.5557 2.0074 0.4 1.0333 1.0579 1.0618 1.0900 1.1254 1.1583 1.2372 1.3816 1.6644 0.5 1.0182 1.0280 1.0360 1.0568 1.0761 1.1048 1.1557 1.2148 1.4661 0.6 1.0113 1.0124 1.0198 1.0231 1.0330 1.0516 1.0695 1.1328 1.2534 0.7 1.0036 1.0025 1.0063 1.0038 1.0119 1.0266 1.0290 1.0614 1.1265 0.8 1.0010 1.0007 0.9987 1.0024 1.0034 1.0052 0.9989 1.0102 1.0415 0.9 0.9985 1.0003 0.9992 0.9991 0.9970 0.9986 0.9969 0.9992 0.9864
n = 400
0.1 1.1759 1.1997 1.2670 1.3370 1.4171 1.5464 1.7130 1.9882 2.6262 0.2 1.1208 1.1496 1.1904 1.2567 1.2944 1.3799 1.5254 1.7701 2.3178 0.3 1.0741 1.0928 1.1180 1.1636 1.2055 1.2606 1.3819 1.5705 2.0247 0.4 1.0470 1.0543 1.0703 1.0997 1.1176 1.1679 1.2432 1.4070 1.7281 0.5 1.0217 1.0339 1.0368 1.0606 1.0651 1.1143 1.1525 1.2538 1.4761 0.6 1.0060 1.0130 1.0239 1.0225 1.0340 1.0552 1.0851 1.1608 1.3101 0.7 1.0023 1.0077 1.0115 1.0100 1.0138 1.0231 1.0389 1.0636 1.1613 0.8 1.0018 1.0032 0.9989 1.0052 1.0064 1.0039 1.0158 1.0177 1.0582 0.9 1.0001 0.9999 0.9997 0.9993 1.0002 0.9990 0.9995 1.0016 1.0038
Notes: All the results are based on 5,000 replications of the simulated data defined in (14), (15), (16), and β = 1.
Table 4. Rejection Percentages of the Shrinkage Estimator under the Null
φZ
φε n 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.1 100 7.02 6.88 6.96 6.44 6.96 7.44 8.00 8.04 10.66 200 6.12 6.02 5.70 6.02 5.88 6.36 6.32 6.82 8.24 400 5.14 6.04 5.54 5.40 5.66 5.88 5.86 6.22 7.34 0.2 100 6.60 7.26 6.60 6.46 7.06 7.10 7.70 8.66 11.24 200 5.60 5.44 5.88 6.22 6.66 6.18 6.88 6.68 7.94 400 5.32 5.58 5.26 5.32 5.96 5.24 6.02 6.28 6.98 0.3 100 6.86 6.38 6.38 6.60 7.06 7.28 7.96 8.48 9.80 200 5.98 5.40 5.54 6.10 5.84 6.26 6.66 7.12 7.84 400 5.10 5.16 5.28 5.08 4.94 5.54 5.64 6.20 7.00 0.4 100 7.08 6.50 6.60 6.62 6.58 6.78 8.06 9.00 11.06 200 5.82 6.10 6.08 5.54 5.76 6.38 6.22 6.90 8.16 400 5.22 5.02 5.22 5.22 5.28 5.32 5.76 5.80 6.48 0.5 100 6.38 6.20 6.74 6.02 7.04 6.76 7.48 8.26 10.48 200 5.84 5.88 5.74 5.60 5.40 6.28 5.92 7.04 8.36 400 4.82 4.66 5.58 5.14 5.58 5.40 6.06 6.18 6.56 0.6 100 5.58 6.30 6.32 6.50 6.40 6.76 7.78 8.38 10.48 200 5.24 5.34 5.52 5.56 5.62 5.60 6.18 6.84 8.62 400 5.82 5.26 5.32 5.08 5.16 5.56 4.98 5.78 6.74 0.7 100 6.02 5.54 5.88 6.82 5.84 5.88 6.78 7.80 10.30 200 5.60 5.66 5.78 5.94 5.80 5.30 6.32 6.66 8.20 400 5.00 5.12 5.08 5.72 5.32 5.74 5.80 5.78 5.98 0.8 100 6.20 6.00 6.52 5.94 6.10 5.88 6.16 7.36 9.78 200 5.58 5.12 5.36 5.84 5.78 5.16 5.72 5.44 6.78 400 5.36 5.38 5.12 5.26 5.48 5.42 5.24 5.86 5.84 0.9 100 5.78 5.76 5.76 5.62 5.60 6.32 6.84 6.64 8.68 200 5.84 5.50 5.86 5.80 5.94 5.64 5.92 5.86 6.48 400 5.42 5.36 5.30 5.32 4.88 5.16 5.44 4.88 5.66
Notes: All the results are based on 5,000 replications of the simulated data defined in (14),
Table 5. Rejection Percentages of the OLS Estimator under the Null
φZ
φε n 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.1 100 6.48 5.84 6.26 5.52 6.44 6.42 7.14 7.38 9.96 200 5.46 5.50 5.34 5.70 5.88 5.92 6.18 6.46 7.88 400 5.08 5.68 5.18 5.24 5.72 5.78 5.70 6.08 7.34 0.2 100 5.64 6.34 6.04 6.32 6.40 6.62 7.34 8.32 10.46 200 5.64 5.04 5.66 5.86 6.28 5.88 6.70 6.68 7.76 400 5.02 5.32 4.74 4.96 6.12 5.22 5.52 6.12 6.80 0.3 100 6.12 5.82 6.24 6.04 6.86 6.94 7.66 8.32 9.76 200 5.88 5.22 5.32 5.78 5.90 6.04 6.50 7.10 7.48 400 5.22 5.12 5.02 5.24 5.06 5.88 5.36 5.86 6.74 0.4 100 6.86 5.76 6.52 6.42 6.52 6.72 7.58 8.94 11.24 200 5.84 5.66 6.42 5.50 5.72 6.70 6.32 6.90 8.20 400 4.50 5.04 5.30 4.78 5.52 5.56 5.62 5.88 6.68 0.5 100 6.44 6.10 5.96 6.18 6.46 7.02 7.78 8.76 10.24 200 5.82 5.96 5.52 5.42 5.40 6.24 6.26 7.38 8.56 400 5.56 4.88 5.68 5.46 5.52 5.48 6.50 6.28 6.76 0.6 100 5.28 5.84 6.66 6.38 6.72 7.82 8.80 9.46 10.98 200 5.40 5.56 5.76 5.76 6.36 5.56 6.86 7.04 8.78 400 5.36 5.74 4.94 5.60 5.70 5.60 5.74 5.64 7.24 0.7 100 5.82 5.60 5.96 7.48 7.04 7.58 8.08 9.30 12.00 200 5.32 5.36 5.52 6.66 6.36 5.92 7.32 7.42 9.84 400 5.42 5.40 5.46 5.54 5.32 5.72 5.84 6.22 6.86 0.8 100 5.86 5.10 6.74 6.18 7.56 7.60 8.40 10.60 13.78 200 5.30 5.76 5.38 6.00 5.90 6.36 7.32 7.96 10.04 400 5.12 5.38 5.78 4.82 5.42 6.12 5.48 6.90 7.02 0.9 100 5.20 5.62 5.68 6.96 7.26 7.38 8.80 11.00 14.90 200 4.78 5.08 5.58 5.78 5.90 6.48 6.94 8.32 10.86 400 4.70 5.16 5.34 5.28 4.96 5.90 6.04 6.50 8.22
Notes: All the results are based on 5,000 replications of the simulated data defined in (14), (15), (16), and β1 = 1. β is the shrinkage estimator defined in (4).b
Table 6. Rejection Percentages of the FD Estimator under the Null
φZ
φε n 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.1 100 6.64 6.28 6.20 5.40 6.16 6.12 6.18 5.56 5.46 200 6.02 6.02 5.44 5.26 5.92 5.58 5.58 5.08 4.74 400 5.38 4.80 5.66 5.30 5.34 5.22 5.18 5.36 5.16 0.2 100 6.26 6.46 5.80 5.44 5.86 6.46 5.98 5.62 6.28 200 5.70 5.62 5.50 6.02 4.96 5.46 5.78 4.28 5.32 400 5.62 5.32 5.14 5.24 5.38 5.08 5.38 5.14 5.20 0.3 100 5.88 6.44 6.00 5.86 6.28 5.62 5.34 5.92 5.32 200 6.06 5.26 5.08 5.34 5.06 5.22 5.48 5.48 5.54 400 5.20 4.98 5.30 5.44 4.98 5.20 5.32 5.80 4.92 0.4 100 6.44 6.14 5.96 6.46 5.60 6.56 6.02 5.74 5.74 200 5.76 6.22 5.72 5.18 5.78 5.34 5.26 5.42 4.84 400 5.30 5.18 4.96 5.12 4.72 4.58 4.70 5.10 5.08 0.5 100 5.96 5.64 5.84 5.66 5.64 5.52 5.32 5.64 5.30 200 5.44 5.42 5.38 5.60 5.32 5.68 5.52 5.20 5.60 400 4.82 4.68 5.26 5.12 5.32 4.78 5.38 5.24 5.12 0.6 100 5.36 5.78 6.02 5.96 5.68 5.82 6.08 5.66 5.74 200 5.22 5.36 5.74 5.62 5.28 5.34 5.50 5.56 5.04 400 5.50 5.20 5.22 4.72 4.94 5.18 4.76 5.52 5.04 0.7 100 6.06 5.28 5.86 6.36 5.54 5.20 5.36 6.20 5.42 200 5.52 5.64 5.46 5.62 5.26 5.22 5.96 5.40 5.46 400 5.10 5.14 4.90 5.80 5.12 5.62 5.34 5.40 5.24 0.8 100 6.24 5.96 6.14 5.62 5.98 5.34 5.66 5.52 6.14 200 5.50 5.28 5.22 5.78 5.74 4.94 5.12 5.18 5.10 400 5.38 5.56 5.04 5.30 5.42 5.34 4.90 5.30 4.82 0.9 100 5.70 5.58 5.58 5.54 5.36 6.12 6.70 5.86 5.68 200 5.68 5.48 5.64 5.64 5.78 5.72 5.52 5.38 5.52 400 5.48 5.36 5.44 5.20 4.74 5.26 5.20 5.18 5.24
Notes: All the results are based on 5,000 replications of the simulated data defined in (14),
Table 7. Emprical Power Performance of Three Estimator: n = 200
φZ
φε Estimator 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.1 OLS 29.36 27.68 28.96 28.44 28.42 28.10 27.66 27.76 29.02 FD 23.56 20.94 19.44 18.94 17.24 14.98 12.60 9.14 7.72 βb 29.98 29.02 29.64 29.48 29.22 28.86 28.38 28.30 29.24 0.2 OLS 28.66 28.38 27.18 26.48 26.80 25.38 25.74 24.54 24.60 FD 24.72 23.80 22.20 20.46 18.76 15.70 13.38 10.38 7.94 βb 30.74 30.62 28.64 29.30 28.24 26.48 26.48 25.58 25.36 0.3 OLS 28.72 26.62 26.42 25.12 24.60 23.58 23.04 22.96 22.76 FD 30.10 27.70 24.18 22.06 20.54 17.20 14.44 11.32 8.86 βb 33.62 31.80 29.94 29.32 27.76 26.74 24.72 23.82 23.26 0.4 OLS 28.86 26.92 25.80 23.88 23.80 21.32 20.32 19.68 20.10 FD 34.92 31.76 27.72 25.06 23.48 19.40 15.82 12.68 8.84 βb 37.38 34.94 32.18 30.10 29.60 26.28 23.16 21.70 21.66 0.5 OLS 29.48 26.58 26.02 22.06 21.14 19.50 19.46 19.28 18.78 FD 40.70 37.56 35.56 30.30 26.54 22.56 18.56 13.80 10.00 βb 42.90 39.86 38.12 33.14 30.44 27.68 24.84 23.18 20.76 0.6 OLS 28.30 26.10 23.98 22.34 20.70 18.54 18.12 17.24 17.14 FD 51.12 45.40 41.98 36.52 32.22 26.98 21.44 17.02 10.88 βb 52.64 46.96 43.68 40.02 35.18 30.42 26.22 23.14 18.92 0.7 OLS 28.62 26.32 23.28 21.18 20.04 17.58 16.74 15.56 15.60 FD 63.16 59.12 52.12 47.18 43.14 34.92 28.52 21.18 12.72 βb 63.60 59.58 53.46 48.38 44.70 37.80 30.98 24.80 19.00 0.8 OLS 29.54 26.42 23.84 21.52 18.28 16.80 16.20 15.00 14.80 FD 81.52 78.10 71.24 67.04 57.90 48.72 39.06 27.80 17.02 βb 81.78 77.90 71.80 67.54 58.66 49.28 40.88 30.64 20.86 0.9 OLS 33.54 29.70 26.54 22.28 19.32 16.86 15.88 14.78 14.78 FD 98.18 97.22 94.74 91.14 87.20 78.24 68.30 50.32 29.22 βb 98.16 97.22 94.74 91.02 87.06 78.36 68.34 50.86 31.12
Notes: All the results are based on 5,000 replications of the simulated data defined in (14), (15), (16), and β1 = 0.9. β is the shrinkage estimator defined in (4).b
Table 8. Emprical Power Performance of Three Estimator: n = 400
φZ
φε Estimator 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.1 OLS 50.36 50.90 49.90 48.82 48.72 46.98 46.46 46.96 46.58 FD 39.36 36.76 34.60 30.86 27.54 24.02 18.60 15.04 9.76 βb 51.82 51.94 50.76 49.90 49.48 47.84 47.28 47.58 47.04 0.2 OLS 51.26 49.40 47.70 46.04 44.72 43.10 42.30 40.62 40.52 FD 45.04 42.04 37.58 34.50 30.72 25.82 22.36 16.38 11.12 βb 54.34 53.08 50.42 48.84 48.22 45.58 44.14 41.76 41.26 0.3 OLS 48.56 48.56 45.40 42.88 40.50 39.96 37.82 36.62 35.00 FD 50.94 47.80 42.46 38.52 33.50 28.82 23.56 18.10 11.60 βb 57.30 55.84 51.90 49.52 46.08 44.50 41.76 39.20 36.38 0.4 OLS 48.82 45.12 44.58 41.60 37.00 35.64 32.58 33.20 30.26 FD 58.74 52.80 48.46 45.34 38.04 32.32 26.78 20.14 13.04 βb 62.40 58.24 55.92 53.20 46.44 44.22 38.82 37.32 32.28 0.5 OLS 47.62 45.52 40.56 38.20 35.28 32.48 29.78 28.04 26.66 FD 66.78 62.92 57.44 52.74 46.14 39.26 31.84 24.20 13.80 βb 69.44 66.00 60.86 58.06 52.48 46.84 41.46 36.20 30.10 0.6 OLS 48.14 43.38 39.70 35.92 33.14 29.40 26.16 24.48 21.58 FD 78.54 72.40 68.66 62.86 57.08 46.24 37.78 28.64 16.44 βb 79.66 73.80 71.06 65.20 60.40 50.70 44.58 36.62 27.68 0.7 OLS 47.60 42.58 38.46 34.02 29.92 26.00 23.94 21.22 18.72 FD 89.62 86.28 81.32 76.10 67.50 59.58 49.20 35.32 20.72 βb 89.96 87.06 82.08 77.22 69.20 62.36 52.32 40.40 28.72 0.8 OLS 46.02 41.84 37.76 33.78 27.62 24.96 21.86 17.80 17.54 FD 98.02 96.54 94.54 91.58 85.84 77.22 66.34 48.30 29.16 βb 98.06 96.70 94.66 91.96 85.90 77.44 67.66 51.50 33.90 0.9 OLS 48.78 44.32 38.14 31.92 28.00 22.62 19.26 15.96 14.38 FD 100.00 99.96 99.94 99.54 99.30 97.42 92.30 79.22 50.80 βb 100.00 99.96 99.94 99.56 99.26 97.44 92.24 78.90 52.10
Notes: All the results are based on 5,000 replications of the simulated data defined in (14),
Table 9. Empirical Power Performance of the Shrinkage Estimator relative to both the OLS and FD Counterparts
φZ
φε 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
n = 200
0.1 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.2 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.3 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.4 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.5 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.6 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.7 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.8 1.0000 0.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9 0.0000 1.0000 1.0000 0.0000 0.0000 1.0000 1.0000 1.0000 1.0000
n = 400
0.1 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.2 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.3 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.4 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.5 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.6 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.7 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.8 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9 1.0000 1.0000 1.0000 1.0000 0.0000 1.0000 0.0000 0.0000 1.0000
Notes: The results are based on the findings in Tables 7 and 8. The value of each entry equals 1 when the power of the shrinkage estimator is better than that of the OLS and that of the FD estimators at the same time. Otherwide, it equals 0.