Figure 1: $$T$$ is small in the literature using LPs.
h | count | bin |
---|---|---|
0 | 7 | (11.0, 27.566666666666666) |
1 | 20 | (27.566666666666666, 44.13333333333333) |
2 | 2 | (44.13333333333333, 60.7) |
3 | 4 | (60.7, 77.26666666666667) |
4 | 2 | (77.26666666666667, 93.83333333333333) |
5 | 4 | (93.83333333333333, 110.4) |
6 | 1 | (110.4, 126.96666666666667) |
7 | 8 | (126.96666666666667, 143.53333333333333) |
8 | 6 | (143.53333333333333, 160.1) |
9 | 2 | (160.1, 176.66666666666666) |
10 | 1 | (176.66666666666666, 193.23333333333332) |
11 | (193.23333333333332, 209.8) | |
12 | 4 | (209.8, 226.36666666666667) |
13 | 1 | (226.36666666666667, 242.93333333333334) |
14 | 2 | (242.93333333333334, 259.5) |
15 | 1 | (259.5, 276.06666666666666) |
16 | 2 | (276.06666666666666, 292.6333333333333) |
17 | (292.6333333333333, 309.2) | |
18 | (309.2, 325.76666666666665) | |
19 | 1 | (325.76666666666665, 342.3333333333333) |
20 | (342.3333333333333, 358.9) | |
21 | (358.9, 375.46666666666664) | |
22 | (375.46666666666664, 392.0333333333333) | |
23 | 1 | (392.0333333333333, 408.6) |
24 | 1 | (408.6, 425.1666666666667) |
25 | (425.1666666666667, 441.73333333333335) | |
26 | (441.73333333333335, 458.3) | |
27 | (458.3, 474.8666666666667) | |
28 | (474.8666666666667, 491.43333333333334) | |
29 | 1 | (491.43333333333334, 508.0) |
Figure 2: LP estimators are biased in empirically-relevant samples when $$y_t$$ is an AR(1) with $$\rho =0.95$$.
Without Controls
h | True Theta | E[hat theta], T = 50 | E[hat theta], T = 100 | E[hat theta], T = 200 |
---|---|---|---|---|
0 | 1.0000 | 0.7449 | 0.8348 | 0.9109 |
1 | 0.9500 | 0.6838 | 0.7830 | 0.8605 |
2 | 0.9025 | 0.6265 | 0.7305 | 0.8123 |
3 | 0.8574 | 0.5675 | 0.6813 | 0.7657 |
4 | 0.8145 | 0.5146 | 0.6341 | 0.7214 |
5 | 0.7738 | 0.4635 | 0.5909 | 0.6801 |
6 | 0.7351 | 0.4194 | 0.5498 | 0.6408 |
7 | 0.6983 | 0.3727 | 0.5128 | 0.6024 |
8 | 0.6634 | 0.3305 | 0.4746 | 0.5679 |
9 | 0.6302 | 0.2881 | 0.4392 | 0.5359 |
10 | 0.5987 | 0.2444 | 0.4062 | 0.5046 |
With Controls
h | True Theta | E[hat theta], T = 50 | E[hat theta], T = 100 | E[hat theta], T = 200 |
---|---|---|---|---|
0 | 1.0000 | 0.9912 | 0.9975 | 0.9990 |
1 | 0.9500 | 0.8836 | 0.9228 | 0.9382 |
2 | 0.9025 | 0.7871 | 0.8505 | 0.8808 |
3 | 0.8574 | 0.6939 | 0.7841 | 0.8256 |
4 | 0.8145 | 0.6093 | 0.7214 | 0.7734 |
5 | 0.7738 | 0.5320 | 0.6642 | 0.7247 |
6 | 0.7351 | 0.4633 | 0.6097 | 0.6792 |
7 | 0.6983 | 0.3973 | 0.5612 | 0.6348 |
8 | 0.6634 | 0.3377 | 0.5129 | 0.5949 |
9 | 0.6302 | 0.2820 | 0.4690 | 0.5578 |
10 | 0.5987 | 0.2272 | 0.4281 | 0.5224 |
Figure 3: The bias approximation is accurate in our LPs.
Without Controls
h | True Theta | E[hat theta], T = 50 (MC) | E[hat theta], T = 50 (Approx) | E[hat theta], T = 100 (MC) | E[hat theta], T = 100 (Approx) |
---|---|---|---|---|---|
0 | 1.0000 | 0.7449 | 0.7626 | 0.8348 | 0.85 |
1 | 0.9500 | 0.6838 | 0.7017 | 0.7830 | 0.79 |
2 | 0.9025 | 0.6265 | 0.6435 | 0.7305 | 0.74 |
3 | 0.8574 | 0.5675 | 0.5879 | 0.6813 | 0.70 |
4 | 0.8145 | 0.5146 | 0.5349 | 0.6341 | 0.65 |
5 | 0.7738 | 0.4635 | 0.4843 | 0.5909 | 0.61 |
6 | 0.7351 | 0.4194 | 0.4361 | 0.5498 | 0.56 |
7 | 0.6983 | 0.3727 | 0.3901 | 0.5128 | 0.52 |
8 | 0.6634 | 0.3305 | 0.3463 | 0.4746 | 0.49 |
9 | 0.6302 | 0.2881 | 0.3047 | 0.4392 | 0.45 |
10 | 0.5987 | 0.2444 | 0.2652 | 0.4062 | 0.42 |
With Controls
h | True Theta | E[hat theta], T = 50 (MC) | E[hat theta], T = 50 (Approx) | E[hat theta], T = 100 (MC) | E[hat theta], T = 100 (Approx) |
---|---|---|---|---|---|
0 | 1.0000 | 0.9912 | 1.0000 | 0.9975 | 1.00 |
1 | 0.9500 | 0.8836 | 0.9102 | 0.9228 | 0.93 |
2 | 0.9025 | 0.7871 | 0.8243 | 0.8505 | 0.86 |
3 | 0.8574 | 0.6939 | 0.7420 | 0.7841 | 0.80 |
4 | 0.8145 | 0.6093 | 0.6630 | 0.7214 | 0.74 |
5 | 0.7738 | 0.5320 | 0.5873 | 0.6642 | 0.69 |
6 | 0.7351 | 0.4633 | 0.5144 | 0.6097 | 0.63 |
7 | 0.6983 | 0.3973 | 0.4443 | 0.5612 | 0.58 |
8 | 0.6634 | 0.3377 | 0.3768 | 0.5129 | 0.53 |
9 | 0.6302 | 0.2820 | 0.3115 | 0.4690 | 0.49 |
10 | 0.5987 | 0.2272 | 0.2484 | 0.4281 | 0.44 |
Figure 4:
$$\widehat{\theta}_{BC}$$ and
$$\widehat{\theta}_{BCC}$$ are closer than
$$\widehat{\theta}_{LS}$$ to $$\theta $$, on average, in our LPs without controls when $$y_t$$ is an AR(1) with $$\rho =0.95$$.
Without Controls
h | True Theta | E[hat theta LS] | E[hat theta BC] | E[hat theta BCC] |
---|---|---|---|---|
0 | 1.0000 | 0.7449 | 0.8412 | 0.9079 |
1 | 0.9500 | 0.6838 | 0.7851 | 0.8543 |
2 | 0.9025 | 0.6265 | 0.7323 | 0.8041 |
3 | 0.8574 | 0.5675 | 0.6776 | 0.7520 |
4 | 0.8145 | 0.5146 | 0.6284 | 0.7054 |
5 | 0.7738 | 0.4635 | 0.5808 | 0.6603 |
6 | 0.7351 | 0.4194 | 0.5396 | 0.6217 |
7 | 0.6983 | 0.3727 | 0.4955 | 0.5801 |
8 | 0.6634 | 0.3305 | 0.4555 | 0.5425 |
9 | 0.6302 | 0.2881 | 0.4148 | 0.5041 |
10 | 0.5987 | 0.2444 | 0.3725 | 0.4639 |
With Controls
h | True Theta | E[hat theta LS] | E[hat theta BC] | E[hat theta BCC] |
---|---|---|---|---|
0 | 1.0000 | 0.8348 | 0.9164 | 0.9451 |
1 | 0.9500 | 0.7830 | 0.8667 | 0.8959 |
2 | 0.9025 | 0.7305 | 0.8162 | 0.8459 |
3 | 0.8574 | 0.6813 | 0.7689 | 0.7992 |
4 | 0.8145 | 0.6341 | 0.7235 | 0.7543 |
5 | 0.7738 | 0.5909 | 0.6819 | 0.7132 |
6 | 0.7351 | 0.5498 | 0.6423 | 0.6741 |
7 | 0.6983 | 0.5128 | 0.6065 | 0.6389 |
8 | 0.6634 | 0.4746 | 0.5696 | 0.6025 |
9 | 0.6302 | 0.4392 | 0.5354 | 0.5687 |
10 | 0.5987 | 0.4062 | 0.5034 | 0.5372 |
Figure 5:
$$\widehat{\theta}_{BC}$$ and
$$\widehat{\theta}_{BCC}$$ are closer than
$$\widehat{\theta}_{LS}$$ to $$\theta $$, on average, in our LPs with controls when $$y_t$$ is an AR(1) with $$\rho =0.95$$.
Without Controls
h | True Theta | E[hat theta LS] | E[hat theta BC] | E[hat theta BCC] |
---|---|---|---|---|
0 | 1.0000 | 0.9912 | 0.9912 | 0.9912 |
1 | 0.9500 | 0.8836 | 0.9218 | 0.9218 |
2 | 0.9025 | 0.7871 | 0.8581 | 0.8596 |
3 | 0.8574 | 0.6939 | 0.7931 | 0.7975 |
4 | 0.8145 | 0.6093 | 0.7326 | 0.7410 |
5 | 0.7738 | 0.5320 | 0.6757 | 0.6893 |
6 | 0.7351 | 0.4633 | 0.6243 | 0.6441 |
7 | 0.6983 | 0.3973 | 0.5728 | 0.5998 |
8 | 0.6634 | 0.3377 | 0.5251 | 0.5603 |
9 | 0.6302 | 0.2820 | 0.4791 | 0.5234 |
10 | 0.5987 | 0.2272 | 0.4318 | 0.4862 |
With Controls
h | True Theta | E[hat theta LS] | E[hat theta BC] | E[hat theta BCC] |
---|---|---|---|---|
0 | 1.0000 | 0.9975 | 0.9975 | 0.9975 |
1 | 0.9500 | 0.9228 | 0.9422 | 0.9422 |
2 | 0.9025 | 0.8505 | 0.8873 | 0.8877 |
3 | 0.8574 | 0.7841 | 0.8364 | 0.8375 |
4 | 0.8145 | 0.7214 | 0.7876 | 0.7897 |
5 | 0.7738 | 0.6642 | 0.7428 | 0.7462 |
6 | 0.7351 | 0.6097 | 0.6994 | 0.7045 |
7 | 0.6983 | 0.5612 | 0.6607 | 0.6676 |
8 | 0.6634 | 0.5129 | 0.6212 | 0.6301 |
9 | 0.6302 | 0.4690 | 0.5851 | 0.5961 |
10 | 0.5987 | 0.4281 | 0.5511 | 0.5645 |
Figure 6: LP estimators without controls are biased in empirically-relevant samples when $$y_{i,t}$$ is an AR(1) with $$\rho =0.95$$.
Without Controls
h | truth | I = 10 | I = 25 | I = 50 |
---|---|---|---|---|
0 | 1.0000 | 0.8372 | 0.8381 | 0.8385 |
1 | 0.9500 | 0.7837 | 0.7853 | 0.7850 |
2 | 0.9025 | 0.7326 | 0.7352 | 0.7346 |
3 | 0.8574 | 0.6845 | 0.6871 | 0.6867 |
4 | 0.8145 | 0.6384 | 0.6410 | 0.6413 |
5 | 0.7738 | 0.5945 | 0.5973 | 0.5979 |
6 | 0.7351 | 0.5528 | 0.5558 | 0.5565 |
7 | 0.6983 | 0.5134 | 0.5161 | 0.5163 |
8 | 0.6634 | 0.4762 | 0.4785 | 0.4788 |
9 | 0.6302 | 0.4409 | 0.4430 | 0.4436 |
10 | 0.5987 | 0.4064 | 0.4088 | 0.4095 |
With Controls
h | truth | I = 10 | I = 25 | I = 50 |
---|---|---|---|---|
0 | 0.9500 | 0.9325 | 0.9342 | 0.9337 |
1 | 0.9025 | 0.8699 | 0.8731 | 0.8727 |
2 | 0.8574 | 0.8107 | 0.8147 | 0.8147 |
3 | 0.8145 | 0.7543 | 0.7586 | 0.7595 |
4 | 0.7738 | 0.7005 | 0.7054 | 0.7067 |
5 | 0.7351 | 0.6493 | 0.6548 | 0.6564 |
6 | 0.6983 | 0.6009 | 0.6063 | 0.6076 |
7 | 0.6634 | 0.5549 | 0.5603 | 0.5617 |
8 | 0.6302 | 0.5114 | 0.5168 | 0.5186 |
9 | 0.5987 | 0.4691 | 0.4748 | 0.4769 |
10 | 0.5688 | 0.4294 | 0.4352 | 0.4374 |
Figure 7: Estimators of standard errors in an LP with $$h=5$$ are biased in empirically-relevant samples when $$y_t$$ is an AR(1) with $$\rho =0.95$$.
Without controls, theta_h = 0
h | True LRV | T = 50 | T = 100 |
---|---|---|---|
0 | 10.2564 | 4.8957 | 7.0704 |
1 | 10.2564 | 4.6998 | 6.9384 |
2 | 10.2564 | 4.5154 | 6.7953 |
3 | 10.2564 | 4.3272 | 6.6523 |
4 | 10.2564 | 4.1423 | 6.5189 |
5 | 10.2564 | 3.9808 | 6.3791 |
6 | 10.2564 | 3.8114 | 6.2513 |
7 | 10.2564 | 3.6585 | 6.1304 |
8 | 10.2564 | 3.5057 | 5.9950 |
9 | 10.2564 | 3.3584 | 5.8733 |
10 | 10.2564 | 3.2103 | 5.7432 |
11 | 10.2564 | 3.0691 | 5.6287 |
12 | 10.2564 | 2.9276 | 5.5155 |
13 | 10.2564 | 2.7982 | 5.3906 |
14 | 10.2564 | 2.6660 | 5.2731 |
15 | 10.2564 | 2.5381 | 5.1470 |
16 | 10.2564 | 2.4091 | 5.0347 |
17 | 10.2564 | 2.2789 | 4.9232 |
18 | 10.2564 | 2.1556 | 4.8123 |
19 | 10.2564 | 2.0333 | 4.6969 |
20 | 10.2564 | 1.9109 | 4.5875 |
Without controls, theta_h = rho^h
h | True LRV | T = 50 | T = 100 |
---|---|---|---|
0 | 12.9507 | 4.7441 | 6.8568 |
1 | 12.9507 | 4.7077 | 7.0079 |
2 | 12.9507 | 4.6650 | 7.1635 |
3 | 12.9507 | 4.6247 | 7.3134 |
4 | 12.9507 | 4.5694 | 7.4575 |
5 | 12.9507 | 4.5225 | 7.6054 |
6 | 12.9507 | 4.2808 | 7.3840 |
7 | 12.9507 | 4.0562 | 7.1731 |
8 | 12.9507 | 3.8464 | 6.9760 |
9 | 12.9507 | 3.6537 | 6.7911 |
10 | 12.9507 | 3.4710 | 6.6020 |
11 | 12.9507 | 3.2910 | 6.4285 |
12 | 12.9507 | 3.1241 | 6.2551 |
13 | 12.9507 | 2.9651 | 6.0841 |
14 | 12.9507 | 2.8133 | 5.9267 |
15 | 12.9507 | 2.6697 | 5.7740 |
16 | 12.9507 | 2.5330 | 5.6255 |
17 | 12.9507 | 2.3957 | 5.4786 |
18 | 12.9507 | 2.2695 | 5.3378 |
19 | 12.9507 | 2.1417 | 5.1950 |
20 | 12.9507 | 2.0076 | 5.0608 |
With controls, theta_h = 0
h | True LRV | T = 50 | T = 100 |
---|---|---|---|
0 | 4.7143 | 3.1556 | 3.9510 |
1 | 4.7143 | 3.0271 | 3.8716 |
2 | 4.7143 | 2.8962 | 3.7917 |
3 | 4.7143 | 2.7650 | 3.7108 |
4 | 4.7143 | 2.6299 | 3.6328 |
5 | 4.7143 | 2.5072 | 3.5529 |
6 | 4.7143 | 2.3851 | 3.4774 |
7 | 4.7143 | 2.2657 | 3.3982 |
8 | 4.7143 | 2.1477 | 3.3192 |
9 | 4.7143 | 2.0366 | 3.2401 |
10 | 4.7143 | 1.9225 | 3.1657 |
11 | 4.7143 | 1.8167 | 3.0903 |
12 | 4.7143 | 1.7108 | 3.0184 |
13 | 4.7143 | 1.6077 | 2.9401 |
14 | 4.7143 | 1.5067 | 2.8675 |
15 | 4.7143 | 1.4117 | 2.7938 |
16 | 4.7143 | 1.3160 | 2.7210 |
17 | 4.7143 | 1.2236 | 2.6493 |
18 | 4.7143 | 1.1347 | 2.5831 |
19 | 4.7143 | 1.0490 | 2.5143 |
20 | 4.7143 | 0.9665 | 2.4442 |
With controls, theta_h = rho^h
h | True LRV | T = 50 | T = 100 |
---|---|---|---|
0 | 4.4149 | 3.0022 | 3.7282 |
1 | 4.4149 | 2.9123 | 3.6871 |
2 | 4.4149 | 2.8207 | 3.6431 |
3 | 4.4149 | 2.7237 | 3.5897 |
4 | 4.4149 | 2.6050 | 3.5313 |
5 | 4.4149 | 2.4676 | 3.4607 |
6 | 4.4149 | 2.3188 | 3.3648 |
7 | 4.4149 | 2.1843 | 3.2698 |
8 | 4.4149 | 2.0602 | 3.1841 |
9 | 4.4149 | 1.9420 | 3.1024 |
10 | 4.4149 | 1.8292 | 3.0244 |
11 | 4.4149 | 1.7178 | 2.9470 |
12 | 4.4149 | 1.6192 | 2.8734 |
13 | 4.4149 | 1.5221 | 2.7975 |
14 | 4.4149 | 1.4252 | 2.7333 |
15 | 4.4149 | 1.3324 | 2.6641 |
16 | 4.4149 | 1.2392 | 2.5947 |
17 | 4.4149 | 1.1506 | 2.5283 |
18 | 4.4149 | 1.0707 | 2.4630 |
19 | 4.4149 | 0.9907 | 2.3965 |
20 | 4.4149 | 0.9076 | 2.3341 |
Figure 8: The effect of monetary policy shocks.
(a) Inflation
h | Least Squares | Bias Corrected |
---|---|---|
0 | 0.2320 | 0.2320 |
1 | 0.2009 | 0.2046 |
2 | 0.0546 | 0.0611 |
3 | -0.0376 | -0.0284 |
4 | -0.0806 | -0.0697 |
5 | -0.0209 | -0.0121 |
6 | -0.1852 | -0.1781 |
7 | -0.2860 | -0.2836 |
8 | -0.6043 | -0.6081 |
9 | -0.8292 | -0.8432 |
10 | -0.9108 | -0.9407 |
11 | -0.9033 | -0.9526 |
12 | -1.1262 | -1.1983 |
13 | -1.3453 | -1.4410 |
14 | -1.1101 | -1.2298 |
15 | -1.3027 | -1.4446 |
16 | -1.2413 | -1.4110 |
17 | -1.0563 | -1.2489 |
18 | -0.9050 | -1.1153 |
19 | -1.0248 | -1.2532 |
20 | -0.7123 | -0.9524 |
(b) Output
h | Least Squares | Bias Corrected |
---|---|---|
0 | 0.6641 | 0.6641 |
1 | 0.8785 | 0.8891 |
2 | -0.1987 | -0.1751 |
3 | -1.1128 | -1.0882 |
4 | -1.4099 | -1.3936 |
5 | -2.7782 | -2.7879 |
6 | -2.7777 | -2.8477 |
7 | -2.5865 | -2.7124 |
8 | -3.3597 | -3.5497 |
9 | -3.9234 | -4.1868 |
10 | -4.0934 | -4.4190 |
11 | -3.3939 | -3.7975 |
12 | -2.6773 | -3.1549 |
13 | -1.9712 | -2.4951 |
14 | -2.0253 | -2.5785 |
15 | -2.3183 | -2.8889 |
16 | -1.2283 | -1.8149 |
17 | 0.3614 | -0.2203 |
18 | 1.0163 | 0.4510 |
19 | 1.5177 | 0.9833 |
20 | 1.8623 | 1.3974 |
Figure C.1: LP estimators are biased in empirically-relevant samples when $$y_t$$ is an AR(1) with $$\rho =0.90$$.
Without Controls
h | True Theta | E[hat theta], T = 50 | E[hat theta], T = 100 | E[hat theta], T = 200 |
---|---|---|---|---|
0 | 1.0000 | 0.8319 | 0.9111 | 0.9529 |
1 | 0.9000 | 0.7287 | 0.8071 | 0.8547 |
2 | 0.8100 | 0.6310 | 0.7145 | 0.7640 |
3 | 0.7290 | 0.5464 | 0.6324 | 0.6818 |
4 | 0.6561 | 0.4650 | 0.5578 | 0.6073 |
5 | 0.5905 | 0.3958 | 0.4898 | 0.5410 |
6 | 0.5314 | 0.3290 | 0.4298 | 0.4817 |
7 | 0.4783 | 0.2723 | 0.3764 | 0.4285 |
8 | 0.4305 | 0.2274 | 0.3275 | 0.3813 |
9 | 0.3874 | 0.1816 | 0.2833 | 0.3384 |
10 | 0.3487 | 0.1426 | 0.2440 | 0.3000 |
With Controls
h | True Theta | E[hat theta], T = 50 | E[hat theta], T = 100 | E[hat theta], T = 200 |
---|---|---|---|---|
0 | 1.0000 | 0.9944 | 1.0002 | 1.0003 |
1 | 0.9000 | 0.8473 | 0.8760 | 0.8925 |
2 | 0.8100 | 0.7151 | 0.7666 | 0.7937 |
3 | 0.7290 | 0.6043 | 0.6710 | 0.7044 |
4 | 0.6561 | 0.5019 | 0.5852 | 0.6242 |
5 | 0.5905 | 0.4152 | 0.5082 | 0.5529 |
6 | 0.5314 | 0.3352 | 0.4409 | 0.4897 |
7 | 0.4783 | 0.2682 | 0.3817 | 0.4331 |
8 | 0.4305 | 0.2157 | 0.3279 | 0.3832 |
9 | 0.3874 | 0.1647 | 0.2801 | 0.3384 |
10 | 0.3487 | 0.1215 | 0.2380 | 0.2982 |
Figure C.2: The bias approximation is accurate in our LPs.
Without Controls
h | True Theta | E[hat theta], T = 50 (MC) | E[hat theta], T = 50 (Approx) | E[hat theta], T = 100 (MC) | E[hat theta], T = 100 (Approx) |
---|---|---|---|---|---|
0 | 1.0000 | 0.8319 | 0.8565 | 0.9111 | 0.9191 |
1 | 0.9000 | 0.7287 | 0.7485 | 0.8071 | 0.8164 |
2 | 0.8100 | 0.6310 | 0.6511 | 0.7145 | 0.7240 |
3 | 0.7290 | 0.5464 | 0.5632 | 0.6324 | 0.6408 |
4 | 0.6561 | 0.4650 | 0.4840 | 0.5578 | 0.5659 |
5 | 0.5905 | 0.3958 | 0.4125 | 0.4898 | 0.4984 |
6 | 0.5314 | 0.3290 | 0.3482 | 0.4298 | 0.4377 |
7 | 0.4783 | 0.2723 | 0.2903 | 0.3764 | 0.3830 |
8 | 0.4305 | 0.2274 | 0.2382 | 0.3275 | 0.3338 |
9 | 0.3874 | 0.1816 | 0.1914 | 0.2833 | 0.2896 |
10 | 0.3487 | 0.1426 | 0.1495 | 0.2440 | 0.2497 |
With Controls
h | True Theta | E[hat theta], T = 50 (MC) | E[hat theta], T = 50 (Approx) | E[hat theta], T = 100 (MC) | E[hat theta], T = 100 (Approx) |
---|---|---|---|---|---|
0 | 1.0000 | 0.9944 | 1.0000 | 1.0002 | 1.0000 |
1 | 0.9000 | 0.8473 | 0.8612 | 0.8760 | 0.8808 |
2 | 0.8100 | 0.7151 | 0.7367 | 0.7666 | 0.7741 |
3 | 0.7290 | 0.6043 | 0.6248 | 0.6710 | 0.6785 |
4 | 0.6561 | 0.5019 | 0.5243 | 0.5852 | 0.5929 |
5 | 0.5905 | 0.4152 | 0.4339 | 0.5082 | 0.5163 |
6 | 0.5314 | 0.3352 | 0.3525 | 0.4409 | 0.4477 |
7 | 0.4783 | 0.2682 | 0.2791 | 0.3817 | 0.3862 |
8 | 0.4305 | 0.2157 | 0.2129 | 0.3279 | 0.3311 |
9 | 0.3874 | 0.1647 | 0.1530 | 0.2801 | 0.2818 |
10 | 0.3487 | 0.1215 | 0.0987 | 0.2380 | 0.2376 |
Figure C.3:
$$\widehat{\theta}_{BC}$$ and
$$\widehat{\theta}_{BCC}$$ are closer than
$$\widehat{\theta}_{LS}$$ to $$\theta $$, on average, in our LPs without controls when $$y_t$$ is an AR(1) with $$\rho =0.90$$.
T = 50
h | True Theta | E[hat theta LS] | E[hat theta BC] | E[hat theta BCC] |
---|---|---|---|---|
0 | 1.0000 | 0.8319 | 0.9058 | 0.9566 |
1 | 0.9000 | 0.7287 | 0.8076 | 0.8603 |
2 | 0.8100 | 0.6310 | 0.7142 | 0.7689 |
3 | 0.7290 | 0.5464 | 0.6333 | 0.6899 |
4 | 0.6561 | 0.4650 | 0.5552 | 0.6138 |
5 | 0.5905 | 0.3958 | 0.4887 | 0.5493 |
6 | 0.5314 | 0.3290 | 0.4242 | 0.4866 |
7 | 0.4783 | 0.2723 | 0.3692 | 0.4335 |
8 | 0.4305 | 0.2274 | 0.3254 | 0.3915 |
9 | 0.3874 | 0.1816 | 0.2804 | 0.3482 |
10 | 0.3487 | 0.1426 | 0.2417 | 0.3112 |
T = 100
h | True Theta | E[hat theta LS] | E[hat theta BC] | E[hat theta BCC] |
---|---|---|---|---|
0 | 1.0000 | 0.9111 | 0.9660 | 0.9856 |
1 | 0.9000 | 0.8071 | 0.8641 | 0.8841 |
2 | 0.8100 | 0.7145 | 0.7734 | 0.7937 |
3 | 0.7290 | 0.6324 | 0.6929 | 0.7136 |
4 | 0.6561 | 0.5578 | 0.6197 | 0.6408 |
5 | 0.5905 | 0.4898 | 0.5531 | 0.5745 |
6 | 0.5314 | 0.4298 | 0.4941 | 0.5159 |
7 | 0.4783 | 0.3764 | 0.4418 | 0.4639 |
8 | 0.4305 | 0.3275 | 0.3937 | 0.4161 |
9 | 0.3874 | 0.2833 | 0.3501 | 0.3729 |
10 | 0.3487 | 0.2440 | 0.3115 | 0.3346 |
Figure C.4:
$$\widehat{\theta}_{BC}$$ and
$$\widehat{\theta}_{BCC}$$ are closer than
$$\widehat{\theta}_{LS}$$ to $$\theta $$, on average, in our LPs with controls when $$y_t$$ is an AR(1) with $$\rho =0.90$$.
T = 50
h | True Theta | E[hat theta LS] | E[hat theta BC] | E[hat theta BCC] |
---|---|---|---|---|
0 | 1.0000 | 0.9944 | 0.9944 | 0.9944 |
1 | 0.9000 | 0.8473 | 0.8848 | 0.8848 |
2 | 0.8100 | 0.7151 | 0.7827 | 0.7842 |
3 | 0.7290 | 0.6043 | 0.6959 | 0.6999 |
4 | 0.6561 | 0.5019 | 0.6125 | 0.6202 |
5 | 0.5905 | 0.4152 | 0.5406 | 0.5527 |
6 | 0.5314 | 0.3352 | 0.4720 | 0.4893 |
7 | 0.4783 | 0.2682 | 0.4134 | 0.4365 |
8 | 0.4305 | 0.2157 | 0.3670 | 0.3965 |
9 | 0.3874 | 0.1647 | 0.3204 | 0.3568 |
10 | 0.3487 | 0.1215 | 0.2801 | 0.3239 |
T = 100
h | True Theta | E[hat theta LS] | E[hat theta BC] | E[hat theta BCC] |
---|---|---|---|---|
0 | 1.0000 | 1.0002 | 1.0002 | 1.0002 |
1 | 0.9000 | 0.8760 | 0.8950 | 0.8950 |
2 | 0.8100 | 0.7666 | 0.8013 | 0.8016 |
3 | 0.7290 | 0.6710 | 0.7186 | 0.7196 |
4 | 0.6561 | 0.5852 | 0.6436 | 0.6455 |
5 | 0.5905 | 0.5082 | 0.5753 | 0.5783 |
6 | 0.5314 | 0.4409 | 0.5151 | 0.5194 |
7 | 0.4783 | 0.3817 | 0.4616 | 0.4672 |
8 | 0.4305 | 0.3279 | 0.4123 | 0.4194 |
9 | 0.3874 | 0.2801 | 0.3681 | 0.3768 |
10 | 0.3487 | 0.2380 | 0.3287 | 0.3390 |
Figure C.5: LP estimators are biased in empirically-relevant samples when $$y_t$$ is an AR(1) with $$\rho =0.99$$.
Without Controls
h | True Theta | E[hat theta], T = 50 | E[hat theta], T = 100 | E[hat theta], T = 200 |
---|---|---|---|---|
0 | 1.0000 | 0.5779 | 0.6316 | 0.7130 |
1 | 0.9900 | 0.5478 | 0.6143 | 0.7001 |
2 | 0.9801 | 0.5240 | 0.5959 | 0.6865 |
3 | 0.9703 | 0.4944 | 0.5761 | 0.6739 |
4 | 0.9606 | 0.4669 | 0.5573 | 0.6623 |
5 | 0.9510 | 0.4387 | 0.5387 | 0.6479 |
6 | 0.9415 | 0.4131 | 0.5190 | 0.6361 |
7 | 0.9321 | 0.3836 | 0.5022 | 0.6238 |
8 | 0.9227 | 0.3598 | 0.4873 | 0.6121 |
9 | 0.9135 | 0.3287 | 0.4713 | 0.6000 |
10 | 0.9044 | 0.3005 | 0.4541 | 0.5876 |
With Controls
h | True Theta | E[hat theta], T = 50 | E[hat theta], T = 100 | E[hat theta], T = 200 |
---|---|---|---|---|
0 | 1.0000 | 0.9839 | 0.9933 | 0.9971 |
1 | 0.9900 | 0.9004 | 0.9499 | 0.9723 |
2 | 0.9801 | 0.8251 | 0.9053 | 0.9468 |
3 | 0.9703 | 0.7485 | 0.8621 | 0.9226 |
4 | 0.9606 | 0.6770 | 0.8203 | 0.8991 |
5 | 0.9510 | 0.6128 | 0.7798 | 0.8732 |
6 | 0.9415 | 0.5525 | 0.7391 | 0.8506 |
7 | 0.9321 | 0.4929 | 0.7027 | 0.8283 |
8 | 0.9227 | 0.4390 | 0.6690 | 0.8060 |
9 | 0.9135 | 0.3826 | 0.6344 | 0.7840 |
10 | 0.9044 | 0.3332 | 0.6015 | 0.7626 |
Figure C.6: The bias approximation is accurate in our LPs.
Without Controls
h | True Theta | E[hat theta], T = 50 (MC) | E[hat theta], T = 50 (Approx) | E[hat theta], T = 100 (MC) | E[hat theta], T = 100 (Approx) |
---|---|---|---|---|---|
0 | 1.0000 | 0.5779 | 0.5917 | 0.6316 | 1 |
1 | 0.9900 | 0.5478 | 0.5644 | 0.6143 | 1 |
2 | 0.9801 | 0.5240 | 0.5371 | 0.5959 | 1 |
3 | 0.9703 | 0.4944 | 0.5097 | 0.5761 | 1 |
4 | 0.9606 | 0.4669 | 0.4822 | 0.5573 | 1 |
5 | 0.9510 | 0.4387 | 0.4546 | 0.5387 | 1 |
6 | 0.9415 | 0.4131 | 0.4269 | 0.5190 | 1 |
7 | 0.9321 | 0.3836 | 0.3993 | 0.5022 | 1 |
8 | 0.9227 | 0.3598 | 0.3716 | 0.4873 | 1 |
9 | 0.9135 | 0.3287 | 0.3441 | 0.4713 | |
10 | 0.9044 | 0.3005 | 0.3166 | 0.4541 |
With Controls
h | True Theta | E[hat theta], T = 50 (MC) | E[hat theta], T = 50 (Approx) | E[hat theta], T = 100 (MC) | E[hat theta], T = 100 (Approx) |
---|---|---|---|---|---|
0 | 1.0000 | 0.9839 | 1.0000 | 0.9933 | 1 |
1 | 0.9900 | 0.9004 | 0.9494 | 0.9499 | 1 |
2 | 0.9801 | 0.8251 | 0.8978 | 0.9053 | 1 |
3 | 0.9703 | 0.7485 | 0.8452 | 0.8621 | 1 |
4 | 0.9606 | 0.6770 | 0.7914 | 0.8203 | 1 |
5 | 0.9510 | 0.6128 | 0.7364 | 0.7798 | 1 |
6 | 0.9415 | 0.5525 | 0.6801 | 0.7391 | 1 |
7 | 0.9321 | 0.4929 | 0.6223 | 0.7027 | 1 |
8 | 0.9227 | 0.4390 | 0.5630 | 0.6690 | 1 |
9 | 0.9135 | 0.3826 | 0.5021 | 0.6344 | 1 |
10 | 0.9044 | 0.3332 | 0.4392 | 0.6015 | 1 |
Figure C.7:
$$\widehat{\theta}_{BC}$$ and
$$\widehat{\theta}_{BCC}$$ are closer than
$$\widehat{\theta}_{LS}$$ to $$\theta $$, on average, in our LPs without controls when $$y_t$$ is an AR(1) with $$\rho =0.99$$.
T= 50
h | True Theta | E[hat theta LS] | E[hat theta BC] | E[hat theta BCC] |
---|---|---|---|---|
0 | 1.0000 | 0.5779 | 0.6785 | 0.7490 |
1 | 0.9900 | 0.5478 | 0.6529 | 0.7262 |
2 | 0.9801 | 0.5240 | 0.6334 | 0.7093 |
3 | 0.9703 | 0.4944 | 0.6079 | 0.6866 |
4 | 0.9606 | 0.4669 | 0.5842 | 0.6656 |
5 | 0.9510 | 0.4387 | 0.5596 | 0.6438 |
6 | 0.9415 | 0.4131 | 0.5372 | 0.6240 |
7 | 0.9321 | 0.3836 | 0.5106 | 0.6001 |
8 | 0.9227 | 0.3598 | 0.4894 | 0.5815 |
9 | 0.9135 | 0.3287 | 0.4607 | 0.5552 |
10 | 0.9044 | 0.3005 | 0.4343 | 0.5311 |
T= 100
h | True Theta | E[hat theta LS] | E[hat theta BC] | E[hat theta BCC] |
---|---|---|---|---|
0 | 1.0000 | 0.6316 | 0.7235 | 0.7558 |
1 | 0.9900 | 0.6143 | 0.7082 | 0.7411 |
2 | 0.9801 | 0.5959 | 0.6918 | 0.7253 |
3 | 0.9703 | 0.5761 | 0.6738 | 0.7080 |
4 | 0.9606 | 0.5573 | 0.6568 | 0.6915 |
5 | 0.9510 | 0.5387 | 0.6399 | 0.6752 |
6 | 0.9415 | 0.5190 | 0.6219 | 0.6577 |
7 | 0.9321 | 0.5022 | 0.6066 | 0.6430 |
8 | 0.9227 | 0.4873 | 0.5932 | 0.6301 |
9 | 0.9135 | 0.4713 | 0.5785 | 0.6160 |
10 | 0.9044 | 0.4541 | 0.5626 | 0.6006 |
Figure C.8:
$$\widehat{\theta}_{BC}$$ and
$$\widehat{\theta}_{BCC}$$ are closer than
$$\widehat{\theta}_{LS}$$ to $$\theta $$, on average, in our LPs with controls when $$y_t$$ is an AR(1) with $$\rho =0.99$$.
T= 50
h | True Theta | E[hat theta LS] | E[hat theta BC] | E[hat theta BCC] |
---|---|---|---|---|
0 | 1.0000 | 0.9839 | 0.9839 | 0.9839 |
1 | 0.9900 | 0.9004 | 0.9388 | 0.9388 |
2 | 0.9801 | 0.8251 | 0.8979 | 0.8994 |
3 | 0.9703 | 0.7485 | 0.8522 | 0.8567 |
4 | 0.9606 | 0.6770 | 0.8082 | 0.8170 |
5 | 0.9510 | 0.6128 | 0.7684 | 0.7830 |
6 | 0.9415 | 0.5525 | 0.7298 | 0.7514 |
7 | 0.9321 | 0.4929 | 0.6896 | 0.7194 |
8 | 0.9227 | 0.4390 | 0.6526 | 0.6919 |
9 | 0.9135 | 0.3826 | 0.6110 | 0.6611 |
10 | 0.9044 | 0.3332 | 0.5741 | 0.6363 |
T= 100
h | True Theta | E[hat theta LS] | E[hat theta BC] | E[hat theta BCC] |
---|---|---|---|---|
0 | 1.0000 | 0.9933 | 0.9933 | 0.9933 |
1 | 0.9900 | 0.9499 | 0.9695 | 0.9695 |
2 | 0.9801 | 0.9053 | 0.9434 | 0.9437 |
3 | 0.9703 | 0.8621 | 0.9176 | 0.9187 |
4 | 0.9606 | 0.8203 | 0.8920 | 0.8943 |
5 | 0.9510 | 0.7798 | 0.8669 | 0.8707 |
6 | 0.9415 | 0.7391 | 0.8407 | 0.8462 |
7 | 0.9321 | 0.7027 | 0.8179 | 0.8256 |
8 | 0.9227 | 0.6690 | 0.7969 | 0.8071 |
9 | 0.9135 | 0.6344 | 0.7742 | 0.7872 |
10 | 0.9044 | 0.6015 | 0.7526 | 0.7687 |
Figure D.1: LP estimators are biased in empirically-relevant samples when $$y_t$$ is an AR(2) with $$\rho =0.95$$ and $$\psi =0.4$$.
Without Controls
h | True Theta | E[hat theta], T = 50 | E[hat theta], T = 100 | E[hat theta], T = 200 |
---|---|---|---|---|
0 | 1.0000 | 0.5897 | 0.7331 | 0.8565 |
1 | 1.3500 | 0.9287 | 1.0833 | 1.2075 |
2 | 1.4425 | 1.0065 | 1.1703 | 1.2992 |
3 | 1.4344 | 0.9779 | 1.1552 | 1.2893 |
4 | 1.3883 | 0.9125 | 1.1019 | 1.2409 |
5 | 1.3291 | 0.8356 | 1.0372 | 1.1800 |
6 | 1.2667 | 0.7617 | 0.9705 | 1.1158 |
7 | 1.2050 | 0.6828 | 0.9061 | 1.0519 |
8 | 1.1454 | 0.6095 | 0.8419 | 0.9918 |
9 | 1.0884 | 0.5358 | 0.7810 | 0.9358 |
10 | 1.0341 | 0.4616 | 0.7226 | 0.8814 |
With Controls
h | True Theta | E[hat theta], T = 50 | E[hat theta], T = 100 | E[hat theta], T = 200 |
---|---|---|---|---|
0 | 1.0000 | 0.9893 | 0.9970 | 0.9990 |
1 | 1.3500 | 1.2624 | 1.3154 | 1.3353 |
2 | 1.4425 | 1.2662 | 1.3677 | 1.4115 |
3 | 1.4344 | 1.1678 | 1.3207 | 1.3862 |
4 | 1.3883 | 1.0382 | 1.2383 | 1.3234 |
5 | 1.3291 | 0.9070 | 1.1474 | 1.2492 |
6 | 1.2667 | 0.7843 | 1.0566 | 1.1737 |
7 | 1.2050 | 0.6671 | 0.9709 | 1.0991 |
8 | 1.1454 | 0.5613 | 0.8879 | 1.0296 |
9 | 1.0884 | 0.4636 | 0.8108 | 0.9649 |
10 | 1.0341 | 0.3694 | 0.7382 | 0.9037 |
Figure D.2: The bias approximation is accurate in our LPs.
Without Controls
h | True Theta | E[hat theta], T = 50 (MC) | E[hat theta], T = 50 (Approx) | E[hat theta], T = 100 (MC) | E[hat theta], T = 100 (Approx) |
---|---|---|---|---|---|
0 | 1.0000 | 0.5897 | 0.5954 | 0.7331 | 0.7418 |
1 | 1.3500 | 0.9287 | 0.9363 | 1.0833 | 1.0902 |
2 | 1.4425 | 1.0065 | 1.0148 | 1.1703 | 1.1787 |
3 | 1.4344 | 0.9779 | 0.9908 | 1.1552 | 1.1657 |
4 | 1.3883 | 0.9125 | 0.9285 | 1.1019 | 1.1144 |
5 | 1.3291 | 0.8356 | 0.8532 | 1.0372 | 1.0502 |
6 | 1.2667 | 0.7617 | 0.7751 | 0.9705 | 0.9828 |
7 | 1.2050 | 0.6828 | 0.6981 | 0.9061 | 0.9163 |
8 | 1.1454 | 0.6095 | 0.6237 | 0.8419 | 0.8521 |
9 | 1.0884 | 0.5358 | 0.5527 | 0.7810 | 0.7906 |
10 | 1.0341 | 0.4616 | 0.4850 | 0.7226 | 0.7320 |
With Controls
h | True Theta | E[hat theta], T = 50 (MC) | E[hat theta], T = 50 (Approx) | E[hat theta], T = 100 (MC) | E[hat theta], T = 100 (Approx) |
---|---|---|---|---|---|
0 | 1.0000 | 0.9893 | 1.0000 | 0.9970 | 1.0000 |
1 | 1.3500 | 1.2624 | 1.3030 | 1.3154 | 1.3265 |
2 | 1.4425 | 1.2662 | 1.3389 | 1.3677 | 1.3904 |
3 | 1.4344 | 1.1678 | 1.2758 | 1.3207 | 1.3542 |
4 | 1.3883 | 1.0382 | 1.1797 | 1.2383 | 1.2821 |
5 | 1.3291 | 0.9070 | 1.0758 | 1.1474 | 1.1993 |
6 | 1.2667 | 0.7843 | 0.9740 | 1.0566 | 1.1155 |
7 | 1.2050 | 0.6671 | 0.8777 | 0.9709 | 1.0345 |
8 | 1.1454 | 0.5613 | 0.7882 | 0.8879 | 0.9575 |
9 | 1.0884 | 0.4636 | 0.7056 | 0.8108 | 0.8849 |
10 | 1.0341 | 0.3694 | 0.6297 | 0.7382 | 0.8167 |
Figure D.3:
$$\widehat{\theta}_{BC}$$ and
$$\widehat{\theta}_{BCC}$$ are closer than
$$\widehat{\theta}_{LS}$$ to $$\theta $$, on average, in our LPs without controls when $$y_t$$ is an AR(1) with $$\rho =0.95$$ and $$\psi =0.4$$.
T = 50
h | True Theta | E[hat theta LS] | E[hat theta BC] | E[hat theta BCC] |
---|---|---|---|---|
0 | 1.0000 | 0.5897 | 0.7578 | 0.8690 |
1 | 1.3500 | 0.9287 | 1.0965 | 1.2122 |
2 | 1.4425 | 1.0065 | 1.1789 | 1.2989 |
3 | 1.4344 | 0.9779 | 1.1566 | 1.2809 |
4 | 1.3883 | 0.9125 | 1.0977 | 1.2263 |
5 | 1.3291 | 0.8356 | 1.0269 | 1.1598 |
6 | 1.2667 | 0.7617 | 0.9585 | 1.0956 |
7 | 1.2050 | 0.6828 | 0.8846 | 1.0259 |
8 | 1.1454 | 0.6095 | 0.8156 | 0.9609 |
9 | 1.0884 | 0.5358 | 0.7454 | 0.8946 |
10 | 1.0341 | 0.4616 | 0.6743 | 0.8270 |
T = 100
h | True Theta | E[hat theta LS] | E[hat theta BC] | E[hat theta BCC] |
---|---|---|---|---|
0 | 1.0000 | 0.9893 | 0.9893 | 0.9893 |
1 | 1.3500 | 1.2624 | 1.3102 | 1.3102 |
2 | 1.4425 | 1.2662 | 1.3696 | 1.3719 |
3 | 1.4344 | 1.1678 | 1.3221 | 1.3295 |
4 | 1.3883 | 1.0382 | 1.2359 | 1.2507 |
5 | 1.3291 | 0.9070 | 1.1401 | 1.1645 |
6 | 1.2667 | 0.7843 | 1.0462 | 1.0820 |
7 | 1.2050 | 0.6671 | 0.9518 | 1.0006 |
8 | 1.1454 | 0.5613 | 0.8638 | 0.9272 |
9 | 1.0884 | 0.4636 | 0.7792 | 0.8585 |
10 | 1.0341 | 0.3694 | 0.6943 | 0.7910 |
Figure D.4:
$$\widehat{\theta}_{BC}$$ and
$$\widehat{\theta}_{BCC}$$ are closer than
$$\widehat{\theta}_{LS}$$ to $$\theta $$, on average, in our LPs with controls when $$y_t$$ is an AR(2) with $$\rho =0.90$$ and $$\psi =0.4$$.
T = 50
h | True Theta | E[hat theta LS] | E[hat theta BC] | E[hat theta BCC] |
---|---|---|---|---|
0 | 1.0000 | 0.7331 | 0.8737 | 0.9213 |
1 | 1.3500 | 1.0833 | 1.2231 | 1.2717 |
2 | 1.4425 | 1.1703 | 1.3118 | 1.3613 |
3 | 1.4344 | 1.1552 | 1.2993 | 1.3496 |
4 | 1.3883 | 1.1019 | 1.2488 | 1.3000 |
5 | 1.3291 | 1.0372 | 1.1869 | 1.2390 |
6 | 1.2667 | 0.9705 | 1.1228 | 1.1758 |
7 | 1.2050 | 0.9061 | 1.0608 | 1.1146 |
8 | 1.1454 | 0.8419 | 0.9989 | 1.0534 |
9 | 1.0884 | 0.7810 | 0.9400 | 0.9954 |
10 | 1.0341 | 0.7226 | 0.8835 | 0.9396 |
T = 100
h | True Theta | E[hat theta LS] | E[hat theta BC] | E[hat theta BCC] |
---|---|---|---|---|
0 | 1.0000 | 0.9970 | 0.9970 | 0.9970 |
1 | 1.3500 | 1.3154 | 1.3393 | 1.3393 |
2 | 1.4425 | 1.3677 | 1.4203 | 1.4208 |
3 | 1.4344 | 1.3207 | 1.4008 | 1.4026 |
4 | 1.3883 | 1.2383 | 1.3434 | 1.3470 |
5 | 1.3291 | 1.1474 | 1.2747 | 1.2806 |
6 | 1.2667 | 1.0566 | 1.2033 | 1.2121 |
7 | 1.2050 | 0.9709 | 1.1348 | 1.1467 |
8 | 1.1454 | 0.8879 | 1.0668 | 1.0824 |
9 | 1.0884 | 0.8108 | 1.0029 | 1.0223 |
10 | 1.0341 | 0.7382 | 0.9417 | 0.9654 |
Figure F.1:
$$\widehat{\beta}_{Bootstrap}$$ provides little bias correction in our LPs $$y_t$$ is an AR(1) with $$\rho =0.95$$.
Without Controls
h | True Theta | E[hat theta LS], T = 50 | E[hat theta bootstrap], T = 50 | E[hat theta LS], T = 100 | E[hat theta bootstrap], T = 100 | E[hat theta LS], T = 200 | E[hat theta bootstrap], T = 200 |
---|---|---|---|---|---|---|---|
0 | 1.0000 | 0.7449 | 0.7487 | 0.8348 | 0.8432 | 0.9109 | 0.9129 |
1 | 0.9500 | 0.6838 | 0.7071 | 0.7830 | 0.7959 | 0.8605 | 0.8668 |
2 | 0.9025 | 0.6265 | 0.6627 | 0.7305 | 0.7500 | 0.8123 | 0.8231 |
3 | 0.8574 | 0.5675 | 0.6113 | 0.6813 | 0.7041 | 0.7657 | 0.7771 |
4 | 0.8145 | 0.5146 | 0.5624 | 0.6341 | 0.6614 | 0.7214 | 0.7327 |
5 | 0.7738 | 0.4635 | 0.5100 | 0.5909 | 0.6186 | 0.6801 | 0.6925 |
6 | 0.7351 | 0.4194 | 0.4638 | 0.5498 | 0.5760 | 0.6408 | 0.6520 |
7 | 0.6983 | 0.3727 | 0.4150 | 0.5128 | 0.5365 | 0.6024 | 0.6153 |
8 | 0.6634 | 0.3305 | 0.3744 | 0.4746 | 0.4997 | 0.5679 | 0.5796 |
9 | 0.6302 | 0.2881 | 0.3302 | 0.4392 | 0.4627 | 0.5359 | 0.5441 |
10 | 0.5987 | 0.2444 | 0.2856 | 0.4062 | 0.4262 | 0.5046 | 0.5118 |
With Controls
h | True Theta | E[hat theta LS], T = 50 | E[hat theta bootstrap], T = 50 | E[hat theta LS], T = 100 | E[hat theta bootstrap], T = 100 | E[hat theta LS], T = 200 | E[hat theta bootstrap], T = 200 |
---|---|---|---|---|---|---|---|
0 | 1.0000 | 0.9912 | 0.9927 | 0.9975 | 0.9978 | 0.9990 | 0.9996 |
1 | 0.9500 | 0.8836 | 0.9198 | 0.9228 | 0.9374 | 0.9382 | 0.9454 |
2 | 0.9025 | 0.7871 | 0.8327 | 0.8505 | 0.8743 | 0.8808 | 0.8916 |
3 | 0.8574 | 0.6939 | 0.7473 | 0.7841 | 0.8144 | 0.8256 | 0.8376 |
4 | 0.8145 | 0.6093 | 0.6658 | 0.7214 | 0.7503 | 0.7734 | 0.7860 |
5 | 0.7738 | 0.5320 | 0.5887 | 0.6642 | 0.6939 | 0.7247 | 0.7350 |
6 | 0.7351 | 0.4633 | 0.5185 | 0.6097 | 0.6373 | 0.6792 | 0.6899 |
7 | 0.6983 | 0.3973 | 0.4547 | 0.5612 | 0.5844 | 0.6348 | 0.6448 |
8 | 0.6634 | 0.3377 | 0.3970 | 0.5129 | 0.5364 | 0.5949 | 0.6046 |
9 | 0.6302 | 0.2820 | 0.3464 | 0.4690 | 0.4936 | 0.5578 | 0.5679 |
10 | 0.5987 | 0.2272 | 0.2935 | 0.4281 | 0.4502 | 0.5224 | 0.5305 |
Figure G.1: LP with lag augmentation when $$y_t$$ is an AR(1) with $$\rho =0.95$$.
LP estimators are biased
h | True Theta | E[hat theta LA], T = 50 | E[hat theta LA], T = 100 | E[hat theta LA], T = 200 |
---|---|---|---|---|
0 | 1.0000 | 0.9913 | 0.9973 | 0.9991 |
1 | 0.9500 | 0.8800 | 0.9215 | 0.9380 |
2 | 0.9025 | 0.7802 | 0.8487 | 0.8806 |
3 | 0.8574 | 0.6839 | 0.7815 | 0.8250 |
4 | 0.8145 | 0.5971 | 0.7185 | 0.7727 |
5 | 0.7738 | 0.5176 | 0.6610 | 0.7237 |
6 | 0.7351 | 0.4473 | 0.6064 | 0.6780 |
7 | 0.6983 | 0.3809 | 0.5576 | 0.6333 |
8 | 0.6634 | 0.3195 | 0.5088 | 0.5934 |
9 | 0.6302 | 0.2641 | 0.4644 | 0.5563 |
10 | 0.5987 | 0.2094 | 0.4235 | 0.5212 |
The bias approximation is accurate
h | True Theta | E[hat theta], T = 50 (MC) | E[hat theta], T = 50 (Approx) | E[hat theta], T = 100 (MC) | E[hat theta], T = 100 (Approx) |
---|---|---|---|---|---|
0 | 1.0000 | 0.9913 | 1.0000 | 0.9973 | 1.0000 |
1 | 0.9500 | 0.8800 | 0.9102 | 0.9215 | 0.9303 |
2 | 0.9025 | 0.7802 | 0.8243 | 0.8487 | 0.8642 |
3 | 0.8574 | 0.6839 | 0.7420 | 0.7815 | 0.8015 |
4 | 0.8145 | 0.5971 | 0.6630 | 0.7185 | 0.7419 |
5 | 0.7738 | 0.5176 | 0.5873 | 0.6610 | 0.6854 |
6 | 0.7351 | 0.4473 | 0.5144 | 0.6064 | 0.6318 |
7 | 0.6983 | 0.3809 | 0.4443 | 0.5576 | 0.5809 |
8 | 0.6634 | 0.3195 | 0.3768 | 0.5088 | 0.5326 |
9 | 0.6302 | 0.2641 | 0.3115 | 0.4644 | 0.4867 |
10 | 0.5987 | 0.2094 | 0.2484 | 0.4235 | 0.4430 |
Figure G.2: Bias correction in an LP with lag augmentation when $$y_t$$ is an AR(1) with $$\rho =0.95$$.
T = 50
h | True Theta | E[hat theta LS] | E[hat theta BC] | E[hat theta BCC] |
---|---|---|---|---|
0 | 1.0000 | 0.9913 | 0.9913 | 0.9913 |
1 | 0.9500 | 0.8800 | 0.9195 | 0.9195 |
2 | 0.9025 | 0.7802 | 0.8537 | 0.8554 |
3 | 0.8574 | 0.6839 | 0.7859 | 0.7906 |
4 | 0.8145 | 0.5971 | 0.7234 | 0.7324 |
5 | 0.7738 | 0.5176 | 0.6638 | 0.6783 |
6 | 0.7351 | 0.4473 | 0.6105 | 0.6315 |
7 | 0.6983 | 0.3809 | 0.5575 | 0.5862 |
8 | 0.6634 | 0.3195 | 0.5075 | 0.5446 |
9 | 0.6302 | 0.2641 | 0.4604 | 0.5070 |
10 | 0.5987 | 0.2094 | 0.4124 | 0.4693 |
T = 100
h | True Theta | E[hat theta LS] | E[hat theta BC] | E[hat theta BCC] |
---|---|---|---|---|
0 | 1.0000 | 0.9973 | 0.9973 | 0.9973 |
1 | 0.9500 | 0.9215 | 0.9413 | 0.9413 |
2 | 0.9025 | 0.8487 | 0.8862 | 0.8866 |
3 | 0.8574 | 0.7815 | 0.8347 | 0.8358 |
4 | 0.8145 | 0.7185 | 0.7857 | 0.7880 |
5 | 0.7738 | 0.6610 | 0.7407 | 0.7443 |
6 | 0.7351 | 0.6064 | 0.6973 | 0.7025 |
7 | 0.6983 | 0.5576 | 0.6583 | 0.6653 |
8 | 0.6634 | 0.5088 | 0.6183 | 0.6274 |
9 | 0.6302 | 0.4644 | 0.5816 | 0.5930 |
10 | 0.5987 | 0.4235 | 0.5475 | 0.5614 |