Finance and Economics Discussion Series: Accessible versions of figures for 2020-010

Bias in Local Projections

Accessible version of figures


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)

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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

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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

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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

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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

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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

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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

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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

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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

Return to text


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

Return to text


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

Return to text


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

Return to text


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

Return to text


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

Return to text


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

Return to text


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

Return to text


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

Return to text


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

Return to text


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

Return to text


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

Return to text


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

Return to text


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

Return to text


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

Return to text