Table 3:
Simulations results (N=50,000) comparing to Average Treatment Effects (ATEs) - %Bias (Coeff. Var.) {Coverage Pr}
| E(Y) | Estimators | Pr(D) = 0.55 | Pr(D) = 0.70 | Pr(D) = 0.85 | Pr(D) = 0.93 | Pr(D) = 0.995 |
|---|---|---|---|---|---|---|
| 0.50~0.60 | Naïve Probit | 248 [.01] {0} | 237 [.01] {0} | 211 [.01] {0} | 187 [.01] {0} | 164 [.01] {0} |
| 2SLS | 28 [.08] {.28} | 18 [.1] {.69} | −11 [.21] {.92} | −43 [.59] {.78} | −80 [4.64] {.86} | |
| 2SRI | −34 [.19] {.28} | −20 [.16] {.66} | 28 [.12] {.55} | 82 [.12] {.03} | 144 [.15] {.09} | |
| 2SRI - sres | 44 [.08] {.05} | 63 [.09] {.01} | 90 [.11] {0} | 119 [.14] {.02} | 151 [.2] {.18} | |
| 2SRI - dres | −108 [−1.45] {0} | −103 [−5.19] {0} | −55 [.42] {.19} | 35 [.19] {.71} | 161 [.12] {.01} | |
| 2SRI - ares | −88 [1.07] {0} | −80 [.68] {0} | −40 [.3] {.42} | 31 [.19] {.74} | 144 [.15] {.05} | |
| 2SRI - gres | −33 [.18] {.3} | −20 [.15] {.63} | 11 [.14] {.88} | 49 [.15] {.42} | 111 [.22] {.36} | |
| Bi.Probit | −1 [.1] {.95} | −1 [.1] {.97} | −1 [.15] {.95} | −3 [.3] {.94} | −25 [1.57] {.85} | |
| 0.80 ~0.90 | Naïve Probit | 244 [.01] {0} | 314 [.01] {0} | 407 [.01] {0} | 489 [.01] {0} | 587 [.02] {0} |
| 2SLS | 3 [.17] {.9} | 45 [.13] {.25} | 98 [.12] {.01} | 125 [.17] {.1} | 107 [.51] {.78} | |
| 2SRI | 2 [.16] {.9} | −10 [.19] {.85} | −49 [.38] {.25} | −68 [.8] {.26} | −55 [1.38] {.72} | |
| 2SRI - sres | 80 [.06] {0} | 149 [.05] {0} | 224 [.06] {0} | 289 [.08] {0} | 390 [.11] {0} | |
| 2SRI - dres | −71 [.69] {.04} | −95 [3.72] {0} | −114 [−1.15] {0} | −103 [−6.45] {.01} | −13 [.65] {.89} | |
| 2SRI - ares | −47 [.34] {.22} | −54 [.39] {.1} | −58 [.42] {.1} | −42 [.42] {.56} | 36 [.45] {.88} | |
| 2SRI - gres | 2 [.15] {.92} | 0 [.17] {.91} | −10 [.2] {.89} | −20 [.3] {.8} | −20 [.69] {.87} | |
| Bi.Probit | 0 [.13] {.94} | 0 [.14] {.91} | 0 [.15] {.93} | 0 [.19] {.94} | 2 [.44] {.93} | |
| 0.9 ~ 0.95 | Naïve Probit | 226 [.02] {0} | 327 [.02] {0} | 484 [.02] {0} | 649 [.02] {0} | 891 [.02] {0} |
| 2SLS | −24 [.29] {.79} | 27 [.18] {.76} | 117 [.13] {.02} | 204 [.15] {0} | 272 [.31] {.38} | |
| 2SRI | 24 [.12] {.6} | 13 [.16] {.89} | −48 [.41] {.36} | −107 [−3.35] {.04} | −131 [−1.19] {.19} | |
| 2SRI - sres | 81 [.06] {0} | 154 [.05] {0} | 268 [.06] {0} | 365 [.08] {0} | 519 [.11] {0} | |
| 2SRI - dres | −37 [.34] {.6} | −72 [.77] {.09} | −107 [−2.91] {0} | −115 [−1.29] {0} | −85 [2.73] {.42} | |
| 2SRI - ares | −18 [.23] {.85} | −31 [.28] {.59} | −37 [.32] {.5} | −32 [.39] {.7} | 19 [.49] {.95} | |
| 2SRI - gres | 21 [.12] {.67} | 14 [.15] {.85} | 4 [.19] {.95} | −17 [.31] {.83} | −39 [.8] {.76} | |
| Bi.Probit | 0 [.16] {.92} | 0 [.15] {.95} | 0 [.17] {.94} | 1 [.2] {.95} | 1 [.38] {.93} | |
| 0.95~0.98 | Naïve Probit | 203 [.02] {0} | 328 [.02] {0} | 549 [.02] {0} | 819 [.03] {0} | 1292 [.04] {0} |
| 2SLS | −50 [.6] {.62} | 0 [.32] {.96} | 111 [.19] {.26} | 259 [.17] {.02} | 482 [.25] {.13} | |
| 2SRI | 40 [.1] {.23} | 33 [.14] {.60} | −29 [.36] {.78} | −128 [−.99] {.03} | −164 [−.33] {.06} | |
| 2SRI - sres | 76 [.06] {0} | 144 [.06] {0} | 301 [.07] {0} | 444 [.09] {0} | 679 [.12] {0} | |
| 2SRI - dres | −4 [.22] {.96} | −42 [.44] {.66} | −89 [2.42] {.1} | −114 [−1.66] {.02} | −112 [−2.92] {.21} | |
| 2SRI - ares | 8 [.17] {.91} | −10 [.23] {.94} | −15 [.29] {.89} | −12 [.32] {.91} | 30 [.49] {.97} | |
| 2SRI - gres | 35 [.1] {.32} | 26 [.14] {.7} | 19 [.21] {.91} | −3 [.3] {.95} | −36 [.82] {.8} | |
| Bi.Probit | −3 [.24] {.94} | −1 [.19] {.96} | 0 [.2] {.96} | 0 [.22] {.97} | 2 [.4] {.94} | |
2SRI – sres: 2SRI with standardized residuals; 2SRI – dres: 2SRI with deviance residuals; 2SRI – ares: 2SRI with Anscombe residuals; 2SRI-gres: 2SRI with generalized residuals; Shaded cells highlight estimator with lowest percentage bias.