Journal of Probability and Statistics

Research Article

Nonparametric Estimation of ATE and QTE: An Application of Fractile Graphical Analysis

Table 2

QTE Monte Carlo simulations.
MSE MAE
Estimator 100 200 500 1000 2000 100 200 500 1000 2000
𝜏 = 0 . 1 0
FGA QTE(a) 0.964 0.585 0.328 0.247 0.155 0.791 0.611 0.461 0.404 0.317
FGA QTE ( 𝑅 × 2 )(a) 0.825 0.433 0.224 0.178 0.117 0.713 0.525 0.379 0.339 0.279
FGA QTE(b) 0.863 0.491 0.243 0.146 0.090 0.802 0.593 0.409 0.312 0.241
FGA QTE ( 𝑅 × 2 )(b) 1.334 0.849 0.391 0.241 0.146 1.074 0.847 0.561 0.433 0.339
Firpo [18] 0.879 0.633 0.438 0.332 0.332 0.733 0.639 0.525 0.467 0.453
Bitler et al. [7] 0.835 0.628 0.432 0.332 0.332 0.733 0.638 0.524 0.467 0.453
PS Matching
Nearest neighbor (1) 0.775 0.649 0.455 0.341 0.383 0.697 0.636 0.532 0.486 0.478
Nearest neighbor (2) 0.551 0.442 0.315 0.269 0.236 0.603 0.555 0.484 0.462 0.444
Nearest neighbor (4) 0.562 0.429 0.309 0.285 0.259 0.591 0.557 0.492 0.494 0.484
Kernel 0.748 0.566 0.467 0.401 0.361 0.691 0.617 0.580 0.587 0.581
Spline 0.626 0.519 0.400 0.389 0.367 0.646 0.622 0.575 0.593 0.591
𝜏 = 0 . 2 5
FGA QTE(a) 0.523 0.333 0.184 0.133 0.083 0.585 0.459 0.336 0.292 0.230
FGA QTE ( 𝑅 × 2 )(a) 0.409 0.246 0.131 0.088 0.058 0.507 0.397 0.291 0.236 0.194
FGA QTE(b) 0.866 0.519 0.270 0.169 0.116 0.823 0.631 0.450 0.352 0.290
FGA QTE ( 𝑅 × 2 )(b) 1.400 0.867 0.462 0.292 0.195 1.126 0.874 0.634 0.503 0.410
Firpo [18] 0.721 0.527 0.360 0.268 0.226 0.641 0.543 0.436 0.377 0.338
Bitler et al. [7] 0.687 0.527 0.360 0.268 0.226 0.635 0.543 0.436 0.377 0.338
PS Matching
Nearest neighbor (1) 0.952 0.757 0.471 0.309 0.209 0.731 0.621 0.517 0.420 0.364
Nearest neighbor (2) 0.563 0.381 0.287 0.195 0.147 0.579 0.472 0.403 0.339 0.306
Nearest neighbor (4) 0.312 0.204 0.153 0.110 0.089 0.438 0.352 0.296 0.255 0.231
Kernel 0.519 0.390 0.299 0.175 0.101 0.553 0.469 0.400 0.313 0.246
Spline 0.491 0.274 0.154 0.104 0.095 0.533 0.406 0.312 0.276 0.278
𝜏 = 0 . 5 0
FGA QTE(a) 0.305 0.191 0.105 0.072 0.046 0.441 0.344 0.259 0.211 0.168
FGA QTE ( 𝑅 × 2 )(a) 0.252 0.143 0.070 0.049 0.029 0.394 0.301 0.212 0.175 0.138
FGA QTE(b) 0.941 0.616 0.349 0.242 0.169 0.871 0.703 0.529 0.435 0.367
FGA QTE ( 𝑅 × 2 )(b) 1.541 0.977 0.567 0.370 0.270 1.187 0.940 0.716 0.576 0.496
Firpo [18] 0.658 0.540 0.358 0.249 0.206 0.606 0.521 0.401 0.332 0.285
Bitler et al. [7] 0.629 0.540 0.358 0.249 0.206 0.603 0.521 0.401 0.332 0.285
PS Matching
Nearest neighbor (1) 1.288 0.748 0.443 0.271 0.184 0.921 0.697 0.534 0.423 0.350
Nearest neighbor (2) 0.993 0.608 0.359 0.229 0.152 0.809 0.624 0.489 0.390 0.317
Nearest neighbor (4) 0.656 0.465 0.292 0.194 0.137 0.654 0.547 0.442 0.360 0.298
Kernel 0.367 0.250 0.130 0.084 0.060 0.476 0.396 0.292 0.234 0.200
Spline 0.648 0.349 0.181 0.121 0.093 0.642 0.476 0.343 0.285 0.235
𝜏 = 0 . 7 5
FGA QTE(a) 0.391 0.256 0.138 0.092 0.069 0.497 0.411 0.295 0.242 0.205
FGA QTE ( 𝑅 × 2 )(a) 0.352 0.218 0.112 0.071 0.051 0.466 0.376 0.267 0.215 0.180
FGA QTE(b) 1.043 0.721 0.452 0.320 0.250 0.919 0.768 0.610 0.515 0.461
FGA QTE ( 𝑅 × 2 )(b) 1.635 1.094 0.675 0.486 0.368 1.229 1.001 0.787 0.670 0.585
Firpo [18] 0.652 0.561 0.356 0.346 0.250 0.615 0.548 0.420 0.390 0.322
Bitler et al. [7] 0.652 0.561 0.356 0.346 0.250 0.615 0.548 0.420 0.390 0.322
PS Matching
Nearest neighbor (1) 1.439 1.138 0.710 0.492 0.317 0.888 0.801 0.624 0.532 0.440
Nearest neighbor (2) 0.768 0.647 0.509 0.396 0.257 0.668 0.599 0.527 0.464 0.387
Nearest neighbor (4) 0.429 0.374 0.326 0.293 0.230 0.507 0.466 0.416 0.389 0.349
Kernel 0.486 0.576 0.407 0.328 0.255 0.534 0.556 0.484 0.445 0.400
Spline 0.713 0.596 0.342 0.284 0.258 0.631 0.578 0.438 0.412 0.408
𝜏 = 0 . 9 0
FGA QTE(a) 0.711 0.398 0.229 0.167 0.125 0.666 0.505 0.385 0.328 0.287
FGA QTE ( 𝑅 × 2 )(a) 0.707 0.333 0.185 0.142 0.100 0.671 0.457 0.345 0.304 0.257
FGA QTE(b) 1.218 0.837 0.560 0.417 0.345 0.979 0.824 0.678 0.588 0.544
FGA QTE ( 𝑅 × 2 )(b) 1.771 1.176 0.809 0.589 0.473 1.269 1.029 0.860 0.736 0.664
Firpo [18] 0.754 0.717 0.444 0.462 0.340 0.651 0.625 0.479 0.461 0.401
Bitler et al. [7] 0.754 0.717 0.444 0.462 0.340 0.651 0.625 0.479 0.461 0.401
PS Matching
Nearest neighbor (1) 0.898 0.930 0.674 0.789 0.631 0.699 0.690 0.566 0.562 0.510
Nearest neighbor (2) 0.424 0.370 0.231 0.287 0.241 0.506 0.449 0.352 0.338 0.310
Nearest neighbor (4) 0.337 0.204 0.112 0.122 0.113 0.456 0.356 0.249 0.228 0.217
Kernel 0.697 0.708 0.581 0.394 0.131 0.613 0.591 0.504 0.395 0.267
Spline 0.432 0.287 0.105 0.078 0.058 0.501 0.411 0.256 0.228 0.209
(a) ̂ 𝛿 𝜏 , (b) ̃ 𝛿 𝜏 . MSE: mean squared error. MAE: mean absolute error. Monte Carlo simulations based on 1000 replications of the baseline model.