Source
: Calonico S, Cattaneo M D, Farrell M H, et al. Regression discontinuity designs using covariates[J]. Review of Economics and Statistics, 2019, 101(3): 442-451.
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use headstart.dta, clear global y mort_age59_related_postHS // y 为因变量 (儿童死亡率) global x povrate60 // x 为分配变量 (forcing variable), 美国县级 1960 年贫困指数 global z census1960_pop census1960_pctsch1417 census1960_pctsch534 /// census1960_pctsch25plus census1960_pop1417 census1960_pop534 /// census1960_pop25plus census1960_pcturban census1960_pctblack // z 为上文提到的协变量
对于 RD 的点估计和偏差估计可以有不同的窗宽，这里分别用 h 和 b 来表示，在实际操作中可以限制 h = b，也可以不加限制。在统计推断时，可以使用 RD 标准的统计推断方法，也可以使用协变量调整的 RD 统计推断方法。因此，又可以分为以下几种情形进行讨论：
情形一
：对 h 和 b 间关系不加限制，选择使得未考虑协变量时，政策估计效应的 MSE 最小化方法来挑选最优窗宽，RD 标准统计推断方法如下：
rdrobust $y $x, c(59.1968) global h = e(h_l) global b = e(b_l) global IL = e(ci_r_rb) - e(ci_l_rb)
其中，
y
为被解释变量，
x
为解释变量，c 括号中为分配变量的值，暂元 h 中存储了 RD 点估计时使用的窗宽，暂元 b 中存储了 RD 偏差估计时使用的窗宽，IL 为 95% 置信区间对应的区间长度。
Sharp RD estimates using local polynomial regression.
Cutoff c = 59.1968| Left of c Right of c Number of obs = 2783 -------------------+---------------------- BW type = mserd Number of obs | 2489 294 Kernel = Triangular Eff. Number of obs | 234 180 VCE method = NN Order est. (p) | 1 1 Order bias (q) | 2 2 BW est. (h) | 6.809 6.809 BW bias (b) | 10.724 10.724 rho (h/b) | 0.635 0.635
Covariate-adjusted sharp RD estimates using local polynomial regression.
Cutoff c = 59.1968| Left of c Right of c Number of obs = 2779 -------------------+---------------------- BW type = Manual Number of obs | 2485 294 Kernel = Triangular Eff. Number of obs | 234 180 VCE method = NN Order est. (p) | 1 1 Order bias (q) | 2 2 BW est. (h) | 6.809 6.809 BW bias (b) | 10.724 10.724 rho (h/b) | 0.635 0.635
情形三
：对 h 和 b 间关系不加限制，选择使得考虑协变量时，政策估计效应的 MSE 最小化方法来挑选最优窗宽，协变量调整的 RD 统计推断方法如下：
rdrobust $y $x, c(59.1968) covs($z)
Covariate-adjusted sharp RD estimates using local polynomial regression.
Cutoff c = 59.1968| Left of c Right of c Number of obs = 2779 -------------------+---------------------- BW type = mserd Number of obs | 2485 294 Kernel = Triangular Eff. Number of obs | 240 184 VCE method = NN Order est. (p) | 1 1 Order bias (q) | 2 2 BW est. (h) | 6.977 6.977 BW bias (b) | 11.636 11.636 rho (h/b) | 0.600 0.600
rdrobust $y $x, c(59.1968) rho(1) global h1 = e(h_l) global b1 = e(b_l)
其中，rho 括号中写 1，表示限制 h 和 b 相同。
Sharp RD estimates using local polynomial regression.
Cutoff c = 59.1968| Left of c Right of c Number of obs = 2783 -------------------+---------------------- BW type = mserd Number of obs | 2489 294 Kernel = Triangular Eff. Number of obs | 234 180 VCE method = NN Order est. (p) | 1 1 Order bias (q) | 2 2 BW est. (h) | 6.809 6.809 BW bias (b) | 6.809 6.809 rho (h/b) | 1.000 1.000
Covariate-adjusted sharp RD estimates using local polynomial regression.
Cutoff c = 59.1968| Left of c Right of c Number of obs = 2779 -------------------+---------------------- BW type = Manual Number of obs | 2485 294 Kernel = Triangular Eff. Number of obs | 234 180 VCE method = NN Order est. (p) | 1 1 Order bias (q) | 2 2 BW est. (h) | 6.809 6.809 BW bias (b) | 6.809 6.809 rho (h/b) | 1.000 1.000
Covariate-adjusted sharp RD estimates using local polynomial regression.
Cutoff c = 59.1968| Left of c Right of c Number of obs = 2779 -------------------+---------------------- BW type = mserd Number of obs | 2485 294 Kernel = Triangular Eff. Number of obs | 240 184 VCE method = NN Order est. (p) | 1 1 Order bias (q) | 2 2 BW est. (h) | 6.977 6.977 BW bias (b) | 6.977 6.977 rho (h/b) | 1.000 1.000
