I consider the estimation of the average treatment effect (ATE), in a population that can be divided into $G$ groups, and such that one has unbiased and uncorrelated estimators of the conditional average treatment effect (CATE) in each group. These conditions are for instance met in stratified randomized experiments. I assume that the outcome is homoscedastic, and that each CATE is bounded in absolute value by $B$ standard deviations of the outcome, for some known constant $B$. I derive, across all linear combinations of the CATEs' estimators, the estimator of the ATE with the lowest worst-case mean-squared error. This estimator assigns a weight equal to group $g$'s share in the population to the most precisely estimated CATEs, and a weight proportional to one over the CATE's variance to the least precisely estimated CATEs. This optimal estimator is feasible: the weights only depend on known quantities. I then allow for positive covariances known up to the outcome's variance between the estimators. This condition is met by differences-in-differences estimators in staggered adoption designs, if potential outcomes are homoscedastic and uncorrelated. Under those assumptions, I show that the minimax estimator is still feasible and can easily be computed. In realistic numerical examples, the minimax estimator can lead to substantial precision and worst-case MSE gains relative to the unbiased estimator.
翻译:我认为对平均治疗效果(ATE)的估计是平均治疗效果(ATE),在可以分为G$组的人口中,可以将平均治疗效果(ATE)分为G$组,而且每个人群都有对每个人群中条件平均治疗效果(CATE)的公正和不相干的估计。例如,这些条件是在分层随机实验中得到满足的。我假设结果是同质的,每个CATE的绝对值是按结果的标准差乘以$B$的标准差,对于某些已知的恒定值为美元。我从CATE的估测者的所有线性组合中得出,ATE的估测者具有最不偏差且最差的平均偏差。这个估测者将赋予相当于美元在人群中的比例与最准确估计的CATETATE的组比重,而这个最优估测值则取决于已知的数量。我随后允许在最不切实际的结果差上知道的ATEAEAD的测算者,如果与相对的精确度值值值差,则该估测算者将得出一个比值。