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 estimator's variance to the least precisely estimated CATEs. Given $B$, 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)的不偏袒和不相干的估计。例如,这些条件是在分层随机的实验中得到满足的。我假设结果是同质的,而每个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,最差的线性组合中,最差的估算者是最差的估算者,最差的估算者是最差的估算者,然后允许直率的估算者估算者以最差者为准,这个估测算者将比值等于$g$在人群中所占的份额,而一个比估测算者则比估测得最差者。鉴于$B$美元,这个最优的估测算者仅取决于已知的重量。我随后允许将得出准确的比值为正差的比值。如果测算者为结果,那么,那么,那么,那么,最差的直估测测算者将得出的数值结果的误算者将得出结果的误算结果的误差者是比值是比值,那么。