Assume that one is interested in estimating an average treatment effect (ATE), equal to a weighted average of $S$ conditional average treatment effects (CATEs). One has unbiased estimators of the CATEs. One could just average the CATE estimators, to form an unbiased estimator of the ATE. However, some CATE estimators may be less precise than others. Then, downweighting the imprecisely estimated CATEs may lead to a lower mean-squared error and/or shorter confidence intervals. This paper investigates this bias-variance trade-off, by deriving minimax linear estimators of, and confidence intervals (CI) for, the ATE, under various restrictions on the CATEs. First, I assume that the magnitude of the CATEs is bounded. Then I assume that their heterogeneity is bounded. I use my results to revisit two empirical applications, and find that minimax-linear estimators and CIs lead to small but non-negligible precision gains.

翻译：假设研究者关注估计平均处理效应（ATE），该效应等于S个条件平均处理效应（CATE）的加权平均值。现有对CATE的无偏估计量。直接平均这些CATE估计量可构成ATE的无偏估计量。然而，部分CATE估计量可能比其他估计量精度更低。此时，降低低精度CATE估计量的权重可能降低均方误差和/或缩短置信区间。本文通过推导不同CATE约束条件下ATE的极小极大线性估计量及置信区间（CI），系统研究这一偏差-方差权衡问题。首先，假设CATE的幅度存在边界约束；其次，假设其异质性存在边界约束。应用本文结论重新分析两项实证研究，发现极小极大线性估计量与置信区间能带来虽小但不可忽略的精度提升。