Trading-off Bias and Variance When the Size of the Treatment Effect is Bounded

TL;DR

This paper explores the bias-variance trade-off in estimating average treatment effects when CATEs are bounded, proposing minimax-linear estimators to improve precision.

Contribution

It introduces minimax-linear estimators for ATE under bounded CATEs and heterogeneity, enhancing estimation accuracy and confidence interval precision.

Findings

Minimax estimators reduce mean-squared error by about 5%.

Confidence intervals become more precise with the proposed method.

Revisiting empirical applications demonstrates practical benefits.

Abstract

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, of around 5\%.