On the Asymptotics of the Minimax Linear Estimator

Abstract

Many causal estimands, such as average treatment effects under unconfoundedness, can be written as continuous linear functionals of an unknown regression function. We study a weighting estimator that sets weights by a minimax procedure: solving a convex optimization problem that trades off worst-case conditional bias against variance. Despite its growing use, general root-$n$ theory for this method has been limited. This paper fills that gap. Under regularity conditions, we show that the minimax linear estimator is root-$n$ consistent and asymptotically normal, and we derive its asymptotic variance. These results justify ignoring worst-case bias when forming large-sample confidence intervals and make inference less sensitive to the scaling of the function class. With a mild variance condition, the estimator attains the semiparametric efficiency bound, so an augmentation step commonly used in the literature is not needed to achieve first-order optimality. Evidence from simulations and three empirical applications, including job-training and minimum-wage policies, points to a simple rule: in designs satisfying our regularity conditions, standard-error confidence intervals suffice; otherwise, bias-aware intervals remain important.