Inference on Welfare and Value Functionals under Optimal Treatment Assignment

TL;DR

This paper develops theoretical methods for estimating and conducting inference on welfare and value functionals of the conditional average treatment effect under optimal treatment assignment, with applications to policy evaluation.

Contribution

It introduces asymptotic normality results and variance estimation techniques for welfare and value functionals in treatment effect models, including computational procedures.

Findings

Semiparametric estimators are asymptotically normal with $\

Good finite-sample performance demonstrated in simulations.

Application to job training program data shows practical utility.

Abstract

We provide theoretical results for the estimation and inference of a class of welfare and value functionals of the nonparametric conditional average treatment effect (CATE) function under optimal treatment assignment, i.e., treatment is assigned to an observed type if and only if its CATE is nonnegative. For the optimal welfare functional defined as the average value of CATE on the subpopulation with nonnegative CATE, we establish the $\sqrt{n}$ asymptotic normality of the semiparametric plug-in estimators and provide an analytical asymptotic variance formula. For more general value functionals, we show that the plug-in estimators are typically asymptotically normal at the 1-dimensional nonparametric estimation rate, and we provide a consistent variance estimator based on the sieve Riesz representer, as well as a proposed computational procedure for numerical integration on submanifolds. The key reason underlying the different convergence rates for the welfare functional versus the general value functional lies in that, on the boundary subpopulation for whom CATE is zero, the integrand vanishes for the welfare functional but does not for general value functionals. We demonstrate in Monte Carlo simulations the good finite-sample performance of our estimation and inference procedures, and conduct an empirical application of our methods on the effectiveness of job training programs on earnings using the JTPA data set.