Inference on Welfare and Value Functionals under Optimal Treatment Assignment

2025-10-29

AI Summary

This paper addresses estimation and statistical inference for welfare and value functionals induced by nonparametric conditional average treatment effect (CATE) functions under optimal treatment assignment—i.e., intervening only when CATE ≥ 0. Methodologically, it introduces a novel framework combining semiparametric plug-in estimation with sieve methods to construct Riesz representers. It establishes, for the first time, √n-asymptotic normality of the optimal welfare functional. Furthermore, it proposes a sieve-based Riesz variance estimator applicable to general value functionals, revealing that boundary subpopulations (where CATE ≈ 0) dominate the convergence rate. The method’s finite-sample robustness is validated via manifold numerical integration and Monte Carlo simulation. Empirical application to the Job Training Partnership Act (JTPA) dataset demonstrates high-precision estimation and reliable inference for income effects of job training programs.

Technology Category

• Machine Learning: Causal Learning
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Learning to Search

Application Category

• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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.

Problem

• Estimating welfare under optimal treatment assignment rules
• Providing inference for general value functionals of CATE
• Analyzing asymptotic normality of semiparametric plug-in estimators

Innovation

• Semiparametric plug-in estimators for welfare functionals
• Sieve Riesz representer for consistent variance estimation
• Numerical integration on submanifolds for value functionals