Partial identification via conditional linear programs: estimation and policy learning

TL;DR

This paper introduces a unified framework for estimating and making decisions based on partially identified parameters using covariate-dependent linear programs, with new de-biased estimators and confidence intervals.

Contribution

It develops two novel de-biased estimators for bounds from conditional linear programs and extends the methodology to policy learning with partial identification.

Findings

Proposed estimators are asymptotically normal and robust to estimation errors.

The methods enable confidence interval construction for partially identified parameters.

Application to Medicaid enrollment demonstrates practical utility.

Abstract

Many important quantities of interest are only partially identified from observable data: the data can limit them to a set of plausible values, but not uniquely determine them. This paper develops a unified framework for covariate-assisted estimation, inference, and decision making in partial identification problems where the parameter of interest satisfies a series of linear constraints, conditional on covariates. In such settings, bounds on the parameter can be written as expectations of solutions to conditional linear programs that optimize a linear function subject to linear constraints, where both the objective function and the constraints may depend on covariates and need to be estimated from data. Examples include estimands involving the joint distributions of potential outcomes, policy learning with inequality-aware value functions, and instrumental variable settings. We propose two de-biased estimators for bounds defined by conditional linear programs. The first directly solves the conditional linear programs with plugin estimates and uses output from standard LP solvers to de-bias the plugin estimate, avoiding the need for computationally demanding vertex enumeration of all possible solutions for symbolic bounds. The second uses entropic regularization to create smooth approximations to the conditional linear programs, trading a small amount of approximation error for improved estimation and computational efficiency. We establish conditions for asymptotic normality of both estimators, show that both estimators are robust to first-order errors in estimating the conditional constraints and objectives, and construct Wald-type confidence intervals for the partially identified parameters. These results also extend to policy learning problems where the value of a decision policy is only partially identified. We apply our methods to a study on the effects of Medicaid enrollment.