Policy Relevant Treatment Effects with Multidimensional Unobserved Heterogeneity

TL;DR

This paper develops a unified, convex relaxation-based framework for bounding policy relevant treatment effects in the presence of multidimensional unobserved heterogeneity, improving robustness and computational simplicity.

Contribution

It introduces a novel convex relaxation approach to derive conservative bounds on treatment effects with multidimensional unobserved heterogeneity, extending prior bounds and allowing shape restrictions.

Findings

Bounds are computationally simple and conservative.

Bounds extend and robustify previous methods under threshold-crossing assumptions.

Numerical results show the bounds are informative and applicable.

Abstract

This paper provides a unified framework for bounding policy relevant treatment effects using instrumental variables. In this framework, the treatment selection may depend on multidimensional unobserved heterogeneity. We derive bilinear constraints on the target parameter by extracting information from identifiable estimands. We apply a convex relaxation method to these bilinear constraints and provide conservative yet computationally simple bounds. Our convex-relaxation bounds extend and robustify the bounds by Mogstad, Santos, and Torgovitsky (2018) which require the threshold-crossing structure for the treatment: if this condition holds, our bounds are simplified to theirs for a large class of target parameters; even if it does not, our bounds include the true parameter value whereas theirs may not and are sometimes empty. Linear shape restrictions can be easily incorporated to narrow the proposed bounds. Numerical and simulation results illustrate the informativeness of our convex-relaxation bounds.