Non-overlap Average Treatment Effect Bounds

Abstract

The average treatment effect (ATE), the mean difference in potential outcomes under treatment and control, is a canonical causal effect. Overlap, which says that all subjects have non-zero probability of either treatment status, is necessary to identify and estimate the ATE. When overlap fails, the standard solution is to change the estimand, and target a trimmed effect in a subpopulation satisfying overlap. When the outcome is bounded, we demonstrate that this compromise is unnecessary. We derive non-overlap bounds: partial identification bounds on the ATE that do not require overlap. The bounds have width proportional to the size of the non-overlap subpopulation, making them informative in common scenarios when overlap violations are limited. Since the bounds are non-smooth functionals, we derive smooth approximations amenable to semiparametric efficiency theory and propose a Targeted Minimum Loss-Based estimator that is $\sqrt{n}$-consistent and asymptotically normal under nonparametric conditions. A multiplier bootstrap procedure yields uniformly valid confidence sets across all non-overlap subpopulation sizes and smoothing parameters, allowing researchers to report the tightest valid interval. Formally, we compare non-overlap confidence intervals to confidence intervals based on point estimation across multiple overlap regimes. We illustrate the method via simulation studies and real-world data applications.

58 pages, 8 figures