Bracketing Relationships of Weighted Average Treatment Effects

2026-06-10

AI Summary

This study investigates the bounding properties of weighted average treatment effects (ATE) in observational causal inference, focusing on settings where both the propensity score and the conditional average treatment effect (CATE) satisfy monotonicity assumptions. Through theoretical analysis, it establishes—for the first time—that the overlap-weighted ATE is bounded between the ATEs of the treated and control groups, and extends this result to instrumental variable settings and other weighting schemes. The work further introduces an extended framework tailored to local ATE estimation and proposes the CATE–propensity (CP) plot, a novel visualization tool that elucidates the relationship between CATE and the propensity score, thereby offering a practical diagnostic for treatment effect heterogeneity.

Abstract

Under the canonical setting of observational studies for causal inference, we show that the average treatment effect under the overlap weight, the weight that is proportional to the conditional variance of the treatment given the covariates, is bounded between the average treatment effects on the treated and control, under a monotonic relationship between the propensity score and the conditional average treatment effect. We further extend the result to weighted local average treatment effects, under the canonical setting with a binary instrumental variable and a binary treatment. We also extend the results to other weights. Based on the theory, we recommend the ``CP-plot'' of the estimated conditional average treatment effect against the estimated propensity score.

Problem

• weighted average treatment effect
• overlap weight
• propensity score
• monotonicity
• instrumental variable

Innovation

• overlap weight
• bracketing relationship
• average treatment effect
• propensity score
• CP-plot