Canonical-weight efficiency theory for overlap-weighted estimands

Develop a canonical-weight efficiency theory for the average treatment effect on the overlap-weighted population by fixing a reference propensity score, such as the propensity score based on all pretreatment covariates, and deriving the corresponding single-functional gradient and efficiency bound for comparing adjustment sets.

Background

For nonaffine propensity-dependent weights such as the overlap weight h(e)=e(1-e), adjusting for an instrument can alter the estimand as well as the efficiency bound. Consequently, efficiency comparisons across adjustment sets are not well-defined until the target functional is fixed independently of the adjustment set. The paper proposes evaluating the overlap weight at a designated reference propensity based on the full pretreatment covariate set, but does not derive the resulting functional’s influence function or efficiency bound.

Resolving this problem would provide a principled efficiency framework for overlap-weighted causal effects in which different valid adjustment sets estimate the same parameter, rather than potentially different adjustment-set-indexed parameters. The authors identify this as the most useful open problem left by the paper.

References

An efficiency theory for the ATO must therefore first fix a canonical weight --- the overlap weight evaluated at a designated reference propensity, such as the full pretreatment covariate propensity --- and compare adjustment sets for the estimation of that fixed functional. We leave this to future work; the functional so defined is a single parameter of the full law, and its gradient and bound are not those of Lemma 2.

— Optimal Covariate Adjustment beyond the Average Treatment Effect: Treated-Population and Overlap-Weighted Estimands  (2609.11222 - Okubo, 10 Sep 2026) in Section 7, immediately following Proposition 1; reiterated in Section 11, “Limitations and extensions”