Doubly robust estimation of average treatment effect revisited

TL;DR

This paper systematically revisits doubly robust estimators for average treatment effect, analyzing their efficiency under various model specifications and misspecifications to guide practical application.

Contribution

It provides a comprehensive comparison of all nine combinations of propensity score and outcome regression models, including under misspecification, revealing efficiency properties and phenomena.

Findings

All correctly specified models achieve the semiparametric efficiency bound.

Misspecified outcome regression increases asymptotic variance beyond the bound.

Misspecified propensity score can lead to super-efficiency in some cases.

Abstract

The research described herewith is to re-visit the classical doubly robust estimation of average treatment effect by conducting a systematic study on the comparisons, in the sense of asymptotic efficiency, among all possible combinations of the estimated propensity score and outcome regression. To this end, we consider all nine combinations under, respectively, parametric, nonparametric and semiparametric structures. The comparisons provide useful information on when and how to efficiently utilize the model structures in practice. Further, when there is model-misspecification, either propensity score or outcome regression, we also give the corresponding comparisons. Three phenomena are observed. Firstly, when all models are correctly specified, any combination can achieve the same semiparametric efficiency bound, which coincides with the existing results of some combinations. Secondly, when the propensity score is correctly modeled and estimated, but the outcome regression is misspecified parametrically or semiparametrically, the asymptotic variance is always larger than or equal to the semiparametric efficiency bound. Thirdly, in contrast, when the propensity score is misspecified parametrically or semiparametrically, while the outcome regression is correctly modeled and estimated, the asymptotic variance is not necessarily larger than the semiparametric efficiency bound. In some cases, the "super-efficiency" phenomenon occurs. We also conduct a small numerical study.