Overlap weight and propensity score residual for heterogeneous effects: A review with extensions

Abstract

Individual responses to a treatment D=0,1 differ, depending on covariates X. Averaging such a heterogeneous effect is usually done with the density of X, but averaging with ‘overlap weight (OW)’ is also often done, where OW is the normalized version of PS×(1-PS) with PS denoting the propensity score. OW attains its maximum at PS=0.5, i.e., when subjects in one group have the best overlap with the other group, and OW accords several advantages to treatment effect estimators as reviewed in this paper. First, matching with OW addresses the non-overlapping support problem in a built-in way, without an arbitrary user intervention. Second, inverse probability weighting with OW overcomes the “too small denominator problem”, and can be efficient as well. Third, regression adjustment with OW is robust to misspecified outcome regression models. Fourth, covariate balance holds exactly, if OW is estimated by the generalized method of moment. In these advantages, the PS residual ‘D−PS’ plays a central role. We also discuss some shortcomings of OW, and show how seemingly unrelated estimators are in fact closely related through OW. Finally, we provide an empirical illustration.