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Ubiquitous Overlap Weight and Propensity Score Residual for Heterogeneous Treatment Effect and Its Estimation

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 the average with `overlap weight (OW)' is done "surreptitiously" with X controlled, where OW is the normalized version of PS×(1-PS) with PS denoting the propensity score. This paper brings OW "out of shadow" to advocate more use of OW. 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. First, matching with OW addresses the well-known 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 become efficient as well. Third, regression adjustment with OW is robust to misspecified outcome regression models. Fourth, covariate balance holds exactly if OW estimated by the method of moment is used. In these advantages, PS residual `D-PS' plays a central role. We also show how seemingly unrelated estimators are in fact closely related through OW, and an empirical analysis illustrates how similar the estimates can be.