Causal inference with observational studies trimmed by the estimated propensity scores

Causal inference with observational studies often relies on the assumptions of unconfoundedness and overlap of covariate distributions in different treatment groups. The overlap assumption is violated when some units have propensity scores close to zero or one, and therefore both theoretical and practical researchers suggest dropping units with extreme estimated propensity scores. We advance the literature in three directions. First, we clarify a conceptual issue of sample trimming by defining causal parameters based on a target population without extreme propensity score. Second, we propose a procedure of smooth weighting, which approximates the existing sample trimming but has better asymptotic properties. The new weighting estimator is asymptotically linear and the bootstrap can be used to construct confidence intervals. Third, we extend the theory to the average treatment effect on the treated, suggesting trimming samples with estimated propensity scores close to one.