On true versus estimated propensity scores for treatment effect estimation with discrete controls

The finite sample variance of an inverse propensity weighted estimator is derived in the case of discrete control variables with finite support. The obtained expressions generally corroborate widely-cited asymptotic theory showing that estimated propensity scores are superior to true propensity scores in the context of inverse propensity weighting. However, similar analysis of a modified estimator demonstrates that foreknowledge of the true propensity function can confer a statistical advantage when estimating average treatment effects.