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The finite sample variance of an inverse propensity weighted estimator isderived in the case of discrete control variables with finite support. Theobtained expressions generally corroborate widely-cited asymptotic theoryshowing that estimated propensity scores are superior to true propensity scoresin the context of inverse propensity weighting. However, similar analysis of amodified estimator demonstrates that foreknowledge of the true propensityfunction can confer a statistical advantage when estimating average treatmenteffects.

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