Semiparametric Efficiency Gains from Parametric Restrictions on the Generalized Propensity Score

Knowledge of the propensity score weakly improves efficiency when estimating causal parameters, but what kind of knowledge is more useful? To examine this, we first derive the semiparametric efficiency bound of multivalued treatment effects when the propensity score is correctly specified by a parametric model. We then reveal which parametric structure on the propensity score enhances the efficiency even when the the model is large. Finally, we apply the general theory we develop to a stratified experiment setup and find that knowing the strata improves the efficiency, especially when the size of each stratum component is small.