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.

翻译：已知倾向得分可以在因果参数估计中微弱地提升效率，但何种知识更具实用性？为探究此问题，我们首先推导了当倾向得分由参数化模型正确指定时多值处理效应的半参数效率界。随后揭示即使模型规模较大，倾向得分的何种参数化结构仍能增强效率。最后，将所发展的通用理论应用于分层实验场景，发现掌握分层信息能提升效率，尤其当各层组成部分的样本量较小时。