We explore how much knowing a parametric restriction on propensity scores improves semiparametric efficiency bounds in the potential outcome framework. For stratified propensity scores, considered as a parametric model, we derive explicit formulas for the efficiency gain from knowing how the covariate space is split. Based on these, we find that the efficiency gain decreases as the partition of the stratification becomes finer. For general parametric models, where it is hard to obtain explicit representations of efficiency bounds, we propose a novel framework that enables us to see whether knowing a parametric model is valuable in terms of efficiency even when it is high-dimensional. In addition to the intuitive fact that knowing the parametric model does not help much if it is sufficiently flexible, we discover that the efficiency gain can be nearly zero even though the parametric assumption significantly restricts the space of possible propensity scores.

翻译：本文探讨在潜在结果框架下，已知倾向得分的参数限制能在多大程度上提升半参数效率界。针对作为参数模型的分层倾向得分，我们推导了已知协变量空间划分方式所带来效率增益的显式公式。基于这些公式，我们发现随着分层划分的细化，效率增益逐渐降低。对于一般参数模型（其效率界难以获得显式表示），我们提出了一个新颖的理论框架，该框架使我们能够判断已知高维参数模型是否在效率层面具有价值。除了“当参数模型足够灵活时，已知其形式对效率提升有限”这一直观结论外，我们还发现：即使参数假设显著限制了可能倾向得分的空间，效率增益仍可能接近于零。