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 very high-dimensional. In addition to the intuitive fact that knowing the parametric model does not help much if it is sufficiently flexible, we reveal that the efficiency gain can be nearly zero even though the parametric assumption significantly restricts the space of possible propensity scores.

翻译：我们探讨了在潜在结果框架中，了解倾向性评分的参数化限制能在多大程度上提高半参数效率边界。对于分层倾向性评分（被视为一种参数模型），我们推导出因知晓协变量空间如何划分而产生的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变得愈加精细，效率增益会逐渐降低。对于难以获得效率边界显式表示的一般参数模型，我们提出了一种新颖的框架，能够判断即使在高维情况下，了解参数模型是否在效率方面具有价值。除了直观上认知的“若参数模型足够灵活则帮助不大”之外，我们揭示出，即便参数假设显著限制了倾向性评分的可能空间，效率增益也可能几乎为零。