This paper considers the effect of a continuous treatment on the entire distribution of outcomes after adjusting for differences in the distribution of covariates across different levels of the treatment. Our methodology encompasses dose response functions, counterfactual distributions, and "distributional policy effects'' depending on the assumptions invoked by the researcher. We propose a three-step estimator that consists of (i) estimating the distribution of the outcome conditional on the treatment and other covariates using quantile regression; (ii) for each value of the treatment, averaging over a counterfactual distribution of the covariates holding the treatment fixed; (iii) manipulating the counterfactual distribution into a parameter of interest. We show that our estimators converge uniformly to Gaussian processes and that the empirical bootstrap can be used to conduct uniformly valid inference across a range of values of the treatment. We use our method to study intergenerational income mobility where we consider distributional effects of parents' income on child's income such as (i) the fraction of children with income below the poverty line, (ii) the variance of child's income, and (iii) the inter-quantile range of child's income -- all as a function of parents' income.

中文翻译：

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连续治疗对代际流动性的应用的分布效应*

本文考虑了在调整了不同治疗水平的协变量分布差异后，连续治疗对整个结果分布的影响。我们的方法包括剂量反应函数、反事实分布和“分布政策效应”，具体取决于研究人员援引的假设。我们提出了一个三步估计器，包括 (i) 估计以治疗为条件的结果分布和使用分位数回归的其他协变量；(ii) 对于每个治疗值，对保持治疗固定的协变量的反事实分布求平均值；(iii) 将反事实分布操纵成感兴趣的参数。我们表明我们的估计量一致地收敛到高斯过程，并且经验引导程序可用于在一系列治疗值上进行一致有效的推理。我们使用我们的方法来研究代际收入流动，我们考虑父母收入对孩子收入的分配影响，例如 (i) 收入低于贫困线的孩子的比例，(ii) 孩子收入的方差，以及 (iii)孩子收入的分位数范围——都是父母收入的函数。