We propose a novel framework for conducting causal inference based on counterfactual densities. While the current paradigm of causal inference is mostly focused on estimating average treatment effects (ATEs), which restricts the analysis to the first moment of the outcome variable, our density-based approach is able to detect causal effects based on general distributional characteristics. Following the Oaxaca-Blinder decomposition approach, we consider two types of counterfactual density effects that together explain observed discrepancies between the densities of the treated and control group. First, the distribution effect is the counterfactual effect of changing the conditional density of the control group to that of the treatment group, while keeping the covariates fixed at the treatment group distribution. Second, the covariate effect represents the effect of a hypothetical change in the covariate distribution. Both effects have a causal interpretation under the classical unconfoundedness and overlap assumptions. Methodologically, our approach is based on analyzing the conditional densities as elements of a Bayes Hilbert space, which preserves the non-negativity and integration-to-one constraints. We specify a flexible functional additive regression model estimating the conditional densities. We apply our method to analyze the German East--West income gap, i.e., the observed differences in wages between East Germans and West Germans. While most of the existing studies focus on the average differences and neglect other distributional characteristics, our density-based approach is suited to detect all nuances of the counterfactual distributions, including differences in probability masses at zero.

翻译：我们提出了一种基于反事实密度的因果推断新框架。当前因果推断范式主要聚焦于平均处理效应（ATE）的估计，这限制了对结果变量一阶矩的分析，而本文基于密度的方法能够通过一般分布特征检测因果效应。遵循Oaxaca-Blinder分解方法，我们考虑两种反事实密度效应，它们共同解释了处理组与对照组密度之间的观测差异。第一，分布效应是改变对照组条件密度至处理组密度（同时保持协变量固定于处理组分布）的反事实效应。第二，协变量效应代表协变量分布假设性变化的影响。在经典的无混淆性和重叠性假设下，这两种效应均具有因果解释。在方法论上，我们的方法将条件密度视为贝叶斯希尔伯特空间的元素，从而保留非负性和积分为一的约束。我们指定了一个灵活的函数型可加回归模型来估计条件密度。我们将该方法应用于分析德国东西部收入差距，即东德人与西德人之间观测到的工资差异。现有研究大多关注平均差异而忽视了其他分布特征，而本文基于密度的方法能够检测反事实分布的所有细微差异，包括零值处的概率质量差异。