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进行因果推断的新框架。当前因果推断范式主要聚焦于平均处理效应（ATEs）的估计，这仅将分析限制在结果变量的第一矩，而我们的基于密度的方法能够基于一般分布特征检测因果效应。遵循Oaxaca-Blinder分解方法，我们考虑两类共同解释处理组与对照组密度之间观测差异的反事实密度效应。其一，分布效应是指在保持协变量固定于处理组分布的情况下，将对照组的条件密度改变为处理组条件密度的反事实效应。其二，协变量效应则代表协变量分布假设性变化的影响。在经典无混淆性假设和重叠假设下，这两种效应均具备因果解释。方法论上，我们的方法将条件密度分析为贝叶斯希尔伯特空间的元素，从而保持了非负性和积分为一的约束。我们设定了一个灵活的函数加性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，即东德人与西德人在工资上的观测差异。尽管多数现有研究侧重于平均差异而忽略其他分布特征，但我们的基于密度的方法能够捕捉反事实分布的所有细微特征，包括零处的概率质量差异。