Counterfactual Density Effects and the German East--West Income Gap

2026-03-30

AI Summary

This study addresses the limitations of traditional causal inference, which focuses primarily on average treatment effects and fails to capture the full distributional structure of income disparities between eastern and western Germany. The authors propose a novel counterfactual density–based causal inference framework that extends causal analysis to the entire outcome distribution. By modeling conditional densities within a Bayesian Hilbert space, the approach guarantees non-negativity and unit integral constraints. Integrating insights from the Oaxaca–Blinder decomposition, the framework identifies distinct distributional and covariate effects. Empirical application reveals multidimensional differences in wage distributions across regions, including disparities in the probability mass at zero income, offering policymakers nuanced insights beyond mean comparisons.

Abstract

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.

Problem

• counterfactual density
• distributional effects
• income gap
• causal inference
• Oaxaca-Blinder decomposition

Innovation

• counterfactual densities
• causal inference
• Bayes Hilbert space
• distributional effects
• functional additive regression