arXiv 30 Mar 2026 · Econometrics
arXiv:2603.28470 · PDF · DOI · OpenAlex · Extracted main text
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.
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The works this paper leans on most, across its whole bibliography — not restricted to papers in our corpus. Ranked by composite intensity, which combines how often a work is mentioned, how many sections mention it, and how much of that falls in the main text rather than the appendix.
| Reference | Intensity | Mentions | Sections | Main text | |
|---|---|---|---|---|---|
| 1 | Maier, Eva-Maria and Fottner, Alexander and Greven, Sonja and Stöcke… (2025) Additive Density Regression self | 1.000 | 11 | 3 | 100% |
| 2 | DiNardo, John and Fortin, Nicole M and Lemieux, Thomas (1996) Labor Market Institutions and the Distribution of Wages, 1973-1992: A Semiparametric Approach | 1.000 | 5 | 3 | 100% |
| 3 | Kennedy, EH and Balakrishnan, S and Wasserman, LA (2023) Semiparametric Counterfactual Density Estimation | 1.000 | 5 | 3 | 100% |
| 4 | Chernozhukov, Victor and Fernández-Val, Iván and Melly, Blaise (2013) Inference on counterfactual distributions | 0.874 | 8 | 2 | 100% |
| 5 | Oaxaca, Ronald (1973) Male-female wage differentials in urban labor markets | 0.737 | 3 | 2 | 100% |
| 6 | Blinder, Alan S (1973) Wage discrimination: reduced form and structural estimates | 0.644 | 2 | 2 | 100% |
| 7 | Fortin, Nicole and Lemieux, Thomas and Firpo, Sergio (2011) Decomposition methods in economics | 0.644 | 2 | 2 | 100% |
| 8 | Maier, Eva-Maria and Stöcker, Almond and Fitzenberger, Bernd and Gre… (2025) Additive density-on-scalar regression in Bayes Hilbert spaces with an application to gender economics self | 0.511 | 2 | 1 | 100% |
| 9 | Van Den Boogart, Karl-Gerald and Egozcue, Juan José and Pawlowsky-Gl… (2010) Bayes linear spaces | 0.405 | 1 | 1 | 100% |
| 10 | van den Boogaart, Karl Gerald and Egozcue, Juan José and Pawlowsky-G… (2014) Bayes Hilbert spaces | 0.405 | 1 | 1 | 100% |
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