This paper introduces a new way to study cause-and-effect that looks at whole outcome distributions instead of just averages. The authors call their idea counterfactual densities. In plain words, they ask what the entire shape of one group’s income distribution would look like if it faced the same conditions as another group. That lets them find differences that a single number like the average wage can miss.

To do this the authors build on a classic tool called the Oaxaca–Blinder decomposition. That tool usually splits differences in averages into parts explained by who the people are (their characteristics) and parts explained by how those characteristics map into outcomes. Here the split is applied to densities. One part measures how the conditional income distribution would change if the relationship between characteristics and wages were those of the other group (the distribution effect). The other part measures how the income distribution would change if the mix of characteristics were different (the covariate effect).

Technically the paper treats conditional densities as objects in a space that enforces two basic rules: densities cannot be negative and must integrate to one. This is done in what the authors call a Bayes Hilbert space. They estimate those conditional densities with a flexible additive regression model built from basis functions like splines. Estimation is carried out using a Poisson approximation to make the calculations practical. The authors also report simulation checks to study how their estimator behaves in finite samples.

The method is applied to the German East–West income gap after reunification. The authors emphasize that income data are often skewed, sometimes have two peaks, and can have a mass point at zero (people with no wages). These features make average-based and quantile-based methods less informative. Using counterfactual densities, they find that the East–West gap has narrowed over about 30 years, but important differences remain. Much of the remaining gap appears to come from differences in the conditional income distribution — how characteristics translate into wages — rather than from differences in the mix of characteristics. The pattern is stronger for men, so the authors describe the gap as to a large extent a male-specific issue.