Additive Density-on-Scalar Regression in Bayes Hilbert Spaces with an Application to Gender Economics

TL;DR

This paper introduces a novel additive density-on-scalar regression model within Bayes Hilbert spaces, enabling flexible analysis of probability densities, including mixed types, with an application to gender income share data in Germany.

Contribution

It develops a new regression framework for densities in Bayes Hilbert spaces, accommodating mixed densities and providing interpretable effects, with a gradient boosting estimation approach.

Findings

East German couples have more symmetric income share distributions.

West German couples show a persistent child penalty.

Differences between East and West German couples decrease over time.

Abstract

Motivated by research on gender identity norms and the distribution of the woman's share in a couple's total labor income, we consider functional additive regression models for probability density functions as responses with scalar covariates. To preserve nonnegativity and integration to one under vector space operations, we formulate the model for densities in a Bayes Hilbert space, which allows to not only consider continuous densities, but also, e.g., discrete or mixed densities. Mixed ones occur in our application, as the woman's income share is a continuous variable having discrete point masses at zero and one for single-earner couples. Estimation is based on a gradient boosting algorithm, allowing for potentially numerous flexible covariate effects and model selection. We develop properties of Bayes Hilbert spaces related to subcompositional coherence, yielding (odds-ratio) interpretation of effect functions and simplified estimation for mixed densities via an orthogonal decomposition. Applying our approach to data from the German Socio-Economic Panel Study (SOEP) shows a more symmetric distribution in East German than in West German couples after reunification and a smaller child penalty comparing couples with and without minor children. These West-East differences become smaller, but are persistent over time.