Additive density-on-scalar regression in Bayes Hilbert spaces with an application to gender economics

Motivated by research on gender identity norms and the distribution of the woman’s share in a couple’s total labor income, we consider additive regression models for densities 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, for example, 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 (linear, nonlinear, categorical, interaction etc.) covariate effects and model selection. We show useful properties of Bayes Hilbert spaces related to subcompositional coherence, also yielding new (odds-ratio) interpretations 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.