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

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 summation and scalar multiplication, we formulate the model for densities in a Bayes Hilbert space with respect to an arbitrary finite measure. This enables us to not only consider continuous densities, but also, e.g., discrete or mixed densities. Mixed densities 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. We discuss interpretation of effect functions in our model in terms of odds-ratios. Estimation is based on a gradient boosting algorithm that allows for a potentially large number of flexible covariate effects. We show how to handle the challenge of estimation for mixed densities within our framework using an orthogonal decomposition. Applying this 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, differences between couples with and without minor children, as well as trends over time.