  We develop a distribution regression model under endogenous sample selection.
This model is a semi-parametric generalization of the Heckman selection model.
It accommodates much richer effects of the covariates on outcome distribution
and patterns of heterogeneity in the selection process, and allows for drastic
departures from the Gaussian error structure, while maintaining the same level
tractability as the classical model. The model applies to continuous, discrete
and mixed outcomes. We provide identification, estimation, and inference
methods, and apply them to obtain wage decomposition for the UK. Here we
decompose the difference between the male and female wage distributions into
composition, wage structure, selection structure, and selection sorting
effects. After controlling for endogenous employment selection, we still find
substantial gender wage gap -- ranging from 21% to 40% throughout the (latent)
offered wage distribution that is not explained by composition. We also uncover
positive sorting for single men and negative sorting for married women that
accounts for a substantive fraction of the gender wage gap at the top of the
distribution.
