We develop a distribution regression model under endogenous sample selection. This model is a semiparametric generalization of the Heckman selection model that accommodates much richer patterns of heterogeneity in the selection process and effect of the covariates. The model applies to continuous, discrete and mixed outcomes. We study the identification of the model, and develop a computationally attractive two-step method to estimate the model parameters, where the first step is a probit regression for the selection equation and the second step consists of multiple distribution regressions with selection corrections for the outcome equation. We construct estimators of functionals of interest such as actual and counterfactual distributions of latent and observed outcomes via plug-in rule. We derive functional central limit theorems for all the estimators and show the validity of multiplier bootstrap to carry out functional inference. We apply the methods to wage decompositions in the UK using new data. Here we decompose the difference between the male and female wage distributions into four effects: composition, wage structure, selection structure and selection sorting. 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 observable labor market characteristics. 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. These findings can be interpreted as evidence of assortative matching in the marriage market and glass-ceiling in the labor market.
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