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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 distributionand patterns of heterogeneity in the selection process, and allows for drasticdepartures from the Gaussian error structure, while maintaining the same leveltractability as the classical model. The model applies to continuous, discreteand mixed outcomes. We provide identification, estimation, and inferencemethods, and apply them to obtain wage decomposition for the UK. Here wedecompose the difference between the male and female wage distributions intocomposition, wage structure, selection structure, and selection sortingeffects. After controlling for endogenous employment selection, we still findsubstantial gender wage gap -- ranging from 21% to 40% throughout the (latent)offered wage distribution that is not explained by composition. We also uncoverpositive sorting for single men and negative sorting for married women thataccounts for a substantive fraction of the gender wage gap at the top of thedistribution.

Distribution Regression with Sample Selection, with an Application to Wage Decompositions in the UK