Conditional independence in sample selection models

In most sample selection models, the conditional mean function for regression error terms is independent of the regressors conditional on the selection-propensity score. Conditioning on the selection-propensity score therefore controls selection bias, a fact that underlies much of the literature on semi-parametric selection models. This note provides a general formulation of the conditional-independence properties that make it possible to control selection bias using the selection-propensity score.