Endogenous selection or treatment model estimation

In a sample selection or treatment effects model, common unobservables may affect both the outcome and the probability of selection in unknown ways. This paper shows that the distribution function of potential outcomes, conditional on covariates, can be identified given an observed variable V$V$ that affects the treatment or selection probability in certain ways and is conditionally independent of the error terms in a model of potential outcomes. Selection model estimators based on this identification are provided, which take the form of simple weighted averages, GMM, or two stage least squares. These estimators permit endogenous and mismeasured regressors. Empirical applications are provided to estimation of a firm investment model and a schooling effects on wages model.