This paper addresses the problem of estimation of a nonparametric regression
function from selectively observed data when selection is endogenous. Our approach
relies on independence between covariates and selection conditionally
on potential outcomes. Endogeneity of regressors is also allowed for. In both
cases, consistent two-step estimation procedures are proposed and their rates of
convergence are derived. Also pointwise asymptotic distribution of the estimators
is established. In addition, we propose a nonparametric specification test
to check the validity of our independence assumption. Finite sample properties
are illustrated in a Monte Carlo simulation study and an empirical illustration.
Endogenous selection, instrumental variable, sieve minimum distance, regression
estimation, convergence rate, asymptotic normality, hypothesis testing, inverse problem.