Sample selection models with monotone control functions

The celebrated Heckman selection model yields a selection correction function (control function) proportional to the inverse Mills ratio, which is monotone. This paper studies a sample selection model that does not impose parametric distributional assumptions on the latent error terms, while maintaining the monotonicity of the control function. We show that a positive (negative) dependence condition on the latent error terms is sufficient for the monotonicity of the control function. The condition is equivalent to a restriction on the copula function of latent error terms. Using the monotonicity, we propose a tuning-parameter-free semiparametric estimation method and establish root n-consistency and asymptotic normality for the estimates of finite-dimensional parameters. A new test for selectivity is also developed in the presence of the shape restriction. Simulations and an empirical application are conducted to illustrate the usefulness of the proposed methods.