Locally robust semiparametric estimation of sample selection models without exclusion restrictions

Existing identification and estimation methods for semiparametric sample selection models rely heavily on exclusion restrictions. However, it is difficult in practice to find a credible excluded variable that has a correlation with selection but no correlation with the outcome. In this paper, we establish a new identification result for a semiparametric sample selection model without the exclusion restriction. The key identifying assumptions are nonlinearity on the selection equation and linearity on the outcome equation. The difference in the functional form plays the role of an excluded variable and provides identification power. According to the identification result, we propose to estimate the model by a partially linear regression with a nonparametrically generated regressor. To accommodate modern machine learning methods in generating the regressor, we construct an orthogonalized moment by adding the first-step influence function and develop a locally robust estimator by solving the cross-fitted orthogonalized moment condition. We prove root-n-consistency and asymptotic normality of the proposed estimator under mild regularity conditions. A Monte Carlo simulation shows the satisfactory performance of the estimator in finite samples, and an application to wage regression illustrates its usefulness in the absence of exclusion restrictions.