Point-Identifying Semiparametric Sample Selection Models with No Excluded Variable

Sample selection is pervasive in applied economic studies. This paper proposes semiparametric selection models that achieve point identification without relying on exclusion restrictions. Our identification conditions require at least one continuously distributed covariate and certain nonlinearity in the selection process. We propose a two-step sieve plug-in estimator that is $\sqrt{n}$-consistent, asymptotically normal, and computationally straightforward, allowing for heteroskedasticity. We further derive the semiparametric efficiency bound for the model and propose a weighted variant of the estimator that attains the bound. Our approach provides a middle ground between Lee (2009)'s nonparametric bounds and Honor\'e and Hu (2020)'s linear selection bounds, while ensuring point identification. Simulation evidence confirms its excellent finite-sample performance. We apply our method to estimate the racial and gender wage disparities using data from the US Current Population Survey. Our estimates often lie outside the Honor\'e and Hu bounds.