Asymptotic variance and extensions of a density-weighted-response semiparametric estimator

Building on some early works, Lewbel (2000) proposed estimators for binary and ordered discrete response models with endogenous regressors. These estimators have been extended for panel data and for truncated and censored models by later papers. The estimators are particularly innovative in that the latent linear regression functions are pulled out of the nonlinear limited dependent variable models, which are then treated as if they were the usual linear models. But understanding the estimators and their applications have been "hampered" by less-than-ideal expositions and assumptions. For this problem, this short note reviews the estimators and makes the following three points. First, the derivation and proper insight of the asymptotic variances are provided. Second, the inefficiency of the ordered discrete response version is pointed out and corrected. Third, assumptions in the panel data extension by Honoré and Lewbel (2002) are relaxed.