Efficient Semiparametric Estimators for Nonlinear Regressions and Models under Sample Selection Bias

Abstract

We study the consistency, robustness and efficiency of parameter estimation in different but related models via semiparametric approach. First, we revisit the second- order least squares estimator proposed in Wang and Leblanc (2008) and show that the estimator reaches the semiparametric efficiency. We further extend the method to the heteroscedastic error models and propose a semiparametric efficient estimator in this more general setting. Second, we study a class of semiparametric skewed distributions arising when the sample selection process causes sampling bias for the observations. We begin by assuming the anti-symmetric property to the skewing function. Taking into account the symmetric nature of the population distribution, we propose consistent estimators for the center of the symmetric population. These estimators are robust to model misspecification and reach the minimum possible estimation variance. Next, we extend the model to permit a more flexible skewing structure. Without assuming a particular form of the skewing function, we propose both consistent and efficient estimators for the center of the symmetric population using a semiparametric method. We also analyze the asymptotic properties and derive the corresponding inference procedures. Numerical results are provided to support the results and illustrate the finite sample performance of the proposed estimators.