Essays on the econometric theory of rank regressions

Several semiparametric estimators recently developed in the econometrics literature are based on the rank correlation between the dependent and explanatory variables. Examples include the maximum rank correlation estimator (MRC) of Han [1987], the monotone rank estimator (MR) of Cavanagh and Sherman [1998], the pairwise-difference rank estimators (PDR) of Abrevaya [2003], and others. These estimators apply to various monotone semiparametric single-index models, such as the binary choice models, the censored regression models, the nonlinear regression models, and the transformation and duration models, among others, without imposing functional form restrictions on the unknown functions and distributions. This work provides several new results on the theory of rank-based estimators. In Chapter 2 we prove that the quantiles and the variances of their asymptotic distributions can be consistently estimated by the nonparametric bootstrap. In Chapter 3 we investigate the accuracy of inference based on the asymptotic normal and bootstrap approximations, and provide bounds on the associated error. In the case of MRC and MR, the bound is a function of the sample size of order close to n^(-1/6). The PDR estimators, however, belong to a special subclass of rank estimators for which the bound is vanishing with the rate close to n^(-1/2). In Chapter 4 we study the efficiency properties of rank estimators and propose weighted rank estimators that improve efficiency. We show that the optimally weighted MR attains the semiparametric efficiency bound in the nonlinear regression model and the binary choice model. Optimally weighted MRC has the asymptotic variance close to the semiparametric efficiency bound in single-index models under independence when the distribution of the errors is close to normal, and is consistent under practically relevant deviations from the single index assumption. Under moderate nonlinearities and nonsmoothness in the data, the efficiency gains from weighting are likely to be small for MRC in the transformation model and for MRC and MR in the binary choice model, and can be large for MRC and MR in the monotone regression model. Throughout, the theoretical results are illustrated with Monte-Carlo experiments and real data examples