Semiparametric and nonparametric estimation of tobit models

This dissertation focuses on the research on the semiparametric and nonparametric estimation of Tobit models such as the truncated and censored regression models, bivariate Tobit models, and censored sample selection models. The applied methods are not traditional compared with the moment-based approaches in the literature on the estimation of the models with limited dependent variables. The dissertation contains four chapters. The first chapter provides a new semiparametric estimator for the coefficients in the truncated regression model based on the least-squares estimation of the conditional survival function for the truncated dependent variable in contrast with the moment-based semiparamtric estimation approach in the literature. Identification conditions about the error term and the regressors for the estimation are also presented, where the non-periodic condition for the hazard function of the error term is much weaker than the log concavity condition for the density function of the error term in the literature. The proposed estimator is shown to be root-n-consistent and asymptotically normal by the theory of U-statistic from Sherman (1994) and the empirical process from Pakes and Pollard (1989). The simulation study shows that our estimator performs well and better than the estimators in the literature in the designs. The method in the chapter can also be generalized to the semiparametric estimation for censored regression models. The second chapter proposes a semiparametric estimation for the bivariate Tobit model. This model was introduced by Amemiya (1974) who gave an estimation based on the joint normality assumption on the error terms. The limitation of the normality assumption and the inefficiency of the Amemiya’s estimator motivate us to semiparametrically estimate the model. So far, this problem has not been studied in the literature. Instead of starting from a simple relationship between the first and second moments of the dependent variables as Amemiya (1974), we begin with the conditional expectation of the joint indicator functions of the dependent variables and construct an integrated least-squares type sample objective function, the minimizer of which is the proposed estimator. The development in the empirical process such as the degenerate U-statistic and the decomposition theory of U-statistic provides us a solution to prove that the estimator is consistent and asymptotically normal. The simulation results show that our estimator performs better than Amemiya's with smaller standard errors and mean squares errors even in the specification of the joint normality for the error terms. More importantly, unlike Amemiya's method based on the normality assumption, our estimator has the good large sample properties for a wide class of error distributions and performs well in small samples in the simulation for other designs for the error terms. The distribution-free property for this estimation makes it more applicable in empirical work. The third chapter considers the nonparametric estimation of the censored regression models. By a location relationship about the conditional survival function of the censored dependent variable, we construct a nonparametric estimator for the regression function, which is the minimizer of an integrated least-squares type sample objective function, based on the kernel estimation for the conditional survival function. Under some regularity conditions and the conditions about the kernel and the bandwidth, we show that the nonparametric estimator is consistent and asymptotically normal. Contrast with other nonparametric estimators for the censored model, our estimator is constructed not based on the mean and variance of the dependent variable and hence is expected to perform better than the moment-based estimators for this kind of models. Simulation studies show that the proposed estimator performs well and better than or similarly to Lewbel and Linton (2002)'s, which is a moment-based estimator. The last chapter studies the nonparametric estimation of the censored model subject to nonparametric sample selection. The estimation is of importance since the variable of interest in application is often subject to sample selection and censoring (such as top-coding) at the same time. So far, there is no nonparametric study on this field in the literature. The special case of this model, where there is no censoring on the limited dependent variable, is the familiar sample selection model for which Das, Newey and Vella (2003) present a nonparametric estimation based on the series estimation by using an additive relationship for the conditional expectation of the dependent variable. By following the insight of the estimation procedure in Chen (2003), this chapter proposes a nonparametric kernel estimation for the general model above through a location relationship about the conditional survival function of the censored and selected dependent variable. Under some regularity conditions and the conditions about the kernel and the bandwidth, the nonparametric estimator is shown to be consistent and asymptotically normal. A simulation study shows that the proposed estimator performs well in the designs for the censored sample selection model. For the special case of sample selection model, our estimator allows for a more general class of regression functions in the outcome equation than the Lipschiz class required in Das, Newey and Vella (2003). We present some designs for this special case to compare the performance of our estimator and the DNV’s estimator