Semi-parametric estimation in Tobit regression models

Abstract

In the classical Tobit regression model, the regression error term is often assumed to have a zero mean normal distribution with unknown variance, and the regression function is assumed to be linear. If the normality assumption is violated, then the commonly used maximum likelihood estimate becomes inconsistent. Moreover, the likelihood function will be very complicated if the regression function is nonlinear even the error density is normal, which makes the maximum likelihood estimation procedure hard to implement. In the full nonparametric setup when both the regression function and the distribution of the error term [epsilon] are unknown, some nonparametric estimators for the regression function has been proposed. Although the assumption of knowing the distribution is strict, it is a widely adopted assumption in Tobit regression literature, and is also confirmed by many empirical studies conducted in the econometric research. In fact, a majority of the relevant research assumes that [epsilon] possesses a normal distribution with mean 0 and unknown standard deviation. In this report, we will try to develop a semi-parametric estimation procedure for the regression function by assuming that the error term follows a distribution from a class of 0-mean symmetric location and scale family. A minimum distance estimation procedure for estimating the parameters in the regression function when it has a specified parametric form is also constructed. Compare with the existing semiparametric and nonparametric methods in the literature, our method would be more efficient in that more information, in particular the knowledge of the distribution of [epsilon], is used. Moreover, the computation is relative inexpensive. Given lots of application does assume that [epsilon] has normal or other known distribution, the current work no doubt provides some more practical tools for statistical inference in Tobit regression model.