Tobit model estimation and sliced inverse regression

It is not unusual for the response variable in a regression model to be subject to censoring or truncation. Tobit regression models are specific examples of such a situation, where for some observations the observed response is not the actual response, but the censoring value (often zero), and an indicator that censoring (from below) has occurred. It is well-known that the maximum likelihood estimator for such a linear model assuming Gaussian errors is not consistent if the error term is not homoscedastic and normally distributed. In this paper, we consider estimation in the Tobit regression context when homoscedasticity and normality of errors do not hold, as well as when the true response is an unspecified nonlinear function of linear terms, using sliced inverse regression (SIR). The properties of SIR estimation for Tobit models are explored both theoretically and based on extensive Monte Carlo simulations. We show that the SIR estimator is a strong competitor to other Tobit regression estimators, in that it has good properties when the usual linear model assumptions hold, and can be much more effective than other Tobit model estimators when those assumptions break down. An example related to household charitable donations demonstrates the usefulness of the SIR estimator.