Properties of the Tobit Maximum Likelihood Estimator under Nonstandard Assumptions
In this section we shall discuss the properties of the Tobit MLE—the estimator that maximizes (10.2.5)—under various types of nonstandard assumptions: heteroscedasticity, serial correlation, and nonnormality. It will be shown that the Tobit MLE remains consistent under serial correlation but not under heteroscedasticity or nonnormality. The same is true of the other estimators considered earlier. This result contrasts with the classical regression model in which the least squares estimator (the MLE under the normality assumption) is generally consistent under all of the three types of nonstandard assumptions mentioned earlier.
Before proceeding with a rigorous argument, we shall give an intuitive explanation of the aforementioned result. By considering (10.4.11) we see that serial correlation of y, should not affect the consistency of the NLLS estimator, whereas heteroscedasticity changes a to at and hence invalidates the estimation of the equation by least squares. If у * is not normal, Eq. (10.4.11) itself is generally invalid, which leads to the inconsistency of the NLLS estimator.
Although the NLLS estimator is different from the ML estimator, we can expect a certain correspondence between the consistency properties of the two estimators.