Estimating variances and covariances in a censored regression model

When the coefficients of a Tobit model are estimated by maximum likelihood their covariance matrix is typically, even if not necessarily, associated with the algorithm employed to maximize the likelihood. Covariance estimators used in practice are derived by: (1) the Hessian (observed information), (2) the matrix of outer products of the first derivatives of the log-likelihood (OPG version), (3) the expected Hessian (estimated information), (4) a mixture of 1 and 2 (White's QML covariance matrix). Significant differences among these estirnates are are usually interpreted as an indication of misspecification. From our Monte Carlo study this seems not to be true, unless the sample size is really very large. Even in absence of misspecification, large differences are encountered in small samples, and the sign of the differences is almost systematic. This suggests that the choice of the covariance estimator is not neutral and the results of hypotheses testing may be strongly affected by such a choice.