An asymptotically efficient likelihood-based semiparametric estimator is derived for the censored regression (tobit) model, based on a new approach for estimating the density function of the residuals in a partially observed regression. Smoothing the self-consistency equation for the nonparametric maximum likelihood estimator of the distribution of the residuals yields an integral equation, which in some cases can be solved explicitly. The resulting estimated density is smooth enough to be used in a practical implementation of the profile likelihood estimator, but is sufficiently close to the nonparametric maximum likelihood estimator to allow estimation of the semiparametric efficient score. The parameter estimates obtained by solving the estimated score equations are then asymptotically efficient. A summary of analogous results for truncated regression is also given. Copyright The Econometric Society 2004.

We derive lower bounds on the asymptotic variances for regular distribution-free estimators of the parameters of the binary choice model and the censored regression (Tobit) model. A distribution-free (or semiparametric) estimator is one that does not require any assumption about the distribution of the stochastic error term in the model, apart from regularity conditions. For the binary choice model, we obtain an explicit lower bound for the asymptotic variance for the slope parameters, or more generally the parameters of a nonlinear regression function in the underlying latent variable model, but we find that there is no regular semiparametric estimator of the constant term (identified by requiring the error distribution to have zero median). Lower bounds are also obtained under the further assumption that the error distribution is symmetric, and in this case there is a finite lower bound for the constant term too. Comparison of the bounds with those for the classical parametric problem shows the loss of information due to lack of a priori knowledge of the functional form of the error distribution. We give the conditions for equality of the parametric and semiparametric lower bounds (in which case adaptive estimation may be possible), both with and without the assumption of a symmetric error distribution. In general, adaptive estimation is not possible, but one special case where these conditions hold is when the regression function is linear and the explanatory variables have a multivariate normal distribution. The Tobit model considered here is the censored nonlinear regression model, with a fixed censoring point. We again give an explicit lower bound for the asymptotic variance for the regression parameters, this time including a constant term (if the error term has zero median). Comparison with the corresponding lower bound for the parametric case shows that adaptive estimation is in general not possible for this model.

is a given function of the exogenous variables z and unknown parameters 9, representing the systematic component of the utility difference, and F is the distribution function of the random component of the utility difference. This paper describes a method of estimating the parameters 9 without assuming any functional form for the distribution function F, and proves that this estimator is consistent. F is also consistently estimated. The method uses maximum likelihood estimation in which the likelihood is maximized not only over the parameter 9 but also over a space which contains all distribution functions.

[A discrete-choice probability model can be estimated from a sample stratified on the choice variable by maximizing the "pseudo-likelihood," a quantity closely related to the log likelihood for a random sample. We investigate the asymptotic properties of the estimator, and show that it is consistent, asymptotically normally distributed, and satisfies a commonly used criterion for asymptotic efficiency.]