# Efficiency Bounds for Distribution-Free Estimators of the Binary Choice and the Censored Regression Models

https://doi.org/0012-9682(198705)55:3<559:EBFDEO>2.0.CO;2-M
p. 559-585

Stephen R. Cosslett

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.