Efficient Estimation of Linear and Type I Censored Regression Models Under Conditional Quantile Restrictions

We consider the linear regression model with censored dependent variable, where the disturbance terms are restricted only to have zero conditional median (or other prespecified quantile) given the regressors and the censoring point. Thus, the functional form of the conditional distribution of the disturbances is unrestricted, permitting heteroskedasticity of unknown form. For this model, a lower bound for the asymptotic covariance matrix for regular estimators of the regression coefficients is derived. This lower bound corresponds to the covariance matrix of an optimally weighted censored least absolute deviations estimator, where the optimal weight is the conditional density at zero of the disturbance. We also show how an estimator that attains this lower bound can be constructed, via nonparametric estimation of the conditional density at zero of the disturbance. As a special case our results apply to the (uncensored) linear model under a conditional median restriction.