Two-Step Quantile Estimation Of The Censored Regression Model

It has previously been shown that consistent estimation of the unknown coefficients of the censored regression (or censored "Tobit") model can be obtained using a quantile estimation approach. Because the functional form of the conditional quantiles of the censored dependent variable does not depend on the parametric form of the distribution of the error terms, the quantile estimators are consistent (and asymptotically normal) under much more general conditions than likelihood-based estimation methods. However, simulation studies have shown that quantile estimates for this model have a small-sample bias in the opposite direction from the well-known least-squares bias for censored data. This bias results from the particular form of the conditional quantile function; quantile estimation in this problem uses only those data points with a positive regression function to estimate the regression coefficients, and the simultaneous estimation of the sign of the regression function and the magnitude of the coefficients induces an asymmetry in the small-sample distribution of the coefficient estimates. The present paper proposes a two-step quantile estimator for this model which is designed to overcome this finite-sample bias. Specifically, the first stage applies quantile estimation to the sign of the observed dependent variable (i.e., the quantile generalization of the "maximum score" estimator for a binary response model) to estimate the sign of the underlying regression function, while the second stage uses the data points with positive (estimated) regression functions to estimate the regression coefficient magnitudes using standard linear quantile estimation. By separating the estimation of the sign of the regression function from estimation of the coefficents themselves, the finite-sample bias of the quantile approach for this model is attenuated. The paper gives conditions under which the two-step estimator is asymptotically equivalent to the one-step version, so that no efficiency loss is incurred from the two-step approach. Also, a small scale efficiency study will document the small-sample benefits of the proposed method.