A Monte Carlo Comparison of Semiparametric Tobit Estimators

SUMMARY
This paper focuses on a performance comparison of semiparametric Tobit estimators. Firstly, a conditional expectation
version of Horowitz's distribution-free least-squares estimator is proposed, together with a short description of the
other estimators considered in the later Monte Carlo experiment. Then, a performance comparison of the following
selected estimators is made through a Monte Carlo experiment: the standard Tobit maximum-likelihood estimator, the
Buckley-James estimator, Horowitz's distribution-free least-squares estimator, a conditional version of Horowitz's
estimator and Powell's least absolute deviations estimator. An empirical example of Engel curve estimation with zero
expenditures follows.
1. INTRODUCTION
In recent years, semiparametric estimation methods have been devised for qualitative and
limited dependent variable models to avoid making restrictive assumptions on the distribution
of the error term. Contributions to the development of these methods include those of Manski
(1975, 1985) and Cosslett (1983) for a qualitative response model, those of Powell
(1984,1986a, 1986b), Duncan (1986), Fernandez (1986), Horowitz (1986) and Ruud (1986) for
a Tobit model, and those of Powell et al. (1987) and Ichimura (1987) for a general model
encompassing the above two types of models as its special cases. In addition, Cosslett (1984)
applies his distribution-free maximum-likelihood estimator to the estimation of the selection
equation in a generalized Tobit model or a type 2 Tobit model (h la Amemiya (1985)) and
thereby introduces a distribution-free two-step estimation method comparable to Heckman's
(1976) two-step estimation. And the Buckley-James estimator (Buckley and James, 1979) is
adapted for a Tobit model by Deaton and Irish (1984), and Brannas and Laitila (1986). '
The model of interest in this paper is the following standard Tobit model set-up without the
standard distributional assumption of normality on the error term ui:
yi=xiP+ui and y i = m a x ( ~ , ~ i * ] f o r i =...l,,n ,
where ui is independently and identically distributed with some distribution function F. In (I),
yi and xi are observables while y? is an unobserved latent variable.
For the estimation of (I), a conditional expectation version of Horowitz's distribution-free
least-squares estimator is introduced in the first part of the paper, together with a short
description of some semiparametric Tobit estimators which will be included in the later Monte
Carlo experiment. Strong consistency follows directly from the strong consistency proofs by
Horowitz (1986) for his estimator under an unconditional expectation model and the almost
sure version of the proof of Lemma 1 by James and Smith (1984), once the global identification
condition of Horowitz (1986) is modified for our conditional expectation model.