Estimation of a Tobit model with unknown censoring threshold

Conventional wisdom suggests that only the estimated intercept is affected by imposition of a zero censoring threshold on a Tobit model. This is true for Heckman-Lee estimation. For maximum likelihood (ML) estimation, however, it is only true if the censoring threshold is known and is subtracted from the dependent variable. Failure to properly transform the dependent variable prior to ML estimation of a zero threshold Tobit model will generally bias the coefficient estimates. A long neglected topic is ML estimation of a Tobit model with common, but unknown, censoring threshold. This paper shows that the ML estimator of the censoring threshold is the minimum order statistic from the observed subsample, and that existing software for estimation of a zero-threshold Tobit model is easily adapted to include estimation of the censoring threshold.