Estimation of a Two-Limit Tobit model with generalized Box–Cox transformation and unknown censoring thresholds

Abstract

This article considers estimation of a Two-Limit Tobit model with generalized Box–Cox transformation and unknown censoring thresholds. The maximum likelihood estimates of the censoring thresholds are the smallest and largest elements of the order statistic of the transformed dependent variable for the uncensored subsample. Conditional on the estimated censoring thresholds and the parameter of the generalized Box–Cox transformation, the model is a standard Tobit model. If the dependent variable is scaled by the geometric mean of its absolute values for the uncensored subsample, then currently available software for estimation of Tobit models may be used in conjunction with a grid search over the Box–Cox parameter to determine the globalmaximum likelihood estimates. The advantage of the models proposed in this article is that: 1) use of estimated censoring thresholds serve to directly eliminate the understatement of tail probabilities that can result from use of fixed thresholds, and 2) use of the generalized Box–Cox transformation allows greater flexibility in the shape of the distribution used to model quantitative variation in the uncensored subsample, as well as greater flexibility in the tail probabilities of the censored subsample.