High Dimensional Tobit Regression

Abstract

High-dimensional regression and regression with a left-censored response are each well-studied topics. In spite of this, few methods have been proposed which deal with both of these complications simultaneously. To fill this gap, we develop extensions of the Tobit model—a standard method for censored regression in economics—for high-dimensional estimation and inference. We focus first on estimation, introducing several penalized Tobit estimators and developing a fast algorithm which combines quadratic majorization with coordinate descent to compute the penalized Tobit solution path. Theoretically, we analyze the Tobit lasso and Tobit with a folded concave penalty, bounding the ℓ2 estimation loss for the former and proving that a local linear approximation estimator for the latter possesses the strong oracle property in an ultra high-dimensional setting. In a thorough simulation study, we assess the prediction, estimation, and selection performance of our penalized Tobit models on high-dimensional left-censored data. We then shift our attention to inference. Few methods have been developed for conducting statistical inference in high-dimensional left-censored regression. Among the methods that do exist, none are flexible enough to test general linear hypotheses—that is, all hypotheses of the form H0: Cβ*M = t. We fill this gap by developing partial penalized Wald, score, and likelihood ratio tests for testing general linear hypotheses in high-dimensional Tobit models. We derive approximate distributions for the partial penalized test statistics under the null hypothesis and local alternatives in an ultra high-dimensional setting. We propose an alternating direction method of multipliers algorithm to compute the partial penalized test statistics. Through an extensive empirical study, we show that the partial penalized Tobit tests achieve their nominal size and are consistent in a finite sample setting