High-dimensional Censored Regression via the Penalized Tobit Likelihood
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The Tobit model has long been the standard method for regression with a left-censored response in economics. In spite of its enduring popularity, the Tobit model has not been extended for high-dimensional regression. To fill this gap, we propose several penalized Tobit models for high-dimensional censored regression. We use Olsen's (1978) convex reparameterization of the Tobit negative log-likelihood as the basis for our models, leveraging the fact that the negative log-likelihood satisfies the quadratic majorization condition to develop a generalized coordinate descent algorithm for computing the solution path. Theoretically, we analyze the Tobit lasso and Tobit with a folded concave penalty, deriving a bound for the ℓ_2 estimation loss for the former and proving that a local linear approximation estimator for the latter possesses the strong oracle property. Through an extensive simulation study, we find that our penalized Tobit models provide more accurate predictions and parameter estimates than their least-squares counterparts on high-dimensional data with a censored response. We demonstrate the superior prediction and variable selection performance of the penalized Tobit models on Mroz's 1975 women's labor supply data.
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