A multiplier bootstrap score test for Tobit specification

Abstract

In this paper, we propose a simple score-based specification test for the Tobit regression model, implemented through a computationally efficient multiplier bootstrap. We test correct specification through the conditional moment restriction that the generalized Tobit residual has zero conditional mean, and construct a feasible score-based empirical process using the integrated conditional moments approach, conditioning on the estimated index so that no smoothing or tuning parameter is required. We derive a first-order uniform representation of the feasible process, showing how its limiting behavior is affected by maximum-likelihood estimation of the structural parameter, and obtain the asymptotic properties of the corresponding Kolmogorov–Smirnov statistic under the null, fixed alternatives, and a sequence of local alternatives. Since the limiting null distribution is non-pivotal, $p$-values are obtained from a multiplier bootstrap that mimics the first-order representation and therefore captures the parameter estimation uncertainty. Monte Carlo experiments, including an unknown-scale design, an index-visible/index-invisible power study, and comparisons with existing specification tests, confirm accurate size and competitive power across a range of designs and covariate dimensions. Two empirical applications further demonstrate the practical utility of our approach.