Bootstrap Methods for Median Regression Models

The least-absolute-deviations (LAD) estimator for a median-regression model does not satisfy the standard conditions for obtaining asymptotic refinements through use of the bootstrap because the LAD objective function is not smooth. This paper overcomes this problem by smoothing the objective function so that it becomes differentiable. The smoothed estimator is asymptotically equivalent to the standard LAD estimator. With bootstrap critical values, the levels of symmetrical t and c2 tests based on the smoothed estimator are correct through O(n-g), where g < 1 but can be arbitrarily close to 1. In contrast, first-order asymptotic approximations make an error of size O(n-g). The bootstrap accounts for terms of size O(n-g) in the asymptotic expansions of the test statistics, whereas first-order approximations ignore these terms. These results also hold for symmetrical t and c2 tests for censored median regression models.