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. The smoothed estimator is asymptotically equivalent to the standard LAD estimator. With bootstrap critical values, the rejection probabilities of symmetrical t and $\chi^2$ tests based on the smoothed estimator are correct through O(n$^{-\gamma}$) under the null hypothesis, where $\gamma$ < 1 but can be arbitrarily close to 1. In contrast, first-order asymptotic approximations make errors of size O(n$^{-\gamma}$). These results also hold for symmetrical t and $\chi^2$ tests for censored median regression models.