Smoothing quantile regressions

We propose to smooth the entire objective function rather than only the check function in a linear quantile regression context. We derive a uniform Bahadur-Kiefer representation for the resulting convolution-type kernel estimator that demonstrates it improves on the extant quantile regression estimators in the literature. In addition, we also show that it is straightforward to compute asymptotic standard errors for the quantile regression coefficient estimates as well as to implement Wald-type tests. Simulations confirm that our smoothed quantile regression estimator performs very well in finite samples