# Smoothing quantile regressions

## TL;DR

This paper introduces a fully smoothed approach to quantile regression that improves estimation accuracy, allows for efficient density estimation without dimensionality issues, and provides practical bandwidth selection rules.

## Contribution

It proposes smoothing the entire objective function in quantile regression, leading to better accuracy, differentiability, and efficient density estimation methods.

## Key findings

- Lower mean squared error compared to standard estimators
- Asymptotic differentiability of the estimator
- Effective bandwidth selection rule

## Abstract

We propose to smooth the entire objective function, rather than only the check function, in a linear quantile regression context. Not only does the resulting smoothed quantile regression estimator yield a lower mean squared error and a more accurate Bahadur-Kiefer representation than the standard estimator, but it is also asymptotically differentiable. We exploit the latter to propose a quantile density estimator that does not suffer from the curse of dimensionality. This means estimating the conditional density function without worrying about the dimension of the covariate vector. It also allows for two-stage efficient quantile regression estimation. Our asymptotic theory holds uniformly with respect to the bandwidth and quantile level. Finally, we propose a rule of thumb for choosing the smoothing bandwidth that should approximate well the optimal bandwidth. Simulations confirm that our smoothed quantile regression estimator indeed performs very well in finite samples.