Smoothing Quantile Regressions

We propose to smooth the objective function, rather than only the indicator on 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 s...

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Bibliographic Details
Published in:Journal of business & economic statistics 2021, Vol.39 (1), p.338-357
Main Authors: Fernandes, Marcelo, Guerre, Emmanuel, Horta, Eduardo
Format: Article
Language:English
Subjects:
Asymptotic expansion
Bahadur-Kiefer representation
Conditional quantile
Convolution-based smoothing
Data-driven bandwidth
Bandwidths
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Summary:We propose to smooth the objective function, rather than only the indicator on 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. Supplementary materials for this article are available online.
ISSN:0735-0015
1537-2707
DOI:10.1080/07350015.2019.1660177