Original Articles

Smoothing Quantile Regressions

Abstract

Abstract

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.

Key Words: Asymptotic expansion; Bahadur–Kiefer representation; Conditional quantile; Convolution-based smoothing; Data-driven bandwidth

ACKNOWLEDGMENTS

We are grateful to Antonio Galvão and Aureo de Paula for valuable comments, as well as to the associate editor and anonymous referees.

Additional information

Funding

Fernandes also thanks the financial support from CNPq (302278/2018-4).

Notes

1 For instance, Hall and Sheather (1988 Hall, P., and Sheather, S. J. (1988), “On the Distribution of a Studentized Quantile,” Journal of the Royal Statistical Society, Series B, 50, 381–391. DOI: 10.1111/j.2517-6161.1988.tb01735.x.[Crossref] , [Google Scholar]) studied higher-order accuracy of confidence intervals. See also Portnoy (2012 Portnoy, S. (2012), “Nearly Root-n Approximation for Regression Quantile Processes,” The Annals of Statistics, 40, 1714–1736. DOI: 10.1214/12-AOS1021.[Crossref], [Web of Science ®] , [Google Scholar]) and references therein.

2 In particular, Roth and Wied (2018 Roth, C., and Wied, D. (2018), “Estimating Derivatives of Function-Valued Parameters in a Class of Moment Condition Models,” University of Mannheim and University of Cologne. [Google Scholar]) developed a qdf estimator based on the estimation of the second derivative of the population QR objective function.

3 Alternatively, one could simply employ the sample analog of $τ (1 - τ) E [X X']$ to estimate $V (τ)$ , which would lead to Powell’s (1991 Powell, J. L. (1991), Estimation of Monotonic Regression Models Under Quantile Restrictions, Cambridge: Cambridge University Press, pp. 357–384. [Google Scholar]) estimator of the asymptotic covariance of the standard QR estimator. See also Angrist, Chernozhukov, and Fernández-Val (2006 Angrist, J., Chernozhukov, V., and Fernández-Val, I. (2006), “Quantile Regression Under Misspecification, With an Application to the U.S. Wage Structure,” Econometrica, 74, 539–563. DOI: 10.1111/j.1468-0262.2006.00671.x.[Crossref], [Web of Science ®] , [Google Scholar]) and Kato (2012). However, we expect that our variance estimator to entail better finite-sample properties and, accordingly, shorter confidence intervals.

4 We omit results using standard errors as in Koenker (2005 Koenker, R. (2005), Quantile Regression, Cambridge: Cambridge University Press.[Crossref] , [Google Scholar], secs. 3.4.2 and 4.10.1) because bootstrapping always entails better empirical coverage. They are available from the authors upon request.

5 It is perhaps worth mentioning that the optimal bandwidth for the QR estimation is not necessarily optimal for the qdf estimator.

Original Articles

Smoothing Quantile Regressions

Original Articles

Smoothing Quantile Regressions

ACKNOWLEDGMENTS

Additional information

Funding

Notes

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