Nonparametric Quantile Regression Estimation With Mixed Discrete and Continuous Data

Abstract

In this article, we investigate the problem of nonparametrically estimating a conditional quantile function with mixed discrete and continuous covariates. A local linear smoothing technique combining both continuous and discrete kernel functions is introduced to estimate the conditional quantile function. We propose using a fully data-driven cross-validation approach to choose the bandwidths, and further derive the asymptotic optimality theory. In addition, we also establish the asymptotic distribution and uniform consistency (with convergence rates) for the local linear conditional quantile estimators with the data-dependent optimal bandwidths. Simulations show that the proposed approach compares well with some existing methods. Finally, an empirical application with the data taken from the IMDb website is presented to analyze the relationship between box office revenues and online rating scores. Supplementary materials for this article are available online.

KEYWORDS:

• Bandwidth selection
• Discrete regressors
• Local linear smoothing
• Nonparametric estimation
• Quantile regression

Supplementary Materials

The document contains: (A) proofs of some supplemental results; (B) weak convergence of quantile regression estimation uniformly over τ; and (C) point-wise confidence intervals in the empirical study.

Acknowledgments

The authors would like to thank a co-editor, an associate editor, and two anonymous reviewers for their insightful comments that substantially improve an early version of the article.