Single-index Thresholding in Quantile Regression

Abstract

Threshold regression models are useful for identifying subgroups with heterogeneous parameters. The conventional threshold regression models split the sample based on a single and observed threshold variable, which enforces the threshold point to be equal for all subgroups of the population. In this article, we consider a more flexible single-index threshold model in the quantile regression setup, in which the sample is split based on a linear combination of predictors. We propose a new estimator by smoothing the indicator function in thresholding, which enables Gaussian approximation for statistical inference and allows characterizing the limiting distribution when the quantile process is interested. We further construct a mixed-bootstrap inference method with faster computation and a procedure for testing the constancy of the threshold parameters across quantiles. Finally, we demonstrate the value of the proposed methods via simulation studies, as well as through the application to an executive compensation data.

Keywords:

• Heterogeneity
• Mixed-bootstrap
• Quantile process
• Smoothed estimator
• Subgroup
• Threshold regression

Supplementary Material

The online supplement contains additional simulation results, and the proofs for the theorems in the article.

Acknowledgments

The authors are grateful to the editor, the AE, and two referees for their helpful suggestions that significantly improved the quality of the article. The authors would like to thank Drs Xiaodong Fan and Ping Yu for sharing the compensation data. Any opinion, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily the views of the NSF.

Additional information

Funding

The research of Zhang is supported by grant KLATASDS200204. The research of Wang was partly supported by the IR/D program from the US National Science Foundation (NSF) and the NSF grant DMS-1712760. The research of Zhu is partially supported by the National Natural Science Foundation of China 11671096, 11731011, 12071087.