A Unified Inference for Predictive Quantile Regression

Abstract

The asymptotic behavior of quantile regression inference becomes dramatically different when it involves a persistent predictor with zero or nonzero intercept. Distinguishing various properties of a predictor is empirically challenging. In this article, we develop a unified predictability test for quantile regression regardless of the presence of intercept and persistence of a predictor. The developed test is a novel combination of data splitting, weighted inference, and a random weighted bootstrap method. Monte Carlo simulations show that the new approach displays significantly better size and power performance than other competing methods in various scenarios, particularly when the predictive regressor contains a nonzero intercept. In an empirical application, we revisit the quantile predictability of the monthly S&P 500 returns between 1980 and 2019. Supplementary materials for this article are available online.

KEYWORD:

• Predictive quantile regression
• Random weighted bootstrap
• Test
• Weighted inference

Supplementary Materials

The supplementary appendix provides the technical proofs of Theorems 1–2 in the main paper and some additional Monte Carlo simulation results.

Acknowledgments

The authors would like to thank Dr. Marina Vannucci (co-editor), an associate editor, and three anonymous reviewers for helpful suggestions that remarkably improve the article. The authors also sincerely thank Dr. Ji Hyung Lee and Dr. Xiaosai Liao for generously sharing the replication codes of the IVX-QR and DW-QR methods.

Notes

1 It is worth mentioning that Zhu, Cai, and Peng (2014) also develop a data splitting strategy, which can handle the intercept effect using the technique of differencing over long-range observations for an ordinary predictive regression model. However, their method can not be applied to heteroscedastic errors.

2 We also examine the scenario, which allows ${\alpha }_{\tau }$ to vary at different quantile levels and find similar results. The detailed results are available upon request. We thank one anonymous reviewer for suggesting this.

3 To apply IVX-QR, we generate the instrument $z_t=R_{nz}{z}_{t-1}+\Delta x_t$ where $\Delta x_t=x_t-x_{t-1},R_{nz}=I_k+n^{\delta }{C}_z$ and $C_z=c_z{I}_k$, with I_k being a k × k identity matrix, $c_z=-5$ as suggested by Lee (2016), and δ selected by the practical rule in Lee (2016). To apply MBB-QR, we generate 1000 random samples in each simulation and set block length $b=\lceil n^{1/4}\rceil$ where $\lceil a\rceil$ denotes the least integer that is greater or equal to a.

4 The result for μ = 0 is similar and documented by Figure A1 in the supplementary appendix.

5 We thank one anonymous reviewer for suggesting a comparison of the two methods under a bivariate predictive QR model.

6 The data can be downloaded from a link given in the article titled “A Comprehensive Look at the Empirical Performance of Equity Premium Prediction” at Professor Amit Goyal’s homepage: https://sites.google.com/view/agoyal145.

Additional information

Funding

Liu’s research is supported by the NSF of China (Grant No.11971208), the National Social Science Foundation of China (21&ZD152), and the Outstanding Youth Fund Project of the Science and Technology Department of Jiangxi Province (No. 20224ACB211003). Long’s research is partly supported by the 2021 Murphy Seed Grant and the SLA Faculty Research Award from Tulane University. Peng’s research is partly supported by the Simons Foundation and the NSF grant of DMS-2012448. Yang’s research is partly supported by the National Natural Science Foundation of China (72173140, 71991474), Humanity and Social Science Youth Foundation of the Ministry of Education of China (19YJC790166), Guangdong Basic and Applied Basic Research Foundation (2022A1515011793).