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A MAX-CORRELATION WHITE NOISE TEST FOR WEAKLY DEPENDENT TIME SERIES

Published online by Cambridge University Press:  12 May 2020

Jonathan B. Hill  and
Kaiji Motegi
Jonathan B. Hill*
Affiliation:
University of North Carolina
Kaiji Motegi
Affiliation:
Kobe University
*
Address correspondence to Jonathan B. Hill, Department of Economics, University of North Carolina at Chapel Hill, Chapel Hill, NC. USA; e-mail: [email protected]; web: https://jbhill.web.unc.edu.
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Abstract

This article presents a bootstrapped p-value white noise test based on the maximum correlation, for a time series that may be weakly dependent under the null hypothesis. The time series may be prefiltered residuals. The test statistic is a normalized weighted maximum sample correlation coefficient $ \max _{1\leq h\leq \mathcal {L}_{n}}\sqrt {n}|\hat {\omega }_{n}(h)\hat {\rho }_{n}(h)|$, where $\hat {\omega }_{n}(h)$ are weights and the maximum lag $ \mathcal {L}_{n}$ increases at a rate slower than the sample size n. We only require uncorrelatedness under the null hypothesis, along with a moment contraction dependence property that includes mixing and nonmixing sequences. We show Shao’s (2011, Annals of Statistics 35, 1773–1801) dependent wild bootstrap is valid for a much larger class of processes than originally considered. It is also valid for residuals from a general class of parametric models as long as the bootstrap is applied to a first-order expansion of the sample correlation. We prove the bootstrap is asymptotically valid without exploiting extreme value theory (standard in the literature) or recent Gaussian approximation theory. Finally, we extend Escanciano and Lobato’s (2009, Journal of Econometrics 151, 140–149) automatic maximum lag selection to our setting with an unbounded lag set that ensures a consistent white noise test, and find it works extremely well in controlled experiments.

Type
ARTICLES
Information
Econometric Theory , Volume 36 , Issue 5 , October 2020 , pp. 907 - 960
Copyright
© The Author(s) 2020. Published by Cambridge University Press

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Footnotes

We thank three referees, Co-Editor Michael Jansson, and Editor Peter C.B. Phillips for helpful comments and suggestions that led to significant improvements to our manuscript. We also thank Eric Ghysels, Shigeyuki Hamori, Peter R. Hansen, Yoshihiko Nishiyama, Kenichiro Tamaki, Kozo Ueda, and Zheng Zhang, seminar participants at the Kyoto Institute of Economic Research, UNC Chapel Hill, the University of Essex, and Kobe University, and conference participants at the 10th Spring Meeting of JSS, 2016 SWET, 2016 AMES, 2016 JJSM, 2016 NBER-NSF Time Series Conference, the 15th International Conference of WEAI, and SETA 2019 for helpful comments. The second author is grateful for financial support from JSPS KAKENHI (Grant Number 16K17104), Kikawada Foundation, Mitsubishi UFJ Trust Scholarship Foundation, Nomura Foundation, and Suntory Foundation.

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