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A GENERAL LIMIT THEORY FOR NONLINEAR FUNCTIONALS OF NONSTATIONARY TIME SERIES

Published online by Cambridge University Press:  25 November 2024

Qiying Wang  and
Peter C. B. Phillips Open the ORCID record for Peter C. B. Phillips [Opens in a new window]
Qiying Wang*
Affiliation:
University of Sydney
Peter C. B. Phillips
Affiliation:
Yale University, University of Auckland, and Singapore Management University
*
Address correspondence to Qiying Wang, University of Sydney, Sydney, NSW, Australia; e-mail: [email protected].
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Abstract

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New limit theory is provided for a wide class of sample variance and covariance functionals involving both nonstationary and stationary time series. Sample functionals of this type commonly appear in regression applications and the asymptotics are particularly relevant to estimation and inference in nonlinear nonstationary regressions that involve unit root, local unit root, or fractional processes. The limit theory is unusually general in that it covers both parametric and nonparametric regressions. Self-normalized versions of these statistics are considered that are useful in inference. Numerical evidence reveals interesting strong bimodality in the finite sample distributions of conventional self-normalized statistics similar to the bimodality that can arise in t-ratio statistics based on heavy tailed data. Bimodal behavior in these statistics is due to the presence of long memory innovations and is shown to persist for very large sample sizes even though the limit theory is Gaussian when the long memory innovations are stationary. Bimodality is shown to occur even in the limit theory when the long memory innovations are nonstationary. To address these complications, new self-normalized versions of the test statistics are introduced that deliver improved approximations that can be used for inference.

Type
ARTICLES
Information
Econometric Theory , First View , pp. 1 - 58
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2024. Published by Cambridge University Press

Footnotes

Wang acknowledges research support from the Australian Research Council (Grant No. DP170104385). Phillips acknowledges research support from the NSF (Grant No. SES 18-50860) and a Kelly Fellowship at the University of Auckland.

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