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LIMIT THEORY FOR LOCALLY FLAT FUNCTIONAL COEFFICIENT REGRESSION

Published online by Cambridge University Press:  14 July 2022

Peter C. B. Phillips Open the ORCID record for Peter C. B. Phillips [Opens in a new window]  and
Ying Wang
Peter C. B. Phillips*
Affiliation:
University of Auckland, Yale University, Singapore Management University, and University of Southampton
Ying Wang
Affiliation:
Renmin University of China
*
Address correspondence to Peter C. B. Phillips, Yale University, New Haven, CT, USA; e-mail: [email protected].
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Abstract

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Functional coefficient (FC) regressions allow for systematic flexibility in the responsiveness of a dependent variable to movements in the regressors, making them attractive in applications where marginal effects may depend on covariates. Such models are commonly estimated by local kernel regression methods. This paper explores situations where responsiveness to covariates is locally flat or fixed. The paper develops new asymptotics that take account of shape characteristics of the function in the locality of the point of estimation. Both stationary and integrated regressor cases are examined. The limit theory of FC kernel regression is shown to depend intimately on functional shape in ways that affect rates of convergence, optimal bandwidth selection, estimation, and inference. In FC cointegrating regression, flat behavior materially changes the limit distribution by introducing the shape characteristics of the function into the limiting distribution through variance as well as centering. In the boundary case where the number of zero derivatives tends to infinity, near parametric rates of convergence apply in stationary and nonstationary cases. Implications for inference are discussed and a feasible pre-test inference procedure is proposed that takes unknown potential flatness into consideration and provides a practical approach to inference.

Type
ARTICLES
Information
Econometric Theory , Volume 39 , Issue 5 , October 2023 , pp. 900 - 949
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), 2022. Published by Cambridge University Press

Footnotes

Our thanks go to the Co-Editor, Liangjun Su, and two referees for most helpful comments on the earlier versions of this paper. Phillips acknowledges research support from the NSF under Grant No. SES 18-50860 at Yale University and a Kelly Fellowship at the University of Auckland. Wang acknowledges support from the National Natural Science Foundation of China (Grant No. 72103197).

References

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