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COINTEGRATION AND REPRESENTATION OF COINTEGRATED AUTOREGRESSIVE PROCESSES IN BANACH SPACES

Published online by Cambridge University Press:  20 April 2022

Won-Ki Seo
Won-Ki Seo*
Affiliation:
University of Sydney
*
Address correspondence to Won-Ki Seo, School of Economics, University of Sydney, Sydney, NSW, Australia; e-mail: won-ki.seo@sydney.edu.au.
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Abstract

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We extend the notion of cointegration for time series taking values in a potentially infinite dimensional Banach space. Examples of such time series include stochastic processes in $C[0,1]$ equipped with the supremum distance and those in a finite dimensional vector space equipped with a non-Euclidean distance. We then develop versions of the Granger–Johansen representation theorems for I(1) and I(2) autoregressive (AR) processes taking values in such a space. To achieve this goal, we first note that an AR(p) law of motion can be characterized by a linear operator pencil (an operator-valued map with certain properties) via the companion form representation, and then study the spectral properties of a linear operator pencil to obtain a necessary and sufficient condition for a given AR(p) law of motion to admit I(1) or I(2) solutions. These operator-theoretic results form a fundamental basis for our representation theorems. Furthermore, it is shown that our operator-theoretic approach is in fact a closely related extension of the conventional approach taken in a Euclidean space setting. Our theoretical results may be especially relevant in a recently growing literature on functional time series analysis in Banach spaces.

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ARTICLES
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Econometric Theory , Volume 39 , Issue 4 , August 2023 , pp. 737 - 788
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

This article benefited from the helpful suggestions for improvement made by the Editor, Peter C.B. Phillips, and three anonymous reviewers, to whom I express my thanks.

References

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