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The Granger–Johansen representation theorem for integrated time series on Banach space

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dc.contributor.authorHowlett, P.
dc.contributor.authorBeare, B.K.
dc.contributor.authorFranchi, M.
dc.contributor.authorBoland, J.
dc.contributor.authorAvrachenkov, K.
dc.date.issued2025
dc.description.abstractWe prove an extended Granger-Johansen representation theorem (GJRT) for finite- or infinite-order integrated autoregressive time series on Banach space. We assume only that the resolvent of the autoregressive polynomial for the series is analytic on and inside the unit circle except for an isolated singularity at unity. If the singularity is a pole of finite order the time series is integrated of the same order. If the singularity is an essential singularity the time series is integrated of order infinity. When there is no deterministic forcing the value of the series at each time is the sum of an almost surely convergent stochastic trend, a deterministic term depending on the initial conditions and a finite sum of embedded white noise terms in the prior observations. This is the extended GJRT. In each case the original series is the sum of two separate autoregressive time series on complementary subspaces - a singular component which is integrated of the same order as the original series and a regular component which is not integrated. The extended GJRT applies to all integrated autoregressive processes irrespective of the spatial dimension, the number of stochastic trends and cointegrating relations in the system and the order of integration.
dc.identifier.citationJournal of Time Series Analysis, 2025; 46(3):432-457
dc.identifier.doi10.1111/jtsa.12766
dc.identifier.issn0143-9782
dc.identifier.issn1467-9892
dc.identifier.orcidBoland, J. [0000-0003-0362-4655] [0000-0003-1132-7589]
dc.identifier.urihttps://hdl.handle.net/11541.2/39866
dc.language.isoen
dc.publisherWILEY
dc.relation.fundingMinistero dell' Universita e della Ricerca 2022372SWE
dc.rightsCopyright 2024 The Author(s). Journal of Time Series Analysis published by John Wiley & Sons Ltd. This is an open access article under the terms of the Creative Commons Attribution-NonCommercial-NoDerivs License. (http://creativecommons.org/licenses/by-nc-nd/4.0/)
dc.source.urihttps://doi.org/10.1111/jtsa.12766
dc.subjectAR time series
dc.subjectresolvent operators
dc.subjectGIRT
dc.subjectcointegrated series
dc.titleThe Granger–Johansen representation theorem for integrated time series on Banach space
dc.typeJournal article
pubs.publication-statusPublished
ror.fileinfo12302245690001831 13302235830001831 Open Access Published Version
ror.mmsid9916894130201831

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