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  • Brendan K. Beare (a1) and Won-Ki Seo (a2)


We develop versions of the Granger–Johansen representation theorems for I(1) and I(2) vector autoregressive processes that apply to processes taking values in an arbitrary complex separable Hilbert space. This more general setting is of central relevance for statistical applications involving functional time series. An I(1) or I(2) solution to an autoregressive law of motion is obtained when the inverse of the autoregressive operator pencil has a pole of first or second order at one. We obtain a range of necessary and sufficient conditions for such a pole to be of first or second order. Cointegrating and attractor subspaces are characterized in terms of the behavior of the autoregressive operator pencil in a neighborhood of one.


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*Address correspondence to Brendan K. Beare, School of Economics, University of Sydney, Sydney, Australia; e-mail:


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We thank Massimo Franchi, Peter Phillips and seminar participants at the Einaudi Institute for Economics and Finance, UC San Diego, the Université libre de Bruxelles, and the 2019 NBER-NSF Time Series Conference in Hong Kong for helpful discussions. Beare also thanks Phil Roberts for his feedback and encouragement. An earlier version of this article titled “Representation of I(1) autoregressive Hilbertian processes” was posted on the preprint repository in January 2017.



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  • Brendan K. Beare (a1) and Won-Ki Seo (a2)


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