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Representation of I(1) and I(2) autoregressive Hilbertian processes

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  • Brendan K. Beare
  • Won-Ki Seo

Abstract

We extend the Granger-Johansen representation theorems for I(1) and I(2) vector autoregressive processes to accommodate processes that take values in an arbitrary complex separable Hilbert space. This more general setting is of central relevance for statistical applications involving functional time series. We first obtain a range of necessary and sufficient conditions for a pole in the inverse of a holomorphic index-zero Fredholm operator pencil to be of first or second order. Those conditions form the basis for our development of I(1) and I(2) representations of autoregressive Hilbertian processes. 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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  • Brendan K. Beare & Won-Ki Seo, 2017. "Representation of I(1) and I(2) autoregressive Hilbertian processes," Papers 1701.08149, arXiv.org, revised Sep 2019.
    • Handle: RePEc:arx:papers:1701.08149
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    Cited by:

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    2. Won-Ki Seo, 2020. "Functional Principal Component Analysis for Cointegrated Functional Time Series," Papers 2011.12781, arXiv.org, revised Apr 2023.
    3. Mario Faliva & Maria Grazia Zoia, 2021. "Cointegrated Solutions of Unit-Root VARs: An Extended Representation Theorem," Papers 2102.10626, arXiv.org.

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