Cointegrated Linear Processes in Hilbert Space |
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| Authors: | Brendan K Beare Juwon Seo Won‐Ki Seo |
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| Affiliation: | 1. Department of Economics, University of California, San Diego, La Jolla, CA, USA;2. Department of Economics, National University of Singapore, Singapore, Singapore |
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| Abstract: | We extend the notion of cointegration for multivariate time series to a potentially infinite‐dimensional setting in which our time series takes values in a complex separable Hilbert space. In this setting, standard linear processes with nonzero long‐run covariance operator play the role of  processes. We show that the cointegrating space for an  process may be sensibly defined as the kernel of the long‐run covariance operator of its difference. The inner product of an  process with an element of its cointegrating space is a stationary complex‐valued process. Our main result is a version of the Granger–Johansen representation theorem: we obtain a geometric reformulation of the Johansen I(1) condition that extends naturally to a Hilbert space setting, and show that an autoregressive Hilbertian process satisfying this condition, and possibly also a compactness condition, admits an  representation. |
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| Keywords: | Cointegration functional time series functional autoregression Granger– Johansen representation theorem |
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