Journal of Time Series Analysis

Cointegrated Linear Processes in Hilbert Space

Journal Article

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 $\mathrm{I}\left(0\right)$ processes. We show that the cointegrating space for an $\mathrm{I}\left(1\right)$ process may be sensibly defined as the kernel of the long‐run covariance operator of its difference. The inner product of an $\mathrm{I}\left(1\right)$ 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 $\mathrm{I}\left(1\right)$ representation.

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