Cointegration and representation of integrated autoregressive processes in function spaces
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We provide a suitable generalization of cointegration for time series taking values in a potentially infinite dimensional Banach space or Hilbert space. Our main result is a generalization of the Granger-Johansen representation theorem: we obtain necessary and sufficient conditions for the existence of I(1) solutions to a given autoregressive law of motion, and a characterization of such solutions. To achieve this goal, we note that such an autoregressive law of motion is characterized by a simple linear operator pencil, so study its spectral properties. Our representation theorem requires a minimal assumption: we do not even assume compactness of autoregressive operators. Based on the results of our I(1) representation theorem, we also provide a representation theorem for I(2) autoregressive processes.
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