We extend the Granger-Johansen representation theory for I(1)  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 obtain necessary and sufficient conditions for the existence of I(1) solutions to a given autoregressive law of motion generalizing the Johansen I(1) condition, and a characterization of such solutions. To accomplish this we obtain necessary and sufficient conditions for a pole in the inverse of a holomorphic index-zero Fredholm operator pencil to be simple, and a formula for its residue. In the case of first order autoregressive dynamics with a unit root our results take a particularly simple form , with the residue associated with the simple pole at one proportional to a Riesz projection .
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 .
R
theory
theorems
A
and I(2)
R
obtain necessary and sufficient conditions for the existence of I(1) solutions to a given autoregressive law of motion generalizing the Johansen I(1) condition, and a characterization of such solutions. To accomplish this we obtain
first obtain a range of
R
simple, and a formula for its residue. In the case of first order autoregressive dynamics with a unit root our results take a particularly simple form , with the residue associated with the simple pole at one proportional to a Riesz projection
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