COINTEGRATION AND REPRESENTATION OF COINTEGRATED AUTOREGRESSIVE PROCESSES IN BANACH SPACES

We extend the notion of cointegration for time series taking values in a potentially infinite dimensional Banach space. Examples such include stochastic processes $C[0,1]$ equipped with supremum distance and those finite vector space non-Euclidean distance. then develop versions Granger–Johansen representation theorems I(1) I(2) autoregressive (AR) To achieve this goal, we first note that an AR( p ) law motion can be characterized by linear operator pencil (an operator-valued map certain properties) via companion form representation, study spectral properties to obtain necessary sufficient condition given admit or solutions. These operator-theoretic results fundamental basis our theorems. Furthermore, it is shown approach fact closely related extension conventional taken Euclidean setting. Our theoretical may especially relevant recently growing literature on functional analysis spaces.