Cointegration and Representation of Cointegrated Autoregressive Processes in Banach Spaces

TL;DR

This paper generalizes cointegration concepts and develops representation theorems for autoregressive processes in infinite-dimensional Banach spaces, extending classical Euclidean results to functional time series analysis.

Contribution

It introduces a novel operator-theoretic framework for cointegration in Banach spaces and extends Granger-Johansen theorems to infinite-dimensional settings.

Findings

Characterization of AR(p) processes via linear operator pencils.

Necessary and sufficient conditions for I(1) and I(2) solutions in Banach spaces.

Extension of classical Euclidean results to functional time series analysis.

Abstract

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