Cointegration in functional autoregressive processes

TL;DR

This paper extends cointegration and unit root theory to infinite-dimensional Hilbert space autoregressive processes, providing a generalized representation theorem and characterizations of their structure and cointegration properties.

Contribution

It introduces a comprehensive framework for cointegration in infinite-dimensional AR processes, generalizing finite-dimensional results and characterizing their structure and relations.

Findings

Derived a generalized Granger-Johansen Representation Theorem for infinite-dimensional ARs.

Showed that solutions with a finite type unit root are integrated of finite order with common trends.

Characterized the structure of cointegrating relations and the process representation in Hilbert spaces.

Abstract

This paper defines the class of $\mathcal{H}$-valued autoregressive (AR) processes with a unit root of finite type, where $\mathcal{H}$ is an infinite dimensional separable Hilbert space, and derives a generalization of the Granger-Johansen Representation Theorem valid for any integration order $d=1,2,\dots$. An existence theorem shows that the solution of an AR with a unit root of finite type is necessarily integrated of some finite integer $d$ and displays a common trends representation with a finite number of common stochastic trends of the type of (cumulated) bilateral random walks and an infinite dimensional cointegrating space. A characterization theorem clarifies the connections between the structure of the AR operators and $(i)$ the order of integration, $(ii)$ the structure of the attractor space and the cointegrating space, $(iii)$ the expression of the cointegrating relations, and $(iv)$ the Triangular representation of the process. Except for the fact that the number of cointegrating relations that are integrated of order 0 is infinite, the representation of $\mathcal{H}$-valued ARs with a unit root of finite type coincides with that of usual finite dimensional VARs, which corresponds to the special case $\mathcal{H}=\mathbb{R}^p$.