The Granger-Johansen representation theorem for integrated time series on Banach space

TL;DR

This paper extends the Granger-Johansen representation theorem to infinite-dimensional Banach spaces, characterizing integrated autoregressive time series through spectral singularities and decomposing them into integrated and non-integrated components.

Contribution

It generalizes the GJRT to Banach space-valued processes, allowing for infinite-dimensional systems and singularities of various types, including essential singularities.

Findings

The theorem applies to all integrated AR processes regardless of dimension or cointegration.

It characterizes the order of integration based on spectral singularities.

The series decomposes into integrated and regular components.

Abstract

We prove an extended Granger-Johansen representation theorem (GJRT) for finite or infinite order integrated autoregressive time series on Banach space. We assume only that the resolvent of the autoregressive polynomial for the series is analytic on and inside the unit circle except for an isolated singularity at unity. If the singularity is a pole of finite order the time series is integrated of the same order. If the singularity is an essential singularity the time series is integrated of order infinity. When there is no deterministic forcing the value of the series at each time is the sum of an almost surely convergent stochastic trend, a deterministic term depending on the initial conditions and a finite sum of embedded white noise terms in the prior observations. This is the extended GJRT. In each case the original series is the sum of two separate autoregressive time series on complementary subspaces--a singular component which is integrated of the same order as the original series and a regular component which is not integrated. The extended GJRT applies to all integrated autoregressive processes irrespective of the spatial dimension, the number of stochastic trends and cointegrating relations in the system, and the order of integration.