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

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.

💡 Research Summary

The paper establishes a fully general Granger‑Johansen representation theorem (GJRT) for autoregressive (AR) processes whose values lie in an arbitrary Banach space. The authors start from the minimal analytic assumption that the resolvent of the AR polynomial, (R(z)=A(z)^{-1}), is analytic on the closed unit disc except possibly at the point (z=1), where it may have an isolated singularity. No further restrictions on the dimension of the space, compactness of the coefficient operators, or complementability of subspaces are imposed.

Under this assumption the nature of the singularity at (z=1) determines the integration order of the process. If the singularity is a pole of finite order (d), the series is integrated of order (d) (denoted (I(d))). If the singularity is essential, the series is integrated of infinite order, (I(\infty)). When there is no singularity the series is stationary ((I(0))).

The core technical contribution is a systematic construction of the Laurent expansion of the resolvent around (z=1): \