# Cointegration in functional autoregressive processes

*Massimo Franchi* () and
*Paolo Paruolo*

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**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 $\mathcal{H}=\mathbb{R}^p$.

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**Date:** 2017-12, Revised 2018-10

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http://arxiv.org/pdf/1712.07522 Latest version (application/pdf)

**Related works:**

Working Paper: Cointegration in functional autoregressive processes (2017)

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**Persistent link:** https://EconPapers.repec.org/RePEc:arx:papers:1712.07522

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