Cointegration in Functional Autoregressive Processes

Abstract

This paper defines the class of H-valued autoregressive (AR) processes with a unit root of finite type, where 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, . . . . 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 H-valued ARs with a unit root of finite type coincides with that of usual finite dimensional VARs, which corresponds to the special case H = R p .

A general inversion theorem for cointegration

“…Finally, the Granger-Johansen Representation Theorems have recently been shown to hold also for infinite dimensional AR processes in Hilbert spaces, see Chang et al (2016), Hu and Park (2016), and Beare et al (2017) for the I(1) case and Beare and Seo (2018) for the I(2) case. Franchi and Paruolo (2017) provide an extension of the present results to the generic I(d) case for infinite dimensional AR processes in Hilbert spaces.…”

Representation of I(1) and I(2) Autoregressive Hilbertian Processes

“…The direct sum appearing in condition (3) of Theorem 3.2 is not in general an orthogonal direct sum. Franchi and Paruolo (2019a) showed that, when (z) is noninvertible at z = 1, condition (3) is equivalent to the following orthogonal direct sum decomposition of H:…”

“…The direct sum appearing in condition (3) of Theorem 4.2 is not in general an orthogonal direct sum. Extending results of Johansen (1992) from the finite dimensional case H = C n to a more general Hilbert space setting, Franchi and Paruolo (2019a) showed that, when (z) is noninvertible at z = 1 and the I(1) condition fails, an equivalent necessary and sufficient condition for a pole of second order is the following tripartite orthogonal direct sum decomposition of H:…”

“…The innovations ε t are referred to as strong white noise due to their being centered (i.e., zero expected value) and iid. We impose these conditions for simplicity, but the results to be developed remain valid if the iid condition is replaced with the weaker requirement that the cross-covariance operators for the ε t 's are all zero, as in Franchi and Paruolo (2019a). In the latter case, the ε t 's are merely said to be white noise.…”

“…It is a consequence of a compactness con-dition we impose on the autoregressive operators. Franchi and Paruolo (2019a) have observed that finite codimensionality of the cointegrating space holds more generally if the autoregressive operator pencil has an eigenvalue of finite type at one.…”