A general inversion theorem for cointegration

Abstract

A generalization of the Granger and the Johansen Representation Theorems valid for any (possibly fractional) order of integration is presented. This Representation Theorem is based on inversion results that characterize the order of the pole and the coefficients of the Laurent series representation of the inverse of a matrix function around a singular point. Explicit expressions of the matrix coefficients of the (polynomial) cointegrating relations, of the Common Trends and of the Triangular representations are provided, either starting from the Moving Average or the Auto Regressive form. This contribution unifies different approaches in the literature and extends them to an arbitrary order of integration. The role of deterministic terms is discussed in detail.

Cointegration in Functional Autoregressive Processes

“…These results show that conditions and properties of AR processes with a unit root of finite type extend those that apply in the usual finite-dimensional VAR case; in fact, setting H = R p in the present results one finds the I (1) and I (2) results in Johansen (1996), and for the generic I (d) case, one finds the results in Franchi and Paruolo (2019). This shows that except for the fact that the dimension of the cointegrating space is infinite when dim H = ∞, the infinite-dimensionality of H does not introduce additional elements in the representation analysis of AR processes with a unit root of finite type.…”

“…Apart from the fact that the dimension of τ 0 is infinite when dim H = ∞, this parallels the finite-dimensional case, see Theorem 4.3 in Franchi and Paruolo (2019).…”

“…This Appendix contains proofs of the results in the text. The proof of Theorem 3.5 makes use of the following fact, which is proven in Franchi and Paruolo (2019)…”

“…Because A P − I is a nilpotent matrix of order d if and only if the largest Jordan block of A P has dimension d, see e.g., Horn and Johnson (2013, p. 181), and the size of the largest Jordan block is equal to d if and only if the POLE(d) condition holds, see Franchi and Paruolo (2019), one has the statement.…”

“…This article introduces the class of AR processes with a unit root of finite type, which contains infinite-dimensional H-valued ARs with compact operators as a special case, and derives a generalization of the Granger-Johansen Representation Theorem valid for any integration order d = 1, 2,... . Necessary and sufficient conditions for AR processes with a unit root of finite type to generate cointegrated I (d) processes are provided, in a parallel way with respect to the I (d) representation results in the finite-dimensional VAR case derived in Franchi and Paruolo (2019). When d > 1, the notion of polynomial cointegration, see Granger and Lee (1989), corresponds to the existence of some linear combination of v-characteristics of x t and other v-characteristics of j x t , j = 1,...,d − 1 that is integrated of order less than d.…”