Cointegrated continuous-time linear state-space and MCARMA models

Abstract

In this paper we define and characterize cointegrated solutions of continuous-time linear state-space models. A main result is that a cointegrated solution of a continuous-time linear state-space model can be represented as a sum of a Lévy process and a stationary solution of a linear state-space model. Moreover, we prove that the class of cointegrated multivariate Lévy-driven autoregressive moving-average (MCARMA) processes, the continuous-time analogues of the classical vector ARMA processes, is equivalent to the class of cointegrated solutions of continuous-time linear state space models. Necessary conditions for MCARMA processes to be cointegrated are given as well extending the results of Comte [11] for MCAR processes. The conditions depend only on the autoregressive polynomial if we have a minimal model. Finally, we investigate cointegrated continuous-time linear state-space models observed on a discrete time-grid and calculate their linear innovations. Based on the representation of the linear innovations we derive an error correction form. The error correction form uses an infinite linear filter in contrast to the finite linear filter for VAR models.

On non-stationary solutions to MSDDEs: Representations and the cointegration space

“…Section 4 discusses how one can rely on the relation between invertible MCARMA equations and MSDDEs to define cointegrated MCARMA processes. In particular, under conditions similar to those imposed in [10,Theorem 4.6], we show existence and uniqueness of a cointegrated solution to the MSDDE associated to the MCARMA(p, p − 1) equation. This complements the result of [10], which ensures existence of cointegrated MCARMA(p, q) processes when p > q + 1.…”

“…In particular, under conditions similar to those imposed in [10,Theorem 4.6], we show existence and uniqueness of a cointegrated solution to the MSDDE associated to the MCARMA(p, p − 1) equation. This complements the result of [10], which ensures existence of cointegrated MCARMA(p, q) processes when p > q + 1. Finally, Section 5 contains the proofs of all the statements presented in the paper together with a few technical results.…”

“…While (X t ) t∈R given by (4.3) is stationary by definition, it does indeed make sense to search for non-stationary, but cointegrated, processes satisfying (i) or (ii) of Proposition 4.2 also when Condition 4.1 does not hold. Fasen-Hartmann and Scholz [10] follow this idea by first characterizing cointegrated solutions to state-space equations and, next, define the cointegrated MCARMA process as a cointegrated solution corresponding to the specific triple (A, B, C). Their definition applies to any MCARMA(p, q) process and they give sufficient conditions on P and Q for the cointegrated MCARMA process to exist when q < p − 1.…”

“…The assumptions (i)-(iii) of Condition 4.3 are also imposed in [10], and (iv) is imposed to ensure that (4.1) admits an MSDDE representation. In [10] they impose an additional assumption, namely that the polynomials P and Q are so-called left coprime, which is used to ensure that the pole of z → P (z) −1 at 0 is also a pole of z → P (z) −1 Q(z). However, in our case this is implied by (iv).…”

“…The remaining part of the proof concerns showing (5.16) or, equivalently, the process ∆ h Y t := Y t − Y t−h , t ∈ R, is unique for any h > 0. We will rely on the same type of ideas as in the proof of [6,Proposition 7] and [10,Proposition 4.5]. Suppose first that Π 0 has reduced rank r ∈ (0, n) and let α, β ∈ R n×r be a rank decomposition of Π 0 as in Remark 3.2.…”