Cointegrated Continuous-time Linear State Space and MCARMA 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.…”