Cointegrated continuous-time linear state-space and MCARMA models

In this paper, we define and characterize cointegrated solutions of continuous-time linear state-space models driven by Levy processes. Cointegrated solutions of such models are shown to be representable as a sum of a Levy process and a stationary solution of a linear state-space model, analogous to the Granger representation for cointegrated VAR models. Moreover, we prove that the class of cointegrated multivariate Levy-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 [Discrete and continuous time cointegration, J. Econometrics 88 (1999), pp. 207-226] 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. Some sufficient identifiable criteria are also given.