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In this paper, we investigate quasi-maximum likelihood (QML) estimation forthe parameters of a cointegrated solution of a continuous-time linear statespace model observed at discrete time points. The class of cointegratedsolutions of continuous-time linear state space models is equivalent to theclass of cointegrated continuous-time ARMA (MCARMA) processes. As a start, somepseudo-innovations are constructed to be able to define a QML-function.Moreover, the parameter vector is divided appropriately in long-run andshort-run parameters using a representation for cointegrated solutions ofcontinuous-time linear state space models as a sum of a L\'evy process plus astationary solution of a linear state space model. Then, we establish theconsistency of our estimator in three steps. First, we show the consistency forthe QML estimator of the long-run parameters. In the next step, we calculateits consistency rate. Finally, we use these results to prove the consistencyfor the QML estimator of the short-run parameters. After all, we derive thelimiting distributions of the estimators. The long-run parameters areasymptotically mixed normally distributed, whereas the short-run parameters areasymptotically normally distributed. The performance of the QML estimator isdemonstrated by a simulation study.

Quasi-maximum likelihood estimation for cointegrated continuous-time state space models observed at low frequencies