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