Quasi-maximum likelihood estimation for cointegrated solutions of continuous-time state space models observed at discrete time points

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. The model is not in innovation form. Therefore we have to construct some pseudo-innovations to be able to define a QML-function. We divide the parameter vector appropriate 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 Levy 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.