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

TL;DR

This paper develops a quasi-maximum likelihood estimation method for cointegrated continuous-time state space models observed at discrete intervals, establishing consistency, asymptotic distributions, and demonstrating performance through simulations.

Contribution

It introduces a novel QML estimation approach for cointegrated continuous-time models, including a decomposition into long- and short-run parameters and proofs of estimator consistency and distribution.

Findings

QML estimators are consistent for both long- and short-run parameters.

Long-run parameters are asymptotically mixed normally distributed.

Short-run parameters are asymptotically normally distributed.

Abstract

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.