Semi-Lévy-Driven CARMA Process: Estimation and Prediction

The Lévy-driven continuous-time ARMA (CARMA) models are restricted for modeling stationary processes. In this paper, we introduce semi-Lévy-driven CARMA (SL-CARMA) process as a generalized form of SL-CAR model which establishes a class of periodically correlated process. By a new representation of the semi-Lévy process, we provide a discretized state-vector process with independent periodically identically distributed noise corresponding to high-frequency data. Then, we estimate the parameters of the SL-CARMA process by Kalman filtering method. By simulation studies, the accuracy of the estimated parameters of a general form of semi-Lévy and a special case of normal inverse Gaussian backdriving processes are evaluated. Finally, the SL-CARMA process has much better fitting to the periodically correlated process in comparison to the retrieved Lévy-driven CARMA models by applying periodic sample from the Apnea-ECG database and the percent log returns of Dow Jones Industrial Average indices by mean absolute error criteria and Diebold–Mariano test.