Calculating the autocovariances and the likelihood for periodic V ARMA models

Abstract

This paper concerns the autocovariance calculation and likelihood evaluation for periodic vector ARMA models (PV ARMA). Based on a state space representation of PV ARMA models, we derive an algorithm for computing the PV ARMA autocovariances. The proposed method computes the autocovariances for distinct seasons separately, thereby facilitating efficient calculation for models with a large period. As a result, the obtained autocovariance calculation procedure is exploited in a periodic Chandrasekhar-type filter to evaluate the exact likelihood for Gaussian PV ARMA series. Empirical evidence shows the superiority of the periodic Chandrasekhar algorithm for likelihood evaluation over the Kalman-based one.

Keywords:

• periodic V ARMA models
• periodic autocovariances
• discrete-time periodic Lyapunov equation
• likelihood evaluation
• periodic Chandrasekhar-type recursions
• Kalman filter

Acknowledgements

The authors are indebted to an anonymous referee for his valuable comments, particularly for suggesting them to adding empirical evidence concerning Algorithm 4.2 and to Professor Mohamed Bentarzi for useful discussions.