Moments of Mixture Periodic Autoregressive Models

Abstract

This article deals with the study of some properties of a mixture periodically correlated autoregressive (MPAR _S ) time series model, which extends the mixture time invariant parameter autoregressive (MAR) model, that has recently received a considerable interest from many economic time series analysts, to mixture periodic parameter autoregressive model. The aim behind this extension is to make the model able to capture, in addition to all features captured by the classical MAR model, the periodicity feature exhibited by the autocovariance structure of many encountered financial and environmental time series with eventual multimodal distributions. Our main contribution here is obtaining of the second moment periodically stationary condition for a MPAR _S (K; 2,…, 2) model, furthermore the closed-form of the second moment is obtained.

Keywords:

• Expectation-Maximization algorithm
• Mixture periodic autoregressive models
• Periodic autoregressive models
• Periodically correlated process

Mathematics Subject Classification:

• 62F12
• 62M10

Acknowledgments

The authors would like to express their deep gratitude to Professor N. Balakrishnan, Editor in Chief, for all his help and encouragement. Also, they present to the anonymous referee their most sincere thanks and profound acknowledgment for the helpful suggestions and very constructive comments which allowed us to improve the quality and readability of the article.