Periodic integer-valued GARCH(1, 1) model

ABSTRACT

This article deals with some probabilistic and statistical properties of a periodic integer-valued GARCH(1,1) model. Necessary and sufficient conditions for the periodical stationary, both in mean and second order, are established. The closed-forms of the mean and the second moment are, under these conditions, obtained. The condition of the existence of higher moment orders and their explicit formula in terms of the parameters are established. The autocovariance structure is studied, while providing the closed-form of the periodic autocorrelation function. The Yule–Walker and the likelihood estimations of the underlying parameters are obtained. A simulation study and an application on real dataset are provided.

KEYWORDS:

• Mean and second moment periodically stationary model
• Periodic INGARCH model
• Periodically correlated integer-valued process

MATHEMATICS SUBJECT CLASSIFICATION:

• 62F12
• 62M10

Acknowledgments

The authors present their most sincere thanks and their deep gratitude to Professor N. Balakrishnan, Editor-in-Chief, for his helps and continued encouragements. Also, they present to the anonymous referees their most sincere thanks and profound acknowledgements for their constructive remarks and suggestions that have allowed us to improve the quality and the readability of the article in its first revised form. Also, we present again our deep thanks and sincere acknowledgements to the referee 1, for his pleasant assistance in the second revision version.