Moment properties and quadratic estimating functions for integer-valued time series models - Mohamad, Nurul Najihah; Mohamed, Ibrahim; Haur, Ng Kok (College of Statistical and Actuarial Sciences, University of Punjab, 2018)
Recently, there has been a growing interest in integer-valued time series models. In this paper, using a martingale difference, we prove a general theorem on the moment properties of a class of integer-valued time series models. This theorem not only contains results in the recent literature as special cases but also has the advantage of a simpler proof. In addition, we derive the closed form expressions for the kurtosis and skewness of the models. The results are very useful in understanding the behaviour of the processes involved and in estimating the parameters of the models using quadratic estimating functions (QEF). Specifically, we derive the optimal function for the integer-valued GARCH (p, q) known as INGARCH (p, q) model. Simulation study is carried out to compare the performance of QEF estimates with corresponding maximum likelihood (ML) and least squares (LS) estimates for the INGARCH (1,1) model with different sets of parameters. Results show that the QEF estimates produce smaller standard errors than the ML and LS estimates for small sample size and are comparable to the ML estimates for larger sample size. For illustration, we fit the 108 monthly strike data to INGARCH (1, 1) models via QEF, ML and LS methods, and show the applicability of QEF method in practice.