Modeling Z-valued time series based on new versions of the Skellam INGARCH model

Abstract

Recently, there has been a growing interest in integer-valued time series models, including integer-valued autoregressive (INAR) models and integer-valued generalized autoregressive conditional heteroscedastic (INGARCH) models, but only a few of them can deal with data on the full set of integers, that is, $\mathbb{Z}=\{\dots ,-2,-1,0,1,2,\dots \}$. Although some attempts have been made to deal with $\mathbb{Z}$-valued time series, these models do not provide enough flexibility in modeling some specific integers (e.g., 0, $\pm 1$). A symmetric Skellam INGARCH$(1,1)$ model was proposed in the literature, but it only considered zero-mean processes, which limits its application. We first extend the symmetric Skellam INGARCH model to an asymmetric version, which can deal with non-zero-mean processes. Then we propose a modified Skellam model which adopts a careful treatment on integers 0 and $\pm 1$ to satisfy a special feature of the data. Our models are easy-to-use and flexible. The maximum likelihood method is used to estimate unknown parameters and the log-likelihood ratio test statistic is provided for testing the asymmetric model against the modified one. Simulation studies are given to evaluate performances of the parametric estimation and log-likelihood ratio test. A real data example is also presented to demonstrate good performances of newly proposed models.