Mixing properties of Skellam-GARCH processes

TL;DR

This paper studies Skellam-GARCH processes, establishing their unique stationary regime and geometric mixing properties, supported by simulations and an application to COVID-19 data.

Contribution

It introduces a novel integer-valued GARCH model with Skellam distribution and proves its stationarity and mixing properties, complemented by empirical analysis.

Findings

Unique stationary regime established

Geometric $eta$-mixing proven

Effective modeling of COVID-19 count data

Abstract

We consider integer-valued GARCH processes, where the count variable conditioned on past values of the count and state variables follows a so-called Skellam distribution. Using arguments for contractive Markov chains we prove that the process has a unique stationary regime. Furthermore, we show asymptotic regularity ($\beta$-mixing) with geometrically decaying coefficients for the count process. These probabilistic results are complemented by a statistical analysis, a few simulations as well as an application to recent COVID-19 data.