On some probabilistic properties of periodic GARCH processes

TL;DR

This paper investigates the probabilistic characteristics of periodic GARCH processes, including stationarity, moments, covariance, and ergodicity, under general assumptions, highlighting their similarities with periodic ARMA models.

Contribution

It provides a comprehensive analysis of probabilistic properties of PGARCH processes, extending understanding of their behavior and establishing conditions for stationarity and mixing.

Findings

PGARCH processes can be strictly and second order periodically stationary.

Conditions for existence of higher-order moments are established.

PGARCH processes exhibit geometric ergodicity and -mixing under certain assumptions.

Abstract

This paper examines some probabilistic properties of the class of periodic GARCH processes (PGARCH) which feature periodicity in conditional heteroskedasticity. In these models, the parameters are allowed to switch between different regimes, so that their structure shares many properties with periodic ARMA process (PARMA). We examine the strict and second order periodic stationarities, the existence of higher-order moments, the covariance structure, the geometric ergodicity and -mixing of the PGARCH(p,q) process under general and tractable assumptions. Some examples are proposed to illustrate the various concepts.