On the stationarity and existence of moments of the periodic EGARCH process

Abstract

Abstract In this paper, we will consider periodic EGARCH ⁡ ( p , p ) {\operatorname{EGARCH}(p,p)} (exponential generalized autoregressive conditional heteroscedastic) processes denoted by PEGARCH ⁡ ( p , p ) {\operatorname{PEGARCH}(p,p)} . These processes are similar to the standard EGARCH processes, but include seasonally varying coefficients. We examine the probabilistic structure of an EGARCH-type stochastic difference equation with periodically-varying parameters. We propose necessary and sufficient conditions ensuring the existence of stationary solutions (in a periodic sense) based on a Markovian representation. The closed forms of higher moments are, under these conditions, established. Furthermore, the expressions for the Kurtosis coefficient and the autocorrelations of squared observations are derived. The general theory is illustrated by considering special cases such as the symmetric and the asymmetric cases of the second order PEGARCH model.