Evolutionary spectral and bispectral analysis of periodic GARCH model

Karima Kimouche; Abdelouahab Bibi

Karima Kimouche Department of Mathematics, 20 ao\^{u}t 1955 University, Skikda,\\ Algeria and\\Department of Mathematics, Larbi ben M-hidi univrsity O.E.B., Algeria
Abdelouahab Bibi Department of Mathematics, 20 ao\^{u}t 1955 University, Skikda,\\ Algeria and\\Department of Mathematics, Larbi ben M-hidi univrsity O.E.B., Algeria

Abstract

This paper defines and analyzes the evolutionary transfer functions for conditional volatility in periodic generalized autoregressive conditional heteroscedasticity $\left( PGARCH\text{ for short}\right) $ model in order
to describe some probabilistic and statistical properties in both frequency and temporal behavior. So, in the first part, we consider a diagonal periodic bilinear representation of $PGARCH\left( p,q\right) $ process with
second$-$order Hermite polynomial with leading coefficient $1$, and we give a necessary and\ sufficient condition for the existence of causal and second order periodically stationarity solution (or also periodically correlated
$\left( PC\right) $). In a second part, we focus on the probabilistic structure of such a representation, more precisely, we analyze the so-called time-dependent\ autocovariance and the associated evolutionary spectral
density function of such a representation and its squared version. As a consequence, it is shown that the second order properties is similar to a $PARMA$ process with uncorrelated innovations and hence the resort to higher-order properties is seems to be necessary to identify the process. Therefore, the explicit expression of evolutionary bispectral density of the process and its squared version are used as useful property that characterizing the non linear models. Several illustrative examples are emphasized.