Dynamical Approach in studying GJR-GARCH (Q,P) Models with Application

This paper deals with finding stationarity Condition of GJR-GARCH(Q,P) model by using a local linearization technique in order to reduce this non-linear model to a linear difference equation with constant coefficients and then obtain the stationarity condition  via a characteristic equation. 
Finally we apply the obtained stationarity conditions of GJR-GARCH(Q,P) model to a real data that represents a monthly Brent Crude oil prices at closing in dollars for period (JUN. 1989-DES. 2018) and we find that GJR-GARCH(3,1) is the best model according to AIC and BIC information criteria.