On GARCH(p,q) convergence

Abstract

The paper deals with symmetric GARCH(p,q) model. Assuming that there exists defined by this model stationary time series, we have proposed the necessary and sufficient condition for exponential mean square convergence of any stochastic recurrent procedure satisfying this model to the above stationary time series. A mathematical background of the proposal approach is based on the derived covariance method for mean square exponential stability analysis of linear stochastic difference equations, which permits one to state a mean square convergence criterion for GARCH(p,q) models with any integer positive p and q in the convenient for application form of an integral inequality involving the model parameters.