A Note on Stochastic Difference Equations and its Application to GARCH Models

In this paper, we give a note to the renewal theory for products of random matrices in Kesten （1973）, and give a feasible way to calculate the tail exponent k₁ in the renewal theory. As an application, the stationarity and the tail behaviour of ARCH（2） and GARCH（2,1） time series models are studied. Our work is a generalization of similar work on ARCH（1） by Embrechts et al. （1997） and on GARCH（1,1） by Mikosch and Stǎricǎ （2000）. Some numerical results are also given in this paper.