On the Geometric Ergodicity of a Non-linear Autoregressive Model with an Autoregressive Conditional Heteroscedastic Term

In this paper, the geometric ergodicity of a non-linear AR model with an ARCH term is discussed. Two non-vacuous and mild sufficient conditions are given. The results obtained modify the vacuous part and reduce the restriction of Masry and Tjφstheim (1995)’s conditions, and lay a foundation for statistical inference of the model (e.g. Mckeague and Zhang (1994) and Masry and Tjφstheim (1995)). It is worth pointing out that the geometric ergodicity of the general β– ARCH(p) model which could not be solved in Guegan and Diebolt (1994) may be easily derived from our results. Compared with Nze (1992), the conditions of this paper may guarantee the existence of the second moments for the stationary solution. A conjecture is also given.