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Statistica Sinica 6(1996), 943-956


THE GEOMETRICAL ERGODICITY OF NONLINEAR

AUTOREGRESSIVE MODELS


H. Z. An and F. C. Huang


Academia Sinica


Abstract: Consider the nonlinear autoregressive model xt=ψ(xt-1,...,xt-p)+εtwhere εt are independent, identically distributed(i.i.d.) random variables with almost everywhere positive density and mean zero. In this paper we discuss the conditions for the geometrical ergodicity of the above nonlinear AR model when there are more than one attractors in the corresponding deterministic dynamical systems, i.e., yt=ψ(yt-1,...yt-p), t>=1. We give several kinds of sufficient conditions for the geometrical ergodicity. By our result, illustrated by many examples, we show that many well-known nonlinear models such as the exponential AR, threshold AR, semi-parametric AR , bounded AR, truncated AR and β-ARCH models are geometrically ergodic under some mild conditions.



Key words and phrases: Exponential autoregressive models, geometrical ergodicity, Markov chains, nonlinear autoregressive models, threshold autoregressive models, β-ARCH models.



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