A note on geometric ergodicity of autoregressive conditional heteroscedasticity (ARCH) model

For the pth-order linear ARCH model, , where [alpha]0 > 0, [ alpha]i [greater-or-equal, slanted] 0, I = 1, 2, ..., p, {[var epsilon]t} is an i.i.d. normal white noise with E[var Epsilon].

Abstract

For the pth-order linear ARCH model, Xt=εtα0+α1Xt−12+α2Xt−22+⋯+αpXt−p2, where α0 > 0, αi ⩾ 0, i = 1, 2, …, p, {εt} is an i.i.d. normal white noise with Eεt = 0, Eεt2 = 1, and εt is independent of {Xs, s < t}, Engle (1982) obtained the necessary and sufficient condition for the second-order stationarity, that is, α1 + α2 + ··· + αp < 1. In this note, we assume that εt has the probability density function p(t) which is positive and lower-semicontinuous over the real line, but not necessarily Gaussian, then the geometric ergodicity of the ARCH(p) process is proved under Eεt2 = 1. When εt has only the first-order absolute moment, a sufficient condition for the geometric ergodicity is also given.