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

Abstract

For the pth-order linear ARCH model, X t=ε t α 0+α 1X t−1 2+α 2X t−2 2+⋯+α pX t−p 2 , where α 0 > 0, α i ⩾ 0, i = 1, 2, …, p, { ε t } is an i.i.d. normal white noise with Eε t = 0, Eε t 2 = 1, and ε t is independent of { X s , s < t}, Engle (1982) obtained the necessary and sufficient condition for the second-order stationarity, that is, α 1 + α 2 + ··· + α p < 1. In this note, we assume that ε t 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 Eε t 2 = 1. When ε t has only the first-order absolute moment, a sufficient condition for the geometric ergodicity is also given.