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]t = 0, E[var epsilon]t2 = 1, and [var epsilon]t is independent of {Xs, s < t}, Engle (1982) obtained the necessary and sufficient condition for the second-order stationarity, that is, [alpha]1 + [alpha]2 + ··· + [alpha]p < 1. In this note, we assume that [var epsilon]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 E[var epsilon]t2 = 1. When [var epsilon]t has only the first-order absolute moment, a sufficient condition for the geometric ergodicity is also given.