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

For the pth-order linear ARCH model, X(t) = epsilon(t) root alpha(0) + alpha(1)X(t-1)(2) + alpha(2)X(t-2)(2) +...+ alpha(p)X(t-p)(2), where alpha(0) > 0, alpha(i) greater than or equal to 0, i = 1, 2,..., p, {epsilon(t)} is an i.i.d. normal white noise with E epsilon(t) = 0, E epsilon(t)(2) = 1, and epsilon t is independent of {X(s), 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 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 epsilon(t)(2) = 1. When epsilon(t) has only the first-order absolute moment, a sufficient condition for the geometric ergodicity is also given.