The tail of the stationary distribution of an autoregressive process with ARCH(1) errors

Abstract

We consider the class of autoregressive processes with ARCH(1) errors given by the stochastic difference equation Xn = αXn-1 + √ β + λX2n-1εn, n ∈ ℕ, where (εn)n∈ℕ i.i.d. random variables. Under general and tractable assumptions we show the existence and uniqueness of a stationary distribution. We prove that the stationary distribution has a Pareto-like tail with a well-specified tail index which depends on α, λ and the distribution of the innovations (εn)n∈ℕ. This paper generalizes results for the ARCH(1) process (the case α = 0). The generalization requires a new method of proof and we invoke a Tauberian theorem.