The Tail of the Stationary Distribution of an Autoregressive Process with Arch(1) Errors

W consider the class of autoregressive processes with ARCH(1)errors given by the stochastic difference equation Xₙ = α Xₙ₋₁ + √β + λ Xₙ₋₁²εₙ, n ∈ N where (εₙ)_n ∈ N are 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.