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

Abstract

W consider the class of autoregressive processes with ARCH(1)errors given by the stochastic difference equation

$$X_n = \alpha X_{n-1} + \sqrt{\beta + \lambda X_{n-1}^2}\varepsilon_n,\quad n \in \mathbb{N}$$

where $(\varepsilon_n)_{n \in \mathbb{N}$ 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 $\alpha, \lambda$ and the distribution of the innovations $(\varepsilon_n)_{n \in \mathbb{N}}$. This paper generalizes results for the ARCH(1) process (the case $\alpha = 0$). The generalization requires a new method of proof and we invoke a Tauberian theorem.