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

Borkovec, Milan ; Klüppelberg, Claudia

Ann. Appl. Probab., Tome 11 (2001) no. 2, p. 1220-1241 / Harvested from Project Euclid

Résumé

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.

Détail

Publié le : 2001-11-14
Classification:  ARCH model,  autoregressive process,  geometric ergodicity,  heavy tail,  heteroscedastic model,  Markov process,  recurrent Harris chain,  regular variation,  Tauberian theorem,  60H25,  60G10,  60J05