Asymptotic behaviour of the sample autocovariance and autocorrelation function of the AR(1) process with ARCH(1) errors

Abstract

We consider a stationary AR(1) process with ARCH(1) errors given by the stochastic difference equation

X_t=α X_{t-1}+\sqrt{β+λ X_{t-1}^2}\,ε_{t}\,, t∈{\mathbb{N}}\,,

where the (ε_t) are independent and identically distributed symmetric random variables. In contrast to ARCH and GARCH processes, AR(1) processes with ARCH(1) errors are not solutions of linear stochastic recurrence equations and there is no obvious way to embed them into such equations. However, we show that they still belong to the class of stationary sequences with regularly varying finite-dimensional distributions and therefore the theory of Davis and Mikosch can be applied. We present a complete analysis of the weak limit behaviour of the sample autocovariance and autocorrelation functions of (X_t), (|X_t|) and (X_t²). The results in this paper can be seen as a natural extension of results for ARCH(1) processes.