Long run behaviour of the autocovariance function of ARCH($\infty$) models

TL;DR

This paper analyzes the long-term behavior of the autocovariance function in ARCH(∞) models, showing its decay properties are directly linked to the kernel of an associated resolvent equation, with implications for understanding their memory structure.

Contribution

It provides a novel analysis connecting the decay of autocovariance in ARCH(∞) models to the properties of a resolvent kernel using Volterra operator theory.

Findings

Autocovariance decays subexponentially or geometrically based on the resolvent kernel.

Decay bounds on autocovariance are equivalent to bounds on the kernel.

The analysis offers insights into the memory structure of ARCH(∞) processes.

Abstract

The asymptotic properties of the memory structure of ARCH($\infty$) equations are investigated. This asymptotic analysis is achieved by expressing the autocovariance function of ARCH($\infty$) equations as the solution of a linear Volterra summation equation and analysing the properties of an associated resolvent equation via the admissibility theory of linear Volterra operators. It is shown that the autocovariance function decays subexponentially (or geometrically) if and only if the kernel of the resolvent equation has the same decay property. It is also shown that upper subexponential bounds on the autocovariance function result if and only if similar bounds apply to the kernel.