Augmented GARCH sequences: Dependence structure and asymptotics

TL;DR

This paper investigates the probabilistic structure and asymptotic behavior of augmented GARCH(1,1) sequences, providing new methods that relax previous restrictions and establish functional CLTs and empirical process results.

Contribution

It introduces a novel approach based on independence properties to analyze augmented GARCH models, extending asymptotic theory without restrictive moment or smoothness assumptions.

Findings

Derived functional CLTs for powers of augmented GARCH variables

Established error rates in the CLT for these processes

Obtained asymptotic results for empirical processes under minimal conditions

Abstract

The augmented GARCH model is a unification of numerous extensions of the popular and widely used ARCH process. It was introduced by Duan and besides ordinary (linear) GARCH processes, it contains exponential GARCH, power GARCH, threshold GARCH, asymmetric GARCH, etc. In this paper, we study the probabilistic structure of augmented $\mathrm {GARCH}(1,1)$ sequences and the asymptotic distribution of various functionals of the process occurring in problems of statistical inference. Instead of using the Markov structure of the model and implied mixing properties, we utilize independence properties of perturbed GARCH sequences to directly reduce their asymptotic behavior to the case of independent random variables. This method applies for a very large class of functionals and eliminates the fairly restrictive moment and smoothness conditions assumed in the earlier theory. In particular, we derive functional CLTs for powers of the augmented GARCH variables, derive the error rate in the CLT and obtain asymptotic results for their empirical processes under nearly optimal conditions.