Remarks on stationary GARCH processes under heavy tail distributions

TL;DR

This paper develops methods for approximating the distribution of sample variances in GARCH processes, especially under heavy tails, providing improved confidence intervals through resampling techniques.

Contribution

It introduces a numerical approximation for the distribution of sample variances in GARCH models and enhances confidence interval accuracy under heavy tail innovations using a new resampling method.

Findings

Numerical approximation of the distribution of sample variances in GARCH processes.

Improved confidence intervals for variance estimates under heavy tails.

Enhanced resampling method for heavy-tailed innovations.

Abstract

Let $(X_n)_{n\in \mathbb Z}$ be a GARCH process with $E(X_0^4)<\infty$, and let $\mu_n$ denote the distribution of $\frac 1{{\sqrt n}}\sum_{i=1}^n [X_i^2-\mathbb E(X_0^2)]$. We derive a numerical approximation of $\mu_n$ when $x_1,...,x_n$ are observed. This yields the derivation of confidence intervals for $\mu= E(X_0^2)$ and we investigate the accuracy of these confidence intervals in comparison with standard ones based on normal approximation. Moreover, when the innovation process has heavy tail distribution, we improve the method using a new resampling method.