Inference in Arch and Garch Models with Heavy-Tailed Errors

TLDR

ARCH and GARCH models capture conditional variance and are widely used for heavy‑tailed financial series, yet their behavior under heavy‑tailed errors is poorly understood and no reliable distributional approximations exist. The authors aim to address this gap by developing percentile‑t, subsample bootstrap methods to approximate the distributions of quasi‑maximum likelihood estimators in heavy‑tailed ARCH and GARCH models. They employ studentization to estimate scale and a subsample bootstrap to capture shape, providing a practical approximation framework. The proposed approach yields accurate, nonnormal distribution estimates, as shown both theoretically and through numerical experiments, whereas standard asymptotics and bootstrap methods fail.

Abstract

ARCH and GARCH models directly address the dependency of conditional second moments, and have proved particularly valuable in modelling processes where a relatively large degree of fluctuation is present. These include financial time series, which can be particularly heavy tailed. However, little is known about properties of ARCH or GARCH models in the heavy–tailed setting, and no methods are available for approximating the distributions of parameter estimators there. In this paper we show that, for heavy–tailed errors, the asymptotic distributions of quasi–maximum likelihood parameter estimators in ARCH and GARCH models are nonnormal, and are particularly difficult to estimate directly using standard parametric methods. Standard bootstrap methods also fail to produce consistent estimators. To overcome these problems we develop percentile–t, subsample bootstrap approximations to estimator distributions. Studentizing is employed to approximate scale, and the subsample bootstrap is used to estimate shape. The good performance of this approach is demonstrated both theoretically and numerically.