Inference in Arch and Garch Models with Heavy-Tailed Errors

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.

Interval estimation of value-at-risk based on GARCH models with heavy-tailed innovations

“…In this case, a subsample bootstrap method is needed (see Hall and Yao (2003a)). However, our methods are valid regardless of finite or infinite fourth moment of the innovation.…”

Persistence and Kurtosis in GARCH and Stochastic Volatility Models

“…Measuring the persistence of volatility correctly is important because it is often estimated very close to one, which has implications for the theoretical models used to value the price of financial stocks. On the other hand, whether the distribution of « t is gaussian or has fat tails has implications mainly for inference on the models fitted to represent the dynamic evolution of volatility [see, e.g., Hall and Yao (2003)]. In this article we show that, although both GARCH and ARSV models are able to explain excess kurtosis and significant autocorrelations of squares with a slow rate of decay, the relationship between the persistence of shocks to the volatility, first-order autocorrelation of squares, and kurtosis implied by each of these models is different.…”

Residual-Based Garch Bootstrap and Second Order Asymptotic Refinement

“…On the other hand, the case in which the distribution of the innovation {ε t } is heavy-tailed is also an important and interesting subject. Berkes and Horváth (2003) show asymptotic properties of QML estimators in the presence of heavy-tails in {ε t }, and Hall and Yao (2003) consider the case of infinite variance such that Eε 2 t = ∞. The rate of convergence and sometimes even the limit distributions for the case of heavytailed innovations differ from those of the usual stationary asymptotics, and we do not cover this case in the present paper.…”