Peter Hall, Australian National University
Qiwei Yao, London School of Economics
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 describe large-sample properties of quasi-maximum likelihood estimators of parameters in ARCH and GARCH models, and develop bootstrap methods for estimating the distributions of parameter estimators. We show that in the case of heavy-tailed errors the range of possible limit distributions is extremely large, and in fact limit laws are not restricted to a class that can be described by a finite number of scalar parameters. Instead, they depend intimately on detailed properties of the error distribution.| This makes it impossible, in the heavy-tailed case, to develop conventional asymptotic approximations to parameter distributions. Standard bootstrap methods also fail to produce consistent estimators. To overcome these problems we develop percentile-$t$, subsample bootstrap approximations to estimator distributions. Studentising 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.