Regular variation of GARCH processes

https://doi.org/10.1016/S0304-4149(01)00156-9Get rights and content
Under an Elsevier user license
open archive

Abstract

We show that the finite-dimensional distributions of a GARCH process are regularly varying, i.e., the tails of these distributions are Pareto-like and hence heavy-tailed. Regular variation of the joint distributions provides insight into the moment properties of the process as well as the dependence structure between neighboring observations when both are large. Regular variation also plays a vital role in establishing the large sample behavior of a variety of statistics from a GARCH process including the sample mean and the sample autocovariance and autocorrelation functions. In particular, if the 4th moment of the process does not exist, the rate of convergence of the sample autocorrelations becomes extremely slow, and if the second moment does not exist, the sample autocorrelations have non-degenerate limit distributions.

MSC

Primary: 62M10
Secondary: 62G20
60G55
62P05
60G10
60G70

Keywords

Point process
Vague convergence
Multivariate regular variation
Mixing condition
Stationary process
Heavy tail
Sample autocovariance
Sample autocorrelation
GARCH
Finance
Markov chain

Cited by (0)

1

This research supported by an NWO Ph.D. Grant.

2

This research supported in part by NSF DMS Grant No. DMS-9972015.

3

This research supported by the European Research Mobility Programme Network DYNSTOCH.