Regular variation of GARCH processes.
type: Item
instance of: scholarly article
title: Regular variation of GARCH processes.
zbMATH Open document ID: 1060.60033
DOI: 10.1016/S0304-4149(01)00156-9
author: Bojan Basrak
author: Richard A. Davis
author: T. Mikosch
published in: Stochastic Processes and their Applications
publication date: 2005-02-25T00:00:00Z
review text: Let a \(d\)-dimensional time series \(({\mathbf X}_t)\) be defined by a stochastic recurrence equation (SRE) \({\mathbf X}_t={\mathbf A}_t{\mathbf X}_{t-1}+{\mathbf B}_t\) where \({\mathbf A}_t\) are \(d\times d\) random matrices and \({\mathbf B}_t\) are \(d\)-dimensional vectors such that \((({\mathbf A}_t,{\mathbf B}_t))\) is an iid sequence. A generalized autoregressive conditionally heteroscedastic process \((X_t)\) of order \((p,q)\) (GARCH\((p,q)\)) is given by \(X_t=\sigma _t Z_t\), \(\sigma _t^2=\alpha _0+\sum _{i=1}^p \alpha _i X_{t-i}^2 + \sum _{j=1}^q \beta _j\sigma _{t-j}^2\) where \((Z_t)\) is an iid sequence and \(\alpha _i\)'s, \(\beta _j\)'s are nonnegative constants. The squared processes \((X_t^2)\) and \((\sigma _t^2)\) satisfy a SRE \({\mathbf X}_t={\mathbf A}_t{\mathbf X}_{t-1}+{\mathbf B}_t\) with \({\mathbf X}_t=(\sigma _{t+1}^2,\dots ,\sigma _{t-q+2}^2, X_t^2, \dots , X_{t-p+2}^2)'\). The authors start with some introduction to basic theory for SRE. This part is a valuable review of results concerning SRE. Using the connection between GARCH and SRE the authors show that the finite-dimensional distributions of a GARCH process are regularly varying. This implies the moment properties of the process and the dependence structure between neighboring observations. Regular variation enables to establish the large sample behavior of statistics from GARCH process such as the sample mean, sample autocovariance and sample autocorrelation function. For example, if the fourth moment of the process does not exist, then the rate of convergence of the sample autocorrelations is extremely slow.
reviewed by: Jiří Anděl
Mathematics Subject Classification ID: 60G10
Mathematics Subject Classification ID: 62M10
Mathematics Subject Classification ID: 60G55
Mathematics Subject Classification ID: 91B28
Mathematics Subject Classification ID: 62G20
Mathematics Subject Classification ID: 62P05
zbMATH DE Number: 2138964
zbMATH Keywords: point process
zbMATH Keywords: vague convergence
zbMATH Keywords: multivariate regular variation
zbMATH Keywords: mixing condition
zbMATH Keywords: stationary process
zbMATH Keywords: heavy tail
zbMATH Keywords: sample autocovariance
zbMATH Keywords: sample autocorrelation
zbMATH Keywords: GARCH
zbMATH Keywords: finance
zbMATH Keywords: Markov chain
MaRDI profile type: Publication
Wikidata QID: Q87452809
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description: scientific article; zbMATH DE number 2138964