Giraitis, Liudas; Leipus, Remigijus; Surgailis, Donatas Recent advances in ARCH modelling. (English) Zbl 1180.62121 Teyssière, Gilles (ed.) et al., Long memory in economics. Berlin: Springer (ISBN 978-3-540-22694-9/hbk). 3-38 (2007). From the introduction: The present paper reviews some recent theoretical findings on ARCH type models. We focus mainly on covariance stationary models which display empirically observed properties known as “stylized facts”. One of the major issues to determine is whether the corresponding model \(r^2_t\) for squares has long memory or short memory, i.e., whether \(\sum^\infty_{k=0}|\text{Cov}(r^2_k,r^2_0)|=\infty\) or \(\sum^\infty_{k=0}|\text{Cov}(r^2_k,r^2_0)|<\infty\) holds. It is pointed out that for several ARCH-type models the behavior of \(\text{Cov}(r^2_k,r^2_0)\) alone is sufficient to derive the limit distribution of \(\sum^N_{j=1}(r^2_j- Er^2_j)\) and statistical inferences, without imposing any additional (e.g., mixing) assumptions on the dependence structure. This review discusses ARCH\((\infty)\) processes and their modifications such as linear ARCH (LARCH), bilinear models, long memory EGARCH and stochastic volatility, regime switching SV models, random coefficient ARCH and aggregation. We give an overview of the theoretical results on the existence of a stationary solution, dependence structure, limit behavior of partial sums, leverage effects and long memory properties of these models. Statistical estimation of ARCH parameters and testing for change-points are also discussed.For the entire collection see [Zbl 1106.91001]. Cited in 20 Documents MSC: 62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH) 62P05 Applications of statistics to actuarial sciences and financial mathematics 60G10 Stationary stochastic processes 62F10 Point estimation PDFBibTeX XMLCite \textit{L. Giraitis} et al., in: Long memory in economics. Berlin: Springer. 3--38 (2007; Zbl 1180.62121)