Journal of Probability and Statistics
Volume 2011 (2011), Article ID 906212, 17 pages
http://dx.doi.org/10.1155/2011/906212
Review Article

Semi- and Nonparametric ARCH Processes

Oliver B. Linton1 and Yang Yan2

1Department of Economics, The London School of Economics and Political Science, Houghton Street, London WC2A 2AE, UK
2Department of Statistics, The London School of Economics and Political Science, Houghton Street, London WC2A 2AE, UK

Received 27 February 2010; Accepted 28 June 2010

Academic Editor: Tak Kuen Siu

Copyright © 2011 Oliver B. Linton and Yang Yan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

ARCH/GARCH modelling has been successfully applied in empirical finance for many years. This paper surveys the semiparametric and nonparametric methods in univariate and multivariate ARCH/GARCH models. First, we introduce some specific semiparametric models and investigate the semiparametric and nonparametrics estimation techniques applied to: the error density, the functional form of the volatility function, the relationship between mean and variance, long memory processes, locally stationary processes, continuous time processes and multivariate models. The second part of the paper is about the general properties of such processes, including stationary conditions, ergodic conditions and mixing conditions. The last part is on the estimation methods in ARCH/GARCH processes.