Modeling volatility dynamics

Many economic and financial time series have been found to exhibit dynamics in variance; that is, the second moment of the time series innovations varies over time. Many possible model specifications are available to capture this phenomena, but to date, the class of models most widely used are autoregressive conditional heteroskedasticity (ARCH) models. ARCH models provide parsimonious approximations to volatility dynamics and have found wide use in macroeconomics and finance. The family of ARCH models is the subject of this paper. In section II, we sketch the rudiments of a rather general univariate time-series model, allowing for dynamics in both the conditional mean and variance. In section III, we provide motivation for the models. In section IV, we discuss the properties of the models in depth, and in section V, we discuss issues related to estimation and testing. In Section VI, we detail various important extensions and applications of the model. We conclude in section VII with speculations on productive directions for future research.