ARCH and GARCH Models

In finance data one often observes so-called volatility clustering, i.e. periods with relatively high volatility and periods with low volatility occur. This is an indication that the (conditional) variance is dependent on past observations. The most common models for the conditional variance are (G)ARCH models and stochastic volatility (SV) models. Both are often used to model financial data, e.g. asset returns. Here we only discuss the GARCH case, since combining AR/ARMA Models with GARCH innovations provides an easy way to model jointly the conditional mean and the conditional variance. A central result in this section is a necessary and sufficient condition for stationary solutions of (G)ARCH systems. In many applications, not only the forecast (conditional mean) is of interest but it is also important to have a measure for the quality of the forecast, in particular, the variance of the prediction errors is of interest. Of course, due to stationarity, the variance of the observed variables as well as the variance of the prediction errors (innovations) is constant (independent of time t). However, the conditional variance, i.e. the variance of the prediction error conditioned on the last observations, may be non-constant. This conditional variance, e.g. may be used for risk analysis in finance. We discuss the most common models for conditional variances such as The ARCH (Autoregressive Conditional Heteroscedasticity) model and the GARCH (Generalized ARCH) model in the scalar case and some of the many multivariate GARCH (MGARCH) models, in particular the VECH model and the BEKK model.