GARCH( p, q ) Processes

In autoregressive conditionally heteroscedastic (ARCH) and their GARCH (generalized ARCH) models, the key concept is the conditional variance. In the classical GARCH models, the conditional variance is expressed as a linear function of the squared past values of the series. The ‘linear’ structure of these models can be displayed through several representations that are studied in this chapter. The chapter presents definitions and representations of GARCH models. Then it establishes the strict and second-order stationarity conditions. Starting with the first-order GARCH model, for which the proofs are easier and the results are more explicit, the chapter extends the study to the general case. It also studies the so-called ARCH(∞) models, which allow for a slower decay of squaredreturn autocorrelations. Then, the chapter considers the existence of moments and the properties of the autocorrelation structure. It concludes by examining forecasting issues. Controlled Vocabulary Terms autocorrelation; conditional variance; forecasting; generalized autoregressive conditional heteroskedasticity