GARCH Models, 2nd Edition

2 GARCH(p, q) Processes

Autoregressive conditionally heteroscedastic (ARCH) models were introduced by Engle (1982), and their GARCH (generalised ARCH) extension is due to Bollerslev (1986). In these models, the key concept is the conditional variance, that is, the variance conditional on the past. In the classical GARCH models, the conditional variance is expressed as a linear function of the squared past values of the series. This particular specification is able to capture the main stylised facts characterising financial series, as described in Chapter 1. At the same time, it is simple enough to allow for a complete study of the solutions. The ‘linear’ structure of these models can be displayed through several representations that will be studied in this chapter.

We first present definitions and representations of GARCH models. Then we establish 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, we extend the study to the general case. We also study the so‐called ARCH(∞) models, which allow for a slower decay of squared‐return autocorrelations. Then, we consider the existence of moments and the properties of the autocorrelation structure. We conclude this chapter by examining forecasting issues.

2.1 Definitions and Representations

We start with a definition of GARCH processes based on the first two conditional moments.

Definition 2.1 GARCH(p, q