Keywords: ARCH, ARMA Model for a Squared GARCH, Autocorrelation, Autocovariance of a Squared GARCH, Ergodicity,
GARCH(p,q), IGARCH,
Infinite ARCH, Lyapounov's Coefficient, Kurtosis, Moments, Prediction of the squares,
Riskmetrics, Shock Persistence,
Strict and second-order stationarity, Strong and Semi-Strong GARCH, Top Lyapounov Exponent, Volatility, Volatility clustering.
Description:
ARCH (AutoRegressive Conditionally Heteroskedastic) models were introduced
by Engle (1982) and their GARCH (Generalized ARCH) extension is due to
Bollerslev (1986). In these models, the key concept is the conditional variance,
that is the variance conditional to 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 stylized
facts characterizing financial series, as described in Chapter 1. In the same time,
it is simple enough to allow for a complete study of the solutions. The 'linear'
structure of these models can be put forward through several representations that
will be studied in this chapter.
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 infinite ARCH models, which allow for a slower decay of squared-returns
autocorrelations. Then, we consider the existence of moments and the properties of
the autocorrelation structure. We conclude this chapter by examining forecasting
issues.
used to plot the stationarity regions in Figures 8 and 10 , Mathematica code 
used for the correlograms of Figure 11, and 
for computing ordinary and partial autocorrelations (of the squares) of an ARMA-GARCH (in F90 with Absoft's compiler) + 
executable program +