GARCH Models, 2nd Edition

4 Alternative Models for the Conditional Variance

Classical GARCH models rely on modelling the conditional variance as a linear function of the squared past innovations. The merits of this specification are its ability to reproduce several important characteristics of financial time series – succession of quiet and turbulent periods, autocorrelation of the squares but absence of autocorrelation of the returns, leptokurticity of the marginal distributions – and the fact that it is sufficiently simple to allow for an extended study of the probability and statistical properties.

The particular functional form of the standard GARCH volatility entails, however, important restrictions. For example, it entails positive autocorrelations of the squares at any lags (see Proposition 2.2). As will see in Part II of this book, the positivity constraints on the GARCH coefficients entail also technical difficulties for the inference. The standard GARCH formulation also does not permit to incorporate exogenous information coming from other time series, for instance macro‐economic variables or intraday realised volatilities, possibly observed at different frequencies.

From an empirical point of view, the symmetric form of the classical GARCH model is one of its most obvious drawbacks. Indeed, by construction, the conditional variance only depends on the modulus of the past variables: past positive and negative innovations have the same effect on the current volatility. This property ...