10
Asymmetries
Classical GARCH models, studied in Parts I and II, rely on modeling 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.
From an empirical point of view, however, the classical GARCH modeling has an important drawback. 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 is in contradiction to many empirical studies on series of stocks, showing a negative correlation between the squared current innovation and the past innovations: if the conditional distribution were symmetric in the past variables, such a correlation would be equal to zero. However, conditional asymmetry is a stylized fact: the volatility increase due to a price decrease is generally stronger than that resulting from a price increase of the same magnitude.
The symmetry property of standard GARCH models has the following interpretation in terms of autocorrelations. If the law of ηt is symmetric, and under the assumption ...
