Keywords: Contemporaneous Aggregation, Semi-Strong GARCH, Stability by Aggregation, Stochastic Volatility, Strong GARCH, Temporal Aggregation, Weak GARCH.
Description: Most financial series are analyzed at different frequencies (day, week, month..). The properties of a series and, as a consequence of the model fitted to this series, often crucially depend on the observation frequency. For instance, empirical studies generally find a stronger persistence, that is α+β closer to 1 in GARCH(1,1) models, when the frequency increases. For a given asset, observed at different frequencies, a natural question is whether strong GARCH models at different frequencies are compatible. If the answer is positive, the class of GARCH will be called stable by temporal aggregation. In this chapter, we consider, more generally, invariance properties of the class of GARCH processes with respect to time transformations frequently encountered in econometrics. It will be seen that, to obtain stability properties, a wider class of GARCH-type models, called weak GARCH and based on the L2 structure of the squared returns, has to be introduced.