Temporal Aggregation and Weak GARCH Models

Most financial series are analyzed at different frequencies (daily, weekly, monthly, …). The properties of a series and, as a consequence, of the model fitted to the 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 models 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.

4.1 Temporal Aggregation of GARCH Processes

Temporal aggregation arises when the frequency of data generation is lower than that of the observations so that the underlying process is only partially observed. The time series resulting from temporal aggregation may of course have very different properties than the original time series. More formally, the temporal aggregation problem can be formulated as follows: given a process (Xt) and an integer m, what are the properties ...

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