Temporal Aggregation of GARCH Models : Conditional Kurtosis and Optimal Frequency

We examine the properties of temporally aggregated distributions when one period changes follow a strong GARCH process. Our main results: (1) We derive explicit expressions for the conditional volatility and kurtosis of the aggregated distribution. (2) As the time horizon gets longer the conditional aggregated kurtosis approaches three (resp. a different constant, for stock variables) or infinity depending on whether or not a simple inequality in term of the GARCH parameters is satisfied. (3) Given that the aggregation of a strong GARCH process is not any more a strong GARCH process, the question arises for which data frequency a description by a strong GARCH process fits the data best. We propose a quasi maximum likelihood method to determine the optimal data frequency for a GARCH description. (4) For models with different basic frequency and with different residual distributions we perform out of sample tests of three months density forecasts on the basis of daily market prices. It turns out that low frequency models with longer basic periods and fewer aggregation steps fare better than high frequency models. This seems to imply that for high frequency models the advantage of having more data available for estimation is outweighed by the disadvantage of aggregation magnifying estimation errors