Marginalization and contemporaneous aggregation in multivariate GARCH processes

We first of all show that contemporaneous aggregation of independent univariate GARCH processes yields a weak GARCH process as introduced by Drost and Nijman (1993). Subsequently we analyze the dependence of the parameters in the aggregate on the parameters in the underlying models and show that the variance parameters after aggregation depend on the underlying variance and kurtosis parameters. Then we generalize the results by showing that a linear combination of variables generated by a multivariate GARCH process will also be weak GARCH. Finally we derive the marginal (weak GARCH) processes implied by several multivariate GARCH processes.