The dynamics of squared returns under contemporaneous aggregation of GARCH models

•We describe the dynamics of the volatility of a portfolio of strong GARCH(1,1) processes with heterogeneous parameters.•We show that the dynamics of the squared return is an ARMA process of infinite order.•We describe an estimation technique that gives consistent estimators of this dynamics.•We compare the aggregation-corrected estimator (ACE) with the usual GARCH(1,1) model for different horizons.•We find that the ACE outperforms the other estimators in forecasting the aggregate variance of a portfolio of U.S. equities.

The paper investigates the properties of a portfolio composed of a large number of assets driven by a strong multivariate GARCH(1,1) process with heterogeneous parameters. The aggregate return is shown to be a weak GARCH process with a (possibly large) number of lags, which reflects the moments of the distribution of the individual persistence parameters. The paper describes a consistent estimator of the aggregate return dynamics, based on nonlinear least squares. The proposed aggregation-corrected estimator (ACE) performs very well and outperforms some competing estimators in forecasting the daily variance of U.S. stocks portfolios at different horizons.