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.