Implied covariances:

In the GARCH based models 4 and 5 we construct covariance forecasts for series i and j by looking at the variance forecasts for the series i and j as well as the forecast for the sum series of i and j (expression ). While applying this procedure we are well aware that formally the sum of two GARCH processes is not a GARCH process. This fact however does not intrinsically pose any caveat to the methodology - since the GARCH parameters for the three series (i, j and the sum thereof) are separately optimized in the in sample for all three - and no implicit constructions are used extending the parameters for the series i and j to their sum series.

What can however be of serious concern is that we have no guarantee that $\Sigma_t$ has positive eigen-values. If $\Sigma_t$ does have one or more negative eigen-values - this implies that there exists a portfolio with a forecast variance which is negative. Unless the dynamics of the market changes drastically between the in-sample and the out-of-sample, it is extremely unlikely that the method of implied covariances will lead to this pathological condition. We can confirm that in our analysis, with the fixed portfolios, we did not run into this situation - but this does not imply that $\Sigma_t$ never had negative eigen-values. In section 7 we again discuss this issues of pathology in $\Sigma_t$.