Building Econometric Models

EWMA Covariance Models - Limitations

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Ch9 slides

This equation can be rewritten as an infinite order function of only the returns by successively substituting out the covariances:
The EWMA model is a restricted version of an integrated GARCH (IGARCH) specification, and it does not guarantee the fitted variance-covariance matrix to be positive definite
EWMA models also cannot allow for the observed mean reversion in the volatilities or covariances of asset returns that is particularly prevalent at lower frequencies.

Multivariate GARCH models are used to estimate and to forecast covariances and correlations.
The basic formulation is similar to that of the GARCH model, but where the covariances as well as the variances are permitted to be time-varying.
There are 3 main classes of multivariate GARCH formulation that are widely used: VECH, diagonal VECH and BEKK.
VECH and Diagonal VECH
e.g. suppose that there are two variables used in the model. The conditional covariance matrix is denoted Ht, and would be 2  2. Ht and VECH(Ht) are

In the case of the VECH, the conditional variances and covariances would each depend upon lagged values of all of the variances and covariances and on lags of the squares of both error terms and their cross products.
In matrix form, it would be written
Writing out all of the elements gives the 3 equations as
Such a model would be hard to estimate. The diagonal VECH is much simpler and is specified, in the 2 variable case, as follows:

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