The model may be estimated in a single stage using ML although this will be difficult. So Engle advocates a two-stage procedure where each variable in the system is first modelled separately as a univariate GARCH
A joint log-likelihood function for this stage could be constructed, which would simply be the sum (over N) of all of the log-likelihoods for the individual GARCH models
In the second stage, the conditional likelihood is maximised with respect to any unknown parameters in the correlation matrix
The log-likelihood function for the second stage estimation will be of the form
where θ1 and θ2 denote the parameters to be estimated in the 1st and 2nd stages respectively.
Asymmetric models have become very popular in empirical applications, where the conditional variances and / or covariances are permitted to react differently to positive and negative innovations of the same magnitude
In the multivariate context, this is usually achieved in the Glosten et al. (1993) framework
Kroner and Ng (1998), for example, suggest the following extension to the BEKK formulation (with obvious related modifications for the VECH or diagonal VECH models)
where zt−1 is an N-dimensional column vector with elements taking the value −ϵt−1 if the corresponding element of ϵt−1 is negative and zero otherwise.
An Example: Estimating a Time-Varying Hedge Ratio for FTSE Stock Index Returns (Brooks, Henry and Persand, 2002).
