Essentials of Time Series for Financial Applications

(6.22)

We call ${\overline{\sigma }}_{ij}\equiv E\left[{\epsilon }_{i,t}{\epsilon }_{j,t}\right]={\omega }_{ij}/(1-\alpha -\beta )$ in Eq. (6.9) the unconditional covariance implied by a stationary GARCH(1,1) model in which $\alpha +\beta <1$. Therefore, Eq. (6.8) can be rewritten as:

${\sigma }_{ij,t+1|t}={\overline{\sigma }}_{ij}+\alpha ({\epsilon }_{i,t}{\epsilon }_{j,t}-{\overline{\sigma }}_{ij})+\beta ({\sigma }_{ij,t|t-1}-{\overline{\sigma }}_{ij}).$ (6.23)

(6.23)

This follows from

$\begin{array}{ll}{\sigma }_{ij,t+1|t}\hfill & ={\omega }_{ij}+\alpha {\epsilon }_{i,t}{\epsilon }_{j,t}+\beta {\sigma }_{ij,t|t-1}={\overline{\sigma }}_{ij}(1-\alpha -\beta )+\alpha {\epsilon }_{i,t}{\epsilon }_{j,t}+\beta {\sigma }_{ij,t|t-1}\hfill \\ ={\overline{\sigma }}_{ij}+\alpha ({\epsilon }_{}\hfill \end{array}$