Essentials of Time Series for Financial Applications

As for the conditions that guarantee that both ${\sigma }_{1,t+1|t}^2$ and ${\sigma }_{2,t+1|t}^2$ are positive at all times, that is, that help ensure semi-positive definiteness of ${\mathit{\Sigma }}_{t+1|t}$, clearly, because under a continuous distribution past squared shocks are unbounded,

$\begin{array}{l}(1-a^{11})T^{-1}\sum _{t=1}^T{\epsilon }_{1,t}^2+a^{11}{\epsilon }_{1,t}^2>0\mathrm{if\; and\; only\; if}{a}^{11}\in (0,1)\hfill \\ (1-a^{33})T^{-1}\sum _{t=1}^T{\epsilon }_{2,t}^2+a^{33}{\epsilon }_{2,t}^2>0\mathrm{if\; and\; only\; if}\hfill \end{array}$