Essentials of Time Series for Financial Applications

(5.46)

where $E\left[I_{\left\{{\epsilon }_t<0\right\}}{\epsilon }_t^2\right]=E\left[I_{\left\{{\epsilon }_t<0\right\}}\right]E\left[{\epsilon }_t^2\right]=0.5{\overline{\sigma }}^2$ holds for all symmetric distributions with zero mean, for which the magnitude of ${\epsilon }_t$ and therefore of its square does not depend on the sign of ${\epsilon }_t$ and

$E\left[I_{\left\{{\epsilon }_t<0\right\}}\right]=\mathrm{Pr}({\epsilon }_t<0)=\frac{1}{2}.$ (5.47)

(5.47)

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