I'm trying to model an EGARCH(1,1). However, I dont understand why the mean from the general to (1,1) becomes $\sqrt{(\frac{2}{\pi})}$.
The following I am refering to is:
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This is because $|z_t|$ is a standard half-normal random variable and have expectation $\sqrt{\frac{2}{\pi}}$.
The expectation, $\mathbb{E}\left[|z_t|\right] = \sqrt{\frac{2}{\pi}}$ is true, when $z_t \overset{iid}{\sim}N(0,1)$. In this case, the absolute value of $z_t$ is called a (standard) half-normal variable that has known expectation. You can verify this from the Wikipedia page. If $z_t \overset{iid}{\sim}N(0,\sigma^2)$ then $\mathbb{E}\left[|z_t|\right]=\sigma\sqrt{\frac{2}{\pi}}$.
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$\begingroup$ Thanks! Didn't actually know, that when you take the absolute value of the expectations, the variable is referred to as a half-normal variable. $\endgroup$ May 13, 2021 at 6:13
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1$\begingroup$ @SebastianStraussHansen, the distribution is actually called Folded normal. $\endgroup$ May 13, 2021 at 7:48
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$\begingroup$ @RichardHardy Indeed. The half-normal distribution is a special case of the Folded normal distribution when $\mu = 0$, which is also described in your link. :-) $\endgroup$– PlebMay 13, 2021 at 10:04