# EGARCH(1,1) mean

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:

## 1 Answer

### 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 $$\mathbb{E}\left[|z_t|\right]=\sigma\sqrt{\frac{2}{\pi}}$$.

• 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. – Sebastian Strauss Hansen May 13 at 6:13
• @SebastianStraussHansen, the distribution is actually called Folded normal. – Richard Hardy May 13 at 7:48
• @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. :-) – Pleb May 13 at 10:04