### Equation on top of page 37

##### October 22, 2011

My FM320 student Yong Bin Ng pointed out a typo in the equation on top of page 37. It should be:$$ E(Y^4)=3E\left[(\omega+\alpha Y_{t-1}^2)^2\right]=3(\omega^2+2\alpha \omega \sigma^2 + \alpha^2 E(Y^4))$$

Also, in case you were wondering how to derive the equation we use previous results on page 36, independence of Y's and Z and properties of the normal distribution, and it's done as follows. \begin{aligned} E(Y^4)&=E(Y_t^4)\ &=E(\sigma_t^4 Z_t^4)\ &=E(\sigma_t^4)E(Z_t^4)\ &=E(\sigma_t^4)3(E(Z_t^2))^2\ &=3E((\sigma_t^2)^2)\ &=3E\left[(\omega+\alpha Y_{t-1}^2)^2\right]\ &=3(\omega^2+2\alpha \omega \sigma^2 + \alpha^2 E(Y_{t-1}^4))\ &=3\omega^2+6\alpha \omega \sigma^2 + 3\alpha^2 E(Y^4)\ &=3\omega^2+6\alpha \omega \frac{\omega}{1-\alpha} + 3\alpha^2 E(Y^4) \end{aligned} then,

$$ E(Y^4)(1-3\alpha^2)(1-\alpha) =3\omega^2(1-\alpha)+6\alpha \omega^2 $$

and $$ E(Y^4)=\frac{3\omega^2(1+\alpha)}{(1-3\alpha^2)(1-\alpha)} $$

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