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:$$E(Y^4)=3E[(\omega +\alpha Y_{t-1}^2{)}^2]=3({\omega }^2+2\alpha \omega {\sigma }^2+{\alpha }^{2}E(Y^4))$$

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. $$\begin{array}{rlrlrlrlrl}E(Y^4)& =E(Y_t^4)& =E({\sigma }_t^4{Z}_t^4)& =E({\sigma }_t^4)E(Z_t^4)& =E({\sigma }_t^4)3(E(Z_t^2){)}^2& =3E(({\sigma }_t^2{)}^2)& =3E[(\omega +\alpha Y_{t-1}^2{)}^2]& =3({\omega }^2+2\alpha \omega {\sigma }^2+{\alpha }^{2}E(Y_{t-1}^4))& =3{\omega }^2+6\alpha \omega {\sigma }^2+3{\alpha }^{2}E(Y^4)& =3{\omega }^2+6\alpha \omega \frac{\omega }{1-\alpha }+3{\alpha }^{2}E(Y^4)\end{array}$$ 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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