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When reading the book of Analysis of Financial Time Series, I have a question on the ARCH model, defined as follows

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Regarding this model, the author also states that.

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I do not understand why does the equation marked with yellow color is satisfied.

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  • $\begingroup$ I don't immediately follow what calculations the author used here, however the fourth central moment of the $N(0,\sigma^2)$ distribution is commonly known to be $3\sigma^4$. See section 2.2 of en.wikipedia.org/wiki/Normal_distribution#Moments. $\endgroup$ – Marc May 27 '14 at 19:23
  • $\begingroup$ See Isserlis' Theorem for more details. $\endgroup$ – user88595 May 27 '14 at 19:35
  • $\begingroup$ Thanks a lot for the hint! $\endgroup$ – user785099 May 29 '14 at 2:12
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$E(a_t^4|F_{t-1}) = E(\sigma_t^4\epsilon_t^4|F_{t-1}) = E(\epsilon_t^4|F_{t-1}) * E[(\sigma_t^2)^2|F_{t-1}] = 3(\alpha_0 + \alpha_1a_{t-1}^2)^2 $

I suggest reading up on the law of iterated expectations.

Ninja edit to your specific question is we know $E(\epsilon_t^4|F_{t-1})$ = 3

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