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(NOTE: also under discussion in sos Math Cyberboard)

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- Aug 15th 2011, 01:33 PMkingwinnerTime Series: ARCH model properties
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(NOTE: also under discussion in sos Math Cyberboard) - Aug 16th 2011, 01:00 AMMooRe: Time Series: ARCH model properties
Hello,

First equality : because depends on , so is independent from , since the Zt are iid, and hence the equality (that's a common property).

Second equality : same stuff, Xt is independent of all these Zt's, because it depends on Zt.

Third equality : ditto - Aug 16th 2011, 08:02 AMkingwinnerRe: Time Series: ARCH model properties
I think you're trying to say that Z_t and X_{t-1} are independent since X_{t-1} is a function of only Z_{t-1} , Z_{t-2}, ..., etc. and Z_i s are iid.

But now my question is: Why is X_{t-1} is function of ONLY Z_{t-1}, Z_{t-2},... ? How can we**prove**this?

The trouble is I think X_{t-1} a function of Z_{t-1}, Z_{t-2},...__AND__some σ_j.

Thanks! - Aug 16th 2011, 02:30 PMMooRe: Time Series: ARCH model properties
Yeah sorry I probably got messed up with the variables' names, well it seems like you understand what I meant (Itwasntme)

Well to see it, you can say that is -measurable (the first sigma is "sigma-algebra generated by...")

is -measurable, etc... So in the end, by independence of the Zt's, (independence).

If you want to prove it, you can just use induction, but it's not necessary in your exercise :)