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Math Help - Time Series: ARCH model properties

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    Time Series: ARCH model properties



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    Moo
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    Re: Time Series: ARCH model properties

    Hello,

    First equality : because \sigma_t depends on Z_{t-1}, so Z_t is independent from \sigma_t, 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
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    Re: Time Series: ARCH model properties

    Quote Originally Posted by Moo View Post
    Hello,

    First equality : because \sigma_t depends on Z_{t-1}, so Z_t is independent from \sigma_t, since the Zt are iid, and hence the equality (that's a common property).
    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!
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    Moo
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    Re: 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

    Well to see it, you can say that X_t is \sigma(\sigma_t,Z_{t-1})-measurable (the first sigma is "sigma-algebra generated by...")
    \sigma_t is \sigma(X_{t-1})=\sigma(\sigma_{t-1},Z_{t-2})-measurable, etc... So in the end, by independence of the Zt's, X_t\perp Z_t (independence).
    If you want to prove it, you can just use induction, but it's not necessary in your exercise
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