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Thread: unconditional expectation for GARCH model with normal inovations

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    unconditional expectation for GARCH model with normal inovations




    Hi i have been asked to find unconditional expectations and then a recursive relation. I think I have started well but i am confused at how the recursive relation is made up.
    this is then to be used with a GARCH model i am working on.

    details of question:

    d(t) = mu + sigma(t)Z(t)
    (sigma(t))^2 = a + ( b(Z(t-1))^2 + c)(sigma(t-1))^2

    Z(t) ~ N(0,1)

    find unconditional moments for: E[d(t)] , Var[d(t)] , Kurtosis[d(t)]
    and E[ (d(t)-mu)^2 * (d(t-1) - mu)^2 ]

    for k = 2, 3, ..., derive a recursive formula for E[ (d(t)-mu)^2 * (d(t-k) - mu)^2 ]
    in terms of E[ (d(t)-mu)^2 * (d(t-(k +1)) - mu)^2 ]

    i found that
    E[d(t)] = mu
    Var[d(t)] = a/(1-b-c)
    Kurtosis[d(t)] = 3(1-(b + c)^2) / (1 - (b + c)^2 - 2b^2)

    its the last bit that eludes me.

    Thank you for any help
    Last edited by grj1; 01-14-2014 at 05:44 PM. Reason: missed information

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    Re: unconditional expectation for GARCH model with normal inovations

    I feel like there is information missing from this - is there more to the question?

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    Re: unconditional expectation for GARCH model with normal inovations


    No that's all. maybe that the Z's are iid.

    like I said I can get started.

    E[d(t)] = mu + E[Z(t)] * E[sigma(t)] = mu since Z~(0,1)


    E[(sigma(t))^2] = E[a + ( b(Z(t-1))^2 + c)(sigma(t-1))^2] (calling the sigma value V for ease and using the fact its unconditional i can disregard the t constraint )
    V = a + bV + cV

    V = a/(1-b-c)

    which if you look at it is the same as Var[d(t)]

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