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Math Help - Using the Law of Iterated Expectations

  1. #1
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    Using the Law of Iterated Expectations

    Hi all,

    Im very desperate at the moment. The situation is im given these equations below:

    Firstly, assume a discrete model for daily price differences as:
     lnP_{t+1} - lnP_{t} = \mu + \sigma_{t+1}e_{t+1}

     e_{t+1} is iid N(0,1)

    Secondly, the GARCH equation is given as:
    \sigma^2_{t+1} = \alpha_{0} + \alpha_{0}\sigma^2_{t}e^2_{t} + \beta_{1}\sigma^2_{t}

    Constraints:

     \alpha_{0} > 0, \alpha_{1}, \beta_{1} \geq 0 and  \alpha_{1} + \beta_{1} < 1

    Use the LIE and the assumption of the variance process is stattionary derive the marginal expected variance:

     E[\sigma^2_{t}]

    The second one is a GARCH(1,1) model. Im not too sure if you are familiar with it. Anyways, I dont have a clue on earth on how to use to the Law if Iterated Expectation and use it one the second equation (ie, i think you use it on the GARCH one).

    It doesn't matter if you know what a GARCH model is, the problem is basically using LIE to solve the second equation and its just a trivial question.

    Can someone work it out for me!? Its driving me nuts!

    Thanks!

    Edit: Consider  \sigma_{t} or  \sigma_{t+1} as the variance at time "t" or something
    Last edited by Redeemer_Pie; October 9th 2010 at 10:41 AM.
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