# Thread: Using the Law of Iterated Expectations

1. ## 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:
$\displaystyle lnP_{t+1} - lnP_{t} = \mu + \sigma_{t+1}e_{t+1}$

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

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

Constraints:

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

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

$\displaystyle E[\sigma^2_{t}]$

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 $\displaystyle \sigma_{t}$ or $\displaystyle \sigma_{t+1}$ as the variance at time "t" or something