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As the above states, I need to find an iterative equation that predicts the variances of a GARCH(1,1) model. Here's how I started:

Let's suppose that we have $h$ as our forecast origin. We know that for a one step ahead forecast,

$$ \sigma_{h}^2(1) = \alpha_0 + \alpha_1a_h^2 +\beta_1 \sigma_h^2$$

Now, for a two step ahead forecast, it can be shown that

$$ \sigma_{h}^2(2) = \alpha_0 + (\alpha_1+\beta_1)\sigma_h^2(1)$$

Is the answer to this question some general $\sigma_h^2 (l)$?

Also, let's suppose that we did not have $a_h, \sigma_h$ prior to this problem. If we had a set of output, how would we determine these properties? Any help would be much appreciated!

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  • $\begingroup$ Are you asking us to clarify the actual question you have (Is the answer to this question some general $\sigma^2_h(l)$?)? Also, what is a set of output, what does it contain? $\endgroup$ – Richard Hardy Mar 16 '16 at 8:26
  • $\begingroup$ I am asking to clarify the question, yes. Also, by a set of output, let's say that the data is a simulated ARMA(2,3) model with GARCH(1,1). $\endgroup$ – K.M. Mar 16 '16 at 17:42
  • $\begingroup$ I was going through my old answers and noticed this one was not accepted. Do you perhaps need further clarification? $\endgroup$ – Richard Hardy Feb 12 '17 at 12:52
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Is the answer to this question some general $\sigma_h^2(l)$?

Given just the limited knowledge provided in the question, I think it could be reasonable to assume that the answer to the question is a general formula for $\sigma_h^2(l)$, something in the spirit of the one you provide for $\sigma_h^2(2)$.

Also, let's suppose that we did not have $a_h$, $\sigma_h$ prior to this problem. If we had a set of output, how would we determine these properties?

If $a_h$ and $\sigma_h$ are not readily given to you but a model form is provided (such as ARMA(2,3)-GARCH(1,1) as suggested in the comments), then you can estimate the model coefficients and obtain fitted values of $a_h$ and $\sigma_h$ from the fitted model. Such output will be available in standard statistical software; e.g. in "rugarch" package in R they will be stored as parts of fitted ARMA-GARCH model object of class uGARCHfit. Then use them as if they were given.

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