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From what I know, the GARCH(p,q) model is estimated via MLE and through an iterative process. Let's say if i wanted to recreate a GARCH(1,1) parameter estimation with excel solver (through maximizing the log-likelihood), how are my initial GARCH terms $ \sigma_t^2$ set?

More specifically, given $ \epsilon_t = v_t\sqrt{\sigma_t^2}$ where $$\sigma_t^2 = \alpha_0 +\alpha_1\epsilon^2_{t-1}+\alpha_2\sigma_{t-1}^2 $$ how does the process of parameter estimation start since we do not know what $\sigma_{t-1}$ is?

One answer I've read from here shows that the program set the initial GARCH term to be the sample variance or its expected value. Is this how we approach it?

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    $\begingroup$ It may depend on the particular software you want to mimic; but sample variance sounds like a good choice in general. $\endgroup$ – Richard Hardy Jan 13 '15 at 20:17
  • $\begingroup$ Let's say it is Matlab, with matlab, i fit the Garch model on the same data set with the estimate() function. Then I inferred the residuals through infer(estmdl, x) and found my fitted model through fitted = y-residuals; On a graph, the estimation was almost spot on. However when I transfer this over to the excel solver, I end up with an almost identical loglikelihood but very different results. I am taking the Loglikelihood function from here: ams.sunysb.edu/~yiyang/research/computational_finance/… $\endgroup$ – Kevin Pei Jan 14 '15 at 4:10
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I know of at least five ways of initializing the volatility process:

1) Set it equal to $\varepsilon_{t-1}^2$,

2) The sample variance,

3) Unconditional variance of the model ($\alpha_0/(1-\alpha_1 - \alpha_2)$),

4) Allow it it to be an parameter to be estimated,

5) Backcasting with an exponential filter.

The topic is discussed in further detail here

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