6
$\begingroup$

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?

Improve this question
$\endgroup$
2
  • 1
    $\begingroup$ 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, 2015 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, 2015 at 4:10

1 Answer 1

Reset to default
6
$\begingroup$

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

Improve this answer
$\endgroup$
1
  • 2
    $\begingroup$ The link from @Johan Stax Jakobsen is now dead. I believe the paper referenced is the following: Pelagatti, Matteo, and Francesco Lisi. "Variance initialisation in GARCH estimation." S. Co. 2009. Sixth Conference. Complex Data Modeling and Computationally Intensive Statistical Methods for Estimation and Prediction. Maggioli Editore, 2009. $\endgroup$
    – agritrader
    Feb 13, 2021 at 19:51

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Not the answer you're looking for? Browse other questions tagged or ask your own question.