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As we all know, the GARCH model is stated as
$\epsilon_t = \sigma_tz_t$
$\sigma_t^2 = w + \sum^q_{i=1}\alpha_i\epsilon_{t-i}^2 + \sum^q_{i=1}\beta_i\sigma_{t-i}^2$

In application, the estimate $\hat{\epsilon}_t^2$ is just the squared of the residuals. My question is what is the estimate of $\hat{\sigma}_t^2$ and how can it be obtained from the time series when using the GARCH estimation?

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    $\begingroup$ I am voting to close this question as it has no direct link with quantitative finance. It is better suited for CrossValidated.SE (migration path not available yet) $\endgroup$ – Quantuple Mar 27 '17 at 19:12
  • $\begingroup$ Maximum likelihood estimation is used to obtain model parameters, from which you can deduce $\sigma^2_t$, $z_t$ and whatever else you need. $\endgroup$ – Richard Hardy Mar 27 '17 at 19:40
  • $\begingroup$ Refer to Richard's answer, thank you very much. However, what is the major difference to interpret $\sigma^2_t$ and $\epsilon^2_t$, are there any formula representing $\sigma^2$ just like $\epsilon_^2$ is the square of the residuals. $\endgroup$ – L.Chau Mar 28 '17 at 3:47
  • $\begingroup$ Well $\sigma_t^2$ is the conditional variance of the residuals $\epsilon_t$ since $\epsilon_t = \sigma_t z_t$ and $z_t$ is a white noise. Hence the "CH" part of "GARCH" = Conditional Heteroskedasticity, meaning that in such a model, the conditional variance $\sigma_t^2$ is not constant. $\endgroup$ – Quantuple Mar 28 '17 at 8:12
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    $\begingroup$ I'm voting to close this question as off-topic because it belongs on Cross Validated. $\endgroup$ – Richard Hardy May 5 '17 at 14:11

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