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I am having trouble calculating the standardized errors of a GARCH(1,1) model. Since $\epsilon_t = \sigma_t z_t$ with $z \sim \text{N}(0,1)$, the standardized error is $z_t = \frac{\epsilon_t}{\sigma_t}$. However, I don't know how to generate the $\sigma_t$. In Stata, I stumbled upon residuals that are described as "predicted innovations". But what are those?

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2 Answers 2

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I don't know how to generate the $\sigma_t$

The fitted standard deviations $\hat\sigma_t$ should be part of the output of a GARCH model. I do not work with Stata so I am not sure how to extract them, but they should be accessible. Given those, the standardized residuals will be $\hat z_t=\frac{\hat\epsilon_t}{\hat\sigma_t}$. Given this, you should not care much about "predicted innovations" which seem to be the residuals of the conditional mean model.

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  • $\begingroup$ Thanks so much! I have to test whether my GARCH model is correctly specificied by calculating the autocorrelation of the standardized errors. Here, you said standardized residuals. Is there a difference? $\endgroup$
    – Taufi
    Commented Mar 1, 2016 at 16:53
  • $\begingroup$ Errors are typically unobservable while residuals are the fitted values for errors. For example, we may have a model $$y=\beta x+\varepsilon$$ where $\varepsilon$ is the (unobservable) error. We fit the model to get $$y=\hat\beta x+\hat\varepsilon$$ or $$y=\hat y+\hat\varepsilon$$ where $\hat y=\hat\beta x$ and $\hat\varepsilon$ is the residual, or the fitted error. Adding the term "standardized" to either the error or the residual does not change the basic intuition. $\endgroup$ Commented Mar 1, 2016 at 16:57
  • $\begingroup$ The terminology in Stata's manuals is a little unorthodox, at least from my experience. "Predicted innovations" may be a misleading term, and it is not so easy to find out how it is defined. I checked a few help files there and got the impression that these are simply the residuals $\hat\varepsilon$ from the conditional mean model. $\endgroup$ Commented Mar 1, 2016 at 17:01
  • $\begingroup$ It helped. I now understand the predicted innovations to be just the predicted residuals of the process. And I found that $\sigma_t$ can be calculated as the square root of the variance which results from the command "predict myvar, variance" in Stata. Thanks a lot, Richard Hardy! $\endgroup$
    – Taufi
    Commented Mar 1, 2016 at 19:12
  • $\begingroup$ I am glad it was helpful! $\endgroup$ Commented Mar 1, 2016 at 19:13
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Just run as below after garch

predict res, resid
predict h, var
gen stdres=res/h
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    $\begingroup$ I didn't downvote but I guess the objection is to posting code only. Note that you divide by the variance, not the standard deviation as required. $\endgroup$
    – Nick Cox
    Commented Dec 16, 2016 at 14:25

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