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I have fit a GARCH(1,) model in Python, assuming the residuals are $t$ distributed. I am checking the standardized residuals. ARCH and Ljung-Box tests don't reject the null hypothesis. However, I am having trouble understanding the behavior of the graphs.

The first graph displays the distribution of standardized residuals. As you can see it is not normally distributed.

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The second one is the QQ plot.

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The last one is a scatter plot of standardized residuals versus estimated volatility.

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How do I interpret these graphs of standardized residuals?

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  1. "As you can see it is not normally distributed." + "I assumed that the residual of the GARCH are t distributed." = What problem do you see in that? I do not see any.
  2. For the second plot, what reference distribution did you use? You should have used Student's $t$ distribution. If you did, you see it is not a great match at the left tail. You probably need an asymmetric distribution (skewed Student's $t$? Johnson's $S_U$?) to fix that. But actually it is easier to just study the probability integral transform of the standardized residuals. For a statistically adequate model, it should be Uniform[0,1] regardless of what distribution you have assumed for the standardized innovations.
  3. I am not sure the third plot is all that informative. I am not used to seeing it as part of GARCH diagnostics.
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  • $\begingroup$ thank you so much for your replies!! $\endgroup$
    – Mattia
    Nov 15, 2023 at 12:46
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    $\begingroup$ @Mattia, you are welcome! $\endgroup$ Nov 15, 2023 at 13:05

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