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I wish to fit the innovations resulting from a GARCH (1,1) process to either a student-t or an NIG distribution. For stability, I had to scale my data before applying GARCH. How will this affect the resulting innovations?

I have taken a look at another post addressing a similar question (Does $\delta$ parameter in GARCH-M stay unchanged when the process is scaled?) but they assume a normal distribution.

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    $\begingroup$ The assumption of normality there is not essential. $\endgroup$
    – Richard Hardy
    Commented Apr 24, 2021 at 6:58

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The fitted standardized innovations from a GARCH model will have (approximately) zero mean and unit variance regardless of the scaling of the original data. This is by construction of the GARCH model which assumes that standardized innovations have (exactly) zero mean and unit variance.

In a simple GARCH(1,1) model, the constant $\omega$ in the conditional variance equation $$ \sigma_t^2=\omega+\alpha_1\varepsilon_{t-1}^2+\beta_1\sigma_{t-1}^2 $$ takes care of that. Scale your original data by $c$, and the (estimate of) $\omega$ will scale by $c^2$, ensuring that the (fitted) standardized innovations have (approximately) unit variance.

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  • $\begingroup$ Thank you, so this applies for non-normal distributions as well? Does it have any affect on distributions with parameters more than the mean and variance? $\endgroup$
    – CBBAM
    Commented Apr 23, 2021 at 15:58
  • $\begingroup$ @CBBAM, this applies to any and all distributions. They are always scaled to unit variance and zero mean, but otherwise retain the shape they would normally have. $\endgroup$
    – Richard Hardy
    Commented Apr 23, 2021 at 16:08
  • $\begingroup$ So in the case of say a innovations with an NIG distribution, the $\alpha$ and $\beta$ parameters remain unchanged if the data fed into the GARCH model is scaled? $\endgroup$
    – CBBAM
    Commented Apr 23, 2021 at 16:13
  • $\begingroup$ @CBBAM, I think so. You can always try it out with some data. $\endgroup$
    – Richard Hardy
    Commented Apr 23, 2021 at 19:16
  • $\begingroup$ From my tests it seems to not be true, but I am having trouble now seeing what the scaling should be. $\endgroup$
    – CBBAM
    Commented Apr 23, 2021 at 19:48

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