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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.

Richard Hardy
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CBBAM
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1 Answers1

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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.

Richard Hardy
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  • 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? – CBBAM Apr 23 '21 at 15:58
  • @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. – Richard Hardy Apr 23 '21 at 16:08
  • 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? – CBBAM Apr 23 '21 at 16:13
  • @CBBAM, I think so. You can always try it out with some data. – Richard Hardy Apr 23 '21 at 19:16
  • From my tests it seems to not be true, but I am having trouble now seeing what the scaling should be. – CBBAM Apr 23 '21 at 19:48
  • @CBBAM, I have experimented with `rugarch` and randomly generated data and think there might be some problems with numerical stability, causing the estimates to differ when they in fact should be the same. – Richard Hardy Apr 24 '21 at 06:38
  • So to confirm, if I have a set of data $Y$ and $cY$ for some scalar $c$, the resulting innovations after applying ARMA GARCH to both sets of data should both have the same parameters? – CBBAM Apr 24 '21 at 07:07
  • Let us [continue this discussion in chat](https://chat.stackexchange.com/rooms/123375/discussion-between-cbbam-and-richard-hardy). – CBBAM Apr 24 '21 at 07:10
  • @CBBAM, let me know if you have any further questions. – Richard Hardy Apr 25 '21 at 19:11