# News impact curves for various GARCH models in the rugarch-package

I'm working with the really nice rugarch-package and currently have an issue with respect to the news impact curves (NIC).

In an attempt to plot several NIC into the same plot I realized that while the NIC for the sGARCH, the gjrGARCH and the eGARCH are given with respect to the $\epsilon_{t-1}$, the NICs for the submodels of the fGARCH model are given in terms of $z_{t-1}$, with $z_t = \frac{r_t-\mu_t}{\sigma_t}= \frac{\epsilon_t}{\sigma_t}$ being the standardized innovations. So they can't be plotted together in a single plot using the same scales.

I am a bit confused since just like the fGARCH model, the eGARCH model (as to the eGARCH-model-setup in the vignette) is also given in terms of the $z_{t-k} , k={1,...,q}$. But nevertheless the NIC of the eGARCH is given in terms of $\epsilon_{t-1}$ . At least this is what the "newsimpact(fit)"-output tells me. I think that this might be a typo and it should mean "$z_{t-1}"$. Can anyone please help?

The eGARCH vola equation is given by

$ln(\sigma_t^2)= \omega + \sum_{k=1}^p \Big( \alpha_k \ z_{t-k} + \gamma_k\ (|z_{t-k}| - E|z_{t-k}|) \Big) + \sum_{k=1}^q \beta_k \ ln(\sigma^2_{t-k})$

And we obviously have

$\alpha_k \ z_{t-k} + \gamma_k\ (|z_{t-k}| - E|z_{t-k}|) = \begin{cases} (\alpha_k + \gamma_k)z_{t-k} + C & z_{t-k} \ge 0 \\ (\alpha_k - \gamma_k)z_{t-k} + C & z_{t-k}<0 \end{cases}$

with the constant $C=-\gamma_k E[|z_{t-k}|]$. So the news impact depends on $\sigma_{t-1}$ which is nonconstant and therefore cannote be ignored or set to a fixed value for the NIC to make a general statement. Rather it has to be considered as being random along with the $\epsilon_{t-1}$ and so we have a NIC that depends on $z_t = \frac{\epsilon_t}{\sigma_t}$. This is why I think that there might be a typo in the rugarch output.

Is there an error in my thinking?

Many thanks and kind regards, Jo

There was a typo indeed, but in the fGARCH-output. The author of the package is already informed. It must be $\epsilon_{t-1}$ in all cases.
This is achieved by plugging the UNCONDITIONAL volatility into the formula as a proxy for the time varying conditional volatility $\sigma_{t-1}$ where needed - e.g. for the eGARCH and fGARCH models.