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I'm reading about the Nonlinear Asymmetric GARCH (NAGARCH) model. If NAGARCH(1, 1) is given by:

$${\displaystyle ~\sigma _{t}^{2}=~\omega +~\alpha (~\epsilon _{t-1}-~\theta ~\sigma _{t-1})^{2}+~\beta ~\sigma _{t-1}^{2}},$$

where ${\displaystyle ~\alpha \geq 0,~\beta \geq 0,~\omega >0}$ and ${\displaystyle ~\alpha (1+~\theta ^{2})+~\beta <1}$. Then what would be its general form — i.e., NAGARCH(p, q)?

Would the following general form of the NAGARCH model be appropriate:

$${\displaystyle ~\sigma _{t}^{2}=~\omega +\sum_{i = 1}^{p}\sum_{j = 1}^{q}~\alpha_i (~\epsilon _{t-i}-~\theta ~\sigma _{t-j})^{2}+\sum_{j = 1}^{q}~\beta_j ~\sigma _{t-j}^{2}}.$$

When searching, I failed to find any paper or article giving the general form of the model.

Thanks in advance.

Blg Khalil
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    I do not think the double sum makes much sense. A single sum should do just fine. – Richard Hardy May 24 '21 at 16:23
  • Thank you @Richard Hardy. This should be now the general form: $${\displaystyle ~\sigma _{t}^{2}=~\omega +\sum_{i = 1}^{p}~\alpha_i (~\epsilon _{t-i}-~\theta ~\sigma _{t-i})^{2}+\sum_{j = 1}^{q}~\beta_j ~\sigma _{t-j}^{2}}$$. And $\theta$ is independent of the number of lags, I.e., there is only one parameter $\theta$ to be estimated rather than 2 for example if $p = 2$. – Blg Khalil May 25 '21 at 20:01
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    I am not entirely sure, but the latter comment generally makes sense. – Richard Hardy May 25 '21 at 20:24

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