I know that by estimating an GARCH model, given by: $$\sigma_t^2 = \omega + \alpha \epsilon_{t-1}^2 + \beta \sigma_{t-1}^2,$$

$\omega, \alpha, \beta >0$ and $\alpha + \beta <1$. But what are the constraints and bounds for a EGARCH and GJR-GARCH model, given by:

$$ln(\sigma_t^2) = \omega + \alpha \Big(\frac{|\epsilon|}{\sigma_t} - \sqrt{\frac{2}{\pi}}\Big) +\gamma \frac{\epsilon_{t-1}}{\sigma_{t-1}} + \beta ln(\sigma_{t-1}^2)$$
$$\sigma_t^2 = \omega + \alpha \epsilon_{t-1}^2 + \gamma \epsilon_{t-1}^2 \cdot 1_{\epsilon_{t-1}<0} + \beta \sigma_{t-1}^2$$

Thanks in advance!


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