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Please how do i interpret the tgarch coefficient of 'RESID(-1)^2*(RESID(-1)<0)' that is negative but significant. All other coefficients in the equation are positive and significant. 'RESID(-1)^2' + 'RESID(-1)^2*(RESID(-1)<0)' will give a positive number.

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  • $\begingroup$ What do you think about my answer? If it is helpful and clear, you may accept it by clicking on the tick mark to the left. Otherwise, you may ask for further clarification. This is how Cross Validated works. $\endgroup$ – Richard Hardy 23 hours ago
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The conditional variance is equation is $$ \sigma_t^2=\omega+(\alpha_1+\gamma_1\mathbb{1}\{\varepsilon_{t-1}<0\})\varepsilon_{t-1}^2+\beta_1\sigma_{t-1}^2 $$ where $\mathbb{1}(\cdot)$ is the indicator function. (There may be higher-order lag terms which I will skip for simplicity.) If $\varepsilon_{t-1}<0$, then $\sigma_t^2=\omega+(\alpha_1+\gamma_1)\varepsilon_{t-1}^2+\beta_1\sigma_{t-1}^2$; otherwise $\sigma_t^2=\omega+\alpha_1\varepsilon_{t-1}^2+\beta_1\sigma_{t-1}^2$. The idea is that a negative error (shock, innovation) will have a different effect on the future conditional variance than a positive one will. According to the equation above, a negative $\gamma_1$ would mean a negative past error $\varepsilon_{t-1}<0$ will lead to smaller $\sigma_t^2$ than a positive past error $\varepsilon_{t-1}>0$ would, and the difference is $\gamma_1\varepsilon_{t-1}^2$. An empirical stylized fact in finance is that this should be reversed. So either your case is atypical or the model is formulated with $\mathbb{1}\{\varepsilon_{t-1}>0\}$ in place of $\mathbb{1}\{\varepsilon_{t-1}<0\}$.

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