**A.2 The probability density function of the student-t distribution is:**

$$f(\mathbf{y}, \boldsymbol{\nu}) = \frac{\Gamma^{\frac{\nu+1}{2}}}{\sqrt{\pi(\boldsymbol{\nu} - \mathbf{2})} \Gamma^{\frac{\nu}{2}} \left(\mathbf{1} + \frac{\boldsymbol{\nu}^2}{\boldsymbol{\nu} - \mathbf{2}}\right)^{\frac{\nu+1}{2}}} \tag{10}$$

Its log likelihood function in GARCH term is:

$$\log\left[\Gamma\left(\frac{\nu+1}{2}\right)\right] - \log\left[\Gamma\left(\frac{\nu}{2}\right)\right] - \frac{1}{2}\log\left(\pi(\nu-2)\right) - \frac{1}{2}\sum\_{j=1}^{n}\left[\log\left(h\_{l}\right) + (\nu+1)\log\left(1 + \frac{e\_{l}^{2}}{h\_{l}(\nu-2)}\right)\right] \tag{11}$$
