**Appendix**

• The probability density function of normal distribution is represented as:

$$f(\mathbf{x}|\mu, \sigma) = \frac{1}{\sigma\sqrt{2\pi}}e^{\frac{-(x-\mu)^2}{2\sigma^2}}\tag{10}$$

• Its log likelihood function in GARCH term is:

$$-\frac{n}{2}\ln\left(2\pi\right) - \frac{n}{2}\ln\left(h\right) - \frac{1}{2h}\sum\_{j=1}^{n}\left(\boldsymbol{x}\_{j} - \boldsymbol{\mu}\right)^{2}\tag{11}$$

• The probability density function for Student-t distribution is illustrated as:

$$f(\boldsymbol{y}, \boldsymbol{\nu}) = \frac{\Gamma^{\frac{\boldsymbol{\nu} + 1}{2}}}{\sqrt{\pi (\boldsymbol{\nu} - 2) \Gamma^{\frac{\boldsymbol{\nu}}{2}} \left(1 + \frac{\boldsymbol{\nu}^{2}}{\boldsymbol{\nu} - 2}\right)^{\frac{\boldsymbol{\nu} + 1}{2}}}} \tag{12}$$

• The log likelihood function in GARCH term is depicted as:

$$\begin{aligned} \log\left[\Gamma\left(\frac{v+1}{2}\right)\right] &- \log\left[\Gamma\left(\frac{v}{2}\right)\right] - \frac{1}{2}\log\left(\pi(v-2)\right) \\ - \frac{1}{2}\sum\_{j=1}^{n} \left[\log\left(h\_t\right) + (v+1)\log\left(1 + \frac{\varepsilon\_t^2}{h\_t(v-2)}\right)\right] &\end{aligned} \tag{13}$$

• The probability density function for generalised error distribution is mathematically represented as:

$$f(\mathbf{x}|\mu, \sigma, k) = \frac{e^{-\frac{1}{2}|\mathbf{x}-\mu|\_{\mathbf{z}}^{\mathbf{1}}}}{2^{k+1}\sigma\Gamma(k+1)}\tag{14}$$

• Its log likelihood function in GARCH term is:

$$-\frac{1}{2}|\boldsymbol{\pi}-\boldsymbol{\mu}|^{\frac{1}{\pi}}-(k+1)\log\left(2\right)-\log\left(h\right)-\log\left(\Gamma\right)-\log\left(k+1\right)\tag{15}$$
