**A.1 The probability density function of normal distribution is written as:**

$$f(\mathfrak{x}|\mu, \sigma) = \frac{1}{\sigma\sqrt{2\pi}}e^{\frac{-(\mathfrak{x}-\mu)^2}{2\sigma^2}}\tag{8}$$

and its log likelihood function in GARCH term is:

$$-\frac{n}{2}\ln\left(2\pi\right) - \frac{n}{2}\ln\left(h\right) - \frac{1}{2h}\sum\_{j=1}^{n}\left(\pi\_j - \mu\right)^2. \tag{9}$$
