# 金融计量经济学 Financial Econometrics ECON60332T

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The distribution of $z_{t}$ is the mixture, with equal weights, of the distributions of $\mathrm{w}{t}$ and $\mathrm{u}{t}$. Ordinary properties of mixtures lead to
\begin{aligned} E\left(\mathrm{z}{l} \otimes \mathrm{z}{l} \otimes \mathrm{z}{l}\right)=& \frac{1}{2} E\left(\mathrm{w}{t} \otimes \mathrm{w}{l} \otimes \mathrm{w}{l}\right)+\frac{1}{2} E\left(\mathrm{u}{l} \otimes \mathrm{u}{t} \otimes \mathrm{u}{t}\right) \ &=\frac{1}{2}\left(\mathrm{~g}{1}+\mathrm{g}{2}\right) \end{aligned} Hence the multivariate skewness of the vector $z{t}$ can be represented as follows:
$$S\left(z_{t}\right)=\frac{1}{4}\left(g_{1}+g_{2}\right)^{T} \Gamma\left(g_{1}+g_{2}\right)$$
The distribution of $w_{t}$ equals that of $-u_{t}$ only when $\gamma=0$. By assumption $\gamma \neq 0$, so that
$$\mathrm{g}{1}+\mathrm{g}{2} \neq 0 \Rightarrow S\left(z_{l}\right)=\frac{1}{4}\left(\mathrm{~g}{1}+\mathrm{g}{2}\right)^{\mathrm{T}} \Gamma\left(\mathrm{g}{1}+\mathrm{g}{2}\right)>0$$

## ECON60332TCOURSE NOTES ：

$$X_{l}=W_{t} \sqrt{a+a_{0} X_{l}^{2}+\sum_{i=1}^{p} a_{i} X_{l-i}^{2}}$$
When the implicit ARCH is fitted to the data, the following residuals ensue:
$$\hat{W}{t}=\frac{X{t}}{\sqrt{\hat{a}+\hat{a}{0} X{l}^{2}+\sum_{i=1}^{p} \hat{a}{i} X{l-i}^{2}}}$$