# 弹性回归 Flexible Regression STATS5052_1/STATS4040_1

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where $G_{0}$ is a standard exponential distribution. The distribution of $\epsilon_{t}$ is therefore
$$f_{v, \xi}\left(\epsilon_{t}\right)=\sum_{j=1}^{\infty} w_{j} v\left(-\xi_{j} e^{-\lambda}, \xi_{j} e^{\lambda}\right)$$
and the conditional return distribution is
$$f_{G, \lambda}\left(\gamma_{t} \mid \sigma_{t}\right)=\sum_{j=1}^{\infty} w_{j} v\left(\gamma_{t} \mid-\xi_{j} \sigma_{t} e^{-\lambda}, \xi_{j} \sigma_{t} e^{\lambda}\right)$$

## STATS5052_1/STATS4040_1 COURSE NOTES ：

Utilising the latent variable, the copula yields
$$P\left(Y_{1}=0, Y_{2} \leq Y_{2}\right)=P\left(Y_{1}^{} \leq 0, Y_{2} \leq Y_{2}\right)=C\left(F_{1}^{}(0), F_{2}\left(\gamma_{2}\right)\right)$$
and
$$P\left(Y_{1}=1, Y_{2} \leq \gamma_{2}\right)=P\left(Y_{1}^{}>0, Y_{2} \leq \gamma_{2}\right)=F_{2}\left(\gamma_{2}\right)-C\left(F_{1}^{}(0), F_{2}\left(\gamma_{2}\right)\right) \text {, }$$
leading to the mixed binary-continuous density
$$p\left(\gamma_{1}, \gamma_{2}\right)=\left(\frac{\partial C\left(F_{1}^{}(0), F_{2}\left(\gamma_{2}\right)\right)}{\partial F_{2}\left(\gamma_{2}\right)}\right)^{1-\gamma_{1}} \cdot\left(1-\frac{\partial C\left(F_{1}^{}(0), F_{2}\left(\gamma_{2}\right)\right)}{\partial F_{2}\left(\gamma_{2}\right)}\right)^{\gamma_{1}} \cdot p_{2}\left(\gamma_{2}\right),$$