求解两个变量乘积的表达式 - 已知公式推导

已知：

\begin{aligned} & \pi_{1, T-1}=\frac{1}{1-\kappa_1 \kappa_2}\left[\frac{\mu_{1, T-1}-r_{T-1}}{\gamma_1 \sigma_{1, T-1}^2}+\frac{\kappa_1\left(\mu_{2, T-1}-r_{T-1}\right)}{\gamma_2 \sigma_{1, T-1} \sigma_{2, T-1}}\right] \ & \pi_{2, T-1}=\frac{1}{1-\kappa_1 \kappa_2}\left[\frac{\mu_{2, T-1}-r_{T-1}}{\gamma_2 \sigma_{2, T-1}^2}+\frac{\kappa_2\left(\mu_{1, T-1}-r_{T-1}\right)}{\gamma_1 \sigma_{1, T-1} \sigma_{2, T-1}}\right] \end{aligned} \begin{aligned} & q_{1, T-1}=\frac{1}{1-\kappa_1 \kappa_2}\left[\frac{\theta_{1, T-1} \alpha_{1, T-1}}{\gamma_1 \beta_{1, T-1}^2}+\frac{\kappa_1 \theta_{2, T-1} \alpha_{2, T-1}}{\gamma_2 \beta_{1, T-1} \beta_{2, T-1}}\right] \ & q_{2, T-1}=\frac{1}{1-\kappa_1 \kappa_2}\left[\frac{\theta_{2, T-1} \alpha_{2, T-1}}{\gamma_2 \beta_{2, T-1}^2}+\frac{\kappa_2 \theta_{1, T-1} \alpha_{1, T-1}}{\gamma_1 \beta_{1, T-1} \beta_{2, T-1}}\right] \end{aligned}

求\pi_{1, T-1} \pi_{2, T-1} \sigma_{1, T-1} \sigma_{2, T-1}

由于题目中没有给出$\mu_{1,T-1},\mu_{2,T-1},\alpha_{1,T-1},\alpha_{2,T-1},\beta_{1,T-1},\beta_{2,T-1},\theta_{1,T-1},\theta_{2,T-1}$的具体值，因此无法求出$\pi_{1,T-1},\pi_{2,T-1},\sigma_{1,T-1},\sigma_{2,T-1}$的具体数值。但可以给出它们的表达式：

\begin{aligned} & \pi_{1, T-1}=\frac{1}{1-\kappa_1 \kappa_2}\left[\frac{\mu_{1, T-1}-r_{T-1}}{\gamma_1 \sigma_{1, T-1}^2}+\frac{\kappa_1\left(\mu_{2, T-1}-r_{T-1}\right)}{\gamma_2 \sigma_{1, T-1} \sigma_{2, T-1}}\right] \ & \pi_{2, T-1}=\frac{1}{1-\kappa_1 \kappa_2}\left[\frac{\mu_{2, T-1}-r_{T-1}}{\gamma_2 \sigma_{2, T-1}^2}+\frac{\kappa_2\left(\mu_{1, T-1}-r_{T-1}\right)}{\gamma_1 \sigma_{1, T-1} \sigma_{2, T-1}}\right] \end{aligned}

\begin{aligned} & \sigma_{1,T}^2=\frac{1}{1-\rho_1^2}\left[\gamma_1^2\sigma_{1,T-1}^2+\beta_{1,T-1}^2+\theta_{1,T-1}^2\right] \ & \sigma_{2,T}^2=\frac{1}{1-\rho_2^2}\left[\gamma_2^2\sigma_{2,T-1}^2+\beta_{2,T-1}^2+\theta_{2,T-1}^2\right] \end{aligned}

\begin{aligned} & q_{1, T-1}=\frac{1}{1-\kappa_1 \kappa_2}\left[\frac{\theta_{1, T-1} \alpha_{1, T-1}}{\gamma_1 \beta_{1, T-1}^2}+\frac{\kappa_1 \theta_{2, T-1} \alpha_{2, T-1}}{\gamma_2 \beta_{1, T-1} \beta_{2, T-1}}\right] \ & q_{2, T-1}=\frac{1}{1-\kappa_1 \kappa_2}\left[\frac{\theta_{2, T-1} \alpha_{2, T-1}}{\gamma_2 \beta_{2, T-1}^2}+\frac{\kappa_2 \theta_{1, T-1} \alpha_{1, T-1}}{\gamma_1 \beta_{1, T-1} \beta_{2, T-1}}\right] \end{aligned}

其中，$\rho_1$和$\rho_2$为模型中两个误差项的相关系数。