I want to compute $Var[A+B+C]$ where $A$, $B$, and $C$ are not independent of each other. In particular, I don't know how to compute $Cov[A,B]$, $Cov[A,C]$, and $Cov[B,C]$. The model specifications follow.
Let:
- $A=X_1(-\lambda_1X_1-\lambda_1\delta_1+\nu_1-\overline{\nu}_1)$
- $B=X_2(-\lambda_2X_2-\lambda_2\delta_2+\nu_2-\overline{\nu}_2)$
- $C=X_3(-\lambda_3X_3-\lambda_3\delta_3+\nu_1+\nu_2-\overline{\nu}_1-\overline{\nu}_2)$
where $\lambda_1$, $\lambda_2$, $\lambda_3$ are constants. Moreover, $\overline{\nu}_1$ is the mean of $\nu_1\sim N(\overline{\nu}_1,\sigma_\nu^2)$ and $\overline{\nu}_2$ is the mean of $\nu_2\sim N(\overline{\nu}_2,\sigma_\nu^2)$. The variables $\nu_1$ and $\nu_2$ are independent of each other. In addition:
- $X_1=\beta_1(\nu_1+\nu_2-\overline{\nu}_1-\overline{\nu}_2)+\alpha_1\delta_1+\alpha_{2s}\delta_2$
- $X_2=\beta_2(\nu_1+\nu_2-\overline{\nu}_1-\overline{\nu}_2)+\alpha_2\delta_2+\alpha_{1s}\delta_1$
- $X_3=\beta_3(\nu_1+\nu_2-\overline{\nu}_1-\overline{\nu}_2)+\alpha_3\delta_3$
where $\beta_1$, $\beta_2$, $\beta_3$, $\alpha_1$, $\alpha_2$, $\alpha_3$, $\alpha_{1s}$, $\alpha_{2s}$ are constants. Moreover, $\delta_1\sim N(0,\sigma_d^2)$, $\delta_2\sim N(0,\sigma_d^2)$, and $\delta_3\sim N(0,\sigma_{d3}^2)$ (note the different variance for $\delta_3$). The variables $\delta_1$, $\delta_2$, and $\delta_3$ are independent of each other. Moreover, $\nu_1$ and $\nu_2$ are independent of $\delta_1$, $\delta_2$, and $\delta_3$.
Now, $A$, $B$, $C$ are all products of the form $WQ$. In general, $Var[WQ]$ can be found by applying a formula* which here reduces to $Var[WQ]=\sigma_W^2\sigma_Q^2+(Cov[W,Q])^2$, where $Cov[W,Q]$ is simply $E[WQ]$ since $E[W]=0$ and $E[Q]=0$ in my three cases. In short, it's relatively straightforward to find $Var[A]$, $Var[B]$, and $Var[C]$. However, I don't know about the covariance terms. How to tackle the covariance terms, i.e., $Cov[A,B]$, $Cov[A,C]$, $Cov[B,C]$?
*See (@whuber's comment in https://stats.stackexchange.com/questions/15978/variance-of-product-of-dependent-variables)