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$ \beta _d$ (Spatially stationary spatial model)

$\displaystyle \beta_d\vert.\sim N(B_{\beta_d}/A_{\beta_d},1/A_{\beta_d})$ (31)

where
$\displaystyle A_{\beta_d}$ $\displaystyle =$ $\displaystyle \sum_{it} \phi_{\epsilon_i} \left( \sum_{j\in{\cal N}_{id}} q_{j(t-1)} \right)^2 +\frac{1}{\sigma_\beta^2}$ (32)
$\displaystyle B_{\beta_d}$ $\displaystyle =$ $\displaystyle \sum_{it} \phi_{\epsilon_i} \left[ \left( q_{it} -\sum_{p=1}^P \alpha_{ip} q_{i(t-p)} \right. \right.$  
    $\displaystyle \left. \left. -\sum_{d' \neq d}\beta_{d'}\sum_{j\in{\cal N}_{id'}} q_{j(t-1)} \right) \sum_{j\in{\cal N}_{id}} q_{j(t-1)} \right]$ (33)