is equivalent to $\Omega$ being a diagonal matrix. Recall that $\eta_i | X_i \sim N(A X,  V)$, where $A$ and $V$ are defined in *Proposition 1*. Since we do not include interactions among the factors, let us just focus on the marginal distribution of the $j^{th}$ factor, i.e $\eta_{i h} | X_i \sim N(\mu_h, \sigma_h^2)$ where $\mu_h = \sum_{j = 1} ^ p a_{hj}X_{i j}$ and $\sigma_h^2 = V_{hh}$. Below we provide an expression for $E( \sum_{q = 1}^Q \omega_h^{(q)} \eta^q_{ih} | X )$, which can be calculated using non-central moments of a Normal distribution, see [@winkelbauer2012] for a reference. $$\begin{aligned}
{\mathbb{E}}( \sum_{q = 1}^Q \omega_j^{(q)} \eta^q_j | X ) = &
\sum_{f = 1}^{\lfloor \frac{Q+1}{2} \rfloor }   \sum_{q = f}^{\lfloor \frac{Q+1}{2} \rfloor}  \omega_h^{(2q-1)} \sigma_h^{2 q -2f}   b_{qf}^{o}  \sum_{k_+= 2 f -1} \binom{2f -1}{k_1 \cdots k_p} \prod_{j = 1}^p (a_{h j} X_j)^{k_j} + \\  & \sum_{f = 0}^{\lceil \frac{Q+1}{2} \rceil }   \sum_{q = f \vee 1}^{\lceil\frac{Q+1}{2} \rceil}  \omega_h^{(2q)} \sigma_h^{2 q  -2f }     b_{qf}^{e}    \sum_{k_+ = 2f} \binom{2f}{k_1 \cdots k_p} \prod_{j = 1}^p (a_{hj} X_j)^{k_j},\end{aligned}$$ where $b_{qf}^{o} = \frac{(2 q -1 )!}{(2 f -1)! (q-f)! 2^{q-f}}   $, $b_{qf}^{e} =  \frac{(2 q  )!}{(2 f)! (q-f)! 2^{q-f}} $ and $k_+ = \sum_{j = 