I estimated a three-level multilevel model in stan but I have some trouble writing it correctly in terms of vectors and matrices. Now, I have the following:

\begin{equation}
\begin{gathered}
y_{ijt} = \boldsymbol{z}_{ijt}'\boldsymbol{\pi}_{ij} + \boldsymbol{a}'\boldsymbol{\theta} + \varepsilon_{ijt} , \\
\boldsymbol{\pi}_{ij} = \boldsymbol{\beta}_{j}'\boldsymbol{x}_{ij} + \boldsymbol{\eta}_{ij}, \\
\boldsymbol{\beta}_{j} = \boldsymbol{\Gamma}'\boldsymbol{w}_j + \boldsymbol{u}_{j}.
\end{gathered}
\end{equation}

With dimensions:

\boldsymbol{z}_{ijt}: p \times 1

\boldsymbol{\pi}_{ij}: p \times 1

\boldsymbol{a}: g \times 1

\boldsymbol{\theta}: g \times 1

\boldsymbol{\beta}_{j}: q \times p

\boldsymbol{x}_{ij}: q \times 1

\boldsymbol{\Gamma}: s\times q

\boldsymbol{w}_j: s \times p

So, there are p variables in the first level, q in the second and s in the third. However, to get the dimensions right my \boldsymbol{w}_j has to have p columns, but I do not know how to incorporate this in stan, since I would think that I only have a \boldsymbol{w}_j with size s \times 1 in the third level (so s variables with only one value each). Now, I could use p equal columns for \boldsymbol{w}_j, but I do not know whether this is right. Can someone explain this to me?

Next to that, I can define the joint posterior to be:

p(\boldsymbol{\pi}_{ij}, \boldsymbol{\beta}_{j}, \boldsymbol{\Gamma}| \boldsymbol{y}) \propto p(\boldsymbol{y}| \boldsymbol{\pi_{ij}})p(\boldsymbol{\pi_{ij}}|\boldsymbol{\beta}_j)p(\boldsymbol{\beta_{j}}|\boldsymbol{\Gamma})p(\boldsymbol{\Gamma}).

But can someone explain how to incorporate the fixed effect part \boldsymbol{a}'\boldsymbol{\theta} in this equation?

Thanks in advance for the help!