# Displaying three-level multilevel model in vector notation

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:

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

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!