# Centring in three levels multilevel models with cross-level interaction

I am relatively new to multi-level models subject and find it challenging to connect theory to modelling. My interest is to examine the moderation influence of green space (at level 3) on the relationship between SES variables at the individual level 1 (education) and the SES variables at the household level (household wealth) with the probability of chronic illness. A hypothesis is: individuals with lower SES and individuals living in lower SES households who live in areas with better greenspace have better health compared to their counterparts who live in an area with lower greenspace.

My problem is that I have three possibilities to test this:

(1) Raw scores or (grand mean centring which have similar interpretation) $$y_{ijk}= γ_{000}+γ_{100} x_{ijk} +γ_{010} h_{j_k} +γ_{001} z_k+γ_{101} (x_{ijk}* z_k)+γ_{011} (h_{jk}*z_k)+V_{10k} x_{ijk}+V_{01k} h_{jk}+V_{00k}+ U _{0jk}+ε_{ijk}$$

The argument by Raudenbush, Enders 2007 and followed by a lot of others is that (1) does not lead to pure estimates because the estimates are composed of a within and a between components, whereby cluster mean centring rule out the between components and leads to pure within estimates which are preferable in cross-level interactions.

(2) Cluster mean centring (hierarchically) $$y_{ijk} = γ_{000} + γ_{100} (x_{ijk}-x ̅_{jk})+ γ_{010}(h_{jk}- h ̅_k)+γ_{001} (z_k-z ̅_.) + γ_{101} (x_{ijk}-x ̅_{jk})*(z_k-z ̅_.))+γ_{011} ((h_{jk}- h ̅_k)*(z_k-z ̅_.))+V_{10k} x_{ijk}+V_{01k} h_{jk}+V_{00k}+U _{0jk}+ε_{ijk}$$ But because my interest is in the moderation effect of green space at level 3 and that individuals and households usually share the same SES, there is another possibility for centring at level 3: (relative SES position in the area)

(3) Cluster mean centring (only at level 3) $$y_{ijk}= γ_{000} +γ_{100} (x_{ijk}-x ̅_k)+γ_{010} (h_{jk}- h ̅_k)+γ_{001} (z_k-z ̅_.)+ γ_{101} ((x_{ijk}-x ̅_k)* (z_k-z ̅_.))+γ_{011} ((h_{jk}- h ̅_k)*(z_k-z ̅_.))+V_{10k} x_{ijk}+V_{01k} h_{jk}+V_{00k}+U _{0jk}+ε_{ijk}$$

These three equations produce different models, my confusion is which one makes more sense and serves better answering my question/hypothesis.

By the way, Kelley et al,.2017 say that (2 and 3) is nonsense and just use 1. Which confused me even more.