I'm familiar with multilevel modelling. However, I'm wondering what would happen if you would include level 2 covariates at level 1. If you have, for example, students at level 1 (age, gender,...) and schools (living area, province,...) at level 2 and your dataset looks like this
studentID age gender schoolID living_area province Y
1 12 1 1 1 1 2.4
2 8 0 1 1 1 3.1
3 10 0 2 0 0 5.2
4 12 0 2 0 0 3.9
5 10 1 3 1 0 4.1
6 9 1 3 1 0 4.8
Why can't I write my model as follows:
$Y_{ij} = \alpha_j + \beta_1*age_{ij} + \beta_2*gender_{ij} + \beta_3*living\_area_{ij} + \beta_4*province_{ij} $ $\alpha_j \sim N(0, \sigma^2_b)$
And why is it better to write my model as follows:
$Y_{ij} = \beta_1*age_{ij} + \beta_2*gender_{ij} + \alpha_j$
$\alpha_j = \alpha_0 + \alpha_1*living\_area_{j} + \alpha_2*province_{j} $
What are the advantages of modelling it that way?