Equation for model with only level 2 covariate (studying ICC) I posted this question in response to an earlier question asked (Is ICC in random-intercept models restricted to the null-model?). I then realized that it is not recommend to write follow-up questions.
I am trying to write the equation of a model with only level 2 covariate. How would you write it? I am looking at the ICC in an unrestricted model and then want to see how the ICC decrease when I control for covariates that all members of a group share.
  = μ + a  + b
is my unrestricted model where  is is the outcome for group member  from group ,  is the population mean,  is a group-specific factor shared by all members from group , and  is an individual-specific factor unique to individual  from group .
If I want to write an equation that shows that I added a level 2 variable (X) in the model (which has the same value for all member of a group), what should it look like?
 A: I am not familiar with the notation you used (and I am not sure it is common or correct, besides usually subscript $i$ refers to individual and $j$ to the group). But I think it won't matter here. You said you already estimated unconstrained model. So we can write the system of equations for this model
$$\begin{align} \text{Level 1: } Y_{ij} & = \beta_{0j} + r_{ij} \\\
\text{Level 2: } \beta_{0j} & = \gamma_{00} + u_{0j} \end{align}$$
If we substitute level-2 equation into level-1:
$$Y_{ij} = \gamma_{00} + u_{0j} + r_{ij}$$
Moving from this unconstrained model to a model with only a level-2 predictor is straightforward:
$$\begin{align} \text{Level 1: } Y_{ij} & = \beta_{0j} + r_{ij} \\\
\text{Level 2: } \beta_{0j} & = \gamma_{00} + \gamma_{01}W_j + u_{0j} \end{align}$$
So, level-1 equation is same, and we add only $W_j$ as a level-2 predictor: 
$$Y_{ij} = \gamma_{00} + \gamma_{01}W_j + u_{0j} + r_{ij}$$
This is a variant of random-intercept model where the intercept ($\beta_{0j}$ in level-1 equation) is a function of level-2 predictor and variability. In your case, there is an additional level-1 predictor ($X_{ij}$). Assuming that its effect is constrained to be invariant across level-2 units:
$$\begin{align} \text{Level 1: } Y_{ij} & = \beta_{0j} + \beta_{1j}X_{ij} + r_{ij} \\\
\text{Level 2: } \beta_{0j} & = \gamma_{00} + \gamma_{01}W_j + u_{0j} 
\\\
\beta_{1j} & = \gamma_{10} \end{align}$$
The model is,
$$Y_{ij} = \gamma_{00} + \gamma_{01}W_j + \gamma_{10}X_{ij} + u_{0j} + r_{ij} $$

Luke, D. A. (2004). Multilevel Modeling. London: Sage Publications.
