I'm reading the following paper to learn about multilevel modeling:
Multilevel Modeling for Binary Data
Guang Guo and Hongxin Zhao
Annual Review of Sociology
Vol. 26, (2000), pp. 441-462
They start by discussing the multilevel linear model. On Page 445, they present Model 1, which seems to be a random coefficients model with a random intercept and a fixed slope.
$Y_{ij} = \beta_0 + \beta_1 x_{ij} + u_j + e_{ij}$
$E(u_j) = E(e_{ij}) = 0$
$Var(u_j) = \sigma_u^2$
$Var(e_{ij}) = \sigma_e^2$
$Cov(u_j, e_{ij}) = 0$
$Cov(u_j, u_{h}) = 0 \ \ for \ \ j \neq h$
They then extend this 2-level model to a 3-level model, and yet it contains a random slope. Thus, the extension seems inconsistent. Here is Model #2.
$Y_{ijk} = \beta_0 + \beta_1 x_{ijk} + u_{1jk}x_{ijk} + v_{0k} + u_{0jk} + e_{0ijk}$
$v_{0k}$ and $u_{0jk}$ are random intercepts for level 3 and level 2, respectively.
$x_{ijk}$ is the observed predictor at level 1.
$u_{1jk}$ is $x_{ijk}$'s random effect at level 2.
There wasn't a random slope in the 2-level model, so why is there a need for $u_{1jk}x_{ijk}$ in the 3-level model?
