Let me start with simpler models (detailed answers below), and let me use a slightly different notation (NB: random effecs at different levels assumed to be uncorrelated).
Model 1 (Two levels, random intercept, fixed slope). There are $N$
observations of $J$ students:
$$\begin{align}y_{ij}&=\pi_{0j}+\beta_1
X_{ij}+\varepsilon_{ij},\quad&\varepsilon_{ij}\sim\mathcal{N}(0,\sigma^2_\varepsilon)\\
\pi_{0j}&=\beta_0+b_{0j},\quad& b_{0j}\sim\mathcal{N}(0,\sigma^2_0)\end{align}$$.
If we substitute for $\pi_0$ and $\pi_1$ we get:
$$y_{ij}=\beta_0+\beta_1X_{ij}+(b_{0j}+\varepsilon_{ij})$$ so:
$$\begin{align}\text{Var}(y_{ij})&=\text{Var}(b_{0j}+\varepsilon_i)=\sigma^2_0+\sigma^2_\varepsilon
\\ \text{Cov}(y_{ij},y_{i'j})&=\sigma^2_0 \quad& i\ne i' \\
\text{Cov}(y_{ij},y_{i'j'})&=0 \quad& i\ne i',\;j\ne j' \end{align}$$ The
intraclass correlation coefficient:
$$\rho=\frac{\sigma^2_0}{\sigma^2_0+\sigma^2_\varepsilon}$$ measures the
proportion of the variance in the outcome that is between students. As to the
overall covariance matrix, it is (Goldstein,
Multilevel Statistical Models,
pg. 20): $$ \begin{bmatrix}
\sigma^2_0\mathbf{J}_{N_1}+\sigma^2_\varepsilon\mathbf{I}_{N_1} & 0 & \dots &
0 \\ \vdots & \vdots & \vdots & \vdots \\ \dots & \dots &
\sigma^2_0\mathbf{J}_{N_j}+\sigma^2_\varepsilon\mathbf{I}_{N_j} & 0 \\ \vdots
& \vdots & \vdots & \vdots \\ 0 & 0 & 0 &
\sigma^2_0\mathbf{J}_{N_J}+\sigma^2_\varepsilon\mathbf{I}_{N_J}
\end{bmatrix}$$ where $N_j$ is the number of observations for student $j$,
$\mathbf{J}_{N_j}$ is the $(N_j\times N_j)$ matrix of ones and
$\mathbf{I}_{N_j}$ is the $(N_j\times N_j)$ identity matrix. For example, if
there are $J=2$ students and $N_1=3,N_1=2$ observations:
$$\begin{bmatrix}
\sigma^2_0+\sigma^2_\varepsilon & \sigma^2_0 & \sigma^2_0 & 0 & 0 \\
\sigma^2_0 & \sigma^2_0+\sigma^2_\varepsilon & \sigma^2_0 & 0 & 0 \\
\sigma^2_0 & \sigma^2_0 & \sigma^2_0+\sigma^2_\varepsilon & 0 & 0 \\ 0 & 0 & 0
& \sigma^2_0+\sigma^2_\varepsilon & \sigma^2_0 \\ 0 & 0 & 0 & \sigma^2_0 &
\sigma^2_0+\sigma^2_\varepsilon
\end{bmatrix}=(\sigma^2_0+\sigma^2_\varepsilon) \begin{bmatrix} 1 & \rho &
\rho & 0 & 0 \\ \rho & 1 & \rho & 0 & 0 \\ \rho & \rho & 1 & 0 & 0 \\ 0 & 0 &
0 & 1 & \rho \\ 0 & 0 & 0 & \rho & 1 \end{bmatrix}$$
Model 2 (Three levels, random intercept, fixed slope). There are $N$ observations, $J$ students, and $K$ schools:
$$\begin{align}
y_{ijk}&=\pi_{0jk}+\beta_1 X_{ijk}+\varepsilon_{ijk},\quad&\varepsilon_{iij}\sim\mathcal{N}(0,\sigma^2_\varepsilon) \\
\pi_{0jk} &= \gamma_{0k}+b_{0jk},\quad& b_{0jk}\sim\mathcal{N}(0,\sigma^2_0) \\
\gamma_{0k} &= \beta_0 + b_{0k},\quad & b_{0k}\sim\mathcal{N}(0,\tau^2_0)
\end{align}$$
The composite model is:
$$y_{ijk}=\beta_0+\beta_1 X_{ijk}+ (b_{0k}+b_{0jk}+\varepsilon_{ijk})$$
and:
$$\text{Var}(y_{ijk})=\tau^2_0+\sigma^2_0+\sigma^2_\varepsilon$$
Instead of one intraclass correlation coefficient, several correlation coefficients may be defined:
- $(\tau^2_0+\sigma^2_0)/(\tau^2_0+\sigma^2_0+\sigma^2_\varepsilon)$: the proportion of variance among students;
- $\tau^2_0/(\tau^2_0+\sigma^2_0+\sigma^2_\varepsilon)$: the proportion of variance among schools;
- $\sigma^2_0/(\tau^2_0+\sigma^2_0+\sigma^2_\varepsilon)$: the proportion of variance among students within schools;
- $\sigma^2_\varepsilon/(\tau^2_0+\sigma^2_0+\sigma^2_\varepsilon)$: the proportion of variance within students.
The covariance matrix is complex. If there are $K=3$ schools, a covariance matrix looks like (Rodriguez):
$$(\tau^2_0+\sigma^2_0+\sigma^2_\varepsilon)\begin{bmatrix}
1 & \rho_1 & \rho_2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
\rho_1 & 1 & \rho_2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
\rho_2 & \rho_2 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & \rho_1 & \rho_2 & \rho_2 & 0 & 0 & 0\\
0 & 0 & 0 & \rho_1 & 1 & \rho_2 & \rho_2 & 0 & 0 & 0 \\
0 & 0 & 0 & \rho_2 & \rho_2 & 1 & \rho_1 & 0 & 0 & 0 \\
0 & 0 & 0 & \rho_2 & \rho_2 & \rho_1 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & \rho_1 & \rho_2 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & \rho_1 & 1 & \rho_2 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & \rho_2 & \rho_2 & 1
\end{bmatrix}$$
where $\rho_1=(\tau^2_0+\sigma^2_0)/(\tau^2_0+\sigma^2_0+\sigma^2_\varepsilon)$ and $\rho_2=\tau^2_0/(\tau^2_0+\sigma^2_0+\sigma^2_\varepsilon)$.
Model 3 (Two levels, random intercept and slope). We let $\pi_1$ vary across the students in Model 1:
$$\begin{align}y_{ij}&=\pi_{0j}+\pi_1 X_{ij}+\varepsilon_{ij},\quad&\varepsilon_i\sim\mathcal{N}(0,\sigma^2_\varepsilon)\\
\pi_{0j}&=\beta_0+b_{0j},\quad& b_{0j}\sim\mathcal{N}(0,\sigma^2_0)\\
\pi_1 &= \beta_1+b_{1j},\quad& b_{1j}\sim\mathcal{N}(0,\sigma^2_1)
\end{align}$$
but now we must represent the dispersion of the level-2 random effects as a covariance matrix:
$$\text{Var}(\mathbf{b}_j)=\text{Var}\begin{bmatrix} b_{0j} \\ b_{1j} \end{bmatrix}
=\begin{bmatrix} \sigma^2_0 & \sigma_{01} \\ \sigma_{01} & \sigma^2_1 \end{bmatrix}$$
The combined model is:
$$y_{ij}=\beta_0+\beta_1X_{ij}+(b_{0j}+b_{1j}X_{ij}+\varepsilon_{ij})$$
so (Goldstein, Multilevel Statistical Models, pg. 22):
$$\begin{align}\text{Var}(y_{ij})&=\begin{bmatrix} \mathbf{1} & \mathbf{X} \end{bmatrix}\begin{bmatrix} \sigma^2_0 & \sigma_{01} \\ \sigma_{01} & \sigma^2_1 \end{bmatrix}\begin{bmatrix} \mathbf{1} & \mathbf{X} \end{bmatrix}'+\sigma^2_\varepsilon \\
\text{Cov}(y_{ij},y_{i'j})&=\sigma^2_0+2\sigma_{01}(X_{ij}+X_{i'j})+\sigma^2_1 X_{ij} X_{i'j} \\
\text{Cov}(y_{ij},y_{i'j'})&=0
\end{align}$$
The overall covariance matrix is still block diagonal but the blocks are: $\left(\begin{bmatrix} \mathbf{1} & \mathbf{X} \end{bmatrix}\text{Var}(\mathbf{b}_j)\begin{bmatrix} \mathbf{1} & \mathbf{X} \end{bmatrix}'\right)\mathbf{J}_{N_j}+\sigma^2_\varepsilon\mathbf{I}_{N_j}$.
We could define:
$$\rho=\frac{\sigma^2_0+2\sigma_{01}X_{ij}+\sigma^2_1 X_{ij}^2}{\sigma^2_0+2\sigma_{01}X_{ij}+\sigma^2_1 X_{ij}^2+\sigma^2_\varepsilon}$$
but it is not a correlation coefficient like the previous ones. The correlation between two observations is given by:
$$\frac{\sigma^2_0+2\sigma_{01}(X_{ij}+X_{i'j})+\sigma^2_1 X_{ij} X_{i'j}}
{\sqrt{(\sigma^2_0+2\sigma_{01}X_{ij}+\sigma^2_1 X_{ij}^2+\sigma^2_\varepsilon)(\sigma^2_0+2\sigma_{01}X_{i'j}+\sigma^2_1 X_{i'j}^2+\sigma^2_\varepsilon)}}$$
Its value is a function of $X$, so there is not a unique correlation coefficient, but several coefficients that can be calculated for any value of the predictor variable(s) (Goldstein et al., Partitioning variation in multilevel models).
Model 4 (Three levels, random intercept and slope). We let $\pi_1$ vary across the students in Model 2:
$$\begin{align}
y_{ijk}&=\pi_{0jk}+\pi_{1jk} X_{ijk}+\varepsilon_{ijk},\quad&\varepsilon_{iij}\sim\mathcal{N}(0,\sigma^2_\varepsilon) \\
\pi_{0jk} &= \gamma_{0k}+b_{0jk},\quad& b_{0jk}\sim\mathcal{N}(0,\sigma^2_0) \\
\pi_{1jk} &= \gamma_{1k}+b_{1jk},\quad& b_{1jk}\sim\mathcal{N}(0,\sigma^2_1) \\
\gamma_{0k} &= \beta_0 + b_{0k},\quad & b_{0k}\sim\mathcal{N}(0,\tau^2_0) \\
\gamma_{1k} &= \beta_1 + b_{1k},\quad & b_{1k}\sim\mathcal{N}(0,\tau^2_1)
\end{align}$$
The covariance matrices for level-2 and level-3 are:
$$\text{Var}(\mathbf{b}_j)=\begin{bmatrix} \sigma^2_0 & \sigma_{01} \\ \sigma_{01} & \sigma^2_1 \end{bmatrix}\qquad\qquad
\text{Var}(\mathbf{b}_k)=\begin{bmatrix} \tau^2_0 & \tau_{01} \\ \tau_{01} & \tau^2_1 \end{bmatrix}$$
The composite model is:
$$y_{ijk}=\beta_0 + \beta_1 X_{ijk} + (b_{0jk}+b_{1jk}
X_{ijk})+(b_{0k}+b_{1k} X_{ijk}) + \varepsilon_{ijk}$$
and the variance at level-1 is:
$$\text{Var}(y_{ijk})=\text{Var}(b_{0jk}+b_{1jk}
X_{ijk})+\text{Var}(b_{0k}+b_{1k} X_{ijk})+\text{Var}(\varepsilon_{ijk})$$
where $\text{Var}(b_{0jk}+b_{1jk} X_{ijk})$ is the variance at level-2 and
$\text{Var}(b_{0k}+b_{1k} X_{ijk})$ is the variance at level-3. They
depend on the value of $X$ as in Model 3.
Now:
Multilevel models including random slopes: how to calculate variance
I hope it is somewhat clear now.
$Y_{ijk}=\beta_0+\beta_1*Time+b_k+b_{jk}+\epsilon_{ijk}$
I'write: $Y_{ijk}=\beta_0+\beta_1\text{Time}+b_{0k}+b_{0jk}$ (Model 2) to
highlight that $b_{0k}$ and $b_{0jk}$ randomize $\beta_0$.
I add a random slope per school:
$Y_{ijk}=\beta_0+\beta_1* Time+b_{0k}+b_{1k}*Time+b_{jk}+\epsilon_{ijk}$
Is in this model the variation of $b_{0k}$ still meaningful?
I'd write: $Y_{ijk}=\beta_0+\beta_1\text{Time}+b_{0k}+b_{1k}\text{Time}+b_{0jk}+\epsilon_{ijk}$, i.e.:
$$Y_{ijk}=(\beta_0+b_{0jk}+b_{0k})+(\beta_1+b_{1k})\text{Time}+\epsilon_{ijk}$$
where $(\beta_0+b_{0jk}+b_{0k})$ is the random intercept per student within a
school and per school, and $(\beta_1+b_{1k})$ is the random slope
per school (Model 4 with $b_{1jk}=E(b_{1jk})=0$, $\sigma^2_1=0$.)
As to $b_{0k}$, the random intercept per school, it is even more meaningful
than $b_{1k}$! "Given that among multilevel modelers random slopes tend to be
more popular than heteroscedasticity, unrecognized heteroscedasticity may show
up in the form of a fitted model with a random slope [...] which then may
disappear if the heteroscedasticity is modeled" (Snijders and Berkhof,
Diagnostic Checks for Multilevel Models, ยง3.2.1).
Is my intuition correct that the variation between different schools
increases over time, as the random slopes drive the schools further apart?
In all the above models I have assumed that all random effects are
uncorrelated and that their variances are constant. Is these assumptions hold,
then covariances increase quadratically with time (see conjectures'
answer). However, if the $\tau$ variances and covariances decrease with time you could
observe a somewhat different pattern.
BTW, your plot does show a different pattern and suggests a bit of
heteroscedasticity more than a random slope.
the covariance between students within the same school is given by:
$\text{Cov}_{school}=\frac{\sigma^2_3}{\sigma^2_1+\sigma^2_2+\sigma^2_3}$
What is this covariance if a random intercept is included?
That is not a covariance, it is the $\rho_2$ correlation coefficient in
Model 2. The covariance between two schools depends on Time.
Best Answer
Are you sure that you have encountered $\beta_{0j}=\beta_0+\beta_{0ij}x_{0ij}+u_{0j}$? Exactly? May be, but the $i$'s can't denote indviduals, because level-2 predictors are individual-invariant.
In general, in a random coefficients model you have: $$\begin{align} \text{Level-1:}&\quad &y_{ij}&=\beta_{0j}+\beta_{1j}x_{ij}+e_{ij}\\ \text{Level-2:}&\quad&\beta_{0j}&=\beta_0+\cdots+u_{0j}\\ &&\beta_{1j}&=\beta_1+\cdots+u_{1j}\end{align}$$ where "..." may be nothing or a set of level-2 predictors. If you add two level-2 predictors to both the intercept and the slope, you could write: $$\begin{align} \beta_{0j}&=\beta_0+\beta_{01}x_{01}+\beta_{02}x_{02}+u_{0j}\\ \beta_{1j}&=\beta_1+\beta_{11}x_{11}+\beta_{12}x_{12}+u_{1j}\end{align}$$ but that would be a mess, because $\beta_{ij}=\beta_{11}$ when $j=1$, etc. Thus you can:
The second notation is better, because you can see that:
In general, there are as many subscripts as levels, i.e. you use three subscripts in three-level models (say, students, classrooms, schools): $$\begin{align} \text{Level-1:}&\quad &y_{ijk}&=\pi_{0jk}+\pi_{1jk}x_{ijk}+e_{ijk}\\ \text{Level-2:}&\quad&\pi_{0jk}&=\beta_{00k}+\beta_{01k}z_{01k}+\beta_{02k}x_{02k}+r_{0jk}\\ &&\pi_{1jk}&=\beta_{10k}+\beta_{11k}z_{11k}+\beta_{12k}x_{12k}+r_{1jk}\\ \text{Level-3:}&\quad &\beta_{00k}&=\gamma_{000}+\gamma_{001}w_{001}+\gamma_{002}w_{002}+u_{00k}\\ &&\dots\\ &&\beta_{12k}&=\gamma_{120}+\gamma_{121}w_{121}+\gamma_{122}w_{122}+u_{12k} \end{align}$$
In three-level models, level-3 coefficients are fixed (population parameters), lower-level coefficients are random.