Suppose you have 8 observations ($i=1,...,8$) from three different states (A, B, C) and you also know that observations for $i=1,2$ are from state A, for $i=3,4,5$ are from state B and for $i=6,7,8$ are from state C. You are trying to estimate parameters with a linear regression model where $\varepsilon_i$ is the error term. The assumptions on this error term are that: $E[\varepsilon_i]=0$, $V[\varepsilon]=\sigma^2$ and:
$$Cov[\varepsilon_i, \varepsilon_j]=\begin{cases} \sigma^2 \rho & \text{ if observation i comes from the same state of observation j} \\ 0 & \text{otherwise} \end{cases}$$
Now you have that:
$$\overline{\varepsilon_h}=\frac{1}{n_h} \sum_{i \in h} \varepsilon_i$$
where $h=A,B,C$
I'm asked to compute the variance-covariance matrix of $\overline{\varepsilon}$ (notated $V[\overline{\varepsilon}]$) so I've started to compute variances and covariances of $\overline{\varepsilon}$ for $h=A, \ B, \ C$.
- $V[\overline{\varepsilon}_A]=...=\frac{\sigma^2}{n_A}$
- $V[\overline{\varepsilon}_B]=...=\frac{\sigma^2}{n_B}$
- $V[\overline{\varepsilon}_C]=...=\frac{\sigma^2}{n_C}$
- $Cov[\overline{\varepsilon}_A, \overline{\varepsilon}_B]= Cov\left [ \frac{1}{n_A} \sum_{i \in A} {\varepsilon}_i, \frac{1}{n_B} \sum_{j \in B} {\varepsilon}_j \right ]=\frac{1}{n_A} \frac{1}{n_B} Cov\left [ \sum_{i = 1}^{2} \varepsilon_i, \sum_{j = 3}^{5} \varepsilon_j \right ] = \frac{1}{n_A} \frac{1}{n_B} \sum_{i = 1}^{2} \sum_{j = 3}^{5} Cov[\varepsilon_i, \varepsilon_j]\underset{i\neq j}=0$
- $Cov[\overline{\varepsilon}_A, \overline{\varepsilon}_C]=...= 0$
- $Cov[\overline{\varepsilon}_B, \overline{\varepsilon}_C]=...= 0$
This leads me to the following variance-covariance matrix:
$$V[\overline{\varepsilon}]=\sigma^2\begin{bmatrix} \frac{1}{n_A} &0 &0 \\ 0 & \frac{1}{n_B} &0 \\ 0 & 0 & \frac{1}{n_C} \end{bmatrix}=\sigma^2 \begin{bmatrix} \frac{1}{2} &0 &0 \\ 0 & \frac{1}{3} &0 \\ 0 & 0 & \frac{1}{3} \end{bmatrix}$$
which apparently is not the right one. I think there is something wrong with the covariances computed above, but I can't see what. Can you please help me?