i have a question and i'm confused with the concepts. Pls aid
Consider the estimation of the population mean in the model: $Y_i = \beta + U_i$ for $i = 1,2,3$. Assume that $E(U_1) = E(U_2) = E(U_3) = 0$ and $Var (U_1) = 1, Var (U_2) = 2$ and $Var (U_3) = 4$. What is the best linear unbiased estimate of variance of $\beta$?
Thanks in advance
I'll write everything in vector notation, i.e. you have $Y = (Y_{1}, Y_{2}, Y_{3})$, $X = (1, 1, 1)$, $U = (U_{1}, U_{2}, U_{3})$ with model being $$ Y = X\beta + U$$
Assuming your population sample is i.i.d. (and since you have no stochastic regressors),
$$ \operatorname{Var} Y = \operatorname{Var} U = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 4 \end{pmatrix}$$ and your model exhibits a case of heteroskedastic residuals.
Redefine your model by premultiplying all the variables by the inverse square root of the covariance matrix, so that $$ \tilde{Y} = \begin{pmatrix} 1 & 0 & 0 \\ 0 & \frac{1}{\sqrt{2}} & 0 \\ 0 & 0 & \frac{1}{2} \end{pmatrix} Y, \quad \tilde{X} = \begin{pmatrix} 1 & 0 & 0 \\ 0 & \frac{1}{\sqrt{2}} & 0 \\ 0 & 0 & \frac{1}{2} \end{pmatrix} X, \quad \tilde{U} = \begin{pmatrix} 1 & 0 & 0 \\ 0 & \frac{1}{\sqrt{2}} & 0 \\ 0 & 0 & \frac{1}{2} \end{pmatrix} U$$
Now you can easily check that the new model, $ \tilde{Y} = \tilde{X}\beta + \tilde{U}$, satisfies all Gauss-Markov assumptions, including homoskedasticity, and all you need to do is to compute the OLS estimates for the new model.