I have attatched an excerpt from my linear modelling lecture notes, this is the statement of the Gauss-Markov theorem, trouble is it goes into no more detail after this (not even explaining what the vector/matrix $l$ is supposed to be). Additionally what does 'the minimum variance linear unbiased estimator of the estimable function' mean? Can anyone shed some light on this explanation?
The vector $\ell$ is whatever you want it to be. That is, if you want to estimate some linear combination of the true coefficients $\beta$, the same linear combination of the estimated coefficients $\hat\beta$ is the minimum variance linear unbiased estimator.
- Unbiased: the expected value of the estimator is the linear combination of the $\beta$ that you are trying to estimate
- Linear: the estimator is a linear combination of the $Y$s
- Minimum variance: out of all estimators that satisfy 1. and 2., the estimator has the smallest variance