The Gauss-Markov Theorem is telling us that in a regression model, where the expected value of our error terms is zero, and variance of the error terms is constant and finite and and are uncorrelated for all and the least squares estimator and are unbiased and have minimum variance among all unbiased linear estimators. Note that there might be biased estimator which have a even lower variance.

You can find the exact proof of the Gauss-Markov Theorem here.

A short reminder of what estimator for and look like:

(1)

(2)

The exact derivation of the least squares estimator in matrix notation can be found here.

The variance estimator for looks the following way:

(3)

The Gauss-Markov Theorem is telling us that has minimum variance among all unbiased linear estimators of the form

(4)

The Gauss-Markov Theorem is further telling us the this estimator must be unbiased. Under the assumption of unbiasedness we know that

(5)

(6)

(7)

(8)

If it holds that we know that must have certain properties/characteristics, i.e. it means that the assumption of unbiasedness imposes some restrictions on the .

You can find the exact proof of the Gauss-Markov Theorem here.

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