What does the Gauss-Markov theorem prove? Without giving mathematical details, explain how the proof proceeds. What is its importance?

What will be an ideal response?

Answer: The Gauss-Markov Theorem proves that in the class of linear and unbiased estimators the OLS estimator has the smallest variance or is BLUE. The proof first establishes the conditions under which a linear estimator is unbiased. It then derives the variance of the estimator. The smallest variance property is then established by showing that the conditional variance of any old linear and unbiased estimator exceeds that of the OLS estimator, unless they are the same. To show this it is assumed that the OLS weights and the weights of any other linear estimator differ by some amount. Substitution of this condition into the conditional variance formula for any linear and unbiased estimator then shows that the resulting variance exceeds that of the OLS estimator unless the difference in the weights is zero. Hence OLS is BLUE. The Gauss-Markov Theorem gave the major justification for the widespread use of the OLS estimator.