The Gauss-Markov theorem

Summary

Setup

The LRM with random sampling

\[y_i=β_1+β_2x_i+ε_i \quad i=1,…,n\]

Linear, efficient and BLUE estimator

• We say that an estimator is linear if it is a linear function of \(y_1,…,y_n\) . The OLS estimators \(b_1,b_2\) are linear estimators.
• We say that an unbiased estimator is more efficient than another unbiased estimator if it has a smaller variance .
• We say that an estimator is BLUE (Best Linear Unbiased Estimator) if it is linear and unbiased and more efficient than any other linear and unbiased estimator.

The Gauss-Markov theorem

• The Gauss-Markov theorem : In the LRM, under the GM assumptions, the OLS estimators will be BLUE .
• Any set of conditions which make the Gauss-Markov theorem true are called Gauss-Markov assumptions.