Gauss-Markov Assumptions › Mathematical Foundation

The Gauss-Markov theorem can be stated mathematically as follows:

Theorem: Under assumptions 1-5, the OLS estimator $\hat{\beta}$ is the best linear unbiased estimator (BLUE) of $\beta$.

This means:

• Unbiased: $E[\hat{\beta}] = \beta$
• Efficient: $\text{Var}(\hat{\beta})$ is the smallest among all linear unbiased estimators
• Consistent: As sample size increases, $\hat{\beta}$ converges to the true $\beta$

The OLS estimator is given by:

$$\hat{\beta} = (X'X)^{-1}X'y$$

Where:

• $X$ is the matrix of independent variables
• $y$ is the vector of dependent variable
• $\hat{\beta}$ is the vector of estimated coefficients