Gauss-Markov Theorem

Definition

Simple Linear Regression Case

Consider a Simple Linear Regression model $Y_i={\beta }_0+{\beta }_1{X}_i+{ϵ}_i$ Let $E({ϵ}_i)=0$ and $\forall i\ne j,\mathrm{Cov}(e_i,e_j)=0$, then the Least Squares Estimator ${\hat{\beta }}_0,{\hat{\beta }}_1$ has the minimum variance among the all unbiased linear estimators, i.e. BLUE

Multiple Linear Regression Case

Consider a Multiple Linear Regression model $\mathbf{y}=\mathbf{X}\mathbf{\beta }+\mathbf{ϵ}$ Let $E(\mathbf{ϵ})=0,\mathrm{Cov}(\mathbf{ϵ})=I_n{\sigma }^2$, then the Least Squares Estimator $\hat{\mathbf{\beta }}$ has the minimum variance among all the unbiased and linear estimators, i.e. BLUE