(Requires Appendix Material)If the Gauss-Markov Conditions Hold, Then OLS Is $\widetilde { \beta } _ { 1 } = \sum _ { i = 1 } ^ { n } a _ { i } Y _ { i }$

(Requires Appendix material)If the Gauss-Markov conditions hold, then OLS is BLUE. In addition, assume here that X is nonrandom. Your textbook proves the Gauss-Markov theorem by using the simple regression model Y_i = β₀ + β₁X_i + u_i and assuming a linear estimator $\widetilde { \beta } _ { 1 } = \sum _ { i = 1 } ^ { n } a _ { i } Y _ { i }$ Substitution of the simple regression model into this expression then results in two conditions for the unbiasedness of the estimator: $\sum _ { i = 1 } ^ { n } a _ { i }$ = 0 and $\sum _ { i = 1 } ^ { n } a _ { i } X _ { i }$ = 1.
The variance of the estimator is var( $\tilde { \beta } _ { 1 }$
| X₁,…, X_n)= $\sigma _ { u } ^ { 2 }$ $\sum _ { i = 1 } ^ { n } a _ { i } ^ { 2 }$ Different from your textbook, use the Lagrangian method to minimize the variance subject to the two constraints. Show that the resulting weights correspond to the OLS weights.