(Requires Appendix Material) If the Gauss-Markov Conditions Hold, Then OLS $X$

(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 $Y _ { i } = \beta _ { 0 } + \beta _ { 1 } X _ { i } + u _ { i }$ and assuming a linear estimator $\tilde { \beta } _ { 1 } = \sum _ { i = 1 } ^ { n } a _ { i } Y _ { i }$ . Substitution of the simple regression model into this expression then results in two conditions for the unbiasedness of the estimator:
$$\sum _ { i = 1 } ^ { n } a _ { i } = 0 \text { and } \sum _ { i = 1 } ^ { n } a _ { i } X _ { i } = 1 .$$
The variance of the estimator is $\operatorname { var } \left( \tilde { \beta } _ { 1 } \mid X _ { 1 } , \ldots , X _ { n } \right) = \sigma _ { u } ^ { 2 } \sum _ { i = 1 } ^ { n } a _ { i } ^ { 2 }$ .
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. .