(Requires Appedix Material and Calculus) Equation (5 $\widetilde { \beta } _ { 1 }$

(Requires Appedix material and Calculus) Equation (5.36) in your textbook derives the conditional variance for any old conditionally unbiased estimator $\widetilde { \beta } _ { 1 }$ to be $\operatorname { var } \left( \widetilde { \beta } _ { 1 } \mid X _ { 1 } , \ldots , X _ { n } \right) = \sigma _ { u } ^ { 2 } \sum _ { i = 1 } ^ { n } a _ { i } ^ { 2 }$ (where the conditions for conditional unbiasedness are $\sum _ { i = 1 } ^ { n } a _ { i } = 0$ and $\sum _ { i = 1 } ^ { n } a _ { i } X _ { i } = 1$ . As an alternative to the BLUE proof presented in your textbook, you recall from one of your calculus courses that you could minimize the variance subject to the two constraints, thereby making the variance as small as possible while the constraints are holding. Show that in doing so you get the OLS weights $\hat { a } _ { i }$ . (You may assume that $X _ { 1 } , \ldots , X _ { n }$ are nonrandom (fixed over repeated samples.)) 22