In the Regression Through the Origin Model $Y _ { i } = \beta _ { 1 } X _ { i } + u _ { i }$

Essay

In the regression through the origin model $Y _ { i } = \beta _ { 1 } X _ { i } + u _ { i }$ the OLS estimator is
$\widehat { \beta } _ { 1 } = \frac { \sum _ { i = 1 } ^ { n } X _ { i } Y _ { i } } { \sum _ { i = 1 } ^ { n } X _ { i } ^ { 2 } }$ Prove that the estimator is a linear function of $Y _ { 1 } , \ldots , Y _ { n }$ and prove
that it is conditionally unbiased.