(Requires Appendix Material)Your Textbook Shows That OLS Is a Linear $\hat { \beta }$

Essay

(Requires Appendix material)Your textbook shows that OLS is a linear estimator $\hat { \beta }$ ₁ = $\sum _ { i = 1 } ^ { n } \hat { a } _ { i } Y _ { i }$ , where $$\hat { a } _ { i } = \frac { \bar { X } _ { i } - \bar { X } } { \sum _ { i = 1 } ^ { n } \left( X _ { i } - \bar { X } \right) ^ { 2 } }$$ For OLS to be conditionally unbiased, the following two conditions must hold: $\sum _ { i = 1 } ^ { n } \hat { a } _ { i } = 0$ and $\sum _ { i = 1 } ^ { n } \hat { a } _ { i } \bar{X} _ { i }$ = 1. Show that this is the case.