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

Question

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

Quizplus · Accepted Answer

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

(Requires Appendix Material) Your Textbook Shows That OLS Is a Linear