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

question 12

Essay

(Requires Appendix material) Your textbook shows that OLS is a linear estimator
$\hat { \beta } _ { 1 } = \sum _ { i = 1 } ^ { n } \hat { a } _ { i } Y _ { i } , \text { where } \hat { a } _ { i } = \frac { X _ { i } - \bar { X } } { \sum _ { i = 1 } ^ { n } \left( X _ { i } - \bar { X } \right) ^ { 2 } }$ .
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.

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Definitions:

Apex

The uppermost part of something, often referring to the tip, peak, or summit.

Cilia

Microscopic, hair-like structures on the surface of cells that move rhythmically to propel substances or cells.

Lateral

Positioned on or relating to the side of an organism or structure.

Proximal

Situated nearer to the center of the body or the point of attachment.