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

question 5

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.

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Definitions:

Thoughts

Mental processes or ideas that are produced by cognitive activities in the brain.

Endless Flow

A continuous, uninterrupted stream or sequence of activity or events.

Brain Wave

Oscillations of electrical potential between parts of the brain, visible on an EEG.

Beta Waves

Brainwaves that are associated with active, alert, and cognitive processing states of consciousness.