Consider the One-Variable Regression Model $Y _ { i } = \beta _ { 0 } + \beta _ { 1 } X _ { i } + u _ { i }$

question 42

Essay

Consider the one-variable regression model, $Y _ { i } = \beta _ { 0 } + \beta _ { 1 } X _ { i } + u _ { i }$
assumptions from Chapter 4 are satisfied. However, suppose that both Y and X are
measured with error, $\widetilde { Y } _ { i } = Y _ { i } + z _ { i } \text { and } \widetilde { X } _ { i } = X _ { i } + w _ { i }$
and independent of both Y and X respectively. If you estimated the regression model
$\widetilde { Y } _ { i } = \beta _ { 0 } + \beta _ { 1 } \widetilde { X } _ { i } + v _ { i }$ \quad using OLS, then show that the slope estimator is not consistent.

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Definitions:

Photosynthesis

The process by which green plants, algae, and some bacteria convert light energy into chemical energy, producing oxygen and glucose.

Light Colors

Hues that reflect a large portion of the visible light spectrum, making them appear bright to the human eye.

Carbon Source

A molecule used by an organism as a source of carbon for growth and development; can be organic, such as glucose, or inorganic, like carbon dioxide.