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:

UV Rays

Ultraviolet rays, a type of electromagnetic radiation emanating from the sun and artificial sources like tanning beds, with wavelengths shorter than visible light but longer than X-rays.

Melanin

A natural pigment found in most organisms, responsible for coloration in skin, hair, and eyes, and provides protection against UV radiation.

Suntanned

The darkening of skin color due to the exposure to sunlight or ultraviolet light which increases melanin production in the skin.

Sunburned

Damage to the skin caused by excessive exposure to ultraviolet radiation from the sun, leading to redness and inflammation.