Consider the One-Variable Regression Model, Y_i = β₀ + β₁X_i $\tilde { Y } _ { i }$

question 25

Essay

Consider the one-variable regression model, Y_i = β₀ + β₁X_i + u_i, where the usual assumptions from Chapter 4 are satisfied. However, suppose that both Y and X are measured with error, $\tilde { Y } _ { i }$ = Y_i + z_i and $\widetilde { X } _ { i }$ = X_i + w_i. Let both measurement errors be i.i.d. and independent of both Y and X respectively. If you estimated the regression model $\tilde { Y } _ { i }$ = β₀ + β₁
$\widetilde { X } _ { i }$ + v_i using OLS, then show that the slope estimator is not consistent.

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Definitions:

Maturation

The process by which an individual or organism develops and reaches full growth or potential, often considered in a biological or psychological context.

Intrinsic Motivation

The drive to engage in an activity for its own sake, enjoying the process without external reward or pressure.

Literacy

The ability to read and write, or having knowledge in a specific area.

Social Status

Social status is the perception of an individual's or group's value within a social hierarchy, influenced by factors such as wealth, occupation, education, and family lineage.