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.

Question

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.

Quizplus · Accepted Answer

The Answer of 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.

Consider the One-Variable Regression Model, Yi = β0 + β1Xi Y~i\tilde { Y } _ { i }Y~i​

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