(5 points) Assuming that the true model is $Y_i = \beta_1 + \beta_2 X + u_i$, it can be shown that the naïve estimator of the slope coefficient is: $\hat{\beta}^*_2 = \frac{Y_n - Y_1}{X_n - X_1}$ And that it is unbiased with variance given as: $\frac{\sigma_u^2}{(X_1 - \bar{X})^2 + (X_n - \bar{X})^2 - 0.5(X_1 + X_n - 2\bar{X})^2}$ Use this information to verify that the naive estimator is less efficient than the OLS estimator. HINT: See Section 2.3 of book for math details.
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5(X_1 + X_n - 2\bar{X})^2}$ The variance of the OLS estimator is given by: $Var(\hat{\beta}_2) = \frac{\sigma_u^2}{\sum_{i=1}^n (X_i - \bar{X})^2}$ Show more…
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