Exam 17: The Theory of Linear Regression With One Regressor

(Requires Appendix material)If the Gauss-Markov conditions hold, then OLS is BLUE. In addition, assume here that X is nonrandom. Your textbook proves the Gauss-Markov theorem by using the simple regression model Y_i = β₀ + β₁X_i + u_i and assuming a linear estimator $\widetilde { \beta } _ { 1 } = \sum _ { i = 1 } ^ { n } a _ { i } Y _ { i }$ Substitution of the simple regression model into this expression then results in two conditions for the unbiasedness of the estimator: $\sum _ { i = 1 } ^ { n } a _ { i }$ = 0 and $\sum _ { i = 1 } ^ { n } a _ { i } X _ { i }$ = 1. The variance of the estimator is var( $\tilde { \beta } _ { 1 }$ | X₁,…, X_n)= $\sigma _ { u } ^ { 2 }$ $\sum _ { i = 1 } ^ { n } a _ { i } ^ { 2 }$ Different from your textbook, use the Lagrangian method to minimize the variance subject to the two constraints. Show that the resulting weights correspond to the OLS weights.

Besides the Central Limit Theorem, the other cornerstone of asymptotic distribution theory is the

Slutsky's theorem combines the Law of Large Numbers

Estimation by WLS

Finite-sample distributions of the OLS estimator and t-statistics are complicated, unless

The OLS estimator is a linear estimator, $\hat { \beta }$ ₁ = $\sum _ { i = 1 } ^ { n } \hat { a } _ { i } Y _ { i }$ , where $\hat a$ _i =

E $\left(\frac { 1 } { n - 2 } \sum _ { i = 1 } ^ { n } \hat { u } _ { i } ^ { 2 }\right )$

Under the five extended least squares assumptions, the homoskedasticity-only t-distribution in this chapter

You need to adjust $S _ {\hat{ u }} ^ { 2 }$ by the degrees of freedom to ensure that $S _ {\hat{ u }} ^ { 2 }$ is