Consider the Model Y_i - β₁X_i + U_i, Where the X_i

Essay

Consider the model Y_i - β₁X_i + u_i, where the X_i and u_i the are mutually independent i.i.d. random variables with finite fourth moment and E(u_i)= 0.
(a)Let $\hat { \beta }$ ₁ denote the OLS estimator of β₁. Show that $\sqrt { n }$ ( $\hat { \beta }$ ₁- β₁)= $\frac { \frac { \sum _ { i = 1 } ^ { n } X _ { i } t _ { i } } { \sqrt { n } } } { \sum _ { i = 1 } ^ { n } X _ { i } ^ { 2 } }$ (b)What is the mean and the variance of $\frac { \sum _ { i = 1 } ^ { n } X _ { i } u _ { i } } { \sqrt { n } }$ ? Assuming that the Central Limit Theorem holds, what is its limiting distribution?
(c)Deduce the limiting distribution of $\sqrt { n }$ ( $\hat { \beta }$ ₁ - β₁)? State what theorems are necessary for your deduction.