Consider the Model $Y _ { i } = \beta _ { 1 } X _ { i } + u _ { i }$

Consider the model $Y _ { i } = \beta _ { 1 } X _ { i } + u _ { i }$ , where the $X _ { i }$ and the $u _ { i }$ are mutually independent i.i.d. random variables with finite fourth moment and $E \left( u _ { i } \right) = 0$ . Let $\widehat { \beta } _ { 1 }$ denote the OLS estimator of $\beta _ { 1 }$ . Show that
$$\sqrt { n } \left( \widehat { \beta } _ { 1 } - \beta _ { 1 } \right) = \frac { \frac { \sum _ { i = 1 } ^ { n } X _ { i } u _ { i } } { \sqrt { n } } } { \frac { \sum _ { i = 1 } ^ { n } X _ { i } ^ { 2 } } { n } } .$$

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