(Requires Calculus)For the Simple Linear Regression Model of Chapter 4 $Y _ { i } = \beta _ { 0 } + \beta _ { 1 } X _ { i } + u _ { i }$

Essay

(Requires Calculus)For the simple linear regression model of Chapter 4, $Y _ { i } = \beta _ { 0 } + \beta _ { 1 } X _ { i } + u _ { i }$ , the OLS estimator for the intercept was $\hat { \beta } _ { 0 } = \bar { Y } - \hat { \beta } _ { 1 } \bar { X }$ , and $\hat { \beta } _ { 1 } = \frac { \sum _ { i = 1 } ^ { n } X _ { i } Y _ { i } - n \overline { X Y } } { \sum _ { i = 1 } ^ { n } X _ { i } ^ { 2 } - n \bar { X } ^ { 2 } }$ Intuitively, the OLS estimators for the regression model $Y _ { i } = \beta _ { 0 } + \beta _ { 1 } X _ { 1 i } + \beta _ { 2 } X _ { 2 i } + u _ { i }$ might be $\hat { \beta } _ { 0 } = \bar { Y } - \hat { \beta } _ { 1 } \bar { X } _ { 1 } - \hat { \beta } _ { 2 } \bar { X } _ { 2 } , \hat { \beta } _ { 1 } = \frac { \sum _ { i = 1 } ^ { n } \bar { X } _ { 1i } Y _ { i } - n \bar { X } _ { 1 } \bar { Y } } { \sum _ { i = 1 } ^ { n } \bar { X } _ { 1 i } ^ { 2 } - n \bar { X } _ { 1 } ^ { 2 } }$ and $\hat { \beta } _ { 2 } = \frac { \sum _ { i = 1 } ^ { n } \bar { X } _ { 2 i } Y _ { i } - n \bar { X } _ { 2 } \bar { Y } } { \sum _ { i = 1 } ^ { n } \bar { X } _ { 2 i } ^ { 2 } - n \bar { X } _ { 2 } ^ { 2 } }$ By minimizing the prediction mistakes of the regression model with two explanatory variables, show that this cannot be the case.