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

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 $\widehat { \beta _ { 0 } } = \bar { Y } - \widehat { \beta _ { 1 } } \bar { X }$ and
$\widehat { \beta } _ { 1 } = \frac { \sum _ { i = 1 } ^ { n } X _ { i } Y _ { i } - n \bar { X } \bar { Y } } { \sum _ { i = 1 } ^ { n } X _ { i } ^ { 2 } - n \bar { X } ^ { 2 } }$ . Intuitively, the OLS estimators for the regression model
$Y _ { i } = \beta _ { 0 } + \beta _ { 1 } X _ { 1 i } + \beta _ { 2 } X _ { 2 i } + u _ { i } \text { might be } \widehat { \beta _ { 0 } } = \bar { Y } - \widehat { \beta } _ { 1 } \bar { X } _ { 1 } - \widehat { \beta } _ { 2 } \bar { X } _ { 2 } , \widehat { \beta } _ { 1 } = \frac { \sum _ { i = 1 } ^ { n } X _ { 1 i } Y _ { i } - n \bar { X } _ { 1 } \bar { Y } } { \sum _ { i = 1 } ^ { n } X _ { 1 i } ^ { 2 } - n \bar { X } _ { 1 } ^ { 2 } }$ and $\widehat { \beta _ { 2 } } = \frac { \sum _ { i = 1 } ^ { n } X _ { 2 i } Y _ { i } - n \bar { X } _ { 2 } \bar { Y } } { \sum _ { i = 1 } ^ { n } X _ { 2 i } ^ { 2 } - n \bar { X } _ { 2 } { } ^ { 2 } }$
By minimizing the prediction mistakes of the regression model with two explanatory variables, show that this cannot be the case.