(Requires Appendix Material) Consider the Following Population Regression Function Model $\widehat { Y } _ { i } = \widehat { \beta } _ { 0 } + \widehat { \beta } _ { 1 } X _ { 1 i } + \widehat { \beta } _ { 2 } X _ { 2 i }$

Essay

(Requires Appendix material) Consider the following population regression function model with two explanatory variables: $\widehat { Y } _ { i } = \widehat { \beta } _ { 0 } + \widehat { \beta } _ { 1 } X _ { 1 i } + \widehat { \beta } _ { 2 } X _ { 2 i }$
It is easy but tedious to
show that $\operatorname { SE } \left( \widehat { \beta _ { 2 } } \right)$ is given by the following formula: $\sigma _ { \widehat { \beta _ { 1 } } } ^ { 2 } = \frac { 1 } { n } \left[ \frac { 1 } { 1 - \rho _ { x _ { 1 } , x _ { 2 } } ^ { 2 } } \right] \frac { \sigma _ { u } ^ { 2 } } { \sigma _ { X _ { 1 } } ^ { 2 } }$ Sketch how
$\operatorname { SE } \left( \widehat { \beta _ { 2 } } \right)$ increases with the correlation between $X _ { 1 i } \text { and } X _ { 2 i } \text {. }$