It Is Not Hard, but Tedious, to Derive the OLS

Essay

It is not hard, but tedious, to derive the OLS formulae for the slope coefficient in the multiple regression case with two explanatory variables. The formula for the first regression slope is $$\hat { \beta } _ { 1 } = \frac { \sum _ { i = 1 } ^ { n } y _ { i } x _ { 1 i } \sum _ { i = 1 } ^ { n } x _ { 2 i } ^ { 2 } - \sum _ { i = 1 } ^ { n } y _ { i } x _ { 2 i } \sum _ { i = 1 } ^ { n } x _ { 1 i } x _ { 2 i } } { \sum _ { i = 1 } ^ { n } x _ { 1 i } ^ { 2 } \sum _ { i = 1 } ^ { n } x _ { 2 i } ^ { 2 } - \left( \sum _ { i = 1 } ^ { n } x _ { 1 i } x _ { 2 i } \right) ^ { 2 } }$$ (small letters refer to deviations from means as in $z _ { i } = Z _ { i } - \bar { Z }$ ).
Show that this formula reduces to the slope coefficient for the linear regression model with one regressor if the sample correlation between the two explanatory variables is zero. Given this result, what can you say about the effect of omitting the second explanatory variable from the regression?