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

question 57

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?

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Definitions:

Self-Esteem

An individual's overall subjective evaluation of their own worth and capabilities.

More Selfish

Displaying an increased focus on one's own needs and desires, often at the expense of considering others'.

More Cooperative

Exhibiting a greater willingness or tendency to work together with others towards common goals or interests.

Adler's Influence

Refers to the impact of Alfred Adler's theories on psychology, emphasizing the importance of feelings of inferiority, the strive for superiority, and the development of social interest.