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:

Generativity

A concern for establishing and guiding the next generation, often considered a stage of personal development in adulthood.

Midlife Crisis

A period of emotional turmoil in middle age characterized by a strong desire for change, often resulting from the realization of aging and mortality.

Middle-Aged Adults

Individuals typically between the ages of 40 and 65, often experiencing significant life and biological changes including career dynamics, family responsibilities, and health transitions.

Psychosocial Generativity

A concept involving the concern for establishing and guiding the next generation, often associated with midlife development.