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

question 51

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 $$\hat { \beta } _ { 1 } = \frac { \sum _ { i = 1 } ^ { n } y _ { i } x _ { 1 i } \sum _ { i = 1 } ^ { n } x _ { 2 i } ^ { 2 } - \sum _ { i = 1 } ^ { n } y _ { i } x _ { 2 i } \sum _ { i = 1 } ^ { n } x _ { 1 i } x _ { 2 i } } { \sum _ { i = 1 } ^ { n } x _ { 1 i } ^ { 2 } \sum _ { i = 1 } ^ { n } x _ { 2 i } ^ { 2 } - \left( \sum _ { i = 1 } ^ { n } x _ { i i } x _ { 2 i } \right) ^ { 2 } }$$
(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:

Enzymes

Biological catalysts that speed up chemical reactions in living organisms without being consumed in the process.

Abiotic Evolution

The process of environmental and geological changes over time, not influenced by biological factors.

Chemical Evolution

The process by which simple chemical compounds spontaneously formed more complex molecules over time, ultimately leading to the emergence of life from non-living matter.

Organic Evolution

The process of change and diversification of living organisms over time through mutations, natural selection, and genetic drift.