The OLS Formula for the Slope Coefficients in the Multiple

question 52

Essay

The OLS formula for the slope coefficients in the multiple regression model become increasingly more complicated, using the "sums" expressions, as you add more regressors. For example, in the regression with a single explanatory variable, the formula is $\frac { \sum _ { i = 1 } ^ { n } \left( X _ { i } - \bar{ X } \right) \left( Y _ { i } - \bar { X } \right) } { \sum _ { i = 1 } ^ { n } \left( X _ { i } - \bar { X } \right) ^ { 2 } }$ whereas this formula for the slope of the first explanatory variable 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 $z _ { i } = Z _ { i } - \bar { Z }$ )
in the case of two explanatory variables. Give an intuitive explanations as to why this is the case.

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