The OLS Formula for the Slope Coefficients in the Multiple

question 9

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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Definitions:

Commercial Substance

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Straight-Line Method

A method of calculating depreciation by spreading the cost of an asset evenly over its useful life.

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The estimated value of an asset at the end of its useful life.

Depreciation

The method of distributing the expense of a physical asset throughout its life span.