In the Multiple Regression Model with Two Regressors, the Formula

question 26

Essay

In the multiple regression model with two regressors, the 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 $z _ { i } = Z _ { i } - \bar { Z }$ ).
An alternative way to derive the OLS estimator is given through the following three step procedure.
Step 1: regress Y on a constant and $X _ { 2 }$ , and calculate the residual (Res1).
Step 2: regress $X _ { 1 }$ on a constant and $X _ { 2 }$ , and calculate the residual (Res2).
Step 3: regress Res1 on a constant and Res2.
Prove that the slope of the regression in Step 3 is identical to the above formula.

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