In the Multiple Regression Model with Two Regressors, the Formula

question 6

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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Definitions:

Paid-In Capital

The amount of capital "paid in" by investors during common or preferred stock issuances, including the par value of the shares themselves plus any additional paid-in capital.

Treasury Stock

Stocks that the original issuing company has repurchased, decreasing the total number of shares available for trade in the public market.

Par Value

The nominal or face value of a bond, share of stock, or coupon as stated by the issuer, which does not necessarily reflect market value.

Cumulative

Relating to an amount that has been gathered or accumulated over time, often used in the context of dividends or other financial aggregates.