Page 263 -

262 Chapter 6 Linear Regression with Multiple Regressors

regressions (steps 1 and 2) remove from Y and X1 their variation associated with the other
X’s, the third regression estimates the effect on Y of X1 using what is left over after remov-
ing (controlling for) the effect of the other X’s. The Frisch-Waugh theorem is proven in

Exercise 18.17.

This theorem suggests how Equation (6.17) can be derived from Equation (5.27).
Because bn1 is the OLS regression coefficient from the regression of Y∼ onto X∼1, Equation (5.27)

suggests that the homoskedasticity-only variance oref gbnr1esissiosn2bno1 f=Xn1ssXo2u21n, twohXer2e(rseX∼c21alils the
variance of X∼1. Because X∼1 is the residual from the that

Equation (6.17) pertains to the model with k = 2 regressors), Equation (6.15) implies that

sX∼2 1 = (1 - R 2 X2)sX2 1, where RsX∼22X1 1¡, Xp2 is the adjusted R2 from the regression of X1 onto X2.
X1, sX21, RX21, X2 ¡p rX21, X2 and sX2 1 ¡p sX21.

Equation (6.17) follows from