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The Frisch–Waugh Theorem 261

Because the errors are homoskedastic, the conditional variance of ui can be written as

var(ui 0 X1i, X2i) = su2. When there are two regressors, X1i and X2i, and the error term is

homoskedastic, in large samples the sampling distribution of bn1 is N(b1, s2bn1) where the
variance of this distribution, sb2n1, is

s2bn1 = 1 - 1 b s2u , (6.17)
na1 rX2 1, sX2 1
X2

where rX1, X2 is the population correlation between the two regressors X1 and X2 and sX2 1
is the population variance of X1.

The variance s2bn1 of the sampling distribution of bn1 depends on the squared correla-
tion between the regressors. If X1 and X2 are highly correlated, either positively or
negatively, then r2X1,X2 is close to 1, and thus the term 1 - r2X1,X2 in the denominator of
Equation (6.17) is small and the variance of bn1 is larger than it would be if rX1, X2 were
close to 0.

Another feature of the joint normal large-sample distribution of the OLS estimators

is that bn1 and bn2 are in general correlated. When the errors are homoskedastic, the correla-
tion between the OLS estimators bn1 and bn2 is the negative of the correlation between the

two regressors:

corr(bn1, bn2) = - rX1, X2. (6.18)

Appendix

6.3 The Frisch–Waugh Theorem

The OLS estimator in multiple regression can be computed by a sequence of shorter

regressions. Consider the multiple regression model in Equation (6.7). The OLS estimator

of b1 can be computed in three steps:

1. Regress X1 on X2, X3, c X, Xk,ka,nadndlelteY∼t X∼d1endoetneottheethreesrideusiadlusafrlsomfrotmhisthreisgrreesgsrieosns;ioann;d
2. Regress YY∼oonnXX∼2,1,X3, c ,
3. Regress

where the regressions include a constant term (intercept). The Frisch-Waugh theorem

states that the OLS coefficient in step 3 equals the OLS coefficient on X1 in the multiple
regression model [Equation (6.7)].

This result provides a mathematical statement of how the multiple regression coeffi-
cient bn1 estimates the effect on Y of X1, controlling for the other X’s: Because the first two