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300 Chapter 7 Hypothesis Tests and Confidence Intervals in Multiple Regression

ui given X1i and X2i—that is, vi = ui - E(ui ͉ X1i, X2i)—so that vi has conditional mean zero:
E(vi ͉ X1i, X2i) = E3ui - E(ui ͉ X1i, X2i) ͉ X1i, X2i4 = E(ui ͉ X1i, X2i) - E(ui ͉ X1i, X2i) = 0 .
Thus,

Yi = b0 + b1X1i + b2X2i + ui

= b0 + b1X1i + b2X2i + E(ui ͉ X1i, X2i) + vi (using the definition of vi)

= b0 + b1X1i + b2X2i + E(ui ͉ X2i) + vi (using conditional mean independence)
= b0 + b1X1i + b2X2i + (g0 + g2X2i) + vi 3using linearity of E(ui ͉ X2i)4

= (b0 + g0) + b1X1i + (b2 + g2)X2i + vi (collecting terms)

= d0 + b1X1i + d2X2i + vi, (7.25)

where d0 = b0 + g0 and d2 = b2 + g2.
The error vi in Equation (7.25) has conditional mean zero; that is, E(vi͉ X1i, X2i) = 0.

Therefore, the first least squares assumption for multiple regression applies to the final line

of Equation (7.25), and if the other three least squares assumptions for multiple regression

also hold, then the OLS regression of Yi on a constant, X1i, and X2i will yield unbiased and
consistent estimators of d0, b1, and d2. Thus the OLS estimator of the coefficient on X1i is
unbiased for the causal effect b1. However, the OLS estimator of the coefficient on X2i is
not unbiased for b2 and instead estimates the sum of the causal effect b2 and the coefficient
g2 arising from the correlation of the control variable X2i with the original error term ui.

The derivation in Equation (7.25) works for any value of b2, including zero. A variable
X2i is a useful control variable if conditional mean independence holds; it need not have a
direct causal effect on Yi.

The fourth line in Equation (7.25) uses the assumption that E(ui͉ X2i) is linear in X2i.
As discussed in Section 2.4, this will be true if ui and X2i are jointly normally distributed.
The assumption of linearity can be relaxed using methods discussed in Chapter 8. Exercise

18.9 works through the steps in Equation (7.25) for multiple variables of interest and mul-

tiple control variables.

In terms of the example in Section 7.5 [the regression in Equation (7.19)], if X2i is
LchPct, then b2 is the causal effect of the subsidized lunch program (b2 is positive if the
program’s nutritional benefits improve test scores), g2 is negative because LchPct is nega-
tively correlated with (controls for) omitted learning advantages that improve test scores,

and d2 = b2 + g2 would be negative if the omitted variable bias contribution through g2
outweights the positive causal effect b2.

To better understand the conditional mean independence assumption, return to the con-

cept of an ideal randomized controlled experiment. As discussed in Section 4.4, if X1i is ran-
domly assigned, then in a regression of Yi on X1i, the conditional mean zero assumption holds.
If, however, X1i is randomly assigned, conditional on another variable X2i, then the conditional
mean independence assumption holds, but if X2i is correlated with ui, the conditional mean zero