The McGraw-Hill Series Economics essentials of economics brue, McConnell, and Flynn Essentials of Economics

(13.3.3)
where 
b
3 2
is the slope in the regression of the excluded variable 
X
3
on the included variable
X
2
(
b
3 2
=
x
3
i
x
2
i
/
x
2
2
i
)
.
As Eq. (13.3.3) shows, 
ˆ
α
2
is biased, unless 
β
3
or 
b
3 2
or both
are zero. We rule out 
β
3 
being zero, because in that case we do not have specification error
to begin with. The coefficient 
b
3 2
will be zero if 
X
2
and 
X
3
are uncorrelated, which is
unlikely in most economic data. 
Generally, however, the extent of the bias will depend on the 
bias term 
β
3
b
3 2
. If, for in-
stance, 
β
3
is positive (i.e., 
X
3
has a positive effect on 
Y
) and 
b
3 2
is positive (i.e., 
X
2
and 
X
3
are positively correlated), 
ˆ
α
2
, on average, will overestimate the true 
β
2
(i.e., positive bias).
But this result should not be surprising, for 
X
2
represents not only its 
direct effect
on 
Y
but
also its 
indirect effect
(via 
X
3
) on 
Y
. In short, 
X
2
gets credit for the influence that is rightly
attributable to 
X
3
, the latter being prevented from showing its effect explicitly because it is
not “allowed” to enter the model. As a concrete example, consider the example discussed
in Chapter 7 (Example 7.1).
7
For an algebraic treatment, see Jan Kmenta, 
Elements of Econometrics,
Macmillan, New York, 1971,
pp. 391–399. Those with a matrix algebra background may want to consult J. Johnston, 
Econometrics
Methods,
4th ed., McGraw-Hill, New York, 1997, pp. 119–112.
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