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

(10.8.1)
then it can be shown that (see Appendix 13A.1)
E
(
b
1 2
)
=
β
2
+
β
3
b
3 2
(10.8.2)
where 
b
3 2
=
slope coefficient in the regression of 
X
3
on 
X
2
. Therefore, it is obvious from
Eq. (10.8.2) that 
b
12
will be a biased estimate of 
β
2
as long as 
b
3 2
is different from zero (it
is assumed that 
β
3
is different from zero; otherwise there is no sense in including 
X
3
in the
original model).
33
Of course, if 
b
3 2
is zero, we have no multicollinearity problem to begin
with. It is also clear from Eq. (10.8.2) that if both 
b
3 2
and 
β
3
are positive (or both are neg-
ative), 
E
(
b
1 2
) will be greater than 
β
2
; hence, on the average 
b
1 2
will overestimate 
β
2
, lead-
ing to a positive bias. Similarly, if the product 
b
3 2
β
3
is negative, on the average 
b
1 2
will
underestimate 
β
2
, leading to a negative bias.
From the preceding discussion it is clear that dropping a variable from the model to
alleviate the problem of multicollinearity may lead to the specification bias. Hence the rem-
edy may be worse than the disease in some situations because, whereas multicollinearity
may prevent precise estimation of the parameters of the model, omitting a variable may
seriously mislead us as to the true values of the parameters. Recall that OLS estimators are
BLUE despite near collinearity.
4.