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

The point is clear: Once a model is formulated on the basis of the relevant
theory, one is ill-advised to drop a variable from such a model.
Inclusion of an Irrelevant Variable (Overfitting a Model)
Now let us assume that
Y
i
=
β
1
+
β
2
X
2
i
+
u
i
(13.3.6)
is the truth, but we fit the following model:
Y
i
=
α
1
+
α
2
X
2
i
+
α
3
X
3
i
+
v
i
(13.3.7)
and thus commit the specification error of including an unnecessary variable in the model.
The consequences of this specification error are as follows:
1. The OLS estimators of the parameters of the “incorrect” model are all 
unbiased and
consistent,
that is, 
E
(
α
1
) 
=
β
1
, 
E
(
ˆ
α
2
) 
=
β
2
, and 
E
(
ˆ
α
3
) 
=
β
3
=
0.
2. The error variance 
σ
2
is correctly estimated.
3. The usual confidence interval and hypothesis-testing procedures remain valid.
4. However, the estimated
α
’s will be generally inefficient, that is, their variances will
be generally larger than those of the
ˆ
β
’s of the true model. The proofs of some of these
statements can be found in Appendix 13A, Section 13A.2. The point of interest here is
the relative inefficiency of the
ˆ
α
’s. This can be shown easily.
11
Note, though, 
ˆ
α
1
is still biased, which can be seen intuitively as follows: We know that
ˆ
β
1
= ¯
Y
− ˆ
β
2
¯
X
2
− ˆ
β
3
¯
X
3
, whereas 
ˆ
α
1
= ¯
Y
− ˆ
α
2
¯
X
2
, and even if 
ˆ
α
2
= ˆ
β
2
, the two intercept estimators
will not be the same.
12
For details, see Adrian C. Darnell, 
A Dictionary of Econometrics,
Edward Elgar Publisher, 1994,
pp. 371–372.
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