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

Additional or new data.
Since multicollinearity is a sample feature, it is possible
that in another sample involving the same variables collinearity may not be so serious as in
the first sample. Sometimes simply increasing the size of the sample (if possible) may
attenuate the collinearity problem. For example, in the three-variable model we saw that
var (
ˆ
β
2
)
=
σ
2
x
2
2
i
1
−
r
2
2 3
guj75772_ch10.qxd 12/08/2008 02:45 PM Page 345

---

346
Part Two
Relaxing the Assumptions of the Classical Model
Now as the sample size increases, 
x
2
2
i
will generally increase. (Why?) Therefore, for any
given 
r
2 3
, the variance of 
ˆ
β
2
will decrease, thus decreasing the standard error, which will
enable us to estimate 
β
2
more precisely.
As an illustration, consider the following regression of consumption expenditure 
Y
on
income 
X
2
and wealth 
X
3
based on 10 observations:
34
ˆ
Y
i
=
24.377
+
0.8716
X
2
i
−
0.0349
X
3
i
t
=
(3.875)
(2.7726)
(
−
1.1595)
R
2
=
0.9682
(10.8.8)
The wealth coefficient in this regression not only has the wrong sign but is also statistically
insignificant at the 5 percent level. But when the sample size was increased to 40 observa-
tions (micronumerosity?), the following results were obtained:
ˆ
Y
i
=
2.0907
+
0.7299
X
2
i
+
0.0605
X
3
i
t
=
(0.8713)
(6.0014)
(2.0014)
R
2
=
0.9672
(10.8.9)
Now the wealth coefficient not only has the correct sign but also is statistically significant
at the 5 percent level.
Obtaining additional or “better” data is not always that easy, for as Judge et al. note:
Unfortunately, economists seldom can obtain additional data without bearing large costs,
much less choose the values of the explanatory variables they desire. In addition, when adding
new variables in situations that are not controlled, we must be aware of adding observations
that were generated by a process other than that associated with the original data set; that is,
we must be sure that the economic structure associated with the new observations is the same
as the original structure.
35
6.