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

Part Two
Relaxing the Assumptions of the Classical Model
standard errors of
ˆ
β
2
and
ˆ
β
3
increase between the two regressions, a usual symptom of
collinearity.
We noted earlier that in the presence of high collinearity one cannot estimate the indi-
vidual regression coefficients precisely but that linear combinations of these coefficients
may be estimated more precisely. This fact can be substantiated from the regressions
(10.5.6) and (10.5.7). In the first regression the sum of the two partial slope coefficients is
0.4493 and in the second it is 0.4284, practically the same. Not only that, their standard
errors are practically the same, 0.1550 vs. 0.1823.
14
Note, however, the coefficient of 
X
3
has
changed dramatically, from 0.003 to 0.027.
Consequences of Micronumerosity
In a parody of the consequences of multicollinearity, and in a tongue-in-cheek manner,
Goldberger cites exactly similar consequences of micronumerosity, that is, analysis based
on small sample size.
15
The reader is advised to read Goldberger’s analysis to see why he
regards micronumerosity as being as important as multicollinearity.
10.6
An Illustrative Example
TABLE 10.3
Hypothetical Data on 
Y
,
X
2
, and 
X
3
Y
X
2
X
3
1
2
4
2
0
2
3
4
12
4
6
0
5
8
16
TABLE 10.4
Hypothetical Data on 
Y
,
X
2
, and 
X
3
Y
X