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

322
Part Two
Relaxing the Assumptions of the Classical Model
As a numerical example, consider the following hypothetical data:
X
2
X
3
X
*
3
10
50
52
15
75
75
18
90
97
24
120
129
30
150
152
It is apparent that 
X
3
i
=
5
X
2
i
.
Therefore, there is perfect collinearity between 
X
2
and 
X
3
since the coefficient of correlation 
r
23
is unity. The variable 
X*
3
was created from 
X
3
by sim-
ply adding to it the following numbers, which were taken from a table of random numbers:
2, 0, 7, 9, 2. Now there is no longer perfect collinearity between 
X
2
and 
X*
3
. However, the
two variables are highly correlated because calculations will show that the coefficient of
correlation between them is 0.9959.
The preceding algebraic approach to multicollinearity can be portrayed succinctly by
the Ballentine (recall Figure 3.8, reproduced in Figure 10.1). In this figure the circles 
Y, X
2
,
and 
X
3
represent, respectively, the variations in 
Y
(the dependent variable) and 
X
2
and 
X
3
(the explanatory variables). The degree of collinearity can be measured by the extent of the
overlap (shaded area) of the 
X
2
and 
X
3
circles. In Figure 10.1
a
there is no overlap between
X
2
and 
X
3
, and hence no collinearity. In Figure 10.1
b
through 10.1
e
there is a “low” to
“high” degree of collinearity—the greater the overlap between 
X
2
and 
X
3
(i.e., the larger the
Y
X
2
X
3
Y
X
2
X
3
Y
X
2
X
3
Y
X
2
X
3
Y
X
2
X
3
(
a
) No collinearity
(
b
) Low collinearity
(
c
) Moderate collinearity
(
d
) High collinearity
(
e
) Very high collinearity