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

FIGURE 10.1
The Ballentine view 
of multicollinearity.
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Chapter 10
Multicollinearity: What Happens If the Regressors Are Correlated?
323
shaded area), the higher the degree of collinearity. In the extreme, if 
X
2
and 
X
3
were to over-
lap completely (or if 
X
2
were completely inside 
X
3
, or vice versa), collinearity would be
perfect.
In passing, note that multicollinearity, as we have defined it, refers only to linear rela-
tionships among the 
X
variables. It does not rule out nonlinear relationships among them.
For example, consider the following regression model:
Y
i
=
β
0
+
β
1
X
i
+
β
2
X
2
i
+
β
3
X
3
i
+
u
i
(10.1.5)
where, say, 
Y
=
total cost of production and 
X
=
output. The variables 
X
2
i
(output squared)
and 
X
3
i
(output cubed) are obviously functionally related to 
X
i
, but the relationship is non-
linear. Strictly, therefore, models such as Eq. (10.1.5) do not violate the assumption of no
multicollinearity. However, in concrete applications, the conventionally measured correla-
tion coefficient will show 
X
i
,
X
2
i
, and 
X
3
i
to be highly correlated, which, as we shall show,
will make it difficult to estimate the parameters of Eq. (10.1.5) with greater precision (i.e.,
with smaller standard errors).
Why does the classical linear regression model assume that there is no multicollinearity
among the 
X
’s? The reasoning is this: