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

Reducing collinearity in polynomial regressions.
In Section 7.10 we discussed
polynomial regression models. A special feature of these models is that the explanatory
variable(s) appears with various powers. Thus, in the total cubic cost function involving the
regression of total cost on output, (output)
2
, and (output)
3
, as in Eq. (7.10.4), the various
output terms are going to be correlated, making it difficult to estimate the various slope co-
efficients precisely.
36
In practice though, it has been found that if the explanatory vari-
able(s) is expressed in the deviation form (i.e., deviation from the mean value),
multicollinearity is substantially reduced. But even then the problem may persist,
37
in
which case one may want to consider techniques such as 
orthogonal polynomials.
38
7.
Other methods of remedying multicollinearity.
Multivariate statistical techniques
such as 
factor analysis
and 
principal components
or techniques such as 
ridge regression
are often employed to “solve” the problem of multicollinearity. Unfortunately, these tech-
niques are beyond the scope of this book, for they cannot be discussed competently with-
out resorting to matrix algebra.
39
34
I am indebted to the late Albert Zucker for providing the results given in the following regressions.
35
Judge et al., op. cit., p. 625. See also Section 10.9.
36
As noted, since the relationship between 
X
, 
X
2
, and 
X
3
is nonlinear, polynomial regressions do not
violate the assumption of no multicollinearity of the classical model, strictly speaking.
37
See R. A. Bradley and S. S. Srivastava, “Correlation and Polynomial Regression,”
American Statisti-
cian,
vol. 33, 1979, pp. 11–14.
38
See Norman Draper and Harry Smith, 
Applied Regression Analysis,
2d ed., John Wiley & Sons, New
York, 1981, pp. 266–274.
39
A readable account of these techniques from an applied viewpoint can be found in Samprit Chatter-
jee and Bertram Price, 
Regression Analysis by Example,
John Wiley & Sons, New York, 1977, Chapters 7
and 8. See also H. D. Vinod, “A Survey of Ridge Regression and Related Techniques for Improvements
over Ordinary Least Squares,” 
Review of Economics and Statistics,
vol. 60, February 1978, pp. 121–131.
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Chapter 10
Multicollinearity: What Happens If the Regressors Are Correlated?