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X
[i.e., data matrix] which would be much preferred to the designs the natural experiment has
provided us [i.e., the sample at hand]. But a complaint about the apparent malevolence of nature
is not at all constructive, and the
ad hoc
cures for a bad design, such as stepwise regression or
ridge regression, can be disastrously inappropriate. Better that we should rightly accept the fact
that our non-experiments [i.e., data not collected by designed experiments] are sometimes not
very informative about parameters of interest.
1
Assumption 8 of the 
classical linear regression model 
(CLRM) is that there is no
multicollinearity
among the regressors included in the regression model. In this chapter
we take a critical look at this assumption by seeking answers to the following questions:
1. What is the nature of multicollinearity?
2. Is multicollinearity really a problem?
3. What are its practical consequences?
4. How does one detect it?
5. What remedial measures can be taken to alleviate the problem of multicollinearity?
In this chapter we also discuss Assumption 6 of the CLRM, namely, that the number of
observations in the sample must be greater than the number of regressors, and Assumption 7,
which requires that there be sufficient variability in the values of the regressors, for they are
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Chapter 10
Multicollinearity: What Happens If the Regressors Are Correlated?
321
intimately related to the assumption of no multicollinearity. Arthur Goldberger has chris-
tened Assumption 6 as the problem of 
micronumerosity,
2
which simply means small sam-
ple size.
10.1
The Nature of Multicollinearity
The term 
multicollinearity
is due to Ragnar Frisch.
3
Originally it meant the existence of a
“perfect,” or exact, linear relationship among some or all explanatory variables of a regres-
sion model.
4
For the 
k
-variable regression involving explanatory variables 
X
1
,
X
2
,
. . .
,
X
k
(where 
X
1
=
1 for all observations to allow for the intercept term), an exact linear rela-
tionship is said to exist if the following condition is satisfied:
λ
1
X
1
+
λ
2
X
2
+ · · · +
λ
k
X
k
=
0