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

Auxiliary regressions.
Since multicollinearity arises because one or more of the
regressors are exact or approximately linear combinations of the other regressors, one way
of finding out which 
X
variable is related to other 
X
variables is to regress each 
X
i
on the
remaining 
X
variables and compute the corresponding 
R
2
, which we designate as 
R
2
i
; each
one of these regressions is called an 
auxiliary regression,
auxiliary to the main regression
of 
Y
on the 
X
’s. Then, following the relationship between 
F
and 
R
2
established in
Eq. (8.4.11), the variable
(10.7.3)
follows the 
F
distribution with 
k
−
2 and 
n
−
k
+
1 df. In Eq. (10.7.3) 
n
stands for the
sample size, 
k
stands for the number of explanatory variables including the intercept term,
and 
R
2
x
i
·
x
2
x
3
···
x
k
is the coefficient of determination in the regression of variable 
X
i
on the
remaining 
X
variables.
23
If the computed 
F
exceeds the critical 
F
i
at the chosen level of significance, it is taken to
mean that the particular 
X
i
is collinear with other 
X
’s; if it does not exceed the critical 
F
i
,
we say that it is not collinear with other 
X
’s, in which case we may retain that variable in the
model. If 
F
i
is statistically significant, we will still have to decide whether the particular 
X
i
should be dropped from the model. This question will be taken up in Section 10.8.
But this method is not without its drawbacks, for
. . . if the multicollinearity involves only a few variables so that the auxiliary regressions do not
suffer from extensive multicollinearity, the estimated coefficients may reveal the nature of the
linear dependence among the regressors. Unfortunately, if there are several complex linear
associations, this curve fitting exercise may not prove to be of much value as it will be difficult
to identify the separate interrelationships.
24
Instead of formally testing all auxiliary 
R
2
values, one may adopt