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

7.8
R
2
and the Adjusted 
R
2
An important property of 
R
2
is that it is a nondecreasing function of the number of
explanatory variables or regressors present in the model, unless the added variable is per-
fectly collinear with the other regressors; as the number of regressors increases, 
R
2
almost
invariably increases and never decreases. Stated differently, an additional 
X
variable will
not decrease 
R
2
. Compare, for instance, regression (7.7.2) or (7.7.3) with (7.6.2). To see
this, recall the definition of the coefficient of determination:
R
2
=
ESS
TSS
=
1
−
RSS
TSS
(7.8.1)
=
1
−
ˆ
u
2
i
y
2
i
Now 
y
2
i
is independent of the number of 
X
variables in the model because it is simply
(
Y
i
− ¯
Y
)
2
.
The RSS, 
ˆ
u
2
i
, however, depends on the number of regressors present in the
model. Intuitively, it is clear that as the number of 
X
variables increases, 
ˆ
u
2
i
is likely to
decrease (at least it will not increase); hence 
R
2
as defined in Eq. (7.8.1) will increase. In
view of this, in comparing two regression models with the 
same dependent variable
but
differing number of 
X
variables, one should be very wary of choosing the model with the
highest 
R
2
.
To compare two 
R
2
terms, one must take into account the number of 
X
variables present
in the model. This can be done readily if we consider an alternative coefficient of determi-
nation, which is as follows:
(7.8.2)
¯
R
2
=
1
−
ˆ
u
2
i
(
n
−
k
)
y
2
i
(
n
−
1)
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