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

Part One
Single-Equation Regression Models
where 
k
=
the number of parameters in the model 
including the intercept term
. (In the
three-variable regression, 
k
=
3
.
Why?) The 
R
2
thus defined is known as the 
adjusted 
R
2
,
denoted by 
¯
R
2
.
The term 
adjusted
means adjusted for the df associated with the sums of
squares entering into Eq. (7.8.1):
ˆ
u
2
i
has 
n
−
k
df in a model involving 
k
parameters,
which include the intercept term, and 
y
2
i
has 
n
−
1 df. (Why?) For the three-variable
case, we know that 
ˆ
u
2
i
has 
n
−
3 df.
Equation (7.8.2) can also be written as
¯
R
2
=
1
−
ˆ
σ
2
S
2
Y
(7.8.3)
where 
ˆ
σ
2
is the residual variance, an unbiased estimator of true 
σ
2
, and 
S
2
Y
is the sample
variance of 
Y
.
It is easy to see that 
¯
R
2
and 
R
2
are related because, substituting Eq. (7.8.1) into
Eq. (7.8.2), we obtain
¯
R
2
=
1
−
(1
−
R
2
)
n
−
1
n
−
k
(7.8.4)
It is immediately apparent from Eq. (7.8.4) that (1) for 
k
>
1,
¯
R
2
<
R
2
which implies that
as the number of 
X
variables increases, the adjusted 
R
2
increases less than the unadjusted
R
2
; and (2) 
¯
R
2
can be negative, although 
R
2
is necessarily nonnegative.
10
In case 
¯
R
2
turns
out to be negative in an application, its value is taken as zero.
Which 
R
2
should one use in practice? As Theil notes:
. . . it is good practice to use
¯
R
2
rather than
R
2
because
R
2
tends to give an overly optimistic picture
of the fit of the regression, particularly when the number of explanatory variables is not very small
compared with the number of observations.
11
But Theil’s view is not uniformly shared, for he has offered no general theoretical justifica-
tion for the “superiority’’ of 
¯
R
2
.
For example, Goldberger argues that the following 
R
2
, call
it