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

Akaike’s Information crite-
rion
and 
Amemiya’s Prediction criteria,
which are used to select between competing
models. We will discuss these criteria when we consider the problem of model selection in
greater detail in a later chapter (see Chapter 13).
Comparing Two 
R
2
Values
It is crucial to note that in comparing two models on the basis of the coefficient of deter-
mination, whether adjusted or not, 
the sample size n and the dependent variable must be the
same;
the explanatory variables may take any form. Thus for the models
ln
Y
i
=
β
1
+
β
2
X
2
i
+
β
3
X
3
i
+
u
i
(7.8.6)
Y
i
=
α
1
+
α
2
X
2
i
+
α
3
X
3
i
+
u
i
(7.8.7)
the computed 
R
2
terms cannot be compared. The reason is as follows: By definition,
R
2
measures the proportion of the variation in the dependent variable accounted for by the
explanatory variable(s). Therefore, in Eq. (7.8.6) 
R
2
measures the proportion of the 
varia-
tion in 
ln
Y
explained by 
X
2
and 
X
3
, whereas in Eq. (7.8.7) it measures the proportion of the
variation in Y,
and the two are not the same thing: As noted in Chapter 6, a change in ln
Y
gives a relative or proportional change in 
Y
, whereas a change in 
Y
gives an absolute
change. Therefore, var
ˆ
Y
i
/
var
Y
i
is not equal to var (
ln
Y
i
)
/
var (ln
Y
i
); that is, the two coef-
ficients of determination are not the same.
13
How then does one compare the 
R
2
’s of two models when the regressand is not in the
same form? To answer this question, let us first consider a numerical example.
13
From the definition of 
R
2
, we know that
1
−
R
2
=
RSS
TSS
=
ˆ
u
2
i
(
Y
i
− ¯
Y
)
2
for the linear model and
1
−
R
2
=
ˆ
u
2
i
(ln
Y
i
−
ln
Y
)
2
for the log model. Since the denominators on the right-hand sides of these expressions are different,
we cannot compare the two 
R
2
terms directly.
As shown in Example 7.2, for the linear specification, the RSS
=
0.1491 (the residual sum of
squares of coffee consumption), and for the log–linear specification, the RSS
=
0.0226 (the residual
sum of squares of log of coffee consumption). These residuals are of different orders of magnitude
and hence are not directly comparable.
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