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

distributed-lag model
(see Chapter 17).
Model B postulates that PPCE depends on current PDPI as well as PPCE in the previous time
period; this model represents what is known as the
autoregressive model
(see Chapter 17
again). The reason for introducing the lagged value of PPCE in this model is to reflect iner-
tia or habit persistence.
The results of estimating these models separately were as follows:
Model A:
PPCE
t
= −
606.6347 
+
0.6170 PDPI
t
+
0.3530 PDPI
t
−
1
t
=
(
−
3.8334)
(2.5706)
(1.4377)
(13.8.9)
R
2
=
0.9983
d
=
0.2161
Model B:
PPCE
t
=
76.8947 
+
0.2074 PDPI
t
+
0.8104 PPCE
t
−
1
t
=
(0.7256)
(2.6734)
(9.7343)
(13.8.10)
R
2
=
0.9996
d
=
0.9732
Year
PPCE
PDPI
Year
PPCE
PDPI
1970
3,162
3,587
1988
13,685
15,297
1971
3,379
3,860
1989
14,546
16,257
1972
3,671
4,140
1990
15,349
17,131
1973
4,022
4,616
1991
15,722
17,609
1974
4,364
5,010
1992
16,485
18,494
1975
4,789
5,498
1993
17,204
18,872
1976
5,282
5,972
1994
18,004
19,555
1977
5,804
6,517
1995
18,665
20,287
1978
6,417
7,224
1996
19,490
21,091
1979
7,073
7,967
1997
20,323
21,940
1980
7,716
8,822
1998
21,291
23,161
1981
8,439
9,765
1999
22,491
23,968
1982
8,945
10,426
2000
23,862
25,472
1983
9,775
11,131
2001
24,722
26,235
1984
10,589
12,319
2002
25,501
27,164
1985
11,406
13,037
2003
26,463
28,039
1986
12,048
13,649
2004
27,937
29,536
1987
12,766
14,241
2005
29,468
30,458
TABLE 13.3
Per Capita Personal
Consumption
Expenditure
(PPCE) and per
Capita Personal
Disposable Income
(PDPI), U.S.,
1970–2005
Source: 
Economic Report of
the President
, 2007.
(
Continued
)
guj75772_ch13.qxd 19/08/2008 12:06 PM Page 491

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492
Part Two
Relaxing the Assumptions of the Classical Model
EXAMPLE 13.4
(
Continued
)
If one were to choose between these two models on the basis of the discrimination
approach, using the highest 
R
2
criterion, one would probably choose Model B (13.8.10)
because it is just slightly higher than Model A (13.8.9). Also, in Model B (13.8.10), both
variables are individually statistically significant, whereas in Model A (13.8.9) only the
current PDPI is statistically significant (there might be a collinearity problem, though). For
predictive purposes, there is not much difference between the two estimated 
R
2
values,
though.
To apply the 
J
test, suppose we assume Model A is the null hypothesis, or the main-
tained model, and Model B is the alternative hypothesis. Following the 
J
test steps
discussed earlier, we use the estimated PPCE values from model (13.8.10) as an additional
regressor in Model A. The following is the outcome from this regression:
PPCE
t
= −
35.17
+
0.2762 PDPI
t
−
0.5141PDPI
t
−
1
+
1.2351
PPCE
B
t
t
=
(
−
0.43)
(2.60)
(
−
4.05) (12.06)
(13.8.11)
R
2
=
1.00
d
=
1.5205
where 
PPCE
B
t
on the right-hand side of Eq. (13.8.11) represents the estimated PPCE values
from the original Model B (13.8.10). Since the coefficient of this variable is statistically
significant with a very high 
t
-statistic of 12.06, following the 
J
test procedure we have to
reject Model A in favor of Model B.
Now we will assume Model B is the maintained hypothesis and Model A is the alterna-
tive. Following the exact same procedure, we obtain the following results:
PPCE
t
= −
823.7
+
1.4309 PDPI
t
+
1.0009 PPCE
t
−
1
−
1.4563
PPCE
A
t
t
=
(
−
3.45)
(4.64)
(12.06)
(
−
4.05)
(13.8.12)
R
2
=
1.00
d
=
1.5205
where 
PPCE
t
A
on the right-hand side of Eq. (13.8.12) represents the estimated PPCE values
from the original Model A (13.8.9). In this regression, the coefficient of 
PPCE
t
A
is also sta-
tistically significant with a 
t
-statistic of 
−
4.05. This result suggests that we should now
reject Model B in favor of Model A.
All this tells us is that neither model is particularly useful in explaining the behavior of
per capita personal consumption expenditure in the United States over the period
1970–2005. Of course, we have considered only two competing models. In reality, there
may be more than two models. The 
J
test procedure can be extended to multiple model
comparisons, although the analysis can quickly become complex.
This example shows very vividly why the CLRM assumes that the regression model
used in the analysis is correctly specified. Obviously, in developing a model it is crucial to
pay very careful attention to the phenomenon being modeled.
Other Tests of Model Selection
The 
J
test just discussed is only one of a group of tests of model selection. There is the