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

449
12.12
Additional Aspects of Autocorrelation
Dummy Variables and Autocorrelation
In Chapter 9 we considered dummy variable regression models. In particular, recall the
U.S. savings–income regression model for 1970–1995 that we presented in Eq. (9.5.1),
which for convenience is reproduced below:
Y
t
=
α
1
+
α
2
D
t
+
β
1
X
t
+
β
2
(
D
t
X
t
)
+
u
t
(12.12.1)
where
Y
=
savings
X
=
income
D
=
1 for observations in period 1982–1995
D
=
0 for observations in period 1970–1981
The regression results based on this model are given in Eq. (9.5.4). Of course, this model
was estimated with the usual OLS assumptions.
But now suppose that 
u
t
follows a first-order autoregressive, AR(1), scheme. That is,
u
t
=
ρ
u
t
−
1
+
ε
t
. Ordinarily, if 
ρ
is known or can be estimated by one of the methods dis-
cussed above, we can use the generalized difference method to estimate the parameters of
the model that is free from (first-order) autocorrelation. However, the presence of the
dummy variable 
D
poses a special problem: Note that the dummy variable simply classifies
an observation as belonging to the first or second period. How do we transform it? One can
follow the following procedure.
46
1. In Eq. (12.12.1), values of 
D
are zero for all observations in the first period; in period
2 the value of 
D
for the 
first
observation is 1
/
(1
−
ρ
) instead of 1, and 1 for all other
observations.
2. The variable
X
t
is transformed as (
X
t
−
ρ
X
t
−
1
). Note that we lose one observation in
this transformation, unless one resorts to