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

first-order coefficient of autocorrelation,
or more accurately, the
coefficient of autocorrelation at lag 1.
9
Given the AR(1) scheme, it can be shown that (see Appendix 12A, Section 12A.2):
var (
u
t
)
=
E
u
2
t
=
σ
2
ε
1
−
ρ
2
(12.2.3)
cov (
u
t
,
u
t
+
s
)
=
E
(
u
t
u
t
−
s
)
=
ρ
s
σ
2
ε
1
−
ρ
2
(12.2.4)
cor (
u
t
,
u
t
+
s
)
=
ρ
s
(12.2.5)
where cov (
u
t
,
u
t
+
s
) means covariance between error terms 
s
periods apart and where
cor (
u
t
,
u
t
+
s
) means correlation between error terms 
s
periods apart. Note that because of
the symmetry property of covariances and correlations, cov (
u
t
,
u
t
+
s
)
=
cov (
u
t
,
u
t
−
s
) and
cor (
u
t
,
u
t
+
s
)
=
cor (
u
t
,
u
t
−
s
).
Since 
ρ
is a constant between 
−
1 and 
+
1, Eq. (12.2.3) shows that under the AR(1)
scheme, the variance of 
u
t
is 
still homoscedastic
, but 
u
t
is correlated not only with its im-
mediate past value but its values several periods in the past. It is 
critical
to note that
|
ρ
|
<
1, that is, the absolute value of 
ρ
is less than 1. If, for example, 
ρ
is 1, the variances
and covariances listed above are not defined. If 
|
ρ
|
<
1, we say that the AR(1) process
given in Eq. (12.2.1) is 
stationary;
that is, the mean, variance, and covariance of 
u
t
do not
change over time. If 
|
ρ
|
is less than 1, then it is clear from Eq. (12.2.4) that the value of the
covariance will decline as we go into the distant past. We will see the utility of the preced-
ing results shortly.
One reason we use the AR(1) process is not only because of its simplicity compared to
higher-order AR schemes, but also because in many applications it has proved to be quite
useful. Additionally, a considerable amount of theoretical and empirical work has been
done on the AR(1) scheme.
Now return to our two-variable regression model: 
Y
t
=
β
1
+
β
2
X
t
+
u
t
. We know from
Chapter 3 that the OLS estimator of the slope coefficient is
ˆ
β
2
=
x
t
y
t
x
2
t