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

Appendix A
for
details of conditional and unconditional variances).
ASSUMPTION 5
No Autocorrelation between the Disturbances:
Given any two 
X
values, 
X
i
and 
X
j
(
i
j
), the correlation between any two 
u
i
and 
u
j
(
i
j
) is zero. In short, the observations
are sampled independently. Symbolically,
cov (
u
i
, 
u
j
|
X
i
, 
X
j
) 
=
0
(3.2.5)
cov (
u
i
, 
u
j
) 
=
0, if 
X
is nonstochastic
where 
i
and 
j
are two different observations and where cov means covariance.
In words, Equation 3.2.5 postulates that the disturbances 
u
i
and 
u
j
are uncorrelated.
Technically, this is the assumption of 
no serial correlation,
or 
no autocorrelation.
This
means that, given 
X
i
, the deviations of any two 
Y
values from their mean value do not
exhibit patterns such as those shown in Figures 3.6(
a
) and (
b
). In Figure 3.6(
a
), we see that
the 
u
’s are positively correlated, a positive 
u
followed by a positive 
u
or a negative 
u
followed by a negative 
u
. In Figure 3.6(
b
), the 
u
’s are 
negatively correlated,
a positive 
u
followed by a negative 
u
and vice versa.
If the disturbances (deviations) follow systematic patterns, such as those shown in Fig-
ures 3.6(
a
) and (
b
), there is auto- or serial correlation, and what Assumption 5 requires is
that such correlations be absent. Figure 3.6(
c
) shows that there is no systematic pattern to
the 
u
’s, thus indicating zero correlation.
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Chapter 3
Two-Variable Regression Model: The Problem of Estimation
67
The full import of this assumption will be explained thoroughly in Chapter 12. But
intuitively one can explain this assumption as follows. Suppose in our PRF (
Y
t
=
β
1
+
β
2
X
t
+
u
t
) that 
u
t
and 
u
t
−
1
are positively correlated. Then 
Y
t
depends not only on 
X
t
but also on 
u
t
−
1
,
for 
u
t
−
1
to some extent determines 
u
t
. At this stage of the development of the subject mat-
ter, by invoking Assumption 5, we are saying that we will consider the systematic effect, if
any, of 
X
t
on 
Y
t
and not worry about the other influences that might act on 
Y
as a result of
the possible intercorrelations among the 
u
’s. But, as noted in Chapter 12, we will see how
intercorrelations among the disturbances can be brought into the analysis and with what
consequences.
But it should be added here that the justification of this assumption depends on the type
of data used in the analysis. If the data are cross-sectional and are obtained as a random
sample from the relevant population, this assumption can often be justified. However, if the
data are time series, the assumption of independence is difficult to maintain, for successive
observations of a time series, such as GDP, are highly correlated. But we will deal with this
situation when we discuss time series econometrics later in the text.
+
u
i
–
u
i
+
u
i
–u
i
+
u
i
–
u
i
+
u
i
–u
i
+
u
i
–
u
i
+
u
i
–u
i
(
a
)
(
b
)
(
c
)
FIGURE 3.6
Patterns of correlation
among the
disturbances.
(
a
) positive serial
correlation;
(
b
) negative serial
correlation; (
c
) zero
correlation.