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

The message is: To establish confidence intervals and to test hypotheses, one should
use GLS and not OLS even though the estimators derived from the latter are unbiased
and consistent.
(However, see Section 12.11 later.)
13
But matrix algebra becomes almost a necessity to avoid tedious algebraic manipulations.
14
See Kmenta, op. cit., pp. 277–278.
GLS 95% interval 
OLS 95% interval
0
H
0
:
2
= 0
β
β
2
b
2
FIGURE 12.4
GLS and OLS 95%
confidence intervals.
OLS Estimation Disregarding Autocorrelation
The situation is potentially very serious if we not only use 
ˆ
β
2
but also continue to use
var (
ˆ
β
2
)
=
σ
2
/
x
2
t
, which completely disregards the problem of autocorrelation, that is,
we mistakenly believe that the usual assumptions of the classical model hold true. Errors
will arise for the following reasons:
1. The residual variance 
ˆ
σ
2
=
ˆ
u
2
t
/
(
n
−
2) is likely to underestimate the true 
σ
2
.
2. As a result, we are likely to overestimate 
R
2
.
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3. Even if 
σ
2
is not underestimated, var (
ˆ
β
2
) may underestimate var (
ˆ
β
2
)
AR1
(Eq. [12.2.8]),
its variance under (first-order) autocorrelation, even though the latter is inefficient com-
pared to var (
ˆ
β
2
)
GLS
.
4. Therefore, the usual 
t
and 
F
tests of significance are no longer valid, and if applied, are
likely to give seriously misleading conclusions about the statistical significance of the
estimated regression coefficients.
To establish some of these propositions, let us revert to the two-variable model. We
know from Chapter 3 that under the classical assumption 
ˆ
σ
2
=
ˆ
u
2
i
(
n
−
2)
provides an unbiased estimator of 
σ
2
, that is, 
E
(
ˆ
σ
2
)
=
σ
2
. But if there is autocorrelation,
given by AR(1), it can be shown that
E
(
ˆ
σ
2
)
=
σ
2
{
n
−
[2
/
(1
−
ρ
)]
−
2
ρ
r
}
n
−
2