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

12.3
The BLUE Estimator in the Presence of Autocorrelation
Continuing with the two-variable model and assuming the AR(1) process, we can show that
the BLUE estimator of 
β
2
is given by the following expression:
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where 
C
is a correction factor that may be disregarded in practice. Note that the subscript 
t
now runs from 
t
=
2 to 
t
=
n
. And its variance is given by
where 
D
too is a correction factor that may also be disregarded in practice. (See Exer-
cise 12.18.)
The estimator 
ˆ
β
GLS
2
, as the superscript suggests, is obtained by the method of GLS. As
noted in Chapter 11, in GLS we incorporate any additional information we have (e.g., the
nature of the heteroscedasticity or of the autocorrelation) directly into the estimating pro-
cedure by transforming the variables, whereas in OLS such side information is not directly
taken into consideration. As the reader can see, the GLS estimator of 
β
2
given in
Eq. (12.3.1) incorporates the autocorrelation parameter 
ρ
in the estimating formula,
whereas the OLS formula given in Eq. (12.2.6) simply neglects it. Intuitively, this is the rea-
son why the GLS estimator is BLUE and not the OLS estimator—the GLS estimator makes
the most use of the available information.
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It hardly needs to be added that if
ρ
=
0, there
is no additional information to be considered and hence both the GLS and OLS estimators
are identical.
In short,
under autocorrelation, it is the GLS estimator given in Eq. (12.3.1) that is
BLUE, and the minimum variance is now given by Eq. (12.3.2) and not by Eq. (12.2.8) and
obviously not by Eq. (12.2.7).
A Technical Note
As we noted in the previous chapter, the Gauss–Markov theorem provides only the suffi-
cient condition for OLS to be BLUE. The necessary and sufficient conditions for OLS to be
BLUE are given by