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

Kruskal’s theorem,
mentioned in the previous chapter. Therefore, in
some cases it can happen that OLS is BLUE despite autocorrelation. But such cases are
infrequent in practice.
What happens if we blithely continue to work with the usual OLS procedure despite
autocorrelation? The answer is provided in the following section.
(12.3.2)
var
ˆ
β
GLS
2
=
σ
2
n
t
=
2
(
x
t
−
ρ
x
t
−
1
)
2
+
D
(12.3.1)
ˆ
β
GLS
2
=
n
t
=
2
(
x
t
−
ρ
x
t
−
1
)(
y
t
−
ρ
y
t
−
1
)
n
t
=
2
(
x
t
−
ρ
x
t
−
1
)
2
+
C
422
Part Two
Relaxing the Assumptions of the Classical Model
11
For proofs, see Jan Kmenta, 
Elements of Econometrics
, Macmillan, New York, 1971, pp. 274–275.
The correction factor 
C
pertains to the first observation, (
Y
1
,
X
1
). On this point see Exercise 12.18.
12
The formal proof that 
ˆ
β
GLS
2
is BLUE can be found in Kmenta, ibid. But the tedious algebraic proof
can be simplified considerably using matrix notation. See J. Johnston, 
Econometric Methods,
3d ed.,
McGraw-Hill, New York, 1984, pp. 291–293.
guj75772_ch12.qxd 14/08/2008 10:40 AM Page 422

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Chapter 12
Autocorrelation: What Happens If the Error Terms Are Correlated?