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

(12.2.9)
If, for example, 
r
=
0.6 and 
ρ
=
0.8, using Eq. (12.2.9) we can check that var (
ˆ
β
2
)
AR1
=
2
.
8461 var (
ˆ
β
2
)
OLS
. To put it another way, var (
ˆ
β
2
)
OLS
=
1
2
.
8461
var (
ˆ
β
2
)
AR1
=
0
.
3513
var (
ˆ
β
2
)
AR1
. That is, the usual OLS formula (i.e., Eq. [12.2.7]) will underestimate the vari-
ance of (
ˆ
β
2
)
AR1
by about 65 percent. As you will realize, this answer is specific for the
given values of 
r
and 
ρ
. But the point of this exercise is to warn you that a blind application
of the usual OLS formulas to compute the variances and standard errors of the OLS
estimators could give seriously misleading results.
Suppose we continue to use the OLS estimator 
ˆ
β
2
and adjust the usual variance for-
mula by taking into account the AR(1) scheme. That is, we use 
ˆ
β
2
given by Eq. (12.2.6)
but use the variance formula given by Eq. (12.2.8). What now are the properties of 
ˆ
β
2
? It
is easy to prove that 
ˆ
β
2
is still linear and unbiased. As a matter of fact, as shown in Ap-
pendix 3A, Section 3A.2, the assumption of no serial correlation, like the assumption of
no heteroscedasticity, is not required to prove that 
ˆ
β
2
is unbiased. Is 
ˆ
β
2
still BLUE? Un-
fortunately, it is not; in the class of linear unbiased estimators, it does not have minimum
variance. In short, 
ˆ
β
2
, although linear-unbiased, is not efficient (relatively speaking, of
course). The reader will notice that this finding is quite similar to the finding that 
ˆ
β
2
is
less efficient in the presence of heteroscedasticity. There we saw that it was the weighted
least-square estimator 
ˆ
β
∗
2
given in Eq. (11.3.8), a special case of the generalized least-
squares (GLS) estimator, that was efficient. In the case of autocorrelation can we find an
estimator that is BLUE? The answer is yes, as can be seen from the discussion in the
following section.
10
Note that the term 
r
=
x
t
x
t
+
1
/
x
2
t
is the correlation between 
X
t
and 
X
t
+
1
(or 
X
t
−
1
, since the
correlation coefficient is symmetric); 
r
2
=
x
t
x
t
+
2
/
x
2
t
is the correlation between the 
X
’s lagged
two periods; and so on.
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