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

weighted
least squares (WLS),
and the estimators thus obtained and given in Eqs. (11.3.8) and (11.3.9)
are known as
WLS estimators.
But WLS is just a special case of the more general estimating
technique, GLS. In the context of heteroscedasticity, one can treat the two terms WLS and
GLS interchangeably. In later chapters we will come across other special cases of GLS.
In passing, note that if 
w
i
=
w
, a constant for all 
i
,
ˆ
β
∗
2
is identical with 
ˆ
β
2
and var (
ˆ
β
∗
2
)
is identical with the usual (i.e., homoscedastic) var (
ˆ
β
2
) given in Eq. (11.2.3), which should
not be surprising. (Why?) (See Exercise 11.8.)
w
i
ˆ
u
2
i
=
w
i
(
Y
i
− ˆ
β
∗
1
X
0
i
− ˆ
β
∗
2
X
i
)
2
var (
ˆ
β
∗
2
)
=
w
i
w
i
w
i
X
2
i
−
w
i
X
i
2
ˆ
β
∗
2
=
w
i
w
i
X
i
Y
i
−
w
i
X
i
w
i
Y
i
w
i
w
i
X
2
i
−
w
i
X
i
2
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374
Part Two
Relaxing the Assumptions of the Classical Model
11.4
Consequences of Using OLS in the Presence 
of Heteroscedasticity
As we have seen, both 
ˆ
β
∗
2
and 
ˆ
β
2
are (linear) unbiased estimators: In repeated sampling, on
the average, 
ˆ
β
∗
2
and 
ˆ
β
2
will equal the true 
β
2
; that is, they are both unbiased estimators. But
we know that it is 
ˆ
β
∗
2
that is efficient, that is, has the smallest variance. What happens to our
confidence interval, hypotheses testing, and other procedures if we continue to use the OLS
estimator 
ˆ
β
2
? We distinguish two cases.
OLS Estimation Allowing for Heteroscedasticity
Suppose we use
ˆ
β
2
and use the variance formula given in Eq. (11.2.2), which takes into
account heteroscedasticity explicitly. Using this variance, and assuming
σ
2
i
are known, can
we establish confidence intervals and test hypotheses with the usual
t
and
F
tests? The
answer generally is no because it can be shown that var (
ˆ
β
∗
2
)
≤
var (
ˆ
β
2
),
5
which means that
confidence intervals based on the latter will be unnecessarily larger. As a result, the
t
and
F
tests are likely to give us inaccurate results in that var (
ˆ
β
2
) is overly large and what appears
to be a statistically insignificant coefficient (because the
t
value is smaller than what is
appropriate) may in fact be significant if the correct confidence intervals were established on
the basis of the GLS procedure.