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

OLS Estimation Disregarding Heteroscedasticity
The situation can become serious if we not only use 
ˆ
β
2
but also continue to use the usual
(homoscedastic) variance formula given in Eq. (11.2.3) even if heteroscedasticity is present
or suspected: Note that this is the more likely case of the two we discuss here, because
running a standard OLS regression package and ignoring (or being ignorant of )
heteroscedasticity will yield variance of 
ˆ
β
2
as given in Eq. (11.2.3). First of all, var (
ˆ
β
2
)
given in Eq. (11.2.3) is a 
biased
estimator of var (
ˆ
β
2
) given in Eq. (11.2.2), that is, on the
X
Y
A
B
Y
i
=
β
β
1
+
2
X
i
u
u
C
0
u
{
FIGURE 11.7
Hypothetical
scattergram.
5
A formal proof can be found in Phoebus J. Dhrymes, 
Introductory Econometrics
, Springer-Verlag, 
New York, 1978, pp. 110–111. In passing, note that the loss of efficiency of 
ˆ
β
2
(i.e., by how much
var [
ˆ
β
2
] exceeds var [
ˆ
β
∗
2
]) depends on the sample values of the 
X
variables and the value of 
σ
2
i
.
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Chapter 11
Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
375
average it overestimates or underestimates the latter, and 
in general
we cannot tell whether
the bias is positive (overestimation) or negative (underestimation) because it depends on
the nature of the relationship between 
σ
2
i
and the values taken by the explanatory variable
X
, as can be seen clearly from Eq. (11.2.2) (see Exercise 11.9). The bias arises from the fact
that 
ˆ
σ
2
, the conventional estimator of 
σ
2
, namely, 
ˆ
u
2
i
/
(
n
−
2) is no longer an unbiased
estimator of the latter when heteroscedasticity is present (see Appendix 11A.3). As a result,
we can no longer rely on the conventionally computed confidence intervals and the
conventionally employed 
t
and 
F
tests.
6