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

consistent,
that is, as the sample size increases indefinitely, the estimators converge on
their true values. If, however, the 
X
’s and 
u
are contemporaneously correlated, the OLS estimators
are biased as well as inconsistent. In Chapter 17 we will show how the method of 
instrumental
variables 
can sometimes be used to obtain consistent estimators in this situation.
4
It is very important to note that this statement is true only if
E
(
u
i
)
=
w
for each
i
. However, if
E
(
u
i
)
=
w
i
,
that is, a different constant for each 
i
, the partial slope coefficients may be biased as well as inconsis-
tent. In this case violation of Assumption 3 will be critical. For proof and further details, see Peter
Schmidt, 
Econometrics
, Marcel Dekker, New York, 1976, pp. 36–39.
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Part Two
Relaxing the Assumptions of the Classical Model
Assumption 10: Normality of u
This assumption is not essential if our objective is estimation only. As noted in Chapter 3,
the OLS estimators are BLUE regardless of whether the 
u
i
are normally distributed or not.
With the normality assumption, however, we were able to establish that the OLS estimators
of the regression coefficients follow the normal distribution, that (
n
−
k
)
ˆ
σ
2
/σ
2
has the 
χ
2
distribution, and that one could use the 
t
and 
F
tests to test various statistical hypotheses re-
gardless of the sample size.
But what happens if the 
u
i
are not normally distributed? We then rely on the following
extension of the central limit theorem; recall that it was the central limit theorem we in-
voked to justify the normality assumption in the first place:
If the disturbances [
u
i
] are independently and identically distributed with zero mean and
[constant] variance 
σ
2
and if the explanatory variables are constant in repeated samples, the
[O]LS coefficient estimators are asymptotically normally distributed with means equal to the
corresponding 
β
’s.
5
Therefore, the usual test procedures—the 
t
and 
F
tests—are still valid 
asymptotically,
that is, in the large sample, but not in the finite or small samples.
The fact that if the disturbances are not normally distributed the OLS estimators are still
normally distributed asymptotically (under the assumption of homoscedastic variance and
fixed
X
’s) is of little comfort to practicing economists, who often do not have the luxury of
large-sample data. Therefore, the normality assumption becomes extremely important for
the purposes of hypothesis testing and prediction. Hence, with the twin problems of estima-
tion and hypothesis testing in mind, and given the fact that small samples are the rule rather
than the exception in most economic analyses, we shall continue to use the normality
assumption.
6 
(But see Chapter 13, Section 13.12.)
Of course, this means that when we deal with a finite sample, we must explicitly test for
the normality assumption. We have already considered the