μiμj
=E[(μ
i
– μu)(μ
j
– μu)] = E(μ
i
μ
j
) = E(μ
i
)E(μ
j
) = 0
…[2.29] where, u is a value in μ as shown in (
1
u
) of Figure 2.0 Note that the population means of μ
i
andμ
j
are 0, by the first Gauss–Markov condition, and that E(μ
i
μ
j
) can be decomposed as E(μ
i
)E(μ
j
) if μ
i
andμ
j
are generated independently. If this condition is not satisfied, OLS will again give inefficient estimates.
2.2.3.2.4 Gauss–Markov Condition 4: u Distributed Independently of the
Explanatory Variables
The final condition comes in two versions, weak and strong. The strong version is that the explanatory variables should be non-stochastic, that is, not have random components. This is very unrealistic for economic variables, and we will eventually switch to the weak version of the condition, where the explanatory variables are allowed to have random components provided that they are distributed independently of the disturbance term. However, the strong version is usually used because it simplifies the analysis of the properties of the estimators.
i
X
i
u
,*
i
X
(
i
X
)+*
i
u
+- (
i
X
i
X
) (
i
u
)
…[2.30]
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