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

random
or 
stochastic regressors.
Now the situation is rather involved. The 
u
i
, by
1
G. Barrie Wetherill, 
Regression Analysis with Applications
, Chapman and Hall, New York, 1986, 
pp. 14–15.
guj75772_ch10.qxd 23/08/2008 04:47 PM Page 316

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Part Two
Relaxing the Assumptions of the Classical Model
317
assumption, are stochastic. If the 
X
’s too are stochastic, then we must specify how the 
X
’s
and 
u
i
are distributed. If we are willing to make Assumption 2 (i.e., the 
X
’s, although ran-
dom, are distributed independently of, or at least uncorrelated with, 
u
i
), then for all practi-
cal purposes we can continue to operate as if the 
X
’s were nonstochastic. As Kmenta notes: 
Thus, 
relaxing the assumption that X is nonstochastic and replacing it by the assumption that
X is stochastic but independent of
[
u
] 
does not change the desirable properties and feasibility
of least squares estimation.
2
Therefore, we will retain Assumption 2 until we come to deal with simultaneous equa-
tions models in Part 4.
3 
Also, a brief discussion of nonstochastic regressors will be given in
Chapter 13.
Assumption 3: Zero Mean Value of u
i
Recall the 
k
-variable linear regression model:
Y
i
=
β
1
+
β
2
X
2
i
+
β
3
X
3
i
+ · · · +
β
k
X
ki
+
u
i
(1)
Let us now assume that
E
(
u
i
|
X
2
i
,
X
3
i
,
. . .
,
X
ki
)
=
w
(2)
where 
w
is a constant; note in the standard model 
w
=
0, but now we let it be any constant.
Taking the conditional expectation of Eq.(1), we obtain
E
(
Y
i
|
X
2
i
,
X
3
i
,
. . .
,
X
ki
)
=
β
1
+
β
2
X
2
i
+
β
3
X
3
i
+ · · · +
β
k
X
ki
+
w
=
(
β
1
+
w
)
+
β
2
X
2
i
+
β
3
X
3
i
+ · · · +
β
k
X
ki
(3)
=
α
+
β
2
X
2
i
+
β
3
X
3
i
+ · · · +
β
k
X
ki
where 
α
=
(
β
1
+
w
) and where in taking the expectations one should note that the 
X
’s are
treated as constants. (Why?)
Therefore, if Assumption 3 is not fulfilled, we see that we cannot estimate the original
intercept 
β
1
; what we obtain is 
α
, which contains 
β
1
and 
E
(
u
i
)
=
w
. In short, we obtain a
biased
estimate of 
β
1
.
But as we have noted on many occasions, in many practical situations the intercept term,
β
1
, is of little importance; the more meaningful quantities are the slope coefficients, which
remain unaffected even if Assumption 3 is violated.
4
Besides, in many applications the
intercept term has no physical interpretation.
2
Jan Kmenta, 
Elements of Econometrics,
2d ed., Macmillan, New York, 1986, p. 338. (Emphasis in the
original.)
3
A technical point may be noted here. Instead of the strong assumption that the 
X
’s and 
u
are inde-
pendent, we may use the weaker assumption that the values of 
X 
variables and
u
are uncorrelated
contemporaneously (i.e., at the same point in time). In this case OLS estimators may be biased but
they are