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

(13.5.4)
Model (13.5.3):
var (
ˆ
β
)
=
σ
2
v
x
2
i
=
σ
2
u
+
σ
2
ε
x
2
i
(13.5.5)
Obviously, the latter variance is larger than the former.
28
Therefore, 
although the errors of
measurement in the dependent variable still give unbiased estimates of the parame-
ters and their variances, the estimated variances are now larger than in the case where
there are no such errors of measurement.
Errors of Measurement in the Explanatory Variable 
X
Now assume that instead of Eq. (13.5.1), we have the following model:
Y
i
=
α
+
β
X
∗
i
+
u
i
(13.5.6)
where 
Y
i
=
current consumption expenditure
X
∗
i
=
permanent income
u
i
=
disturbance term (equation error)
27
This phrase is due to Milton Friedman. See also Exercise 13.8.
28
But note that this variance is still unbiased because under the stated conditions the composite error
term 
v
i
=
u
i
+
ε
i
still satisfies the assumptions underlying the method of least squares.
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Part Two
Relaxing the Assumptions of the Classical Model
Suppose instead of observing 
X
∗
i
, we observe
X
i
=
X
∗
i
+
w
i