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

Part Two
Relaxing the Assumptions of the Classical Model
transform data to reflect specific types of heteroscedasticity. To illustrate this, let us revert
to the two-variable regression model:
Y
i
=
β
1
+
β
2
X
i
+
u
i
We now consider several assumptions about the pattern of heteroscedasticity.
The error variance is proportional to 
X
2
i
:
E
u
2
i
=
σ
2
X
2
i
(11.6.5)
38
If, as a matter of “speculation,” graphical methods, or Park and Glejser approaches, it is
believed that the variance of 
u
i
is proportional to the square of the explanatory variable 
X
(see Figure 11.10), one may transform the original model as follows. Divide the original
model through by 
X
i
:
Y
i
X
i
=
β
1
X
i
+
β
2
+
u
i
X
i
=
β
1
1
X
i
+
β
2
+
v
i
(11.6.6)
where 
v
i
is the transformed disturbance term, equal to 
u
i
/
X
i
. Now it is easy to verify that
E
v
2
i
=
E
u
i
X
i
2
=
1
X
2
i
E
u
2
i
=
σ
2
using (11.6.5)
Hence the variance of 
v
i
is now homoscedastic, and one may proceed to apply OLS to the
transformed equation (11.6.6), regressing 
Y
i
/
X
i
on 1
/
X
i
.
38
Recall that we have already encountered this assumption in our discussion of the Goldfeld–Quandt
test.