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

Part Two
Relaxing the Assumptions of the Classical Model
Now assume that the heteroscedastic variances 
σ
2
i
are 
known.
Divide Eq. (11.3.2)
through by 
σ
i
to obtain
Y
i
σ
i
=
β
1
X
0
i
σ
i
+
β
2
X
i
σ
i
+
u
i
σ
i
(11.3.3)
which for ease of exposition we write as
Y
∗
i
=
β
∗
1
X
∗
0
i
+
β
∗
2
X
∗
i
+
u
∗
i
(11.3.4)
where the starred, or transformed, variables are the original variables divided by (the known)
σ
i
. We use the notation
β
∗
1
and
β
∗
2
, the parameters of the transformed model, to distinguish
them from the usual OLS parameters
β
1
and
β
2
.
What is the purpose of transforming the original model? To see this, notice the follow-
ing feature of the transformed error term 
u
∗
i
:
var (
u
∗
i
)
=
E
(
u
∗
i
)
2
=
E
u
i
σ
i
2
since
E
(
u
∗
i
)
=
0
=
1
σ
2
i
E
u
2
i
since
σ
2
i
is known
=
1
σ
2
i
σ
2
i
since
E
u
2
i
=
σ
2
i
=
1
which is a constant. That is, the variance of the transformed disturbance term 
u
∗
i
is now ho-
moscedastic. Since we are still retaining the other assumptions of the classical model, the
finding that it is 
u
∗
that is homoscedastic suggests that if we apply OLS to the transformed
model (11.3.3) it will produce estimators that are BLUE. In short, the estimated 
β
∗
1
and 
β
∗
2
are now BLUE and not the OLS estimators 
ˆ
β
1
and 
ˆ
β
2
.
This procedure of transforming the original variables in such a way that the transformed
variables satisfy the assumptions of the classical model and then applying OLS to them is
known as the method of generalized least squares (GLS). 
In short, GLS is OLS on the trans-
formed variables that satisfy the standard least-squares assumptions.
The estimators thus
obtained are known as