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

248
Part One
Single-Equation Regression Models
8.6
Restricted Least Squares: Testing Linear Equality Restrictions
There are occasions where economic theory may suggest that the coefficients in a regression
model satisfy some linear equality restrictions. For instance, consider the Cobb–Douglas
production function:
Y
i
=
β
1
X
β
2
2
i
X
β
3
3
i
e
u
i
(7.9.1) 
=
(8.6.1)
where 
Y
=
output, 
X
2
=
labor input, and 
X
3
=
capital input. Written in log form, the equa-
tion becomes
ln
Y
i
=
β
0
+
β
2
ln
X
2
i
+
β
3
ln
X
3
i
+
u
i
(8.6.2)
where 
β
0
=
ln
β
1
.
Now if there are constant returns to scale (equiproportional change in output for an
equiproportional change in the inputs), economic theory would suggest that
β
2
+
β
3
=
1
(8.6.3)
which is an example of a linear equality restriction.
12
How does one find out if there are constant returns to scale, that is, if the restriction
(8.6.3) is valid? There are two approaches.
EXAMPLE 8.2
The Cubic Cost
Function
Revisited
Recall the cubic total cost function estimated in Example 7.4, Section 7.10, which for con-
venience is reproduced below:
ˆ
Y
i
=
141.7667
+
63.4777
X
i
−
12.9615
X
2
i
+
0.9396
X
i
3
se
=
(6.3753)
(4.7786)
(0.9857)
(0.0591)