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

(7.10.6)
cov (
ˆ
β
3
,
ˆ
β
4
)
= −
0
.
0576
;
R
2
=
0
.
9983
where 
Y
is total cost and 
X
is output, and where the figures in parentheses are the esti-
mated standard errors.
Suppose we want to test the hypothesis that the coefficients of the 
X
2
and 
X
3
terms in
the cubic cost function are the same, that is, 
β
3
=
β
4
or (
β
3
−
β
4
)
=
0. In the regression
(7.10.6) we have all the necessary output to conduct the 
t
test of Eq. (8.5.5). The actual
mechanics are as follows:
t
=
ˆ
β
3
− ˆ
β
4
var (
ˆ
β
3
)
+
var (
ˆ
β
4
)
−
2 cov (
ˆ
β
3
,
ˆ
β
4
)
=
−
12
.
9615
−
0
.
9396
(0
.
9867)
2
+
(0
.
0591)
2
−
2(
−
0
.
0576)
(8.5.6)
=
−
13
.
9011
1
.
0442
= −
13
.
3130
The reader can verify that for 6 df (why?) the observed 
t
value exceeds the critical 
t
value
even at the 0.002 (or 0.2 percent) level of significance (two-tail test); the 
p
value is ex-
tremely small, 0.000006. Hence we can reject the hypothesis that the coefficients of 
X
2
and 
X
3
in the cubic cost function are identical.
12
If we had 
β
2
+
β
3
<
1, this relation would be an example of a linear inequality restriction. To handle
such restrictions, one needs to use mathematical programming techniques.
guj75772_ch08.qxd 12/08/2008 10:03 AM Page 248

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Chapter 8
Multiple Regression Analysis: The Problem of Inference
249
The 
t
-Test Approach
The simplest procedure is to estimate Eq. (8.6.2) in the usual manner without taking into
account the restriction (8.6.3) explicitly. This is called the 
unrestricted
or 
unconstrained
regression.
Having estimated 
β
2
and 
β
3
(say, by the OLS method), a test of the hypothesis
or restriction (8.6.3) can be conducted by the 
t
test of Eq. (8.5.3), namely,
(8.6.4)
where (
β
2
+
β
3
)
=
1 under the null hypothesis and where the denominator is the standard
error of (
ˆ
β
2
+ ˆ
β
3
). Then following Section 8.5, if the 
t
value computed from Eq. (8.6.4) ex-
ceeds the critical 
t
value at the chosen level of significance, we reject the hypothesis of con-
stant returns to scale; otherwise we do not reject it.
The 
F
-Test Approach: Restricted Least Squares
The preceding 
t
test is a kind of postmortem examination because we try to find out whether
the linear restriction is satisfied after estimating the “unrestricted’’ regression. A direct ap-
proach would be to incorporate the restriction (8.6.3) into the estimating procedure at the
outset. In the present example, this procedure can be done easily. From (8.6.3) we see that
β
2
=
1
−
β
3