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

8.5
Testing the Equality of Two Regression Coefficients
Suppose in the multiple regression
Y
i
=
β
1
+
β
2
X
2
i
+
β
3
X
3
i
+
β
4
X
4
i
+
u
i
(8.5.1)
10
For proof, see Dennis J. Aigner, 
Basic Econometrics,
Prentice Hall, Englewood Cliffs, NJ, 1971,
pp. 91–92.
guj75772_ch08.qxd 12/08/2008 10:03 AM Page 246

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Chapter 8
Multiple Regression Analysis: The Problem of Inference
247
we want to test the hypotheses
H
0
:
β
3
=
β
4
or
(
β
3
−
β
4
)
=
0
H
1
:
β
3
=
β
4
or
(
β
3
−
β
4
)
=
0
(8.5.2)
that is, the two slope coefficients 
β
3
and 
β
4
are equal.
Such a null hypothesis is of practical importance. For example, let Eq. (8.5.1) represent
the demand function for a commodity where
Y
=
amount of a commodity demanded,
X
2
=
price of the commodity,
X
3
=
income of the consumer, and
X
4
=
wealth of the consumer.
The null hypothesis in this case means that the income and wealth coefficients are the same.
Or, if
Y
i
and the
X
’s are expressed in logarithmic form, the null hypothesis in Eq. (8.5.2) im-
plies that the income and wealth elasticities of consumption are the same. (Why?)
How do we test such a null hypothesis? Under the classical assumptions, it can be shown
that
(8.5.3)
follows the
t
distribution with (
n
−
4) df because Eq. (8.5.1) is a four-variable model or,
more generally, with (
n
−
k
) df, where
k
is the total number of parameters estimated,
including the constant term. The se (
ˆ
β
3
− ˆ
β
4
) is obtained from the following well-known
formula (see