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

Appendix A
for details):
(8.5.4)
If we substitute the null hypothesis and the expression for the se (
ˆ
β
3
− ˆ
β
4
) into 
Eq. (8.5.3), our test statistic becomes
t
=
ˆ
β
3
− ˆ
β
4
var (
ˆ
β
3
)
+
var (
ˆ
β
4
)
−
2 cov (
ˆ
β
3
,
ˆ
β
4
)
(8.5.5)
Now the testing procedure involves the following steps:
1. Estimate 
ˆ
β
3
and 
ˆ
β
4
. Any standard computer package can do that.
2. Most standard computer packages routinely compute the variances and covariances of
the estimated parameters.
11
From these estimates the standard error in the denominator
of Eq. (8.5.5) can be easily obtained.
3. Obtain the 
t
ratio from Eq. (8.5.5). Note the null hypothesis in the present case is
(
β
3
−
β
4
)
=
0.
4. If the 
t
variable computed from Eq. (8.5.5) exceeds the critical 
t
value at the designated
level of significance for given df, then you can reject the null hypothesis; otherwise, you
do not reject it. Alternatively, if the 
p
value of the 
t
statistic from Eq. (8.5.5) is reason-
ably low, one can reject the null hypothesis. Note that the lower the 
p
value, the greater
the evidence against the null hypothesis. Therefore, when we say that a 
p
value is low or
reasonably low, we mean that it is less than the significance level, such as 10, 5, or 1 per-
cent. Some personal judgment is involved in this decision.
se (
ˆ
β
3
− ˆ
β
4
)
=
var (
ˆ
β
3
)
+
var (
ˆ
β
4
)
−
2 cov (
ˆ
β
3
,
ˆ
β
4
)
t
=
(
ˆ
β
3
− ˆ
β
4
)
−
(
β
3
−
β
4
)
se (
ˆ
β
3
− ˆ
β
4
)
11
The algebraic expression for the covariance formula is rather involved. 
Appendix C
provides a
compact expression for it, however, using matrix notation.
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