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

Part One
Single-Equation Regression Models
we established a 95 percent confidence interval for 
β
2
. But if we use the same sample data
to establish a confidence interval for
β
3
, say, with a confidence coefficient of 95 percent, we
cannot assert that both 
β
2
and 
β
3
lie in their respective confidence intervals with a proba-
bility of (1
−
α
)(1
−
α
)
=
(0
.
95)(0
.
95).
In other words, although the statements
Pr [
ˆ
β
2
−
t
α/
2
se (
ˆ
β
2
)
≤
β
2
≤ ˆ
β
2
+
t
α/
2
se (
ˆ
β
2
)]
=
1
−
α
Pr [
ˆ
β
3
−
t
α/
2
se (
ˆ
β
3
)
≤
β
3
≤ ˆ
β
3
+
t
α/
2
se (
ˆ
β
3
)]
=
1
−
α
are individually true, 
it is not true 
that the probability that the intervals
[
ˆ
β
2
±
t
α/
2
se (
ˆ
β
2
),
ˆ
β
3
±
t
α/
2
se (
ˆ
β
3
)]
simultaneously include 
β
2
and 
β
3
is (1
−
α
)
2
, because the intervals may not be indepen-
dent when the same data are used to derive them. To state the matter differently,
. . . testing a series of single [individual] hypotheses is 
not
equivalent to testing those same
hypotheses jointly. The intuitive reason for this is that in a joint test of several hypotheses any
single hypothesis is “affected’’ by the information in the other hypotheses.
5
The upshot of the preceding argument is that for a given example (sample) only one con-
fidence interval or only one test of significance can be obtained. How, then, does one test
the simultaneous null hypothesis that 
β
2
=
β
3
=
0? The answer follows.