Part One
Single-Equation Regression Models
This example provides us an opportunity to decide whether we want to use a one-tail
or a two-tail
t
test. Since a priori child mortality and per capita GNP are expected to be
negatively related (why?), we should use the one-tail test. That is, our null and alternative
hypothesis should be:
H
0
:
β
2
<
0
and
H
1
:
β
2
≥
0
As the reader knows by now, we can reject the null hypothesis on the basis of the one-tail
t
test in the present instance. If we can reject the null hypothesis in a two-sided test, we will
have enough evidence to reject in the one-sided scenario as long as the statistic is in the
same direction as the test.
In Chapter 5 we saw the intimate connection between hypothesis testing and confidence
interval estimation. For our example, the 95 percent confidence interval for
β
2
is:
which in our example becomes
−
0
.
0056
−
2(0
.
0020)
≤
β
2
≤ −
0
.
0056
+
2(0
.
0020)
that is,
−
0
.
0096
≤
β
2
≤ −
0
.
0016
(8.3.2)
that is, the interval,
−
0.0096 to
−
0.0016 includes the true
β
2
coefficient with 95 percent
confidence coefficient. Thus, if 100 samples of size 64 are selected and 100 confidence in-
tervals like Eq. (8.3.2) are constructed, we expect 95 of them to contain the true population
parameter
β
2
. Since the interval (8.3.2) does not include the null-hypothesized value of
zero, we can reject the null hypothesis that the true
β
2
is zero with 95 percent confidence.
Thus, whether we use the
t
test of significance as in (8.3.1) or the confidence interval
estimation as in (8.3.2), we reach the same conclusion. However, this should not be
surprising in view of the close connection between confidence interval estimation and
hypothesis testing.
Following the procedure just described, we can test hypotheses about the other parame-
ters of our child mortality regression model. The necessary data are already provided in
Eq. (8.1.4). For example, suppose we want to test the hypothesis that, with the influence of
PGNP held constant, the female literacy rate has no effect whatsoever on child mortality. We
can confidently reject this hypothesis, for under this null hypothesis the
p
value of obtaining
an absolute
t
value of as much as 10.6 or greater is practically zero.
Before moving on, remember that the
t
-testing procedure is based on the assumption
that the error term
u
i
follows the normal distribution. Although we cannot directly observe
ˆ
β
2
−
t
α/
2
se (
ˆ
β
2
)
≤
β
2
≤ ˆ
β
2
+
t
α/
2
se (
ˆ
β
2
)
Do'stlaringiz bilan baham: |