Pr [Pi tai se Pi Pi Pi tai se Pi 1 a

which gives the interval in which p2 will fall with 1 — a probability, given p2 = p2. In the language of hypothesis testing, the 100(1 — a)% confidence interval established in (5.7.2) is known as the region of acceptance (of

9Details may be found in E. L. Lehman, Testing Statistical Hypotheses, John Wiley & Sons, New York, 1959.

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the null hypothesis) and the region(s) outside the confidence interval is (are) called the region(s) of rejection (of H0) or the critical region(s). As noted previously, the confidence limits, the endpoints of the confidence interval, are also called critical values.

The intimate connection between the confidence-interval and test-of-significance approaches to hypothesis testing can now be seen by comparing (5.3.5) with (5.7.2). In the confidence-interval procedure we try to establish a range or an interval that has a certain probability of including the true but unknown ß2, whereas in the test-of-significance approach we hypothesize some value for ß2 and try to see whether the computed fa lies within reasonable (confidence) limits around the hypothesized value.

Once again let us revert to our consumption-income example. We know that fa = 0.5091, se (fa) = 0.0357, and df = 8. If we assume a = 5 percent, ta/2 = 2.306. If we let H:ß2 = ß2* = 0.3 and Hi:ß2 = 0.3, (5.7.2) becomes

as shown diagrammatically in Figure 5.3. Since the observed fa lies in the critical region, we reject the null hypothesis that true ß2 = 0.3.

In practice, there is no need to estimate (5.7.2) explicitly. One can compute the t value in the middle of the double inequality given by (5.7.1) and see whether it lies between the critical t values or outside them. For our example, t= ^^ = 586 <5.7.4,

FIGURE 5.3 The 95% confidence interval for ¡2 under the hypothesis that ¡2 = 0.3.

10In Sec. 5.2, point 4, it was stated that we cannot say that the probability is 95 percent that the fixed interval (0.4268, 0.5914) includes the true ¡2. But we can make the probabilistic statement given in (5.7.3) because ¡¡¡2, being an estimator, is a random variable.

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FIGURE 5.4 The 95% confidence interval for f(8 df).

which clearly lies in the critical region of Figure 5.4. The conclusion remains the same; namely, we reject H0.

Notice that if the estimated ft2 (= ft2) is equal to the hypothesized ft2, the t value in (5.7.4) will be zero. However, as the estimated ft2 value departs from the hypothesized ft2 value, |t| (that is, the absolute t value; note: t can be positive as well as negative) will be increasingly large. Therefore, a "large" |t| value will be evidence against the null hypothesis. Of course, we can always use the t table to determine whether a particular t value is large or small; the answer, as we know, depends on the degrees of freedom as well as on the probability of Type I error that we are willing to accept. If you take a look at the t table given in Appendix D, you will observe that for any given value of df the probability of obtaining an increasingly large | t| value becomes progressively smaller. Thus, for 20 df the probability of obtaining a |t| value of 1.725 or greater is 0.10 or 10 percent, but for the same df the probability of obtaining a |t| value of 3.552 or greater is only 0.002 or 0.2 percent.

Since we use the t distribution, the preceding testing procedure is called appropriately the t test. In the language of significance tests, a statistic is said to be statistically significant if the value of the test statistic lies in the critical region. In this case the null hypothesis is rejected. By the same token, a test is said to be statistically insignificant if the value of the test statistic lies in the acceptance region. In this situation, the null hypothesis is not rejected. In our example, the t test is significant and hence we reject the null hypothesis.

Before concluding our discussion of hypothesis testing, note that the testing procedure just outlined is known as a two-sided, or two-tail, test-of-significance procedure in that we consider the two extreme tails of the relevant probability distribution, the rejection regions, and reject the null hypothesis if it lies in either tail. But this happens because our H1 was a

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two-sided composite hypothesis; fa = 0.3 means fa is either greater than or less than 0.3. But suppose prior experience suggests to us that the MPC is expected to be greater than 0.3. In this case we have: H0: fa < 0.3 and Hi: fa > 0.3. Although Hi is still a composite hypothesis, it is now one-sided. To test this hypothesis, we use the one-tail test (the right tail), as shown in Figure 5.5. (See also the discussion in Section 5.6.)

The test procedure is the same as before except that the upper confidence limit or critical value now corresponds to ta = t,05, that is, the 5 percent level. As Figure 5.5 shows, we need not consider the lower tail of the t distribution in this case. Whether one uses a two- or one-tail test of significance will depend upon how the alternative hypothesis is formulated, which, in turn, may depend upon some a priori considerations or prior empirical experience. (But more on this in Section 5.8.)

We can summarize the t test of significance approach to hypothesis testing as shown in Table 5.1.

FIGURE 5.5 One-tail test of significance.

FIGURE 5.5 One-tail test of significance.

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TABLE 5.1 THE t TEST OF SIGNIFICANCE: DECISION RULES

Type of

H0: the null

H1: the alternative

Decision rule:

hypothesis

hypothesis

hypothesis

reject H0 if

Two-tail

f 2 = f2

f 2 = f 2

lt 1 > ta/2,df

Right-tail

f 2 < f2

f 2 > f2

t > ta,df

Left-tail

f 2 > f2

f 2 < 02

t < ta,df

Notes: f* is the hypothesized numerical value of f2. if| means the absolute value of t.

ta or f„/2 means the critical t value at the a or a/2 level of significance. df: degrees of freedom, (n - 2) for the two-variable model, (n - 3) for the three-variable model, and so on.

The same procedure holds to test hypotheses about f 1.

Notes: f* is the hypothesized numerical value of f2. if| means the absolute value of t.

ta or f„/2 means the critical t value at the a or a/2 level of significance. df: degrees of freedom, (n - 2) for the two-variable model, (n - 3) for the three-variable model, and so on.

The same procedure holds to test hypotheses about f 1.

Testing the Significance of a2: The x2 Test

As another illustration of the test-of-significance methodology, consider the following variable:

which, as noted previously, follows the x 2 distribution with n — 2 df. For the hypothetical example, a2 = 42.1591 and df = 8. If we postulate that H0: a2 = 85 vs. H1: a2 = 85, Eq. (5.4.1) provides the test statistic for H0. Substituting the appropriate values in (5.4.1), it can be found that under H0, x2 = 3.97. If we assume a = 5%, the critical x2 values are 2.1797 and 17.5346. Since the computed x 2 lies between these limits, the data support the null hypothesis and we do not reject it. (See Figure 5.1.) This test procedure is called the chi-square test of significance. The x2 test of significance approach to hypothesis testing is summarized in Table 5.2.

TABLE 5.2 A SUMMARY OF THE z2 TEST

H0: the null

H1: the alternative

Critical region:

hypothesis

hypothesis

reject H0 if

22 a2 = a0

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