## Hypothesis Testing About Individual Regression Coefficients

If we invoke the assumption that ui ~ N(0, a2), then, as noted in Section 8.1, we can use the t test to test a hypothesis about any individual partial regression coefficient. To illustrate the mechanics, consider the child mortality regression, (8.2.1). Let us postulate that

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The null hypothesis states that, with X3 (female literacy rate) held constant, X2 (PGNP) has no (linear) influence on Y(child mortality).2 To test the null hypothesis, we use the t test given in (8.1.2). Following Chapter 5 (see Table 5.1), if the computed t value exceeds the critical t value at the chosen level of significance, we may reject the null hypothesis; otherwise, we may not reject it. For our illustrative example, using (8.1.2) and noting that 02 = 0 under the null hypothesis, we obtain t=-00026=-28187 <-')

Notice that we have 64 observations. Therefore, the degrees of freedom in this example are 61 (why?). If you refer to the t table given in Appendix D, we do not have data corresponding to 61 df. The closest we have are for 60 df. If we use these df, and assume a, the level of significance (i.e., the probability of committing a Type I error) of 5 percent, the critical t value is 2.0 for a two-tail test (look up ta/2 for 60 df) or 1.671 for a one-tail test (look up ta for 60 df).

For our example, the alternative hypothesis is two-sided. Therefore, we use the two-tail t value. Since the computed t value of 2.8187 (in absolute terms) exceeds the critical t value of 2, we can reject the null hypothesis that PGNP has no effect on child mortality. To put it more positively, with the female literacy rate held constant, per capita GNP has a significant (negative) effect on child mortality, as one would expect a priori. Graphically, the situation is as shown in Figure 8.1.

In practice, one does not have to assume a particular value of a to conduct hypothesis testing. One can simply use the p value given in (8.2.2), 2In most empirical investigations the null hypothesis is stated in this form, that is, taking the extreme position (a kind of straw man) that there is no relationship between the dependent variable and the explanatory variable under consideration. The idea here is to find out whether the relationship between the two is a trivial one to begin with.

2In most empirical investigations the null hypothesis is stated in this form, that is, taking the extreme position (a kind of straw man) that there is no relationship between the dependent variable and the explanatory variable under consideration. The idea here is to find out whether the relationship between the two is a trivial one to begin with.

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which in the present case is 0.0065. The interpretation of this p value (i.e., the exact level of significance) is that if the null hypothesis were true, the probability of obtaining a t value of as much as 2.8187 or greater (in absolute terms) is only 0.0065 or 0.65 percent, which is indeed a small probability, much smaller than the artificially adopted value of a = 5%.

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:

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.

In Chapter 5 we saw the intimate connection between hypothesis testing and confidence interval estimation. For our example, the 95% confidence interval for fa is:

that is, the interval, -0.0096 to -0.0016 includes the true fa coefficient with 95% confidence coefficient. Thus, if 100 samples of size 64 are selected and 100 confidence intervals like (8.4.2) are constructed, we expect 95 of them to contain the true population parameter fa. Since the interval (8.4.2) does not include the null-hypothesized value of zero, we can reject the null hypothesis that the true fa is zero with 95% confidence.

Thus, whether we use the t test of significance as in (8.4.1) or the confidence interval estimation as in (8.4.2), 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 parameters of our child mortality regression model. The necessary data are already provided in Eq. (8.2.1). 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 ui follows the normal distribution. Although we cannot directly observe ui, we can observe their proxy, the uui, that is, the CHAPTER EIGHT: MULTIPLE REGRESSION ANALYSIS: THE PROBLEM OF INFERENCE 253 