follows the x2 distribution with n — 2 df.5 Therefore, we can use the x2 distribution to establish a confidence interval for a2

Pr (xi2— a/2 < X2 < x2/2) = 1 — « (5.4.2)

where the x 2 value in the middle of this double inequality is as given by (5.4.1) and where x?—a/2 and xOt/2 are two values of x2 (the critical x2 values) obtained from the chi-square table for n — 2 df in such a manner that they cut off 100(a/2) percent tail areas of the x2 distribution, as shown in Figure 5.1.

Substituting x2 from (5.4.1) into (5.4.2) and rearranging the terms, we obtain

which gives the 100(1 — a)% confidence interval for a2.

To illustrate, consider this example. From Chapter 3, Section 3.6, we obtain a2 = 42.1591 and df = 8. If a is chosen at 5 percent, the chi-square table for 8 df gives the following critical values: x<2025 = 17.5346, and x<2 975 = 2.1797. These values show that the probability of a chi-square value exceeding 17.5346 is 2.5 percent and that of 2.1797 is 97.5 percent. Therefore, the interval between these two values is the 95% confidence interval for x2, as shown diagrammatically in Figure 5.1. (Note the skewed characteristic of the chi-square distribution.)

5For proof, see Robert V. Hogg and Allen T. Craig, Introduction to Mathematical Statistics, 2d ed., Macmillan, New York, 1965, p. 144.

5. Two-Variable Regression: Interval Estimation and Hypothesis Testing

126 PART ONE: SINGLE-EQUATION REGRESSION MODELS

Substituting the data of our example into (5.4.3), the reader should verify that the 95% confidence interval for a2 is as follows:

The interpretation of this interval is: If we establish 95% confidence limits on a2 and if we maintain a priori that these limits will include true a2, we shall be right in the long run 95 percent of the time.

Having discussed the problem of point and interval estimation, we shall now consider the topic of hypothesis testing. In this section we discuss briefly some general aspects of this topic; Appendix A gives some additional details.

The problem of statistical hypothesis testing may be stated simply as follows: Is a given observation or finding compatible with some stated hypothesis or not? The word "compatible," as used here, means "sufficiently" close to the hypothesized value so that we do not reject the stated hypothesis. Thus, if some theory or prior experience leads us to believe that the true slope coefficient ft of the consumption-income example is unity, is the observed ft = 0.5091 obtained from the sample of Table 3.2 consistent with the stated hypothesis? If it is, we do not reject the hypothesis; otherwise, we may reject it.

In the language of statistics, the stated hypothesis is known as the null hypothesis and is denoted by the symbol H0. The null hypothesis is usually tested against an alternative hypothesis (also known as maintained hypothesis) denoted by H1, which may state, for example, that true ft is different from unity. The alternative hypothesis may be simple or compos-ite.6 For example, H1: ft = 1.5 is a simple hypothesis, but H1: ft = 1.5 is a composite hypothesis.

The theory of hypothesis testing is concerned with developing rules or procedures for deciding whether to reject or not reject the null hypothesis. There are two mutually complementary approaches for devising such rules, namely, confidence interval and test of significance. Both these approaches predicate that the variable (statistic or estimator) under consideration has some probability distribution and that hypothesis testing involves making statements or assertions about the value(s) of the parameter(s) of such distribution. For example, we know that with the normality assumption ft is normally distributed with mean equal to ft and variance given by (4.3.5). If we hypothesize that ft = 1, we are making an assertion about one

6A statistical hypothesis is called a simple hypothesis if it specifies the precise value(s) of the parameter(s) of a probability density function; otherwise, it is called a composite hypothesis. For example, in the normal pdf(1/aV2^") exp{—^[(X — i)/a]2}, if we assert that Hi:/ = 15 and a = 2, it is a simple hypothesis; but if H1:/i = 15 and a > 15, it is a composite hypothesis, because the standard deviation does not have a specific value.

5.5 HYPOTHESIS TESTING: GENERAL COMMENTS

5. Two-Variable Regression: Interval Estimation and Hypothesis Testing

CHAPTER FIVE: TWO VARIABLE REGRESSION: INTERVAL ESTIMATION AND HYPOTHESIS TESTING 127

of the parameters of the normal distribution, namely, the mean. Most of the statistical hypotheses encountered in this text will be of this type—making assertions about one or more values of the parameters of some assumed probability distribution such as the normal, F, t, or x2. How this is accomplished is discussed in the following two sections.

To illustrate the confidence-interval approach, once again we revert to the consumption-income example. As we know, the estimated marginal propensity to consume (MPC), j32, is 0.5091. Suppose we postulate that that is, the true MPC is 0.3 under the null hypothesis but it is less than or greater than 0.3 under the alternative hypothesis. The null hypothesis is a simple hypothesis, whereas the alternative hypothesis is composite; actually it is what is known as a two-sided hypothesis. Very often such a two-sided alternative hypothesis reflects the fact that we do not have a strong a priori or theoretical expectation about the direction in which the alternative hypothesis should move from the null hypothesis.

Is the observed /32 compatible with H0? To answer this question, let us refer to the confidence interval (5.3.9). We know that in the long run intervals like (0.4268, 0.5914) will contain the true fa2 with 95 percent probability. Consequently, in the long run (i.e., repeated sampling) such intervals provide a range or limits within which the true fa2 may lie with a confidence coefficient of, say, 95%. Thus, the confidence interval provides a set of plausible null hypotheses. Therefore, if fa2 under H0 falls within the 100(1 — a)% confidence interval, we do not reject the null hypothesis; if it lies outside the interval, we may reject it.7 This range is illustrated schematically in Figure 5.2.

Decision Rule: Construct a 100(1 — a)% confidence interval for fa2. If the fa2 under H0 falls within this confidence interval, do not reject H0, but if it falls outside this interval, reject H0.

Following this rule, for our hypothetical example, H0: fa2 = 0.3 clearly lies outside the 95% confidence interval given in (5.3.9). Therefore, we can reject

7Always bear in mind that there is a 100a percent chance that the confidence interval does not contain fa2 under H0 even though the hypothesis is correct. In short, there is a 100a percent chance of committing a Type I error. Thus, if a = 0.05, there is a 5 percent chance that we could reject the null hypothesis even though it is true.

5.6 HYPOTHESIS TESTING:

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