econometrics i chapter 5: two-variable regression: interval estimation and hypothesis testing...

ECONOMETRICS I

CHAPTER 5: TWO-VARIABLE REGRESSION: INTERVAL ESTIMATION

AND HYPOTHESIS TESTING

Textbook: Damodar N. Gujarati (2004) Basic Econometrics, 4th edition, The McGraw-Hill Companies

5.2 Interval Estimation: Some Basic Ideas

• Because of sampling fluctuations, a single estimate is likely to differ from the true value, although in repeated sampling its mean value is expected to be equal to the true value.

5.2 Interval Estimation: Some Basic Ideas

• In statistics the reliability of a point estimator is measured by its standard error. Therefore, instead of relying on the point estimate alone, we may construct an interval around the point estimator, say within two or three standard errors on either side of the point estimator, such that this interval has, say, 95 percent probability of including the true parameter value. This is roughly the idea behind interval estimation.

5.2 Interval Estimation: Some Basic Ideas

5.2 Interval Estimation: Some Basic Ideas

• Confidence coefficient= 0.95 = 95 %• Level of significance= 0.05 = 5 %

If α = 0.05, or 5 percent, (5.2.1) would read: The probability that the (random) interval shown there includes the true β2 is 0.95, or 95 percent. The interval estimator thus gives a range of values within which the true β2 may lie.

5.2 Interval Estimation: Some Basic Ideas

• It is very important to know the following aspects of interval estimation:

5.2 Interval Estimation: Some Basic Ideas

5.2 Interval Estimation: Some Basic Ideas

5.3 CONFIDENCE INTERVALS FOR REGRESSION COEFFICIENTS β1 AND β2

• CONFIDENCE INTERVAL FOR β2

With the normality assumption for ui, the OLS estimator is normally distributed.

CONFIDENCE INTERVAL FOR β2

• We can use the normal distribution to make probabilistic statements about β2 provided the true population variance σ2 is known. If σ2 is known, an important property of a normally distributed variable with mean μ and variance σ2 is that the area under the normal curve between μ ± σ is about 68 percent, that between the limits μ ± 2σ is about 95 percent, and that between μ ± 3σ is about 99.7 percent.

CONFIDENCE INTERVAL FOR β2

CONFIDENCE INTERVAL FOR β2

• The t value in the middle of this double inequality is the t value given by (5.3.2) and where tα/2 is the value

of the t variable obtained from the t distribution for α/2 level of significance and n − 2 df. it is often called the critical t value at α/2 level of significance.

CONFIDENCE INTERVAL FOR β2

CONFIDENCE INTERVAL FOR β2

CONFIDENCE INTERVAL FOR β2

CONFIDENCE INTERVAL FOR β1

CONFIDENCE INTERVAL FOR β1

5.4 CONFIDENCE INTERVAL FOR σ2

5.4 CONFIDENCE INTERVAL FOR σ2

5.4 CONFIDENCE INTERVAL FOR σ2

5.4 CONFIDENCE INTERVAL FOR σ2

5.5 HYPOTHESIS TESTING: GENERAL COMMENTS

5.5 HYPOTHESIS TESTING: GENERAL COMMENTS

1. Confidence interval approach2. Test of significance approach

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.

5.6 HYPOTHESIS TESTING: THE CONFIDENCE-INTERVAL APPROACH

95 % CI for Beta-2 is (0.4268, 0.5914).

• In statistics, when we reject the null hypothesis, we say that our finding is statistically significant. On the other hand, when we do not reject the null hypothesis, we say that our finding is not statistically significant.Two-sided test vs. one-sided test

• → two-sided test

• → one-sided test

5.7 HYPOTHESIS TESTING: THE TEST-OF-SIGNIFICANCE APPROACH

• Broadly speaking, a test of significance is a procedure by

which sample results are used to verify the truth or falsity of a

null hypothesis. The key idea behind tests of significance is

that of a test statistic (estimator) and the sampling

distribution of such a statistic under the null hypothesis. The

decision to accept or reject H0 is made on the basis of the

value of the test statistic obtained from the data at hand.

5.7 HYPOTHESIS TESTING: THE TEST-OF-SIGNIFICANCE APPROACH

• This variable follows the t distribution with n−2 df.

5.7 HYPOTHESIS TESTING: THE TEST-OF-SIGNIFICANCE APPROACH

• is the value of β2 under H0 and where −tα/2 and tα/2 are the

values of t (the critical t values) obtained from the t table for

(α/2) level of significance and n − 2 df.

5.7 HYPOTHESIS TESTING: THE TEST-OF-SIGNIFICANCE APPROACH

5.7 HYPOTHESIS TESTING: THE TEST-OF-SIGNIFICANCE APPROACH

5.7 HYPOTHESIS TESTING: THE TEST-OF-SIGNIFICANCE APPROACH

5.7 HYPOTHESIS TESTING: THE TEST-OF-SIGNIFICANCE APPROACH

5.7 HYPOTHESIS TESTING: THE TEST-OF-SIGNIFICANCE APPROACH

• 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.

5.7 HYPOTHESIS TESTING: THE TEST-OF-SIGNIFICANCE APPROACH

• To test this hypothesis, we use the one-tail test (the right tail), as shown in Figure 5.5.

• The test procedure is the same as before except that the upper

confidence limit or critical value now corresponds to tα = t0.05,

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.

• CI = (- ∞, 0.3664)

TABLE 5.1 (page 133)

Testing the Significance of σ2: The χ2 Test

Testing the Significance of σ2: The χ2 Test

The Meaning of “Accepting” or “Rejecting” a Hypothesis

The Exact Level of Significance: The p Value

• Once a test statistic (e.g., the t statistic) is obtained in a given

example, why not simply go to the appropriate statistical table

and find out the actual probability of obtaining a value of the

test statistic as much as or greater than that obtained in the

example? This probability is called the p value (i.e., probability

value), also known as the observed or exact level of significance

or the exact probability of committing a Type I error. More

technically, the p value is defined as the lowest significance level

at which a null hypothesis can be rejected.

5.9 REGRESSION ANALYSIS AND ANALYSIS OF VARIANCE

• TSS = ESS + RSS

• A study of these components of TSS is known as the analysis of variance (ANOVA) from the regression viewpoint.

5.9 REGRESSION ANALYSIS AND ANALYSIS OF VARIANCE

5.9 REGRESSION ANALYSIS AND ANALYSIS OF VARIANCE

5.9 REGRESSION ANALYSIS AND ANALYSIS OF VARIANCE

5.11 REPORTING THE RESULTS OF REGRESSION ANALYSIS

5.11 REPORTING THE RESULTS OF REGRESSION ANALYSIS

In Eq. (5.11.1) the figures in the first set of parentheses are the estimated standard errors of the regression coefficients, the figures in the second set are estimated t values computed from (5.3.2) under the null hypothesis that the true population value of each regression coefficient individually is zero (e.g., 3.8128 = 24.4545 ÷ 6.4138), and the figures in the third set are the estimated p values. Thus, for 8 df the probability of obtaining a t value of 3.8128 or greater is 0.0026 and the probability of obtaining a t value of 14.2605 or larger is about 0.0000003.