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
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.
• Confidence coefficient= 0.95 = 95 %
• Level of significance= 0.05 = 5 %
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.
2.
Confidence interval approach
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.