CHAPTER 5: TWO-VARIABLE REGRESSION: INTERVAL ESTIMATION AND HYPOTHESIS TESTING

Interval Estimation: Some Basic Ideas
1 – α – confidence coefficient
α (0<α<1) – level of significance
Confidence limits – endpoints of the confidence interval
β2 -      - lower confidence limit
β1 -      - upper confidence limit
Following aspects of interval estimation:
1.      Since β2, although an unknown, is assumed to be some fixed number, either it lies in the interval or it does not.
2.      Random interval will vary from one sample to the next because it is based on β2, which is random.
3.      Since the confidence interval is random, the probability statements attached to it should be understood in the long-run sense, that is, repeated sampling.
Confidence Intervals for Regression Coefficients β1 and β2

• Confidence Interval for β2
• Confidence Interval for β1
• Confidence Interval for
•  Hypothesis Testing: the Confidence-Interval Approach

• Two-sided or Two-tail test

Decision Rule: Construct a 100(1 – α)% confidence interval for β2. If the β2 under H0 falls within this confidence interval, do not reject H0, but if it falls outside the interval, reject H0.

• One-sided or One-tail test

Hypothesis Testing: The Test-of-Significance Approach

• Testing the significance of Regression Coefficients: the t – Test

Broadly speaking, the test of significance is a procedure by which sample results are used to verify the truth of falsify of null hypothesis.
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.
Testing the Significance of        : The r2 Test
Hypothesis Testing: Some Practical Aspects

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

On the basis of a test of significance, say, the t-test, we decide to “accept” the null hypothesis, all we are saying is that on the basis of the sample evidence we have no reason to reject it; we are not saying that the null hypothesis is time beyond any doubt.

• The “Zero” Null Hypothesis and the “2 – t” Rule of Thumb

“2 – t” Rule of Thumb – if the number of degrees of freedom is 20 or more and if α, the level of significance is set at 0.05, then the null hypothesis β2 = 0 can be rejected if the t-value [ = β2/se(β2)] exceeds 2 in absolute value.

• Forming the Null and Alternative Hypotheses

It is extremely important that the researcher establish these hypotheses before carrying out the empirical investigation.

• The Exact Level of Significance: The P Value

P Value (probability value) – also known as the observed or exact level of significance or the exact probability of committing a Type I error. It is defined as the lowest significance level at which a null hypothesis can be rejected.
Application of Regression Analysis: The Problem of Prediction

• Individual Prediction

Therefore, one should exercise great caution in “extrapolating” the historical regression line to predict E(Y/X0) or Y0 associated with a given X0 that is far removed from the sample mean X.
Reporting the Results of Regression Analysis
Evaluating the Results of Regression Analysis

• Normality Tests

Histogram of Residuals – simple graphic device that is used to learn something about the shape of the PDF of a random variable.
Normal Probability Plot – comparatively simply graphical device to study the shape of the probability density function of a random variable.
Jacque-Bera Test of Normality – is an asymptotic or large-sample test. It is also based on OLS residuals. This test first computes the skewness and kurtosis.


JB = n     S2  +   (k – 3)2
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