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Lecture 4

Lecture 4. Main Tasks Today 1. Review of Lecture 3. 2. Accuracy of the LS estimators 3. Significance Tests of the Parameters 4. Confidence Interval. 5. Prediction and Prediction Interval. Review of Lecture 3. The LS- Estimators of and :. Properties :.

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Lecture 4

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  1. Lecture 4 Main Tasks Today 1. Review of Lecture 3 2. Accuracy of the LS estimators 3. Significance Tests of the Parameters 4. Confidence Interval 5. Prediction and Prediction Interval

  2. Review of Lecture 3 The LS- Estimators of and : Properties: (1).Linearity (2).Unbiased (3). Normality

  3. Review of Lecture 3 (cont.) (4).Best Linear Unbiased Estimators.For any other Unbiased Linear Estimators and of and , we have

  4. Accuracy of the LS-Estimates Noise Variance is usually unknown. It has a natural estimator using residuals, given by SSE=Sum of Squared Errors (Residuals) n-2= degrees of freedom of SSE=Sample size- # of Coefficients Using the estimated noise variance, we can obtain the measures of the accuracy (Standard Errors, s.e.) of the estimators and

  5. Significance Test of Parameters In the SLR model, we often want to know if (1) “ Y and X are linearly uncorrelated” (X has no effect on Y) ( ) (2) “ the linear regression line passes the origin” (no intercept is needed).( ) A hypothesis test has two hypotheses: H0 (Null) and H1 ( Alternative). A general test about the slope (intercept ) can be written as versus (a) versus (b) where ( or ) is a predetermined constant (belief of the experts). To test (1) [or (2)] is equivalent to test =0 [or =0]

  6. Test Statistics and 2-Sided Tests To test the hypotheses, we need construct test statistics. Test statistic can usually be constructed as: For example, to test (1) and (2) , we have test statistics respectively Which have t-distributions with n-2 degrees of freedom (dfs). Let be the Upper - percentile of t-distribution with n-2 dfs. Then the level two- sided test for test (1) or (2) is can be found in the Appendix (Table A.2, Page 339)

  7. P-value of the Test In another way, we can compute the p-value of the test statistic to perform the test: If the p-value< , then the test is level significant. The smaller the p-value, the higher significant level the test is. In general, we say 1) when p-value<.05, the test is significant; 2) when p-value < .01, the test is very significant; 3) when p-value>.20, the test is not significant. Example For computer repair data, to test Since 6.948>2.18, the test is significant at 5%-level significance. The <.005. So the test is highly significant

  8. One-Sided Tests: (a) ( (b) ) (c) ( ) (d)

  9. Minitab Output Results for: P027.txt Regression Analysis: Minutes versus Units The regression equation is Minutes = 4.16 + 15.5 Units Predictor Coef SE Coef T P-value Constant 4.162 3.355 1.24 0.239 Units 15.5088 0.5050 30.71 0.000 S = 5.392 R-Sq = 98.7% R-Sq(adj) = 98.6% Note that the T-values calculated here are for testing`` the true parameters are zeros”.

  10. A Test Using Correlation Coefficient Since =0 is equivalent to Cor(Y,X)=0. So the following two tests are equivalent to each other: (3) (4) where denotes the true correlation between Y and X. To test (4), we can use the following test statistic We can also compute to perform the test

  11. Example: For the computer repair data, since Cor(Y,X)=.996, we have Which is much larger than . So the correlation is highly Significantly different from 0. That is X has high impact on Y. We reject H0 of test (4) or (3) at 5%-significant level. The corresponding p-value much less than .0001. So the significance is very high.

  12. Confidence Interval Since For SLR Model We have (similarly for ):

  13. Example Interpretation of CI:

  14. Prediction Interval For SLR Model

  15. Prediction Interval (cont.)

  16. Standard Errors Reasons: (1). is a random variable (2). is a constant.

  17. Prediction Interval/Prediction Limits/Forecast Intervals Remark:

  18. Example

  19. Note that the PI bands (blue) are wider than the CI bands (red)

  20. Remark

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