1. © 2001 Prentice-Hall, Inc.Chap 13-1 BA 201 Lecture 21 Autocorrelation and Inferences about the Slope

2. © 2001 Prentice-Hall, Inc. Chap 13-2 Topics Measuring Autocorrelation Inferences about the Slope

3. © 2001 Prentice-Hall, Inc. Chap 13-3 Autocorrelation What is Autocorrelation? The error term in one time period is related (correlated, autocorrelated) to the error term in a different time period Can happen only in time-series data

4. © 2001 Prentice-Hall, Inc. Chap 13-4 Residual Analysis for Independence (Graphical Approach) Not Independent Independent e e Time Residual Is Plotted Against Time to Detect Any Autocorrelation No Particular PatternCyclical Pattern Time Y Y 00

5. © 2001 Prentice-Hall, Inc. Chap 13-5 Residual Analysis for Independence The Durbin-Watson Statistic Used when data is collected over time to detect autocorrelation (residuals in one time period are related to residuals in another period) Measures violation of independence assumption Should be close to 2 for independence of errors. If not, examine the model for autocorrelation.

6. © 2001 Prentice-Hall, Inc. Chap 13-6 Durbin-Watson Statistic in PHStat PHStat | Regression | Simple Linear Regression … Check the box for Durbin-Watson Statistic An example in Excel spreadsheet Y is the DJIA (measured in % change from previous day’s closing number to current day’s closing number) X is the U.S. Treasury 30-year bond rates (measured in % change from previous day’s closing rate to current day’s closing rate)

7. © 2001 Prentice-Hall, Inc. Chap 13-7 Accept H 0 (no autocorrelatin) Using the Durbin-Watson Statistic : No autocorrelation (error terms are independent) : There is autocorrelation (error terms are not independent) 042 dLdL 4-d L dUdU 4-d U Reject H 0 (positive autocorrelation) Inconclusive Reject H 0 (negative autocorrelation)

8. © 2001 Prentice-Hall, Inc. Chap 13-8 Sample Linear Regression (continued) Y X Observed Value p.462

9. © 2001 Prentice-Hall, Inc. Chap 13-9 Inference about the Slope: t Test t Test for a Population Slope Is there a linear dependency of Y on X ? Null and Alternative Hypotheses H 0 :  1 = 0(No Linear Dependency) H 1 :  1  0(Linear Dependency) Test Statistic Assumption Needed Normality pp.483-484

10. © 2001 Prentice-Hall, Inc. Chap 13-10 Example: Produce Store Data for 7 Stores: Estimated Regression Equation: The slope of this model is 1.487. Is Square Footage of the store affecting its Annual Sales at 5% level of significance? Annual Store Square Sales Feet($000) 1 1,726 3,681 2 1,542 3,395 3 2,816 6,653 4 5,555 9,543 5 1,292 3,318 6 2,208 5,563 7 1,313 3,760

11. © 2001 Prentice-Hall, Inc. Chap 13-11 Inferences about the Slope: t Test Example H 0 :  1 = 0 H 1 :  1  0  .05 df  7 - 2 = 5 Critical Value(s): Test Statistic: Decision: Conclusion: There is evidence that square footage affects annual sales. t 02.5706-2.5706.025 Reject.025 From Excel Printout Reject H 0 p-value b 0

12. © 2001 Prentice-Hall, Inc. Chap 13-12 Inferences about the Slope: Confidence Interval Example Confidence Interval Estimate of the Slope: Excel Printout for Produce Stores At 95% level of confidence the confidence interval for the slope is (1.062, 1.911). Does not include 0. Conclusion: There is a significant linear dependency of annual sales on the size of the store. p.486

13. © 2001 Prentice-Hall, Inc. Chap 13-13 Inferences about the Slope: F Test F Test for a Population Slope Is there a linear dependency of Y on X ? Null and Alternative Hypotheses H 0 :  1 = 0(No Linear Dependency) H 1 :  1  0(Linear Dependency) Test Statistic Numerator d.f.=1, denominator d.f.=n-2 Assumption Needed Normality pp.484-485

14. © 2001 Prentice-Hall, Inc. Chap 13-14 Inferences about the Slope: F Test Example Test Statistic: Decision: Conclusion: H 0 :  1 = 0 H 1 :  1  0  .05 numerator df = 1 denominator df  7 - 2 = 5 There is evidence that square footage affects annual sales. From Excel Printout Reject H 0 06.61 Reject  =.05 p-value

15. © 2001 Prentice-Hall, Inc. Chap 13-15 Relationship between a t Test and an F Test Null and Alternative Hypotheses H 0 :  1 = 0(No Linear Dependency) H 1 :  1  0(Linear Dependency) The p –value of a t Test and the p –value of an F Test Are Exactly the Same The Rejection Region of an F Test is always in the upper tail p.???

16. © 2001 Prentice-Hall, Inc. Chap 13-16 Summary Addressed Measuring Autocorrelation Described Inference about the Slope