1. Copyright © 2005 by the McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin Managerial Economics Thomas Maurice eighth edition Chapter 4 Basic Estimation Techniques

2. Managerial Economics 2 Copyright © 2005 by the McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin 2 Simple Linear Regression Simple linear regression model relates dependent variable Y to one independent (or explanatory) variable X Slope parameter ( b ) gives the change in Y associated with a one-unit change in X,

3. Managerial Economics 3 Copyright © 2005 by the McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin 3 Method of Least Squares The sample regression line is an estimate of the true regression line

4. Managerial Economics 4 Copyright © 2005 by the McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin 4 Sample Regression Line (Figure 4.2) A 0 8,0002,000 10,000 4,000 6,000 10,000 20,000 30,000 40,000 50,000 60,000 70,000 Advertising expenditures (dollars) Sales (dollars) S eiei

5. Managerial Economics 5 Copyright © 2005 by the McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin 5 The distribution of values the estimates might take is centered around the true value of the parameter An estimator is unbiased if its average value (or expected value) is equal to the true value of the parameter Unbiased Estimators

6. Managerial Economics 6 Copyright © 2005 by the McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin 6 Relative Frequency Distribution (Figure 4.3) 0 8210 4 6 1 1 3 57 9

7. Managerial Economics 7 Copyright © 2005 by the McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin 7 Must determine if there is sufficient statistical evidence to indicate that Y is truly related to X (i.e., b  0) Statistical Significance Test for statistical significance using t -tests or p -values

8. Managerial Economics 8 Copyright © 2005 by the McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin 8 First determine the level of significance Probability of finding a parameter estimate to be statistically different from zero when, in fact, it is zero Probability of a Type I Error 1 – level of significance = level of confidence Performing a t -Test

9. Managerial Economics 9 Copyright © 2005 by the McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin 9 Performing a t -Test Use t -table to choose critical t -value with n – k degrees of freedom for the chosen level of significance n = number of observations k = number of parameters estimated

10. Managerial Economics 10 Copyright © 2005 by the McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin 10 Performing a t -Test If absolute value of t -ratio is greater than the critical t, the parameter estimate is statistically significant

11. Managerial Economics 11 Copyright © 2005 by the McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin 11 Using p -Values Treat as statistically significant only those parameter estimates with p -values smaller than the maximum acceptable significance level p -value gives exact level of significance Also the probability of finding significance when none exists

12. Managerial Economics 12 Copyright © 2005 by the McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin 12 Coefficient of Determination R 2 measures the percentage of total variation in the dependent variable that is explained by the regression equation Ranges from 0 to 1 High R 2 indicates Y and X are highly correlated

13. Managerial Economics 13 Copyright © 2005 by the McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin 13 F -Test Used to test for significance of overall regression equation Compare F -statistic to critical F - value from F -table Two degrees of freedom, n – k & k – 1 Level of significance If F -statistic exceeds the critical F, the regression equation overall is statistically significant

14. Managerial Economics 14 Copyright © 2005 by the McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin 14 Multiple Regression Uses more than one explanatory variable Coefficient for each explanatory variable measures the change in the dependent variable associated with a one-unit change in that explanatory variable

15. Managerial Economics 15 Copyright © 2005 by the McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin 15 Quadratic Regression Models Use when curve fitting scatter plot is  -shaped or  -shaped

16. Managerial Economics 16 Copyright © 2005 by the McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin 16 Log-Linear Regression Models