1. Copyright © 2014 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education. Chapter 9 Heteroskedasticity

2. 9-2 Learning Objectives Understand methods for detecting heteroskedasticity Correct for heteroskedasticity

3. 9-3 What is Heteroskedasticity?

4. 9-4 A Picture of Homoskedasticity Versus Heteroskedasticity

5. 9-5 The Issues And Consequences Associated With Heteroskedastic Data Problem: Heteroskedasticity violates assumption M6, which states that the error term must have constant variance. Consequences: Under heteroskedasticity parameter estimates are unbiased. Parameter estimates are not minimum variance among all unbiased estimators. Estimated standard errors are incorrect and all measures of precision based on the estimated standard errors are also incorrect.

6. 9-6 Goals of this Chapter

7. 9-7 An Important Caveat before Continuing With more advanced statistical packages, many researchers include a very simple command asking their chosen statistical program to provides standard error estimates that automatically correct for heteroskedasticity (White’s heteroskedastic consistent standard errors) Even though correcting for heteroskedasticity is straightforward, it important to first work through the more “old-school” examples that we do below before learning how to calculate White’s heteroskedastic consistent standard errors.

8. 9-8 Understand Methods For Detecting Heteroskedasticity Informal methods - Graphs Formal methods using statistical tests - Breusch-Pagan test - General White’s Test - Modified White’s Test - Goldfeld-Quandt Test

9. 9-9 Informal Method Either graph: (1)The dependent variable against each independent variable… (2)The residuals against each independent variable… (3)The residuals squared against each independent variable… (4)The standardized residuals against each independent variable… and look for a pattern in the dispersion of the observations. If a pattern exists then that is evidence of heteroskedasticity.

10. 9-10 Regression of Number of Olympic Medals on per capita GDP by Country

11. 9-11 Notice how the variance increases as the independent variable increases. This is evidence of heteroskedasticity.

12. 9-12 This residual plot is obtained by checking the residual plot option in Excel when running a regression. As in the previous slide, notice how the variance increases as the independent variable (GDP per Capita) increases. This is evidence of heteroskedasticity.

13. 9-13 The primary drawback of the informal method is that it is not clear how much of a pattern needs to exist to lead us to the conclusion that the model is heteroskedastic. This leads us to the need for formal tests of heteroskedasticity.

14. 9-14 Formal Methods for Detecting Heteroskedasticity

15. 9-15 Testing for Heteroskedasticity (1)Breusch - Pagan (2)Modified White’s Test (3)Goldfeld-Quandt Test

16. 9-16 Breusch-Pagan Test

17. 9-17 Breusch-Pagan Test Why It Works: If the squared residuals are found to be statistically related to the independent variables then we conclude that the data are heteroskedastic and we should take the appropriate steps to correct for the problem.

18. 9-18 Breusch-Pagan Test for Olympic Medal vs GDP per Capita Data Dependent Variable is Residuals Squared The significant F is much less than 0.05 (or 0.01 for that matter) so we reject the null hypothesis of homoskedasticity and conclude model is heteroskedastic.

19. 9-19 Modified White’s Test

20. 9-20 Modified White’s Test

21. 9-21 Modified White’s Test for Olympic Medal vs GDP per Capita Data Dependent Variable is Residuals Squared The significant F is much less than 0.05 (or 0.01 for that matter) so we reject the null hypothesis of homoskedasticity and conclude the model is heteroskedastic.

22. 9-22 Goldfeld-Quandt Test

23. 9-23 Goldfeld-Quandt Test Why It Works: This test works when the suspected heteroskedasticity is of the type that the error variances either increase (or decrease) with the value of a given independent variable. If we find that the unexplained sum of squares for the largest values is “large” relative to the unexplained sum of squares for the smallest values, then we conclude that the error variance changes significantly with the value of the independent variable, suggesting that the data are heteroskedastic.

24. 9-24 Goldfeld-Quandt Test How to do it: For the Olympic Medal Data, there are 408 observations. Dividing the data into thirds, the first regression should contain the smallest 136 (408/3) GDP per capita data, and the second regression should contain the largest 136 GDP per capita data.

25. 9-25 USS 1

26. 9-26 USS 2

27. 9-27 Goldfeld-Quandt Test Example

28. 9-28 Correcting for Heteroskedasticity (1)Weighted least squares (2)White’s heteroskedastic consistent standard errors

29. 9-29 Weighted Least Squares

30. 9-30 Weighted Least Squares

31. 9-31 Weighted Least Squares Example

32. 9-32 Weighted Least Squares Example

33. 9-33 Weighted Least Squares Example Excel Results

34. 9-34 Breusch-Pagan Test of Transformed Weighted Least Squares Data Unfortunately, even after the transformation this model still suffers from heteroskedasticity

35. 9-35 Robust Standard Errors The preferred method to correct for heteroskedasticity is to use White’s heteroskedastic consistent standard errors. The coefficient estimates are still unbiased so the only thing that needs to be corrected are the standard errors. In STATA, the command is reg y x1 x2 x3, robust The,robust (or even,r) is the portion of the command that corrects the standard errors.

36. 9-36 STATA Results with Original Standard Errors STATA Results with Robust Standard Errors