1. Jennifer Lehman Hossein Salehi Jacob Tenney

2. Story Definition Detection Consequences Solutions Heteroscedasticity Presentation Agenda

3. This is Our Story In this presentation, we will use an example from Personal Financial Planning. Two variables: Annuities (Y) and Accumulated Wealth (X)

4. This is Our Story (Dr. Westfall) http://stats.stackexchange.com/questions/86788/why-do-we-say-that-the-variance-of-the-error-terms-is-constant/86792#86792

5. This is Our Story

6. Assumptions in Linear Regression Models The data generating process has: 1. Correct Functional Specification (Linearity) 2. Normality 3. Uncorrelated Error Terms (Conditional Independence) 4. Constant Variance (Homoscedasticity) Definition (Dr. Westfall)

7. Homoscedasticity (Dr. Westfall)

8. Under the homoscedasticity assumption, the variance in amount annuitized (Y) is the same regardless of accumulated wealth (X). However, in real life this assumption is badly violated. A Homoscedastic Example Photo Courtesy: http://www.statsblogs.com/2015/09/10/plot-the-conditional-distribution-of-the-response-in-a-linear-regression-model/

9. The distributions of amount annuitized at different accumulated wealth levels are more likely to look like this: A Heteroscedastic Example Photo Courtesy: http://www.statsblogs.com/2015/09/16/error-distributions-and-exponential-regression-models/

10. Heteroscedasticity

11. Covariance Matrix Comparison HomoscedasticityHeteroscedasticity

12. Plots Use scatterplot to show the absolute residuals In our “amount annuitized” simulation, heteroscedasticity is evident from the data. However, this is not always the case. Detection

14. TESTSGRAPHS BENEFITS Objective measure of what is explainable by chance alone Sometimes required by journals Transparency “Practical Importance” is easily determined Larger sample sizes point us closer to the best answer when well-chosen graphs are used CONCERNS All assumptions are violated, but the question is how badly. Assumptions are null hypotheses, which cannot be proven true. Sample size: 1. Small sample: Low power, likely to fail to reject the null. 2. Large sample: Everything is significant Tests are not perfectly objective. Interpretation requires practice, judgment and statistical knowledge Generating good graphs requires skill and practice (Dr. Westfall, Feb. 2 reading material) Detection

15. 1. Wrong Standard Error: Heteroscedasticity causes the OLS estimates of the SE to be biased, leading to unreliable hypothesis testing. The variance formula tends to underestimate the true variance of the OLS estimate. 2. Inefficient OLS Estimate: Heteroscedasticity typically causes OLS to no longer be the minimum variance estimator (of all the linear unbiased estimators) (Lung-fei Lee, Ohio State University, Lectures 2001) The Consequences of Heteroscedasticity (Blunch, 2011) (Walter Sosa-Escudero, 2009)

16. Ordinary Least Squares (Paul Johnson, Oct. 2005)

17. Ordinary Least Squares

18. The Consequences of Heteroscedasticity in Examples

19. 2. Inefficient OLS Estimate: e.g. So OLS is no longer BLUE (Best Linear Unbiased Estimator). Possible Solutions: WLS MLE Note: Correcting S.E. and using OLS will NOT help. The Consequences of Heteroscedasticity in Examples

20.  Transformations  Weighted Least Squares  OLS with corrected standard errors  MLE Solutions

21.  Transformations  Weighted Least Squares  OLS with corrected standard errors  MLE Solutions

22. Solutions (Transformations)

23.  Journals may not like it  Harder to interpret  It might not work, so we need to go back to the other solutions. Concerns with Transformations

24.  Transformations  Weighted Least Squares  OLS with corrected standard errors  MLE Solutions

25. Covariance Matrix Comparison  Recall from what we discussed earlier, HomoscedasticityHeteroscedasticity

26. OLSWLS Variance Function Sum of Squared Residuals Beta Estimates Comparing OLS and WLS (Ingo Ruczinski, Chapter 5) (Paul Johnson, Oct. 2005)

27. Comparing OLS & WLS (Maria L. Durban Reguera, UC3M reading material)

28. Solutions (Weighted Least Squares)

29. Accumulated Wealth & Annuitization

30. We can use simulated data to see that adding weights improves the accuracy of the estimate, as measured by standard error. Let’s simulate and then translate what we learn to a real dataset. Simulating Weighted Least Squares

31.  With the right weight function, the estimates are efficient and MLE. Benefits of WLS

32.  BUT … How do we determine the weights when we do not know the variances? Problems with WLS (Cosma Shalizi, Oct. 2009)

33.  What if … The Oracle may be out or too creepy to visit. Problems with WLS (Cosma Shalizi, Oct. 2009)

34. Weighted Least Squares

35.  Transformations  Weighted Least Squares  OLS with corrected standard errors  MLE Solutions

36. Ordinary Least Squares (Paul Johnson, Oct. 2005)

37. OLS With Corrected Standard Errors (Consistent Covariance Matrix)

38. Other Solutions

39. Story Definition Detection Consequences Solutions Wrapping Up

40. Dr. Westfall Niels-Hugo Blunch, 2011, Using Econometrics A Practical Guide, Pearson. Walter Sosa-Escudero, 2009, Heteroscedasticity and WLS http://halweb.uc3m.es/esp/Personal/personas/durban/esp/web/notes/gls.pdf Ingo Ruczinski webpage: http://www.biostat.jhsph.edu/~iruczins/teaching/jf/ch5.pdfhttp://www.biostat.jhsph.edu/~iruczins/teaching/jf/ch5.pdf Paul Johnson, Oct. 2005. Maria L. Durban Reguera, reading material: http://halweb.uc3m.es/esp/Personal/personas/durban/esp/web/notes/gls.pdf http://halweb.uc3m.es/esp/Personal/personas/durban/esp/web/notes/gls.pdf Cosma Shalizi, Oct. 2009, Extending Linear Regression: Weighted Least Squares, Heteroskedasticity, Local Polynomial Regression Westfall and Henning, 2013, Understanding Advanced Statistical Methods Drew Dimmery, April 2012, Robust SEs in R, http://www.drewdimmery.com/robust- ses-in-r/ Reference Page

41. Slide 4: http://stats.stackexchange.com/questions/86788/why-do-we-say-that-the-variance-of- the-error-terms-is-constant/86792#86792 Slide 5: http://www.statsblogs.com/2015/09/10/plot-the-conditional-distribution-of-the- response-in-a-linear-regression-model/ Slide 6: http://www.statsblogs.com/2015/09/16/error-distributions-and-exponential- regression-models/ Slides 33 and 34: Cosma Shalizi, Oct. 2009. Photo Credit Page

42. Thank you & Enjoy Your Spring Break!