1. Multiple Regression & OLS violations Week 4 Lecture MG461 Dr. Meredith Rolfe

2. Which group are you in? Which group are you in? Group 1 Group 2 Group 3 Group 4 Group 5 Group 6 Group 7 Group 8

3. Key Goals of the Week • What is multiple regression? • How to interpret regression results: • estimated regression coefficients • significance tests for coefficients • Violations of OLS assumptions • Diagnostics • What to do MG461, Week 3 Seminar

4. Multiple Regression

5. When to use Regression We want to know whether the outcome, y, varies depending on x Continuous variables (but many exceptions) Observational data (mostly) The relationship between x and y is linear MG461, Week 3 Seminar

6. Simple Linear Model MG461, Week 3 Seminar

7. Regression is a set of statistical tools to model the conditional expectation… of one variable on another variable. of one variable on one or more other variables.

8. Multiple Regression

9. Which best accounts for variation in supervisor ratings? Does not allow special privileges. Opportunity to learn. Too critical of poor performance. Handles employee complaints.

10. Simple linear model: Rating vs. No Special Privileges Source: Chatterjee et al, Regression Analysis by Example • Note on significance of coefficients: • ***p < 0.001 • **p < 0.01 • *p < 0.05 • . p < 0.1

11. SPSS output -> Regression Table βhat0 βhat1 se(βhat0) se(βhat1) t(βhat0-0) t(βhat1-0) ignore x variable

12. 42% of employees value supervisors who don’t grant special privileges? • Yes • No 32% 68%

13. Simple linear model #2:Rating vs. Opportunity to Learn • Note on significance of coefficients: • ***p < 0.001 • **p < 0.01 • *p < 0.05 • . p < 0.1

15. Are these good estimates of the relationship between x and y? Yes No

16. Multiple potential explanations… • Experimental Controls: • Random assignment • Experimental Design • Observational data analysis: • Statistical Controls

17. Multiple Regression Model Observation or data point, i, goes from 1…n Error Intercept Coefficients Dependent Variable Independent Variables MG461, Week 3 Seminar

18. Which model parameter do we NOT need to estimate? Β0 x1,i βp σ2

19. Multiple RegressionOLS Estimates (matrix) Y = Xβ +ε

21. Significance of Results Model Significance Coefficient Significance H0: ß1=0, there is no relationship (covariation) between x and y HA: ß1≠0, there is a relationship (covariation) between x and y Application: a single estimated coefficient Test: t-test **assumes errors (ei) are normally distributed • H0: None of the 1 (or more) independent variables covary with the dependent variable • HA: At least one of the independent variables covaries with d.v. • Application: compare two fitted models • Test: Anova/F-Test • **assumes errors (ei) are normally distributed MG461, Week 3 Seminar

22. Comparing Models: Anova Anova Model Comparison All Variables (Full) vs. Complaints & Learn: F=0.53 p=0.72 Complaints & Learn vs. Complaints: F=2.47 p=0.13

23. 1) p-values & significance 2) Coefficients significant from tables 2) substantive interpretation of coefficients Speed Practice: Interpreting Regression Results

24. Does “Critical” have an effect on supervisor ratings? 33% 67% 0% 0% • Yes • No Countdown

25. Does Income have an effect on Immigration Rate? 50% 50% 0% 0% • Yes • No Countdown

26. Does having a HS Degree affect salary? 0% 0% • Yes • No 10 Countdown

27. Do strike outs affect salary? 95% 5% 0% 0% • Yes • No Countdown

28. Does %Female affect Cigarette Sales? 11% 89% 0% 0% • Yes • No Countdown

29. Practice 2:Significant Coefficients in Tables

30. Does Total Employment affect CEO Compensation? • Yes • No 86% 14% Countdown

31. Does Restructuring Affect Firm ROA? • Yes • No 14% 86% Countdown

32. Does firm sales growth affect the length of CEO tenure? • Yes • No 75% 25% Countdown

33. Does Total Employment affect CEO Compensation? • Yes • No 82% 18% Countdown

34. Are employees more aggressive when their job is stressful? • Yes • No 44% 56% Countdown

35. Does employee turnover affect Firm Productivity? • Yes • No 91% 9% Countdown

36. Practice 3:Interpreting Coefficients

37. High values of 1983 centralization product a(n) ….. in current centralization • Increase • Decrease 2% 98% Countdown

38. Corporations are more likely to enter petitions when their market share is… • High • Low 81% 19% Countdown

39. Starting compensation is a good predictor of current compensation? • True • False 68% 32% Countdown

40. Managers at larger firms get paid more? • True • False 18% 82% Countdown

41. More centralized companies invest more in Research? • True • False 60% 40% Countdown

42. Participant Scores

43. Fastest Responders (in seconds)

44. Team Scores

45. Team MVP

46. OLS Violations & Other Issues

47. Assumptions of OLS Regression • . • correctly specified model • linear relationship • Errors are normally distributed • Errors have mean of 0: E(εi)=0 • Homoscedastic: Var(εi)=σ2 • Uncorrelated Errors: Cov(εi,εi)=0 • No multicollinearity MG461, Week 3 Seminar

48. When is a model linear? • Linear in the parameters • Transformations of x and/or y variables can turn a relationship that isn’t linear initially into one that is linear in the parameters

49. Example: The Challenger disaster

50. Example: Challenger Shuttle disaster 30°