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Multiple Regression & OLS violations

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  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

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

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

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

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

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

  19. 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

  20. 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

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

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

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

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

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

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

  27. Practice 2:Significant Coefficients in Tables

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

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

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

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

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

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

  34. Practice 3:Interpreting Coefficients

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

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

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

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

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

  40. Participant Scores

  41. Fastest Responders (in seconds)

  42. Team Scores

  43. Team MVP

  44. OLS Violations & Other Issues

  45. 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

  46. 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

  47. Example: The Challenger disaster

  48. Example: Challenger Shuttle disaster 30°