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## Multiple Regression & OLS violations

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**Multiple Regression & OLS violations**Week 4 Lecture MG461 Dr. Meredith Rolfe**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**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**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**Simple Linear Model**MG461, Week 3 Seminar**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.**Which best accounts for variation in supervisor ratings?**Does not allow special privileges. Opportunity to learn. Too critical of poor performance. Handles employee complaints.**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**SPSS output -> Regression Table**βhat0 βhat1 se(βhat0) se(βhat1) t(βhat0-0) t(βhat1-0) ignore x variable**42% of employees value supervisors who don’t grant special**privileges? • Yes • No 32% 68%**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**Are these good estimates of the relationship between x and**y? Yes No**Multiple potential explanations…**• Experimental Controls: • Random assignment • Experimental Design • Observational data analysis: • Statistical Controls**Multiple Regression Model**Observation or data point, i, goes from 1…n Error Intercept Coefficients Dependent Variable Independent Variables MG461, Week 3 Seminar**Which model parameter do we NOT need to estimate?**Β0 x1,i βp σ2**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**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**1) p-values & significance**2) Coefficients significant from tables 2) substantive interpretation of coefficients Speed Practice: Interpreting Regression Results**Does “Critical” have an effect on supervisor ratings?**33% 67% 0% 0% • Yes • No Countdown**Does Income have an effect on Immigration Rate?**50% 50% 0% 0% • Yes • No Countdown**Does having a HS Degree affect salary?**0% 0% • Yes • No 10 Countdown**Do strike outs affect salary?**95% 5% 0% 0% • Yes • No Countdown**Does %Female affect Cigarette Sales?**11% 89% 0% 0% • Yes • No Countdown**Does Total Employment affect CEO Compensation?**• Yes • No 86% 14% Countdown**Does Restructuring Affect Firm ROA?**• Yes • No 14% 86% Countdown**Does firm sales growth affect the length of CEO tenure?**• Yes • No 75% 25% Countdown**Does Total Employment affect CEO Compensation?**• Yes • No 82% 18% Countdown**Are employees more aggressive when their job is stressful?**• Yes • No 44% 56% Countdown**Does employee turnover affect Firm Productivity?**• Yes • No 91% 9% Countdown**High values of 1983 centralization product a(n) ….. in**current centralization • Increase • Decrease 2% 98% Countdown**Corporations are more likely to enter petitions when their**market share is… • High • Low 81% 19% Countdown**Starting compensation is a good predictor of current**compensation? • True • False 68% 32% Countdown**Managers at larger firms get paid more?**• True • False 18% 82% Countdown**More centralized companies invest more in Research?**• True • False 60% 40% Countdown**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**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