1. Feb 21, 2006Lecture 6Slide #1 Adjusted R 2, Residuals, and Review Adjusted R 2 Residual Analysis Stata Regression Output revisited –The Overall Model –Analyzing Residuals Review for Exam 2

2. Feb 21, 2006Lecture 6Slide #2 Homework Review Use variables 87-89 to make an “egalitarian” index for your dependent variable (Y) Use p98_ideo (ideology) as the independent variable (X) to predict egalitarianism. Fully interpret the results

3. Feb 21, 2006Lecture 6Slide #3 Adjusted R 2 : An Alternative “Goodness of Fit” Measure Recall that R 2 is calculated as: Hypothetically, as K approaches n, R 2 approaches one (why?) – “degrees of freedom” Adjusted R 2 compensates for that tendency “explained sum of squares”“total sum of squares”

4. Feb 21, 2006Lecture 6Slide #4 Calculating Adjusted R 2 The bigger the sample size (n), the smaller the adjustment The more complex the model (the bigger K is), the larger the adjustment The bigger R 2 is, the smaller the adjustment

5. Feb 21, 2006Lecture 6Slide #5 Residual Analysis: Trouble Shooting Conceptual use of residuals –e, or what the model can’t explain Visual Diagnostics –Ideal: a “Sneeze plot” –Diagnostics using Residual Plots: Checking for heteroscedasticity Checking for non-linearity Checking for outliers Saving and Analyzing Residuals in Stata

6. Feb 21, 2006Lecture 6Slide #6 Review: Assumptions Necessary for Estimating Linear Models 1.Errors have identical distributions Zero mean, same variance, across the range of X 2.Errors are independent of X and other  i 3.Errors are normally distributed   i =0 X

7. Feb 21, 2006Lecture 6Slide #7 The Ideal: Sneeze Splatter e Predicted Y Problems: It is possible to “over-interpret” residual plots; it is also possible to miss patterns when there are large numbers of observations

8. Feb 21, 2006Lecture 6Slide #8 Heteroscedasticity e Predicted Y Problem: Standard errors are not constant; hypothesis tests invalid

9. Feb 21, 2006Lecture 6Slide #9 Non-Linearity e Predicted Y Problem: Biased estimated coefficients, inefficient model

10. Feb 21, 2006Lecture 6Slide #10 Checking for Outliers e Predicted Y Problem: Under-specified model; measurement error Residuals for model using all data Possible Outliers Residuals for model with outliers deleted

11. Feb 21, 2006Lecture 6Slide #11 Stata Regression Model: Regressing “Militant” onto “Egal” Y is index of 81, 82 (rev’d), 84-86: 1=low militant; 7=high militant X is “Egal” -- an index of 87-89 1=reject egal; 7=accept egal

12. Feb 21, 2006Lecture 6Slide #12 Regression Output

13. Feb 21, 2006Lecture 6Slide #13 Regression Plot

14. Feb 21, 2006Lecture 6Slide #14 Predicted Case Confidence Bands

15. Feb 21, 2006Lecture 6Slide #15 Residual Plot Probable Outliers

16. Feb 21, 2006Lecture 6Slide #16 Examination of Residuals gsort e (or you can use “-e”) list resp_id egal militant yhat e in 1/5 Use the respondent ID number to find the relevant observation in the data set

17. Feb 21, 2006Lecture 6Slide #17 Residuals v. Predicted Values Using an “ocular test,” neither non-linearity nor heteroscedasticity are obvious here. But should we trust our eyeballs?

18. Feb 21, 2006Lecture 6Slide #18 Formal Test for Non-linearity: Omitted Variables Tests whether adding 2nd, 3rd and 4th powers of X will improve the fit of the model: Y=b 0 +b 1 X+b 2 X 2 +b 3 X 3 +b 4 X 4 +e

19. Feb 21, 2006Lecture 6Slide #19 Formal Tests for Heteroscedasticity Tests to see whether the squared standardized residuals are linearly related to the predicted value of Y: std(e 2 )=b 0 +b 1 (Predicted Y)

20. Feb 21, 2006Lecture 6Slide #20 Case-wise Influence Analysis The Leverage versus Squared Residual Plot

21. Feb 21, 2006Lecture 6Slide #21 What to Do? Nonlinearity –Polynomial regression: try X and X 2 –Variable transformation: logged variables –Use non-OLS regression Heteroscedasticity –Re-specify model Omitted variables? Use non-OLS regression (WLS) Influential and Deviant Cases –Evaluate the cases –Run with controls (multivariate model) –Omit cases (last option)

22. Feb 21, 2006Lecture 6Slide #22 Review Basic statistics and functions Calculus –Derivatives and critical values Deriving b 1 and b 0 –Intuitive meaning of formulas Interpreting OLS bivariate regression output Basic regression diagnostics