1. Linear Regression Basics III Violating Assumptions Fin250f: Lecture 7.2 Spring 2010 Brooks, chapter 4(skim) 4.1-2, 4.4, 4.5, 4.7, 4.9-13

2. Outline  Violating assumptions  Parameter stability  Model building

3. OLS Assumptions  Error variances  Error correlations  Error normality  Functional forms and linearity  Omitting variables  Adding irrelevant variables

4. Error Variance

5. Error Variance Which is a bigger error? * * * * * * Y X

6. Error Correlations  Patterns in residuals  Plot residuals/residual diagnostics  Further modeling necessary If you can forecast u(t+1), need to work harder

7. Error Normality  Skewness and kurtosis in residuals  Testing Plots Bera-Jarque test  How can this impact results?

8. Bera-Jarque Test for Normality

9. Nonnormal Errors: Impact  For some theory: No  In practice can be big problem  Many extreme data points  Forecasting models work hard to fit these extreme outliers  Some solutions: Drop data points Robust forecast objectives (absolute errors)

10. Functional Forms  Y=a+bX  Actual function is nonlinear  Several types of diagnostics Higher order (squared) terms (RESET) Think about specific nonlinear models  Neural networks  Threshold models  Tricky: More later

11. Omitting Variables Leave out x(2) If it is correlated with x(1) this is a problem. Beta(1) will be biased and inconsistent. Forecast will not be optimal

12. Irrelevant Variables  Overfitting/data snooping Model fits to noise  Impacts standard errors for coefficients  Coefficients still consistent and unbiased

13. Parameter Stability  Known break point Chow test Predictive failure test  Unknown break Quant likelihood ratio test Recursive least squares

14. Chow Test

16. Predictive Failure

18. Unknown Breaks  Search for break  Look for maximum Chow level  Distribution is tricky Monte-carlo/bootstrap

19. Recursive/rolling estimation  Recursive Estimate (1,T1) move T1 to full sample T See if parameters converge  Rolling Roll bands (t-T,t) through data Watch parameters move through time  We’ll use some of these

20. Pure Out of Sample Tests  Estimate parameters over (1,T1)  Get errors over (T1+1,T)

21. Model Construction  General -> specific Less financial theory More statistics Problems: large unwieldy models  Simple -> general More theory at the start Problems: can leave out important stuff