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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. Linear Regression Basics III Violating Assumptions. Outline. Violating assumptions Parameter stability Model building. OLS Assumptions. Error variances Error correlations Error normality

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Linear Regression Basics III Violating Assumptions

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

  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 VarianceWhich 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

  15. Chow Test

  16. Predictive Failure

  17. 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

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