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Hypothesis Testing and Confidence Intervals in Multiple Regression

This chapter discusses hypothesis testing and confidence intervals for single coefficients, joint hypotheses, and multiple coefficients in multiple regression. It includes examples using the California class size data and explains the F-statistic and its distribution. The chapter also covers testing single restrictions on multiple coefficients and constructing joint confidence sets.

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Hypothesis Testing and Confidence Intervals in Multiple Regression

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  1. Chapter 7 Hypothesis Tests and Confidence Intervals in Multiple Regression

  2. Outline

  3. Hypothesis Tests and Confidence Intervals for a Single Coefficient in Multiple Regression (SW Section 7.1)

  4. Example: The California class size data

  5. Standard errors in multiple regression in STATA

  6. Tests of Joint Hypotheses(SW Section 7.2)

  7. Tests of joint hypotheses, ctd.

  8. Why can’t we just test the coefficients one at a time?

  9. Suppose t1and t2 are independent (for this calculation).

  10. The F-statistic

  11. The F-statistic testing 1 and 2:

  12. Large-sample distribution of the F-statistic

  13. Computing the p-value using the F-statistic:

  14. F-test example, California class size data:

  15. The “restricted” and “unrestricted” regressions

  16. Simple formula for the homoskedasticity-only F-statistic:

  17. Example:

  18. The homoskedasticity-only F-statistic – summary

  19. Digression: The F distribution

  20. The Fq,n–k–1 distribution:

  21. Another digression: A little history of statistics…

  22. A little history of statistics, ctd…

  23. Summary: the homoskedasticity-only F-statistic and the F distribution

  24. Summary: testing joint hypotheses

  25. Testing Single Restrictions on Multiple Coefficients (SW Section 7.3)

  26. Testing single restrictions on multiple coefficients, ctd.

  27. Method 1:Rearrange (“transform”) the regression

  28. Rearrange the regression, ctd.

  29. Method 2: Perform the test directly

  30. Confidence Sets for Multiple Coefficients (SW Section 7.4)

  31. Joint confidence sets ctd.

  32. Confidence set based on inverting the F-statistic

  33. An example of a multiple regression analysis – and how to decide which variables to include in a regression…

  34. A general approach to variable selection and “model specification”

  35. Digression about measures of fit…

  36. Back to the test score application:

  37. More California data…

  38. Digression on presentation of regression results

  39. Summary: Multiple Regression

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