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

2. 2 Outline

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

4. 4 Example: The California class size data

5. 5 Standard errors in multiple regression in STATA

6. 6 Tests of Joint Hypotheses (SW Section 7.2)

7. 7 Tests of joint hypotheses, ctd.

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

9. 9 Suppose t 1 and t 2 are independent (for this calculation).

10. 10

11. 11 The F-statistic

12. 12 The F-statistic testing  1 and  2 :

13. 13 Large-sample distribution of the F-statistic

14. 14

15. 15 Computing the p-value using the F-statistic:

16. 16 F-test example, California class size data:

17. 17

18. 18 The “restricted” and “unrestricted” regressions

19. 19 Simple formula for the homoskedasticity-only F-statistic:

20. 20 Example:

21. 21 The homoskedasticity-only F-statistic – summary

22. 22 Digression: The F distribution

23. 23 The F q,n–k–1 distribution:

24. 24 Another digression: A little history of statistics…

25. 25 A little history of statistics, ctd…

26. 26 Summary: the homoskedasticity-only F- statistic and the F distribution

27. 27 Summary: testing joint hypotheses

28. 28 Testing Single Restrictions on Multiple Coefficients (SW Section 7.3)

29. 29 Testing single restrictions on multiple coefficients, ctd.

30. 30 Method 1: Rearrange (“transform”) the regression

31. 31 Rearrange the regression, ctd.

32. 32 Method 2: Perform the test directly

33. 33 Confidence Sets for Multiple Coefficients (SW Section 7.4)

34. 34 Joint confidence sets ctd.

35. 35

36. 36 Confidence set based on inverting the F-statistic

37. 37 An example of a multiple regression analysis – and how to decide which variables to include in a regression…

38. 38 A general approach to variable selection and “model specification”

39. 39 Digression about measures of fit…

40. 40 Back to the test score application:

41. 41 More California data…

42. 42 Digression on presentation of regression results

43. 43

44. 44 Summary: Multiple Regression