1. Chapter 8 Nonlinear Regression Functions

2. 2 Nonlinear Regression Functions (SW Chapter 8)

3. 3 The TestScore – STR relation looks linear (maybe)…

4. 4 But the TestScore – Income relation looks nonlinear...

5. 5 Nonlinear Regression Population Regression Functions – General Ideas (SW Section 8.1)

6. 6 The general nonlinear population regression function

7. 7

8. 8 Nonlinear Functions of a Single Independent Variable (SW Section 8.2)

9. 9 1. Polynomials in X

10. 10 Example: the TestScore – Income relation

11. 11 Estimation of the quadratic specification in STATA

12. 12 Interpreting the estimated regression function:

13. 13 Interpreting the estimated regression function, ctd:

14. 14

15. 15 Estimation of a cubic specification in STATA

16. 16

17. 17 Summary: polynomial regression functions

18. 18 2. Logarithmic functions of Y and/or X

19. 19 The three log regression specifications:

20. 20 I. Linear-log population regression function

21. 21 Linear-log case, continued

22. 22 Example: TestScore vs. ln(Income)

23. 23 The linear-log and cubic regression functions

24. 24 II. Log-linear population regression function

25. 25 Log-linear case, continued

26. 26 III. Log-log population regression function

27. 27 Log-log case, continued

28. 28 Example: ln( TestScore) vs. ln( Income)

29. 29 Example: ln( TestScore) vs. ln( Income), ctd.

30. 30 The log-linear and log-log specifications:

31. 31 Summary: Logarithmic transformations

32. 32 Other nonlinear functions (and nonlinear least squares) (SW App. 8.1)

33. 33 Negative exponential growth

34. 34 Nonlinear Least Squares

35. 35

36. 36

37. 37 Interactions Between Independent Variables (SW Section 8.3)

38. 38 (a) Interactions between two binary variables

39. 39 Interpreting the coefficients

40. 40 Example: TestScore, STR, English learners

41. 41 (b) Interactions between continuous and binary variables

42. 42 Binary-continuous interactions: the two regression lines

43. 43 Binary-continuous interactions, ctd.

44. 44 Interpreting the coefficients

45. 45 Example: TestScore, STR, HiEL (=1 if PctEL  10)

46. 46 Example, ctd: Testing hypotheses

47. 47 (c) Interactions between two continuous variables

48. 48 Interpreting the coefficients:

49. 49 Example: TestScore, STR, PctEL

50. 50 Example, ctd: hypothesis tests

51. 51 Application: Nonlinear Effects on Test Scores of the Student-Teacher Ratio (SW Section 8.4)

52. 52 Strategy for Question #1 (different effects for different STR?)

53. 53 Strategy for Question #2 (interactions between PctEL and STR?)

54. 54 What is a good “base” specification?

55. 55

56. 56 Tests of joint hypotheses:

57. 57 Interpreting the regression functions via plots:

58. 58 Next, compare the regressions with interactions:

59. 59 Summary: Nonlinear Regression Functions