The "Instrumental Variables" webinar, presented by Peter Lance, was the fifth and final webinar in a series of discussions on the popular MEASURE Evaluation manual, How Do We Know If a Program Made a Difference? A Guide to Statistical Methods for Program Impact Evaluation.

1. Peter M. Lance, PhD
MEASURE Evaluation
University of North Carolina at
Chapel Hill
April 13, 2017
Instrumental Variables

2. Peter M. Lance, PhD
MEASURE Evaluation
University of North Carolina at
Chapel Hill
April 13, 2017
Instrumental Variables:
Too LATE for Experimental Compliance?

3. Peter M. Lance, PhD
MEASURE Evaluation
University of North Carolina at
Chapel Hill
April 13, 2017
Instrumental Variables:
Too LATE for Experimental Compliance?

4. Peter M. Lance, PhD
MEASURE Evaluation
University of North Carolina at
Chapel Hill
April 13, 2017
Instrumental Variables:
Too LATE for Experimental Compliance?

5. Peter M. Lance, PhD
MEASURE Evaluation
University of North Carolina at
Chapel Hill
April 13, 2017
Instrumental Variables:
Too LATE for Experimental Compliance?

6. Peter M. Lance, PhD
MEASURE Evaluation
University of North Carolina at
Chapel Hill
April 13, 2017
Instrumental Variables:
Too LATE for Experimental Compliance?

7. Peter M. Lance, PhD
MEASURE Evaluation
University of North Carolina at
Chapel Hill
April 13, 2017
Instrumental Variables:
Too LATE for Experimental Compliance?

8. Peter M. Lance, PhD
MEASURE Evaluation
University of North Carolina at
Chapel Hill
April 13, 2017
Instrumental Variables:
Too LATE for Experimental Compliance?

9. Peter M. Lance, PhD
MEASURE Evaluation
University of North Carolina at
Chapel Hill
April 13, 2017
Instrumental Variables:
Too LATE for Experimental Compliance?

10. Peter M. Lance, PhD
MEASURE Evaluation
University of North Carolina at
Chapel Hill
April 13, 2017
Instrumental Variables:
Too LATE for Experimental Compliance?

11. Peter M. Lance, PhD
MEASURE Evaluation
University of North Carolina at
Chapel Hill
April 13, 2017
Instrumental Variables:
Too LATE for Experimental Compliance?

12. Global, five-year, $180M cooperative agreement
Strategic objective:
To strengthen health information systems – the
capacity to gather, interpret, and use data – so
countries can make better decisions and sustain good
health outcomes over time.
Project overview

13. Improved country capacity to manage health
information systems, resources, and staff
Strengthened collection, analysis, and use of
routine health data
Methods, tools, and approaches improved and
applied to address health information challenges
and gaps
Increased capacity for rigorous evaluation
Phase IV Results Framework

14. Global footprint (more than 25 countries)

15. • The program impact evaluation challenge
• Randomization
• Selection on observables
• Within estimators
• Instrumental variables

16. • The program impact evaluation challenge
• Randomization
• Selection on observables
• Within estimators
• Instrumental variables

19. YP
μ
X

20. YP
μ
X

21. YP
μ
X

22. YP
μ
X

23. YP
μ
X

24. YP
μ
X

25. YP
μ
Z
X

26. YP
μ
Z
X

27. YP
μ
Z
X

28. YP
μ
Z
X

29. YP
μ
Z
X

30. YP
μ
Z
X

31. YP
μ
Z
X

32. YP
μ
Z
X

33. Fascinating, right???
So what the heck does it
all mean, and how would
you implement it?

34. Fascinating, right???
So what the heck does it
all mean, and how would
you implement it?

35. 𝑌0 = 𝛽0 + 𝛽2 ∙ 𝑥 + 𝛽3 ∙ 𝜇 + 𝜀
𝑌1 = 𝛽0 + 𝛽1 + 𝛽2 ∙ 𝑥 + 𝛽3 ∙ 𝜇 + 𝜀

36. 𝑌0 = 𝛽0 + 𝛽2 ∙ 𝒙 + 𝛽3 ∙ 𝜇 + 𝜀
𝑌1 = 𝛽0 + 𝛽1 + 𝛽2 ∙ 𝒙 + 𝛽3 ∙ 𝜇 + 𝜀

37. 𝑌0 = 𝛽0 + 𝛽2 ∙ 𝑥 + 𝛽3 ∙ 𝝁 + 𝜀
𝑌1 = 𝛽0 + 𝛽1 + 𝛽2 ∙ 𝑥 + 𝛽3 ∙ 𝝁 + 𝜀

38. 𝑌0 = 𝛽0 + 𝛽2 ∙ 𝑥 + 𝛽3 ∙ 𝜇 + 𝜺
𝑌1 = 𝛽0 + 𝛽1 + 𝛽2 ∙ 𝑥 + 𝛽3 ∙ 𝜇 + 𝜺

39. 𝑌1 − 𝑌0
= 𝛽0 + 𝛽1 + 𝜖 − 𝛽0 + 𝜖
= 𝛽0 + 𝛽1 + 𝜖 − 𝛽0 − 𝜖
= 𝛽1

40. 𝑌1 − 𝑌0
= 𝛽0 + 𝛽1 + 𝛽2 ∙ 𝑥 + 𝛽3 ∙ 𝜇 + 𝜀
− 𝛽0 + 𝛽2 ∙ 𝑥 + 𝛽3 ∙ 𝜇 + 𝜀
= 𝛽0 + 𝛽1 + 𝜖 − 𝛽0 − 𝜖
= 𝛽1

41. 𝑌1 − 𝑌0
= 𝛽1

42. 𝑌 = 𝑃 ∙ 𝑌1
+ 1 − 𝑃 ∙ 𝑌0
= 𝑃 ∗ 𝛽0 + 𝛽1 + 𝜖 + 1 − 𝑃 ∗ 𝛽0 + 𝜖
= 𝑃 ∗ 𝛽0 + 𝑃 ∗ 𝛽1 + 𝑃 ∗ 𝜖
+𝛽0 + 𝜖 − 𝑃 ∗ 𝛽0 − 𝑃 ∗ 𝜖
= 𝛽0 + 𝑃 ∗ 𝛽1 + 𝜖

43. 𝑌 = 𝑃 ∙ 𝑌1
+ 1 − 𝑃 ∙ 𝑌0
= 𝑃 ∙ 𝛽0 + 𝛽1 + 𝛽2 ∙ 𝑥 + 𝛽3 ∙ 𝜇 + 𝜀
+ 1 − 𝑃 ∙ 𝛽0 + 𝛽2 ∙ 𝑥 + 𝛽3 ∙ 𝜇 + 𝜀
= 𝑃 ∗ 𝛽0 + 𝑃 ∗ 𝛽1 + 𝑃 ∗ 𝜖
+𝛽0 + 𝜖 − 𝑃 ∗ 𝛽0 − 𝑃 ∗ 𝜖

44. 𝑌 = 𝛽0 + 𝛽1 ∙ 𝑃 + 𝛽2 ∙ 𝑥 + 𝛽3 ∙ 𝜇 + 𝜀

45. 𝑌 = 𝛽0 + 𝛽1 ∙ 𝑃 + 𝛽2 ∙ 𝑥 + 𝛽3 ∙ 𝜇 + 𝜀

46. 𝑌 = 𝛽0 + 𝛽1 ∙ 𝑃 + 𝛽2 ∙ 𝑥 + 𝛽3 ∙ 𝜇 + 𝜀
𝑌 = 𝛾0 + 𝛾1 ∙ 𝑃 + 𝛾2 ∙ 𝑥 + 𝜀
𝐸 𝛾1 = 𝛽1

47. 𝑌 = 𝛽0 + 𝛽1 ∙ 𝑃 + 𝛽2 ∙ 𝑥 + 𝛽3 ∙ 𝜇 + 𝜀
𝑌 = 𝛾0 + 𝛾1 ∙ 𝑃 + 𝛾2 ∙ 𝑥 + 𝜀
𝐸 𝛾1 = 𝛽1
?

48. 𝑌 = 𝛽0 + 𝛽1 ∙ 𝑃 + 𝛽2 ∙ 𝑥 + 𝛽3 ∙ 𝜇 + 𝜀
𝑌 = 𝛾0 + 𝛾1 ∙ 𝑃 + 𝛾2 ∙ 𝑥 + 𝜀
𝐸 𝛾1 = 𝛽1
?

49. 𝑌 = 𝛽0 + 𝛽1 ∙ 𝑃 + 𝛽2 ∙ 𝑥 + 𝛽3 ∙ 𝜇 + 𝜀
𝑌 = 𝛾0 + 𝛾1 ∙ 𝑃 + 𝛾2 ∙ 𝑥 + 𝜀
𝐸 𝛾1 = 𝛽1
?

50. YP
μ
X

51. YP
μ
X

52. 𝑌 = 𝛽0 + 𝛽1 ∙ 𝑃 + 𝛽2 ∙ 𝑥 + 𝛽3 ∙ 𝜇 + 𝜀
𝑌 = 𝛾0 + 𝛾1 ∙ 𝑃 + 𝛾2 ∙ 𝑥 + 𝜀
𝐸 𝛾1 = 𝛽1
?

53. 𝑌 = 𝛽0 + 𝛽1 ∙ 𝑃 + 𝛽2 ∙ 𝑥 + 𝛽3 ∙ 𝜇 + 𝜀
𝑌 = 𝛾0 + 𝛾1 ∙ 𝑃 + 𝛾2 ∙ 𝑥 + 𝜀
𝐸 𝛾1 = 𝛽1
?

54. 𝑌 = 𝛽0 + 𝛽1 ∙ 𝑃 + 𝛽2 ∙ 𝑥 + 𝛽3 ∙ 𝜇 + 𝜀
𝑌 = 𝛾0 + 𝛾1 ∙ 𝑃 + 𝛾2 ∙ 𝑥 + 𝜀
𝐸 𝛾1 = 𝛽1
?

55. 𝑌 = 𝛽0 + 𝛽1 ∙ 𝑃 + 𝛽2 ∙ 𝑥 + 𝛽3 ∙ 𝜇 + 𝜀
𝑌 = 𝛾0 + 𝛾1 ∙ 𝑃 + 𝛾2 ∙ 𝑥 + 𝜀
𝐸 𝛾1 ≠ 𝛽1

56. 𝑌 = 𝛽0 + 𝛽1 ∙ 𝑃 + 𝛽2 ∙ 𝑥 + 𝛽3 ∙ 𝜇 + 𝜀
𝑌 = 𝛾0 + 𝛾1 ∙ 𝑃 + 𝛾2 ∙ 𝑥 + 𝜀
𝐸 𝛾1 ≠ 𝛽1

57. 𝑌 = 𝛽0 + 𝛽1 ∙ 𝑃 + 𝛽2 ∙ 𝑥 + 𝛽3 ∙ 𝜇 + 𝜀
𝑌 = 𝛾0 + 𝛾1 ∙ 𝑃 + 𝛾2 ∙ 𝑥 + 𝜀
𝐸 𝛾1 ≠ 𝛽1

58. 𝑌 = 𝛽0 + 𝛽1 ∙ 𝑃 + 𝛽2 ∙ 𝑥 + 𝛽3 ∙ 𝜇 + 𝜀
𝑌 = 𝛾0 + 𝛾1 ∙ 𝑃 + 𝛾2 ∙ 𝑥 + 𝜀
𝐸 𝛾1 ≠ 𝛽1

59. Cost of Participation
𝐶 = 𝜌0 + 𝜌1 ∙ 𝑥 + 𝜌2 ∙ 𝜇 + 𝜌3 ∙ 𝑧 + 𝜂

60. Cost of Participation
𝐶 = 𝜌0 + 𝜌1 ∙ 𝒙 + 𝜌2 ∙ 𝜇 + 𝜌3 ∙ 𝑧 + 𝜂

61. Cost of Participation
𝐶 = 𝜌0 + 𝜌1 ∙ 𝑥 + 𝜌2 ∙ 𝝁 + 𝜌3 ∙ 𝑧 + 𝜂

62. Cost of Participation
𝐶 = 𝜌0 + 𝜌1 ∙ 𝑥 + 𝜌2 ∙ 𝜇 + 𝜌3 ∙ 𝒛 + 𝜂

63. Cost of Participation
𝐶 = 𝜌0 + 𝜌1 ∙ 𝑥 + 𝜌2 ∙ 𝜇 + 𝜌3 ∙ 𝑧 + 𝜼

64. Benefit-Cost>0
𝑌1
− 𝑌0
− C > 0
𝛽1 − 𝐶 > 0
𝛽1 − 𝜌0 + 𝜌1 ∙ 𝑥 + 𝜌2 ∙ 𝜇 + 𝜌3 ∙ 𝑧 + 𝜂 > 0

65. Benefit-Cost>0
𝑌1
− 𝑌0
− C > 0
𝛽1 − 𝐶 > 0
𝛽1 − 𝜌0 + 𝜌1 ∙ 𝑥 + 𝜌2 ∙ 𝜇 + 𝜌3 ∙ 𝑧 + 𝜂 > 0

66. Benefit-Cost>0
𝑌1
− 𝑌0
− C > 0
𝛽1 − 𝐶 > 0
𝛽1 − 𝜌0 + 𝜌1 ∙ 𝑥 + 𝜌2 ∙ 𝜇 + 𝜌3 ∙ 𝑧 + 𝜂 > 0

67. Benefit-Cost>0
𝑌1
− 𝑌0
− C > 0
𝛽1 − 𝐶 > 0
𝛽1 − 𝜌0 + 𝜌1 ∙ 𝑥 + 𝜌2 ∙ 𝜇 + 𝜌3 ∙ 𝑧 + 𝜂 > 0

68. 𝑥 𝑃
𝜇 𝑃
𝑧 𝑃

69. 𝑥 𝑃
𝜇 𝑃
𝑧 𝑃

70. YP
μ
X

71. Benefit-Cost>0
𝑌1
− 𝑌0
− C > 0
𝛽1 − 𝐶 > 0
𝛽1 − 𝜌0 + 𝜌1 ∙ 𝑥 + 𝜌2 ∙ 𝜇 + 𝜌3 ∙ 𝑧 + 𝜂 > 0

72. 𝑥 𝑃
𝜇 𝑃
𝑧 𝑃

73. 𝑌0 = 𝛽0 + 𝛽2 ∙ 𝑥 + 𝛽3 ∙ 𝜇 + 𝜀
𝑌1 = 𝛽0 + 𝛽1 + 𝛽2 ∙ 𝑥 + 𝛽3 ∙ 𝜇 + 𝜀

74. 𝑌 = 𝑃 ∙ 𝑌1
+ 1 − 𝑃 ∙ 𝑌0
= 𝑃 ∗ 𝛽0 + 𝛽1 + 𝜖 +

75. 𝑌 = 𝑷 ∙ 𝑌1
+ 1 − 𝑷 ∙ 𝑌0
= 𝑃 ∗ 𝛽0 + 𝛽1 + 𝜖 +

76. YP
μ
Z
X

77. Benefit-Cost>0
𝑌1
− 𝑌0
− C > 0
𝛽1 − 𝐶 > 0
𝛽1 − 𝜌0 + 𝜌1 ∙ 𝑥 + 𝜌2 ∙ 𝜇 + 𝜌3 ∙ 𝑧 + 𝜂 > 0

78. 𝛽1 − 𝜌0 + 𝜌1 ∙ 𝑥 + 𝜌2 ∙ 𝜇 + 𝜌3 ∙ 𝑧 + 𝜂 > 0
𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧 + 𝜉

79. 𝛽1 − 𝜌0 + 𝜌1 ∙ 𝑥 + 𝜌2 ∙ 𝜇 + 𝜌3 ∙ 𝑧 + 𝜼 > 0
𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧 + 𝜉

80. 𝛽1 − 𝜌0 + 𝜌1 ∙ 𝑥 + 𝜌2 ∙ 𝜇 + 𝜌3 ∙ 𝑧 + 𝜂 > 0
𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧 + 𝜉

81. 𝛽1 − 𝜌0 + 𝜌1 ∙ 𝑥 + 𝜌2 ∙ 𝜇 + 𝜌3 ∙ 𝑧 + 𝜂 > 0
𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧 + 𝝃

82. 𝛽1 − 𝜌0 + 𝜌1 ∙ 𝑥 + 𝜌2 ∙ 𝜇 + 𝜌3 ∙ 𝑧 + 𝜂 > 0
𝑃∗ = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧 + 𝜉

83. 𝛽1 − 𝜌0 + 𝜌1 ∙ 𝑥 + 𝜌2 ∙ 𝜇 + 𝜌3 ∙ 𝑧 + 𝜂 > 0
𝑃𝑟 𝑃 = 1 =
𝑒𝑥𝑝 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧
1 + 𝑒𝑥𝑝 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧

84. 𝛽1 − 𝜌0 + 𝜌1 ∙ 𝑥 + 𝜌2 ∙ 𝜇 + 𝜌3 ∙ 𝑧 + 𝜂 > 0
𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧 + 𝜉

85. 𝑌 = 𝛾0 + 𝛾1 ∙ 𝑃 + 𝛾2 ∙ 𝑥 + 𝜀
𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧 + 𝜉

86. 𝒀 = 𝜸 𝟎 + 𝜸 𝟏 ∙ 𝑷 + 𝜸 𝟐 ∙ 𝒙 + 𝜺
𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧 + 𝜉

87. 𝑌 = 𝛾0 + 𝛾1 ∙ 𝑃 + 𝛾2 ∙ 𝑥 + 𝜀
𝑷 = 𝝓 𝟎 + 𝝓 𝟏 ∙ 𝒙 + 𝝓 𝟐 ∙ 𝒛 + 𝝃

88. 𝑌 = 𝛾0 + 𝛾1 ∙ 𝑃 + 𝛾2 ∙ 𝑥 + 𝜀
𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧 + 𝜉

89. 𝑌 = 𝛾0 + 𝛾1 ∙ 𝑃 + 𝛾2 ∙ 𝑥 + 𝜀
𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧 + 𝜉
𝝁

90. 𝑌 = 𝛾0 + 𝛾1 ∙ 𝑃 + 𝛾2 ∙ 𝑥 + 𝜺
𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧 + 𝝃
𝝁

91. 𝑌 = 𝛾0 + 𝛾1 ∙ 𝑃 + 𝛾2 ∙ 𝑥 + 𝜺
𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧 + 𝝃
𝝁

92. 𝑌 = 𝛾0 + 𝛾1 ∙ 𝑃 + 𝛾2 ∙ 𝑥 + 𝜺
𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧 + 𝝃
𝝁

93. 𝑌 = 𝛾0 + 𝛾1 ∙ 𝑃 + 𝛾2 ∙ 𝑥 + 𝜺
𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧 + 𝝃
𝑬( 𝜸 𝟏) ≠ 𝜷 𝟏

94. Two-Stage Least Squares

95. Two-Stage Least Squares
1. Regress 𝑃 on 𝑥 and 𝑧 (i.e. estimate 𝜙s
from 𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧 + 𝜉)
2. Use the fitted/estimated model to predict
participation:
𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧
3. Regress 𝑌 on 𝑃 and 𝑥

96. Two-Stage Least Squares
1. Regress 𝑃 on 𝑥 and 𝑧 (i.e. estimate 𝜙s
from 𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧 + 𝜉)
2. Use the fitted/estimated model to predict
participation:
𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧
3. Regress 𝑌 on 𝑃 and 𝑥

97. Two-Stage Least Squares
1. Regress 𝑃 on 𝑥 and 𝑧 (i.e. estimate 𝜙s
from 𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧 + 𝜉)
2. Use the fitted/estimated model to predict
participation:
𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧
3. Regress 𝑌 on 𝑃 and 𝑥

98. Two-Stage Least Squares
1. Regress 𝑃 on 𝑥 and 𝑧 (i.e. estimate 𝜙s
from 𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧 + 𝜉)
2. Use the fitted/estimated model to predict
participation:
𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧
3. Regress 𝑌 on 𝑃 and 𝑥

99. Two-Stage Least Squares
1. Regress 𝑃 on 𝑥 and 𝑧 (i.e. estimate 𝜙s
from 𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧 + 𝜉)
2. Use the fitted/estimated model to predict
participation:
𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧
3. Regress 𝑌 on 𝑃 and 𝑥

100. 𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧 + 𝜉
𝜙2 = 0
𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜉
𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥
𝑌 = 𝛾0 + 𝛾1 ∙ 𝑃 + 𝛾2 ∙ 𝑥
-+ 𝜀

101. 𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧 + 𝜉
𝜙2 = 0
𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜉
𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥
𝑌 = 𝛾0 + 𝛾1 ∙ 𝑃 + 𝛾2 ∙ 𝑥
-+ 𝜀

102. 𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧 + 𝜉
𝜙2 = 0
𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜉
𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥
𝑌 = 𝛾0 + 𝛾1 ∙ 𝑃 + 𝛾2 ∙ 𝑥
-+ 𝜀

103. 𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧 + 𝜉
𝜙2 = 0
𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜉
𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥
𝑌 = 𝛾0 + 𝛾1 ∙ 𝑃 + 𝛾2 ∙ 𝑥
-+ 𝜀

104. 𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧 + 𝜉
𝜙2 = 0
𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜉
𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥
𝑌 = 𝛾0 + 𝛾1 ∙ 𝑃 + 𝛾2 ∙ 𝑥 + 𝜀
-+ 𝜀

105. 𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧 + 𝜉
𝜙2 = 0
𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜉
𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥
𝑌 = 𝛾0 + 𝛾1 ∙ ( 𝜙0 + 𝜙1 ∙ 𝑥) + 𝛾2 ∙ 𝑥 + 𝜀
-+ 𝜀

106. 𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧 + 𝜉
𝜙2 = 0
𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜉
𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥
𝑌 = 𝛾0 + 𝛾1 ∙ 𝜙0 + 𝛾1 ∙ 𝜙1 ∙ 𝑥 + 𝛾2 ∙ 𝑥 + 𝜀
-+ 𝜀

107. 𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧 + 𝜉
𝜙2 = 0
𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜉
𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥
𝑌 = 𝛾0 + 𝛾1 ∙ 𝜙0 + 𝛾1 ∙ 𝜙1 + 𝛾2 ∙ 𝑥 + 𝜀
-+ 𝜀

108. Two-Stage Least Squares
𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧 + 𝜉
𝑌 = 𝛿0 + 𝛿1 ∙ 𝑥 + 𝜀
where
𝛿0 = 𝛾0 + 𝛾1 ∙ 𝜙0
and
𝛿1 = 𝛾1 ∙ 𝜙1 + 𝛾2

109. Two-Stage Least Squares
𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧 + 𝜉
𝑌 = 𝛿0 + 𝛿1 ∙ 𝑥 + 𝜀
where
𝛿0 = 𝛾0 + 𝛾1 ∙ 𝜙0
and
𝛿1 = 𝛾1 ∙ 𝜙1 + 𝛾2

110. Two-Stage Least Squares
𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧 + 𝜉
𝑌 = 𝛿0 + 𝛾1 ∙ 𝜙1 + 𝛾2 ∙ 𝑥 + 𝜀
where
𝛿0 = 𝛾0 + 𝛾1 ∙ 𝜙0
and
𝛿1 = 𝛾1 ∙ 𝜙1 + 𝛾2

111. Two-Stage Least Squares
𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧 + 𝜉
𝑌 = 𝛿0 + 𝛾1 ∙ 𝜙1 ∙ 𝑥 + 𝛾2 ∙ 𝑥 + 𝜀
where
𝛿0 = 𝛾0 + 𝛾1 ∙ 𝜙0
and
𝛿1 = 𝛾1 ∙ 𝜙1 + 𝛾2

112. Two-Stage Least Squares
𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧 + 𝜉
𝑌 = 𝛿0 + 𝛿11 ∙ 𝑥 + 𝛿12 ∙ 𝑥 + 𝜀
where
𝛿0 = 𝛾0 + 𝛾1 ∙ 𝜙0
and
𝛿1 = 𝛾1 ∙ 𝜙1 + 𝛾2

113. Two-Stage Least Squares
𝑌0
= 𝛽0 + 𝛽2 ∙ 𝑥 + 𝛽3 ∙ 𝜇 + 𝜀
𝑌1
= 𝛽0 + 𝛽1 + 𝛽2 ∙ 𝑥 + 𝛽3 ∙ 𝜇 + 𝜀
𝐶
= 𝜌0 + 𝜌1 ∙ 𝑥 + 𝜌2 ∙ 𝜇 + 𝜌3 ∙ 𝑧1 + 𝜌4 ∙ 𝑧2 + 𝜂

114. Two-Stage Least Squares
1. Regress 𝑃 on 𝑥 and 𝑧 (i.e. estimate 𝜙s
from
𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧1 + 𝜙3 ∙ 𝑧2 + 𝜉
2. Use the fitted/estimated model to predict
participation:
𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧1 + 𝜙3 ∙ 𝑧2
3. Regress 𝑌 on 𝑃 and 𝑥

115. Two-Stage Least Squares
𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧 + 𝜉
𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧
𝑌 = 𝛾0 + 𝛾1 ∙ 𝑃 + 𝛾2 ∙ 𝑥 + 𝜀
𝛾0, 𝛾1, and 𝛾2

116. Two-Stage Least Squares
𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧 + 𝜉
𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧
𝑌 = 𝛾0 + 𝛾1 ∙ 𝑃 + 𝛾2 ∙ 𝑥 + 𝜀
𝛾0, 𝛾1, and 𝛾2

117. Two-Stage Least Squares
𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧 + 𝜉
𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧
𝑌 = 𝛾0 + 𝛾1 ∙ 𝑃 + 𝛾2 ∙ 𝑥 + 𝜀
𝛾0, 𝛾1, and 𝛾2

118. Two-Stage Least Squares
𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧 + 𝜉
𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧
𝑌 = 𝛾0 + 𝛾1 ∙ 𝑃 + 𝛾2 ∙ 𝑥 + 𝜀
𝛾0, 𝛾1, and 𝛾2

119. Two-Stage Least Squares
𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧 + 𝜉
𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧
𝑌 = 𝛾0 + 𝛾1 ∙ 𝑃 + 𝛾2 ∙ 𝑥 + 𝜀
𝛾0, 𝛾1, and 𝛾2

120. Two-Stage Least Squares
𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧 + 𝜉
𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧
𝑌 = 𝛾0 + 𝛾1 ∙ 𝑃 + 𝛾2 ∙ 𝑥 + 𝜀
𝛾0, 𝛾1, and 𝛾2

121. Two-Stage Least Squares
𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧 + 𝜉
𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧
𝑌 = 𝛾0 + 𝛾1 ∙ 𝑃 + 𝛾2 ∙ 𝑥 + 𝜀
𝛾0, 𝛾1, and 𝛾2

122. Two-Stage Least Squares
𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧 + 𝜉
𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧
𝑌 = 𝛾0 + 𝛾1 ∙ 𝑃 + 𝛾2 ∙ 𝑥 + 𝜀
𝛾0, 𝛾1, and 𝛾2

123. Two-Stage Least Squares
1. Regress 𝑃 on 𝑥 and 𝑧 (i.e. estimate 𝜙s
from 𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧 + 𝜉)
2. Use the fitted/estimated model to predict
participation:
𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧
3. Regress 𝑌 on 𝑃 and 𝑥

124. Two-Stage Least Squares
𝑌 = 𝛾0 + 𝛾1 ∙ 𝑃 + 𝛾2 ∙ 𝑥 + 𝜀
𝐸 𝛾0 = 𝛽1

126. Two-Stage Least Squares
𝑌 = 𝛾0 + 𝛾1 ∙ 𝑃 + 𝛾2 ∙ 𝑥 + 𝜀
𝐸 𝛾0 ≠ 𝛽1

127. Two-Stage Least Squares
𝑌 = 𝛾0 + 𝛾1 ∙ 𝑃 + 𝛾2 ∙ 𝑥 + 𝜀
𝐸 𝛾0 ≠ 𝛽1

128. ????!

129. Consistency
𝑝𝑙𝑖𝑚 𝛾1 = 𝛽1
𝑙𝑖𝑚 𝑛→∞ 𝑃𝑟 𝛾1 − 𝛽1 ≥ 𝑐 = 0

130. Consistency
𝑝𝑙𝑖𝑚 𝛾1 = 𝛽1
𝑙𝑖𝑚 𝑛→∞ 𝑃𝑟 𝛾1 − 𝛽1 ≥ 𝑐 = 0
= 0

133. 𝑌 = 𝛽0 + 𝛽1 ∙ 𝑃 + 𝛽2 ∙ 𝑥 + 𝛽3 ∙ 𝜇 + 𝜀

135. Some Big Take-Aways About IV/TSLS
1. More powerful instruments are better
2. Over-identification is (generally) your friend
3. Be careful about getting fancy with binary
endogenous regressors and outcomes
4. Be careful about being LATE

136. Some Big Take-Aways About IV/TSLS
1. More powerful instruments are better
2. Over-identification is (generally) your friend
3. Be careful about getting fancy with binary
endogenous regressors and outcomes
4. Be careful about being LATE

139. Anything

140. Literally anything

141. Instrumentation and It’s
Discontents

142. Two-Stage Least Squares
𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧 + 𝜉
𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧
𝑌 = 𝛾0 + 𝛾1 ∙ 𝑃 + 𝛾2 ∙ 𝑥 + 𝜀
𝛾0, 𝛾1, and 𝛾2

143. Two-Stage Least Squares
𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧 + 𝜉
𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧
𝑌 = 𝛾0 + 𝛾1 ∙ 𝑃 + 𝛾2 ∙ 𝑥 + 𝜀
𝛾0, 𝛾1, and 𝛾2

144. Two-Stage Least Squares
𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧1 + 𝜙3 ∙ 𝑧2 + 𝜉
𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧
𝑌 = 𝛾0 + 𝛾1 ∙ 𝑃 + 𝛾2 ∙ 𝑥 + 𝜀
𝛾0, 𝛾1, and 𝛾2

145. Two-Stage Least Squares
𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧1 + 𝜙3 ∙ 𝑧2 + 𝜉
𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧
𝑌 = 𝛾0 + 𝛾1 ∙ 𝑃 + 𝛾2 ∙ 𝑥 + 𝜀
𝛾0, 𝛾1, and 𝛾2

147. Two-Stage Least Squares
𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧 + 𝜉
𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧
𝑌 = 𝛾0 + 𝛾1 ∙ 𝑃 + 𝛾2 ∙ 𝑥 + 𝜀
𝛾0, 𝛾1, and 𝛾2

148. 𝑎𝑏𝑠 𝛽1 − 𝛾1
𝑛

149. Two-Stage Least Squares
𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧1 + 𝜙3 ∙ 𝑧2 + 𝜉
𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧
𝑌 = 𝛾0 + 𝛾1 ∙ 𝑃 + 𝛾2 ∙ 𝑥 + 𝜀
𝛾0, 𝛾1, and 𝛾2

150. Some Big Take-Aways About IV/TSLS
1. More powerful instruments are better
2. Over-identification is (generally) your
friend
3. Be careful about getting fancy with binary
endogenous regressors and outcomes
4. Be careful about being LATE

151. Two-Stage Least Squares
𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧 + 𝜉
𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧
𝑌 = 𝛾0 + 𝛾1 ∙ 𝑃 + 𝛾2 ∙ 𝑥 + 𝜀
𝛾0, 𝛾1, and 𝛾2

152. Two-Stage Least Squares
𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧1 + 𝜙3 ∙ 𝑧2 + 𝜉
𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧
𝑌 = 𝛾0 + 𝛾1 ∙ 𝑃 + 𝛾2 ∙ 𝑥 + 𝜀
𝛾0, 𝛾1, and 𝛾2

153. Two-Stage Least Squares
𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧1 + 𝜙3 ∙ 𝑧2 + 𝜉
𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧
𝑌 = 𝛾0 + 𝛾1 ∙ 𝑃 + 𝛾2 ∙ 𝑥 + 𝜀
𝛾0, 𝛾1, and 𝛾2

155. Some Big Take-Aways About IV/TSLS
1. More powerful instruments are better
2. Over-identification is (generally) your friend
3. Be careful about getting fancy with
binary endogenous regressors and
outcomes
4. Be careful about being LATE

156. Two-Stage Least Squares
𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧 + 𝜉
𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧
𝑌 = 𝛾0 + 𝛾1 ∙ 𝑃 + 𝛾2 ∙ 𝑥 + 𝜀
𝛾0, 𝛾1, and 𝛾2

157. Two-Stage Least Squares
𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧 + 𝜉
𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧
𝑌 = 𝛾0 + 𝛾1 ∙ 𝑃 + 𝛾2 ∙ 𝑥 + 𝜀
𝛾0, 𝛾1, and 𝛾2

158. Two-Stage Least Squares
𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧 + 𝜉
𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧
𝑌 = 𝛾0 + 𝛾1 ∙ 𝑃 + 𝛾2 ∙ 𝑥 + 𝜀
𝛾0, 𝛾1, and 𝛾2

159. Two-Stage Least Squares
𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧 + 𝜉
𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧
𝑌 = 𝛾0 + 𝛾1 ∙ 𝑃 + 𝛾2 ∙ 𝑥 + 𝜀
𝛾0, 𝛾1, and 𝛾2

160. Two-Stage Least Squares
𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧 + 𝜉
𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧
𝑌 = 𝛾0 + 𝛾1 ∙ 𝑃 + 𝛾2 ∙ 𝑥 + 𝜀
𝛾0, 𝛾1, and 𝛾2

161. Some Big Take-Aways About IV/TSLS
1. More powerful instruments are better
2. Over-identification is (generally) your friend
3. Be careful about getting fancy with binary
endogenous regressors and outcomes
4. Be careful about being LATE

162. 𝑌0 = 𝛽0 + 𝛽2 ∙ 𝑥 + 𝛽3 ∙ 𝜇 + 𝜀
𝑌1 = 𝛽0 + 𝛽1 + 𝛽2 ∙ 𝑥 + 𝛽3 ∙ 𝜇 + 𝜀

163. 𝑌𝑖
0
= 𝛽0 + 𝛽2 ∙ 𝑥𝑖 + 𝛽3 ∙ 𝜇𝑖 + 𝜀𝑖
𝑌𝑖
1
= 𝛽0 + 𝛽1 + 𝛽2 ∙ 𝑥𝑖 + 𝛽3 ∙ 𝜇𝑖 + 𝜀𝑖

164. 𝑌𝑖
0
= 𝛽0 + 𝛽2 ∙ 𝑥𝑖 + 𝛽3 ∙ 𝜇𝑖 + 𝜀𝑖
𝑌𝑖
1
= 𝛽0 + 𝛽1𝑖 + 𝛽2 ∙ 𝑥𝑖 + 𝛽3 ∙ 𝜇𝑖 + 𝜀𝑖

165. 𝑌𝑖
0
= 𝛽0 + 𝛽2 ∙ 𝑥𝑖 + 𝛽3 ∙ 𝜇𝑖 + 𝜀𝑖
𝑌𝑖
1
= 𝛽0 + 𝛽1𝑖 + 𝛽2 ∙ 𝑥𝑖 + 𝛽3 ∙ 𝜇𝑖 + 𝜀𝑖

166. 𝑌𝑖
1
− 𝑌𝑖
0
= 𝛽0 + 𝛽1 + 𝜖 − 𝛽0 + 𝜖
= 𝛽0 + 𝛽1 + 𝜖 − 𝛽0 − 𝜖
= 𝛽1

167. 𝑌𝑖
1
− 𝑌𝑖
0
= 𝛽0 + 𝛽1𝑖 + 𝛽2 ∙ 𝑥𝑖 + 𝛽3 ∙ 𝜇𝑖
+ 𝜀𝑖
− 𝛽0 + 𝛽2 ∙ 𝑥𝑖 + 𝛽3 ∙ 𝜇𝑖 + 𝜀𝑖
= 𝛽0 + 𝛽1 + 𝜖 − 𝛽0 − 𝜖
= 𝛽1

168. 𝑌𝑖
1
− 𝑌𝑖
0
= 𝛽1𝑖

169. 𝑌𝑖
1
− 𝑌𝑖
0
= 𝛽1𝑖

170. Two-Stage Least Squares
𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧 + 𝜉
𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧
𝑌 = 𝛾0 + 𝛾1 ∙ 𝑃 + 𝛾2 ∙ 𝑥 + 𝜀
𝛾0, 𝛾1, and 𝛾2

171. Two-Stage Least Squares
𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧 + 𝜉
𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧
𝑌 = 𝛾0 + 𝛾1 ∙ 𝑃 + 𝛾2 ∙ 𝑥 + 𝜀
𝛾0, 𝛾1, and 𝛾2

172. Two-Stage Least Squares
𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝝓 𝟐 ∙ 𝑧 + 𝜉
𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧
𝑌 = 𝛾0 + 𝛾1 ∙ 𝑃 + 𝛾2 ∙ 𝑥 + 𝜀
𝛾0, 𝛾1, and 𝛾2

173. The Three LATE Types
1. Always Takers- Always Participate
2. Never Takers-Never Participate
3. Compliers- Comply with their
instrumental “assignment”

174. The Three LATE Types
1. Always Takers- Always Participate
2. Never Takers-Never Participate
3. Compliers- Comply with their
instrumental “assignment”

175. Benefit-Cost>0
𝑌𝑖
1
− 𝑌𝑖
0
− 𝐶𝑖 > 0
𝛽1𝑖 − 𝐶𝑖 > 0
𝛽1𝑖 − 𝜌0 + 𝜌1 ∙ 𝑥𝑖 + 𝜌2 ∙ 𝜇𝑖 + 𝜌3 ∙ 𝑧𝑖 + 𝜂𝑖
> 0

176. 𝑧𝑖 = 1
𝑌𝑖
1
− 𝑌𝑖
0
− 𝐶𝑖 > 0
𝑧𝑖 = 0
𝑌𝑖
1
− 𝑌𝑖
0
− 𝐶𝑖 ≤ 0

177. 𝑧𝑖 = 1
𝑌𝑖
1
− 𝑌𝑖
0
− 𝐶𝑖 > 0
𝑧𝑖 = 0
𝑌𝑖
1
− 𝑌𝑖
0
− 𝐶𝑖 > 0

178. 𝑧𝑖 = 1
𝑌𝑖
1
− 𝑌𝑖
0
− 𝐶𝑖 ≤ 0
𝑧𝑖 = 0
𝑌𝑖
1
− 𝑌𝑖
0
− 𝐶𝑖 ≤ 0

179. Two-Stage Least Squares
𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧 + 𝜉
𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧
𝑌 = 𝛾0 + 𝛾1 ∙ 𝑃 + 𝛾2 ∙ 𝑥 + 𝜀
𝛾0, 𝛾1, and 𝛾2

181. Month Day 1969 1970 Month Day 1969 1970
3 7 122 141 9 11 158 288
8 22 339 250 11 1 19 243
4 18 90 138 6 4 20 42
7 12 15 257 7 13 42 349
5 9 197 357 12 30 3 192

182. Month Day 1969 1970 Month Day 1969 1970
3 7 122 141 9 11 158 288
8 22 339 250 11 1 19 243
4 18 90 138 6 4 20 42
7 12 15 257 7 13 42 349
5 9 197 357 12 30 3 192

183. The Three LATE Types
1. Always Takers- Always Participate
2. Never Takers-Never Participate
3. Compliers- Comply with their
instrumental “assignment”

185. The Three LATE Types
1. Always Takers- Always Participate
2. Never Takers-Never Participate
3. Compliers- Comply with their
instrumental “assignment”

187. Month Day 1969 1970 Month Day 1969 1970
3 7 122 141 9 11 158 288
8 22 339 250 11 1 19 243
4 18 90 138 6 4 20 42
7 12 15 257 7 13 42 349
5 9 197 357 12 30 3 192

190. Regression Discontinuity
Designs

191. 𝑥 ≤ 𝑒
𝑥 > 𝑒

193. A “Discontinuity”

194. 𝐸 𝑌|𝑥
𝑥𝑒

195. 𝐸 𝑌|𝑥
𝑥𝑒
Another
“Discontinuity”

196. A “Discontinuity”

197. 𝐸 𝑌|𝑥
𝑥𝑒
Another
“Discontinuity”

198. 𝐸 𝑌|𝑥
𝑥𝑒
Impact

199. 𝑌0
= 𝛽0 + 𝛽2 ∙ 𝑥 + 𝛽3 ∙ 𝜇 + 𝜀
𝑌1
= 𝛽0 + 𝛽1 + 𝛽2 ∙ 𝑥 + 𝛽3 ∙ 𝜇 + 𝜀
𝑌 = 𝑃 ∙ 𝑌1 + 1 − 𝑃 ∙ 𝑌0
𝑌 = 𝛽0 + 𝛽1 ∙ 𝑃 + 𝛽2 ∙ 𝑥 + 𝛽3 ∙ 𝜇 + 𝜀

200. 𝑌0
= 𝛽0 + 𝛽2 ∙ 𝑥 + 𝛽3 ∙ 𝜇 + 𝜀
𝑌1
= 𝛽0 + 𝛽1 + 𝛽2 ∙ 𝑥 + 𝛽3 ∙ 𝜇 + 𝜀
𝑌 = 𝑃 ∙ 𝑌1 + 1 − 𝑃 ∙ 𝑌0
𝑌 = 𝛽0 + 𝛽1 ∙ 𝑃 + 𝛽2 ∙ 𝑥 + 𝛽3 ∙ 𝜇 + 𝜀

201. 𝑌0
= 𝛽0 + 𝛽2 ∙ 𝑥 + 𝛽3 ∙ 𝜇 + 𝜀
𝑌1
= 𝛽0 + 𝛽1 + 𝛽2 ∙ 𝑥 + 𝛽3 ∙ 𝜇 + 𝜀
𝑌 = 𝑃 ∙ 𝑌1 + 1 − 𝑃 ∙ 𝑌0
𝑌 = 𝛽0 + 𝛽1 ∙ 𝑃 + 𝛽2 ∙ 𝑥 + 𝛽3 ∙ 𝜇 + 𝜀

202. A “Discontinuity”

203. A “Discontinuity”
𝐼 𝑥 ≤ 𝑒 = P

204. A “Discontinuity”
𝐼 𝑥 ≤ 𝑒 = P

205. 𝑌0
= 𝛽0 + 𝛽2 ∙ 𝑥 + 𝛽3 ∙ 𝜇 + 𝜀
𝑌1
= 𝛽0 + 𝛽1 + 𝛽2 ∙ 𝑥 + 𝛽3 ∙ 𝜇 + 𝜀
𝑌 = 𝑃 ∙ 𝑌1
+ 1 − 𝑃 ∙ 𝑌0
𝑌 = 𝛽0 + 𝛽1 ∙ 𝑃 + 𝛽2 ∙ 𝑥 + 𝛽3 ∙ 𝜇 + 𝜀

206. 𝑌0
= 𝛽0 + 𝛽2 ∙ 𝑥 + 𝛽3 ∙ 𝜇 + 𝜀
𝑌1
= 𝛽0 + 𝛽1 + 𝛽2 ∙ 𝑥 + 𝛽3 ∙ 𝜇 + 𝜀
𝑌 = 𝑃 ∙ 𝑌1
+ 1 − 𝑃 ∙ 𝑌0
𝑌 = 𝛽0 + 𝛽1 ∙ 𝑰(𝒙 ≤ 𝒆) + 𝛽2 ∙ 𝑥 + 𝛽3 ∙ 𝜇 + 𝜀

207. A “Discontinuity”

208. Cost of Participation
𝐶 = 𝜌0 + Ω ∙ 1 − 𝐼 𝑥 ≤ 𝑒 + 𝜌1 ∙ 𝑥
+𝜌2 ∙ 𝜇 + 𝜌3 ∙ 𝑧 + 𝜂

209. Cost of Participation
𝐶 = 𝜌0 + 𝜴 ∙ 𝟏 − 𝑰 𝒙 ≤ 𝒆 + 𝜌1 ∙ 𝑥
+𝜌2 ∙ 𝜇 + 𝜌3 ∙ 𝑧 + 𝜂

210. Cost of Participation
𝐶 = 𝜌0 + 𝛺 ∙ 1 − 𝐼 𝑥 ≤ 𝑒 + 𝜌1 ∙ 𝑥
+𝜌2 ∙ 𝜇 + 𝜌3 ∙ 𝑧 + 𝜂
𝐶 = 𝜌0 + 𝜌1 ∙ 𝑥 + 𝜌2 ∙ 𝜇 + 𝜌3 ∙ 𝑧 + 𝜂

211. Cost of Participation
𝐶 = 𝜌0 + 𝛺 ∙ 1 − 𝐼 𝑥 ≤ 𝑒 + 𝜌1 ∙ 𝑥
+𝜌2 ∙ 𝜇 + 𝜌3 ∙ 𝑧 + 𝜂
𝐶 = 𝜌0 + 𝜌1 ∙ 𝑥 + 𝜌2 ∙ 𝜇 + 𝜌3 ∙ 𝑧 + 𝜂

212. Benefit-Cost>0
𝑌1 − 𝑌0 − 𝐶 > 0
𝛽1 − 𝐶 > 0

213. 𝑌 = 𝛽0 + 𝛽1 ∙ 𝑃 + 𝛽2 ∙ 𝑥 + 𝛽3 ∙ 𝜇 + 𝜀

214. 𝑌 = 𝛽0 + 𝛽1 ∙ 𝑃 + 𝛽2 ∙ 𝑥 + 𝛽3 ∙ 𝜇 + 𝜀
𝑌 = 𝛾0 + 𝛾1 ∙ 𝑃 + 𝛾2 ∙ 𝑥 + 𝜀

215. 𝑌 = 𝛽0 + 𝛽1 ∙ 𝑃 + 𝛽2 ∙ 𝑥 + 𝛽3 ∙ 𝜇 + 𝜀
𝑌 = 𝛾0 + 𝛾1 ∙ 𝑃 + 𝜀

216. 𝑌 = 𝛽0 + 𝛽1 ∙ 𝑃 + 𝛽2 ∙ 𝑥 + 𝛽3 ∙ 𝜇 + 𝜀
𝑌 = 𝛾0 + 𝛾1 ∙ 𝑃 + 𝛾2 ∙ 𝑥 + 𝜀

217. Benefit-Cost>0
𝑌1 − 𝑌0 − 𝐶 > 0
𝛽1 − 𝐶 > 0

218. Benefit-Cost>0
𝑌1 − 𝑌0 − 𝐶 > 0
𝜷 𝟏 − 𝑪 > 0

219. Cost of Participation
𝐶 = 𝜌0 + 𝛺 ∙ 1 − 𝐼 𝑥 ≤ 𝑒 + 𝜌1 ∙ 𝑥
+𝜌2 ∙ 𝜇 + 𝜌3 ∙ 𝑧 + 𝜂
𝐶 = 𝜌0 + 𝜌1 ∙ 𝑥 + 𝜌2 ∙ 𝜇 + 𝜌3 ∙ 𝑧 + 𝜂

220. Cost of Participation
𝐶 = 𝜌0 + 𝜴 ∙ 1 − 𝐼 𝑥 ≤ 𝑒 + 𝜌1 ∙ 𝑥
+𝜌2 ∙ 𝜇 + 𝜌3 ∙ 𝑧 + 𝜂
𝐶 = 𝜌0 + 𝜌1 ∙ 𝑥 + 𝜌2 ∙ 𝜇 + 𝜌3 ∙ 𝑧 + 𝜂

221. Cost of Participation
𝐶 = 𝜌0 + 𝜴 ∙ 1 − 𝐼 𝑥 ≤ 𝑒 + 𝜌1 ∙ 𝑥
+𝜌2 ∙ 𝜇 + 𝜌3 ∙ 𝑧 + 𝜂
𝑪 = 𝝆 𝟎 + 𝝆 𝟏 ∙ 𝒙 + 𝝆 𝟐 ∙ 𝝁 + 𝝆 𝟑 ∙ 𝒛 + 𝜼

222. Two-Stage Least Squares
1. Regress 𝑃 on 𝑥 and 𝑧 (i.e. estimate 𝜙s
from 𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧 + 𝜉)
2. Use the fitted/estimated model to predict
participation:
𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧
3. Regress 𝑌 on 𝑃 and 𝑥

223. 𝑧 = 𝐼 𝑥 ≤ 𝑒

224. 𝑧 = 𝐼 𝑥 ≤ 𝑒

225. Links:
The manual:
http://www.measureevaluation.org/resources/publications/ms-
14-87-en
The webinar introducing the manual:
http://www.measureevaluation.org/resources/webinars/metho
ds-for-program-impact-evaluation
My email:
pmlance@email.unc.edu

226. MEASURE Evaluation is funded by the U.S. Agency
for International Development (USAID) under terms
of Cooperative Agreement AID-OAA-L-14-00004 and
implemented by the Carolina Population Center,
University of North Carolina at Chapel Hill in
partnership with ICF International, John Snow, Inc.,
Management Sciences for Health, Palladium Group,
and Tulane University. The views expressed in this
presentation do not necessarily reflect the views of
USAID or the United States government.
www.measureevaluation.org