Instrumental Variables

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?

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

17. YP μ X

18. YP μ X

19. YP μ X

20. YP μ X

21. YP μ X

22. YP μ X

23. YP μ Z X

24. YP μ Z 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. Fascinating, right??? So what the heck does it all mean, and how would you implement it?

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

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

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

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

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

39. 𝑌1 − 𝑌0 = 𝛽1

40. 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

41. 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

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

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

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

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

48. YP μ X

49. YP μ X

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

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

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

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

66. 𝑥 𝑃 𝜇 𝑃 𝑧 𝑃

68. YP μ X

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

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

74. YP μ Z X

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

77. 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 𝑃𝑟 𝑃 = 1 = 𝑒𝑥𝑝 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧 1 + 𝑒𝑥𝑝 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧

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

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

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

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

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

88. 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 𝑬( 𝜸 𝟏) ≠ 𝜷 𝟏

92. Two-Stage Least Squares

93. 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. 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 -+ 𝜀

103. 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

104. 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

105. 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

106. 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

107. 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

108. 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

109. 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

110. 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

111. 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

112. 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

113. 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 𝑌 = 𝛾0 + 𝛾1 ∙ 𝑃 + 𝛾2 ∙ 𝑥 + 𝜀 𝐸 𝛾0 = 𝛽1

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

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

125. ????!

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

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

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

129. 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

131. Anything

132. Literally anything

133. Instrumentation and It’s Discontents

136. 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

139. 𝑎𝑏𝑠 𝛽1 − 𝛾1 𝑛

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

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

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

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

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

158. 𝑌𝑖 1 − 𝑌𝑖 0 = 𝛽1𝑖

159. 𝑌𝑖 1 − 𝑌𝑖 0 = 𝛽1𝑖

162. 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

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

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

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

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

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

170. 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

175. Regression Discontinuity Designs

176. 𝑥 ≤ 𝑒 𝑥 > 𝑒

177. A “Discontinuity”

178. 𝐸 𝑌|𝑥 𝑥𝑒

179. 𝐸 𝑌|𝑥 𝑥𝑒 Another “Discontinuity”

181. 𝐸 𝑌|𝑥 𝑥𝑒 Another “Discontinuity”

182. 𝐸 𝑌|𝑥 𝑥𝑒 Impact

183. 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

187. A “Discontinuity” 𝐼 𝑥 ≤ 𝑒 = P

188. A “Discontinuity” 𝐼 𝑥 ≤ 𝑒 = P

190. 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

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

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

194. 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

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

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

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

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

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

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

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

204. 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

205. 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

207. 𝑧 = 𝐼 𝑥 ≤ 𝑒

208. 𝑧 = 𝐼 𝑥 ≤ 𝑒

209. 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

210. 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