Successfully reported this slideshow.
We use your LinkedIn profile and activity data to personalize ads and to show you more relevant ads. You can change your ad preferences anytime.

Selection on Observables

938 views

Published on

This webinar by Peter Lance considered impact evaluation estimation methods based on an identification strategy that assumes we can observe all factors that influence both program participation and the outcome of interest. It was the third 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.

Published in: Education
  • Be the first to comment

  • Be the first to like this

Selection on Observables

  1. 1. Peter M. Lance, PhD MEASURE Evaluation University of North Carolina at Chapel Hill September 13 and 15, 2016 Selection on Observables
  2. 2. 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
  3. 3. 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
  4. 4. Global footprint (more than 25 countries)
  5. 5. • The program impact evaluation challenge • Randomization • Selection on observables • Within estimators • Instrumental variables
  6. 6. • The program impact evaluation challenge • Randomization • Selection on observables • Within estimators • Instrumental variables
  7. 7. X Y P
  8. 8. X Y P 𝑌 = 𝑃 ∙ 𝑌1 + 1 − 𝑃 ∙ 𝑌0
  9. 9. X Y P
  10. 10. X Y P
  11. 11. 𝐸 𝑌1 − 𝑌0 = 𝐸 𝑌1 − 𝐸 𝑌0 Average Y across sample of participants − Average Y across sample of non−participants
  12. 12. 𝐸 𝑌1 − 𝑌0 = 𝐸 𝑌1 − 𝐸 𝑌0 Average Y across sample of participants − Average Y across sample of non−participants
  13. 13. 𝐸 𝑌1 − 𝑌0 = 𝐸 𝑌1 − 𝐸 𝑌0 Average Y across sample of participants − Average Y across sample of non−participants
  14. 14. 𝐸 𝑌1 − 𝑌0 = 𝐸 𝑌1 − 𝐸 𝑌0 Average Y across sample of participants − Average Y across sample of non−participants
  15. 15. 𝐸 𝑌1 − 𝑌0 = 𝐸 𝑌1 − 𝐸 𝑌0 Average Y across sample of participants − Average Y across sample of non−participants
  16. 16. 𝐸 𝑌1 − 𝑌0 = 𝐸 𝑌1 − 𝐸 𝑌0 Average Y across sample of participants − Average Y across sample of non−participants
  17. 17. 𝐸 𝑌1 − 𝑌0 = 𝐸 𝑌1 − 𝐸 𝑌0 Average Y across sample of participants − Average Y across sample of non−participants
  18. 18. 𝐸 𝑌1 𝑃 = 1 = 𝐸 𝑌1 𝑃 = 0 = 𝐸 𝑌1
  19. 19. X Y P
  20. 20. 𝐸 𝑋 𝑃 = 1 ≠ 𝐸 𝑋 𝑃 = 0
  21. 21. X Y P
  22. 22. X Y P
  23. 23. 𝐸 𝑋 𝑃 = 1 ≠ 𝐸 𝑋 𝑃 = 0
  24. 24. 𝐸 𝑋 𝑃 = 1 ≠ 𝐸 𝑋 𝑃 = 0 𝐸 𝑌1 𝑃 = 1 ≠ 𝐸 𝑌1 𝑃 = 0
  25. 25. 𝐸 𝑋 𝑃 = 1 ≠ 𝐸 𝑋 𝑃 = 0 𝐸 𝑌0 𝑃 = 1 ≠ 𝐸 𝑌0 𝑃 = 0
  26. 26. 𝐸 𝑌1 − 𝑌0 = 𝐸 𝑌1 − 𝐸 𝑌0 Average Y across sample of participants − Average Y across sample of non−participants
  27. 27. 𝐸 𝑌1 𝑃 = 1 ≠ 𝐸 𝑌1 𝑃 = 0 𝑋 = 𝑥∗ 𝐸 𝑌1 𝑃 = 1, 𝑋 = 𝑥∗ = 𝐸 𝑌1 𝑃 = 0, 𝑋 = 𝑥∗ 𝐸 𝑌0 𝑃 = 1, 𝑋 = 𝑥∗ = 𝐸 𝑌0 𝑃 = 0, 𝑋 = 𝑥∗
  28. 28. 𝐸 𝑌1 𝑃 = 1 ≠ 𝐸 𝑌1 𝑃 = 0 𝑋 = 𝑥∗ 𝐸 𝑌1 𝑃 = 1, 𝑋 = 𝑥∗ = 𝐸 𝑌1 𝑃 = 0, 𝑋 = 𝑥∗ 𝐸 𝑌0 𝑃 = 1, 𝑋 = 𝑥∗ = 𝐸 𝑌0 𝑃 = 0, 𝑋 = 𝑥∗
  29. 29. X Y P
  30. 30. 𝐸 𝑌1 𝑃 = 1 ≠ 𝐸 𝑌1 𝑃 = 0 𝑋 = 𝑥∗ 𝐸 𝑌1 𝑃 = 1, 𝑋 = 𝑥∗ = 𝐸 𝑌1 𝑃 = 0, 𝑋 = 𝑥∗ 𝐸 𝑌0 𝑃 = 1, 𝑋 = 𝑥∗ = 𝐸 𝑌0 𝑃 = 0, 𝑋 = 𝑥∗
  31. 31. 𝐸 𝑌1 𝑃 = 1 ≠ 𝐸 𝑌1 𝑃 = 0 𝑋 = 𝑥∗ 𝐸 𝑌1 𝑃 = 1, 𝑋 = 𝑥∗ = 𝐸 𝑌1 𝑃 = 0, 𝑋 = 𝑥∗ 𝐸 𝑌0 𝑃 = 1, 𝑋 = 𝑥∗ = 𝐸 𝑌0 𝑃 = 0, 𝑋 = 𝑥∗
  32. 32. 𝐸 𝑌1 𝑃 = 1 ≠ 𝐸 𝑌1 𝑃 = 0 𝑋 = 𝑥∗ 𝐸 𝑌1 𝑃 = 1, 𝑋 = 𝑥∗ = 𝐸 𝑌1 𝑃 = 0, 𝑋 = 𝑥∗ 𝐸 𝑌0 𝑃 = 1, 𝑋 = 𝑥∗ = 𝐸 𝑌0 𝑃 = 0, 𝑋 = 𝑥∗
  33. 33. 𝐸 𝑌1 − 𝑌0 = 𝐸 𝑌1 − 𝐸 𝑌0 Average Y across sample of participants − Average Y across sample of non−participants
  34. 34. 𝐸 𝑌1 − 𝑌0 |𝑋 = 𝑥∗ = 𝐸 𝑌1 |𝑋 = 𝑥∗ − 𝐸 𝑌0 |𝑋 = 𝑥∗ Average Y across sample of participants for whom 𝑋 = 𝑥∗ − Average Y across sample of participants of non−participants 𝑋 = 𝑥∗
  35. 35. 𝐸 𝑌1 − 𝑌0 = 𝐸 𝑌1 − 𝐸 𝑌0 Population Participation Rate Poor .4 .7 Middle .5 .3 Rich .1 .1
  36. 36. 𝐸 𝑌1 − 𝑌0 = 𝐸 𝑌1 − 𝐸 𝑌0 Population Participation Rate Poor .4 .7 Middle .5 .3 Rich .1 .1
  37. 37. 𝐸 𝑌1 − 𝑌0 = 𝐸 𝑌1 − 𝐸 𝑌0 Population Participation Rate Poor .4 .7 Middle .5 .3 Rich .1 .1
  38. 38. 𝐸 𝑌1 − 𝑌0 = 𝐸 𝑌1 − 𝐸 𝑌0 Participants (𝑷 = 𝟏) Non- participants (𝑷 = 𝟎) Poor .64 .21 Middle .34 .63 Rich .02 .16
  39. 39. 𝐸 𝑌1 − 𝑌0 = 𝐸 𝑌1 − 𝐸 𝑌0 Participants (𝑷 = 𝟏) Non- participants (𝑷 = 𝟎) Poor .64 .21 Middle .34 .63 Rich .02 .16
  40. 40. 𝐸 𝑌1 − 𝑌0 = 𝐸 𝑌1 − 𝐸 𝑌0 Participants (𝑷 = 𝟏) Non- participants (𝑷 = 𝟎) Poor .64 .21 Middle .34 .63 Rich .02 .16
  41. 41. 𝐸 𝑌1 − 𝑌0 = 𝐸 𝑌1 − 𝐸 𝑌0 Participants (𝑷 = 𝟏) Weight Poor .64 .63 Middle .34 1.47 Rich .02 4.4
  42. 42. 𝐸 𝑌1 − 𝑌0 = 𝐸 𝑌1 − 𝐸 𝑌0 Participants (𝑷 = 𝟏) Weight Poor .64 .63 Middle .34 1.47 Rich .02 4.4
  43. 43. 𝐸 𝑌1 − 𝑌0 = 𝐸 𝑌1 − 𝐸 𝑌0 Participants (𝑷 = 𝟏) Weight Poor .64 .63 Middle .34 1.47 Rich .02 4.4
  44. 44. 𝐸 𝑌1 − 𝑌0 = 𝐸 𝑌1 − 𝐸 𝑌0 Participants (𝑷 = 𝟏) Weight Poor .64 .63 Middle .34 1.47 Rich .02 4.5
  45. 45. 𝐸 𝑌1 − 𝑌0 = 𝐸 𝑌1 − 𝐸 𝑌0 Average of Y across a sample of participants = 𝑖=1 𝑛 𝑃 𝑤𝑖 ∗ 𝑌𝑖 𝑛 𝑃 𝒘𝒊=.63 if individual i is poor 𝒘𝒊=1.47 if individual i is middle class 𝒘𝒊=4.4 if individual i is rich
  46. 46. 𝐸 𝑌1 − 𝑌0 = 𝐸 𝑌1 − 𝐸 𝑌0 Average of Y across a sample of participants = 𝑖=1 𝑛 𝑃 𝑌𝑖 𝑛 𝑃 𝒘𝒊=.63 if individual i is poor 𝒘𝒊=1.47 if individual i is middle class 𝒘𝒊=4.4 if individual i is rich
  47. 47. 𝐸 𝑌1 − 𝑌0 = 𝐸 𝑌1 − 𝐸 𝑌0 Average of Y across a sample of participants = 𝑖=1 𝑛 𝑃 𝑤𝑖 ∙ 𝑌𝑖 𝑖=1 𝑛 𝑃 𝑤𝑖 𝒘𝒊=.63 if individual i is poor 𝒘𝒊=1.47 if individual i is middle class 𝒘𝒊=4.4 if individual i is rich
  48. 48. 𝐸 𝑌1 − 𝑌0 = 𝐸 𝑌1 − 𝐸 𝑌0 Average of Y across a sample of participants = 𝑖=1 𝑛 𝑃 𝒘𝒊 ∙ 𝑌𝑖 𝑖=1 𝑛 𝑃 𝑤𝑖 𝒘𝒊=.63 if individual i is poor 𝒘𝒊=1.47 if individual i is middle class 𝒘𝒊=4.4 if individual i is rich
  49. 49. 𝐸 𝑌1 − 𝑌0 = 𝐸 𝑌1 − 𝐸 𝑌0 Participants (𝑷 = 𝟏) Weight Poor .64 .63 Middle .34 1.47 Rich .02 4.5
  50. 50. 𝐸 𝑌1 − 𝑌0 = 𝐸 𝑌1 − 𝐸 𝑌0 Average of Y across a sample of participants = 𝑖=1 𝑛 𝑃 𝑤𝑖 ∙ 𝑌𝑖 𝑖=1 𝑛 𝑃 𝑤𝑖 𝒘𝒊=.63 if individual i is poor 𝒘𝒊=1.47 if individual i is middle class 𝒘𝒊=4.5 if individual i is rich
  51. 51. 𝐸 𝑌1 − 𝑌0 = 𝐸 𝑌1 − 𝐸 𝑌0 . 4 ∗ Average Y across sample of participants who are poor + .5 ∗ Average Y across sample of non−participants who are middle class +.1 ∗ Average Y across sample of participants who are rich
  52. 52. 𝐸 𝑌1 − 𝑌0 = 𝐸 𝑌1 − 𝐸 𝑌0 Average of Y across a sample of participants = 𝑖=1 𝑛 𝑃 𝑤𝑖 ∙ 𝑌𝑖 𝑖=1 𝑛 𝑃 𝑤𝑖 𝒘𝒊=.63 if individual i is poor 𝒘𝒊=1.47 if individual i is middle class 𝒘𝒊=4.5 if individual i is rich
  53. 53. 𝑌0 = 𝛽0 + 𝜖 𝑌1 = 𝛽0 + 𝛽1 + 𝜖
  54. 54. 𝑌1 − 𝑌0 = 𝛽0 + 𝛽1 + 𝜖 − 𝛽0 + 𝜖 = 𝛽0 + 𝛽1 + 𝜖 − 𝛽0 − 𝜖 = 𝛽1
  55. 55. 𝑌1 − 𝑌0 = 𝛽0 + 𝛽1 + 𝜖 − 𝛽0 + 𝜖 = 𝛽0 + 𝛽1 + 𝜖 − 𝛽0 − 𝜖 = 𝛽1
  56. 56. 𝑌1 − 𝑌0 = 𝛽0 + 𝛽1 + 𝜖 − 𝛽0 + 𝜖 = 𝛽0 + 𝛽1 + 𝜖 − 𝛽0 − 𝜖 = 𝛽1
  57. 57. 𝑌1 − 𝑌0 = 𝛽0 + 𝛽1 + 𝜖 − 𝛽0 + 𝜖 = 𝛽0 + 𝛽1 + 𝜖 − 𝛽0 − 𝜖 = 𝛽1
  58. 58. 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
  59. 59. 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
  60. 60. 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
  61. 61. 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
  62. 62. 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
  63. 63. 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
  64. 64. 𝑌 = 𝛽0 + 𝑃 ∙ 𝛽1 + 𝜖
  65. 65. 𝑌 = 𝛽0 + 𝛽1 ∙ 𝑃 + 𝜖
  66. 66. 𝑌0 = 𝛽0 + 𝜖 𝑌1 = 𝛽0 + 𝛽1 + 𝜖 𝑌 = 𝑃 ∙ 𝑌1 + 1 − 𝑃 ∙ 𝑌0
  67. 67. 𝑌0 = 𝛽0 + 𝜖 𝑌1 = 𝛽0 + 𝛽1 + 𝜖 𝑌 = 𝑃 ∙ 𝑌1 + 1 − 𝑃 ∙ 𝑌0
  68. 68. 𝑌0 = 𝛽0 + 𝜖 𝑌1 = 𝛽0 + 𝛽1 + 𝜖 𝑌 = 𝑃 ∙ 𝑌1 + 1 − 𝑃 ∙ 𝑌0
  69. 69. 𝑌 = 𝛽0 + 𝛽1 ∙ 𝑃 + 𝜖
  70. 70. 𝑌0 = 𝛽0 + 𝜖 𝑌1 = 𝛽0 + 𝛽1 + 𝜖 𝑌 = 𝑃 ∙ 𝑌1 + 1 − 𝑃 ∙ 𝑌0
  71. 71. 𝑌 = 𝛽0 + 𝛽1 ∙ 𝑃 + 𝜖
  72. 72. 𝑌𝑖 = 𝛽0 + 𝛽1 ∙ 𝑃𝑖 + 𝜖𝑖 𝑖 = 1, … , 𝑛
  73. 73. 𝜖𝑖 = 𝑌𝑖 − 𝛽0 − 𝛽1 ∙ 𝑃𝑖 𝑚𝑖𝑛 𝛽0, 𝛽1 𝑖=1 𝑛 𝜀𝑖 2 = 𝑚𝑖𝑛 𝛽0, 𝛽1 𝑖=1 𝑛 𝑌𝑖 − 𝛽0 − 𝛽1 ∙ 𝑃𝑖 2 𝜖𝑖
  74. 74. 𝑌 = 𝛽0 + 𝛽1 ∙ 𝑃 + 𝜖
  75. 75. 𝑌 = 𝛽0 + 𝛽1 ∙ 𝑃 + 𝜖
  76. 76. 𝜖𝑖 = 𝑌𝑖 − 𝛽0 − 𝛽1 ∙ 𝑃𝑖 𝑚𝑖𝑛 𝛽0, 𝛽1 𝑖=1 𝑛 𝜀𝑖 2 = 𝑚𝑖𝑛 𝛽0, 𝛽1 𝑖=1 𝑛 𝑌𝑖 − 𝛽0 − 𝛽1 ∙ 𝑃𝑖 2 𝜖𝑖
  77. 77. 𝛽1 = 𝑖=1 𝑛 𝑃𝑖 − 𝑃 ∙ 𝑌𝑖 𝑖=1 𝑛 𝑃𝑖 − 𝑃 2 where 𝑃 = 𝑖=1 𝑛 𝑃𝑖 𝑛
  78. 78. 𝐸 𝛽1 = 𝛽1
  79. 79. An expected value for a random variable is the average value from a large number of repetitions of the experiment that random variable represents An expected value is the true average of a random variable across a population Expected value
  80. 80. An expected value for a random variable is the average value from a large number of repetitions of the experiment that random variable represents An expected value is the true average of a random variable across a population Expected value
  81. 81. An expected value is the true average of a random variable across a population 𝐸 𝑋 = some true value Expected value
  82. 82. 𝑌 = 𝛽0 + 𝛽1 ∙ 𝑃 + 𝜖
  83. 83. 𝐸 𝑐 = 𝑐 𝐸 𝑐 ∙ 𝑊 = 𝑐 ∙ 𝐸 𝑊 𝐸 𝑊 + 𝑍 = 𝐸 𝑊 + 𝐸 𝑍 𝐸 𝑊 − 𝑍 = 𝐸 𝑊 − 𝐸 𝑍 𝐸 𝑎 ∙ 𝑊 ± 𝑏 ∙ 𝑍 = 𝑎 ∙ 𝐸 𝑊 ± 𝑏 ∙ 𝐸 𝑍 Expectations: Properties
  84. 84. 𝑬 𝒄 = 𝒄 𝐸 𝑐 ∙ 𝑊 = 𝑐 ∙ 𝐸 𝑊 𝐸 𝑊 + 𝑍 = 𝐸 𝑊 + 𝐸 𝑍 𝐸 𝑊 − 𝑍 = 𝐸 𝑊 − 𝐸 𝑍 𝐸 𝑎 ∙ 𝑊 ± 𝑏 ∙ 𝑍 = 𝑎 ∙ 𝐸 𝑊 ± 𝑏 ∙ 𝐸 𝑍 Expectations: Properties
  85. 85. 𝐸 𝑐 = 𝑐 𝑬 𝒄 ∙ 𝑾 = 𝒄 ∙ 𝑬 𝑾 𝐸 𝑊 + 𝑍 = 𝐸 𝑊 + 𝐸 𝑍 𝐸 𝑊 − 𝑍 = 𝐸 𝑊 − 𝐸 𝑍 𝐸 𝑎 ∙ 𝑊 ± 𝑏 ∙ 𝑍 = 𝑎 ∙ 𝐸 𝑊 ± 𝑏 ∙ 𝐸 𝑍 Expectations: Properties
  86. 86. 𝐸 𝑐 = 𝑐 𝐸 𝑐 ∙ 𝑊 = 𝑐 ∙ 𝐸 𝑊 𝑬 𝑾 + 𝒁 = 𝑬 𝑾 + 𝑬 𝒁 𝐸 𝑊 − 𝑍 = 𝐸 𝑊 − 𝐸 𝑍 𝐸 𝑎 ∙ 𝑊 ± 𝑏 ∙ 𝑍 = 𝑎 ∙ 𝐸 𝑊 ± 𝑏 ∙ 𝐸 𝑍 Expectations: Properties
  87. 87. 𝐸 𝑐 = 𝑐 𝐸 𝑐 ∙ 𝑊 = 𝑐 ∙ 𝐸 𝑊 𝐸 𝑊 + 𝑍 = 𝐸 𝑊 + 𝐸 𝑍 𝑬 𝑾 − 𝒁 = 𝑬 𝑾 − 𝑬 𝒁 𝐸 𝑎 ∙ 𝑊 ± 𝑏 ∙ 𝑍 = 𝑎 ∙ 𝐸 𝑊 ± 𝑏 ∙ 𝐸 𝑍 Expectations: Properties
  88. 88. 𝐸 𝑐 = 𝑐 𝐸 𝑐 ∙ 𝑊 = 𝑐 ∙ 𝐸 𝑊 𝐸 𝑊 + 𝑍 = 𝐸 𝑊 + 𝐸 𝑍 𝐸 𝑊 − 𝑍 = 𝐸 𝑊 − 𝐸 𝑍 𝑬 𝒂 ∙ 𝑾 ± 𝒃 ∙ 𝒁 = 𝒂 ∙ 𝑬 𝑾 ± 𝒃 ∙ 𝑬 𝒁 Expectations: Properties
  89. 89. 𝐸 𝑊 ∙ 𝑍 ≠ 𝐸 𝑊 ∙ 𝐸 𝑍 𝐸 𝑊 𝑍 ≠ 𝐸 𝑊 𝐸 𝑍 𝐸 𝑓 𝑊 ≠ 𝑓 𝐸 𝑊 Expectations: Properties
  90. 90. 𝑬 𝑾 ∙ 𝒁 ≠ 𝑬 𝑾 ∙ 𝑬 𝒁 𝐸 𝑊 𝑍 ≠ 𝐸 𝑊 𝐸 𝑍 𝐸 𝑓 𝑊 ≠ 𝑓 𝐸 𝑊 Expectations: Properties
  91. 91. 𝐸 𝑊 ∙ 𝑍 ≠ 𝐸 𝑊 ∙ 𝐸 𝑍 𝑬 𝑾 𝒁 ≠ 𝑬 𝑾 𝑬 𝒁 𝐸 𝑓 𝑊 ≠ 𝑓 𝐸 𝑊 Expectations: Properties
  92. 92. 𝐸 𝑊 ∙ 𝑍 ≠ 𝐸 𝑊 ∙ 𝐸 𝑍 𝐸 𝑊 𝑍 ≠ 𝐸 𝑊 𝐸 𝑍 𝑬 𝒇 𝑾 ≠ 𝒇 𝑬 𝑾 Expectations: Properties
  93. 93. 𝑌𝑖 = 𝛽0 + 𝛽1 ∙ 𝑃𝑖 + 𝜖𝑖 𝛽1 = 𝑖=1 𝑛 𝑃𝑖 − 𝑃 ∙ 𝑌𝑖 𝑖=1 𝑛 𝑃𝑖 − 𝑃 2
  94. 94. 𝑌𝑖 = 𝛽0 + 𝛽1 ∙ 𝑃𝑖 + 𝜖𝑖 𝛽1 = 𝑖=1 𝑛 𝑃𝑖 − 𝑃 ∙ 𝑌𝑖 𝑖=1 𝑛 𝑃𝑖 − 𝑃 2
  95. 95. 𝑌𝑖 = 𝛽0 + 𝛽1 ∙ 𝑃𝑖 + 𝜖𝑖 𝛽1 = 𝑖=1 𝑛 𝑃𝑖 − 𝑃 ∙ 𝑌𝑖 𝑖=1 𝑛 𝑃𝑖 − 𝑃 2
  96. 96. 𝐸 𝛽1 = 𝛽1
  97. 97. 𝐸 𝛽1 = 𝛽1
  98. 98. 𝛽1 = 𝑖=1 𝑛 𝑃𝑖 − 𝑃 ∙ 𝑌𝑖 𝑖=1 𝑛 𝑃𝑖 − 𝑃 2
  99. 99. 𝐸( 𝛽1) = 𝐸 𝑖=1 𝑛 𝑃𝑖 − 𝑃 ∙ 𝑌𝑖 𝑖=1 𝑛 𝑃𝑖 − 𝑃 2
  100. 100. 𝑌𝑖 = 𝛽0 + 𝛽1 ∙ 𝑃𝑖 + 𝜖𝑖
  101. 101. 𝐸( 𝛽1) = 𝐸 𝑖=1 𝑛 𝑃𝑖 − 𝑃 ∙ 𝑌𝑖 𝑖=1 𝑛 𝑃𝑖 − 𝑃 2
  102. 102. 𝐸 𝛽1 = 𝐸 𝑖=1 𝑛 𝑃𝑖 − 𝑃 ∙ 𝛽0 + 𝛽1 ∙ 𝑃𝑖 + 𝜖𝑖 𝑖=1 𝑛 𝑃𝑖 − 𝑃 2
  103. 103. 𝐸 𝛽1 = 𝐸 𝑖=1 𝑛 𝛽0 ∙ 𝑃𝑖 − 𝑃 𝑖=1 𝑛 𝑃𝑖 − 𝑃 2 + 𝑖=1 𝑛 𝛽1 ∙ 𝑃𝑖 ∙ 𝑃𝑖 − 𝑃 𝑖=1 𝑛 𝑃𝑖 − 𝑃 2
  104. 104. 𝐸 𝑊 + 𝑍 = 𝐸 𝑊 + 𝐸 𝑍 𝐸 𝑊 + 𝑍 + 𝑄 = 𝐸 𝑊 + 𝐸 𝑍 + 𝐸 𝑄
  105. 105. 𝐸 𝑊 + 𝑍 = 𝐸 𝑊 + 𝐸 𝑍 𝐸 𝑊 + 𝑍 + 𝑄 = 𝐸 𝑊 + 𝐸 𝑍 + 𝐸 𝑄
  106. 106. 𝐸 𝛽1 = 𝐸 𝑖=1 𝑛 𝛽0 ∙ 𝑃𝑖 − 𝑃 𝑖=1 𝑛 𝑃𝑖 − 𝑃 2 + 𝑖=1 𝑛 𝛽1 ∙ 𝑃𝑖 ∙ 𝑃𝑖 − 𝑃 𝑖=1 𝑛 𝑃𝑖 − 𝑃 2
  107. 107. 𝐸 𝛽1 = 𝐸 𝑖=1 𝑛 𝛽0 ∙ 𝑃𝑖 − 𝑃 𝑖=1 𝑛 𝑃𝑖 − 𝑃 2 + 𝑖=1 𝑛 𝛽1 ∙ 𝑃𝑖 ∙ 𝑃𝑖 − 𝑃 𝑖=1 𝑛 𝑃𝑖 − 𝑃 2
  108. 108. 𝐸 𝛽1 = 𝐸 𝑖=1 𝑛 𝛽0 ∙ 𝑃𝑖 − 𝑃 𝑖=1 𝑛 𝑃𝑖 − 𝑃 2 + 𝑖=1 𝑛 𝛽1 ∙ 𝑃𝑖 ∙ 𝑃𝑖 − 𝑃 𝑖=1 𝑛 𝑃𝑖 − 𝑃 2
  109. 109. 𝐸 𝛽1 = 𝐸 𝑖=1 𝑛 𝛽0 ∙ 𝑃𝑖 − 𝑃 𝑖=1 𝑛 𝑃𝑖 − 𝑃 2 + 𝑖=1 𝑛 𝛽1 ∙ 𝑃𝑖 ∙ 𝑃𝑖 − 𝑃 𝑖=1 𝑛 𝑃𝑖 − 𝑃 2
  110. 110. 𝐸 𝛽1 = 𝐸 𝑖=1 𝑛 𝛽0 ∙ 𝑃𝑖 − 𝑃 𝑖=1 𝑛 𝑃𝑖 − 𝑃 2 + 𝑖=1 𝑛 𝛽1 ∙ 𝑃𝑖 ∙ 𝑃𝑖 − 𝑃 𝑖=1 𝑛 𝑃𝑖 − 𝑃 2
  111. 111. 𝐸 𝛽1 = 𝐸 𝑖=1 𝑛 𝛽0 ∙ 𝑃𝑖 − 𝑃 𝑖=1 𝑛 𝑃𝑖 − 𝑃 2 + 𝐸 𝑖=1 𝑛 𝛽1 ∙ 𝑃𝑖 ∙ 𝑃𝑖 − 𝑃 𝑖=1 𝑛 𝑃𝑖 − 𝑃 2 + 𝐸 𝑖=1 𝑛 𝜖𝑖 ∙ 𝑃𝑖 − 𝑃 𝑖=1 𝑛 𝑃𝑖 − 𝑃 2
  112. 112. 𝐸 𝛽1 = 𝐸 𝑖=1 𝑛 𝛽0 ∙ 𝑃𝑖 − 𝑃 𝑖=1 𝑛 𝑃𝑖 − 𝑃 2 + 𝐸 𝑖=1 𝑛 𝛽1 ∙ 𝑃𝑖 ∙ 𝑃𝑖 − 𝑃 𝑖=1 𝑛 𝑃𝑖 − 𝑃 2 + 𝐸 𝑖=1 𝑛 𝜖𝑖 ∙ 𝑃𝑖 − 𝑃 𝑖=1 𝑛 𝑃𝑖 − 𝑃 2
  113. 113. 𝐸 𝑐 ∙ 𝑊 = 𝑐 ∙ 𝐸 𝑊
  114. 114. 𝐸 𝛽1 = 𝐸 𝑖=1 𝑛 𝛽0 ∙ 𝑃𝑖 − 𝑃 𝑖=1 𝑛 𝑃𝑖 − 𝑃 2 + 𝐸 𝑖=1 𝑛 𝛽1 ∙ 𝑃𝑖 ∙ 𝑃𝑖 − 𝑃 𝑖=1 𝑛 𝑃𝑖 − 𝑃 2 + 𝐸 𝑖=1 𝑛 𝜖𝑖 ∙ 𝑃𝑖 − 𝑃 𝑖=1 𝑛 𝑃𝑖 − 𝑃 2
  115. 115. 𝐸 𝛽1 = 𝛽0 ∙ 𝐸 𝑖=1 𝑛 𝑃𝑖 − 𝑃 𝑖=1 𝑛 𝑃𝑖 − 𝑃 2 + 𝐸 𝑖=1 𝑛 𝛽1 ∙ 𝑃𝑖 ∙ 𝑃𝑖 − 𝑃 𝑖=1 𝑛 𝑃𝑖 − 𝑃 2 + 𝐸 𝑖=1 𝑛 𝜖𝑖 ∙ 𝑃𝑖 − 𝑃 𝑖=1 𝑛 𝑃𝑖 − 𝑃 2
  116. 116. 𝐸 𝛽1 = 𝛽0 ∙ 𝐸 𝑖=1 𝑛 𝑃𝑖 − 𝑃 𝑖=1 𝑛 𝑃𝑖 − 𝑃 2 + 𝐸 𝑖=1 𝑛 𝛽1 ∙ 𝑃𝑖 ∙ 𝑃𝑖 − 𝑃 𝑖=1 𝑛 𝑃𝑖 − 𝑃 2 + 𝐸 𝑖=1 𝑛 𝜖𝑖 ∙ 𝑃𝑖 − 𝑃 𝑖=1 𝑛 𝑃𝑖 − 𝑃 2
  117. 117. 𝐸 𝛽1 = 𝛽0 ∗ 𝐸 0 𝑖=1 𝑛 𝑃𝑖 − 𝑃 2 + 𝐸 𝑖=1 𝑛 𝛽1 ∙ 𝑃𝑖 ∙ 𝑃𝑖 − 𝑃 𝑖=1 𝑛 𝑃𝑖 − 𝑃 2 + 𝐸 𝑖=1 𝑛 𝜖𝑖 ∙ 𝑃𝑖 − 𝑃 𝑖=1 𝑛 𝑃𝑖 − 𝑃 2
  118. 118. 𝐸 𝛽1 = 𝛽0 ∗ 0 +𝐸 𝑖=1 𝑛 𝛽1 ∙ 𝑃𝑖 ∙ 𝑃𝑖 − 𝑃 𝑖=1 𝑛 𝑃𝑖 − 𝑃 2 + 𝐸 𝑖=1 𝑛 𝜖𝑖 ∙ 𝑃𝑖 − 𝑃 𝑖=1 𝑛 𝑃𝑖 − 𝑃 2
  119. 119. 𝐸 𝛽1 = 𝐸 𝑖=1 𝑛 𝛽1 ∙ 𝑃𝑖 ∙ 𝑃𝑖 − 𝑃 𝑖=1 𝑛 𝑃𝑖 − 𝑃 2 + 𝐸 𝑖=1 𝑛 𝜖𝑖 ∙ 𝑃𝑖 − 𝑃 𝑖=1 𝑛 𝑃𝑖 − 𝑃 2
  120. 120. 𝐸 𝛽1 = 𝐸 𝑖=1 𝑛 𝛽1 ∙ 𝑃𝑖 ∙ 𝑃𝑖 − 𝑃 𝑖=1 𝑛 𝑃𝑖 − 𝑃 2 + 𝐸 𝑖=1 𝑛 𝜖𝑖 ∙ 𝑃𝑖 − 𝑃 𝑖=1 𝑛 𝑃𝑖 − 𝑃 2
  121. 121. 𝐸 𝛽1 = 𝛽1 ∙ 𝐸 𝑖=1 𝑛 𝑃𝑖 ∙ 𝑃𝑖 − 𝑃 𝑖=1 𝑛 𝑃𝑖 − 𝑃 2 + 𝐸 𝑖=1 𝑛 𝜖𝑖 ∙ 𝑃𝑖 − 𝑃 𝑖=1 𝑛 𝑃𝑖 − 𝑃 2
  122. 122. 𝐸 𝛽1 = 𝛽1 ∙ 𝐸 𝑖=1 𝑛 𝑃𝑖 ∙ 𝑃𝑖 − 𝑃 𝑖=1 𝑛 𝑃𝑖 − 𝑃 2 + 𝐸 𝑖=1 𝑛 𝜖𝑖 ∙ 𝑃𝑖 − 𝑃 𝑖=1 𝑛 𝑃𝑖 − 𝑃 2
  123. 123. 𝑖=1 𝑛 𝑃𝑖 ∙ 𝑃𝑖 − 𝑃 = 𝑖=1 𝑛 𝑃𝑖 − 𝑃 2
  124. 124. 𝐸 𝛽1 = 𝛽1 ∗ 𝐸 𝑖=1 𝑛 𝑃𝑖 ∙ 𝑃𝑖 − 𝑃 𝑖=1 𝑛 𝑃𝑖 − 𝑃 2 + 𝐸 𝑖=1 𝑛 𝜖𝑖 ∙ 𝑃𝑖 − 𝑃 𝑖=1 𝑛 𝑃𝑖 − 𝑃 2
  125. 125. 𝐸 𝛽1 = 𝛽1 ∙ 𝐸 𝑖=1 𝑛 𝑃𝑖 − 𝑃 2 𝑖=1 𝑛 𝑃𝑖 − 𝑃 2 + 𝐸 𝑖=1 𝑛 𝜖𝑖 ∙ 𝑃𝑖 − 𝑃 𝑖=1 𝑛 𝑃𝑖 − 𝑃 2
  126. 126. 𝐸 𝛽1 = 𝛽1 ∙ 𝐸 1 +𝐸 𝑖=1 𝑛 𝜖𝑖 ∙ 𝑃𝑖 − 𝑃 𝑖=1 𝑛 𝑃𝑖 − 𝑃 2
  127. 127. 𝐸 𝛽1 = 𝛽1 ∙ 1 +𝐸 𝑖=1 𝑛 𝜖𝑖 ∙ 𝑃𝑖 − 𝑃 𝑖=1 𝑛 𝑃𝑖 − 𝑃 2
  128. 128. 𝐸 𝛽1 = 𝛽1 +𝐸 𝑖=1 𝑛 𝜖𝑖 ∙ 𝑃𝑖 − 𝑃 𝑖=1 𝑛 𝑃𝑖 − 𝑃 2
  129. 129. 𝐸 𝛽1 = 𝛽1 +𝐸 𝑖=1 𝑛 𝜖𝑖 ∙ 𝑃𝑖 − 𝑃 𝑖=1 𝑛 𝑃𝑖 − 𝑃 2
  130. 130. 𝐸 𝛽1 = 𝛽1 +0
  131. 131. 𝐸 𝛽1 = 𝛽1
  132. 132. 𝐸 𝛽1 = 𝛽1 +𝐸 𝑖=1 𝑛 𝜖𝑖 ∙ 𝑃𝑖 − 𝑃 𝑖=1 𝑛 𝑃𝑖 − 𝑃 2
  133. 133. 𝐸 𝑊 ∙ 𝑍 ≠ 𝐸 𝑊 ∙ 𝐸 𝑍 Expectations: Properties
  134. 134. 𝐸 𝛽1 = 𝛽1 +𝐸 𝑖=1 𝑛 𝜖𝑖 ∙ 𝑃𝑖 − 𝑃 𝑖=1 𝑛 𝑃𝑖 − 𝑃 2
  135. 135. 𝐸 𝛽1 = 𝛽1 + 𝑖=1 𝑛 𝐸 𝜖𝑖 ∙ 𝑃𝑖 − 𝑃 𝑖=1 𝑛 𝑃𝑖 − 𝑃 2
  136. 136. 𝐸 𝛽1 = 𝛽1 + 𝑖=1 𝑛 𝐸 𝑖=1 𝑛 𝑃𝑖 − 𝑃 𝑖=1 𝑛 𝑃𝑖 − 𝑃 2 ∙ 𝐸 𝜖𝑖
  137. 137. 𝐸 𝛽1 = 𝛽1 + 𝑖=1 𝑛 𝐸 𝑖=1 𝑛 𝑃𝑖 − 𝑃 𝑖=1 𝑛 𝑃𝑖 − 𝑃 2 ∙ 0
  138. 138. 𝐸 𝛽1 = 𝛽1 +0
  139. 139. 𝐸 𝛽1 = 𝛽1
  140. 140. 𝐸 𝛽1 = 𝛽1 𝑌𝑖 = 𝛽0 + 𝛽1 ∙ 𝑃𝑖 + 𝜖𝑖
  141. 141. 𝐸 𝛽1 = 𝛽1 𝑌𝑖 = 𝛽0 + 𝛽1 ∙ 𝑃𝑖 + 𝜖𝑖
  142. 142. 𝐸 𝛽1 = 𝛽1 𝑌𝑖 = 𝛽0 + 𝛽1 ∙ 𝑃𝑖 + 𝜖𝑖
  143. 143. Peter M. Lance, PhD MEASURE Evaluation University of North Carolina at Chapel Hill September 13 and 15, 2016 Selection on Observables: Part Deux
  144. 144. 𝑌0 = 𝛽0 + 𝜖 𝑌1 = 𝛽0 + 𝛽1 + 𝜖 𝑌 = 𝑃 ∙ 𝑌1 + 1 − 𝑃 ∙ 𝑌0 𝑌 = 𝛽0 + 𝛽1 ∙ 𝑃 + 𝜖
  145. 145. 𝑌0 = 𝛽0 + 𝜖 𝑌1 = 𝛽0 + 𝛽1 + 𝜖 𝑌 = 𝑃 ∙ 𝑌1 + 1 − 𝑃 ∙ 𝑌0 𝑌 = 𝛽0 + 𝛽1 ∙ 𝑃 + 𝜖
  146. 146. 𝑌0 = 𝛽0 + 𝜖 𝑌1 = 𝛽0 + 𝛽1 + 𝜖 𝑌 = 𝑃 ∙ 𝑌1 + 1 − 𝑃 ∙ 𝑌0 𝑌 = 𝛽0 + 𝛽1 ∙ 𝑃 + 𝜖
  147. 147. 𝐸 𝛽1 = 𝛽1 +𝐸 𝑖=1 𝑛 𝜖𝑖 ∙ 𝑃𝑖 − 𝑃 𝑖=1 𝑛 𝑃𝑖 − 𝑃 2
  148. 148. 𝐸 𝛽1 = 𝛽1 +𝐸 𝑖=1 𝑛 𝜖𝑖 ∙ 𝑃𝑖 − 𝑃 𝑖=1 𝑛 𝑃𝑖 − 𝑃 2
  149. 149. 𝐸 𝛽1 = 𝛽1 𝑌𝑖 = 𝛽0 + 𝛽1 ∙ 𝑃𝑖 + 𝜖𝑖
  150. 150. 𝐸 𝛽1 = 𝛽1 𝑌𝑖 = 𝛽0 + 𝛽1 ∙ 𝑃𝑖 + 𝜖𝑖
  151. 151. 𝐸 𝛽1 = 𝛽1 𝑌𝑖 = 𝛽0 + 𝛽1 ∙ 𝑃𝑖 + 𝜖𝑖
  152. 152. 𝑌0 = 𝛽0 + 𝜀 𝑌1 = 𝛽0 + 𝛽1 + 𝜀
  153. 153. 𝑌0 = 𝛽0 + 𝛽2 ∙ 𝑥 + 𝜀 𝑌1 = 𝛽0 + 𝛽1 + 𝛽2 ∙ 𝑥 + 𝜀
  154. 154. 𝑌1 − 𝑌0 = 𝛽0 + 𝛽1 + 𝜖 − 𝛽0 + 𝜖 = 𝛽0 + 𝛽1 + 𝜖 − 𝛽0 − 𝜖 = 𝛽1
  155. 155. 𝑌1 − 𝑌0 = 𝛽0 + 𝛽1 + 𝛽2 ∗ 𝑥 + 𝜀 − 𝛽0 + 𝛽2 ∗ 𝑥 + 𝜀 = 𝛽0 + 𝛽1 + 𝜖 − 𝛽0 − 𝜖 = 𝛽1
  156. 156. 𝑌1 − 𝑌0 = 𝛽1
  157. 157. 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
  158. 158. 𝑌 = 𝑃 ∙ 𝑌1 + 1 − 𝑃 ∙ 𝑌0 = 𝑃 ∙ 𝛽0 + 𝛽1 + 𝛽2 ∙ 𝑥 + 𝜀 + 1 − 𝑃 ∙ 𝛽0 + 𝛽2 ∙ 𝑥 + 𝜀 = 𝑃 ∗ 𝛽0 + 𝑃 ∗ 𝛽1 + 𝑃 ∗ 𝜖 +𝛽0 + 𝜖 − 𝑃 ∗ 𝛽0 − 𝑃 ∗ 𝜖
  159. 159. 𝑌 = 𝛽0 + 𝛽1 ∙ 𝑃 + 𝛽2 ∙ 𝑥 + 𝜀
  160. 160. Cost of participation 𝐶 = 𝜌0 + 𝜌1 ∙ 𝑥
  161. 161. Cost of participation 𝐶 = 𝜌0 + 𝜌1 ∙ 𝑥
  162. 162. Benefit-Cost>0 𝑌1 − 𝑌0 − C > 0 𝛽1 − 𝐶 > 0 𝛽1 − 𝛾0 + 𝛾1 ∗ 𝑥 > 0
  163. 163. Benefit-Cost>0 𝑌1 − 𝑌0 − C > 0 𝛽1 − 𝐶 > 0 𝛽1 − 𝛾0 + 𝛾1 ∗ 𝑥 > 0
  164. 164. Benefit-Cost>0 𝑌1 − 𝑌0 − C > 0 𝛽1 − 𝐶 > 0 𝛽1 − 𝛾0 + 𝛾1 ∗ 𝑥 > 0
  165. 165. Benefit-Cost>0 𝑌1 − 𝑌0 − C > 0 𝛽1 − 𝐶 > 0 𝛽1 − 𝜌0 + 𝜌1 ∙ 𝑥 > 0
  166. 166. 𝑥 𝑃
  167. 167. X Y P
  168. 168. Benefit-Cost>0 𝛽1 − 𝜌0 + 𝜌1 ∙ 𝑥 > 0 𝜀
  169. 169. Benefit-Cost>0 𝛽1 − 𝜌0 + 𝜌1 ∙ 𝑥 > 0 𝜀
  170. 170. Benefit-Cost>0 𝛽1 − 𝜌0 + 𝜌1 ∙ 𝑥 > 0 𝜀
  171. 171. P and 𝜺 are independent
  172. 172. True model: 𝑌 = 𝛽0 + 𝛽1 ∙ 𝑃 + 𝛽2 ∙ 𝑥 + 𝜀 We actually attempt to estimate: 𝑌 = 𝜏0 + 𝜏1 ∙ 𝑃 + 𝜖 Error term now contains: 𝛽2 ∙ 𝑥
  173. 173. True model: 𝑌 = 𝛽0 + 𝛽1 ∙ 𝑃 + 𝛽2 ∙ 𝑥 + 𝜀 We actually attempt to estimate: 𝑌 = 𝜏0 + 𝜏1 ∙ 𝑃 + 𝜖 Error term now contains: 𝛽2 ∙ 𝑥
  174. 174. True model: 𝑌 = 𝛽0 + 𝛽1 ∙ 𝑃 + 𝛽2 ∙ 𝑥 + 𝜀 We actually attempt to estimate: 𝑌 = 𝜏0 + 𝜏1 ∙ 𝑃 + 𝜖 Error term now contains: 𝛽2 ∙ 𝑥
  175. 175. 𝐸( 𝜏1) = 𝐸 𝑖=1 𝑛 𝑃𝑖 − 𝑃 ∙ 𝑌𝑖 𝑖=1 𝑛 𝑃𝑖 − 𝑃 2
  176. 176. 𝐸 𝜏1 = 𝐸 𝑖=1 𝑛 𝑃𝑖 − 𝑃 ∙ 𝛽0 + 𝛽1 ∙ 𝑃𝑖 + 𝛽2 ∙ 𝑥𝑖 + 𝜀𝑖 𝑖=1 𝑛 𝑃𝑖 − 𝑃 2
  177. 177. 𝐸 𝜏1 = 𝐸 𝑖=1 𝑛 𝛽0 ∙ 𝑃𝑖 − 𝑃 𝑖=1 𝑛 𝑃𝑖 − 𝑃 2 + 𝑖=1 𝑛 𝛽1 ∙ 𝑃𝑖 ∙ 𝑃𝑖 − 𝑃 𝑖=1 𝑛 𝑃𝑖 − 𝑃 2 + 𝑖=1 𝑛 𝛽2 ∙ 𝑥𝑖 ∙ 𝑃𝑖 − 𝑃 𝑖=1 𝑛 𝑃𝑖 − 𝑃 2 + 𝑖=1 𝑛 𝜀𝑖 ∙ 𝑃𝑖 − 𝑃 𝑖=1 𝑛 𝑃𝑖 − 𝑃 2
  178. 178. 𝐸 𝜏1 = 𝐸 𝑖=1 𝑛 𝛽0 ∙ 𝑃𝑖 − 𝑃 𝑖=1 𝑛 𝑃𝑖 − 𝑃 2 + 𝐸 𝑖=1 𝑛 𝛽1 ∙ 𝑃𝑖 ∙ 𝑃𝑖 − 𝑃 𝑖=1 𝑛 𝑃𝑖 − 𝑃 2 +𝐸 𝑖=1 𝑛 𝛽2 ∙ 𝑥𝑖 ∙ 𝑃𝑖 − 𝑃 𝑖=1 𝑛 𝑃𝑖 − 𝑃 2 +𝐸 𝑖=1 𝑛 𝜀𝑖 ∙ 𝑃𝑖 − 𝑃 𝑖=1 𝑛 𝑃𝑖 − 𝑃 2
  179. 179. 𝐸 𝜏1 = 𝐸 𝑖=1 𝑛 𝛽0 ∙ 𝑃𝑖 − 𝑃 𝑖=1 𝑛 𝑃𝑖 − 𝑃 2 + 𝐸 𝑖=1 𝑛 𝛽1 ∙ 𝑃𝑖 ∙ 𝑃𝑖 − 𝑃 𝑖=1 𝑛 𝑃𝑖 − 𝑃 2 +𝐸 𝑖=1 𝑛 𝛽2 ∙ 𝑥𝑖 ∙ 𝑃𝑖 − 𝑃 𝑖=1 𝑛 𝑃𝑖 − 𝑃 2 +𝐸 𝑖=1 𝑛 𝜀𝑖 ∙ 𝑃𝑖 − 𝑃 𝑖=1 𝑛 𝑃𝑖 − 𝑃 2
  180. 180. 𝐸 𝜏1 = 0 +𝐸 𝑖=1 𝑛 𝛽1 ∙ 𝑃𝑖 ∙ 𝑃𝑖 − 𝑃 𝑖=1 𝑛 𝑃𝑖 − 𝑃 2 +𝐸 𝑖=1 𝑛 𝛽2 ∙ 𝑥𝑖 ∙ 𝑃𝑖 − 𝑃 𝑖=1 𝑛 𝑃𝑖 − 𝑃 2 +𝐸 𝑖=1 𝑛 𝜀𝑖 ∙ 𝑃𝑖 − 𝑃 𝑖=1 𝑛 𝑃𝑖 − 𝑃 2
  181. 181. 𝐸 𝜏1 = 𝐸 𝑖=1 𝑛 𝛽1 ∙ 𝑃𝑖 ∙ 𝑃𝑖 − 𝑃 𝑖=1 𝑛 𝑃𝑖 − 𝑃 2 +𝐸 𝑖=1 𝑛 𝛽2 ∙ 𝑥𝑖 ∙ 𝑃𝑖 − 𝑃 𝑖=1 𝑛 𝑃𝑖 − 𝑃 2 +𝐸 𝑖=1 𝑛 𝜀𝑖 ∙ 𝑃𝑖 − 𝑃 𝑖=1 𝑛 𝑃𝑖 − 𝑃 2
  182. 182. 𝐸 𝜏1 = 𝐸 𝑖=1 𝑛 𝛽1 ∙ 𝑃𝑖 ∙ 𝑃𝑖 − 𝑃 𝑖=1 𝑛 𝑃𝑖 − 𝑃 2 +𝐸 𝑖=1 𝑛 𝛽2 ∙ 𝑥𝑖 ∙ 𝑃𝑖 − 𝑃 𝑖=1 𝑛 𝑃𝑖 − 𝑃 2 +𝐸 𝑖=1 𝑛 𝜀𝑖 ∙ 𝑃𝑖 − 𝑃 𝑖=1 𝑛 𝑃𝑖 − 𝑃 2
  183. 183. 𝐸 𝜏1 = 𝛽1 +𝐸 𝑖=1 𝑛 𝛽2 ∙ 𝑥𝑖 ∙ 𝑃𝑖 − 𝑃 𝑖=1 𝑛 𝑃𝑖 − 𝑃 2 +𝐸 𝑖=1 𝑛 𝜀𝑖 ∙ 𝑃𝑖 − 𝑃 𝑖=1 𝑛 𝑃𝑖 − 𝑃 2
  184. 184. 𝐸 𝜏1 = 𝛽1 +𝐸 𝑖=1 𝑛 𝛽2 ∙ 𝑥𝑖 ∙ 𝑃𝑖 − 𝑃 𝑖=1 𝑛 𝑃𝑖 − 𝑃 2 +𝐸 𝑖=1 𝑛 𝜀𝑖 ∙ 𝑃𝑖 − 𝑃 𝑖=1 𝑛 𝑃𝑖 − 𝑃 2
  185. 185. 𝐸 𝜏1 = 𝛽1 +𝐸 𝑖=1 𝑛 𝛽2 ∙ 𝑥𝑖 ∙ 𝑃𝑖 − 𝑃 𝑖=1 𝑛 𝑃𝑖 − 𝑃 2 +0
  186. 186. 𝐸 𝜏1 = 𝛽1 +𝐸 𝑖=1 𝑛 𝛽2 ∙ 𝑥𝑖 ∙ 𝑃𝑖 − 𝑃 𝑖=1 𝑛 𝑃𝑖 − 𝑃 2
  187. 187. 𝐸 𝜏1 = 𝛽1 +𝐸 𝑖=1 𝑛 𝛽2 ∙ 𝑥𝑖 ∙ 𝑃𝑖 − 𝑃 𝑖=1 𝑛 𝑃𝑖 − 𝑃 2
  188. 188. 𝐸 𝑐 ∙ 𝑊 = 𝑐 ∙ 𝐸 𝑊
  189. 189. 𝐸 𝜏1 = 𝛽1 +𝛽2 ∙ 𝐸 𝑖=1 𝑛 𝑥𝑖 ∙ 𝑃𝑖 − 𝑃 𝑖=1 𝑛 𝑃𝑖 − 𝑃 2
  190. 190. 𝐸 𝜏1 = 𝛽1 +𝛽2 ∙ 𝐸 𝑖=1 𝑛 𝑥𝑖 ∙ 𝑃𝑖 − 𝑃 𝑖=1 𝑛 𝑃𝑖 − 𝑃 2
  191. 191. 𝐸 𝜏1 = 𝛽1 +𝛽2 ∙ 𝐸 𝑖=1 𝑛 𝑥𝑖 ∙ 𝑃𝑖 − 𝑃 𝑖=1 𝑛 𝑃𝑖 − 𝑃 2 𝑥𝑖= 𝛾0 + 𝛾1 ∙ 𝑃𝑖 + 𝜗𝑖
  192. 192. 𝐸 𝜏1 = 𝛽1 +𝛽2 ∙ 𝐸 𝑖=1 𝑛 𝑥𝑖 ∙ 𝑃𝑖 − 𝑃 𝑖=1 𝑛 𝑃𝑖 − 𝑃 2 𝑥𝑖= 𝛾0 + 𝛾1 ∙ 𝑃𝑖 + 𝜗𝑖
  193. 193. 𝐸 𝜏1 = 𝛽1 +𝛽2 ∙ 𝐸 𝑖=1 𝑛 𝑥𝑖 ∙ 𝑃𝑖 − 𝑃 𝑖=1 𝑛 𝑃𝑖 − 𝑃 2 𝑥𝑖= 𝛾0 + 𝛾1 ∙ 𝑃𝑖 + 𝜗𝑖
  194. 194. 𝐸 𝜏1 = 𝛽1 +𝛽2 ∙ 𝛾1 𝑖=1 𝑛 𝑥 𝑖=1 𝑛 𝑥 𝑥𝑖= 𝛾0 + 𝛾1 ∙ 𝑃𝑖 + 𝜗𝑖
  195. 195. True model: 𝑌 = 𝛽0 + 𝛽1 ∙ 𝑃 + 𝛽2 ∙ 𝑥 + 𝜀 We actually attempt to estimate: 𝑌 = 𝜏0 + 𝜏1 ∙ 𝑃 + 𝜖 Error term now contains: 𝛽2 ∙ 𝑥
  196. 196. 𝐸 𝜏1 = 𝛽1 + 𝛽2 ∙ 𝐸 𝑖=1 𝑛 𝑥𝑖 ∙ 𝑃𝑖 − 𝑃 𝑖=1 𝑛 𝑃𝑖 − 𝑃 2
  197. 197. 𝐸 𝜏1 = 𝛽1 + 𝛽2 ∙ 𝛾1
  198. 198. 𝐸 𝜏1 = 𝛽1 + 𝛽2 ∙ 𝛾1 The Actual Causal Effect of P on y
  199. 199. 𝐸 𝜏1 = 𝛽1 + 𝛽2 ∙ 𝛾1 The actual causal effect of P on y The actual causal effect of the omitted variable X on Y
  200. 200. 𝐸 𝜏1 = 𝛽1 + 𝛽2 ∙ 𝛾1 The actual causal effect of P on y Th”Effect” of P on x: 𝑥𝑖= 𝛾0 + 𝛾1 ∙ 𝑃𝑖 + 𝜗𝑖 The actual causal effect of the omitted variable X on Y
  201. 201. True model: 𝑌 = 𝛽0 + 𝛽1 ∙ 𝑃 + 𝛽2 ∙ 𝑥 + 𝜀 We actually attempt to estimate: 𝑌 = 𝜏0 + 𝜏1 ∙ 𝑃 + 𝜖 Error term now contains: 𝛽2 ∙ 𝑥
  202. 202. True model: 𝑌 = 𝛽0 + 𝛽1 ∙ 𝑃 + 𝛽2 ∙ 𝑥 + 𝜀 We actually attempt to estimate: 𝑌 = 𝜏0 + 𝜏1 ∙ 𝑃 + 𝜖 Error term now contains: 𝛽2 ∙ 𝑥
  203. 203. True model: 𝑌 = 𝛽0 + 𝛽1 ∙ 𝑃 + 𝛽2 ∙ 𝑥 + 𝜀 We actually attempt to estimate: 𝑌 = 𝜏0 + 𝜏1 ∙ 𝑃 + 𝜖 Error term now contains: 𝛽2 ∙ 𝑥
  204. 204. True model: 𝑌 = 𝛽0 + 𝛽1 ∙ 𝑃 + 𝛽2 ∙ 𝑥 + 𝜀 We actually attempt to estimate: 𝑌 = 𝜏0 + 𝜏1 ∙ 𝑃 + 𝜖 Error term now contains: 𝛽2 ∙ 𝑥
  205. 205. True model: 𝑌 = 𝛽0 + 𝛽1 ∙ 𝑃 + 𝛽2 ∙ 𝑥 + 𝜀 We actually attempt to estimate: 𝑌 = 𝜏0 + 𝜏1 ∙ 𝑃 + 𝜖 𝑬 𝝉 𝟏 ≠ 𝜷
  206. 206. 𝐸 𝜏1 = 𝛽1 + 𝛽2 ∙ 𝛾1 The actual causal effect of P on y Th”Effect” of P on x: 𝑥𝑖= 𝛾0 + 𝛾1 ∙ 𝑃𝑖 + 𝜗𝑖 The actual causal effect of the omitted variable X on Y
  207. 207. 𝐸 𝜏1 = 𝛽1 + 𝛽2 ∙ 𝛾1 The actual causal effect of P on y Th”Effect” of P on x: 𝑥𝑖= 𝛾0 + 𝛾1 ∙ 𝑃𝑖 + 𝜗𝑖 The actual causal effect of the omitted variable X on Y
  208. 208. 𝑌0 = 𝛽0 + 𝛽2 ∙ 𝑥1 + 𝛽3 ∙ 𝑥2 + 𝛽4 ∙ 𝑥2 + 𝜀 𝑌1 = 𝛽0 + 𝛽1 + 𝛽2 ∙ 𝑥1 + 𝛽3 ∙ 𝑥2 + 𝛽4 ∙ 𝑥2 + 𝜀
  209. 209. 𝑌0 = 𝛽0 + 𝛽2 ∙ 𝑥1 + 𝛽3 ∙ 𝑥2 + 𝛽4 ∙ 𝑥2 + 𝜀 𝑌1 = 𝛽0 + 𝛽1 + 𝛽2 ∙ 𝑥1 + 𝛽3 ∙ 𝑥2 + 𝛽4 ∙ 𝑥3 + 𝜀
  210. 210. 𝑌1 − 𝑌0 = 𝛽0 + 𝛽1 + 𝜖 − 𝛽0 + 𝜖 = 𝛽0 + 𝛽1 + 𝜖 − 𝛽0 − 𝜖 = 𝛽1
  211. 211. 𝑌1 − 𝑌0 = 𝛽0 + 𝛽1 + 𝛽2 ∙ 𝑥1 + 𝛽3 ∙ 𝑥2 + 𝛽4 ∙ 𝑥2 + 𝜀 − 𝛽0 + 𝛽2 ∙ 𝑥1 + 𝛽3 ∙ 𝑥2 + 𝛽4 ∙ 𝑥2 + 𝜀 = 𝛽0 + 𝛽1 + 𝜖 − 𝛽0 − 𝜖 = 𝛽1
  212. 212. 𝑌1 − 𝑌0 = 𝛽1
  213. 213. 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
  214. 214. 𝑌 = 𝑃 ∙ 𝑌1 + 1 − 𝑃 ∙ 𝑌0 = 𝑃 ∙ 𝛽0 + 𝛽1 + 𝛽2 ∙ 𝑥1 + 𝛽3 ∙ 𝑥2 + 𝛽4 ∙ 𝑥3 + 𝜀 + 1 − 𝑃 ∙ 𝛽0 + 𝛽2 ∙ 𝑥 + 𝛽2 ∙ 𝑥1 + 𝛽3 ∙ 𝑥2 + 𝛽4 ∙ 𝑥3 + 𝜀
  215. 215. 𝑌 = 𝛽0 + 𝛽1 ∙ 𝑃 + 𝛽2 ∙ 𝑥1 + 𝛽3 ∙ 𝑥2 + 𝛽4 ∙ 𝑥3 + 𝜀
  216. 216. Cost of participation 𝐶 = 𝜌0 + 𝜌1 ∙ 𝑥1 + 𝜌2 ∙ 𝑥2 + 𝜌3 ∙ 𝑥3
  217. 217. True model: 𝑌 = 𝛽0 + 𝛽1 ∙ 𝑃 + 𝛽2 ∙ 𝑥1 + 𝛽3 ∙ 𝑥2 + 𝛽4 ∙ 𝑥3 + 𝜀 We actually attempt to estimate: 𝑌 = 𝜏0 + 𝜏1 ∙ 𝑃 + 𝜖
  218. 218. 𝐸 𝜏1 = 𝛽1 + 𝛽2 ∙ 𝛾11 + 𝛽3 ∙ 𝛾21 + 𝛽4 ∙ 𝛾31
  219. 219. 𝐸 𝜏1 = 𝛽1 + 𝛽2 ∙ 𝛾11 + 𝛽3 ∙ 𝛾21 + 𝛽4 ∙ 𝛾31 The actual causal effect of P on y
  220. 220. 𝐸 𝜏1 = 𝛽1 + 𝛽2 ∙ 𝛾11 + 𝛽3 ∙ 𝛾21 + 𝛽4 ∙ 𝛾31 The actual causal effect of P on y Th”Effect” of P on x1: 𝑥1𝑖= 𝛾10 + 𝛾11 ∙ 𝑃𝑖 + 𝜗1𝑖 The actual causal effect of the omitted variable X1 on Y
  221. 221. 𝐸 𝜏1 = 𝛽1 + 𝛽2 ∙ 𝛾11 + 𝛽3 ∙ 𝛾21 + 𝛽4 ∙ 𝛾31 The actual causal effect of P on y ”Effect” of P on x1: 𝑥1𝑖= 𝛾10 + 𝛾11 ∙ 𝑃𝑖 + 𝜗1𝑖 The actual causal effect of the omitted variable X1 on Y The actual causal effect of the omitted variable X2 on Y Th”Effect” of P on x2: 𝑥2𝑖= 𝛾20 + 𝛾21 ∙ 𝑃𝑖 + 𝜗2𝑖
  222. 222. 𝐸 𝜏1 = 𝛽1 + 𝛽2 ∙ 𝛾11 + 𝛽3 ∙ 𝛾21 + 𝛽4 ∙ 𝛾31 The actual causal effect of P on y ”Effect” of P on x1: 𝑥1𝑖= 𝛾10 + 𝛾11 ∙ 𝑃𝑖 + 𝜗1𝑖 The actual causal effect of the omitted variable X1 on Y The actual causal effect of the omitted variable X2 on Y ”Effect” of P on x2: 𝑥2𝑖= 𝛾20 + 𝛾21 ∙ 𝑃𝑖 + 𝜗2𝑖 Th”Effect” of P on x1: 𝑥3𝑖= 𝛾30 + 𝛾31 ∙ 𝑃𝑖 + 𝜗3𝑖 The actual causal effect of the omitted variable X3 on Y
  223. 223. 𝐸 𝜏1 = 𝛽1 + 𝛽2 ∙ 𝛾11 + 𝛽3 ∙ 𝛾21 + 𝛽4 ∙ 𝛾31 The actual causal effect of P on y ”Effect” of P on x1: 𝑥1𝑖= 𝛾10 + 𝛾11 ∙ 𝑃𝑖 + 𝜗1𝑖 The actual causal effect of the omitted variable X1 on Y The actual causal effect of the omitted variable X2 on Y ”Effect” of P on x2: 𝑥2𝑖= 𝛾20 + 𝛾21 ∙ 𝑃𝑖 + 𝜗2𝑖 Th”Effect” of P on x3: 𝑥3𝑖= 𝛾30 + 𝛾31 ∙ 𝑃𝑖 + 𝜗3𝑖 The actual causal effect of the omitted variable X3 on Y
  224. 224. 𝐸 𝜏1 = 𝛽1 + 𝛽2 ∙ 𝛾11 + 𝛽2 ∙ 𝛾21 + 𝛽3 ∙ 𝛾31
  225. 225. Potential outcomes 𝑌0 = 𝛽0 + 𝛽2 ∙ 𝑥1 + 𝛽3 ∙ 𝑥2 + 𝜀 𝑌1 = 𝛽0 + 𝛽1 + 𝛽2 ∙ 𝑥1 + 𝛽3 ∙ 𝑥2 + 𝜀 Costs of participation 𝐶 = 𝜌0 + 𝜌2 ∙ 𝑥2 + 𝜌3 ∙ 𝑥3
  226. 226. X2 Y P X1 X3
  227. 227. 𝐸 𝜏1 = 𝛽1 + 𝛽2 ∙ 𝛾11 + 𝛽2 ∙ 𝛾21 + 𝛽3 ∙ 𝛾31
  228. 228. 𝐸 𝜏1 = 𝛽1 + 𝛽2 ∙ 𝛾11 + 𝛽2 ∙ 𝛾21 + 𝛽3 ∙ 𝛾31
  229. 229. 𝐸 𝜏1 = 𝛽1 + 𝛽2 ∙ 𝛾21 + 𝛽3 ∙ 𝛾31
  230. 230. X2 Y P X1 X3
  231. 231. 𝐸 𝜏1 = 𝛽1 + 𝛽2 ∙ 𝛾21 + 𝛽3 ∙ 𝛾31
  232. 232. 𝐸 𝜏1 = 𝛽1 + 𝛽2 ∙ 𝛾21
  233. 233. X2 Y P X1 X3
  234. 234. 𝐸 𝜏1 = 𝛽1 + 𝛽2 ∙ 𝛾21
  235. 235. X2 Y P X1 X3
  236. 236. 𝑌 = 𝛽0 + 𝛽1 ∙ 𝑃 + 𝛽2 ∙ 𝑥1 + 𝛽3 ∙ 𝑥2 + 𝛽4 ∙ 𝑥3 + 𝜀
  237. 237. 𝑌 = 𝛽0 + 𝛽1 ∙ 𝑃 + 𝛽2 ∙ 𝑥 + 𝜀 𝐸 𝛽1 = 𝛽1
  238. 238. 𝑌 = 𝛽0 + 𝛽1 ∙ 𝑃 + 𝛽2 ∙ 𝑥 + 𝜀 Regress 𝑌 on 𝑃 and 𝑥 𝐸 𝛽1 = 𝛽1
  239. 239. 𝑌 = 𝛽0 + 𝛽1 ∙ 𝑃 + 𝛽2 ∙ 𝑥 + 𝜀 Regress 𝑌 on 𝑃 and 𝑥 𝐸 𝛽1 = 𝛽1
  240. 240. 𝑌 = 𝛽0 + 𝛽1 ∙ 𝑃 + 𝛽2 ∙ 𝑥 + 𝜀 Regress 𝑌 on 𝑃 and 𝑥 and 𝑍 𝐸 𝛽1 ≠ 𝛽1
  241. 241. 𝑌 = 𝛽0 + 𝛽1 ∙ 𝑃 + 𝛽2 ∙ 𝑥 + 𝜀 Regress 𝑌 on 𝑃 and 𝑥 and 𝑍 𝐸 𝛽1 ≠ 𝛽1
  242. 242. Bad controls
  243. 243. X2 Y P X1 X3
  244. 244. X2 Y P X1 X3 Z
  245. 245. 𝑌 = 𝛽0 + 𝛽1 ∙ 𝑃 + 𝛽𝑧 ∙ 𝑍 + 𝜔 1.Is correlated with the regressor of interest 2.Is correlated with the error term
  246. 246. 𝑌 = 𝛽0 + 𝛽1 ∙ 𝑃 + 𝛽𝑧 ∙ 𝑍 + 𝜔 1.Is correlated with the regressor of interest 2.Is correlated with the error term
  247. 247. 𝑌 = 𝛽0 + 𝛽1 ∙ 𝑃 + 𝛽𝑧 ∙ 𝑍 + 𝜔 1.Is correlated with the regressor of interest 2.Is correlated with the error term
  248. 248. 𝑌 = 𝛽0 + 𝛽1 ∙ 𝑃 + 𝛽𝑧 ∙ 𝑍 + 𝜔 1.Is correlated with the regressor of interest 2.Is correlated with the error term
  249. 249. Matching
  250. 250. X Y P
  251. 251. 1. 𝐸 𝑌1 |𝑋 = 𝑐 ≠ 𝐸 𝑌1 |𝑋 = 𝑘 𝐸 𝑌0 |𝑋 = 𝑐 ≠ 𝐸 𝑌0 |𝑋 = 𝑘 2. 𝑃𝑟 𝑃 = 1|𝑋 = 𝑐 ≠ 𝑃𝑟 𝑃 = 1|𝑋 = 𝑘 𝐸 𝑋|𝑃 = 1 ≠ 𝐸 𝑋|𝑃 = 0
  252. 252. 𝑋 = 0 if male 1 if female
  253. 253. 𝐸 𝑌1 |𝑋 𝐸 𝑌0 |𝑋 𝐸 𝑌1 |𝑋 − 𝐸 𝑌 0 |𝑋 𝑋 = 0 6 4 2 𝑋 = 1 5 1 4
  254. 254. 𝐸 𝑌1 |𝑋 𝐸 𝑌0 |𝑋 𝐸 𝑌1 |𝑋 − 𝐸 𝑌 0 |𝑋 𝑋 = 0 6 4 2 𝑋 = 1 5 1 4 𝐸 𝑌1 − 𝑌0 |𝑋 = 0 ∙ 𝑃𝑟 𝑋 = 0 +𝐸 𝑌1 − 𝑌0 |𝑋 = 1 ∙ 𝑃𝑟 𝑋 = 1 = 2 ∙ .5 + 4 ∙ .5 = 3
  255. 255. 𝐸 𝑌1 |𝑋 𝐸 𝑌0 |𝑋 𝐸 𝑌1 |𝑋 − 𝐸 𝑌 0 |𝑋 𝑋 = 0 6 4 2 𝑋 = 1 5 1 4 𝐸 𝑌1 |𝑋 = 0 ∙ 𝑃𝑟 𝑋 = 0 +𝐸 𝑌1 |𝑋 = 1 ∙ 𝑃𝑟 𝑋 = 1 = 6 ∙ .5 + 5 ∙ .5 = 5.5
  256. 256. 𝐸 𝑌1 |𝑋 𝐸 𝑌0 |𝑋 𝐸 𝑌1 |𝑋 − 𝐸 𝑌 0 |𝑋 𝑋 = 0 6 4 2 𝑋 = 1 5 1 4 𝐸 𝑌0 |𝑋 = 0 ∙ 𝑃𝑟 𝑋 = 0 +𝐸 𝑌0 |𝑋 = 1 ∙ 𝑃𝑟 𝑋 = 1 = 4 ∙ .5 + 1 ∙ .5 = 2.5
  257. 257. 𝐸 𝑃 = 1|𝑋 𝐸 𝑃 = 0|𝑋 𝑋 = 0 .1 .9 𝑋 = 1 .5 .5 𝐸 𝑃 = 1|𝑋 = 0 ∙ 𝑃𝑟 𝑋 = 0 +𝐸 𝑃 = 1|𝑋 = 1 ∙ 𝑃𝑟 𝑋 = 1 = .1 ∙ .5 + .5 ∙ .5 = .3
  258. 258. 𝐸 𝑃 = 1|𝑋 𝐸 𝑃 = 0|𝑋 𝑋 = 0 .1 .9 𝑋 = 1 .5 .5 𝐸 𝑃 = 1|𝑋 = 0 ∙ 𝑃𝑟 𝑋 = 0 +𝐸 𝑃 = 1|𝑋 = 1 ∙ 𝑃𝑟 𝑋 = 1 = .1 ∙ .5 + .5 ∙ .5 = .3
  259. 259. 𝑛 = 1000 𝑖=1 𝑛 𝑋𝑖 𝑛 = 𝑖=1 1000 𝑋𝑖 1000 ≈ 500 𝑖=1 𝑛 𝑃𝑖 𝑛 = 𝑖=1 1000 𝑃𝑖 1000 ≈ 300
  260. 260. 𝑛 = 1000 𝑖=1 𝑛 𝑋𝑖 𝑛 = 𝑖=1 1000 𝑋𝑖 1000 ≈ .5 𝑖=1 𝑛 𝑃𝑖 𝑛 = 𝑖=1 1000 𝑃𝑖 1000 ≈ 300
  261. 261. 𝑛 = 1000 𝑖=1 𝑛 𝑋𝑖 𝑛 = 𝑖=1 1000 𝑋𝑖 1000 ≈ .5 𝑖=1 𝑛 𝑃𝑖 𝑛 = 𝑖=1 1000 𝑃𝑖 1000 ≈ .3
  262. 262. 𝑗=1 300 𝑌𝑗 300 : 6 ∙ .16667 + 5 ∙ .8333 = 5.16652
  263. 263. 𝑗=1 300 𝑌𝑗 300 : 6 ∙ .16667 + 5 ∙ .8333 = 5.16652 𝑗=1 700 𝑌𝑗 700 : 4 ∙ .6428 + 1 ∙ .3571 = 2.9286
  264. 264. ?
  265. 265. 𝑖 = 237 𝑋237 = 0 𝑃237 = 1 𝑌237 = 𝑌237 1 = 5.3
  266. 266. 𝑖 = 237 𝑋237 = 0 𝑃237 = 1 𝑌237 = 𝑌237 1 = 5.3
  267. 267. 𝑖 = 237 𝑋237 = 0 𝑃237 = 1 𝑌237 = 𝑌237 1 = 5.3
  268. 268. 𝑖 = 237 𝑋237 = 0 𝑃237 = 1 𝑌237 = 𝑌237 1 = 5.3
  269. 269. 𝑖 = 237 𝑋237 = 0 𝑃237 = 1 𝑌237 = 𝑌237 1 = 6.3
  270. 270. 𝐼237 = 𝑌237 1 − 𝑌237 0 = 6.3−?
  271. 271. 𝑌237 0
  272. 272. 𝑌237 0 = 𝑌766 0
  273. 273. 𝑌237 0 = 𝑌48 0 + 𝑌109 0 + 𝑌418 0 + 𝑌505 0 + 𝑌919 0 5
  274. 274. 𝐼𝑖 = 𝑌𝑖 1 − 𝑌𝑖 0
  275. 275. 𝐴𝑇𝐸 = 𝑖=1 𝑛 𝐼𝑖 𝑛
  276. 276. 𝑘=0 𝐾 𝐸 𝐼|𝑋 = 𝑘 ∙ 𝑃𝑟 𝑋 = 𝑘 = 𝑘=0 𝐾 𝐸 𝑌1 − 𝑌0 |𝑋 = 𝑘 ∙ 𝑃𝑟 𝑋 = 𝑘
  277. 277. 𝑘=0 𝐾 𝐸 𝐼|𝑋 = 𝑘, 𝑃 = 1 ∙ 𝑃𝑟 𝑋 = 𝑘|𝑃 = 1 = 𝑘=0 𝐾 𝐸 𝑌1 − 𝑌0 |𝑋 = 𝑘, 𝑃 = 1 ∙ 𝑃𝑟 𝑋 = 𝑘|𝑃 = 1
  278. 278. 𝐼237 = 𝑌237 1 − 𝑌237 0 = 6.3−?
  279. 279. 𝑘=0 𝐾 𝐸 𝐼|𝑋 = 𝑘 ∙ 𝑃𝑟 𝑋 = 𝑘 = 𝑘=0 𝐾 𝐸 𝑌1 − 𝑌0 |𝑋 = 𝑘 ∙ 𝑃𝑟 𝑋 = 𝑘
  280. 280. X Y P
  281. 281. 2X Gender-male/female 5X SES-5 quintiles 4X Age-broken up into 4 categories 2X Sector-Urban/Rural 6X Religion-Christian, Muslim, Hindu, Buddhist, Jewish, Traditional. 6X Occupation-6 occupation types 2 Insurance-Insured/not insured ____ 5,760 potential “types”
  282. 282. 2X Gender-male/female 5X SES-5 quintiles 4X Age-broken up into 4 categories 2X Sector-Urban/Rural 6X Religion-Christian, Muslim, Hindu, Buddhist, Jewish, Traditional. 6X Occupation-6 occupation types 2 Insurance-Insured/not insured ____ 2 potential “types”
  283. 283. 2X Gender-male/female 5X SES-5 quintiles 4X Age-broken up into 4 categories 2X Sector-Urban/Rural 6X Religion-Christian, Muslim, Hindu, Buddhist, Jewish, Traditional. 6X Occupation-6 occupation types 2 Insurance-Insured/not insured ____ 10 potential “types”
  284. 284. 2X Gender-male/female 5X SES-5 quintiles 4X Age-broken up into 4 categories 2X Sector-Urban/Rural 6X Religion-Christian, Muslim, Hindu, Buddhist, Jewish, Traditional. 6X Occupation-6 occupation types 2 Insurance-Insured/not insured ____ 40 potential “types”
  285. 285. 2X Gender-male/female 5X SES-5 quintiles 4X Age-broken up into 4 categories 2X Sector-Urban/Rural 6X Religion-Christian, Muslim, Hindu, Buddhist, Jewish, Traditional. 6X Occupation-6 occupation types 2 Insurance-Insured/not insured ____ 80 potential “types”
  286. 286. 2X Gender-male/female 5X SES-5 quintiles 4X Age-broken up into 4 categories 2X Sector-Urban/Rural 6X Religion-Christian, Muslim, Hindu, Buddhist, Jewish, Traditional. 6X Occupation-6 occupation types 2 Insurance-Insured/not insured ____ 480 potential “types”
  287. 287. 2X Gender-male/female 5X SES-5 quintiles 4X Age-broken up into 4 categories 2X Sector-Urban/Rural 6X Religion-Christian, Muslim, Hindu, Buddhist, Jewish, Traditional. 6X Occupation-6 occupation types 2 Insurance-Insured/not insured ____ 2,880 potential “types”
  288. 288. 2X Gender-male/female 5X SES-5 quintiles 4X Age-broken up into 4 categories 2X Sector-Urban/Rural 6X Religion-Christian, Muslim, Hindu, Buddhist, Jewish, Traditional. 6X Occupation-6 occupation types 2 Insurance-Insured/not insured ____ 5,760 potential “types”
  289. 289. 2X Gender-male/female 5X SES-5 quintiles 4X Age-broken up into 4 categories 2X Sector-Urban/Rural 6X Religion-Christian, Muslim, Hindu, Buddhist, Jewish, Traditional. 6X Occupation-6 occupation types 2 Insurance-Insured/not insured ____ 5,760 potential “types”
  290. 290. Pr(𝑃 = 1|𝑋)
  291. 291. 𝐼237 = 𝑌237 1 − 𝑌237 0 = 6.3−?
  292. 292. 𝐼237 = 𝑌237 1 − 𝑌237 0 = 6.3−?
  293. 293. Pr(𝑃 = 1|𝑋)
  294. 294. 𝑃𝑟(𝑃 = 1|𝑋)
  295. 295. 𝑃𝑟 𝑃 = 1 𝑋 = 𝑒𝑥𝑝 𝛽0 + 𝛽1 ∙ 𝑋 1 + 𝑒𝑥𝑝 𝛽0 + 𝛽1 ∙ 𝑋
  296. 296. The basics of the method 1. Pool your sample of participants and non-participants and define various characteristics 𝑋. 2. Estimate the probability of program participation conditional on Pr 𝑃 = 1|𝑋 3. Compute the propensity score using the fitted binary participation model. 𝑃𝑟 𝑃𝑖 = 1|𝑋𝑖 4. Find a counterfactual outcome for each individual by identifying some individual who did experience the counterfactual conditions and had a similar propensity score
  297. 297. Matching approaches 1. Nearest neighbor 2. Caliper 3. (Budget) Caliper 4. Weighting
  298. 298. Matching approaches 1. Nearest neighbor 2. Caliper 3. (Budget) Caliper 4. Weighting
  299. 299. Balancing property Pr 𝑃 = 1|𝑋
  300. 300. Propensity score 0 1 P=0 P=1 Common support
  301. 301. Propensity score 0 1 P=0 P=1 Failure of common support
  302. 302. Other applications of the propensity score 1.Weighting 2.Regression
  303. 303. Other applications of the propensity score 1.Weighting 2.Regression
  304. 304. 𝐸 𝑃 = 1|𝑋 𝐸 𝑃 = 0|𝑋 𝑋 = 0 .1 .9 𝑋 = 1 .5 .5 𝐸 𝑃 = 1|𝑋 = 0 ∙ 𝑃𝑟 𝑋 = 0 +𝐸 𝑃 = 1|𝑋 = 1 ∙ 𝑃𝑟 𝑋 = 1 = .1 ∙ .5 + .5 ∙ .5 = .3 1,000 observations
  305. 305. 𝐸 𝑃 = 1|𝑋 𝐸 𝑃 = 0|𝑋 𝑋 = 0 .1 .9 𝑋 = 1 .5 .5 𝐸 𝑃 = 1|𝑋 = 0 ∙ 𝑃𝑟 𝑋 = 0 +𝐸 𝑃 = 1|𝑋 = 1 ∙ 𝑃𝑟 𝑋 = 1 = .1 ∙ .5 + .5 ∙ .5 = .3 1,000 observations 300 participants
  306. 306. 𝐸 𝑃 = 1|𝑋 𝐸 𝑃 = 0|𝑋 𝑋 = 0 .1 .9 𝑋 = 1 .5 .5 𝐸 𝑃 = 1|𝑋 = 0 ∙ 𝑃𝑟 𝑋 = 0 +𝐸 𝑃 = 1|𝑋 = 1 ∙ 𝑃𝑟 𝑋 = 1 = .1 ∙ .5 + .5 ∙ .5 = .3 1,000 observations 300 participants 700 non-participants
  307. 307. 𝐸 𝑃 = 1|𝑋 𝐸 𝑃 = 0|𝑋 𝑋 = 0 .1 .9 𝑋 = 1 .5 .5 𝐸 𝑃 = 1|𝑋 = 0 ∙ 𝑃𝑟 𝑋 = 0 +𝐸 𝑃 = 1|𝑋 = 1 ∙ 𝑃𝑟 𝑋 = 1 = .1 ∙ .5 + .5 ∙ .5 = .3 1,000 observations 300 participants 700 non-participants 50 men 250 women
  308. 308. 𝐸 𝑃 = 1|𝑋 𝐸 𝑃 = 0|𝑋 𝑋 = 0 .1 .9 𝑋 = 1 .5 .5 𝐸 𝑃 = 1|𝑋 = 0 ∙ 𝑃𝑟 𝑋 = 0 +𝐸 𝑃 = 1|𝑋 = 1 ∙ 𝑃𝑟 𝑋 = 1 = .1 ∙ .5 + .5 ∙ .5 = .3 1,000 observations 300 participants 700 non-participants 50 men 250 women 450 men 250 women
  309. 309. 𝐸 𝑃 = 1|𝑋 𝐸 𝑃 = 0|𝑋 𝑋 = 0 .1 .9 𝑋 = 1 .5 .5 𝐸 𝑃 = 1|𝑋 = 0 ∙ 𝑃𝑟 𝑋 = 0 +𝐸 𝑃 = 1|𝑋 = 1 ∙ 𝑃𝑟 𝑋 = 1 = .1 ∙ .5 + .5 ∙ .5 = .3 1,000 observations 300 participants 700 non-participants 500 men 500 women 450 men 250 women
  310. 310. 𝐸 𝑃 = 1|𝑋 𝐸 𝑃 = 0|𝑋 𝑋 = 0 .1 .9 𝑋 = 1 .5 .5 𝐸 𝑃 = 1|𝑋 = 0 ∙ 𝑃𝑟 𝑋 = 0 +𝐸 𝑃 = 1|𝑋 = 1 ∙ 𝑃𝑟 𝑋 = 1 = .1 ∙ .5 + .5 ∙ .5 = .3 1,000 observations 300 participants 700 non-participants 500 men 500 women 500 men 500 women
  311. 311. 𝑊𝑌𝑖 = 𝑃𝑖 ∙ 𝑌𝑖 𝑃𝑟 𝑃𝑖 = 1|𝑋𝑖 − 1 − 𝑃𝑖 ∙ 𝑌𝑖 1 − 𝑃𝑟 𝑃𝑖 = 1|𝑋𝑖 𝐴𝑇𝐸 = 𝑖=1 𝑛 𝑊𝑌𝑖 𝑖=1 𝑛 𝑃𝑖 𝑃𝑟 𝑃𝑖 = 1|𝑋𝑖 + 1 − 𝑃𝑖 1 − 𝑃𝑟 𝑃𝑖 = 1|𝑋𝑖
  312. 312. 𝑊𝑌𝑖 = 𝑃𝑖 ∙ 𝑌𝑖 𝑃𝑟 𝑃𝑖 = 1|𝑋𝑖 − 1 − 𝑃𝑖 ∙ 𝑌𝑖 1 − 𝑃𝑟 𝑃𝑖 = 1|𝑋𝑖 𝐴𝑇𝐸 = 𝑖=1 𝑛 𝑊𝑌𝑖 𝒊=𝟏 𝒏 𝑷𝒊 𝑷𝒓 𝑷𝒊 = 𝟏|𝑿𝒊 + 𝟏 − 𝑷𝒊 𝟏 − 𝑷𝒓 𝑷𝒊 = 𝟏|𝑿𝒊
  313. 313. 𝑊𝑌𝑖 = 𝑃𝑖 ∙ 𝑌𝑖 𝑃𝑟 𝑃𝑖 = 1|𝑋𝑖 − 1 − 𝑃𝑖 ∙ 𝑌𝑖 1 − 𝑃𝑟 𝑃𝑖 = 1|𝑋𝑖 𝐴𝑇𝐸 = 𝑖=1 𝑛 𝑊𝑌𝑖 𝑖=1 𝑛 𝑃𝑖 𝑃𝑟 𝑃𝑖 = 1|𝑋𝑖 + 1 − 𝑃𝑖 1 − 𝑃𝑟 𝑃𝑖 = 1|𝑋𝑖
  314. 314. Other applications of the propensity score 1.Weighting 2.Regression
  315. 315. Regress 𝑌𝑖 on 𝑃𝑖 and 𝑃𝑟 𝑃𝑖 = 1|𝑋𝑖
  316. 316. Regress 𝑌𝑖 on 𝑃𝑖 and 𝑃𝑟 𝑃𝑖 = 1|𝑋𝑖
  317. 317. Conclusion
  318. 318. 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
  319. 319. 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

×