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Peter M. Lance, PhD
MEASURE Evaluation
University of North Carolina at
Chapel Hill
April 13, 2017
Instrumental Variables
Peter M. Lance, PhD
MEASURE Evaluation
University of North Carolina at
Chapel Hill
April 13, 2017
Instrumental Variables:
...
Peter M. Lance, PhD
MEASURE Evaluation
University of North Carolina at
Chapel Hill
April 13, 2017
Instrumental Variables:
...
Peter M. Lance, PhD
MEASURE Evaluation
University of North Carolina at
Chapel Hill
April 13, 2017
Instrumental Variables:
...
Peter M. Lance, PhD
MEASURE Evaluation
University of North Carolina at
Chapel Hill
April 13, 2017
Instrumental Variables:
...
Peter M. Lance, PhD
MEASURE Evaluation
University of North Carolina at
Chapel Hill
April 13, 2017
Instrumental Variables:
...
Peter M. Lance, PhD
MEASURE Evaluation
University of North Carolina at
Chapel Hill
April 13, 2017
Instrumental Variables:
...
Peter M. Lance, PhD
MEASURE Evaluation
University of North Carolina at
Chapel Hill
April 13, 2017
Instrumental Variables:
...
Peter M. Lance, PhD
MEASURE Evaluation
University of North Carolina at
Chapel Hill
April 13, 2017
Instrumental Variables:
...
Peter M. Lance, PhD
MEASURE Evaluation
University of North Carolina at
Chapel Hill
April 13, 2017
Instrumental Variables:
...
Peter M. Lance, PhD
MEASURE Evaluation
University of North Carolina at
Chapel Hill
April 13, 2017
Instrumental Variables:
...
Global, five-year, $180M cooperative agreement
Strategic objective:
To strengthen health information systems – the
capacit...
Improved country capacity to manage health
information systems, resources, and staff
Strengthened collection, analysis, an...
Global footprint (more than 25 countries)
• The program impact evaluation challenge
• Randomization
• Selection on observables
• Within estimators
• Instrumental va...
• The program impact evaluation challenge
• Randomization
• Selection on observables
• Within estimators
• Instrumental va...
YP
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YP
μ
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YP
μ
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YP
μ
Z
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YP
μ
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YP
μ
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YP
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Fascinating, right???
So what the heck does it
all mean, and how would
you implement it?
Fascinating, right???
So what the heck does it
all mean, and how would
you implement it?
𝑌0 = 𝛽0 + 𝛽2 ∙ 𝑥 + 𝛽3 ∙ 𝜇 + 𝜀
𝑌1 = 𝛽0 + 𝛽1 + 𝛽2 ∙ 𝑥 + 𝛽3 ∙ 𝜇 + 𝜀
𝑌0 = 𝛽0 + 𝛽2 ∙ 𝒙 + 𝛽3 ∙ 𝜇 + 𝜀
𝑌1 = 𝛽0 + 𝛽1 + 𝛽2 ∙ 𝒙 + 𝛽3 ∙ 𝜇 + 𝜀
𝑌0 = 𝛽0 + 𝛽2 ∙ 𝑥 + 𝛽3 ∙ 𝝁 + 𝜀
𝑌1 = 𝛽0 + 𝛽1 + 𝛽2 ∙ 𝑥 + 𝛽3 ∙ 𝝁 + 𝜀
𝑌0 = 𝛽0 + 𝛽2 ∙ 𝑥 + 𝛽3 ∙ 𝜇 + 𝜺
𝑌1 = 𝛽0 + 𝛽1 + 𝛽2 ∙ 𝑥 + 𝛽3 ∙ 𝜇 + 𝜺
𝑌1 − 𝑌0
= 𝛽0 + 𝛽1 + 𝜖 − 𝛽0 + 𝜖
= 𝛽0 + 𝛽1 + 𝜖 − 𝛽0 − 𝜖
= 𝛽1
𝑌1 − 𝑌0
= 𝛽0 + 𝛽1 + 𝛽2 ∙ 𝑥 + 𝛽3 ∙ 𝜇 + 𝜀
− 𝛽0 + 𝛽2 ∙ 𝑥 + 𝛽3 ∙ 𝜇 + 𝜀
= 𝛽0 + 𝛽1 + 𝜖 − 𝛽0 − 𝜖
= 𝛽1
𝑌1 − 𝑌0
= 𝛽1
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 ∗ 𝛽...
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
+...
𝑌 = 𝛽0 + 𝛽1 ∙ 𝑃 + 𝛽2 ∙ 𝑥 + 𝛽3 ∙ 𝜇 + 𝜀
𝑌 = 𝛽0 + 𝛽1 ∙ 𝑃 + 𝛽2 ∙ 𝑥 + 𝛽3 ∙ 𝜇 + 𝜀
𝑌 = 𝛽0 + 𝛽1 ∙ 𝑃 + 𝛽2 ∙ 𝑥 + 𝛽3 ∙ 𝜇 + 𝜀
𝑌 = 𝛾0 + 𝛾1 ∙ 𝑃 + 𝛾2 ∙ 𝑥 + 𝜀
𝐸 𝛾1 = 𝛽1
𝑌 = 𝛽0 + 𝛽1 ∙ 𝑃 + 𝛽2 ∙ 𝑥 + 𝛽3 ∙ 𝜇 + 𝜀
𝑌 = 𝛾0 + 𝛾1 ∙ 𝑃 + 𝛾2 ∙ 𝑥 + 𝜀
𝐸 𝛾1 = 𝛽1
?
𝑌 = 𝛽0 + 𝛽1 ∙ 𝑃 + 𝛽2 ∙ 𝑥 + 𝛽3 ∙ 𝜇 + 𝜀
𝑌 = 𝛾0 + 𝛾1 ∙ 𝑃 + 𝛾2 ∙ 𝑥 + 𝜀
𝐸 𝛾1 = 𝛽1
?
𝑌 = 𝛽0 + 𝛽1 ∙ 𝑃 + 𝛽2 ∙ 𝑥 + 𝛽3 ∙ 𝜇 + 𝜀
𝑌 = 𝛾0 + 𝛾1 ∙ 𝑃 + 𝛾2 ∙ 𝑥 + 𝜀
𝐸 𝛾1 = 𝛽1
?
YP
μ
X
YP
μ
X
𝑌 = 𝛽0 + 𝛽1 ∙ 𝑃 + 𝛽2 ∙ 𝑥 + 𝛽3 ∙ 𝜇 + 𝜀
𝑌 = 𝛾0 + 𝛾1 ∙ 𝑃 + 𝛾2 ∙ 𝑥 + 𝜀
𝐸 𝛾1 = 𝛽1
?
𝑌 = 𝛽0 + 𝛽1 ∙ 𝑃 + 𝛽2 ∙ 𝑥 + 𝛽3 ∙ 𝜇 + 𝜀
𝑌 = 𝛾0 + 𝛾1 ∙ 𝑃 + 𝛾2 ∙ 𝑥 + 𝜀
𝐸 𝛾1 = 𝛽1
?
𝑌 = 𝛽0 + 𝛽1 ∙ 𝑃 + 𝛽2 ∙ 𝑥 + 𝛽3 ∙ 𝜇 + 𝜀
𝑌 = 𝛾0 + 𝛾1 ∙ 𝑃 + 𝛾2 ∙ 𝑥 + 𝜀
𝐸 𝛾1 = 𝛽1
?
𝑌 = 𝛽0 + 𝛽1 ∙ 𝑃 + 𝛽2 ∙ 𝑥 + 𝛽3 ∙ 𝜇 + 𝜀
𝑌 = 𝛾0 + 𝛾1 ∙ 𝑃 + 𝛾2 ∙ 𝑥 + 𝜀
𝐸 𝛾1 ≠ 𝛽1
𝑌 = 𝛽0 + 𝛽1 ∙ 𝑃 + 𝛽2 ∙ 𝑥 + 𝛽3 ∙ 𝜇 + 𝜀
𝑌 = 𝛾0 + 𝛾1 ∙ 𝑃 + 𝛾2 ∙ 𝑥 + 𝜀
𝐸 𝛾1 ≠ 𝛽1
𝑌 = 𝛽0 + 𝛽1 ∙ 𝑃 + 𝛽2 ∙ 𝑥 + 𝛽3 ∙ 𝜇 + 𝜀
𝑌 = 𝛾0 + 𝛾1 ∙ 𝑃 + 𝛾2 ∙ 𝑥 + 𝜀
𝐸 𝛾1 ≠ 𝛽1
𝑌 = 𝛽0 + 𝛽1 ∙ 𝑃 + 𝛽2 ∙ 𝑥 + 𝛽3 ∙ 𝜇 + 𝜀
𝑌 = 𝛾0 + 𝛾1 ∙ 𝑃 + 𝛾2 ∙ 𝑥 + 𝜀
𝐸 𝛾1 ≠ 𝛽1
Cost of Participation
𝐶 = 𝜌0 + 𝜌1 ∙ 𝑥 + 𝜌2 ∙ 𝜇 + 𝜌3 ∙ 𝑧 + 𝜂
Cost of Participation
𝐶 = 𝜌0 + 𝜌1 ∙ 𝒙 + 𝜌2 ∙ 𝜇 + 𝜌3 ∙ 𝑧 + 𝜂
Cost of Participation
𝐶 = 𝜌0 + 𝜌1 ∙ 𝑥 + 𝜌2 ∙ 𝝁 + 𝜌3 ∙ 𝑧 + 𝜂
Cost of Participation
𝐶 = 𝜌0 + 𝜌1 ∙ 𝑥 + 𝜌2 ∙ 𝜇 + 𝜌3 ∙ 𝒛 + 𝜂
Cost of Participation
𝐶 = 𝜌0 + 𝜌1 ∙ 𝑥 + 𝜌2 ∙ 𝜇 + 𝜌3 ∙ 𝑧 + 𝜼
Benefit-Cost>0
𝑌1
− 𝑌0
− C > 0
𝛽1 − 𝐶 > 0
𝛽1 − 𝜌0 + 𝜌1 ∙ 𝑥 + 𝜌2 ∙ 𝜇 + 𝜌3 ∙ 𝑧 + 𝜂 > 0
Benefit-Cost>0
𝑌1
− 𝑌0
− C > 0
𝛽1 − 𝐶 > 0
𝛽1 − 𝜌0 + 𝜌1 ∙ 𝑥 + 𝜌2 ∙ 𝜇 + 𝜌3 ∙ 𝑧 + 𝜂 > 0
Benefit-Cost>0
𝑌1
− 𝑌0
− C > 0
𝛽1 − 𝐶 > 0
𝛽1 − 𝜌0 + 𝜌1 ∙ 𝑥 + 𝜌2 ∙ 𝜇 + 𝜌3 ∙ 𝑧 + 𝜂 > 0
Benefit-Cost>0
𝑌1
− 𝑌0
− C > 0
𝛽1 − 𝐶 > 0
𝛽1 − 𝜌0 + 𝜌1 ∙ 𝑥 + 𝜌2 ∙ 𝜇 + 𝜌3 ∙ 𝑧 + 𝜂 > 0
𝑥 𝑃
𝜇 𝑃
𝑧 𝑃
𝑥 𝑃
𝜇 𝑃
𝑧 𝑃
YP
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Benefit-Cost>0
𝑌1
− 𝑌0
− C > 0
𝛽1 − 𝐶 > 0
𝛽1 − 𝜌0 + 𝜌1 ∙ 𝑥 + 𝜌2 ∙ 𝜇 + 𝜌3 ∙ 𝑧 + 𝜂 > 0
𝑥 𝑃
𝜇 𝑃
𝑧 𝑃
𝑌0 = 𝛽0 + 𝛽2 ∙ 𝑥 + 𝛽3 ∙ 𝜇 + 𝜀
𝑌1 = 𝛽0 + 𝛽1 + 𝛽2 ∙ 𝑥 + 𝛽3 ∙ 𝜇 + 𝜀
𝑌 = 𝑃 ∙ 𝑌1
+ 1 − 𝑃 ∙ 𝑌0
= 𝑃 ∗ 𝛽0 + 𝛽1 + 𝜖 +
𝑌 = 𝑷 ∙ 𝑌1
+ 1 − 𝑷 ∙ 𝑌0
= 𝑃 ∗ 𝛽0 + 𝛽1 + 𝜖 +
YP
μ
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Benefit-Cost>0
𝑌1
− 𝑌0
− C > 0
𝛽1 − 𝐶 > 0
𝛽1 − 𝜌0 + 𝜌1 ∙ 𝑥 + 𝜌2 ∙ 𝜇 + 𝜌3 ∙ 𝑧 + 𝜂 > 0
𝛽1 − 𝜌0 + 𝜌1 ∙ 𝑥 + 𝜌2 ∙ 𝜇 + 𝜌3 ∙ 𝑧 + 𝜂 > 0
𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧 + 𝜉
𝛽1 − 𝜌0 + 𝜌1 ∙ 𝑥 + 𝜌2 ∙ 𝜇 + 𝜌3 ∙ 𝑧 + 𝜼 > 0
𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧 + 𝜉
𝛽1 − 𝜌0 + 𝜌1 ∙ 𝑥 + 𝜌2 ∙ 𝜇 + 𝜌3 ∙ 𝑧 + 𝜂 > 0
𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧 + 𝜉
𝛽1 − 𝜌0 + 𝜌1 ∙ 𝑥 + 𝜌2 ∙ 𝜇 + 𝜌3 ∙ 𝑧 + 𝜂 > 0
𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧 + 𝝃
𝛽1 − 𝜌0 + 𝜌1 ∙ 𝑥 + 𝜌2 ∙ 𝜇 + 𝜌3 ∙ 𝑧 + 𝜂 > 0
𝑃∗ = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧 + 𝜉
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
𝛽1 − 𝜌0 + 𝜌1 ∙ 𝑥 + 𝜌2 ∙ 𝜇 + 𝜌3 ∙ 𝑧 + 𝜂 > 0
𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧 + 𝜉
𝑌 = 𝛾0 + 𝛾1 ∙ 𝑃 + 𝛾2 ∙ 𝑥 + 𝜀
𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧 + 𝜉
𝒀 = 𝜸 𝟎 + 𝜸 𝟏 ∙ 𝑷 + 𝜸 𝟐 ∙ 𝒙 + 𝜺
𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧 + 𝜉
𝑌 = 𝛾0 + 𝛾1 ∙ 𝑃 + 𝛾2 ∙ 𝑥 + 𝜀
𝑷 = 𝝓 𝟎 + 𝝓 𝟏 ∙ 𝒙 + 𝝓 𝟐 ∙ 𝒛 + 𝝃
𝑌 = 𝛾0 + 𝛾1 ∙ 𝑃 + 𝛾2 ∙ 𝑥 + 𝜀
𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧 + 𝜉
𝑌 = 𝛾0 + 𝛾1 ∙ 𝑃 + 𝛾2 ∙ 𝑥 + 𝜀
𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧 + 𝜉
𝝁
𝑌 = 𝛾0 + 𝛾1 ∙ 𝑃 + 𝛾2 ∙ 𝑥 + 𝜺
𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧 + 𝝃
𝝁
𝑌 = 𝛾0 + 𝛾1 ∙ 𝑃 + 𝛾2 ∙ 𝑥 + 𝜺
𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧 + 𝝃
𝝁
𝑌 = 𝛾0 + 𝛾1 ∙ 𝑃 + 𝛾2 ∙ 𝑥 + 𝜺
𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧 + 𝝃
𝝁
𝑌 = 𝛾0 + 𝛾1 ∙ 𝑃 + 𝛾2 ∙ 𝑥 + 𝜺
𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧 + 𝝃
𝑬( 𝜸 𝟏) ≠ 𝜷 𝟏
Two-Stage Least Squares
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/est...
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/est...
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/est...
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/est...
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/est...
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
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
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
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
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
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
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
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
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
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
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 ∙ 𝜙...
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...
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 ∙ ...
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 ∙ 𝑧...
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 ...
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
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
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
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
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
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
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
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
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/est...
Two-Stage Least Squares
𝑌 = 𝛾0 + 𝛾1 ∙ 𝑃 + 𝛾2 ∙ 𝑥 + 𝜀
𝐸 𝛾0 = 𝛽1
Two-Stage Least Squares
𝑌 = 𝛾0 + 𝛾1 ∙ 𝑃 + 𝛾2 ∙ 𝑥 + 𝜀
𝐸 𝛾0 ≠ 𝛽1
Two-Stage Least Squares
𝑌 = 𝛾0 + 𝛾1 ∙ 𝑃 + 𝛾2 ∙ 𝑥 + 𝜀
𝐸 𝛾0 ≠ 𝛽1
????!
Consistency
𝑝𝑙𝑖𝑚 𝛾1 = 𝛽1
𝑙𝑖𝑚 𝑛→∞ 𝑃𝑟 𝛾1 − 𝛽1 ≥ 𝑐 = 0
Consistency
𝑝𝑙𝑖𝑚 𝛾1 = 𝛽1
𝑙𝑖𝑚 𝑛→∞ 𝑃𝑟 𝛾1 − 𝛽1 ≥ 𝑐 = 0
= 0
𝑌 = 𝛽0 + 𝛽1 ∙ 𝑃 + 𝛽2 ∙ 𝑥 + 𝛽3 ∙ 𝜇 + 𝜀
Some Big Take-Aways About IV/TSLS
1. More powerful instruments are better
2. Over-identification is (generally) your frien...
Some Big Take-Aways About IV/TSLS
1. More powerful instruments are better
2. Over-identification is (generally) your frien...
Anything
Literally anything
Instrumentation and It’s
Discontents
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
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
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, ...
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, ...
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
𝑎𝑏𝑠 𝛽1 − 𝛾1
𝑛
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, ...
Some Big Take-Aways About IV/TSLS
1. More powerful instruments are better
2. Over-identification is (generally) your
frien...
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
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, ...
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, ...
Some Big Take-Aways About IV/TSLS
1. More powerful instruments are better
2. Over-identification is (generally) your frien...
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
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
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
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
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
Some Big Take-Aways About IV/TSLS
1. More powerful instruments are better
2. Over-identification is (generally) your frien...
𝑌0 = 𝛽0 + 𝛽2 ∙ 𝑥 + 𝛽3 ∙ 𝜇 + 𝜀
𝑌1 = 𝛽0 + 𝛽1 + 𝛽2 ∙ 𝑥 + 𝛽3 ∙ 𝜇 + 𝜀
𝑌𝑖
0
= 𝛽0 + 𝛽2 ∙ 𝑥𝑖 + 𝛽3 ∙ 𝜇𝑖 + 𝜀𝑖
𝑌𝑖
1
= 𝛽0 + 𝛽1 + 𝛽2 ∙ 𝑥𝑖 + 𝛽3 ∙ 𝜇𝑖 + 𝜀𝑖
𝑌𝑖
0
= 𝛽0 + 𝛽2 ∙ 𝑥𝑖 + 𝛽3 ∙ 𝜇𝑖 + 𝜀𝑖
𝑌𝑖
1
= 𝛽0 + 𝛽1𝑖 + 𝛽2 ∙ 𝑥𝑖 + 𝛽3 ∙ 𝜇𝑖 + 𝜀𝑖
𝑌𝑖
0
= 𝛽0 + 𝛽2 ∙ 𝑥𝑖 + 𝛽3 ∙ 𝜇𝑖 + 𝜀𝑖
𝑌𝑖
1
= 𝛽0 + 𝛽1𝑖 + 𝛽2 ∙ 𝑥𝑖 + 𝛽3 ∙ 𝜇𝑖 + 𝜀𝑖
𝑌𝑖
1
− 𝑌𝑖
0
= 𝛽0 + 𝛽1 + 𝜖 − 𝛽0 + 𝜖
= 𝛽0 + 𝛽1 + 𝜖 − 𝛽0 − 𝜖
= 𝛽1
𝑌𝑖
1
− 𝑌𝑖
0
= 𝛽0 + 𝛽1𝑖 + 𝛽2 ∙ 𝑥𝑖 + 𝛽3 ∙ 𝜇𝑖
+ 𝜀𝑖
− 𝛽0 + 𝛽2 ∙ 𝑥𝑖 + 𝛽3 ∙ 𝜇𝑖 + 𝜀𝑖
= 𝛽0 + 𝛽1 + 𝜖 − 𝛽0 − 𝜖
= 𝛽1
𝑌𝑖
1
− 𝑌𝑖
0
= 𝛽1𝑖
𝑌𝑖
1
− 𝑌𝑖
0
= 𝛽1𝑖
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
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
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
The Three LATE Types
1. Always Takers- Always Participate
2. Never Takers-Never Participate
3. Compliers- Comply with thei...
The Three LATE Types
1. Always Takers- Always Participate
2. Never Takers-Never Participate
3. Compliers- Comply with thei...
Benefit-Cost>0
𝑌𝑖
1
− 𝑌𝑖
0
− 𝐶𝑖 > 0
𝛽1𝑖 − 𝐶𝑖 > 0
𝛽1𝑖 − 𝜌0 + 𝜌1 ∙ 𝑥𝑖 + 𝜌2 ∙ 𝜇𝑖 + 𝜌3 ∙ 𝑧𝑖 + 𝜂𝑖
> 0
𝑧𝑖 = 1
𝑌𝑖
1
− 𝑌𝑖
0
− 𝐶𝑖 > 0
𝑧𝑖 = 0
𝑌𝑖
1
− 𝑌𝑖
0
− 𝐶𝑖 ≤ 0
𝑧𝑖 = 1
𝑌𝑖
1
− 𝑌𝑖
0
− 𝐶𝑖 > 0
𝑧𝑖 = 0
𝑌𝑖
1
− 𝑌𝑖
0
− 𝐶𝑖 > 0
𝑧𝑖 = 1
𝑌𝑖
1
− 𝑌𝑖
0
− 𝐶𝑖 ≤ 0
𝑧𝑖 = 0
𝑌𝑖
1
− 𝑌𝑖
0
− 𝐶𝑖 ≤ 0
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
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 25...
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 25...
The Three LATE Types
1. Always Takers- Always Participate
2. Never Takers-Never Participate
3. Compliers- Comply with thei...
The Three LATE Types
1. Always Takers- Always Participate
2. Never Takers-Never Participate
3. Compliers- Comply with thei...
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 25...
Regression Discontinuity
Designs
𝑥 ≤ 𝑒
𝑥 > 𝑒
A “Discontinuity”
𝐸 𝑌|𝑥
𝑥𝑒
𝐸 𝑌|𝑥
𝑥𝑒
Another
“Discontinuity”
A “Discontinuity”
𝐸 𝑌|𝑥
𝑥𝑒
Another
“Discontinuity”
𝐸 𝑌|𝑥
𝑥𝑒
Impact
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...
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...
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...
A “Discontinuity”
A “Discontinuity”
𝐼 𝑥 ≤ 𝑒 = P
A “Discontinuity”
𝐼 𝑥 ≤ 𝑒 = P
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...
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 +...
A “Discontinuity”
Cost of Participation
𝐶 = 𝜌0 + Ω ∙ 1 − 𝐼 𝑥 ≤ 𝑒 + 𝜌1 ∙ 𝑥
+𝜌2 ∙ 𝜇 + 𝜌3 ∙ 𝑧 + 𝜂
Cost of Participation
𝐶 = 𝜌0 + 𝜴 ∙ 𝟏 − 𝑰 𝒙 ≤ 𝒆 + 𝜌1 ∙ 𝑥
+𝜌2 ∙ 𝜇 + 𝜌3 ∙ 𝑧 + 𝜂
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
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
Benefit-Cost>0
𝑌1 − 𝑌0 − 𝐶 > 0
𝛽1 − 𝐶 > 0
𝑌 = 𝛽0 + 𝛽1 ∙ 𝑃 + 𝛽2 ∙ 𝑥 + 𝛽3 ∙ 𝜇 + 𝜀
𝑌 = 𝛽0 + 𝛽1 ∙ 𝑃 + 𝛽2 ∙ 𝑥 + 𝛽3 ∙ 𝜇 + 𝜀
𝑌 = 𝛾0 + 𝛾1 ∙ 𝑃 + 𝛾2 ∙ 𝑥 + 𝜀
𝑌 = 𝛽0 + 𝛽1 ∙ 𝑃 + 𝛽2 ∙ 𝑥 + 𝛽3 ∙ 𝜇 + 𝜀
𝑌 = 𝛾0 + 𝛾1 ∙ 𝑃 + 𝜀
𝑌 = 𝛽0 + 𝛽1 ∙ 𝑃 + 𝛽2 ∙ 𝑥 + 𝛽3 ∙ 𝜇 + 𝜀
𝑌 = 𝛾0 + 𝛾1 ∙ 𝑃 + 𝛾2 ∙ 𝑥 + 𝜀
Benefit-Cost>0
𝑌1 − 𝑌0 − 𝐶 > 0
𝛽1 − 𝐶 > 0
Benefit-Cost>0
𝑌1 − 𝑌0 − 𝐶 > 0
𝜷 𝟏 − 𝑪 > 0
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
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
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
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/est...
𝑧 = 𝐼 𝑥 ≤ 𝑒
𝑧 = 𝐼 𝑥 ≤ 𝑒
Links:
The manual:
http://www.measureevaluation.org/resources/publications/ms-
14-87-en
The webinar introducing the manual...
MEASURE Evaluation is funded by the U.S. Agency
for International Development (USAID) under terms
of Cooperative Agreement...
Instrumental Variables
Instrumental Variables
Instrumental Variables
Instrumental Variables
Instrumental Variables
Instrumental Variables
Instrumental Variables
Instrumental Variables
Instrumental Variables
Instrumental Variables
Instrumental Variables
Instrumental Variables
Instrumental Variables
Instrumental Variables
Instrumental Variables
Instrumental Variables
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Instrumental Variables

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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.

Published in: Health & Medicine
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Instrumental Variables

  1. 1. Peter M. Lance, PhD MEASURE Evaluation University of North Carolina at Chapel Hill April 13, 2017 Instrumental Variables
  2. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 14. Global footprint (more than 25 countries)
  15. 15. • The program impact evaluation challenge • Randomization • Selection on observables • Within estimators • Instrumental variables
  16. 16. • The program impact evaluation challenge • Randomization • Selection on observables • Within estimators • Instrumental variables
  17. 17. YP μ X
  18. 18. YP μ X
  19. 19. YP μ X
  20. 20. YP μ X
  21. 21. YP μ X
  22. 22. YP μ X
  23. 23. YP μ Z X
  24. 24. YP μ Z X
  25. 25. YP μ Z X
  26. 26. YP μ Z X
  27. 27. YP μ Z X
  28. 28. YP μ Z X
  29. 29. YP μ Z X
  30. 30. YP μ Z X
  31. 31. Fascinating, right??? So what the heck does it all mean, and how would you implement it?
  32. 32. Fascinating, right??? So what the heck does it all mean, and how would you implement it?
  33. 33. 𝑌0 = 𝛽0 + 𝛽2 ∙ 𝑥 + 𝛽3 ∙ 𝜇 + 𝜀 𝑌1 = 𝛽0 + 𝛽1 + 𝛽2 ∙ 𝑥 + 𝛽3 ∙ 𝜇 + 𝜀
  34. 34. 𝑌0 = 𝛽0 + 𝛽2 ∙ 𝒙 + 𝛽3 ∙ 𝜇 + 𝜀 𝑌1 = 𝛽0 + 𝛽1 + 𝛽2 ∙ 𝒙 + 𝛽3 ∙ 𝜇 + 𝜀
  35. 35. 𝑌0 = 𝛽0 + 𝛽2 ∙ 𝑥 + 𝛽3 ∙ 𝝁 + 𝜀 𝑌1 = 𝛽0 + 𝛽1 + 𝛽2 ∙ 𝑥 + 𝛽3 ∙ 𝝁 + 𝜀
  36. 36. 𝑌0 = 𝛽0 + 𝛽2 ∙ 𝑥 + 𝛽3 ∙ 𝜇 + 𝜺 𝑌1 = 𝛽0 + 𝛽1 + 𝛽2 ∙ 𝑥 + 𝛽3 ∙ 𝜇 + 𝜺
  37. 37. 𝑌1 − 𝑌0 = 𝛽0 + 𝛽1 + 𝜖 − 𝛽0 + 𝜖 = 𝛽0 + 𝛽1 + 𝜖 − 𝛽0 − 𝜖 = 𝛽1
  38. 38. 𝑌1 − 𝑌0 = 𝛽0 + 𝛽1 + 𝛽2 ∙ 𝑥 + 𝛽3 ∙ 𝜇 + 𝜀 − 𝛽0 + 𝛽2 ∙ 𝑥 + 𝛽3 ∙ 𝜇 + 𝜀 = 𝛽0 + 𝛽1 + 𝜖 − 𝛽0 − 𝜖 = 𝛽1
  39. 39. 𝑌1 − 𝑌0 = 𝛽1
  40. 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. 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. 42. 𝑌 = 𝛽0 + 𝛽1 ∙ 𝑃 + 𝛽2 ∙ 𝑥 + 𝛽3 ∙ 𝜇 + 𝜀
  43. 43. 𝑌 = 𝛽0 + 𝛽1 ∙ 𝑃 + 𝛽2 ∙ 𝑥 + 𝛽3 ∙ 𝜇 + 𝜀
  44. 44. 𝑌 = 𝛽0 + 𝛽1 ∙ 𝑃 + 𝛽2 ∙ 𝑥 + 𝛽3 ∙ 𝜇 + 𝜀 𝑌 = 𝛾0 + 𝛾1 ∙ 𝑃 + 𝛾2 ∙ 𝑥 + 𝜀 𝐸 𝛾1 = 𝛽1
  45. 45. 𝑌 = 𝛽0 + 𝛽1 ∙ 𝑃 + 𝛽2 ∙ 𝑥 + 𝛽3 ∙ 𝜇 + 𝜀 𝑌 = 𝛾0 + 𝛾1 ∙ 𝑃 + 𝛾2 ∙ 𝑥 + 𝜀 𝐸 𝛾1 = 𝛽1 ?
  46. 46. 𝑌 = 𝛽0 + 𝛽1 ∙ 𝑃 + 𝛽2 ∙ 𝑥 + 𝛽3 ∙ 𝜇 + 𝜀 𝑌 = 𝛾0 + 𝛾1 ∙ 𝑃 + 𝛾2 ∙ 𝑥 + 𝜀 𝐸 𝛾1 = 𝛽1 ?
  47. 47. 𝑌 = 𝛽0 + 𝛽1 ∙ 𝑃 + 𝛽2 ∙ 𝑥 + 𝛽3 ∙ 𝜇 + 𝜀 𝑌 = 𝛾0 + 𝛾1 ∙ 𝑃 + 𝛾2 ∙ 𝑥 + 𝜀 𝐸 𝛾1 = 𝛽1 ?
  48. 48. YP μ X
  49. 49. YP μ X
  50. 50. 𝑌 = 𝛽0 + 𝛽1 ∙ 𝑃 + 𝛽2 ∙ 𝑥 + 𝛽3 ∙ 𝜇 + 𝜀 𝑌 = 𝛾0 + 𝛾1 ∙ 𝑃 + 𝛾2 ∙ 𝑥 + 𝜀 𝐸 𝛾1 = 𝛽1 ?
  51. 51. 𝑌 = 𝛽0 + 𝛽1 ∙ 𝑃 + 𝛽2 ∙ 𝑥 + 𝛽3 ∙ 𝜇 + 𝜀 𝑌 = 𝛾0 + 𝛾1 ∙ 𝑃 + 𝛾2 ∙ 𝑥 + 𝜀 𝐸 𝛾1 = 𝛽1 ?
  52. 52. 𝑌 = 𝛽0 + 𝛽1 ∙ 𝑃 + 𝛽2 ∙ 𝑥 + 𝛽3 ∙ 𝜇 + 𝜀 𝑌 = 𝛾0 + 𝛾1 ∙ 𝑃 + 𝛾2 ∙ 𝑥 + 𝜀 𝐸 𝛾1 = 𝛽1 ?
  53. 53. 𝑌 = 𝛽0 + 𝛽1 ∙ 𝑃 + 𝛽2 ∙ 𝑥 + 𝛽3 ∙ 𝜇 + 𝜀 𝑌 = 𝛾0 + 𝛾1 ∙ 𝑃 + 𝛾2 ∙ 𝑥 + 𝜀 𝐸 𝛾1 ≠ 𝛽1
  54. 54. 𝑌 = 𝛽0 + 𝛽1 ∙ 𝑃 + 𝛽2 ∙ 𝑥 + 𝛽3 ∙ 𝜇 + 𝜀 𝑌 = 𝛾0 + 𝛾1 ∙ 𝑃 + 𝛾2 ∙ 𝑥 + 𝜀 𝐸 𝛾1 ≠ 𝛽1
  55. 55. 𝑌 = 𝛽0 + 𝛽1 ∙ 𝑃 + 𝛽2 ∙ 𝑥 + 𝛽3 ∙ 𝜇 + 𝜀 𝑌 = 𝛾0 + 𝛾1 ∙ 𝑃 + 𝛾2 ∙ 𝑥 + 𝜀 𝐸 𝛾1 ≠ 𝛽1
  56. 56. 𝑌 = 𝛽0 + 𝛽1 ∙ 𝑃 + 𝛽2 ∙ 𝑥 + 𝛽3 ∙ 𝜇 + 𝜀 𝑌 = 𝛾0 + 𝛾1 ∙ 𝑃 + 𝛾2 ∙ 𝑥 + 𝜀 𝐸 𝛾1 ≠ 𝛽1
  57. 57. Cost of Participation 𝐶 = 𝜌0 + 𝜌1 ∙ 𝑥 + 𝜌2 ∙ 𝜇 + 𝜌3 ∙ 𝑧 + 𝜂
  58. 58. Cost of Participation 𝐶 = 𝜌0 + 𝜌1 ∙ 𝒙 + 𝜌2 ∙ 𝜇 + 𝜌3 ∙ 𝑧 + 𝜂
  59. 59. Cost of Participation 𝐶 = 𝜌0 + 𝜌1 ∙ 𝑥 + 𝜌2 ∙ 𝝁 + 𝜌3 ∙ 𝑧 + 𝜂
  60. 60. Cost of Participation 𝐶 = 𝜌0 + 𝜌1 ∙ 𝑥 + 𝜌2 ∙ 𝜇 + 𝜌3 ∙ 𝒛 + 𝜂
  61. 61. Cost of Participation 𝐶 = 𝜌0 + 𝜌1 ∙ 𝑥 + 𝜌2 ∙ 𝜇 + 𝜌3 ∙ 𝑧 + 𝜼
  62. 62. Benefit-Cost>0 𝑌1 − 𝑌0 − C > 0 𝛽1 − 𝐶 > 0 𝛽1 − 𝜌0 + 𝜌1 ∙ 𝑥 + 𝜌2 ∙ 𝜇 + 𝜌3 ∙ 𝑧 + 𝜂 > 0
  63. 63. Benefit-Cost>0 𝑌1 − 𝑌0 − C > 0 𝛽1 − 𝐶 > 0 𝛽1 − 𝜌0 + 𝜌1 ∙ 𝑥 + 𝜌2 ∙ 𝜇 + 𝜌3 ∙ 𝑧 + 𝜂 > 0
  64. 64. Benefit-Cost>0 𝑌1 − 𝑌0 − C > 0 𝛽1 − 𝐶 > 0 𝛽1 − 𝜌0 + 𝜌1 ∙ 𝑥 + 𝜌2 ∙ 𝜇 + 𝜌3 ∙ 𝑧 + 𝜂 > 0
  65. 65. Benefit-Cost>0 𝑌1 − 𝑌0 − C > 0 𝛽1 − 𝐶 > 0 𝛽1 − 𝜌0 + 𝜌1 ∙ 𝑥 + 𝜌2 ∙ 𝜇 + 𝜌3 ∙ 𝑧 + 𝜂 > 0
  66. 66. 𝑥 𝑃 𝜇 𝑃 𝑧 𝑃
  67. 67. 𝑥 𝑃 𝜇 𝑃 𝑧 𝑃
  68. 68. YP μ X
  69. 69. Benefit-Cost>0 𝑌1 − 𝑌0 − C > 0 𝛽1 − 𝐶 > 0 𝛽1 − 𝜌0 + 𝜌1 ∙ 𝑥 + 𝜌2 ∙ 𝜇 + 𝜌3 ∙ 𝑧 + 𝜂 > 0
  70. 70. 𝑥 𝑃 𝜇 𝑃 𝑧 𝑃
  71. 71. 𝑌0 = 𝛽0 + 𝛽2 ∙ 𝑥 + 𝛽3 ∙ 𝜇 + 𝜀 𝑌1 = 𝛽0 + 𝛽1 + 𝛽2 ∙ 𝑥 + 𝛽3 ∙ 𝜇 + 𝜀
  72. 72. 𝑌 = 𝑃 ∙ 𝑌1 + 1 − 𝑃 ∙ 𝑌0 = 𝑃 ∗ 𝛽0 + 𝛽1 + 𝜖 +
  73. 73. 𝑌 = 𝑷 ∙ 𝑌1 + 1 − 𝑷 ∙ 𝑌0 = 𝑃 ∗ 𝛽0 + 𝛽1 + 𝜖 +
  74. 74. YP μ Z X
  75. 75. Benefit-Cost>0 𝑌1 − 𝑌0 − C > 0 𝛽1 − 𝐶 > 0 𝛽1 − 𝜌0 + 𝜌1 ∙ 𝑥 + 𝜌2 ∙ 𝜇 + 𝜌3 ∙ 𝑧 + 𝜂 > 0
  76. 76. 𝛽1 − 𝜌0 + 𝜌1 ∙ 𝑥 + 𝜌2 ∙ 𝜇 + 𝜌3 ∙ 𝑧 + 𝜂 > 0 𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧 + 𝜉
  77. 77. 𝛽1 − 𝜌0 + 𝜌1 ∙ 𝑥 + 𝜌2 ∙ 𝜇 + 𝜌3 ∙ 𝑧 + 𝜼 > 0 𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧 + 𝜉
  78. 78. 𝛽1 − 𝜌0 + 𝜌1 ∙ 𝑥 + 𝜌2 ∙ 𝜇 + 𝜌3 ∙ 𝑧 + 𝜂 > 0 𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧 + 𝜉
  79. 79. 𝛽1 − 𝜌0 + 𝜌1 ∙ 𝑥 + 𝜌2 ∙ 𝜇 + 𝜌3 ∙ 𝑧 + 𝜂 > 0 𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧 + 𝝃
  80. 80. 𝛽1 − 𝜌0 + 𝜌1 ∙ 𝑥 + 𝜌2 ∙ 𝜇 + 𝜌3 ∙ 𝑧 + 𝜂 > 0 𝑃∗ = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧 + 𝜉
  81. 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
  82. 82. 𝛽1 − 𝜌0 + 𝜌1 ∙ 𝑥 + 𝜌2 ∙ 𝜇 + 𝜌3 ∙ 𝑧 + 𝜂 > 0 𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧 + 𝜉
  83. 83. 𝑌 = 𝛾0 + 𝛾1 ∙ 𝑃 + 𝛾2 ∙ 𝑥 + 𝜀 𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧 + 𝜉
  84. 84. 𝒀 = 𝜸 𝟎 + 𝜸 𝟏 ∙ 𝑷 + 𝜸 𝟐 ∙ 𝒙 + 𝜺 𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧 + 𝜉
  85. 85. 𝑌 = 𝛾0 + 𝛾1 ∙ 𝑃 + 𝛾2 ∙ 𝑥 + 𝜀 𝑷 = 𝝓 𝟎 + 𝝓 𝟏 ∙ 𝒙 + 𝝓 𝟐 ∙ 𝒛 + 𝝃
  86. 86. 𝑌 = 𝛾0 + 𝛾1 ∙ 𝑃 + 𝛾2 ∙ 𝑥 + 𝜀 𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧 + 𝜉
  87. 87. 𝑌 = 𝛾0 + 𝛾1 ∙ 𝑃 + 𝛾2 ∙ 𝑥 + 𝜀 𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧 + 𝜉 𝝁
  88. 88. 𝑌 = 𝛾0 + 𝛾1 ∙ 𝑃 + 𝛾2 ∙ 𝑥 + 𝜺 𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧 + 𝝃 𝝁
  89. 89. 𝑌 = 𝛾0 + 𝛾1 ∙ 𝑃 + 𝛾2 ∙ 𝑥 + 𝜺 𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧 + 𝝃 𝝁
  90. 90. 𝑌 = 𝛾0 + 𝛾1 ∙ 𝑃 + 𝛾2 ∙ 𝑥 + 𝜺 𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧 + 𝝃 𝝁
  91. 91. 𝑌 = 𝛾0 + 𝛾1 ∙ 𝑃 + 𝛾2 ∙ 𝑥 + 𝜺 𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧 + 𝝃 𝑬( 𝜸 𝟏) ≠ 𝜷 𝟏
  92. 92. Two-Stage Least Squares
  93. 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
  94. 94. 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
  95. 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. 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. 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. 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
  99. 99. 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
  100. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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
  114. 114. 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
  115. 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. 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. 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. 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. 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. 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. 121. 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
  122. 122. Two-Stage Least Squares 𝑌 = 𝛾0 + 𝛾1 ∙ 𝑃 + 𝛾2 ∙ 𝑥 + 𝜀 𝐸 𝛾0 = 𝛽1
  123. 123. Two-Stage Least Squares 𝑌 = 𝛾0 + 𝛾1 ∙ 𝑃 + 𝛾2 ∙ 𝑥 + 𝜀 𝐸 𝛾0 ≠ 𝛽1
  124. 124. Two-Stage Least Squares 𝑌 = 𝛾0 + 𝛾1 ∙ 𝑃 + 𝛾2 ∙ 𝑥 + 𝜀 𝐸 𝛾0 ≠ 𝛽1
  125. 125. ????!
  126. 126. Consistency 𝑝𝑙𝑖𝑚 𝛾1 = 𝛽1 𝑙𝑖𝑚 𝑛→∞ 𝑃𝑟 𝛾1 − 𝛽1 ≥ 𝑐 = 0
  127. 127. Consistency 𝑝𝑙𝑖𝑚 𝛾1 = 𝛽1 𝑙𝑖𝑚 𝑛→∞ 𝑃𝑟 𝛾1 − 𝛽1 ≥ 𝑐 = 0 = 0
  128. 128. 𝑌 = 𝛽0 + 𝛽1 ∙ 𝑃 + 𝛽2 ∙ 𝑥 + 𝛽3 ∙ 𝜇 + 𝜀
  129. 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
  130. 130. 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. 131. Anything
  132. 132. Literally anything
  133. 133. Instrumentation and It’s Discontents
  134. 134. 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
  135. 135. 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
  136. 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
  137. 137. 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
  138. 138. 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
  139. 139. 𝑎𝑏𝑠 𝛽1 − 𝛾1 𝑛
  140. 140. 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
  141. 141. 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
  142. 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. 143. 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
  144. 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. 145. 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
  146. 146. 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
  147. 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. 148. 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
  149. 149. 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
  150. 150. 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
  151. 151. 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
  152. 152. 𝑌0 = 𝛽0 + 𝛽2 ∙ 𝑥 + 𝛽3 ∙ 𝜇 + 𝜀 𝑌1 = 𝛽0 + 𝛽1 + 𝛽2 ∙ 𝑥 + 𝛽3 ∙ 𝜇 + 𝜀
  153. 153. 𝑌𝑖 0 = 𝛽0 + 𝛽2 ∙ 𝑥𝑖 + 𝛽3 ∙ 𝜇𝑖 + 𝜀𝑖 𝑌𝑖 1 = 𝛽0 + 𝛽1 + 𝛽2 ∙ 𝑥𝑖 + 𝛽3 ∙ 𝜇𝑖 + 𝜀𝑖
  154. 154. 𝑌𝑖 0 = 𝛽0 + 𝛽2 ∙ 𝑥𝑖 + 𝛽3 ∙ 𝜇𝑖 + 𝜀𝑖 𝑌𝑖 1 = 𝛽0 + 𝛽1𝑖 + 𝛽2 ∙ 𝑥𝑖 + 𝛽3 ∙ 𝜇𝑖 + 𝜀𝑖
  155. 155. 𝑌𝑖 0 = 𝛽0 + 𝛽2 ∙ 𝑥𝑖 + 𝛽3 ∙ 𝜇𝑖 + 𝜀𝑖 𝑌𝑖 1 = 𝛽0 + 𝛽1𝑖 + 𝛽2 ∙ 𝑥𝑖 + 𝛽3 ∙ 𝜇𝑖 + 𝜀𝑖
  156. 156. 𝑌𝑖 1 − 𝑌𝑖 0 = 𝛽0 + 𝛽1 + 𝜖 − 𝛽0 + 𝜖 = 𝛽0 + 𝛽1 + 𝜖 − 𝛽0 − 𝜖 = 𝛽1
  157. 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. 158. 𝑌𝑖 1 − 𝑌𝑖 0 = 𝛽1𝑖
  159. 159. 𝑌𝑖 1 − 𝑌𝑖 0 = 𝛽1𝑖
  160. 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. 161. 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
  162. 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. 163. The Three LATE Types 1. Always Takers- Always Participate 2. Never Takers-Never Participate 3. Compliers- Comply with their instrumental “assignment”
  164. 164. The Three LATE Types 1. Always Takers- Always Participate 2. Never Takers-Never Participate 3. Compliers- Comply with their instrumental “assignment”
  165. 165. Benefit-Cost>0 𝑌𝑖 1 − 𝑌𝑖 0 − 𝐶𝑖 > 0 𝛽1𝑖 − 𝐶𝑖 > 0 𝛽1𝑖 − 𝜌0 + 𝜌1 ∙ 𝑥𝑖 + 𝜌2 ∙ 𝜇𝑖 + 𝜌3 ∙ 𝑧𝑖 + 𝜂𝑖 > 0
  166. 166. 𝑧𝑖 = 1 𝑌𝑖 1 − 𝑌𝑖 0 − 𝐶𝑖 > 0 𝑧𝑖 = 0 𝑌𝑖 1 − 𝑌𝑖 0 − 𝐶𝑖 ≤ 0
  167. 167. 𝑧𝑖 = 1 𝑌𝑖 1 − 𝑌𝑖 0 − 𝐶𝑖 > 0 𝑧𝑖 = 0 𝑌𝑖 1 − 𝑌𝑖 0 − 𝐶𝑖 > 0
  168. 168. 𝑧𝑖 = 1 𝑌𝑖 1 − 𝑌𝑖 0 − 𝐶𝑖 ≤ 0 𝑧𝑖 = 0 𝑌𝑖 1 − 𝑌𝑖 0 − 𝐶𝑖 ≤ 0
  169. 169. 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
  170. 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
  171. 171. 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
  172. 172. The Three LATE Types 1. Always Takers- Always Participate 2. Never Takers-Never Participate 3. Compliers- Comply with their instrumental “assignment”
  173. 173. The Three LATE Types 1. Always Takers- Always Participate 2. Never Takers-Never Participate 3. Compliers- Comply with their instrumental “assignment”
  174. 174. 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. 175. Regression Discontinuity Designs
  176. 176. 𝑥 ≤ 𝑒 𝑥 > 𝑒
  177. 177. A “Discontinuity”
  178. 178. 𝐸 𝑌|𝑥 𝑥𝑒
  179. 179. 𝐸 𝑌|𝑥 𝑥𝑒 Another “Discontinuity”
  180. 180. A “Discontinuity”
  181. 181. 𝐸 𝑌|𝑥 𝑥𝑒 Another “Discontinuity”
  182. 182. 𝐸 𝑌|𝑥 𝑥𝑒 Impact
  183. 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
  184. 184. 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
  185. 185. 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
  186. 186. A “Discontinuity”
  187. 187. A “Discontinuity” 𝐼 𝑥 ≤ 𝑒 = P
  188. 188. A “Discontinuity” 𝐼 𝑥 ≤ 𝑒 = P
  189. 189. 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
  190. 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
  191. 191. A “Discontinuity”
  192. 192. Cost of Participation 𝐶 = 𝜌0 + Ω ∙ 1 − 𝐼 𝑥 ≤ 𝑒 + 𝜌1 ∙ 𝑥 +𝜌2 ∙ 𝜇 + 𝜌3 ∙ 𝑧 + 𝜂
  193. 193. Cost of Participation 𝐶 = 𝜌0 + 𝜴 ∙ 𝟏 − 𝑰 𝒙 ≤ 𝒆 + 𝜌1 ∙ 𝑥 +𝜌2 ∙ 𝜇 + 𝜌3 ∙ 𝑧 + 𝜂
  194. 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
  195. 195. 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. 196. Benefit-Cost>0 𝑌1 − 𝑌0 − 𝐶 > 0 𝛽1 − 𝐶 > 0
  197. 197. 𝑌 = 𝛽0 + 𝛽1 ∙ 𝑃 + 𝛽2 ∙ 𝑥 + 𝛽3 ∙ 𝜇 + 𝜀
  198. 198. 𝑌 = 𝛽0 + 𝛽1 ∙ 𝑃 + 𝛽2 ∙ 𝑥 + 𝛽3 ∙ 𝜇 + 𝜀 𝑌 = 𝛾0 + 𝛾1 ∙ 𝑃 + 𝛾2 ∙ 𝑥 + 𝜀
  199. 199. 𝑌 = 𝛽0 + 𝛽1 ∙ 𝑃 + 𝛽2 ∙ 𝑥 + 𝛽3 ∙ 𝜇 + 𝜀 𝑌 = 𝛾0 + 𝛾1 ∙ 𝑃 + 𝜀
  200. 200. 𝑌 = 𝛽0 + 𝛽1 ∙ 𝑃 + 𝛽2 ∙ 𝑥 + 𝛽3 ∙ 𝜇 + 𝜀 𝑌 = 𝛾0 + 𝛾1 ∙ 𝑃 + 𝛾2 ∙ 𝑥 + 𝜀
  201. 201. Benefit-Cost>0 𝑌1 − 𝑌0 − 𝐶 > 0 𝛽1 − 𝐶 > 0
  202. 202. Benefit-Cost>0 𝑌1 − 𝑌0 − 𝐶 > 0 𝜷 𝟏 − 𝑪 > 0
  203. 203. 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
  204. 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. 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
  206. 206. 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
  207. 207. 𝑧 = 𝐼 𝑥 ≤ 𝑒
  208. 208. 𝑧 = 𝐼 𝑥 ≤ 𝑒
  209. 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. 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

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