Instrumental Variables

MEASURE Evaluation
MEASURE EvaluationMEASURE Evaluation works to improve collection, analysis and presentation of data to promote better use of data in planning, policymaking, managing, monitoring and evaluating population, health and nutrition programs.
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:
Too LATE for Experimental Compliance?
Peter M. Lance, PhD
MEASURE Evaluation
University of North Carolina at
Chapel Hill
April 13, 2017
Instrumental Variables:
Too LATE for Experimental Compliance?
Peter M. Lance, PhD
MEASURE Evaluation
University of North Carolina at
Chapel Hill
April 13, 2017
Instrumental Variables:
Too LATE for Experimental Compliance?
Peter M. Lance, PhD
MEASURE Evaluation
University of North Carolina at
Chapel Hill
April 13, 2017
Instrumental Variables:
Too LATE for Experimental Compliance?
Peter M. Lance, PhD
MEASURE Evaluation
University of North Carolina at
Chapel Hill
April 13, 2017
Instrumental Variables:
Too LATE for Experimental Compliance?
Peter M. Lance, PhD
MEASURE Evaluation
University of North Carolina at
Chapel Hill
April 13, 2017
Instrumental Variables:
Too LATE for Experimental Compliance?
Peter M. Lance, PhD
MEASURE Evaluation
University of North Carolina at
Chapel Hill
April 13, 2017
Instrumental Variables:
Too LATE for Experimental Compliance?
Peter M. Lance, PhD
MEASURE Evaluation
University of North Carolina at
Chapel Hill
April 13, 2017
Instrumental Variables:
Too LATE for Experimental Compliance?
Peter M. Lance, PhD
MEASURE Evaluation
University of North Carolina at
Chapel Hill
April 13, 2017
Instrumental Variables:
Too LATE for Experimental Compliance?
Peter M. Lance, PhD
MEASURE Evaluation
University of North Carolina at
Chapel Hill
April 13, 2017
Instrumental Variables:
Too LATE for Experimental Compliance?
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
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
Global footprint (more than 25 countries)
• The program impact evaluation challenge
• Randomization
• Selection on observables
• Within estimators
• Instrumental variables
• The program impact evaluation challenge
• Randomization
• Selection on observables
• Within estimators
• Instrumental variables
Instrumental Variables
Instrumental Variables
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YP
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YP
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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 ∗ 𝛽1 + 𝜖
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
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
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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/estimated model to predict
participation:
𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧
3. Regress 𝑌 on 𝑃 and 𝑥
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
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
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
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
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 ∙ 𝜙1 + 𝛾2
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
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
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
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
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/estimated model to predict
participation:
𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧
3. Regress 𝑌 on 𝑃 and 𝑥
Two-Stage Least Squares
𝑌 = 𝛾0 + 𝛾1 ∙ 𝑃 + 𝛾2 ∙ 𝑥 + 𝜀
𝐸 𝛾0 = 𝛽1
Instrumental Variables
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
Instrumental Variables
Instrumental Variables
𝑌 = 𝛽0 + 𝛽1 ∙ 𝑃 + 𝛽2 ∙ 𝑥 + 𝛽3 ∙ 𝜇 + 𝜀
Instrumental Variables
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
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
Instrumental Variables
Instrumental Variables
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, 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, 𝛾1, and 𝛾2
Instrumental Variables
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, 𝛾1, and 𝛾2
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
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, 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, 𝛾1, and 𝛾2
Instrumental Variables
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
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 friend
3. Be careful about getting fancy with binary
endogenous regressors and outcomes
4. Be careful about being LATE
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 their
instrumental “assignment”
The Three LATE Types
1. Always Takers- Always Participate
2. Never Takers-Never Participate
3. Compliers- Comply with their
instrumental “assignment”
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
Instrumental Variables
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
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
The Three LATE Types
1. Always Takers- Always Participate
2. Never Takers-Never Participate
3. Compliers- Comply with their
instrumental “assignment”
Instrumental Variables
The Three LATE Types
1. Always Takers- Always Participate
2. Never Takers-Never Participate
3. Compliers- Comply with their
instrumental “assignment”
Instrumental Variables
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
Instrumental Variables
Instrumental Variables
Regression Discontinuity
Designs
𝑥 ≤ 𝑒
𝑥 > 𝑒
Instrumental Variables
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 + 𝜀
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
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
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 + 𝜀
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
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/estimated model to predict
participation:
𝑃 = 𝜙0 + 𝜙1 ∙ 𝑥 + 𝜙2 ∙ 𝑧
3. Regress 𝑌 on 𝑃 and 𝑥
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:
http://www.measureevaluation.org/resources/webinars/metho
ds-for-program-impact-evaluation
My email:
pmlance@email.unc.edu
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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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?
  • 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
  • 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
  • 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 ?
  • 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
  • 71. Benefit-Cost>0 𝑌1 − 𝑌0 − C > 0 𝛽1 − 𝐶 > 0 𝛽1 − 𝜌0 + 𝜌1 ∙ 𝑥 + 𝜌2 ∙ 𝜇 + 𝜌3 ∙ 𝑧 + 𝜂 > 0
  • 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 +
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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