Econometrics for Policy Analysis
Max Batstra
March 29, 2026
1
, Lecture 1
Slides 1
Outline
• Causal inference and potential outcome model
• The selection problem
• Random assignment
• Regression analysis of experiments
In this part of the course we’ll go over:
• Social (field) experiments
• Controlling for observed covariates (multiple linear regression model, matching estimators)
• Instrumental variables regression models
• Regression discontinuity models
• Difference-in-differences estimation
Consider the following linear regression model:
yi = β + αdi + ui
Where:
• yi outcome person i
(
1 if treated
• di treatment indicator person i, di =
0 otherwise
• β intercept parameter
• α slope parameter
• ui error term/ unobserved component of person i
Frequent assumptions for estimation
MLR1 Linear in parameters
MLR2 Random sample (observations are i.i.d.)
MLR3 No perfect collinearity
MLR4 E[ui | di ]=0 ∀di (exogeneity)
2
,OLS Interpretation
How to interpret α̂
• MLR1-MLR3
Cov(d
d i , y)
α̂ = = Ê[yi | di = 1] − Ê[yi | di = 0]
Var(d
d i)
– Interpretation as correlation
– Suppose α̂ = −0.25, this implies that persons who have been in a hospital in the last year are
25% less likely to be in self-reported good health compared to people who have not been in the
hospital.
• MLR1 - MLR4
α̂ and β̂ are unbiased: E[α̂] = α and E[β̂] = β
– Causal interpretation
– A hospital stay decreases the likelihood to be in self-reported good health by 25%.
Causality formalized
Let di is an indicator function for treatment allocation
(
1 if unit i is exposed to treatment
di =
0 if unit i is not exposed to treatment
yi is health status of unit i and is the outcome of interest. For each unit there are two potential outcomes:
(
yi1 outcome when treated, so if di = 1
yi =
yi0 outcome when not treated, so if di = 0
Based on this we can define yi
yi = di · yi1 + (1 − di ) · yi0
Potential Outcome
A potential outcome are different possible outcomes.
yi0 and yi1
are all potential outcomes, if you can either be treated (1) or not (0).
Counterfactual
A counterfactual is a hypothetical scenario that describes what would have happened under different condi-
tions. For example:
E[yi0 |di = 1] and E[yi1 |di = 0] are counterfactuals
Participation Effect/ Individual Treatment Effect α = yi1 − yi0
3
,Average Treatment Effect[ATE]
αAT E = E[αi ] = E[yi1 − yi0 ] = E[yi1 ] − E[yi0 ]
• Compares potential outcomes when all units receive treatment (E[yi1 ]) with potential outcomes when
no units receive treatment (E[yi0 ])
• Note that E[α̂] = E[yi |di = 1] − E[yi |di = 0] = β + E[αi ] + E[ui |di = 1] − β − E[ui |di = 0] =
E[α] + E[ui |di = 1] − E[ui |di = 0]
So ATE = E[α̂] under MLR4
Average Treatment effect on the Treated[ATT]
αAT T = E (αi | di = 1) = E yi1 | di = 1 − E yi0 | di = 1
• Describes how much on average treated individuals benefit from treatment. (difference between treat-
ment outcome and what would have happened if they did not receive treatment)
• E[yi0 | di = 1] is a counterfactual and E[yi1 | di = 1] can be directly estimated.
Treatment parameters
(
yi0 = β + ui
• We assume the following form for potential outcomes: yi =
yi1 = β + αi + ui
– αi individual treatment effect
– β intercept parameter
– ui unobserved component
• Observable outcome:
yi = di yi1 + (1 − di ) yi0 = β + di αi + ui ⇐⇒ yi = β + di αi + αAT E di − αAT E di + ui
⇐⇒ yi = β + αAT E di + ui + di (αi − αAT E )
• For α̂OLS :
E[α̂OLS ] = E[yi | di = 1] − E[yi | di = 0]
= β + αAT E + E[ui | di = 1] + E[αi − αAT E | di = 1] − β − αAT E · 0 − E[ui | di = 0] − 0
= αAT E + E[ui | di = 1] − E[ui | di = 0] + E[αi − αAT E | di = 1]
From this we obtain two selection effects:
– Selection on nontreated outcomes: E[ui | di = 1] = E[ui | di = 0]
– Selection on gains from treatment: E[αi − αAT E ]
4
, Social experiments
An example of social experiments in medicine are randomized clinical trials. Randomization rules out selec-
tion. Patients are randomly assigned to a treatment group and a control group.
Under randomization the following assumptions holds:
R1 : E[ui | di = 1] = E[ui | di = 0] = E[ui ]
R2 : E[αi | di = 1] = E[αi | di = 0] = E[αi ]
Under R1 and R2, OLS regression for the equation below allows for consistent and unbiased estimators:
yi = β + αAT E di + ui + di αi − αAT E
Or
yi = β + αAT E di + ei , where ei = ui + di αi − αAT E
Alternative ways to state R1 and R2 are:
E[yi0 | di = 1] = E[yi0 | di = 0] = E[yi0 ] (equal to R1)
E[yi1 | di = 1] = E[yi1 | di = 0] = E[yi1 ] (based on R1 and R2)
A stronger form of this assumption is yi0 , yi1 ⊥ di , which implies independence of yi0 and di and yi1 and di .
Slides 2
Outline
• Review potential outcome model
• Random assignment in social experiments
• Conditional Independence Assumption
• Potential problems with social experiments
Review Questions
• What is potential outcome?
A: A potential outcome is an outcome that occurs if treatment is allocated or not
• What is counterfactual outcome?
A: Counterfactual outcome is an unobserved outcome that describes what would have happened oth-
erwise. For example, given that you’re being allocated treatment, a counterfactual outcome describes
what would have happened if you did not received treatment.
• How are observable outcomes and potential outcomes related?
A: yi = di · yi1 + (1 − di ) · yi0
5
Max Batstra
March 29, 2026
1
, Lecture 1
Slides 1
Outline
• Causal inference and potential outcome model
• The selection problem
• Random assignment
• Regression analysis of experiments
In this part of the course we’ll go over:
• Social (field) experiments
• Controlling for observed covariates (multiple linear regression model, matching estimators)
• Instrumental variables regression models
• Regression discontinuity models
• Difference-in-differences estimation
Consider the following linear regression model:
yi = β + αdi + ui
Where:
• yi outcome person i
(
1 if treated
• di treatment indicator person i, di =
0 otherwise
• β intercept parameter
• α slope parameter
• ui error term/ unobserved component of person i
Frequent assumptions for estimation
MLR1 Linear in parameters
MLR2 Random sample (observations are i.i.d.)
MLR3 No perfect collinearity
MLR4 E[ui | di ]=0 ∀di (exogeneity)
2
,OLS Interpretation
How to interpret α̂
• MLR1-MLR3
Cov(d
d i , y)
α̂ = = Ê[yi | di = 1] − Ê[yi | di = 0]
Var(d
d i)
– Interpretation as correlation
– Suppose α̂ = −0.25, this implies that persons who have been in a hospital in the last year are
25% less likely to be in self-reported good health compared to people who have not been in the
hospital.
• MLR1 - MLR4
α̂ and β̂ are unbiased: E[α̂] = α and E[β̂] = β
– Causal interpretation
– A hospital stay decreases the likelihood to be in self-reported good health by 25%.
Causality formalized
Let di is an indicator function for treatment allocation
(
1 if unit i is exposed to treatment
di =
0 if unit i is not exposed to treatment
yi is health status of unit i and is the outcome of interest. For each unit there are two potential outcomes:
(
yi1 outcome when treated, so if di = 1
yi =
yi0 outcome when not treated, so if di = 0
Based on this we can define yi
yi = di · yi1 + (1 − di ) · yi0
Potential Outcome
A potential outcome are different possible outcomes.
yi0 and yi1
are all potential outcomes, if you can either be treated (1) or not (0).
Counterfactual
A counterfactual is a hypothetical scenario that describes what would have happened under different condi-
tions. For example:
E[yi0 |di = 1] and E[yi1 |di = 0] are counterfactuals
Participation Effect/ Individual Treatment Effect α = yi1 − yi0
3
,Average Treatment Effect[ATE]
αAT E = E[αi ] = E[yi1 − yi0 ] = E[yi1 ] − E[yi0 ]
• Compares potential outcomes when all units receive treatment (E[yi1 ]) with potential outcomes when
no units receive treatment (E[yi0 ])
• Note that E[α̂] = E[yi |di = 1] − E[yi |di = 0] = β + E[αi ] + E[ui |di = 1] − β − E[ui |di = 0] =
E[α] + E[ui |di = 1] − E[ui |di = 0]
So ATE = E[α̂] under MLR4
Average Treatment effect on the Treated[ATT]
αAT T = E (αi | di = 1) = E yi1 | di = 1 − E yi0 | di = 1
• Describes how much on average treated individuals benefit from treatment. (difference between treat-
ment outcome and what would have happened if they did not receive treatment)
• E[yi0 | di = 1] is a counterfactual and E[yi1 | di = 1] can be directly estimated.
Treatment parameters
(
yi0 = β + ui
• We assume the following form for potential outcomes: yi =
yi1 = β + αi + ui
– αi individual treatment effect
– β intercept parameter
– ui unobserved component
• Observable outcome:
yi = di yi1 + (1 − di ) yi0 = β + di αi + ui ⇐⇒ yi = β + di αi + αAT E di − αAT E di + ui
⇐⇒ yi = β + αAT E di + ui + di (αi − αAT E )
• For α̂OLS :
E[α̂OLS ] = E[yi | di = 1] − E[yi | di = 0]
= β + αAT E + E[ui | di = 1] + E[αi − αAT E | di = 1] − β − αAT E · 0 − E[ui | di = 0] − 0
= αAT E + E[ui | di = 1] − E[ui | di = 0] + E[αi − αAT E | di = 1]
From this we obtain two selection effects:
– Selection on nontreated outcomes: E[ui | di = 1] = E[ui | di = 0]
– Selection on gains from treatment: E[αi − αAT E ]
4
, Social experiments
An example of social experiments in medicine are randomized clinical trials. Randomization rules out selec-
tion. Patients are randomly assigned to a treatment group and a control group.
Under randomization the following assumptions holds:
R1 : E[ui | di = 1] = E[ui | di = 0] = E[ui ]
R2 : E[αi | di = 1] = E[αi | di = 0] = E[αi ]
Under R1 and R2, OLS regression for the equation below allows for consistent and unbiased estimators:
yi = β + αAT E di + ui + di αi − αAT E
Or
yi = β + αAT E di + ei , where ei = ui + di αi − αAT E
Alternative ways to state R1 and R2 are:
E[yi0 | di = 1] = E[yi0 | di = 0] = E[yi0 ] (equal to R1)
E[yi1 | di = 1] = E[yi1 | di = 0] = E[yi1 ] (based on R1 and R2)
A stronger form of this assumption is yi0 , yi1 ⊥ di , which implies independence of yi0 and di and yi1 and di .
Slides 2
Outline
• Review potential outcome model
• Random assignment in social experiments
• Conditional Independence Assumption
• Potential problems with social experiments
Review Questions
• What is potential outcome?
A: A potential outcome is an outcome that occurs if treatment is allocated or not
• What is counterfactual outcome?
A: Counterfactual outcome is an unobserved outcome that describes what would have happened oth-
erwise. For example, given that you’re being allocated treatment, a counterfactual outcome describes
what would have happened if you did not received treatment.
• How are observable outcomes and potential outcomes related?
A: yi = di · yi1 + (1 − di ) · yi0
5
