📉
Microeconometrics
Potential Outcome Framework
Framework
Group Level Effect
Perfect randomization
Best-Predictor Regressions
Regressions
Conditional Prediction Problem
Loss Functions
How to choose the loss function?
Extrapolation problem
Estimation of Best-Predictor Regressions (in Sample Land)
Parametric model specification
Method of moments
Nonparametric estimation of continuous regressions under random sampling
1. Basic idea, when P (x = ξ) > 0 [P is discrete]
2. Main case, when P (ξ) = 0 [P (x)is continuous and ξon the support of x]
Local-Weighted Average with Kernel Weighting [Kernel Regression]
Tradeoffs between nonparametric and parametric
Parametric Mean Regression
Three propoerties of the CEF - Conditional Expectation Functions - E(y|X)
1. Law of Iterated Expectations (LIE)
2. CEF Decomposition Property
3. Conditional Expectation Funcitons Prediction Property:
Linear Regression Model for Population Conditional Mean
Mean Independence or Zero Conditional Mean (and implications)
Moment Conditions and Method of Moments (MM)
Bivariate Case: “Simple Regression”
Mean Linear Regression Anatomy: Frisch-Waugh-Lovell
Partialling Out Procedure
Linearity and Specifications (parametric model)
Log(y)-log(x) specification:
Log(y)-level(x) specification:
Polynomials
Dummies
Slope dummies
Dummies to incorporate ordinal X
Saturated Regressor Models
Linear Regression with Binary y: the Linear Probability Model (LPM)
Microeconometrics 1
, Linear Probability Model
Appendix
Mean Linear Regression & Gauss-Markov Assumptions
POF Meets Regression
Mean Regression Formulation for Realized Outcomes & Individual Treatment Effects
Mean Regression Formulation for Realized Outcomes & Average Treatment Effects
Selection & Heterogeneity Terms for Average Treatment Effect
Polar Case 1 - Unbiasedness under Randomization
Polar Case 2 - Unbiasedness under Mean Independence
Appendix: ATT Case
Bias for ATT
Sample-Land Implications
Selection on Observables and the Conditional (Mean) Independence Assumption
Conditioning on X via regression
Identification of CATE under CMI
Selection on Unobservables
Traditional Instrumental Variables
Adding Covariates X
Multiple IVs for One Endogenous Treatment (Regressor)
Difference-in-Difference
Before-After Comparison (B-A)
Identification Condition for ATT:
NEW Identification Condition for ATT
DiD Definition and Estimator
ATU - ATE slide
How to read and interpret BA and DID tables
Threats to Parallel Trends
DiD Regression with Repeated Cross Sections
Garbage Incinerator on Housing Prices Example
DiD Regression with Panel Data
DiD and IV(s)
DiD approach (no IVs)
Family CEOs vs External CEOs for profitability of firms
Instrumental Variables approach
Requirements meeting
IV as 2SLS: First Stage
IV as 2SLS: Second Stage
IVs as “Natural” or “Quasi” Experiments
A-I-R Framework
Assumptions of the model
Assumption 1:
Stable Unit Treatment Value Assumption (SUTVA)
Assumption 2
Random Assignment
Average Intention-to-Treat Effects
Assumption 3
Nonzero ATE of Z and D (Relevance)
Assumption 4
Exclusion Restriction (Validity)
Assumption 5
Monotonicity
Deriving the LATE
Potential Outcome Framework
Framework
1. Business analytics: Does internet advertising increase sales?
Δi = Yi (1) − Yi (0)
Microeconometrics 2
, Since the counterfactual is not observable, the individual level effect is not measurable.
Fundamental Problem of Causal Inference, after Holland
1. Corporate finance: Does family succession hurt firm performance?
2. Human capital: Does getting an advanced university degree increase lifetime earnings?
3. Social policy: Do summer jobs reduce youth crime?
4. Yesterday: Do hospitals improve health?
Formal framework to study causality
1. Population (sample): units, indexed (i=1,2,3…)
2. Observables
a. Treatment (D)
b. Outcome (Y)
3. Objective: to understand whether, in what sense and under what conditions one can conclude that "D causes Y"
Three Treatments Notions within POF Three Outcomes Notions
1. EX ANTE: feasible or potential treatments: 1. EX ANTE: hypothetical outcome
(D = 0, 1) Yi (1) or Y1i hypothetically under the treatment
2. EX POST 1 realized or actual treatment:
Yi (0) or Y0i hypothetically under no treatment
Di = 1for those who received the treatment
2. EX POST 1: realized outcome
Di = 0for those who did not receive the treatment
Yi (1)∣D = 1
1. EX POST 2:
Yi (0)∣D = 0
The ex ante treatment not realized is the ex post 3. EX POST 2: Counterfactual outcomes
counterfactual. Yi (1)∣D = 0
Yi (0)∣D = 1
Outcomes not experienced by
i.
Referring to the hospital experiment
i Di Yi (Di )
T Di Yi (T Di )
Y belongs to
1 1 5 0
[1,2,3,4,5]
2 0 4 1 ?
n … … … …
We know that person 1 has been treated so his/her health is 5, but what could be the counterfactual situation (i.e. if no
treatment)?
This equation links the potential outcomes to the realized outcomes
Yi = Di ⋅ Yi (1) + (1 − Di ) ⋅ Yi (0) =
Yi (0) + (Yi (1) − Yi (0)) ⋅ Di
Ex post only the outcome corresponding to the realized treatment is observable. This means that one of the two values
of Y will be 0 depending on the treatment.
Assumptions
1. Treatments (D) are binary, outcomes (Y) can be either discrete or continuous.
2. Treatments can be more than two, mutually exclusive.
3. In the model the treatment response is
individualistic (not bound to others' responses), the assumption is called SUTVA (Stable Unit Treatment Value
Assignment).
Microeconometrics 3
, D belongs to [0,1]
Y belongs to:
1,1 1,2 1,3 1,4 1,5
2,1 2,2 2,3 2,4 2,5
3,1 3,2 3,3 3,4 3,5
4,1 4,2 4,3 4,4 4,5
5,1 5,2 5,3 5,4 5,5
You can only see one value for each combination patient-treatment.
All other combinations are counterfactuals.
Observing more data for treatment and outcomes wouldn’t help in predicting the counterfactuals
Observing the same individual in different points in time for the same treatment might help, after having made
several assumptions or restrictions for the counterfactuals.
Useful in difference in difference and before/after.
Assuming/Making up the counterfactuals is possible if not random and under some assumptions.
Individual level causal effect
D has a causal effect on Y for a unit if exposure to treatment level 1 instead of treatment level 0 implies Yi = Yi (1)
instead of Yi
= Yi (0).
The individual level causal effect of D on Y for a unit
i is:
Group Level Effect
We need to set rules or regimes for all i’s in a group.
Same hypothetical treatment for a group:
Regime 1: all units are treated
Regime 2: all units are untreated
1. How should outcomes be compared?
Compare the probability density functions of potential outcomes of the different regimes (not groups)
f[(Yi(1)] and f[Yi(0)].
Compare the Δof the two distributions: f[(Yi(1)] − f[Yi(0)]
Compare the means of the two functions (ATE - Average Treatment Effects):
⁍
Compare median or quantile treatment effect (αQTE
The perfect randomization must be tested, usually through t tests, to check whether the distribution (and covariates
such as mean and variance) are the same in the T and C group.
⁍
2. What is the relevant group?
The δATE measures the difference in the whole population:
δATE = E[Δi] = E[Yi (1) − Yi (0)]
= E[Yi (1)] − E[Yi (0)]
The expectation in unconditional since it is taken over all the population.
The δATT (Average Treatment Effect on the Treated) only the treated:
δATT = E[Δi ∣Di = 1]
Microeconometrics 4
Microeconometrics
Potential Outcome Framework
Framework
Group Level Effect
Perfect randomization
Best-Predictor Regressions
Regressions
Conditional Prediction Problem
Loss Functions
How to choose the loss function?
Extrapolation problem
Estimation of Best-Predictor Regressions (in Sample Land)
Parametric model specification
Method of moments
Nonparametric estimation of continuous regressions under random sampling
1. Basic idea, when P (x = ξ) > 0 [P is discrete]
2. Main case, when P (ξ) = 0 [P (x)is continuous and ξon the support of x]
Local-Weighted Average with Kernel Weighting [Kernel Regression]
Tradeoffs between nonparametric and parametric
Parametric Mean Regression
Three propoerties of the CEF - Conditional Expectation Functions - E(y|X)
1. Law of Iterated Expectations (LIE)
2. CEF Decomposition Property
3. Conditional Expectation Funcitons Prediction Property:
Linear Regression Model for Population Conditional Mean
Mean Independence or Zero Conditional Mean (and implications)
Moment Conditions and Method of Moments (MM)
Bivariate Case: “Simple Regression”
Mean Linear Regression Anatomy: Frisch-Waugh-Lovell
Partialling Out Procedure
Linearity and Specifications (parametric model)
Log(y)-log(x) specification:
Log(y)-level(x) specification:
Polynomials
Dummies
Slope dummies
Dummies to incorporate ordinal X
Saturated Regressor Models
Linear Regression with Binary y: the Linear Probability Model (LPM)
Microeconometrics 1
, Linear Probability Model
Appendix
Mean Linear Regression & Gauss-Markov Assumptions
POF Meets Regression
Mean Regression Formulation for Realized Outcomes & Individual Treatment Effects
Mean Regression Formulation for Realized Outcomes & Average Treatment Effects
Selection & Heterogeneity Terms for Average Treatment Effect
Polar Case 1 - Unbiasedness under Randomization
Polar Case 2 - Unbiasedness under Mean Independence
Appendix: ATT Case
Bias for ATT
Sample-Land Implications
Selection on Observables and the Conditional (Mean) Independence Assumption
Conditioning on X via regression
Identification of CATE under CMI
Selection on Unobservables
Traditional Instrumental Variables
Adding Covariates X
Multiple IVs for One Endogenous Treatment (Regressor)
Difference-in-Difference
Before-After Comparison (B-A)
Identification Condition for ATT:
NEW Identification Condition for ATT
DiD Definition and Estimator
ATU - ATE slide
How to read and interpret BA and DID tables
Threats to Parallel Trends
DiD Regression with Repeated Cross Sections
Garbage Incinerator on Housing Prices Example
DiD Regression with Panel Data
DiD and IV(s)
DiD approach (no IVs)
Family CEOs vs External CEOs for profitability of firms
Instrumental Variables approach
Requirements meeting
IV as 2SLS: First Stage
IV as 2SLS: Second Stage
IVs as “Natural” or “Quasi” Experiments
A-I-R Framework
Assumptions of the model
Assumption 1:
Stable Unit Treatment Value Assumption (SUTVA)
Assumption 2
Random Assignment
Average Intention-to-Treat Effects
Assumption 3
Nonzero ATE of Z and D (Relevance)
Assumption 4
Exclusion Restriction (Validity)
Assumption 5
Monotonicity
Deriving the LATE
Potential Outcome Framework
Framework
1. Business analytics: Does internet advertising increase sales?
Δi = Yi (1) − Yi (0)
Microeconometrics 2
, Since the counterfactual is not observable, the individual level effect is not measurable.
Fundamental Problem of Causal Inference, after Holland
1. Corporate finance: Does family succession hurt firm performance?
2. Human capital: Does getting an advanced university degree increase lifetime earnings?
3. Social policy: Do summer jobs reduce youth crime?
4. Yesterday: Do hospitals improve health?
Formal framework to study causality
1. Population (sample): units, indexed (i=1,2,3…)
2. Observables
a. Treatment (D)
b. Outcome (Y)
3. Objective: to understand whether, in what sense and under what conditions one can conclude that "D causes Y"
Three Treatments Notions within POF Three Outcomes Notions
1. EX ANTE: feasible or potential treatments: 1. EX ANTE: hypothetical outcome
(D = 0, 1) Yi (1) or Y1i hypothetically under the treatment
2. EX POST 1 realized or actual treatment:
Yi (0) or Y0i hypothetically under no treatment
Di = 1for those who received the treatment
2. EX POST 1: realized outcome
Di = 0for those who did not receive the treatment
Yi (1)∣D = 1
1. EX POST 2:
Yi (0)∣D = 0
The ex ante treatment not realized is the ex post 3. EX POST 2: Counterfactual outcomes
counterfactual. Yi (1)∣D = 0
Yi (0)∣D = 1
Outcomes not experienced by
i.
Referring to the hospital experiment
i Di Yi (Di )
T Di Yi (T Di )
Y belongs to
1 1 5 0
[1,2,3,4,5]
2 0 4 1 ?
n … … … …
We know that person 1 has been treated so his/her health is 5, but what could be the counterfactual situation (i.e. if no
treatment)?
This equation links the potential outcomes to the realized outcomes
Yi = Di ⋅ Yi (1) + (1 − Di ) ⋅ Yi (0) =
Yi (0) + (Yi (1) − Yi (0)) ⋅ Di
Ex post only the outcome corresponding to the realized treatment is observable. This means that one of the two values
of Y will be 0 depending on the treatment.
Assumptions
1. Treatments (D) are binary, outcomes (Y) can be either discrete or continuous.
2. Treatments can be more than two, mutually exclusive.
3. In the model the treatment response is
individualistic (not bound to others' responses), the assumption is called SUTVA (Stable Unit Treatment Value
Assignment).
Microeconometrics 3
, D belongs to [0,1]
Y belongs to:
1,1 1,2 1,3 1,4 1,5
2,1 2,2 2,3 2,4 2,5
3,1 3,2 3,3 3,4 3,5
4,1 4,2 4,3 4,4 4,5
5,1 5,2 5,3 5,4 5,5
You can only see one value for each combination patient-treatment.
All other combinations are counterfactuals.
Observing more data for treatment and outcomes wouldn’t help in predicting the counterfactuals
Observing the same individual in different points in time for the same treatment might help, after having made
several assumptions or restrictions for the counterfactuals.
Useful in difference in difference and before/after.
Assuming/Making up the counterfactuals is possible if not random and under some assumptions.
Individual level causal effect
D has a causal effect on Y for a unit if exposure to treatment level 1 instead of treatment level 0 implies Yi = Yi (1)
instead of Yi
= Yi (0).
The individual level causal effect of D on Y for a unit
i is:
Group Level Effect
We need to set rules or regimes for all i’s in a group.
Same hypothetical treatment for a group:
Regime 1: all units are treated
Regime 2: all units are untreated
1. How should outcomes be compared?
Compare the probability density functions of potential outcomes of the different regimes (not groups)
f[(Yi(1)] and f[Yi(0)].
Compare the Δof the two distributions: f[(Yi(1)] − f[Yi(0)]
Compare the means of the two functions (ATE - Average Treatment Effects):
⁍
Compare median or quantile treatment effect (αQTE
The perfect randomization must be tested, usually through t tests, to check whether the distribution (and covariates
such as mean and variance) are the same in the T and C group.
⁍
2. What is the relevant group?
The δATE measures the difference in the whole population:
δATE = E[Δi] = E[Yi (1) − Yi (0)]
= E[Yi (1)] − E[Yi (0)]
The expectation in unconditional since it is taken over all the population.
The δATT (Average Treatment Effect on the Treated) only the treated:
δATT = E[Δi ∣Di = 1]
Microeconometrics 4
