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K. Tatsiramos - Topics in Econometrics, University of Nottingham
Topics in EconometricsHandout 1
Konstantinos TatsiramosUniversity of Nottingham
January 2015
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K. Tatsiramos - Topics in Econometrics, University of Nottingham
Outline
I IntroductionI The Fundamental Problem of Causal InferenceI Treatment Effects of InterestI Naive Estimation of Treatment Effects
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K. Tatsiramos - Topics in Econometrics, University of Nottingham
Introduction
I The methods studied in this course are usually employed inmicroeconometric analysis.
I Microeconometric analysis is about the analysis of micro-leveldata on the economic behaviour of agents (individuals,households, firms).
I Data include cross-sectional surveys, longitudinal surveys andcensuses.
I Often concerned with the effect of policy interventions oneconomic behaviour.
I Our emphasis will be on methods and assumptions we need tomaintain to determine causation rather than just measurecorrelations.
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K. Tatsiramos - Topics in Econometrics, University of Nottingham
Causality versus Correlation
I Running a regression of Y on X will in general be informativeabout patterns of correlation.
I However, correlation is not causation.
I The ceteris paribus effect (i.e. holding all other relevantfactors fixed) may be given a causal interpretation if we havefully controlled for all other relevant factors.
I In most cases this is not possible (unobserved heterogeneity)and may also not be sufficient.
I We need to develop the methods and specify the maintainedassumptions for identifying the causal effect of X on Y .
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K. Tatsiramos - Topics in Econometrics, University of Nottingham
Classic Example - Schooling and Earnings
I Suppose schooling (x1) is the source of variation in earnings(y).
I Suppose ability (x2) is another source of variation but it isignored in the model (unobserved).
I By ignoring ability, the part of the variation of earnings that isdue to ability is incorrectly attributed to schooling.
I So the relative importance of schooling and ability inexplaining earnings is confounded.
I The leading source of confounding bias is the omission offactors that affect the outcome of interest and are correlatedwith other relevant factors.
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K. Tatsiramos - Topics in Econometrics, University of Nottingham
Program Evaluation or Treatment Evaluation
I Program Evaluation is a new strand in the microeconometricliterature which provides a statistical framework for theestimation of causal parameters.
I In medical studies, a new drug is tested by comparing theresponse to the drug between a group which is treated andanother which is untreated.
I In economics, we are interested in evaluating the effect ofexposure of individuals, households, or firms to a program(treatment), on some outcome.
I Unlike medical studies, it is not always possible to useexperimental variation so we may need to rely onobservational data.
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K. Tatsiramos - Topics in Econometrics, University of Nottingham
Examples of Programs/Treatments
I Labor market training programs.
I School types (e.g. catholic/private vs. public schools).
I Being a member of a trade union.
I Receipt of a transfer payment.
I Changes in regulation for receiving a transfer payment.
I Changes in rules and regulations in financial transactions.
I etc.
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K. Tatsiramos - Topics in Econometrics, University of Nottingham
Training programs and Earnings
I Training programs for disadvantaged individuals (vocationaleducation, on-the-job training etc.).
I Participants include heterogenous individuals such as welfarerecipients, displaced workers etc.
I Evaluating such programs poses significant challenges.
I Trainees tend to experience a downward spiral in earnings justbefore receiving training (Ashenfelters Dip).
I Even in the absence of training, the wages of trainees wouldrise to some degree.
I Comparing earnings before and after program participationmay lead to significantly biased results.
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K. Tatsiramos - Topics in Econometrics, University of Nottingham
School Type and Educational Outcomes
I Early research has shown that Catholic schools are moreeffective than public schools in teaching maths and reading.
I Criticism: Students in the two types of schools areinsufficiently comparable even after adjusting for familybackground and motivation.
I Self-Selection: those who are more likely to benefit fromCatholic schools will enroll net of all observable characteristics.
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K. Tatsiramos - Topics in Econometrics, University of Nottingham
Outline
I IntroductionI The Fundamental Problem of Causal InferenceI Treatment Effects of InterestI Naive Estimation of Treatment Effects
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K. Tatsiramos - Topics in Econometrics, University of Nottingham
Potential Outcomes or Counterfactuals
I Consider the existence of two well-defined causal states inwhich individuals can be exposed to.
I We can think of one state as the treatment and the otheras the control.
I The key assumption of the counterfactual framework is thefollowing:I Each individual in the population of interest has a potential
outcome under each treatment state.
I However, each individual can be observed in only onetreatment state at any point in time.
I Despite this fundamental problem, the potential outcomesmodel allows to conceptualise observational studies as if theywere experimental designs.
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K. Tatsiramos - Topics in Econometrics, University of Nottingham
Example
I Consider, for example, the effect of having a college degreerather than only a high school degree on earnings.
I Those who have high school degrees have theoretical what-ifearnings under the state have a college degree.
I Those who have completed college degrees have theoreticalwhat-if earnings under the state have only a high schooldegree.
I These what-if potential outcomes are counterfactual, theyare not actually observed.
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K. Tatsiramos - Topics in Econometrics, University of Nottingham
The Fundamental Problem of Causal Inference I
I The evaluation is based on the comparison of outcomes underthe two possible states of treatment.
I The problem is that we can at most observe one of theseoutcomes because each unit can be exposed to only one levelof treatment.
I We cannot observe the outcome of a unit both when treatedand during the counterfactual situation of not being treated.
I Causal effects cannot be calculated at the individual level.
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K. Tatsiramos - Topics in Econometrics, University of Nottingham
The Fundamental Problem of Causal Inference II
I To perform the evaluation we need to compare distinct unitsreceiving different levels of the treatment (treated vs.control).
I This comparison can involve different physical units or thesame physical unit at different times.
I In the training program example we could compare theoutcomes before and after participating in training.
I In the Catholic school example we could compare studentswho attend Catholic vs. public schools.
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K. Tatsiramos - Topics in Econometrics, University of Nottingham
The Fundamental Problem of Causal Inference III
I The threat of identification is that the treated are bydefinition different from the non-treated.
I If the differences that determine treatment also influence theoutcomes, then this may invalidate the comparison ofoutcomes by treatment status.
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K. Tatsiramos - Topics in Econometrics, University of Nottingham
The Potential Outcomes Model
I Consider program participation over a population of interest.
I We denote with the indicator Di the causal exposure variable:I Di = 1 for those exposed to the treatment stateI Di = 0 for those exposed to the control state
I For each individual i there are two potential outcomes:Yi (0) and Yi (1).
I The potential outcome Yi (0) denotes the outcome that wouldbe realized if i does not participate in the program.
I The potential outcome Yi (1) denotes the outcome that wouldbe realized if i participates in the program.
I The individual-level causal effect of the treatment is definedas:
i = Yi (1) Yi (0)
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K. Tatsiramos - Topics in Econometrics, University of Nottingham
Treatment Groups and Observed Outcomes
I We can define the observed outcome variable Yi in terms ofthe potential outcomes and the treatment variable.
I The observed outcome is defined as follows:
Yi = Yi (1) if Di = 1
Yi = Yi (0) if Di = 0
I This can be written compactly as:
Yi = Yi (Di ) = DiYi (1) + (1 Di )Yi (0)
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K. Tatsiramos - Topics in Econometrics, University of Nottingham
Table: The Fundamental Problem of Causal Inference
Group Yi (1) Yi (0)
Treatment Group (Di = 1) Observable as Yi CounterfactualControl Group (Di = 0) Counterfactual Observable as Yi
I Individual causal effects are defined within rows but are notobservable.
I Only the diagonal of the table is observable.I The outcome variable Yi reveals only half of the information
contained in the underlying potential outcome variables.
I Individuals contribute outcome information only from theobserved treatment state.
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K. Tatsiramos - Topics in Econometrics, University of Nottingham
Outline
I IntroductionI The Fundamental Problem of Causal InferenceI Treatment Effects of InterestI Naive Estimation of Treatment Effects
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K. Tatsiramos - Topics in Econometrics, University of Nottingham
The Average Treatment Effect
I Because we cannot estimate individual treatment effects, wefocus on averages.
I The average treatment effect in the population is:
ATE E [] = E [Yi (1) Yi (0)]
= E [Yi (1)] E [Yi (0)]I It measures the expected causal effect of randomly assigning a
person in the population to the program.
I It is relevant if the policy under consideration would expose allunits of the population to treatment or none at all.
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K. Tatsiramos - Topics in Econometrics, University of Nottingham
Example of ATE
I In the Catholic school example the individual treatment effecti is the what-if difference in outcomes if a person i waseducated both in a Catholic and in a public school.
I The average treatment effect (E []) is the mean valueamong all students in the population of these what-ifdifferences in outcomes.
I It is equal to the expected value of the what-if difference inoutcomes for a randomly selected student from thepopulation.
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K. Tatsiramos - Topics in Econometrics, University of Nottingham
Conditional Average Treatment Effects - ATT
Sometimes a treatment is available to individuals without forcingthem to take the treatment. We would like to know how effectivethe treatment is for those who choose to take the treatment.
I The Average Treatment Effect for the Treated (ATT) is themean effect for those who typically take the treatment.
ATT E [Yi (1) Yi (0)|Di = 1]
= E [Yi (1)|Di = 1] E [Yi (0)|Di = 1]I ATT in many cases is the more interesting parameter: is a
program beneficial for participants?
I Catholic school example: ATT is the average effect for thosewho attend Catholic schools; not across all students whocould potentially attend.
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K. Tatsiramos - Topics in Econometrics, University of Nottingham
Conditional Average Treatment Effects - ATC
I The Average Treatment Effect for the Controls (ATC) is themean effect for those who typically do not take the treatment.
ATC E [Yi (1) Yi (0)|Di = 0]
= E [Yi (1)|Di = 0] E [Yi (0)|Di = 0]I ATC is of interest if the goal is to determine the effect of a
potential policy intervention.
I Catholic school example: a new school voucher programwhich moves students from public to Catholic schools.
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K. Tatsiramos - Topics in Econometrics, University of Nottingham
Taking Stock
I The potential outcomes model allows to conceptualizeobservational studies as if they were experimental designs.
I Each individual in the population of interest has a potentialoutcome under each treatment state.
I Because of the fundamental problem of causal inference wefocus on non-individual causal effects (ATE, ATT etc.).
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K. Tatsiramos - Topics in Econometrics, University of Nottingham
Outline
I IntroductionI The Fundamental Problem of Causal InferenceI Treatment Effects of InterestI Naive Estimation of Treatment Effects
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K. Tatsiramos - Topics in Econometrics, University of Nottingham
Naive Estimation of ATE
I The naive estimator of ATE is the following:
NAIVE = E [Yi |Di = 1] E [Yi |Di = 0]= E [Yi (1)|Di = 1] E [Yi (0)|Di = 0]
I With observational data, in general, the naive estimator doesnot converge to a consistent estimate of the ATE.
I To see this, consider the following decomposition of ATE:
E [] = {piE [Yi (1)|Di = 1] + (1 pi)E [Yi (1)|Di = 0]}{piE [Yi (0)|Di = 1] + (1 pi)E [Yi (0)|Di = 0]}
where pi is the proportion of individuals in the population ofinterest that takes the treatment.
I We do not observe the counterfactual conditionalexpectations.
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K. Tatsiramos - Topics in Econometrics, University of Nottingham
Expected Bias of Naive Estimator
I The decomposition of ATE can be also written as follows:
E [Yi (1)|Di = 1] E [Yi (0)|Di = 0] =
E []
+{E [Yi (0)|Di = 1] E [Yi (0)|Di = 0]}+(1 pi){E [|Di = 1] E [|Di = 0]}.
I The naive estimator converges to the LHS of this equation,which is not equal only to the ATE.
I There are two sources of bias: baseline bias and differentialtreatment effect bias.
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K. Tatsiramos - Topics in Econometrics, University of Nottingham
Sources of Bias
I The baseline bias is equal to the difference in the averageoutcome in the absence of the treatment, Yi (0), betweenthose in the treatment group and those in the control group.
I The differential treatment effect bias is equal to the expecteddifference in the treatment effect, E [], between those in thetreatment and those in the control group.
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K. Tatsiramos - Topics in Econometrics, University of Nottingham
Example for the Bias of the Naive Estimator
Consider the treatment to be college education and a labor marketoutcome.
Group E [Yi (1)|.] E [Yi (0)|.]Treatment Group (Di = 1) 10 6Control Group (Di = 0) 8 5
I The naive comparison between college (COL) and high school(HS) graduates gives a difference of 5 (10-5).
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K. Tatsiramos - Topics in Econometrics, University of Nottingham
Example (continued)
Group E [Yi (1)|.] E [Yi (0)|.]Treatment Group (Di = 1) 10 6Control Group (Di = 0) 8 5
I COL would have done better than HS even if had not gone tocollege (6 vs. 5) - baseline bias. Note this is opposite for HS(8 vs. 10).
I ATT is higher (10 6 = 4) than ATC (8 5 = 3)differential treatment effect bias.
I If the proportion of COL is 30% then:ATE = 0.3(10 6) + (1 0.3)(8 5) = 3.3
I Total Bias=baseline bias (6 5)+differential treatmenteffect bias (1 0.3)(4 3) = 1.7
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K. Tatsiramos - Topics in Econometrics, University of Nottingham
Maintained Assumptions and Causal Effects
I Under what assumptions the naive estimator will identify theATE?
Assumption 1 : E [Yi (1)|Di = 1] = E [Yi (1)|Di = 0]
Assumption 2 : E [Yi (0)|Di = 1] = E [Yi (0)|Di = 0]I If both hold then we know the counterfactuals and there is no
bias : ATE = ATT = ATC .
I When these assumptions are reasonable?
I When treatment is assigned randomly: Yi (1),Yi (0) D(independence assumption).
I With observational data is rarely justified.
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K. Tatsiramos - Topics in Econometrics, University of Nottingham
Naive Estimation Assuming Independence
I Consider the observed outcome Yi :
Yi = DiYi (1) + (1 Di )Yi (0) =
Yi (0) + Di [Yi (1) Yi (0)]I Taking expectations:
E [Yi |Di ] =E [Yi (0)|Di ] + Di{E [Yi (1)|Di ] E [Yi (0)|Di ]} =
E [Yi (0)] + Di{E [Yi (1)] E [Yi (0)]}I The last equality follows by assuming independence.I Then:
E [Yi |Di = 1] E [Yi |Di = 0] =E [Yi (1)] E [Yi (0)] ATE
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K. Tatsiramos - Topics in Econometrics, University of Nottingham
Maintained Assumptions and Causal Effects
I If only Ass1 holds then the naive estimator is consistent forATC:
NAIVE E [Yi (1)|Di = 1] E [Yi (0)|Di = 0]
= E [Yi (1)|Di = 0] E [Yi (0)|Di = 0] ATCI Use the observed outcome of the treated as the counterfactual
for the non-treated.
I If only Ass2 holds then the naive estimator is consistent forATT:
NAIVE E [Yi (1)|Di = 1] E [Yi (0)|Di = 0]
= E [Yi (1)|Di = 1] E [Yi (0)|Di = 1] ATTI Use the observed outcome of the controls as the appropriate
counterfactual for the treated.
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K. Tatsiramos - Topics in Econometrics, University of Nottingham
The Stable Unit Treatment Value Assumption (SUTVA)
I SUTVA is a basic assumption of causal effect stability.
I SUTVA requires that the potential outcomes of individuali do not depend on the treatments received by otherindividuals.
I In economics, it is referred to as a no-macro effect or partialequilibrium assumption.
I It is violated when the underlying causal effects are a functionof the treatment assignment patterns.
I That is, when the treatment effect differs (e.g. be lesseffective) when more individuals are assigned to it.
I In the Catholic schooling example, school effectiveness shouldnot depend on the number/composition of students who enterCatholic schools.
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K. Tatsiramos - Topics in Econometrics, University of Nottingham
SUTVA (continued)
I If SUTVA is maintained but there is some doubt about itsvalidity, then certain types of marginal effects can still bedefended.
I The estimates of ATE hold only for what-if movements of avery small number of individuals from one hypotheticaltreatment state to another.
I For more extensive interventions when SUTVA is clearlyviolated the variation of the causal effect as a function oftreatment assignment patterns need to be modeled explicitly.
I This requires to develop a full structural model and it isbeyond the scope of this course.
I In what follows we will maintain SUTVA.
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K. Tatsiramos - Topics in Econometrics, University of Nottingham
Taking Stock
I With observational data, if we do not impose independence,then we need to find solutions for correcting the bias.
I We need to understand what determines treatment selection.
I An ideal possibility is to rely on experimental variation intreatment selection (assignment).
I If that is not available we can consider subgroups for whichthe assumptions for identification can be defended.
I This strategy involves conditioning on observed variables fromthe population.
I This leads us to the idea of the regression in the potentialoutcomes framework.
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K. Tatsiramos - Topics in Econometrics, University of Nottingham
References
I The material in this handout follows:
Stephen L. Morgan and Christopher Winship Counterfactualsand Causal Inference. Methods and Principles for SocialResearch. 2007. Cambridge University Press.
I Required reading: Sections 1.3, 1.4, 2.1-2.6
I Suggested reading: Sections 1.1-1.2 provide a nice historicaldiscussion of the framework.
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