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The Economics of European Regions:

Theory, Empirics, and Policy

Dipartimento di Economia e Management

Davide Fiaschi Angela Parenti1

[email protected], and [email protected] The Economics of European Regions 1 / 50

Introduction

Introduction

Many empirical question in economics and other social science depend oncausal effects of policies, as for example:

Fiaschi-Parenti The Economics of European Regions 2 / 50

Introduction

Introduction


Medical therapies: does aspirin reduce the headache? Do vaccinesincreased life expectancy?


Introduction

Introduction



Educational policies: does college increase future earnings? Dostudents’ loans increase college education and earnings of poor highschool graduates?


Introduction

Introduction




Labor market programs: does training increase the employmentprobability of unemployed? Does Employment Protection Legislation(EPL) increase unemployment?


Introduction

Introduction





Cohesion policy: do EU Structural and Cohesion Funds boostregional economic growth in Europe?


Introduction

Introduction





Cohesion policy: do EU Structural and Cohesion Funds boostregional economic growth in Europe?

Free movement policy: does the implementation of the SchengenAgreement boost labour mobility?


Introduction

Policy intervention

A policy is an intervention targeted to a specific population with thepurpose of inducing a change in a defined state and/or behaviour.


Introduction

Policy intervention


A policy intervention is characterised by:

a target population: a well-defined set of units (e.g. persons,households, firms, geographic areas) upon which the intervention willoperate at a particular time;


Introduction

Policy intervention



a target population: a well-defined set of units (e.g. persons,households, firms, geographic areas) upon which the intervention willoperate at a particular time;an intervention: an action, or a set of actions, whose effect on theoutcome the analyst wishes to assess relative to non-intervention;


Introduction

Policy intervention



a target population: a well-defined set of units (e.g. persons,households, firms, geographic areas) upon which the intervention willoperate at a particular time;an intervention: an action, or a set of actions, whose effect on theoutcome the analyst wishes to assess relative to non-intervention;an outcome variable: an observable and measurable characteristic ofthe units of the population on which the intervention may have aneffect.


Introduction

Policy evaluations

Public policies or interventions are implemented with the expectationof improving the situation of the individuals affected by them.


Introduction

Policy evaluations


The extent to which they do so can only be assessed by undertakingan adequate policy evaluation exercise.


Introduction

Policy evaluations



Consider the example of publicly-provided training programmesoffered to unemployed individuals, aiming at increasing their skills andcontributing to their probability of finding a job.


Introduction

Policy evaluations




Measuring the effect of these programmes on an individual’s futureemployability it is not trivial:


Introduction

Policy evaluations





Should the outcomes of the participants be compared to theirpre-programme situations?


Introduction

Policy evaluations





Should the outcomes of the participants be compared to theirpre-programme situations?Or should the outcomes of the participants be compared with those ofthe non-participants?


Introduction

Policy evaluations





Should the outcomes of the participants be compared to theirpre-programme situations?Or should the outcomes of the participants be compared with those ofthe non-participants?What is the best approach to measure the effect of the programme onthe outcomes of participants?


Introduction

Policy evaluation (cont.)

“Policy evaluation uses a range of research methods to systematicallyinvestigate the effectiveness of policy interventions, implementationand processes, and to determine their merit, worth, or value in termsof improving the social and economic conditions of differentstakeholders.”


Introduction



“Policy evaluation is conducted for checking the effects of the policiesof respective ministries and for evaluating the policies in terms ofnecessity, efficiency, validity, etc. to improve the planning andimplementation process.”


Introduction




“Evaluation is judgement of interventions according to their results,impacts and needs they aim to satisfy.”


Introduction




“Evaluation is judgement of interventions according to their results,impacts and needs they aim to satisfy.”

⇒ Policy evaluation applies evaluation principles and methods to examinethe content, implementation or impact of a policy, to understand themerit, worth, and utility of a policy.


Introduction

Why evaluate?

To contribute to the design of interventions, including providing inputfor setting political priorities.


Introduction

Why evaluate?


To assist in an efficient allocation of resources.


Introduction

Why evaluate?



To improve the quality of the intervention.


Introduction

Why evaluate?



To improve the quality of the intervention.

To report on the achievements of the intervention (i.e.accountability).


Introduction

Conducting evaluation: causality

The aim of policy evaluation is to measure the causal effect of apolicy on outcomes of interest, on which it is expected to have animpact.


Introduction

Conducting evaluation: causality

The aim of policy evaluation is to measure the causal effect of apolicy on outcomes of interest, on which it is expected to have animpact.

The policy’s causal effect is defined as the difference between theoutcome of the units affected by the policy (the actual situation) andthe outcome that these same units would experience had the policynot been implemented.


Definition of causality

Causality: the basic framework

Imbens and Rubin (2015)

“In everyday life, causal language is widely used in an informal way”. Onemight say: “My headache went away because I took an aspirin,” or “Shegot a good job last year because she went to college,” or “She has longhair because she is a girl.”

Imbens and Rubin (2015). Causal Inference for Statistics, Social, and Biomedical Sciences.



Causality: the basic framework (cont.)

Such comments are typically informed by observations on pastexposures of:





headache outcomes after taking aspirin or not;





headache outcomes after taking aspirin or not;characteristics of jobs of people with or without college educations;





headache outcomes after taking aspirin or not;characteristics of jobs of people with or without college educations;the typical hair length of boys and girls.





headache outcomes after taking aspirin or not;characteristics of jobs of people with or without college educations;the typical hair length of boys and girls.

Therefore, these observations generally involve informal statisticalanalyses, drawing conclusions from associations betweenmeasurements of different quantities that vary from individual toindividual (i.e., random variables).



Correlation vs causation

Correlation (or association) is not the same as causation!





Causality is tied to an action (or manipulation, treatment, orintervention), applied to a unit.






A unit can be a physical object, a firm, an individual person, orcollection of objects or persons, such as a classroom or a market, at aparticular point in time.






A unit can be a physical object, a firm, an individual person, orcollection of objects or persons, such as a classroom or a market, at aparticular point in time.

The same physical object or person at a different time is a different

unit.



Correlation vs causation (cont.)

A causal statement presumes that, although a unit was at aparticular point in time exposed to a particular action (or treatment),the same unit could have been exposed to an alternative action (ortreatment) at the same point in time.



Correlation vs causation (cont.)

A causal statement presumes that, although a unit was at aparticular point in time exposed to a particular action (or treatment),the same unit could have been exposed to an alternative action (ortreatment) at the same point in time.

For instance, we could have decided to take an aspirin to relieve theheadache, or not to take the aspirin, or to take an alternativemedicine.



Treatment and control

In most of the cases the possible actions are two and often:





one of these actions corresponds to a more active treatment (e.g.,taking an aspirin);






in contrast to a more passive action (e.g., not taking the aspirin).







Formally the two treatments are viewed symmetrically although theliterature usually labels:








treatment the more active treatment; and,








treatment the more active treatment; and,

control the other.



Potential outcome

Given a unit and a set of actions, we associate each action-unit pair

with a potential outcome.



Potential outcome



These outcomes are potential outcomes because only one willultimately be realized and therefore possibly observed (i.e., the onecorresponding to the action actually taken).



Potential outcome




Ex post, the other potential outcomes cannot be observed becausethe corresponding actions were not taken.



Potential outcome





The causal effect of one action or treatment relative to anotherinvolves the comparison of these potential outcomes, one realized(and perhaps, though not necessarily, observed), and the others notrealized and therefore not observable.



Potential outcome





The causal effect of one action or treatment relative to anotherinvolves the comparison of these potential outcomes, one realized(and perhaps, though not necessarily, observed), and the others notrealized and therefore not observable.

Any treatment must occur temporally before the observation ofany associated potential outcome is possible.



Example 1: headache

In the headache example the action is clear: “I took an aspirin, but at thetime that I took the aspirin, I could have followed the alternate course ofnot taking an aspirin.”



Example 1: headache

In the headache example the action is clear: “I took an aspirin, but at thetime that I took the aspirin, I could have followed the alternate course ofnot taking an aspirin.”

⇒ The “because” statement is causal as it reflects the comparison ofthose two potential outcomes.



Example 2: college

In the college example it is less clear what the treatment and itsalternative are: “she went to college, and at the point in time when shedecided to go to college, she could have decided not to go to college”.



Example 2: college


In the latter case, she might have had a different job a year ago.



Example 2: college



⇒ The implied causal statement compares the quality of the job sheactually had then, to the quality of the job she would have had a year agowithout going to college.



Example 2: college



⇒ The implied causal statement compares the quality of the job sheactually had then, to the quality of the job she would have had a year agowithout going to college.

However, in this example, the potential outcome under the alternativeaction (the job obtained a year ago without enrolling in college) issomewhat murky: if not enrolled in college what would she have done?



Example 3: long hair

The informal statement is “she has long hair because she is a girl.”





The implicit treatment is being a girl, and the implicit alternative is beinga boy.






But there is no action articulated that would have made her a boy andallowed us to observe the alternative potential outcome of hair length forthis person as a boy!






But there is no action articulated that would have made her a boy andallowed us to observe the alternative potential outcome of hair length forthis person as a boy!

⇒ This “because” statement is ill-defined as a causal statement.



Counterfactual outcome

To better understand the differences between the three previous “causal”statements consider, after the application of the treatment, thecounterfactual value of the potential outcome corresponding to thetreatment not applied.





1 The treatment applied is “aspirin taken”, and the counterfactualpotential outcome is the state of the headache under “aspirin nottaken” → unambiguous.






2 The counterfactual outcome is her job a year ago had she decided notto go to college → not as well defined.







3 The counterfactual outcome, the persons hair length if she were a boyrather than a girl → not at all well defined (no action!).







3 The counterfactual outcome, the persons hair length if she were a boyrather than a girl → not at all well defined (no action!).

⇒ Causal statements become more clearly defined by more preciselyarticulating the intervention that would have made the alternativepotential outcome the realized one.



Definition of causal effect

Single unit at a particular point in time.





Two treatment levels: taking an aspirin, and not taking an aspirin.






If I take the aspirin, my headache may remain or not one hour later:we denote this outcome by Y (Aspirin), which can be either“Headache” or “No Headache”.







If I do not take the aspirin, my headache may remain or not one hourlater: we denote this potential outcome by Y (No Aspirin), which alsocan be either “Headache”, or “No Headache”.








There are therefore two potential outcomes, Y (Aspirin) and Y (NoAspirin), one for each level of the treatment.








There are therefore two potential outcomes, Y (Aspirin) and Y (NoAspirin), one for each level of the treatment.

The causal effect of the treatment involves the comparison ofthese two potential outcomes.



Definition of causal effect (cont.)

Because in this example each potential outcome can take on only twovalues, the unit-level causal effect (i.e., the comparison of these twooutcomes for the same unit) involves one of four (two-by-two) possibilities:





1 Headache gone only with aspirin:Y (Aspirin) = No Headache, Y (No Aspirin) = Headache






2 No effect of aspirin, with a headache in both cases:Y (Aspirin) = Headache, Y (No Aspirin) = Headache







3 No effect of aspirin, with the headache gone in both cases:Y (Aspirin) = No Headache, Y (No Aspirin) = No Headache







3 No effect of aspirin, with the headache gone in both cases:Y (Aspirin) = No Headache, Y (No Aspirin) = No Headache

4 Headache gone only without aspirin:Y (Aspirin) = Headache, Y (No Aspirin) = No Headache




• There is a zero causal effect of taking aspirin in the second and thirdpossibilities.




• There is a zero causal effect of taking aspirin in the second and thirdpossibilities.• In the other two cases the aspirin has a causal effect, making theheadache go away in one case and not allowing it to go away in the other.




• There is a zero causal effect of taking aspirin in the second and thirdpossibilities.• In the other two cases the aspirin has a causal effect, making theheadache go away in one case and not allowing it to go away in the other.

Illustration of case 1. Source: Imbens and Rubin (2015).




The definition of the causal effect depends on the potentialoutcomes, but it does not depend on which outcome is actuallyobserved.




The definition of the causal effect depends on the potentialoutcomes, but it does not depend on which outcome is actuallyobserved.Whether I take an aspirin (and am therefore unable to observe the state of myheadache with no aspirin) or do not take an aspirin (and am thus unable to observethe outcome with an aspirin) does not affect the definition of the causal effect.





The causal effect is the comparison of potential outcomes, for thesame unit, at the same moment in time post-treatment.





The causal effect is the comparison of potential outcomes, for thesame unit, at the same moment in time post-treatment.The causal effect is not defined in terms of comparisons of outcomes at differenttimes, as in a before-and-after comparison of my headache before and afterdeciding to take or not to take the aspirin.


Estimation of causality The fundamental problem of causal inference

The fundamental problem of causal inference




Rubin (1978, p. 38)

“The fundamental problem facing inference for causal effects is that iftreatment t is assigned to the ith experimental unit (i.e., Wi = t), onlyvalues in Y t can be observed, Y j for j 6= t being unobservable (ormissing).”In Rubin’s notation, i denotes the unit, and Y t and Y j denote the two potentialoutcomes for unit i under the two levels of the treatment t and j .




Rubin (1978, p. 38)

“The fundamental problem facing inference for causal effects is that iftreatment t is assigned to the ith experimental unit (i.e., Wi = t), onlyvalues in Y t can be observed, Y j for j 6= t being unobservable (ormissing).”In Rubin’s notation, i denotes the unit, and Y t and Y j denote the two potentialoutcomes for unit i under the two levels of the treatment t and j .

Holland (1986, p. 947)

“It is impossible to observe the value of Yt(u) and Yc(u) on the same unitand, therefore, it is impossible to observe the effect of t on u” (emphasisin original).In Holland’s notation, u denotes the unit, and Yt(u) and Yc(u) denote the two potentialoutcomes for unit u under the two levels of the treatment t and c.



The fundamental problem of causal inference(cont.)

Assuming the action taken was that you took the aspirin. Source: Imbens and Rubin (2015).





• If the action you take is Aspirin, you observe Y (Aspirin) and will neverknow the value of Y (No Aspirin).





• If the action you take is Aspirin, you observe Y (Aspirin) and will neverknow the value of Y (No Aspirin).• If the action is No Aspirin, you observe Y (No Aspirin) but cannot knowthe value of Y (Aspirin).





• If the action you take is Aspirin, you observe Y (Aspirin) and will neverknow the value of Y (No Aspirin).• If the action is No Aspirin, you observe Y (No Aspirin) but cannot knowthe value of Y (Aspirin).

⇒ Even though the unit-level causal effect (the comparison of the twopotential outcomes) may be well defined, by definition we cannot learn itsvalue from just the single realized potential outcome.



Estimation of causal effect

For the estimation of causal effects, as opposed to the definition ofcausal effects, we will need to make different comparisons from thecomparisons made for their definitions.





For estimation and inference, we need to compare observedoutcomes, that is, observed realizations of potential outcomes.






Because there is only one realized potential outcome per unit, we willneed to consider multiple units.







For estimation it will also be critical to know about, or makeassumptions about, the reason why certain potential outcomes wererealized and not others.







For estimation it will also be critical to know about, or makeassumptions about, the reason why certain potential outcomes wererealized and not others.

That is, we will need to think about the assignment mechanism.


Estimation of causality Multiple units

Multiple units

Because with a single unit we can at most observe a single potentialoutcome, we must rely on multiple units to make causal inferences.



Multiple units


More specifically, we must observe multiple units, some exposed tothe active treatment, some exposed to the alternative (control)treatment.



Multiple units



One option is to observe the same physical object under differenttreatment levels at different points in time.Myself at different times, taking and not taking aspirin.



Multiple units



One option is to observe the same physical object under differenttreatment levels at different points in time.Myself at different times, taking and not taking aspirin.

Alternatively, one might observe different physical objects atapproximately the same time.If both you and I have headaches, but only one of us takes an aspirin, we mayattempt to infer the efficacy of taking aspirin by comparing our subsequentheadaches.



Solutions to fundamental problem of causality

Holland (1986)




Holland (1986)

Scientific solution

The scientific solution is to exploit various homogeneity or invarianceassumptions.




Holland (1986)

Scientific solution

The scientific solution is to exploit various homogeneity or invarianceassumptions.

Statistical solution

The statistical solution makes use of the population in a typicallystatistical way.



Notation

The focus is a population of units, indexed by i = 1, ...,N.



Notation


Each unit in this population can be exposed to one of a set oftreatments T = {0, 1}.



Notation


Each unit in this population can be exposed to one of a set oftreatments T = {0, 1}.

For each unit we therefore have two potential outcomes (one for eachlevel of the treatment):

Yi(0) if T = 0Yi(1) if T = 1



Scientific solution

Temporal stability

Assume that the value of Yi(0) does not depend on when the sequence“apply T = 0 to i then measure Y on i” occurs.The same assumption holds for treatment T = 1.



Scientific solution

Temporal stability


Causal transience

The value of Y (1) is not affected by the prior exposure of unit i to T = 0.



Scientific solution

Temporal stability


Causal transience


⇒ If these two assumptions can be met, then causal inference can bemeasured by sequential exposure of i to T = 0 then T = 1, measuringY after each exposure.



Scientific solution

Temporal stability


Causal transience


⇒ If these two assumptions can be met, then causal inference can bemeasured by sequential exposure of i to T = 0 then T = 1, measuringY after each exposure.

Headache example: myself at different times, taking and not taking aspirin.I may become more or less sensitive to aspirin, evenings may differ from mornings, etc.



Scientific solution (cont.)

Unit homogeneity

Assume that Yi(0) = Yh(0) and Yi(1) = Yh(1) for i 6= h.The causal effect equals Yi(1) − Yh(0).




Unit homogeneity


⇒ If this two assumption can be met, then a comparison between twounits can be used to estimate the causal inference on any one unit.




Unit homogeneity


⇒ If this two assumption can be met, then a comparison between twounits can be used to estimate the causal inference on any one unit.

Headache example: if both you and I have headaches, but only one of us takes anaspirin, we may attempt to infer the efficacy of taking aspirin by comparing oursubsequent headaches.I may be more or less sensitive to aspirin, or I may have started with a more or lesssevere headache.




The statistical solution replaces the impossible-to-observe causal effect ofT = 1 on a specific unit i with the possible-to-estimate average causaleffect of T = 1 over a population of unit.





The average causal effect is the expected value of the differenceYi(1)− Yi(0) over the population of units i = 1, ...,N, i.e.:

E [Y (1)− Y (0)] = E [Y (1)] − E [Y (0)]






E [Y (1)− Y (0)] = E [Y (1)] − E [Y (0)]

⇒ Information on different units that can be observed can be used to gainknowledge about the causal effect.






E [Y (1)− Y (0)] = E [Y (1)] − E [Y (0)]

⇒ Information on different units that can be observed can be used to gainknowledge about the causal effect.

If some units are exposed to T = 0 they may be used to give information aboutE [Y (0)], and if other units are exposed to T = 1 they may be used to give informationabout E [Y (1)].


Estimation of causality Problems with multiple units: SUTVA

The Stable Unit Treatment Value Assumption (SUTVA)

In many situations it may be reasonable to assume that treatment appliedto one unit do not affect the outcome for another unit.




In many situations it may be reasonable to assume that treatment appliedto one unit do not affect the outcome for another unit.If we are in different locations and have no contact with each other, it is reasonable to assumethat whether you take an aspirin has no effect on the status of my headache.

But, this assumption might not hold if we are in the same location, and your behaviour affected

by whether you take an aspirin, may affect the status of my headache.




In many situations it may be reasonable to assume that treatment appliedto one unit do not affect the outcome for another unit.If we are in different locations and have no contact with each other, it is reasonable to assumethat whether you take an aspirin has no effect on the status of my headache.

But, this assumption might not hold if we are in the same location, and your behaviour affected

by whether you take an aspirin, may affect the status of my headache.

SUTVA, Rubin (1978)

The potential outcomes for any unit do not vary with the treatmentsassigned to other units, and, for each unit, there are no different forms orversions of each treatment level, which lead to different potentialoutcomes.



SUTVA (cont.)

Therefore, the SUTVA implies that:



SUTVA (cont.)


units do not interfere with one another;



SUTVA (cont.)



for each unit there is only a single version of each treatment level(ruling out, for example, that a particular individual could take aspirintablets of varying efficacy).



SUTVA (cont.)



for each unit there is only a single version of each treatment level(ruling out, for example, that a particular individual could take aspirintablets of varying efficacy).

⇒ These two elements of the stability assumption enable us to exploit thepresence of multiple units for estimating causal effects.



Exclusion restrictions

SUTVA is the first of a number of assumptions that are referred togenerally as exclusion restrictions: assumptions that rely on external,substantive, information to rule out the existence of a causal effect of aparticular treatment relative to an alternative.





In the aspirin example:





In the aspirin example:• we could exclude the possibility that your taking or not taking aspirin has any effecton my headache;





In the aspirin example:• we could exclude the possibility that your taking or not taking aspirin has any effecton my headache; • we could exclude the possibility that the aspirin tablets available tome are of different strengths.





In the aspirin example:• we could exclude the possibility that your taking or not taking aspirin has any effecton my headache; • we could exclude the possibility that the aspirin tablets available tome are of different strengths.

Note, that these assumptions are not directly informed by observationsthey are assumptions!! Therefore, they rely on previously acquiredknowledge for their justification.Causal inference is generally impossible without such assumptions, andthus it is critical to be explicit about their content and their justifications.



SUTVA: No Interference

The no-interference component of SUTVA is the assumption that thetreatment applied to one unit does not affect the outcome for other units.





Example on its reliability:






The study of the effect of different types of fertilizers in agriculturalexperiments on plot yields: take care to separate plots using “guardrows”;







in the headache example: people are paired, with each pair placed ina separate room;








immunizations against infectious diseases: redefine the unit to be thecommunity within which individuals interact;








immunizations against infectious diseases: redefine the unit to be thecommunity within which individuals interact;

large-scale job training programs ⇒ general equilibrium effect.



SUTVA: No Interference (cont.)

In economics the general equilibrium effect might be a seriousconcern.





A labour market program that affects the labour market outcomes foran individual potentially has an effect also on the labour marketoutcomes for others.






In a world with a fixed number of jobs, a training program could onlyredistribute the jobs.







In practice these general equilibrium effects may or not be a seriousproblems depending on the size of the indirect effect compared to thedirect one.








Solutions:








Solutions:

redefine the unit of interest;








Solutions:

redefine the unit of interest;

directly model the interactions (Manski, 1993).



SUTVA: No Hidden Variations of Treatments

The second component of SUTVA requires that an individual receiving aspecific treatment level cannot receive different forms of that treatment.






In the aspirin example: for the potential outcome with both of ustaking aspirin suppose that one of the tablets is old and no longercontains a fully effective dose, whereas the other is new and at fullstrength⇒ redefine the treatment as taking a randomly selected aspirin;






In the aspirin example: for the potential outcome with both of ustaking aspirin suppose that one of the tablets is old and no longercontains a fully effective dose, whereas the other is new and at fullstrength⇒ redefine the treatment as taking a randomly selected aspirin;

differences in the method of administering the treatment matter: theeffect of taking a drug for a particular individual may differ dependingon whether the individual was assigned to receive it or chose to takeit.



SUTVA: No Hidden Variations of Treatments (cont.)

The second component of SUTVA is again an exclusion restriction.





The label of the aspirin tablet cannot alter the potential outcome forany unit.






This assumption does not require that all forms of each level of thetreatment are identical across all units.







It requires only that unit i exposed to treatment level w specifies awell-defined potential outcome, Yi(w), for all i and w .








Solutions:








Solutions:

redefining the represented treatment levels to comprise a larger set oftreatments (Aspirin, Aspirin+, and no-aspirin);








Solutions:

redefining the represented treatment levels to comprise a larger set oftreatments (Aspirin, Aspirin+, and no-aspirin);

coarsening the outcome (dead or alive).



Summarizing the SUTVA

Summarizing, assessing the causal effect of a binary treatment requiresobserving more than a single unit, because we must have observations ofpotential outcomes under both treatments (those associated with thereceipt of the treatment on some units and those associated with noreceipt of it on some other units).





However, with more than one unit, we face two immediate complications.






1 there exists the possibility that the units interfere with one another;







2 because in multi-unit settings, we must have available more than onecopy of each treatment, we may face circumstances in which a unit’spotential outcome when receiving the same nominal level of atreatment could vary with different versions of that treatment.







2 because in multi-unit settings, we must have available more than onecopy of each treatment, we may face circumstances in which a unit’spotential outcome when receiving the same nominal level of atreatment could vary with different versions of that treatment.

Unless we restrict them by assumptions, combined with careful studydesign to make these assumptions more realistic, any causal inference willhave only limited credibility!



Example of potential outcomes and causal effects underSUTVA

Example with Two Units. Source: Imbens and Rubin (2015).



Notation




Notation


The treatment indicator Wi takes on the values 0 (the controltreatment, e.g., no aspirin) and 1 (the active treatment, e.g., aspirin).



Notation



For each unit we have one realized (and possibly observed) potentialoutcome.



Notation




For unit i , now i ∈ 1, ...,N , let Y obsi denote this realized (and possibly

observed) outcome:

Y obsi = Yi(Wi ) =

{

Yi(0) if Wi = 0Yi(1) if Wi = 1



Notation




For unit i , now i ∈ 1, ...,N , let Y obsi denote this realized (and possibly

observed) outcome:

Y obsi = Yi(Wi ) =

{


For each unit we also have one missing potential outcome, for unit idenoted by Ymis

i :

Ymisi = Yi(1−Wi ) =

{



Estimation of causality Problems with multiple units: assignment mechanism

The assignment mechanism

The information on observed and missing potential outcome does notallow us to infer the causal effect.





In the headache example:






Suppose that the person who chose not to take the aspirin did so because he hadonly a minor headache.







Suppose then that an hour later both headaches have faded: the headache for thefirst person possibly faded because of the aspirin, and the headache of the secondperson faded simply because it was not a serious headache.








When comparing these two observed potential outcomes, we might conclude thatthe aspirin had no effect, whereas in fact it may have been the cause of easing themore serious headache.








When comparing these two observed potential outcomes, we might conclude thatthe aspirin had no effect, whereas in fact it may have been the cause of easing themore serious headache.

⇒ The key piece of information that we lack is how each individual cameto receive the treatment level actually received, i.e. the assignmentmechanism.



The assignment mechanism (cont.)

Because causal effects are defined by comparing potential outcomes(only one of which can ever be observed), they are well definedirrespective of the actions actually taken.





But, because we observe at most half of all potential outcomes, andnone of the unit-level causal effects, there is an inferential problemassociated with assessing causal effects.






Given any treatment assigned to an individual unit, the potentialoutcome associated with any alternate treatment is missing.







A key role is therefore played by the missing data mechanism, that isthe assignment mechanism.







A key role is therefore played by the missing data mechanism, that isthe assignment mechanism.

How is it determined which units get which treatments or,equivalently, which potential outcomes are realized and which are not?




Consider the example in Imbens and Rubin (2015):





4 patients (units).





4 patients (units).

Two possible medical procedures (treatment) labelled 0 (Drug) and 1(Surgery).





4 patients (units).


Years of post-treatment survival. Source: Imbens and Rubin (2015).





4 patients (units).


Years of post-treatment survival. Source: Imbens and Rubin (2015).

On average, Surgery is better than Drug by two years’ life expectancy.Fiaschi-Parenti The Economics of European Regions 43 / 50



Suppose now that the doctor knows enough about these potentialoutcomes and so assigns each patient to the treatment that is more

beneficial to that patient.






Assignment mechanism. Source: Imbens and Rubin (2015).







• Average observed outcome with surgery: (7+5)/2=6







• Average observed outcome with surgery: (7+5)/2=6• Average observed outcome with the drug treatment: (6+8)/2=7




The average observed outcome with surgery (6) is one year less thanthe average observed outcome with the drug treatment (7).





This might be led to believe that, on average, the drug treatment issuperior to surgery.






In fact, the opposite is true!







If the drug treatment were uniformly applied to a population of thesefour patients, the average survival would be four years, as opposed tosix years if all patients were treated with surgery!







If the drug treatment were uniformly applied to a population of thesefour patients, the average survival would be four years, as opposed tosix years if all patients were treated with surgery!

⇒ we cannot simply look at the observed values of potential outcomesunder different treatments and reach valid causal conclusions irrespectiveof the assignment mechanism!


Estimation of causality Problems with multiple units: pre-treatment variables

Pre-treatment variables

Consider a study of causal effects involving many units, satisfying theSUTVA.





At least half of all potential outcomes will be unobserved or missing,because only one potential outcome can be observed for each unit(i.e., the one corresponding to the realized level of the treatment).






To estimate the causal effect for any particular unit, we need topredict the missing potential outcome.







Comparing the predicted missing outcome to the realized and

observed outcome for this unit allows us to estimate the unit-levelcausal effect.







Comparing the predicted missing outcome to the realized and

observed outcome for this unit allows us to estimate the unit-levelcausal effect.

To make these predicted values we generally use unit-specificpre-treatment variables, or covariates, Xi (1× K ).



Pre-treatment variables (cont.)

Examples:




Examples:

In the headache example pre-treatment variables could include the intensity of theheadache before making the decision to take aspirin or not.




Examples:


In the evaluation of job training on future earnings pre-treatment variables couldinclude age, previous educational achievement, family, and socio-economic status,or pre-training earnings.




Examples:



Sometimes a covariate (e.g., pre-training earnings) differs from thepotential outcome (post-training earnings) solely in the timing ofmeasurement, in which case the covariates can be highly predictive of thepotential outcomes.




Examples:



Sometimes a covariate (e.g., pre-training earnings) differs from thepotential outcome (post-training earnings) solely in the timing ofmeasurement, in which case the covariates can be highly predictive of thepotential outcomes.

The key characteristic of these covariates is that they are a priori knownto be unaffected by the treatment assignment.

They are permanent characteristics of units, or they took on their values prior to thetreatment being assigned




The information available in these covariates can be used in three ways.





1 Covariates commonly serve to make estimates more precise byexplaining some of the variation in outcomes.Holding constant the intensity of the headache before receiving the treatment by

studying units with the same initial headache intensity should give more precise

estimates of the effect of aspirin.





1 Covariates commonly serve to make estimates more precise byexplaining some of the variation in outcomes.Holding constant the intensity of the headache before receiving the treatment by

studying units with the same initial headache intensity should give more precise

estimates of the effect of aspirin.

2 The researcher may be interested in the typical (e.g., average) causaleffect of the treatment on subgroups (as defined by a covariate) inthe population of interest.To evaluate the effects of a job-training program separately for people withdifferent education levels.




3 They have an effect on the assignment mechanism: assumptionsabout the assignment mechanism and its independence on potentialoutcomes are more plausible within subpopulations that arehomogeneous with respect to some covariates, that is, conditionallygiven the covariates, rather than unconditionally.




3 They have an effect on the assignment mechanism: assumptionsabout the assignment mechanism and its independence on potentialoutcomes are more plausible within subpopulations that arehomogeneous with respect to some covariates, that is, conditionallygiven the covariates, rather than unconditionally.

Young unemployed individuals may be more interested in training programs aimedat acquiring new skills.As a result, those taking the active treatment may differ in the values of theirbackground characteristics from those taking the control treatment.At the same time, these characteristics may be associated with the potentialoutcomes.


Summary

Summarizing: the framework for causal inference

In this lectures we have seen the three basic concepts for causal inference:


Summary



1 The potential outcomes, one for each unit for each level of thetreatment. Causal estimands are defined in terms of these potentialoutcomes, possibly also involving the treatment assignments andpre-treatment variables.


Summary




2 Because at most only one of the potential outcomes can be observed,there is a need for observing multiple units to be able to conductcausal inference. In order to exploit the presence of multiple units, weuse the stability assumption, SUTVA.


Summary




2 Because at most only one of the potential outcomes can be observed,there is a need for observing multiple units to be able to conductcausal inference. In order to exploit the presence of multiple units, weuse the stability assumption, SUTVA.

3 Finally the assignment mechanism, which determines which unitsreceive which treatment.


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