Introduction to Econometrics- Stock & Watson -Ch 11 Slides.doc

8/14/2019 Introduction to Econometrics- Stock & Watson -Ch 11 Slides.doc

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Experiments and Quasi-Experiments

(SW Chapter 11)

Why study experiments? Ideal randomized controlled experiments provide a benchmark for assessing observational studies.

Actual experiments are rare ($$$ but influential. !xperiments can solve the threats to internal validity

of observational studies" but they have their o#n

threats to internal validity.

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&hinking about experiments helps us to understand'uasi%experiments" or natural experiments") in #hichthere some variation is as if) randomly assigned.

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Terminology: experiments and quasi-experiments An experiment is designed and implemented

consciously by human researchers. An experimententails conscious use of a treatment and control group#ith random assignment (e.g. clinical trials of a drug

A quasi-experiment or natural experiment has a

source of randomization that is as if) randomlyassigned" but this variation #as not part of a consciousrandomized treatment and control design.

Program evaluation is the field of statistics aimed atevaluating the effect of a program or policy" forexample" an ad campaign to cut smoking.

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Di erent types o experiments: three examples

,linical drug trial- does a proposed drug lo#ercholesterol?

o Y cholesterol levelo X treatment or control group (or dose of drug

/ob training program (/ob &raining 0artnership Acto Y has a 1ob" or not (or Y #age incomeo X #ent through experimental program" or not

,lass size effect (&ennessee class size experimento Y test score (2tanford Achievement &est

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o X class size treatment group (regular" regular 4

aide" small

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!ur treatment o experiments: "rie outline

Why (precisely do ideal randomized controlledexperiments provide estimates of causal effects?

What are the main threats to the validity (internal andexternal of actual experiments 6 that is" experiments

actually conducted #ith human sub1ects? 7la#s in actual experiments can result in X and u being

correlated (threats to internal validity .

2ome of these threats can be addressed using theregression estimation methods #e have used so far-multiple regression" panel data" I8 regression.

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#deali$ed Experiments and Causal E e%ts(SW Se%tion 11&1)

An ideal randomized controlled experiment randomlyassigns sub1ects to treatment and control groups.

:ore generally" the treatment level X is randomlyassigned-

Y i ; 4 X i 4 ui

If X is randomly assigned (for example by computer

then u and X are independently distributed and E (ui< X i ;" so =>2 yields an unbiased estimator of .

&he causal effect is the population value of in an

ideal randomized controlled experiment %


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Estimation o %ausal e e%ts in an ideal randomi$ed%ontrolled experiment

@andom assignment of X implies that E (ui< X i ;. &hus the =>2 estimator $A is unbiased.

When the treatment is binary" $A is 1ust the difference

in mean outcome ( Y in the treatment vs. controlgroup ( treated Y 6 control Y .

&his differences in means is sometimes called the

differences estimator &

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'otential 'ro"lems ith Experiments in 'ra%ti%e(SW Se%tion 11& )

Threats to #nternal *alidity. Failure to randomize (or imperfect randomization

for example" openings in 1ob treatment program arefilled on first%come" first%serve basisC latecomers arecontrols

result is correlation bet#een X and u

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Threats to internal +alidity, %td&

*. Failure to follow treatment protocol (or partialcompliance

some controls get the treatment some treated) get controls

errors%in%variables) bias- corr( X "u ; Attrition (some sub1ects drop out suppose the controls #ho get 1obs move out of to#nC

then corr( X "u ;

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Threats to internal +alidity, %td&.& Experimental effects

experimenter bias (conscious or subconscious -treatment X is associated #ith extra effort) or

extra care") so corr( X "u ; sub1ect behavior might be affected by being in an

experiment" so corr( X "u ; (Ea#thorne effect

/ust as in regression analysis #ith observational data"

threats to the internal validity of regression #ithexperimental data implies that corr( X "u ; so =>2 (thedifferences estimator is biased./eorge Elton 0ayo and the a thorne Experiment

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2ub1ects in the Ea#thorne plant experiments" D*3 6 D+*

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Threats to External *alidity

. Fonrepresentative sample*. Fonrepresentative treatment) (that is" program or

policy+. General e'uilibrium effects (effect of a program can

depend on its scaleC admissions counseling 3. &reatment v. eligibility effects (#hich is it you #ant

to measure- effect on those #ho take the program" orthe effect on those are eligible

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2egression Estimators o Causal E e%ts 3singExperimental Data

(SW Se%tion 11&.) 7ocus on the case that X is binary (treatmentHcontrol . =ften you observe sub1ect characteristics" W i" " W ri. !xtensions of the differences estimator-

o can improve efficiency (reduce standard errorso can eliminate bias that arises #hen-

treatment and control groups differ

there is conditional randomization)there is partial compliance

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&hese extensions involve methods #e have alreadyseen 6 multiple regression" panel data" I8 regression

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Estimators o the Treatment E e%t 1 using

Experimental Data ( X 4 1 i treated, 5 i %ontrol)

Dep&+"le

#nd&+"le(s)

method

differences Y X =>2

differences%in%differences

Y

Y after 6 Y before

X =>2 ad1usts for initialdifferences bet#eentreatment and control

groups

differences #ithaddJl regressors

Y X "W "" W n

=>2 controls foradditional sub1ectcharacteristics W

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Estimators ith experimental data, %td&Dep&+"le

#nd&+"le(s)

method

differences%in%differences #ithaddJl regressors

Y Y after 6 Y before

X "W "" W n

=>2 ad1usts for groupdifferences 4 controlsfor sub1ect charJs W

Instrumentalvariables

Y X &2>2 Z initial randomassignmentC

eliminates bias from partial compliance

&2>2 #ith Z initial random assignment also can beapplied to the differences%in%differences estimator andthe estimators #ith additional regressors ( W Js

The di eren%es-in-di eren%es estimator%


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2uppose the treatment and control groups differsystematicallyC maybe the control group is healthier(#ealthierC better educatedC etc.

&hen X is correlated #ith u" and the differencesestimator is biased.

&he differences%in%differences estimator ad1usts for pre%experimental differences by subtracting off eachsub1ectJs pre%experimental value of Y

o beforeiY value of Y for sub1ect i before the expt

o after iY value of Y for sub1ect i after the expt

o Y i after iY 6before

iY change over course of expt

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$diffs in diffs ( "treat after Y 6 "treat beforeY 6 ( "control after Y 6 "control beforeY

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The di eren%es-in-di eren%es estimator, %td .

( Kifferences) formulation-

Y i ; 4 X i 4 ui

#hereY i after iY 6

beforeiY

X i if treated" ; other#ise

$

A is the diffs%in%diffs estimator

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The di eren%es-in-di eren%es estimator, %td .

(* !'uivalent panel data) version-

Y it ; 4 X it 4 * D it 4 +G it 4 vit " i " " n

#here

t (before experiment " * (after experiment D it ; for t " for t *G it ; for control group" for treatment group

X it if treated" ; other#ise

D it G it interaction effect of being in treatment group in the second period

$

A is the diffs%in%diffs estimator %*


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#n%luding additional su"6e%t %hara%teristi%s ( W 7s)

&ypically you observe additional sub1ect characteristics"W i" " W ri

Kifferences estimator #ith addJl regressors-

Y i ; 4 X i 4 *W i 4 4 r 4 W ri 4 ui

Kifferences%in%differences estimator #ith W Js-

Y i ; 4 X i 4 *W i 4 4 r 4 W ri 4 ui

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#here Y i after iY 6before

iY .

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Why in%lude additional su"6e%t %hara%teristi%s ( W 7s)8

. Efficiency - more precise estimator of (smallerstandard errors

*. Check for randomization . If X is randomly assigned"then the =>2 estimators #ith and #ithout the W Js

should be similar 6 if they arenJt" this suggests that X #asnJt randomly designed (a problem #ith the expt.

Note - &o check directly for randomization"regress

X on the

W Js and do a

%test.

+. !d"ust for conditional randomization (#e$ll return tothis later%

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Estimation hen there is partial %omplian%e,onsider diffs%in%diffs estimator" X actual treatment

Y i ; 4 X i 4 ui 2uppose there is partial compliance- some of the

treated donJt take the drugC some of the controls go to 1ob training any#ay

&hen X is correlated #ith u" and =>2 is biased 2uppose initial assignment" Z " is random &hen ( corr( Z " X ; and (* corr( Z "u ;

&hus can be estimated by &2>2" #ith instrumentalvariable Z initial assignment

&his can be extended to W Js (included exog. variables

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Experimental Estimates o the E e%t o 2edu%tion: The Tennessee Class Si$e Experiment

(SW Se%tion 11&9)0ro1ect 2&A@ (2tudent%&eacher Achievement @atio

3%year study" $ * million Lpon entering the school system" a student #as

randomly assigned to one of three groups-o regular class (** 6 *5 studentso regular class 4 aide

o small class ( + 6 students regular class students re%randomized after first year to

regular or regular4aide Y 2tanford Achievement &est scores

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De+iations rom experimental design

0artial compliance-o ;M of students s#itched treatment groups because

of incompatibility) and behavior problems) 6 ho#much of this #as because of parental pressure?

o Fe#comers- incomplete receipt of treatment forthose #ho move into district after grade

Attrition

o students move out of districto students leave for privateHreligious schools

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2egression analysis

&he differences) regression model-Y i ; 4 &mallClass i 4 * 'eg!ide i 4 ui

#here&mallClass i if in a small class

'eg!ide i if in regular class #ith aide

Additional regressors ( W Js

o teacher experienceo free lunch eligibilityo gender" race

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Kifferences estimates (no W Js

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o "ig are these estimated e e%ts8 0ut on same basis by dividing by std. dev. of Y Lnits are no# standard deviations of test scores

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Eo# do these estimates compare to those from the,alifornia" :ass. observational studies? (,h. 3 6

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Summary: The Tennessee Class Si$e Experiment

@emaining threats to internal validity partial complianceHincomplete treatment

o can use &2>2 #ith Z initial assignmento &urns out" &2>2 and =>2 estimates are similar

(Nrueger ( DDD " so this bias seems not to be large

:ain findings-

&he effects are small 'uantitatively (same size asgender difference

!ffect is sustained but not cumulative or increasing biggest effect at the youngest grades

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What is the Di eren%e et een a Control *aria"leand the *aria"le o #nterest8

(SW ;pp& 11&.)

!xample- free lunch eligible) in the 2&A@ regressions ,oefficient is large" negative" statistically significant

0olicy interpretation- (aking students ineligible for a free school lunch #ill im)rove their test scores*

Is this really an estimate of a causal effect?

Is the =>2 estimator of its coefficient unbiased? ,an it be that the coefficient on free lunch eligible)

is biased but the coefficient on &mallClass is not?

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E+am)le - free lunch eligible") ctd. ,oefficient on free lunch eligible) is large" negative"

statistically significant 0olicy interpretation- (aking students ineligible for a free school lunch #ill im)rove their test scores*

Why (precisely can #e interpret the coefficient on

&mallClass as an unbiased estimate of a causal effect" but not the coefficient on free lunch eligible)?

&his is not an isolated exampleO

o =ther control variables) #e have used- gender"race" district income" state fixed effects" time fixedeffects" city (or state population"

What is a control variable) any#ay?%+


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Simplest %ase: one X , one %ontrol +aria"le W

Y i ; 4 X i 4 *W i 4 ui


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7urther suppose W is correlated #ith u

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Y i ; 4 X i 4 *W i 4 ui&u))ose- &he control variable W is correlated #ith u Given W ; (ineligible " X is randomly assigned Given W (eligible " X is randomly assigned.

,hen- Given the value of W " X is randomly assignedC &hat is" controlling for W " X is randomly assignedC

&hus" controlling for W " X is uncorrelated #ith u :oreover" E (u< X "W doesnJt depend on X &hat is" #e have conditional mean independence:

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E( X"W E( W


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E (u< X "W E (u


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#mpli%ations o %onditional mean independen%e

Y i ; 4 X i 4 *W i 4 ui

2uppose E (u


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#mpli%ations o %onditional mean independen%e: &he conditional mean of Y given X and W is

E (Y i< X i"W i ( ; 4 ; 4 X i 4 ( 4 * W i &he effect of a change in X under conditional mean

independence is the desired causal effect-

E (Y i< X i +4 +"W i 6 E (Y i< X i +"W i +or

( < " ( < "i i i i i i E Y X + + W E Y X + W

+

= + =

If X is binary (treatmentHcontrol " this becomes-

( < " ( < ;"i i i i i i E Y X W E Y X W

+

= =

#hich is the desired treatment effect.%3+

# li% i % di i l i d d % % d&


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#mpli%ations o %onditional mean independen%e, %td&

Y i ; 4 X i 4 *W i 4 ui

,onditional mean independence says- E (u< X "W E (u


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So, hat is a %ontrol +aria"le8

A control variable W is a variable that results in X satisfying the conditional mean independence condition-

E (u< X "W E (u


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Example : E e%t o tea%her experien%e on test s%ores:ore on the design of 0ro1ect 2&A@-

&eachers didnJt change school because of the expt. Within their normal school" teachers #ere randomly

assigned to smallHregularHreg4aide classrooms. What is the effect of X years of teacher education?

,he design im)lies conditional mean inde)endence - W school binary indicator

Given W (school " X is randomly assigned &hat is" E (u< X "W E (u


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W is plausibly correlated #ith u (nonzero school fixedeffects- some schools are betterHricherHetc than others

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%3B

Example : tea%her experien%e %td&


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Example : tea%her experien%e, %td& Without school fixed effects (* " the estimated effect of

an additional year of experience is .3 ( &E . 0Controlling for the school ) (+ " the estimated effect of

an additional year of experience is . 3 ( &E . Kirection of bias makes sense-

o less experienced teachers at #orse schoolso years of experience picks up this school effect

=>2 estimator of coefficient on years of experience is

biased up #ithout school effectsC #ith school effects"=>2 yields unbiased estimator of causal effect

2chool effect coefficients donJt have a causalinterpretation (effect of student changing schools

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Quasi Experiments


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Quasi-Experiments(SW Se%tion 11&=)

A quasi-experiment or natural experiment has a sourceof randomization that is as if) randomly assigned" butthis variation #as not part of a conscious randomized

treatment and control design.

o cases-(a &reatment ( X is as if) randomly assigned (=>2(b A variable ( Z that influences treatment ( X is

as if) randomly assigned (I8

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T o types o quasi experiments


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T o types o quasi-experiments

(a &reatment ( X is as if) randomly assigned (perhaps

conditional on some control variables W E+- !ffect of marginal tax rates on labor supply

o X marginal tax rate (rate changes in one state"

not anotherC state is as if) randomly assigned

(b A variable ( Z that influences treatment ( X isas if) randomly assigned (I8 !ffect on survival of cardiac catheterization

X cardiac catheterizationC Z differential distance to ,, hospital

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E%onometri% methods


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E%onometri% methods(a &reatment ( X is as if) randomly assigned (=>2

Kiffs%in%diffs estimator using panel data methods-

Y it ; 4 X it 4 * D it 4 +G it 4 uit " i " " n#here

t (before experiment " * (after experiment D it ; for t " for t *G it ; for control group" for treatment group

X it if treated" ; other#ise

D it G it interaction effect of being in treatmentgroup in the second period

$

A is the diffs%in%diffs estimator%5*

The panel data di s in di s estimator simpli ies to


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The panel data di s-in-di s estimator simpli ies tothe >%hanges di s-in-di s estimator hen T 4

Y it ; 4 X it 4 * D it 4 +G it 4 uit " i " " n (P

7or t - D i ; and X i ; (nobody treated " so

Y i ; 4 +G i 4 ui

7or t *- D i* and X i* if treated" ; if not" soY i* ; 4 X i* 4 * 4 +G i* 4 ui*

so

Y i Y i* 6 Y i ( ; 4 X i*4 *4 +G i*4ui* 6 ( ; 4 +G i 4ui X i 4 * 4 ( ui 6 ui* (since G i G i*

or

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Y 4 X 4 v " #here v u 6 u (PP


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Y i * 4 X i 4 vi #here vi ui 6 ui* (PP

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Di eren%es-in-di eren%es ith %ontrol +aria"les


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Di eren%es-in-di eren%es ith %ontrol +aria les

Y it ; 4 X it 4 * D it 4 +G it 4 3W it 4 4 +4r W rit 4 uit "

X it if the treatment is received" ; other#ise

G it D it ( for treatment group in second period

If the treatment ( X is as if) randomly assigned"given W " then u is conditionally mean indep. of X -

E (u< X " D"G"W E (u< D"G"W

=>2 is a consistent estimator of " the causal effectof a change in X In general" the =>2 estimators of the other

coefficients do not have a causal interpretation.%55

(b A variable ( Z that influences treatment ( X is


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(b A variable ( Z that influences treatment ( X isas if) randomly assigned (I8

Y it ; 4 X it 4 * D it 4 +G it 4 3W it 4 4 +4r W rit 4 uit "

X it if the treatment is received" ; other#ise

G it D it ( for treatment group in second period Z it variable that influences treatment but is

uncorrelated #ith uit (given W Js&2>2-

X endogenous regressor D"G"W " " W r included exogenous variables Z instrumental variable

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'otential Threats to Quasi-Experiments


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otential Threats to Quasi Experiments(SW Se%tion 11&?)

&he threats to the internal +alidity of a 'uasi%

experiment are the same as for a true experiment" #ithone addition.3. Failure to randomize (imperfect randomization

Is the as if) randomization really random" so that X (or Z is uncorrelated #ith u?

=&Failure to follow treatment protocol attrition9. Experimental effects (not applicable

. !nstrument invalidit" (relevance 4 exogeneity(:aybe healthier patients do live closer to ,, hospitals

6they might have better access to care in general

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&he threats to the external +alidity of a 'uasi%


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&he threats to the external +alidity of a uasi%experiment are the same as for an observational study.5. Fonrepresentative sample

9. Fonrepresentative treatment) (that is" program or policy

E+am)le - ,ardiac catheterization &he ,, study has better external validity thancontrolled clinical trials because the ,, study usesobservational data based on real%#orldimplementation of cardiac catheterization.

Eo#ever that study used data from the early D;Js 6 do itsfindings apply to ,, usage today?

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Experimental and Quasi-Experiments Estimates in


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Experimental and Quasi Experiments Estimates ineterogeneous 'opulations

(SW Se%tion 11&@)

We have discussed the) treatment effect Qut the treatment effect could vary across individuals-

o !ffect of 1ob training program probably depends oneducation" years of education" etc.

o !ffect of a cholesterol%lo#ering drug could depend

other health factors (smoking" age" diabetes" If this variation depends on observed variables" thenthis is a 1ob for interaction variablesO

Qut #hat if the source of variation is unobserved?%5D

eterogeneity o %ausal e e%ts


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eterogeneity o %ausal e e%tsWhen the causal effect (treatment effect varies amongindividuals" the population is said to be #eterogeneous .

When there are heterogeneous causal effects that are notlinked to an observed variable-

What do #e #ant to estimate?o =ften" the average causal effect in the populationo Qut there are other choices" for example the average

causal effect for those #ho participate (effect oftreatment on the treated

What do #e actually estimate?o using =>2? using &2>2?

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'opulation regression model ith heterogeneous


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opulation regression model ith heterogeneous%ausal e e%ts-

Y i ; 4 i X i 4 ui" i " " n

i is the causal effect (treatment effect for the ith individual in the sample

7or example" in the /&0A experiment" i could be zeroif person i already has good 1ob search skills

What do #e #ant to estimate?o

effect of the program on a randomly selected person(the average causal effect ) 6 our main focus

o effect on those most (least? benefitedo effect on those #ho choose to go into the program?

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The ;+erage Causal E e%t


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The ; erage Causal E e%t

Y i ; 4 i X i 4 ui" i " " n

&he average causal effect (or average treatment effect is the mean value of i in the population.

We can think of as a random variable- it has adistribution in the population" and dra#ing a different

person yields a different value of (1ust like X and Y

7or example" for person R+3 the treatment effect is notrandom 6 it is her true treatment effect 6 but before she

%9*

is selected at random from the population" her value of


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p p

can be thought of as randomly distributed.

The a+erage %ausal e e%t, %td&

Y i ; 4 i X i 4 ui" i " " n

&he average causal effect is E ( . What does =>2 estimate-

(a When the conditional mean of u given X is zero?

(b Lnder the stronger assumption that X is randomlyassigned (as in a randomized experiment ?

1n this case2 34& is a consistent estimator ofthe average causal effect .

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!AS ith eterogeneous Causal E e%ts


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g

Y i ; 4 i X i 4 ui" i " " n (a 2uppose E (ui< X i ; so cov( ui" X i ;.

If X is binary (treatedHuntreated " $A treated Y 6 control Y

estimates the causal effect among those #ho receivethe treatment.

Why? 7or those treated" treated Y reflects the effect ofthe treatment on them . Qut #e donJt kno# ho# theuntreated #ould have responded had they beentreatedO

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T#e mat# - suppose X is binary and E (ui< X i ;.


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pp y ( ;&hen

$A treated Y 6 control Y

7or the treated-

E (Y i< X i ; 4 E ( i X i< X i 4 E (ui< X i

; 4 E ( i< X i

7or the controls- E (Y i< X i ; ; 4 E ( i X i< X i ; 4 E (ui< X i ;

;

&hus-$

A )

E (Y i< X i 6 E (Y i< X i ; E ( i< X i

average effect of the treatment on the treated

%95

!AS ith heterogeneous treatment e e%ts: general X


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g gith E (u i B X i ) 4 5

$

A *

XY

X

s

s

)

* XY

X

; $cov( "

var(

i i i i

i

X u X

X

+ +

; $cov( " cov( " cov( "

var(i i i i i i

i

X X X u X

X

+ +

$cov( "

var(i i i

i

X X

X

(because cov( ui" X i ;

If X is binary" this simplifies to the effect oftreatment on the treated)

Without heterogeneity" i and $A )

In general" the treatment effects of individuals #ithlarge values of X are given the most #eight

%99

(b Fo# make a stronger assumption- that X is randomly


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( g p yassigned (experiment or 'uasi%experiment . &hen#hat does =>2 actually estimate?

I X i is randomly assigned" it is distributedindependently of i" so there is no difference

bet#een the population of controls and the

population in the treatment group &hus the effect of treatment on the treated the

average treatment effect in the population.

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&he math-


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$A

)

$cov( "

var(i i i

i

X X

X

$ $

cov( "2 estimates the average treatment effect.

If X i is not randomly assigned but E (ui< X i ;" =>2estimates the effect of treatment on the treated.

%9B

Without heterogeneity2 the effect of treatment on the


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g y

treated and the average treatment effect are the same#* 2egression ith eterogeneous Causal E e%ts

2uppose the treatment effect is heterogeneous and theeffect of the instrument on X is heterogeneous-

Y i ; 4 i X i 4 ui (e'uation of interest

X i ; 4 i Z i 4 vi (first stage of &2>2

In general" &2>2 estimates the causal effect for those#hose value of X (probability of treatment is mostinfluenced by the instrument.

%9D

#* ith heterogeneous %ausal e e%ts, %td&


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Intuition-

2uppose iJs #ere kno#n. If for some people i ;" then their predicted value of X i #ouldnJt dependon Z " so the I8 estimator #ould ignore them.

&he I8 estimator puts most of the #eight onindividuals for #hom Z has a large influence on X .

&2>2 measures the treatment effect for those #hose probability of treatment is most influenced by X .

% ;

,he math


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&o simplify things" suppose- i and i are distributed independently of ( ui"vi" Z i

E (ui< Z i ; and E (vi< Z i ; E ( i ;

&hen $,&4&

)

$ $

$

(

(

i i

i

E

E

(derived in 2W App. .3

&2>2 estimates the causal effect for those individualsfor #hom Z is most influential (those #ith large i .

%

When there are heterogeneous %ausal e e%ts, hat


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TSAS estimates depends on the %hoi%e o instruments With different instruments" &2>2 estimates different

#eighted averagesOOO 2uppose you have t#o instruments" Z and Z *.

o In general these instruments #ill be influential for

different members of the population.o Lsing Z " &2>2 #ill estimate the treatment effect for

those people #hose probability of treatment ( X ismost influenced by Z

o &he treatment effect for those most influenced by Z

might differ from the treatment effect for those mostinfluenced by Z *

% *

When does TSAS estimate the a+erage %ausal e e%t8


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$,&4&

)

$ $

$

((

i i

i

E E

&2>2 estimates the average causal effect (that is"$,&4&

) E ( i if-

o If i and i are independent

o If i (no heterogeneity in e'uation of interesto If i (no heterogeneity in first stage e'uation

Qut in general $,&4& does not estimate E ( i O

% +

Example - ,ardiac catheterization


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Y i survival time (days for A:I patients X i received cardiac catheterization (or not

Z i differential distance to ,, hospital

!'uation of interest-

&urvivalDays i ; 4 iCardCath i 4 ui7irst stage ( linear )robability model -

CardCath i ; 4 i Distance i 4 vi

7or #hom does distance have the great effect on the probability of treatment?

7or those patients" #hat is their causal effect i?% 3

!'uation of interest-


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&urvivalDays i ; 4 iCardCath i 4 ui7irst stage ( linear )robability model -

CardCath i ; 4 i Distance i 4 vi

&2>2 estimates the causal effect for those #hose

value of X i is most heavily influenced by Z i &2>2 estimates the causal effect for those for #hom

distance most influences the probability of treatment

What is their causal effect? ( We might as #ell go tothe ,, hospital" its not too much farther)

&his is one explanation of #hy the &2>2 estimate issmaller than the clinical trial =>2 estimate.

% 5

eterogeneous Causal E e%ts: Summary


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Eeterogeneous causal effects means that the causal (ortreatment effect varies across individuals.

When these differences depend on observable variables"heterogeneous causal effects can be estimated using

interactions (nothing ne# here . When these differences are unobserved ( i the average

causal (or treatment effect is the average value in the

population" E ( i . When causal effects are heterogeneous" =>2 and &2>2

estimate .

% 9

!AS ith eterogeneous Causal E e%ts


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X is: 2elation "et een X i andu i :

Then !AS estimates:

binary E (ui< X i ; effect of treatment on thetreated- E ( i< X i

X randomly assigned (so X i and ui are independent

average causal effect E ( i

general E (ui< X i ; #eighted average of i" placing most #eight on

those #ith large < X i 6 X 2 also estimates a causal effect- the average

effect of treatment on those most influenced by theinstrument

o In general" this is neither the average causal effect

nor the effect of treatment on the treated% B

Summary: Experiments and Quasi-Experiments


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(SW Se%tion 11& )

Experiments: Average causal effects are defined as expected values

of ideal randomized controlled experiments

Actual experiments have threats to internal validity &hese threats to internal validity can be addressed (in part by-

o panel methods (differences%in%differenceso multiple regressiono I8 (using initial assignment as an instrument

% D

Summary, %td&


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Quasi-experiments:

Suasi%experiments have an as%if) randomly assignedsource of variation.

&his as%if random variation can generate-

o X i #hich satisfies E (ui< X i ; (so estimation proceeds using =>2 C or

o instrumental variable(s #hich satisfy E (ui< Z i ;

(so estimation proceeds using &2>2 Suasi%experiments also have threats to internal vaidity

%B;

Summary, %td&


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T o additional su"tle issues:

What is a control variable?o A variable W for #hich X and u are uncorrelated"

given the value of W (conditional mean

independence- E (ui< X i"W i E (ui


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What do =>2 and &2>2 estimate #hen there is

unobserved heterogeneity of causal effects? In general" #eighted averages of causal effects-

o If X is randomly assigned" then =>2 estimates the

average causal effect.o If X i is not randomly assigned but E (ui< X i ;" =>2

estimates the average effect of treatment on thetreated.

o If E (ui< Z i ;" &2>2 estimates the average effect of

treatment on those most influenced by Z i.

Documents

Introduction to Econometrics- Stock & Watson -Ch 11 Slides.doc