Efficient Semiparametric Estimation of Quantile Treatment Effects
Author(s): Sergio Firpo
Reviewed work(s):
Source: Econometrica, Vol. 75, No. 1 (Jan., 2007), pp. 259-276
Published by: The Econometric Society
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Vol.75,No. 1 (January,
Econometrica,
2007),259-276
EFFICIENT SEMIPARAMETRIC ESTIMATION OF
QUANTILE TREATMENT EFFECTS
BY SERGIOFIRPO'
forquantiletreatment
Thispaperdevelopsestimators
effects
undertheidentifying
is based on observablecharacteristics.
restriction
thatselectionto treatment
Identifiof theconditionalquantilesof the
cationis achievedwithoutrequiring
computation
resultsforthemarginalquantileslead
potentialoutcomes.Instead,theidentification
to an estimation
effectparameters
thathas two
procedureforthequantiletreatment
estimation
of thepropensity
scoreand computation
of thedifsteps:nonparametric
ferencebetweenthesolutionsoftwoseparateminimization
Root-Nconsisproblems.
and achievement
ofthesemiparametric
bound
tency,asymptotic
normality,
efficiency
A consistent
are shownforthatestimator.
estimation
procedureforthevarianceis also
themethoddevelopedhereis appliedto evaluation
ofa job training
presented.
Finally,
andto a MonteCarloexercise.Resultsfromtheempirical
indicate
program
application
thatthemethodworksrelatively
wellevenfora datasetwithlimitedoverlapbetween
treatedand controlsin thesupportofcovariates.The MonteCarlostudyshowsthat,
fora relatively
smallsamplesize,themethodproducesestimates
withgood precision
and lowbias,especially
formiddlequantiles.
KEYWORDS:
Quantiletreatment
effects,
score,semiparametric
propensity
efficiency
estimation.
bounds,efficient
estimation,
semiparametric
1. INTRODUCTION
IN PROGRAM EVALUATION STUDIES, it is oftenimportant
to learnaboutdis-
tributional
of theprogram.For example,
impactsbeyondthe averageeffects
a policy-maker
in theeffectof a treatment
on thedispermightbe interested
on thelowertailoftheoutcomedistribution.
sionofan outcomeor itseffect
One wayto capturethiseffectin a setting
withbinarytreatment
and scalar
outcomesis to computethequantilesofthedistribution
oftreatedandcontrol
outcomes.Usingquantiles,discretized
versionsof the distribution
functions
of treatedand controlcan be calculated.Quantilescan also be used to obtainmanyinequality
forinstance,
measurements,
quantileratios,interquantile
concentration
and
functions, theGinicoefficient.
ranges,
Finally,differences
'This paperis based on thesecondchapterofmyPh.D. dissertation
at UC-Berkeley,written underthe supervision
of Guido Imbensand JimPowell.I am indebtedto themfortheir
I am grateful
forhelpfulcommentsfromtwo anonyadvice,support,and manysuggestions.
mousrefereesand the co-editoron earlierdrafts.I have also benefited
fromcomments
from
MarceloFernandes,
FernandoFerreira,
CarlosFlores,GustavoGonzaga,Jinyong
Hahn,Michael
ShakeebKhan,DavidLee, ThomasLemieux,Thierry
Jansson,
Magnac,Rosa Matzkin,Dan McJamesRobins,Paul Ruud,Jeffrey
Fadden,JodoPinhode Mello,Deb Nolan,David Reinstein,
and participants
of seminarsat EPGE-FGV,Georgetown,
MSU, PUC-Rio,UBC,
Wooldridge,
ofMiami,University
ofToronto,
ofWashingUC-Berkeley,UC-Davis, University
University
ton,CESG-2004,andUSP-RP.Specialthanksgo to GeertRidderformanydetailedsuggestions.
All errorsare mine.
FinancialsupportfromCAPES-Brazil is acknowledged.
259
260
SERGIO FIRPO
in quantilesare important
becausetheeffects
ofa treatment
maybe heterogethe
outcome
distribution.
neous,varying
along
in thepresenceofheterogeneous
A parameter
ofinterest
treatment
effects
is thequantiletreatment
As
defined
Doksum
effect
originally
by
(QTE).
(1974)
foranyfixedpercentile,
to the
and Lehmann(1974), the QTE corresponds,
In definfunctions.
horizontaldistancebetweentwocumulativedistribution
effectat the individuallevel,bothDoksum and
ing QTE as the treatment
Lehmannimplicitly
arguedthatan observedindividualwould maintainhis
ofhistreatment
status.Thispaperwillrefer
rankinthedistribution
regardless
as a rankpreservation
to thistypeofassumption
assumption.
because it requiresthe relative
Rank preservation
is a strongassumption
value (rank)of the potentialoutcomefora givenindividualto be the same
or in the control
regardlessof whetherthatindividualis in the treatment
in
rank
are
two
to
deal
with
cases
which
There
ways
preservation
groups.
is an unreasonableassumption.
First,as suggestedbyHeckman,Smith,and
oftreatment
Clements(1997),one couldcomputeboundsforthedistribution
of reordering
of the ranks.Second,
effects,
allowingforseveralpossibilities
is violated,a meaningful
evenwhenrankpreservation
parameterforpolicy
inparameters
oftwomarginaldistripurposesmightbe thesimpledifference
and the distribution
of
butions:thedistribution
of outcomeundertreatment
outcomeundernontreatment.
inlearnis interested
Considerthelattercase,inwhichall thepolicy-maker
ofthepotentialoutcomes.A convenient
ingaboutthemarginaldistributions
is by computing
wayto summarizeinteresting
aspectsof thesedistributions
effects
are simpledifferences
theirquantiles.In thiscase, quantiletreatment
ofpotentialoutcomes.2
Howbetweenquantilesofthemarginaldistributions
in quantilesturn
ever,ifrankpreservation
holds,thenthesimpledifferences
effect.3
outto be thequantilesofthetreatment
withtheselectionon
ofquantiletreatment
Thisdefinition
effects,
together
that
allowsidentification
ofvariousQTE parameters
observablesassumption,
to whichtheyrefer.Followingthe approachof
differby the subpopulation
Heckmanand Robb (1986),Hahn (1998),Hirano,Imbens,and Ridder(2003;
willbe theprimary
henceforth
HIR), andImbens(2004),twoQTE parameters
in
are
labeled
the
overall
of
this
quantiletreatment
paper.They
object study
and
treatment
the
treated
the
on
effect
effect
quantile
(QTT); theformer
(QTE)
and the
is theQTE parameterforthewholepopulationunderconsideration
to
treatment.
Before
we
latteris theparameterforthoseindividuals
subject
introduce
some
useful
notation.
define
those
let
us
formally
parameters,
DefineT as theindicator
variableoftreatment.
Foran individual
i,ifTi = 1,
ifTi = 0,we observeYi(0). Here Yi(1) and Yi(0)
we observeYi(1); otherwise,
whicharetheobjectsofinterest
inquantileregression
meth2Notethatconditional
quantiles,
ods,are notapproachedinthispaper.
becausedifferences
oftheaveragetreatment
3Thereis no similarprobleminestimation
effect,
inmeansalwayscoincidewithmeansofdifferences.
QUANTILE TREATMENT EFFECTS
261
the potentialoutcomesof receivingand not receivingthe
are, respectively,
Whereasa givenindividuali is eithertreatedor not,we definethe
treatment.
observedoutcomeas Yi = Yi(1) - Ti + Yi(O) - (1 - Ti). Assumethatwe also
observea randomvectorXi ofcovariateswithsupportX c R'.
Let r be a real in (0, 1). The QTE and QTT parameterscan thenbe expressedas follows:
* The QTE is writtenas A, = q1,, - q0,7,whereqj,,- infqPr[Y(j)< q] > 7,
j = 0, 1.
* The QTT is written
as
= q1,7iT=1 - qo,71T=1, where qj,7lT=l
j= 0, 1.
infqPr[Y(j)< qIT = 1] > 7, AT=1
As is thecase foranytreatment
effectparameter,
identification
restrictions
In thispaper,therelevantrestriction
is
are necessary
forconsistent
estimation.
theassumption
thatselectiontotreatment
isbasedon observable
variables(exIn otherwords,itis assumedthatgivena setofobserved
ogeneity
assumption).
individuals
are randomly
covariates,
assignedeitherto thetreatment
groupor
to thecontrolgroup.Thatassumption
was termedbyRubin(1977) theunconanditcharacterizes
theselection
on observables
branch
assumption
foundedness
of the programevaluationliterature.
and
Barnow,Cain,
Goldberger(1980),
Heckman,Ichimura,Smith,andTodd(1998),Dehejia and Wahba(1999),and
HIR are important
examples.Furtherdiscussionoftheseidentifying
assumptionswillbe providedin a latersection.
Estimationof averagetreatment
effects(ATE) underthisexogeneity
asis
first
often
a
conditional
treatby
sumption
performed
computing
average
menteffectand thenintegrating
overthedistribution
ofcovariatesto recover
theunconditional
effect.
However,becausethemeanofthe
averagetreatment
to
is
not
the
of
the
a first-stage
commean,integrating
quantiles
equal
quantile
the
of
conditional
the
treated
and
the
control
putation
quantiles(of
outcomes)
willnotyieldthemarginalquantiles.Instead,thispaperdemonstrates
howto
use the selectionon observablesassumption
to calculatethe marginalquantilesforthetreatedand forthecontroloutcomeswithoutcomputing
thecorconditionalquantiles.The rolethattheobservablecovariatesplay
responding
in identifying
bothATE and QTE is made clearerin theQTE case, because
forthelatter,thecovariatesserveonlyto removetheselectionbias.
Despite the relevanceof QTE, the programevaluationliteratureon this
ATE. Traditionally,
extopic is not as vast as thatof its main competitor,
have
received
more
attention
in
the
literature
thanquantiles.Pipectations
suchas thosebyKoenkerandBassett
oneeringpaperson quantileestimation,
in
an
instrumental
variables
setting,
byAmemiya(1982) andPowell
(1978) and,
In
have
to
this
the
treatment
effects
some
literature,
helped bridge gap.
(1983),
havealso been madeto thestudyofthedistributional
recentcontributions
effectsofthetreatment.
and
Imbens
and
Amongthem,Abadie,Angrist,
(2002)
deal withthefactthattreatment
Chernozhukov
and Hansen (2005) explicitly
effects
andtheypropose
alongtheoutcomedistribution
maybe nonmonotonic
methodsto estimateQTE parameters.
262
SERGIO FIRPO
There are two important
differences
betweenthe approachesof Abadie,
and
Imbens
and
Chernozhukov
and Hansen (2005), and our
Angrist,
(2002)
The
first
difference
is
that
consider
a conditionalon covariapproach.
they
ates versionof QTE. As statedbefore,conditionalquantilesare notdirectly
usefulif we are interestedin quantilesof the marginaldistribution;
thereit
is
not
clear
how
to
recover
the
unconditional
from
their
QTE
fore,
settings.
The second difference
involvesthe choiceof the identifying
set of assumptions.In thispaperwe showhowto identify
underunconQTE parameters
whereas
and
Imbens
and
Chernozhukov
foundedness,
Abadie,Angrist,
(2002)
and Hansen (2005) showhowinstrumental
variablescan be used to identify
is based on unobservable
variables.HowQTE whenselectionto treatment
there
is
one
betweenAbadie,Angrist,
and Imbens
ever,
important
similarity
(2002) and ourmethod.Bothmethodsuse Koenkerand Bassett's(1978) representation
forquantilesas minimizers
ofexpectations
ofcheckfunctions
and
bothshowhowto identify
functions
to
QTE byintroducing
properweighting
thoseexpectations.
Because the QTE parametersidentified
in our workare
different
fromthosein Abadie,Angrist,and Imbens(2002), the
essentially
functions
willreflect
thisdifference.
weighting
an extension
to identification
andestimation
ofa classofparameRecently,
tersmoregeneralthanQTE, termedstructural
was proquantilefunctions,
posed by Imbensand Newey(2003) and appliedto the case of continuous
treatments.
UnlikeAbadie,Angrist,and Imbens(2002) and Chernozhukov
and Hansen (2005), who use instrumental
variables,Imbensand Newey's
identification
is
based
on
control
functions.
For a generalconstrategy
(2003)
trolvariable,thekeyidentifying
is
that
observable
and unobservrequirement
able factorsthatexplainthe responsevariableare independent,
giventhat
controlvariable.Withan additionalcommonsupportassumption,
theyshow
howto identify
theconditional
cumulative
distribution
function
(c.d.f.)ofthe
and
to
obtain
of
its
variable,
covariates,
response
given
quantiles
marginaldiswhichresultsfromintegrating
theconditional
c.d.f.and inverting
it.
tribution,
UnlikeImbensandNewey(2003),ouridentification
resultsareapplieddirectly
to thequantilesofthemarginaldistribution
we do notneed to
and,therefore,
workwithc.d.f.'sas a first
step.
Imbensand Rubin (1997) and Abadie (2002) proposedmethodsto estimate some distributional
featuresfora subsetof the treatedunits.Their
in
also
was
used
an instrumental
variablessettingand looked at
proposal
c.d.f.'s,not quantiles.Atheyand Imbens(2006) focusedon the effectsat
quantilesin the situationin whichrepeatedcross sectionsor longitudinal
data are available.Distributional
effects
havealso been studiedempirically
in
Card (1996),DiNardo,Fortin,and Lemieux(1996), and Bitler,Gelbach,and
in the paper by DiNardo,Fortin,and Lemieux,
Hoynes(2006). Particularly
effects
havebeen indirectly
quantiletreatment
computedas an estimation
byforthetreatedsubpopproductofnonparametric
potentialoutcomesdensities
ulation.
QUANTILE TREATMENT EFFECTS
263
is preeach QTE parameter
methodofestimating
Herein,a semiparametric
in
a
first
sented.Thisestimation
techniquerequires nonparametric step which
willbe equal to thedifThe finalestimators
scoreis estimated.
thepropensity
ferencesbetweentwoquantiles,whichcan be expressedas solutionsofminiwhich
wheretheminimands
are sumsofcheckfunctions,
mizationproblems,
the
consisare convexempiricalprocesses.Using empirical
processliterature,
effiresultsare derived.The semiparametric
normality
tencyand asymptotic
inNewey(1990) and
suggested
ciencyboundis computedusingthetechniques
Bickel,Klaassen,Ritov,andWellner(1993),anditisshownthattheasymptotic
varianceoftheQTE estimator
equals thatbound.
2. IDENTIFICATION OF QTE PARAMETERS
as p(x) and let the
Let thepropensity
score,Pr[T = 1IX = x], be written
=
=
as p.
of
marginalprobability beingtreated,Pr[T 1] E[p(X)], be written
is
used
here.
The following
assumption
identifying
ASSUMPTION
1-Strong Ignorability
(Rosenbaumand Rubin(1983)): Let
Then,forall x in X, thesupport
(Y(1), Y(0), T, X) havea jointdistribution.
hold:
ofX, thefollowing
conditions
Given X, (Y(1), Y(0)) is jointlyindependent
(i) Unconfoundedness:
fromT.
For somec > 0, c < p(x) < 1 - c.
(ii) Commonsupport:
it has been used
Althoughpart(i) ofAssumption1 is a strongassumption,
or programs.
examin severalstudieson the effectof treatments
Prominent
Smith,and Todd(1998) and Dehejia andWahba
ples are Heckman,Ichimura,
assign(1999). Part(ii) statesthatforalmostall valuesof X, bothtreatment
ofoccurrence.
mentlevelshavea positiveprobability
Now considereach one of the fourtypesof quantilesdefinedpreviously:
andqo,7Tr=1.
Wewillassume
that
forsomevalues
of7 E (0, 1),
q1,7, q0,T,
q1,-IT=1,
are welldefinedand thattherespective
thesequantiles
quantilesare unique.
thatthedistribution
functions
ofthepotentialoutcomes
We do so byassuming
are continuousand notflatat theT-percentile.
These conditions
are statedin
thefollowing
assumption:
ASSUMPTION
2-Uniqueness of Quantiles:For j = 0, 1, Y(j) is a continstatements
uous randomvariablewithsupportin R and wherethefollowing
apply:
(i) There are nonemptysets Y, and Y0o,such that Yj = {r e (0, 1);
Pr[Y(j) < qj,r- c] < Pr[Y(j) < qj,I + c], V c E R, c > 0}.
(ii) Therearenonempty
setsYr7l=land
= {7 E (0, 1);
suchthat
Yjlr=l
Y0or=l,
Pr[Y(j) < qj,, - cIT = 1] < Pr[Y(j) < qj,71T=l
+ clT = 1],
Vce R, c > 0}.
7=a
264
SERGIO FIRPO
1 and2,bothQTE andQTT becomeestimablefromthe
UnderAssumptions
data on (Y, T, X). To showthis,we first
provethatthequantilesofthepotentialoutcomedistributions
can be written
oftheobserved
as implicit
functions
data.4
LEMMA1-Identification
ofQuantiles:UnderAssumptions
1 and 2, ql,r,qo,,,
data: (i) 7 =
can
be
written
as
observed
and
of
implicit
functions
qo,,lT=1
ql,71T=l,
E[ TQ)-ff{Y q1,7] T E Y; (ii) = E 1T
1{Y<< qo,7, EE Y0;(iii) 7 =
E[ P 1{Y < ql,,1T=11],
p(X) . 1{ < qo,IT=11
q
Vr E Y1T=1;and(iv)7=
r E[1-T
p (1-p(X))
V7 E Y01T=1
Note that Assumption1 plays no role in the identification
of ql,71T=1.
Heckman,Ichimura,and Todd (1997) have alreadyshownan analogousreeffects
sultinthesearchforidentification
conditions
fortheaveragetreatment
is a straighton thetreated.Finally,identification
ofquantiletreatment
effects
forward
consequenceofLemma1,as statedinthenextcorollary.
COROLLARY1-Identificationof QuantileTreatment
EffectParameters:
1 and 2, thefollowing
UnderAssumptions
are
fromdata
parameters identified
n
on (Y, T, X): (i) theoverallquantiletreatment
A,for7 E Yj Y0;(ii) the
effect
on thetreated
A,7T=1
forr e Y1IT= n Yo0T=..5
effect
quantiletreatment
Estimationof the QTT parameterbased on the resultsof Lemma 1 has
beenusedintheappliedliterature.
DiNardo,Fortin,and Lemieux(1996) procounterfactual
of Y(0)|IT = 1 usinga reweighting
the
density
posedestimating
thatis similarto theresultforthe
methodthathas a populationcounterpart
of qo,I7T=1.Theyarguedthatonce thecounterfactual
identification
densityis
itis possibleto recoverthecounterfactual
and,
therefore,
estimated,
quantiles
betweenthequantilesofthetreatedgroupand thecounterfacthedifference
tual quantilesof the controlgroup.However,as is made clearby Lemma 1
iftheultimate
thereis no needto computedensitiesfirst
goal is theestimation
ofquantiles.
3. ESTIMATION
In thissection,we use thesampleanalogyprinciple(Manski(1988)) to moin solutionsof miniforA, and Ad7T=1 thatare differences
tivateestimators
and showhowto
We presenttheirlargesampleproperties
mizationproblems.
variancesconsistently.
estimatetheirasymptotic
to thisarticle(Firpo(2007)). The
4Allproofsof theresultscan be foundin thesupplement
whichwe briefly
also includesdetailsabouta MonteCarlostudy,
reportinSection5
supplement
ofthisarticle.
assumethatwhenwe considerA,,thepossible
therestofthepaper,we implicitly
5Throughout
for
7E
valuesof7willbe suchthatr E Y n Y0;analogously,
YoIrT=.
Ar7T=1,
Yir=1
QUANTILE TREATMENT EFFECTS
265
The estimation
inthesensethat
techniqueproposedhereis semiparametric
it does notimposeanyrestriction
on thejointdistribution
of (Y, T, X). It extendsto quantiletreatment
effects
thecharacteristics
previously
proposedfor
treatment
effects.
of
estimation
forATE
average
Examples semiparametric
can be foundin Hahn (1998), Heckman,Ichimura,Smith,and Todd (1998),
and HIR.
3.1. Two-Step
Approach
Usingtheidentification
expressionfromLemma 1,we presentherean estimationmethodthatis a reweighedversionof the procedureproposedby
Koenkerand Bassett(1978) forthequantileestimation
problem.Let theestimators
of functionals
of (Y, T, X) be denotedbya "hat."For example,the
estimator
ofthepropensity
scoreis p(x). The estimator
forthe
nonparametric
- q0,,, where,forj = 0, 1,
QTE parameterA, is A, 1,r
N
(1)
j,,
- arg min
i=1
,i
(Yi
-
q),
wherethecheckfunction
p,(.) evaluatedat a realnumbera is p,(a) = a -(7 wheretheweightsW2,ji
and w0,jare
?1{a< 0}) and,finally,
(2)
Ti
) 1,i
and UOO,i
=
1- T
N. p(Xi)
The definition
oftheestimator
inEquation(1) relieson thefactthatsample
can
be
found
a sumofcheckfunctions,
as pointedout
quantiles
byminimizing
case,we havea weightedsum
byKoenkerandBassett(1978). In ourparticular
ofcheckfunctions,
whichreflects
thefactthatthedistribution
ofthecovariates
differs
inthetwogroups.
We focuson the samplequantileof the Y(1) distribution
q1,,.This object is definedas the minimizerof a weightedsum,wherethe weightof
each unitis givenby W"1,;.
To get some intuitionon whyq1,, is consistent
forql,,, note thatan approximate
firstderivative
of Equation (1) usingthe
weightdefinedin thefirstpartof Equation(2) and evaluatedat q1,,is equal
to (1/N) ii,(Ti/(p(Xi)))) (1{Yi,< ,,I}- 7). Whereas q1,, is the minimizerof theconvexfunction
expressedin Equation(1) usingtheweight&2^,
tozero
(1{Yi < i1,,}- 7) willconvergeinprobability
(1/N)
as N increases.
iN(Ti/(p(Xi))).
The same line of reasoningcan be applied to estimationof AIT=l. The
betweenthe solutionsof
proposedestimatoris definedas the difference
two minimizations
of sumsof weightedcheckfunctions:A,IT=l
1,TIT=l
where
and
LN
argminq j 1,ilr=1 PT-(Yi
q)
qo,71T=1,
=-
q1,T=l
q0,7T=1
SERGIO FIRPO
266
argminqEZiN
givenby
(3)
(O,ilT=1
(-1,i(T=I
OO,ilT=
P(Yi
are
- q). The weightsused in thosedefinitions
Ti
and
E=
1
Ti
1-p(X;)
1=,
/
I
p(Xi)
1- Ti
ENT
-
For the remainderof the paper,we restrict
the discussionto the estimator
for
of
because
extensions
immediately.
ql,,
qo0,,q1,TIT=1, andq0,Tir=l follow
qa,,thatforthederivation
Also note
oftheasymptotic
it is enoughto
properties,
concentrate
on
becausethedifferences
involveindependent
A, and
q1,,, thecovariancetermwillbe ArT=1
termsand,therefore,
zero.6 Focusingthuson the
In thefirst
estimator1,,,we see thatit is a two-step
estimator.
step,we estimatethepropensity
scorenonparametrically.
In thesecondstage,we minimize
(4)
G,N(q; p)'=
1
Si=l
y
T
(Xi)
Ti-
(Yi - q) -(7 - 1{Yi q}),
<
whichis a weighted
sumofcheckfunctions
whoseweightsareintroduced
here
to correctfortheselection(on observables)problem.
to computation
of theweightsused in this
Let us now turnour attention
In particular,
on thecalculationof
subsection.
letus concentrate
^,&i.
3.2. FirstStep
scoreestimation
byHIR, we use
Followingthepropensity
strategy
employed
of X is used
a logisticpowerseriesapproximation,
i.e., a seriesof functions
to approximate
thelog-oddsratioofthepropensity
score.The log-oddsratio
of p(x) is equal to log(p(x)/(1 - p(x))). These functions
are chosento be
of x and thecoefficients
thatcorrespond
to thosefunctions
are
polynomials
likelihoodmethod.
estimated
bya pseudo-maximum
Startbydefining
HK(x) = [HK,j(x)] (j = 1, ..., K), a vectoroflengthK of
the following
of x e X thatsatisfy
polynomialfunctions
properties:(i) HK :
X --+RK, (ii) HK,1(x) = 1, and (iii) ifK > (n + 1)', thenHK(x) includesall
we assumethatK is a function
of
polynomials
up to ordern.7 In whatfollows,
thesamplesizeN suchthatK(N) -- oo as N -- o.
and (o are independent,
thesamesituationoccurswith
6BecausetheweightsW&I
(01T=1*
(llT-1T
and
thechoiceofHK(x) and itsasymptotic
detailsregarding
can be foundin
properties
7Further
thesupplement
andinHIR.
267
QUANTILE TREATMENT EFFECTS
score is estimated.Let p(x) = L(HK(X)'rK), where
Next,the propensity
L:R -+ 1R,L(z) = (1 + exp(-z))-l, and
(5)
1
=
rK argmaxN
N
i=1
-log(L(HK(Xi)'Fr))
(Ti"
+ (1 - Ti) log(1 - L(HK(Xi)'%r))).
Afterestimating
the propensity
score,we minimizeG,,N(q; ^) withrespect
to q, obtaining
q1,,.
3.3. LargeSampleProperties
In thissubsectionwe showthe mainasymptotic
resultof the paper: A, is
for
and
normal.
Thisresultis shown
root-N
consistent
A,
(ii) asymptotically
(i)
on
of
and
that
the
sametypeofresult
the
case
byarguing
byconcentrating
q1,,
a
Before
the
result
as
letus define
for
theorem,
holds,byanalogy, qo,,.
stating
=
For
i
someimportant
...,
N,
1,
quantities.
(6)
(7)
(8)
t - p(x)
t=
p
E[gl,,(Y)|x, T = 1],
l,r(y,t,x) p(x) 1(y)
p(x)
gl
t
1- t
*go0,(y)+ t- p(x) E[go,-(Y)|x, T = 0],
t,x) =
qjo,C(y,
1 - p(x)
1- p(x)
g,(yY)
1 {y < qj,,} - 7
y< qjG)
(j
fj(qj,,
=
0,
1)
ofthesamplequantiles
functions
wherefunctions
gj,,(.) wouldbe theinfluence
=
if
were
ofthepotentialoutcomesY(j), j 0, 1, they
fullyobservable.
from
and go,,to be, respectively,
different
There are tworeasonsfor
gi,7
and
The firstreasonis relatedto thepartialunobservability
of the
q/1,,
qf0o,.
outcomes.Because we cannotobservetwopotentialoutcomesfor
potential
the same unit,the momentconditionsassociatedwithql,, and qo0,,as seen
fromLemma1,are,respectively,
(9)
t
1-t
p
t,x) p=
g~,,(y) and ?o,,(y, t,x) =
1- p(x)
p,,7(y,
p(x)
go,,(y)
is
because
T, X)] =
T, X)] = 0.8 The second difference
E[0po,,(Y,
E[(Ip,,(Y,
the potentialoutrelatedto the factthat,in additionto not fullyobserving
themoment
conditions
associated
withthequanthepotential
8Ifwecouldobserve
outcomes,
wouldsimply
be E[1I{Y(1)< ql,,}- 7]= E[1{Y(O) < qo,,7tilesofthepotential
outcomes
7]= 0.
268
SERGIO FIRPO
score. The effectof escomes,we also do not knowthe truepropensity
of
the
on
influence
functions
the
sample quantilesis given
timatingp(.)
the
of pj,, and o0, withreof
derivatives
conditional
the
expectation
by
=
to
which
to
correspond al,,(t, x)
spect p(.),
-E[gl,,(Y)lx, T = 1] (t =
=
T
and
0]
(t
p(x))/(1 p(x)). Dep(x))/p(x)
ao,7(t,x) E[go,,(Y)lx,
finenow qfi,(y,
=
t,x)
1,7(y,t,x) qfo,(y, t,x), p,(y,t,x) = ~i,,7(y,t,x) define
t,x), anda,(t, x) = al,,(t,x) - ao,,(t,x). Finally,
9
0o,O(Y,
V,= E[(qi~(Y, T, X))2]
=E[Vl[g17,(Y)IX,
p(X)
= E[( o,(Y, T, X) + a,(T, X))2]
T=
V[go,(Y)IX, T =
11+
1 - p(X)
01
+ (E[gl,,(Y)IX, T = 1] - E[go,7(Y)IX, T = 0])
result:
We nowstatethemainasymptotic
THEOREM1--Asymptotic
1 and 2
Propertiesof A,: UnderAssumptions
A.1 and A.2 insupplements,
hereinandAssumptions
=
(a, - A,)
C
i=1
X1)+ o,(1) - N(0, V,).
(Yi, TT,
We showa sketchof theproofof Theorem1. For thecompleteproof,see
thesupplement.
We startwiththeresultsderivedinHIR, inturnbasedon Newey(1995),for
of thepropensity
estimation
of thenonparametric
the asymptotic
properties
Theirapscore in the firststepby meansof a powerseriesapproximation.
under
certain
score
thepropensity
regularity
guarantees,
proachto estimating
conofthepropensity
that 3(x), theestimator
score,is uniformly
conditions,
is listedin thesupconditions
sistentforthetruep(x). Thatsetofregularity
resultis presented
A.1 and theuniform
convergence
plementas Assumption
thereas LemmaA.1.
We thenfocuson the case of q1,, The next stepin the proofis to define
= N - (G,,N(q;p) =
G,,N(qi,,;3)), whereq q,, + u//N and
Q,,N(u; 3)
=
reaches
u is somerealnumber.Therefore,
- ql,,),
at u, I/NV(q1,,
P)
Q,,N(';
value.We thusshowthatQ,,N(u;P) - Q7,N(U) is Op(1) forany
itsminimum
fixedu,whereQ7,N(u),whichdoes notdependon k(x), is a quadraticrandom
that
For thatresultto be true,we imposethetechnicalrequirement
function.
=
in
x
is
differentiable (AssumptionA.2).
Pr[Y(1)< ql,,lX x] continuously
V[-]is thevarianceoperator.
9Inwhatfollows,
269
QUANTILE TREATMENT EFFECTS
of the firstderivative
of the condiThis is analogousto imposingcontinuity
of Y(1) givenX, as HIR do forATE. The resultthatshows
tionalexpectation
thatQr,N(U; p) - QT,N(U) = Op(1) appearsin LemmaA.2.
Let i, be theargument
thatminimizes
Q,,N(').We showinLemmaA.3 that
- N(0,
=
and
and
we
derivean expression
forV1,,.To get
ii, Op(1)
i,
VI,A),
aboutq^,, we use a resultin Hj6rtand
results
about u, and, consequently,
In
ofconvexrandomfunctions.
Pollard(1993) on thenearnessofminimizers
Lemma
2
to
our
we
and
Pollard's
showcase,
directly
particular, applyHj6rt
ingin LemmaA.4 thati^, - ii, = op(l). Thisconcludestheproofthatq1,,is
forql,T and (ii) asymptotically
normal,whichare shown
(i) root-Nconsistent
in LemmaA.5. Extending
thisresultto qO,T, Theorem1 is easilyshowntohold.
oftheQTT parameter,
Finally,notethatestimation
AIT=1,willyielda similarresult,whichcouldhavebeenobtainedusingstepsanalogousto thoseused
fortheQTE parameterA, to getresultssimilarto Theorem1. Forthesake of
we presentnowthenormalized
varianceoftheQTT
completeness,
asymptotic
estimator,
ATIT=1:
D
(10)
VrT =
p
E[P(X)
p(X)
p
p
V[gl,,rlT=(Y)IX, T= 1]
V[go,7ITr=(Y)IX,T =0]
1 - p(X)
T=
T= (E[gl,7,rT=(Y)fX, 1] E[go,,1rT=(Y)IX, 0])2
p
1{YY < qj,71T=l} - 7
gj,,r= (Y) = fjlT=l(qj,rlr=l)
(j = 0, 1),
and fliT=l(') and flTr=l(') are the densitiesof Y(1)IT = 1 and Y(0)IT = 1,
respectively.
3.4. Variance
Estimation
In thissectionwe presentan estimator
forV,,the normalizedasymptotic
forthatestimator
varianceA,. We thenstatesomesufficient
conditions
to be
in
is
the
that
which
Note
the
same
consistent,
proved
argument
supplement.
showcouldbe easilyextendedto theestimation
the
we subsequently
of
VIDT=,,
varianceA,7T=1.
normalizedasymptotic
= E[(p,(Y, T, X) +
The normalizedasymptoticvariance of A, is
V,
A
termis byusnatural
to
estimate
that
variance
a,(Y, T, X))2].
procedure
=
ingVT
+ a 7,iw)2, where
-i=(&,
1- Ti
Ti
p(Xi)
1- P(Xi)
SERGIO FIRPO
270
gj,T(y)
1{y< j,,l}- r
(j
, 1),
and fj(.) is an estimator
of the densityof thepotentialoutcomeY(j). Note
thata simpleapplicationof iteratedexpectations
to thedefinition
of a,(t, x)
yields
a(t, x) = -E
g Y)
=p(X))2
FT
. 1,7(
(p(X))2
( 1- T) go, Y )
(1 . -((t-7( X
-p(X))21I-
1
p(x)),
whichleadsto itsestimation
through
T -1,T(Y)
- P(Xi)).
- (1- T) -9o,(Y)X= X=X;i(Tp
=
]
(X))
)
-Ei-+i=
(X))2I
G
(P"j(X))2
Bi--[
Further
detailsoftheestimation
can
procedureofsuchconditional
expectation
be foundinthesupplement.
we showthatunderthesameassumptions
usedtoprovethemainasFinally,
result
a
set
of
conditions
plus
regularity
ymptotic
presentedin thesupplement
as Assumption
forV,:
A.3, V,willbe consistent
a7j
oftheAsymptotic
VarianceofA,: UnTHEOREM2-ConsistentEstimation
=
and
1,2,A.1,A.2,
A.3, V, V, op(l).
derAssumptions
4. SEMIPARAMETRIC EFFICIENCY BOUNDS
We nowshowthattherespective
estimators
A, and AIT=1ofQTE and QTT
intheclassofsemiparametric
estimators.
To showthis,we
are indeedefficient
boundsforQTE and QTT parameters
calculatethesemiparametric
efficiency
underunconfoundedness
ofthetreatment
and unknown
score.
propensity
boundsforsemiparametric
modelswerefirst
introduced
Efficiency
byStein
(1956), but have become more popularin the econometricliteratureonly
recentlyafterthe systematic
presentations
by Bickel,Klaassen,Ritov,and
Wellner(1993), and Newey(1990, 1994). For semiparametric
modelswith
data
under
the
at
random"
Robins,Rotnitzky,
missing
"missing
assumption,
and Zhao (1994), Robinsand Rotnitzky
and Robins
(1995), and Rotnitzky
have
shown
how
to
the
bounds
theef(1995)
compute efficiency
byderiving
ficientscoreof thatmodelfromthe efficient
scoreunderrandomsampling.
Underunconfoundedness,
Hahn (1998) and HIR havecomputedthebounds
fortheaveragetreatment
effect(ATE) and foraveragetreatment
for
effects
with
on
the
bounds
for
the
givensubpopulations, particular
emphasis
average
treatment
effect
on thetreatedATT.
Forthequantiletreatment
effects
we computedboundsfortwoparasetting,
1 and 2, thesemiparametric
and
With
meters,namely,A,
Assumptions
Ad1T=1.
for
bounds
and
can
be
calculated:
A,
efficiency
A,7=1
QUANTILE TREATMENT EFFECTS
271
1 and 2, the
THEOREM3-Bounds forA, and A,71T1:UnderAssumptions
and
bounds
A7T=1are,respectively,
equal to
for A,
efficiency
semiparametric
and
V,
V7IT=1.
Note thattheboundsV, and V71T= are similarto theboundscomputedby
Hahn (1998) forthe mean case. There are two reasonsforthe similarities.
model
First,both QTE and ATE are parametersfromthe same statistical
of
distribution
of
the
same
be
as
functionals
can
and, therefore,
expressed
in solutionsto moment
as differences
the data. Second,bothcan be written
forthe QTE case) over the same density.The latter
conditions(implicitly
ofgj,,(.) and gj,r1= 1(') (j = 0, 1) as theincanbe seenfromthedefinitions
of the samplequantilesof Y(1) and Y(0) underfullobfluencefunctions
ofpotentialoutcomes.Because,forexample,E[gj,,(Y(j))] = 0, we
servability
knowthatA, = argzeroqE[1{Y(1) < q} - 7] - argzeroqE[I { Y(0) < q} - 7].
Also, by defining
P1 = E[Y(1)], 8o = E[Y(0)], and 3 = 31- 3o,then/ =
argzerobE[Y(1) b] - argzerobE[Y(0) - b].10
5. SUMMARY OF RESULTS
We have consideredtwoapplicationsforthe methoddevelopedhere:the
firstis the evaluationof a well studiedjob training
program;the secondis a
as wellas all figures
MonteCarlo study.Furtherdetailsofbothapplications,
and tablescan be foundin thesupplement"(Firpo(2007)).
5.1. Empirical
Application
The empiricalexerciseused the job trainingprogramdata set firstanalyzedbyLaLonde (1986) and laterbymanyothers,includingHeckmanand
Hotz (1989), Dehejia and Wahba(1999), Smithand Todd (2001,2005), and
thecausal effects
ofjob training
on
is to identify
Imbens(2003). The interest
we haveused one ofthemanydata subfutureearnings.For our application,
based on theoriginalsample.We haveboththeexsetsLaLonde constructed
from
data sets,thelatterusinginformation
and
the
observational
perimental
thePanel Studyof IncomeDynamics(PSID) and theformerusinginformationfromtheNationalSupportedWorkProgram(NSW). FollowingDehejia
theanalysisto a subsetof 185 treatedunits,
and Wahba(1999),we restricted
unitsfromthePSID.12 Somesummary
260controlunits,and2,490comparison
are presentedinTableI inthesupplement.
statistics
betweenthe"on the
couldverywellbe appliedto thecomparison
10Notethatthisargument
andE[Y(1)IT = 1] - E[Y(O)IT = 1].
treated"parameters
A,IT1=l
andintheMonteCarlostudyis availusedintheempirical
"1TheMatlabprogram
application
Materials
website(Firpo(2007)).
able on theEconometrica
Supplementary
to thecontrolsamplelabeledbyLaLonde (1986) and Dehejia andWahba
12Thiscorresponds
(1999) as PSID-1.
272
SERGIO FIRPO
oftheQTT. We also performed
Usingthatdata set,we generatedestimates
an experimental
betweenthe
bysimply
takingthedifference
QTE estimation,
and
the
without
of
the
treated
controls,
anyreweighing.
quantiles
experimental
Our resultsarepresentedin TableII andFigures1-5 inthesupplement.
showthatwhenexperimental
controlsare
Figures1-3 (in thesupplement)
in
effectstendto be morehomogeneousthan the observaused,treatment
effects
tionalsetting.Witha nonexperimental
comparisongroup,treatment
aroundthe medianuntilalseem to increasealongthe distribution,
starting
ATT is
mosttheupperendofthedistribution.
Also,although
nonexperimental
notstatistically
different
fromzero (pointestimateof$1,163with
significantly
a standarderrorof $1,736),as shownin TableII, QTT estimatesare positive
and significant
fromthemedianto the85thpercentile.
the
fact
thatobservational
treatment
effects
seemto be morehetDespite
than
the
distribution
the
confidence
effects,
erogeneousalong
experimental
bandsthesizeof2 standarderrorsaroundthedifference
betweentheQTT estimatesincludethezeroforalmostall quantiles,as shownbyFigure4. TableII
betweentheexperimenrevealsthesamepatternforthemean.The difference
talATT ($1,794)and nonexperimental
ATT ($1,163)estimatesis $631,which
is relatively
smallgiventhemagnitude
itsstandarderrors($1,860).
We also checkedthecommonsupportassumption.
As TableI reveals,comformostofthecovariates.
The
parisonandtreatedgroupsare largelydifferent
scoreis, however,boundedawayfrom0 and 1, which
estimatedpropensity
allowsus to proceedwiththe analysisand to computeweightsthatare well
inthesample.'3A closerinspection
definedforall observations
ofthedata reoftheestimatedpropensity
veals thatthedistributions
scoresfortreatedand
different.
Figure5 showsthe histograms
comparisongroupsare importantly
Mostof thedata in thecomparisongrouppresent
of thesetwodistributions.
scorebelow 0.1. In fact,thereare only
values forthe estimatedpropensity
132 out of 2,490comparisonobservations
thathave an estimatedpropensity
scoreabove 0.1. This meansthatalthoughwe are usingall 2,490comparison
forcomparisons
be important
with
units,onlya fewof themwilleffectively
the treatedgroup,because theweightsj01T=1 used to estimatethe quantiles
of thatcounterfactual
distribution
are proportional
to p(x)/(1 - p(x)) and,
will
in
we
therefore,
weighdownobservations thecomparison
groupwithestimatedpropensity
scoresclose to zero.We shouldexpectthatbecausewe are
fromPSID, thestandarderrorsofQTT
effectively
usingjustfewobservations
Our findings
do indicatethatthisis exactly
what
estimatesshouldbe affected.
in
the
there
are
some
estimates
the
middle
of
distribution
that
happens:
QTT
havelargerstandarderrorsthanthoseQTT estimatesat theupperend ofthe
As we showin moredetailin thesupplement,
distribution.
thisis due to the
13Aswe are considering
the quantiletreatment
effecton the treated,a sufficient
common
is thatp(x) < 1.
supportassumption
QUANTILE TREATMENT EFFECTS
273
presenceof some comparisonunitsthathave largevalues of the estimated
ofearnings.
scoreinthemiddleoftheempiricaldistribution
propensity
The mainfindings
oftheempiricalapplicationare thatobservational
QTT
closeto theexperimental
ones,butthattheyare
pointestimatesare relatively
estimatedat some partsof the earningsdistribution.
The large
imprecisely
standarderrorswe reportfortheobservational
can
be
attributed
to the
QTT
limitedoverlapin thecovariatesdistribution
betweentreatedand comparison
ofcovarigroups.Limitedoverlapin thesupportoftheempiricaldistribution
thattheproposedmethodis inadequateforevalatesdoes notmean,however,
uationofthisprogram.In fact,as discussedin Imbens(2004),thereweighting
methodappliedto ATT estimation
is welldesignedto cope withlimitedoverbecause
estimates
are
not
affected
lap
point
substantially
bytheinclusionof
controlsthathave propensity
scoreclose to 0, but standarderrorswouldincreasewiththeinclusionofcontrolsthathavepropensity
scorecloseto 1. The
sameis truefortheQTT estimation
in
methodpresented thispaper.
5.2. MonteCarlo
In the Monte Carlo study,1,000 replicationswithsample sizes of 500
and 5,000were considered.The data designimpliedthatassignment
to the
treatment
was notcompletely
random,but satisfiedthe ignorability
assumption.Hence, estimationof treatment
effectsthatdo not take the selection
intoaccountwouldinevitably
estimates.We have actuproduceinconsistent
suchcalculations
without
fortheselectionproballyperformed
anycorrection
lem:theyare calledthe"naive"estimators.
Because Y(1) and Y(0) are known
foreach observation
of parai, we can also compute"unfeasible"estimators
metersofthemarginaldistributions
of Y(1) and Y(0). Finally,we compared
thenaiveand theunfeasibleestimators
withtheone proposedin thisarticle
in termsof bias, rootmean squarederror(RMSE), medianabsoluteerror
intervals.
As we expected,
(MAE), and coveragerate(CR) of90% confidence
theunfeasibleestimator
had thebestperformance,
butitwas closelyfollowed
estimatorin all thosemeasures.The asymptotic
variance
bythe reweighting
was estimated
usingtheanalytical
expression
providedinthispaperanditwas
comparedto thevarianceestimategeneratedthroughthe 1,000experiment
replications.
The resultsindicatethatthereweighting
estimator
wellaccording
performs
to RMSE and MAE criteria.
estimator
were
Coverageratesofthereweighting
close to thosefromtheunfeasibleestimator
and to 90%. The naiveestimator
in all thosemeasures.Also,looking
had,as expected,theworstperformance
separatelyat bias and varianceterms,it is clear thatthe bias vanishesrelativelyfastas thesampleincreasesforall of the quantilesbeingestimatedby
thereweighting
does notoccurwiththenaiveestimethod;thesamesituation
mator.Analytical
standarderrorstendto be (eitherlookingat theaverageor
at themedian)close to theMonteCarlo standarderrorsforall samplesizes
274
SERGIO FIRPO
and quantiles.Thisindicatesthatbootstrapping
to
maybe a good alternative
standard
errors
estimation.
analytical
6. CONCLUSION
Forapplications
inwhichtheexogeneity
islikelytohold,thispaassumption
effects
perhas shownhowto estimatethequantiletreatment
usinga two-step
is shownto be root-Nconsistent
and asymptotically
procedure.The estimator
normal.We also calculatedthe semiparametric
boundand proved
efficiency
thatthisquantiletreatment
effects
estimator
achievesit.
We investigated
and inference
forQTE and QTT
identification,
estimation,
which
are
in
differences
solutions
of
minimization
parameters,
problems.Undertheassumptions
used in thisarticle,themethoddevelopedherecouldbe
extendedto quantities
thatare solutionsto thegeneralproblem
(11)
51- o = argminE[m(Y(1); )] -argminE[m(Y(O); [)]
= argminE
p(X)
.m(Y; )]
p(X)
S1-
- arg
minE1-
p(X
m(Y)
forsomeknownreal-valuedfunction
m. Thereare severalexamplesoffunctionsm, buttwoof themostimportant
ones are m(Y; ?) = (Y - )2, which
allowsus to use theidentification
resultofEquation(11) to computeaverage
treatment
and m(Y; ?) = p,(Y - ?), the checkfunction,
whichwas
effects,
used inthispaperto computequantiletreatment
effects.
Finally,notethatifa
randomsampleof (Y, T, X) wereavailable,estimation
of (1 and ?owouldfollowbyreplacingpopulationmoments
their
sumsand replacingtheweights
by
byW&and o,respectively.
Whereastherearemanyinequality
measuresthatarefunctions
ofquantiles,
an obviousextensionofthemethodpresentedherewouldinvolveestimation
ofinequality
measuresforthepotentialoutcomes.Severalrelevantinequality
measuresare of interestin theappliedliterature.
The framework
developed
herecouldbe extendedto estimateand infertheresponseof suchinequality
measuresto a treatment.
Anotherextension
woulddeal withthefactthatunderthecurrent
assumpeffects
distribution
arenotpointidentified
tions,thequantilesofthetreatment
a strategy
to computebounds,in thespiritof Manski(2003),
and,therefore,
couldbe an alternative.
is to provideconditions
for
Thus,a naturalextension
interval
identification
forthosequantilesandto provideestimates
ofthoseinA comparison
tervals.
betweenthetwomethodswouldbe usefulinthesenseof
how
close
inquantilesofpotentialoutcomesare to the
differences
establishing
QUANTILE TREATMENT EFFECTS
275
effects
evenwhenno rankpreservation
assumptions
quantilesofthetreatment
are invoked.
Cat6lica do Rio de Janeiro,
Dept. de Economia,PontificiaUniversidade
Rua Marquis de Sdo Vicente,225-Gdvea,22453 Rio de Janeiro,RJ,Brazil;
[email protected],
http://www.econ.puc-rio.br/firpo/homepagepuc.htm.
received
received
2004;finalrevision
June,2006.
Manuscript
August,
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