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IntroductionQuantile Regressions

Quantile Treatment Effect

Quantiles regression

Pauline Givord

CREST, INSEE

2011/2012

Pauline Givord Evaluation of Public Policies
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Quantiles regression

Pauline Givord

CREST, INSEE

2011/2012

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Introduction

95% of applied econometrics is concerned with averagesbut many variables (earnings, test scores...) have continuousdistributions: they can change in a way not revealed by anexamination of averages

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Moving Beyond Average

Growing interest on distributional outcomes beyond simple averages(what is happening to the entire distribution) :

inequalities analysis

ex: at average real wages in US since the 80s, but upperearnings quantiles have been increasing while lower quantileshave been falling (Buchinsky, 1998,...)policy maker might be interested in the effect of treatment ondispersion of the outcome, or its effect on lower tail of theoutcome distribution:

Heckman, Smith and Clements (1997): many persons would judgeprograms to be successful if (...) enough of the right kinds of persons, reaped benets from them even if the average participant

did not.Pauline Givord Evaluation of Public Policies
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Quantile Regression

rapidly expanding empirical (and theoretical) quantile regression

literature in economicsquantile regression:provides a convenient linear framework for examining how thequantiles of a dependent variables change in response to a setof independent variablesallows the estimation of linear conditional quantile functions

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Course Overview

This course :1. (briey) introduces quantile regression

for a more detailed presentation see Buchinsky (1998) andKoenker et Hallock (2002)

2. and discuss some major application(s) to the evaluation

framework : quantile treatment effect

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Denition and Notation

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Denition and NotationQuantile RegressionEstimationInterpretationDiscussion

Minimization

Helpful to think about quantile as the solution to a minimizationproblem.

simplest case : Medianit solves argmin b i |Y i b | : Least Absolute Deviation (LAD)


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Intuition

First order conditions

S ( y , b )

b =

i

(1(u i > 0)(+ 1) + 1(u i < 0)( 1))

with u i = y i b .minimum for b such as we have as many negative and positiveresiduals.i.e. we have as many y i higher than b as we have y i smallerthan b : denition of median.

Note that the location of the median depends only on the signs of the residuals ( y i b ), so robust to outliers



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Check Function

This formula can be generalized to any other quantile :the th sample quantile solves

argmin b i :Y i b

|Y i b | + i :Y i < b (1 )|Y i b |

or : argmin b i (Y i b )solution to a problem that minimizes the weighted sum of the

absolute value of the residualsthe weight function is called the check function:

(u ) = u ( 1(u < 0))



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Quantile Regression

We are interested in the conditional distribution according toobservable X.e.g. : Quetelets growth chart (distribution of weight/heightconditionally on age)


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Quantile RegressionEstimationInterpretationDiscussion

Quantile Regression

Motivation : observables X affect the entire shape of thedistributionthe impact on tail quantile can be very different than the impact on

the central quantilethe quantile regression assumes a linear dependence of thequantile in these observables.quantile regression model replaces the b in the precedingprogram by a linear function of the covariateswe estimate :

= arg min (Y i X i )


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Estimation

the check function is not differentiable, so common gradientprocedure cannot be usedcan be written as the solution of linear programming model(Koenker and Bassett, 1978).implementation by stata (qreg, sqreg) or R (rq), sas

(quantreg).


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Results

we could run a quantile regression for each different of :

Q Y |X ( ) = X

the impact is summarized by the function coefficient:

i.e. we will get a different coefficient vector for every value of presentation of results: usually graph of the coefficientestimates as a function of the quantiles


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Example : Koenker and Hallock

Birthweight


I d iDenition and NotationQ il R i
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Interpretation

we measure how a quantile of the conditional distributionchanges with change in the covariates

parameter of interest: EQ Y | X ( ) X j i.e. marginal change in the th conditional quantile after amarginal change in X j .if x j is entered linearly, it is just j not individual interpretation: does not imply that a subject inthe th quantile of one conditional distribution would still ndhimself there with different value of X


I t d tiDenition and NotationQ til R g i
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Conditional versus Unconditional Quantile

we obtained estimates of the impact on a covariate on theconditional quantilein average estimation framework (OLS), it is sufficient toobtain consistent estimates of the impact of X on thepopulation unconditional mean of the outcome Y :a linear model for conditional meansE [Y |X ] = X impliesthat E [Y ] = E [X ] (law of iterated expectations)BUT conditional quantiles do not average up to their

population counterparts:q Y ( ) = E [q Y |X ( )]

estimates cannot be used to estimate the impact of X on thecorresponding unconditional quantile


IntroductionDenition and NotationQuantile Regression
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Conditional versus Unconditional Quantile : Example

Firpo, Fortin and Lemieux (2009) Unconditional quantileregressions , Econometrica

empirical question:what is the impact on median earnings of increasingproportion of unionized workers, holding everything elseconstant?estimates obtained by running a quantile regression could not

answer this simple questionFFL propose an estimation of the unconditional quantiletreatment effect (under exogeneity assumption for T )


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cond. versus uncond. quantile impact of union status onearnings


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cond. versus uncond. quantile impact of union status onearnings

unions reduce within-group dispersion, where groupconsists of workers with sameX and increase conditional mean of wages of union workers (i.e.between group inequalities)so tend to increase wages for low unconditional quantiles (both

effects go in same direction)but can decrease wages for high unconditional quantiles(effects go in opposite directions)


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Advantages on OLS

estimate is less sensitive to outliers values of the outcomeequivariance to monotone transformation :if h is a monotone function, P (T < t |X ) = P (h(T ) < h(t )|X )then Q h(Y ) |X ( ) = h(Q Y |X ( ))The mean does not share this property:

E (h(Y )|X ) = h(E (Y |X ))

useful for censored data (cf Buchinsky, 1998), as thequantiles of the censored conditional quantiles functioncorrespond to the quantiles of the uncensored conditionalquantile function


Introduction Denition
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Quantile RegressionsQuantile Treatment Effect

Identication : CIAIdentication: IV

Denition

Evaluation framework : we are interested in the effect of a binarytreatment T on an outcome Y .

Let Y 0 and Y 1 the potential outcomes with and withouttreatmentF Y 0 and F Y 1 the corresponding distributions.


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we dene the th quantile treatment effect (QTE):

= F 1Y 1

( ) F 1Y 0

( )

Horizontal distance between both distributions (Lehmann,1974 and Doksum, 1974).Similarly, we could dene the restriction to the treated

(QTET): |T = 1 = F

1Y 1 |T = 1 ( ) F

1Y 0 |T = 1 ( )


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Interpretation

without further assumption on the joint distribution of potential outcomes, we estimate the difference of the quantilesand not the quantile of the difference (i.e. the treatmenteffect) Y 1 Y 0 no individual interpretation : a nding of a treatment effectof at the th quantile says nothing about the treatmenteffect for the person at the th quantile of the untreatedoutcome distribution

In many applications, this is sufficient to answer economicallymeaningful questionschanges in the median, in the lower tails of the distribution, of the Gini coefficient...otherwise, need to make assumption on joint distributions


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Rank Invariance Assumption

rank invariance: implies that the treatment does not alter the

ranking of the units:If i as Y 0 i < Q Y 0 ( ), then Y 1 i < Q Y 1 ( )when it is not likely to be satised for all observations,Heckman, Smith and Clements (1997) propose bounds for thequantile treatment effect under several assumptions on theranking


IntroductionQ il R i

DenitionId i i CIA
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Identication

preceding quantile regression methods could be used (we justemphasize the impact of a particular explanatory variable T )

but so far we have (implicitly) ignored potentially endogeneityexcept in case of experimental data, we have to deal with thesame selection effects as usual

Extension of the classical identication methods toestimate counterfactual distributions.still ongoing research eld...

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DenitionIdentication : CIA

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Unconditional Quantile Treatment Effect

we face the same problem as before : usual quantile regressionsestimate treatment effect at different conditional quantilesbut as stated before, it cannot be used to estimate the impactof the treatment on corresponding unconditional quantilesFirpo proposes a (two stages) semi-parametric direct

estimation of unconditional quantile treatment effect



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Identication

Firpo shows that under preceding assumptions the quantile Q Y 1 ( )

could be expressed as an implicit function of theobserved (Y , T , X ):

= E T

p (X )1(Y Q Y 1 ( ))

in this expression, the onlyconditional function to estimate is thescore p (X ) = P (T = 1|X ).



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Conditional Quantiles

Similarly, we have:

= E 1 T

1 p (X )1(Y Q Y 0 ( ))

and for conditional quantiles Q Y 1 ( |T = 1) and Q Y 0 ( |T = 1):

= E T p

1(Y Q Y 1 |T = 1 ( ))

and = E

p (X )1 p (X )

(1 T )p

1(Y Q Y 0 |T = 1 ( ))

with p = P (T = 1)



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Q gQuantile Treatment Effect Identication: IV

Estimation

Two-stages procedure :1. (non parametric) estimation of p (X )2. Estimates of Q Y 1 and Q Y 0 are obtained by a reweighted

version of the standard quantile regression procedure:consistent estimators of Q Y t ( ) (t = 0, 1) are given by:argmin b t ,i (Y i b )with 1 ,i = T i N p (X ) (i.e. sample analogue of T / p (X )) and

0 ,i =1 T

i N (1 p (X i )) .QTE = Q Y 1 ( ) Q Y 0 ( )

We could similarly estimate QTET.



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Q gQuantile Treatment Effect Identication: IV

Empirical Application

identication of the causal impact of a training program onfuture earningsLalonde data set (1986): use bot experimental data set(National Supported Work Program) and observational dataset (PSID)see also Dehejia and Whaba (1999): less destructive resultsthan Lalonde for non experimental data, when one correctly

corrects for differences in observables variables in treatmentand control groupsestimation of the score that balances covariates betweentreated and control groups



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Quantile Treatment Effect Identication: IV

Observed and Counterfactual Distributions of PotentialOutcomes (Treatment Group)



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Quantile Treatment Effect for the Treated

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Q il T Eff

DenitionIdentication : CIAId i i IV

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Instrumental Variable Estimate of the Quantile TreatmentEffect

Abadie, Angrist et Imbens (2002), Instrumental Variables

Estimates of the Effect of Subsidized training on the quantiles of Trainee Earnings, Econometricaextension of AIR framework to quantile regressionestimation of the QTE with an instrument

as Firpo, AAI show that it could be obtained as a weightedversion of the standard quantile regressionapplication to a randomized experiment evaluation




DenitionIdentication : CIAIdentication: IV
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Notation

Random affectation to treatment : Z = 0, 1treatment T = 0, 1, depends on instrument (denoted by T 0and T 1 )outcome Y depends on treatment Y t (ie Y 0 and Y 1 )observable characteristics X




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Assumptions

1. independence: (Y 1 , Y 0 , T 1 , T 0 ) indep of Z cond. X 2. non trivial assignment : 0< P (Z = 1|X ) < 1

3. rst stage E [T 1 |X ] = E [T 0 |X ]4. monotonicity : P (T 1 T 0 |X ) = 1 (no deers)

compliers still dened as individuals who change their treatmentstatus with instrument : T 1 > T 0

independence of the potential outcome with treatment forcompliers:(Y 1 , Y 0 ) T |X , T 1 > T 0




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Estimation of QTEc

Identiable parameter:

Q (

Y |X

,T

,T 1

>T 0

) = T

+X

i.e. estimation of the QTE for compliers estimation of

( , ) = argminE ( (Y T X )|T 1 > T 0 )

Pb: population of compliers is not identied




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AAD show that we could use the weight function:

(T , Z , X ) = 1 T (1 Z )

(1 0 (X ))

(1 T )Z 0 (X )

with 0 (X ) = P (Z = 1|X ).note that equals 1 if T = Z (Compliers).AAD show that for all functionh(Y , T , X ):

E [h(Y , T , X )|T 1 > T 0 ] =1

P (T 1 > T 0 ) E [h(Y , T , X )]




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Q

the could estimate using

( , ) = argminE [ (Y T X )]

in practice, pb as could be negative, so they use instead thenonnegative weight = E [ |Y , T , X ] = P (T 1 > T 0 |Y , T , X )




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Q

Application : JTPA

experimental data

Job Training Partnership Act (JTPA) : offer services forindividuals facing barriers to employmentrandom assignment (20 000 individuals), but only about 60%of those offered training actually received JTPAnote that very few individuals in the control group receivedJTPA services: less than 2%

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Results





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Extension of classical evaluation methods to quantile treatmenteffect estimation :

Lamarche (2007) : xed effects to deal with endogeneity biasin a evaluation of the vouchers experiment (Milwaukee).

Froelich and Melly (2008) extension to discontinuity regressiondesignAthey et Imbens (2003): application to differences indifferences.

application to duration models : Koenker and Bilias (2002),evaluation of the Bonus Experiment


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