Identification and estimation of interventions using changes in inequality measures

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DENTIFICATION AND

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STIMATION OF

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ISTRIBUTIONAL

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MPACTS

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NTERVENTIONS

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SING

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NEQUALITY

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EASURES

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TEXTO PARA DISCUSSÃO 214 • OUTUBRO DE 2009 • 1

Os artigos dos Textos para Discussão da Escola de Economia de São Paulo da Fundação Getulio Vargas são de inteira responsabilidade dos autores e não refletem necessariamente a opinião da

FGV-EESP. É permitida a reprodução total ou parcial dos artigos, desde que creditada a fonte. Escola de Economia de São Paulo da Fundação Getulio Vargas FGV-EESP

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Identi…cation and Estimation of Distributional Impacts

of Interventions Using Changes in Inequality Measures

Sergio Firpo

y

August 2008

Abstract

This paper presents semiparametric estimators of changes in inequality measures of a dependent variable distribution taking into account the possible changes on the distribu-tions of covariates. When we do not impose parametric assumpdistribu-tions on the conditional distribution of the dependent variable given covariates, this problem becomes equivalent to estimation of distributional impacts of interventions (treatment) when selection to the pro-gram is based on observable characteristics. The distributional impacts of a treatment will be calculated as di¤erences in inequality measures of the potential outcomes of receiving and not receiving the treatment. These di¤erences are called here Inequality Treatment E¤ects (ITE). The estimation procedure involves a …rst non-parametric step in which the probability of receiving treatment given covariates, the propensity-score, is estimated. Using the inverse probability weighting method to estimate parameters of the marginal dis-tribution of potential outcomes, in the second step weighted sample versions of inequality measures are computed. Root-N consistency, asymptotic normality and semiparametric e¢ciency are shown for the semiparametric estimators proposed. A Monte Carlo exercise is performed to investigate the behavior in …nite samples of the estimator derived in the paper. We also apply our method to the evaluation of a job training program.

JEL_{: C1, C3.} Keywords_{: Inequality Measures, Treatment E¤ects, Semiparametric}

E¢ciency, Reweighting Estimator.

An earlier version of this paper circulated under the names “Treatment E¤ects on Some Inequality Measures” and “Inequality Treatment E¤ects”. I have bene…ted from comments from David Green, Jim Heckman, Guido Imbens, Je¤ Smith and participants at the 2004 Meeting of the Sociedade Brasileira de Econometria, 2005 World Congress of the Econometric Society, seminars at UBC, USC, and EPGE-FGV. Rafael Cayres and Breno Braga helped me with superb research assistance. Financial support from CNPq-Brazil is acknowledged. All errors are mine.

y_{Escola de Economia de São Paulo, Fundação Getúlio Vargas. Mailing address: R. Itapeva 474/1215, São}

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1 In

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odu

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ion

For the evaluation of a social program the policy-maker may want to learn about the distrib-utional e¤ects of the program, going beyond the progra s mean impact. For example, it is

reasonable to assume that the policy-maker is interested in the e¤ect of the treatment on the dispersion of the outcome, which can be captured by commonly used inequality measures such as the Gini coe¢cient, the interquartile range or other inequality indices, as those belonging to the Generalized Entropy Class.1

The distributional impact of the program on the outcome can be measured by what we call hereInequality Treatment E¤ects (ITE), which are de…ned as di¤erences in inequality measures of the distributions of the potential outcome of joining the program (receiving the treatment) and not joining it (not receiving the treatment).

We follow an increasing part of the literature of program evaluation that is interested in distributional impacts. Some recent examples are the papers by Heckman (199 Imbens and

Rubin ( 99, Heckman, Smith and Clements (199 Heckman and Smith ( 998), Abadie,

Angrist and Imbens (2002), Carneiro, Hansen and Heckman (2001, 2003), Cunha, Heckman and Navarro (2005), Aakvik, Heckman and Vytlacil (2005), Dehejia (2005) and Firpo (2007). In the applied literature a recent paper that focuses on the distributional e¤ects of a social program is Bitler, Gelbach and Hoynes( .

We discuss identi…cation of inequality treatment e¤ects parameters under the assumption termed by Rubin (977) as treatment unconfoundedness, which is also known as the selection

on observables assumption. Important examples where this assumption has been used are Barnow, Cain and Goldberger (19 Rosenbaum and Rubin( 983), Heckman, Ichimura, Smith

and Todd (9 9 , Dehejia and Wahba ( 99 9 and Hirano, Imbens and Ridder (2003). The

unconfoundedness assumption is a conditional independence assumption: Given observable characteristics, the decision to be treated is independent of the potential outcome of being treated and the one of not being treated. This assumption is crucial as it allows that functionals of the potential outcome distributions be identi…ed from the observed data.

A two step estimation procedure is proposed. In the …rst step, weighting functions are nonparametrically estimate in the second step inequality measures are calculated using the weighted data. The e¤ect of the program is estimated, therefore, as a simple di¤erence in weighted inequality measures. Under unconfoundedness assumption we show that those esti-mators are consistent for the ITE parameters. For the class of estiesti-mators of inequality measures that are asymptotically linear we also show that the ITE estimators are asymptotically normal and semiparametrically e¢cient.

The key methodological contribution of this paper is to provide a detailed estimation pro-cedure (with asymptotically valid inference) for treatment e¤ects on inequality measures. We consider four popular inequality measures: the coe¢cient of variation, the interquartile range, the Theil index and the Gini coe¢cient. Our discussion is made on a very general level treating

those measures as functionals of the distribution. In this sense, our approach can also be seen as a generalization of the discussion of identi…cation and estimation of average and quantile treatment e¤ects as both are di¤erences in functionals of the distribution.

Recently, Tarozzi (2007) and Chen, Hong and Tarozzi (2008) have shown how to generalize treatment e¤ects identi…cation and estimation under unconfoundedness for a class of parame-ters that satisfy certain moment conditions. Their discussion encompasses a broader class of

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problems, such as missing data and non-classical errors in variables. n inequality treatment

e¤ects see also the recent work of Thuysbaert (2007).

Under failure of unconfoundedness, the estimation method we develop here can be seen as a way to compare inequality measures taking into account the role of the covariates (observables). Applied researchers are very often interested in comparing features of two or more outcome distributions. For example, we might be interested in comparing the Gini coe¢cient of two di¤erent wage distributions (e.g. two di¤erent countries). Acknowledging for the fact that there are many observed factors whose distributions di¤er across countries, such as schooling and job experience, leads us to try to control for these factors when comparing Gini coe¢cients. By doing so, we will be able to identify how systematic di¤erences in the pay structure of the two countries a¤ect the Gini coe¢cient, …xing the distribution of covariates to be the same.

course, we could make parametric and functional form assumptions in order to relate covariates to the wage distribution in each economy. However, if we are not willing to impose restrictive parametric assumptions, it is not clear how to compare Gini coe¢cients …xing the distribution of observed covariates.

Note that even though in the example of a cross-country comparison of Gini coe¢cients we do not have a clear “treatment” involved, the problem of estimation of changes in distributions of a dependent variable when the distribution of covariates is …xed can be equivalently stated as the estimation problem of distributional impacts of social programs (treatments) when selection to the program is based on observable characteristics. Although we lose causal interpretation, our method is easily implementable and robust to misspeci…cation of the conditional wage distributions.

This paper is divided as follows: In the next section we present more formally the ITE class of parameters. Section 3 presents the main identi…cation result. Section 4 discusses estimation and derives the large sample properties of the inequality treatment e¤ects estimators. Section 5 discusses …nite-sample behavior through a Monte Carlo exercise. We present in section 6 a

small empirical exercise that uses data on a Brazilian job training program of the late s. Although the training program had been designed to be a randomized experiment, in fact it serves as an interesting example where there is failure in the randomization to treatment and control groups. Finally, section 7 concludes. Proofs of results are left to the Appendix.

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We start by assuming that there is an available random sample ofN individuals (units). For each unit, letXibe a random vector of observed covariates with support R

r_{. De…ne}

Yi(1)

as the potential outcome for individual if she enters in the program, and Yi(0) the potential

outcome for the same individual if she does not enter. L the treatment assignment be de…ned

asi, which equals one if individual is exposed to the program and equals zero otherwise. As

we only observe each unit at one treatment status, we say that the unobserved outcome is the counterfactual outcome. Thus, the observed outcome can be expressed as:

Yi=iYi(1) + (1 i)Yi(0); 8.

A legitimate way to introduce inequality measures is to assume that there is a social welfare function, W, that depends on a vector of functionals of the outcome distribution. Suppose in

particular thatW assumes the following form:

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whereis the outcome mean,is the inequality measure and is a distribution function.

2 _We

de…ne the inequality measure as a functional of the distribution, :! "R. where #! if

( )<+$. A particular example of% andis the case whereis the Gini coe¢cient and%

is decreasing in. Under this setting, a natural parameter used to compare two distributions

andG#! is the simple di¤erence( ) (G). We discuss three comparisons of distributions

that give rise to three di¤erent inequality treatment e¤ect parameters.3

The …rst case arises when we want to compare the situation in which everyone is exposed to the program with the situation in which no one is exposed to it. Under the …rst scenario, the distribution of the outcome equals _Y₍₁₎, the distribution of&(1)'while in the second scenario,

the outcome distribution equals _Y₍₀₎. The di¤erence in a given inequality measure between

these two hypothetical cases is the)*+, -.. Inequality Treatment E¤ect(ITE), , de…ned

as:

= Y(1) Y(0)

= 1 0

/0her parameters could be de…ned for subpopulations. In particular, consider the

Inequal-ity Treatment E¤ect on the Treated (ITT), _T:

T = Y(1)

jT=1 Y(0)

jT=1

= 11 01

where _Y₍₁₎_j_T₌₁ and _Y₍₀₎_j_T₌₁ are respectively the conditional distributions of the potential outcomes of being in the program and of not being in the program for the subpopulation that was actually exposed to the program.

We …nally consider a parameter which is a comparison between the current inequality( Y)

and the inequality that we would encounter if there were no program Y(0) . We call this

parameter theCurrent Inequality Treatment E¤ect(CIT):4

C = ( Y)

Y(0)

= Y 0

3 Identi…cation of Inequality Treatment E¤ects

This section is divided up into four subsections. In the …rst one, we introduce some weighting functions and de…ne the concept of weighted distributions. Subsection 2 presents the identi…-cation assumptions, while in subsection 3 we present the main identi…identi…-cation results. Finally, subsection 4 brings some examples of inequality measures and shows how they …t into the framework just presented.

2_{This is the reduced-form social welfare function discussed by Champernowne and Cowell (1}

3 3 3) and Cowell

(2000).

3_{Alternative setups to what follows can be found in Manski (1}

3 37) and would lead to the de…nition of some

other possible treatment e¤ects parameters. That includes allowing individuals to choose their treatment status and assigning them to treatment based on observed characteristics.

4_If _{is not decomposable, we cannot write the CIT as linear combination of the previous parameters. Note that}

in general, (FY)6= FYjT=1 Pr [T = 1]+ FYjT=0 Pr [T = 0]. Note also that many other parameters could

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3.1 Weighted Distributions

We now set up assumptions for identi…cation of . Remember that because 4(1) and 4(0)

are never fully observable, we need to impose some identifying assumptions in order to be able to express functionals of their marginal distributions as functionals of the joint distribution of observable variables (4,5,7). Thus, identi…cation of will follow after we establish

condi-tions for identi…cation of functionals of the distribucondi-tions of 4(1) and 4(0), as the parameters

are de…ned as di¤erences between functionals of those distributions.

We start by de…ning the propensity-score (or simply, p-score),p(x), as the probability that

given a valuex:= an individual will be in the treatment group, that is,p(x)>Pr[5 = 1j7 =

x]. The unconditional probability,Pr [5 = 1], isp, which will be assumed to be positive. ?@AB C[0D1]be the image set of the mapping

p( H)D

p:

= JB. A restriction onB will be

made later on Assumption 2. Next, de…ne the following four “weighting functions”, generally written asK, such thatK :f0D1g B JR:

K1(tDp(x)) =

t

p(x) K0(t Dp(x)) =

1 t

1 p(x)

K11(tDp(x)) =

t

p K01(tDp(x)) =

1 t

1 p(x) H

p(x)

p

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?@A the data be de…ned by the sequence f4iD5iD7ig

N

i=1 where each element (4iD5iD7i) is a

random draw fromMY;T;X, the joint distribution of(4D5D7):R f0D1g =. Assuming that

MY;T;X is absolutely continuous, the joint density function of (

4D5D7) is:

NY;T;X(QDtDx) = p(x)HRM

YjT;X(Q; 1Dx)

t

H (1 p(x))HRM

YjT;X(Q; 0Dx)

(1 t)

HRMX(x) (2)

where MY

jT;X(Q;tDx) is the cumulative distribution function (c.d.f.) of 4 given 5 = t and 7=x evaluated atQSand MX(x)is the c.d.f. of7 atx. For simplicity, assume that those two

distributions are absolutely continuous, which allows their respective densities to be well de…ned asNY

jT;X(Q;tDx) andNX(x). After straight integration we obtain the c.d.f. of4 evaluated at

a real numberU:

MY (U) =

Z a

y2R

Z

x2X

(NY;T;X(QD1Dx) +NY;T;X(QD0Dx))HRx

=

Z

x2X

p(x)H

Z a

y2R

NY

jT;X(Q; 1Dx)HR Q+ (1 p(x))H

Z a

y2R

NY

jT;X(Q; 0Dx)HRQ HNX(x)HRx

Introduction of proper weights that depend ontandp(x)only is straightforward. Assuming

that the following integrals exist, we can write the weighted density and the weighted c.d.f. of

4 (at U) as:

N

! Y (Q) =

Z

x2X

fK(1Dp(x))Hp(x)HNY

jT;X(Q; 1Dx)

+K(0Dp(x))H(1 p(x))HNY

jT;X(Q; 0Dx)gHNX(x)HRx

M

! Y (U) =

Z

x2X

fK(1Dp(x))Hp(x)H

Z a

y2R

N

YjT;X(Q; 1Dx)HR Q

+K(0Dp(x))H(1 p(x))H

Z a

y2R

NY

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where V(Z [\(])) =V(1[\(])) if Z = 1 and V(Z[\(])) = V(0[\(])) if Z = 0. Simple algebra

allows us to show that:

^

!

Y (_) =`[V(b[\(c))h1Ikr s_v] (3)

where1Ikhvis the indicator function. Note that the de…nitions of weighted c.d.f and p.d.f. ofr

encompass the case of the simple c.d.f. and p.d.f. ofr by makingV= 1. It is worthwhile also

noting that the c.d.f. may be seen as a particular case of a weighted expectation. In general, for an integrable function w(r) of the random variable r, we write its weighted mean using

weightsV as:

`[V(b[\(c))hw(r)] =

Z

w(r)hz

!

Y ({)h|{} (4)

3.2 Identifying Assumptions

We now invoke the set of identifying restrictions that will permit that we write the distribution of the unobserved potential outcomes in terms of observable data. Moreover, those distributions will actually fall into the category of the weighted distributions just de…ned.

~ [Unconfoundedness]Let (r(1)[r(0)[b[c)have a joint distribution. For all ]in : (r(1)[r(0)) is jointly independent fromb givenc=], that is,(r(1)[r(0))b c = ].

Assumption 1 is sometimes a strong assumption and its plausibility has to be analyzed in a case by case basis. It has been used, however, in several studies of the e¤ect of treatments or programs. Prominent examples are Rosenbaum and Rubin 3 and Heckman and

Robb

onde

Card and Sullivan ( Heckman, Ichimura, and Todd (

Heckman, Ichimura, Smith, and Todd (1 Hahn 8),echner ), Dehejia and Wahba

and Becker and Ichino (2002). We present in the empirical section an example where

there is evidence that Assumption 1 is valid. We also make an overlap assumption:

~ [Common Support]For all ] in, 0\(])1}

Assumption 2 states that with probability one there will be no particular value ]in that

belongs to either the treated group or the control group. Such assumption is important as it allows that groups (b = 1 andb = 0) become fully comparable in terms of c. Assumptions 1

and 2 are termed together as strong unconfoundedness.

3.3 Identi…cation Results

Finally, the main identi…cation result will follow as a corollary of the next theorem. We therefore write the ITE parameters as functions of the observable variables (r,b,c).

Let r(1)

^Y(1), r(0)

^Y(0), r(1) b = 1

^Y(1)

jT=1, r(0)b = 1 ^Y(0)

jT=1 and r ^Y. Under Assumptions 1 and 2, for _ R, the c.d.f.s associated with

those distributions can be written as follows:

^Y(1)(_) =

Z

x2X Z a

y2R

zY

jT;X({; 1[])h|{hzX(])h|]

= ^

!=!1

Y (_)

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Y(0)() =

Z

x2X Z a

y2R

Y

jT;X(¡; 0¢£)¥¦¡¥ X(£)¥¦£

=

!=!0

Y ()

= §[¨0(©¢ª(«))¥1I¬ ®¯]

Y(1)jT=1() =

Z

x2X

ª(£) ª

¥

Z a

y2R

YjT;X(¡; 1¢£)¥¦¡¥ X(£)¥¦£

=

!=!11

Y ()

= §[¨11( ©¢

ª(«)) ¥1I

¬ ® ¯]

Y(0)jT=1() =

Z

x2X

ª(£) ª

¥

Z a

y2R

YjT;X(¡; 0¢£)¥¦¡¥ X(£)¥¦£

=

!=!01

Y ()

= §[¨01(©¢ª(«))¥1I¬ ®¯]

° ± ²o³³´ ²y µ Under Assumptions 1 and 2 ,

T, and C are identi…able.

¶·¸ ¹ we know that the inequality treatment e¤ects are identi…able, we can now turn our

attention to estimation and inference. Before doing so, let us be explicit about the inequality measures that are considered in this article.

3.4 Some Inequality Measures

We now turn our attention to some concrete examples of inequality measures and express them as functionals of a weighted distribution of.

Comparison of inequality measures is often performed on the basis of the attainment of some desirable properties for inequality measures. There is no clear ranking among the measures, but it is common in the welfare literature to check which of the usual properties an inequality measure possesses. Among those properties, the most common and important ones are the

principle of transfers,invariance,decomposability and anonymity. For a detailed discussion on this topic, see Cowell (2000) and Cowell (2003).5

We now consider four popular inequality measures: the coe¢cient of variation, the in-terquartile range, the Theil index and the Gini coe¢cient. As discussed in Cowell (2000), the coe¢cient of variation will satisfy all properties listed before but invariance. The interquar-tile range will not satisfy any of those properties besides anonymity. The Theil index, being a member of the Generalized Entropy class, will satisfy all four properties, whereas the Gini coe¢cient, probably the most used inequality measure, is known to be non-decomposable.

We proceed treating those four measures as functionals of a weighted outcome distribu-tion. By doing that, we gain theºexibility necessary to further de…ne the treatment e¤ects as

di¤erences in functionals of weighted distributions:»

5_{An interesting result in the income distribution literature establishes that any continuous inequality measure}

that satis…es the principle of transfers, scale invariance, decomposability and the anonymity must be ordinally equivalent to the Generalized Entropy class, which is indexed by a single scalar parameter. See Cowell (2003), Theorem 2.

¼

In what follows we assume that ! Y

R

y dFW

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1. Coe¢cient of Variation(CV):

¾

CV ₍

¿

! Y) =

R

À

R

ÁÂÃ¿

! Y (Á)

2

ÂÃ¿

! Y (À)

1=2

R

ÀÂÃ¿

! Y (À)

=

r

Ä

h

Å(ÆÇÈ(É))Â(Ê Ä[Å(ÆÇÈ(É))ÂÊ])

2i

Ä[Å(ÆÇÈ(É))ÂÊ]

2. Interquartile RangeË ÌÍR):

¾

IQR₍

¿

!

Y) = ¾

Q:75₍

¿

! Y) ¾

Q:25₍

¿ ! Y) = inf q Z q 1 Î !

Y (À)ÂÃ ÀÏ

3

4 infq

Z q

1

Î

!

Y (À)ÂÃ ÀÏ

1 4

3. Theil Index (TI):7

¾

T I₍

¿

! Y) =

R

ÀÂ log (À) log

R

ÁÂÃ¿

!

Y (Á) ÂÃ¿

! Y (À)

R

ÀÂÃ¿

! Y (À)

= Ä Å(ÆÇÈ(É))Â

Ê

Ä[Å(ÆÇÈ(É))ÂÊ] Âlog

Ê

4. Gini Coe¢cient(GC):

¾

GC₍

¿

!

Y) = 1 2

Ð(¿

! Y)

R

ÀÂÃ¿

! Y (À;Å)

= 1 2 Ð(¿

! Y)

where

Ð(¿

! Y) =

Z 1

0

Z Q ₍_F! Y)

1

ÀÂÃ¿

!

Y (À;Å)ÂÃÑ

=

Z 1

0

Ä Å(ÆÇÈ(É))Â1IÒÊ Óinf

q

Z q

1

Î

!

Y (Á)ÂÃÁÏÑ ÔÂ Ê ÂÃÑ

4 Estimation and Large Sample Inference

We now focus our attention to estimation of ¾(¿

!

Y), the inequality measure of a weighted

outcome distribution. We …rst show how to estimate and derive the asymptotic distribution of the estimator of¾(¿

!

Y) with a generalÅ, and later show how to use these results to estimation

and inference regarding , _T and _C.

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4.1 Estimation

Estimation of Õ(Ö

!

Y) follows from the sample analogy principle. We replace the population

distribution Ö

!

Y, by its empirical distribution counterpart with estimated weights,Öb

!=!b

Y , and

plug it into the functional Õ. The estimator will therefore be:

b

Õ

b

!

Y =Õ Öb

!=!b

Y

Note that we take advantage of the fact that the weighted c.d.f. is expressed as Ö

! Y (×) = Ø[Ù(ÚÛÜ(Ý))Þ1Iß à á

×â], and we write its sample analog as:

b

Ö

!=!b

Y (×) =

N

X

i=1

b

Ù(ÚiÛÜb(Ýi))Þ1Iß ài á×â

and it is clear that we have to consider carefully the estimation of weights Ù(ãÛÜ(ä)) by

b

Ù(ã ÛÜb(ä)).

4.1.1 Weights Estimation

We have four weighting functions to consider: Ù1, Ù0,Ù11, and Ù01. Three of them depend on

the propensity-score Ü(ä), the exception beingÙ11.

For the propensity-score estimation we do not impose any parametric assumption on the conditional distribution of Ú given Ý nor assume that the propensity-score has a given

func-tional form. We follow the sieve åæ approach proposed by Hirano, Imbens and Ridder

(2003). They approximate the log odds ratio of the propensity score, ç(Ü(ä)) by a series

of polynomial functions of ä.

8 _{Stacking all these polynomials in a vector, we end up with}

èK( ä) = [

èK; j( ä)] (

é = 1 Û

êê êÛë), a vector of length ë of polynomial functions of ä

ì í.

The estimation procedure will therefore involve computation of the vector of lengthë of

coef-…cientsî^K:

ç(Üb(ä)) = èK(ä)

0_b

îK

b

Ü(ä) = ç

1

èK(ä)

0_b

îK = èK(ä)

0_b

îK

where :_RïR, (ð) = (1 +ñäÜ( ð))

1 _{is the the c.d.f. of a logistic distribution evaluated}

atð. The nonparametric òavor of such procedure comes from the fact that ë is a function of

the sample size N such that ë(N) ï ó as N ï ó. Therefore, the vector î^K increases in

length as the sample size increases. The actual calculation ofî^K follows by a pseudo-maximum

likelihood approach:

^

îK =arg max K2RK

N

X

i=1

ÚiÞlog( ( èK(

Ýi)

0

îK)) + (1

Úi)Þlog(1 ( èK(

Ýi)

0

îK)) ê

In the implementation of this procedure, following Hirano, Imbens and Ridder (2003), we restrict the choice of èK(

Þ) to the class of polynomial vectors satisfying at least the following

three properties: (ô) èK : í ï R

K

õ(ôô) èK;1(ä) = 1, and (ôô ô) if ë ö(÷+ 1)

r_{, then}

èK(ä)

includes all polynomials up order ÷. ø

8_{The log odds ratio of}_z_,_L₍_z₎_{, is}_L₍_z_{) = log (}_z=₍₁ _z₎₎_.

ù

Further details regarding the choice ofHK(x)and its asymptotic properties can be found in appendix and

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We propose an estimator ú_b (normalized to sum up exactly one) for the weighting function

ú:

b

úi =

1 N!

ûú(üiýþb(ÿi))

whereN!, which is chosen to converge in probability to one, equals

N!= 1 N û

N

X

i=1

ú(üiýþb(ÿi))

Speci…cally for the weighting functions ú1,ú0,ú11, andú01 we have

b

ú1;i =

1 N1

ûú1(üiýþb(ÿi))

= 1 N₁ û

üi

b

þ(ÿi)

= 0 @1 N û N X j=1 üj b

þ(ÿj)

1 A 1 û üi b

þ(ÿi)

b

ú0;i =

1 N0

= 1 N0

û

1 üi

1 þ_b(ÿi)

= 0 @ N X j=1

1 üj

1 þ_b(ÿj)

1 A

1

û

1 üi

1 þ_b(ÿi)

b

ú11;i =

1 N11

= 1 N11 û üi þ = 0 @ 1 N û N X j=1 üj 1 A 1

ûüi

b

ú01;i =

1 N01

= 1 N01

û

b

þ(ÿi) þ

û

1 üi

1 þ_b(ÿi)

= 0 @1 N û N X j=1 b

þ(ÿj)û

1 üj

1 þ_b(ÿj)

1 A

1

ûþb(ÿi)û

1 üi

1 þ_b(ÿi)

where

N1 = _N1 û

PN

i=1pb(TXii) N0=

1

N û

PN

i=1 11pb(TXii)

N11= _N1 û

PN

i=1 Tpi N01= N1 û

PN i=1 b

p(Xi)

p û

1 Ti

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4.1.2 Estimation of inequality treatment e¤ects

O the weights have been computed, the three ITE parameters are easily estimated by the

plug-in method. De…ne the corresponding estimators of , _T, and _C as:

b = _b1 b0 = Fb

!=b!1

Y Fb

!=!b0

Y

b_T = _b11 b01= Fb

!=!b11

Y Fb

!=!b01

Y

b_C = _bY b0 = FbY Fb

!=!b0

Y

As an illustration, we consider in details the estimation of three inequality treatment e¤ect parameters: (i) the coe¢cient of variation CIT, (i i) the Theil index ITE and (i ii) the Gini

coe¢cient ITT:

Example 1: The estimator for the coe¢cient of variation CIT:

bCV

C =

PN

i=1 N1 Yi

PN

i=1N1 Yi

2 1=2

PN

i=1N1 Yi

PN

i=1!b0;i Yi

PN

i=1!b0;iYi

2 1=2

PN

i=1!b0;iYi

Example 2: The estimator for the Theil index ITE:

bT I ₌

PN

i=1!b1;iYi log (Yi) log

PN

i=1!b1;iYi

PN

i=1!b1;iYi

PN

i=1!b0;iYi log (Yi) log

PN

i=1!b0;iYi

PN i=1!b0;i

Yi

Example 3: The estimator for the Gini coe¢cient ITT:

bGC

T = 2

Z 1

0

f

PN i=1!b01;i

1I

fYibq

01

gYi

PN

i=1!b01;iYi

PN

i=1!b11;i1IfYi qb

11

gYi

PN

i=1!b11;iYi

gd

where

b

q

01 ₌

Q _b

F

!=b!01

Y = arg min_q N

X

i=1

b

!01;i

( Yi

q)

b

q

11 ₌

Q _b

F

!=b!11

Y = arg min_q N

X

i=1

b

!11;i (Yi q)

and, following Koenker and Bassett ( , (u) = u( 1Ifu0g) is the check function

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4.2 Large Sample Inference

We now devote our attention to the asymptotic behavior of our estimators. We derive the asymptotic distribution for inequality treatment e¤ect parameters based on inequality measures that are asymptotically linear. We then invoke known results established by Hahn (1

Hirano, Imbens and Ridder (2003) and Chen, Hong and Tarozzi (2008) to consider e¢ciency of our estimators.10

4.2.1 Asymptotic Linearity

We will need to establish some extra conditions to be able to derive the asymptotic normality of the inequality estimators just proposed. We restrict the discussion to the class of asymptotically linear estimators.

First, we recall the usual de…nition of the uence function of a general functional

of the distribution (Hampel, 14). L ng that functional be an inequality measure , its

uence function is the directional derivative of evaluated in in the direction of y, the

degenerate distribution that put mass only at a single pointy:

(y;) = lim

s!0

(s( y) +) () s

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For an alternative distribution G, close enough to , a von Mises type expansion of the

functional is allowable (Hampel, Ronchetti, Rousseeuw, and Stahel, 1 :

(G) =() +

Z

(y;)G(y) +r(;G)

wherer(;G) is a remainder term that depends on the distributions and G.

Definition 1[Asymptotic Linearity] Using the general weighted distribution framework,

we de…ne a functional estimate as being asymptotically linear if, after expanding the functional at the empirical distribution b

!=b!

Y around the population distribution

!

Y we get a remainder

term that converges at the parametric rateN 1=2. Another way to say that the estimator_b

b

!

Y = b

!=!b

Y is asymptotically linear is by using

directly Equationreplacing

Gby b

!=!b

Y and byb

!

Y. The estimatorb

b

!

Y will be

asymptot-ically linear if the following assumption holds.

Assumption 3 [Asymptotic Linearity of Inequality Estimators]:

b

!=b!

Y (

! Y) =

N

X

i=1

b

i (i;

!

Y) +op(1= p

N)

This assumption holds for all four inequality measures considered in this article. Treating the weights as …xed, we can verify that three out of four of our estimators are non-linear functions of exactly linear functionals (expectations) evaluated at the empirical distribution"therefore, a

simple application of the so-called delta method su¢ces to show asymptotic linearity. The only one that does not fall into that category, the interquartile range, is however just a di¤erence

1 0_{Similar e¢ciency results can be found in the missing data literature. Robins, Rotnitzky, and}

Zhao (1# #4),

Robins and Rotnitzky (1# #5) and Rotnitzky and Robins (1# #5) provide calculations of the semiparametric

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in quant$% &') a proof of asymptotic linearity of the sample quantile is found in van der Vaart

*+, ,8), for example.

Proofs that the four inequality measures considered here are asymptotically linear are not provided since they are simple applications of the delta-method. Nevertheless, we list the actual format of the $ - .uence function of each one of the four inequality measures, /

CV_,

/

IQR_,

/

T I_,

and/

GC_:11

Coe¢cient of Variation:

We write /

CV ₍

0

! Y) = 2

CV

/

V ₍

0

! Y)34

!

Y , where /

V ₍

0

!

Y) is the variance and the mean is

4

!

Y = E[5(T36(X))78]. : &t 2

CV

1 (737) and 2

CV

2 (737) be the partial derivatives with respect

to the …rst and the second arguments respectively. Then, 2

CV ₍

a 3b) =a

1=2

7b

1_,

2

CV

1 (a 3b) =

1

27a

1=2

7b

1_,

2

CV

2 (a3b) = a

1=2

7b

2_.

:&t<

V _{be the in}

.uence of the function of the variance

and< be the$-.uence of the function of the mean (see, for example,: &> ?@-n,+, ,, A p. 30, BC

< (D;0

!

Y) =D 4

! Y

<

V ₍

D;0

!

Y) = (D 4

! Y)2 /

V ₍

0

! Y)

By a simple application of the delta-method, we obtain <

CV_{, the in}

.uence of the function of

the coe¢cient of variation:

<

CV ₍

D;0

! Y) = 2 CV 1 / V ₍ 0 ! Y)34

! Y 7<

V ₍

D;0

! Y) +2

CV

2 /

V ₍

0

! Y)34

!

Y 7< (D;0

! Y)

= 1 27

(D E[5(T36(X))78])

2

E 5(T36(X))78

2 ₍

E[5(T36(X))78])

2

E[5(T36(X))78]7

q

E[5(T36(X))78

2_] ₍

E[5(T36(X))78])

2

q

E[5(T36(X))78

2_] ₍

E[5(T36(X))78])

2

(E[5(T36(X))78])

2 7(D E[5(T36(X))78])

Interquartile Range:

Following, for example, Ferguson * +, ,H,I J, + B and van der Vaart, *+,,M, p.307), we de…ne

the$-.uence function of theN-th quantile of the weighted distribution of 8 as:

12

<

Q ₍

D;0

!

Y) = (N 1IP D QR

!

S)U V

! Y(R

!₎

where V

! Y(R

!₎ _{is the density evaluated at the quantile}

R

!_{. The}

$ - .uence function of the

in-terquartile range is simply the di¤erence between two quantile in.uence functions:

<

IQR₍

D;0

!

Y) = <

Q:75₍

D;0

! Y) <

Q:25₍

D;0

! Y)

= W75 1IP D Q/

IQR₊

R

! :25S V

! Y(/

IQR₊

R

! :25)

(W25 1IPDQR

! :25S) V

! Y(R

! :25)

.

1 1_{We are assuming in what follows that all relevant integrals exist and denominators are non-zero.}

1 2_{For notational simplicity, we denote} Q _y_;_FW

Y (!) by qW. The number is a real in the open interval

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For \ ^_1`4c3`4e we can write the quantile as the minimizer of the expected check function

hjoenker and Basset,kl78):

m ! ₌ n Q ₍ v !

Y) = arg min_q w[x(zc{(|))}(~ m)}(\ 1I_ ~ me)]

Theil index:

We write n

T I₍

v

! Y) =

T I

n

L₍

v

! Y)c

!

Y , where n

L₍

v

!

Y) = w[x(zc{(|))}~ }log (~)].

T I

1 (}c}) and

T I

2 (}c}) be the partial derivatives with respect to the …rst and the second

arguments respectively. Then,

T I₍

c) =}

1 _{log (}

),

T I

1 (c) =

1_,

T I

2 (c) =

1

}

(1 +`).

L_{be the in}

uence of the function of the n

L_:

L₍

;v

!

Y) =}log () n

L₍

v

! Y)

then by another application of the delta-method, we obtain

T I_:

T I₍

;v

!

Y) =

T I

1 n

L₍

v

! Y)c

! Y }

L₍

;v

! Y) +

T I

2 n

L₍

v

! Y)c

!

Y } (;v

! Y) = (w[x(zc{(|))}~])

1

}(}log () w[x(zc{(|))}~ }log (~)])

(w[x(zc{(|))}~])

2

}(w[x(zc{(|))}~ }log (~)] +w[x(zc{(|))}~]) }( w[x(zc{(|))}~])

Gini Coe¢cient:

Theuence function of the Gini coe¢cient was derived by Hoe¤dingh kl ) as an example

of results on the asymptotic distribution of-statistics. We follow a more recent literature on

the statistical properties of inequality measures (see the survey paper by Cowell, 2000). For example, following Schluter and Trede (2003), we write the inuence function of the Gini

coe¢cient as:

GC₍

;v

! Y) =

1

! Y

}_ n

GC₍

v

!

Y) 1 }(

! Y +)

2}} 1

Z y

1

v

!

Y (;x) 2}

Z Q ₍_F! Y)

1

}v

!

Y (;x)e

= 1

w[x(zc{(|))}~] _ n

GC₍

v

!

Y) 1 }(w[x(zc{(|))}~] +)

2}}(1 w[x(zc{(|))}1I_ ~ e])

2}w x(zc{(|))}1I_~ inf

q

Z q

1

!

Y ()}\ e} ~ e

where

n

GC₍

v

!

Y) = 1 2

R1

0 w

h

x(zc{(|))}1I_ ~ infq

nRq

1

!

Y ()}\

o

e} ~

i

}\ w[x(zc{(|))}~]

4.2.2 Asymptotic Normality and E¢ciency

We now derive the limiting distribution of estimators of inequality measures for weighted dis-tributions. We …rst show that under an additional set of mild regularity conditions and if our estimators are asymptotically linear, then they will be asymptotically equivalent to a sum of terms that do not depend on the estimated weights, but instead, on the true ones.

The additional regularity conditions that we need to impose are directly on the uence

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Assumption 4 [In‡uence Function] For all weighting functions considered,

(i) R ( (;

! Y))2

!

Y () , and

(ii) R (;

! Y)

!

YjX(;) is continuously di¤erentiable for all x in ,

where

! YjX(;

) =(1 (

)) (

)

YjT;X(; 1

) +(0 (

))

(1 ( ))

YjT;X(; 0

) (7)

Part ( ) of Assumption 4 is a condition of …nite variance of the in¡uence function. It is

de…nitely important because it allows for a central limit theorem to be used. Part ( ) is

analogous to the more technical requirement imposed by Hirano, Imbens and Ridder (2003, assumption 3) that the conditional expectation of¢ be continuously di¤erentiable. As we have

a more general framework, we need that the conditional expectation of the£¤¡uence function

be continuously di¤erentiable.

We are now able to state the central result. We …rst consider a proposition that establishes asymptotic normality for the estimators of inequality measures using the empirical distribution

b

!=b!

Y . The assumptions required for the proposition have been stated along the ¥ ¦§¥ ¨ we also

invoke an assumption presented in the appendix (Assumption A.1) that guarantees uniform convergence of the estimated propensity-score to the true one. Asymptotic properties of our estimators of the inequality treatment e¤ects will follow after that as a direct consequence.

Proposition 1 Let ©( (

))ª«( (

)) ¬ «(

). Then, under Assumptions 1-4 and A.1:

N ® b

!=b!

Y ®(

! Y) =

1 N N X i=1 (

¯i(°i)) ( ¢i;

! Y)

+1

N N X i=1 ±[©(¯(°)) (¢; !

Y)²°=°i](¯i (°i)) +³p(1)

D

´N(0µ )

where

µ =±

h

((¯(°)) (¢;

!

Y) +±[©(¯(°)) (¢;

!

Y)²°](¯ (°)))

2i

Proposition 1 follows immediately after the results by Newey¶ ·¸¸ ¹ ºand, particularly, those

by Hirano, Imbens and Ridder (2003) and Chen, Hong and Tarozzi (2008). Proposition 1 is a general result that specializes for the four weighting functions considered here. It can be easily seen that the derivatives of the four weighting functions with respect to() are

©1(»())ª

@!1(t;p(x))

@p(x) =

t p2

(x) ©0(»())ª

@!0(t;p(x))

@p(x) =

1 t

(1 p(x))2

©11( »(

))ª

@!11(t;p(x))

@p(x) = 0 ©01( »(

))ª

@!01(t;p(x))

@p(x) = _p₍₁1_p₍t_x₎₎2

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Proposition 2 Under Assumptions 1-4 and A.1:

¼

N ½ b

= ¼1

N N

X

i=1

¾1(¿iÀÁ(Âi))½Ã Äi;Å

!=!1

Y ¾0(¿iÀÁ(Âi))½Ã Äi;Å

!=!0

Y

+¼1

N N

X

i=1

(Æ Ç1(¿ÀÁ(Â))½Ã Ä;Å

!=!1

Y ÈÂ =Âi

Æ Ç0(¿ÀÁ(Â))½Ã Ä;Å

!=!0

Y ÈÂ =Âi )½(¿i Á(Âi)) +Ép(1)

D

ÊN(0ÀË)

¼

N ½ b

T T

= ¼1

N N

X

i=1

¾11(¿iÀÁ(Âi))½Ã Äi;Å

!=!11

Y ¾01(¿iÀÁ(Âi))½Ã Äi;Å

!=!01

Y 1 ¼ N N X i=1

Æ Ç01(¿ÀÁ(Â)) ½

Ã Ä;Å

!=!01

Y È Â =Âi ½(

¿i Á(Âi)) + Ép(1)

D

ÊN(0ÀËT)

¼

N ½ b

C C =

1 ¼ N N X i=1

Ã (Äi;ÅY) ¾0(¿iÀÁ(Âi)) ½

Ã Äi;Å

!=!0

Y 1 ¼ N N X i=1

!=!0

Y ÈÂ=Âi ½(¿i Á(Âi)) +Ép(1)

D

ÊN(0ÀËC)

whereË,ËT andËC , whose formulas are given below, are the semiparametric e¢ciency bounds

for, respectively, , _T, and _C

Ë =Æ[(¾1(¿ÀÁ(Â))½Ã Ä;Å

!=!1

Y ¾0(¿ÀÁ(Â))½Ã Ä;Å

!=!0

Y + (Æ Ç1(¿ÀÁ(Â))½Ã Ä;Å

!=!1

Y ÈÂ

!=!0

Y ÈÂ )½(¿ Á(Â)) )

2_]

ËT = Æ[(¾11(¿ÀÁ(Â))½Ã Ä;Å

!=!11

Y ¾01(¿ÀÁ(Â))½Ã Ä;Å

!=!01

Y

!=!01

Y ÈÂ ½(¿ Á(Â)) )

2

]

ËC = Æ[(Ã (Ä;ÅY) ¾0(¿ÀÂ)½Ã Ä;Å

!=!0

Y

Æ Ç0(¿ÀÁ(Â)) ½

Ã Ä;Å

!=!0

Y È Â

½(

¿ Á(Â)) )

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Valid inference for inequality treatment e¤ect parameters can be implemented either by estimation of the analytical expressions for the variance terms presented in Proposition 2 or by resampling methods, such as the bootstrapping. Given the asymptotic normality of the estimators, the bootstrap may be a valid and is surely an easier alternative for calculation of standard errors. In the next sections, we present a Monte Carlo exercise and an empirical application that use bootstrapped standard errors.

5 A Monte Carlo Exercise

In this section we report the results of Monte Carlo exercises. The interest is in learning how the estimators for the overall inequality treatment e¤ect (ITE) and estimators of their asymptotic variances behave in …nite samples. Ìne thousand replications of the experiment with sample

sizes of 500 and 2,500 observations were considered.

We design the data generation process to produce “selection on observables”, that is, the conditional distribution of Í given Î will di¤er from the marginal distribution of Í, but

marginal distributions of the potential outcomes will be independent ofÎ given Í. Note that

asÏ(1)andÏ(0)are known for each observation Ð, we can compute “unfeasible” estimators of

parameters of the marginal distributions ofÏ(1)and Ï(0).

The generated data follows a very simple speci…cation. Starting with Í = [Í1ÑÍ2]

| _we

set Í1 ÒUnif

h

Ó

X1

p

12

2 ÑÓ

X1+

p

12 2

i

and Í2 Ò Unif

h

Ó

X2

p

12

2 ÑÓ

X2 +

p

12 2

i

, which will be independent random variables with the following means and variances: Ô[Í1] =Ó

X1,Ô[Í2] = Ó

X2 andÕ [Í1] =Õ [Í2] = 1. The treatment indicator is set to be

Î = 1IÖ 0+ 1Í1+ 2Í2+ 3Í

2

1 + 4Í

2

2 + 5Í1Í2+×Ø0Ù

where×has a standard logistic c.d.f. Ú (Û) = 1 + exp ÜÛÝ Þ

3 1. The potential outcomes are

Ï (0) = exp

00+ 01Í1+

02Í2+

03Í

2

1 + 04Í

2

2 + 05Í1Í2+ß0

Ï (1) = exp

10+ 11Í1+

12Í2+

13Í

2

1 + 14Í

2

2 + 15Í1Í2+ß1

where

ß0 =

s

00+ s01Í1+

s

02Í2+

s

03Í

2

1 + s04Í

2

2 + s05Í1Í2

àá0 ß1 =

s

10+ s11Í1+

s

12Í2+

s

13Í

2

1 + s14Í

2

2 + s15Í1Í2 àá1

and where á0 and á1 are distributed as standard normals. The variables Í,×, á0 and á1 are

mutually independent. Under this speci…cation, Ï(1) and Ï(0) will not have a closed form

distribution. We present in the Appendix a characterization of their conditional densities given

Í1 =â1 and Í2 =â2 as functions of the parameters. The unconditional pã äãåã æs are obtained

through numerical integration.

The parameters were chosen to be Ó

X1 = 1,Ó

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Table 1: Parameter speci…cation for Monte Carlo Exercise

coef f:é| 0 1 2 3 4 5

j 1 10 2 10 3 10

0j 0:01 0:01 0:01 0:01 0:01 0:02

1j 0:1 0:01 0:01 0:01 0:01 0:01

s

0j 0:01 0:01 0:01 0:01 0:01 0:02

s

1j 0:01 0:01 0:01 0:01 0:01 0:01

The values of some functionals of the distributions of are listed below:

Table 2: Features of the Distributions of Potential êëì íî ïðs

éDistribution Y(0) Y(1)

Mean ñ òó ô 1.83

Standard Deviation (s.d.) ñ òõ ö 1.08

Mean of ÷î øùrithm úñ òõô 0.û ô

S.D. of ÷îøù üithm 0.35 0.45

10th Percentile 0.47 ñòô ô

1stý ëùüìþle ñ òöÿ 1.22

Median 0.80 1.55 3rdý ëùüìile ñ ò ôû 2.ñô

ôñì9Percentile 1.07 õòôõ

Table 3: Inequality Measures of Potential êëì íî ïðs

éDistribution Y(0) Y(1)

Coe¢cient of Variation 0.33 0.5 ô

Interquartile Range 0.31 0.87 Theil Index 0.05 0.13 Gini Coe¢cient 0.18 ñ òõ ö

We provide in Tables 4-7 results for the unfeasible estimator and also for three types of estimators that do not use information from Y(1) and Y(0) but instead use information from usually available data (Y; T; X). The …rst one is the estimator proposed here and labelled “feasible estimator”. The second is the one based on the empirical distributions of YjT = 1

and YjT = 0. We call this estimator the “naive estimator”. Given that there is selection into

treatment based on observables, the naive estimator will be inconsistent to the ITE parameters. Finally we consider what we call the “counterfactual distribution (CD) estimator”. This is constructed in the following way. We …rst run two linear regressions (with intercept) ofY on

X1 and X2 forT = 1and of Y onX1 and X2 forT = 0. Save residualsubj, coe¢cient estimates

b0j,b1j,b2j, ands2j wherej = 0;1 indexes treatment groups,sj2 = (Nj 3) 1PiNj(ubji)2 and N1 =PNi Ti and N0 =N N1. Then, let Yij be counterfactual of observationi:

Y_i₁ = Yi

b01+b11X1i+b

21X2i+

p

s2

1=s20ub0i

ifTi = 1

ifTi = 0

Y_i₀ = Yi

b00+b10X1i+b

20X2i+

p

s2

0=s21ub1i

ifTi = 0

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and since Y_i₁ and Y_i₀ are well de…ned for all i, we compute the inequality measures for these two distributions. Note that this is a way of “controlling” for covariates.

Results in Tables 4 and 5 show distribution features for each one of the estimators of in-equality treatment e¤ect parameters. We report average, standard deviation and quantiles (10th percentile, lower quartile, median, upper quartile and 90th percentile) for the four types of treatment e¤ects on inequality measures here considered (coe¢cient of variation, interquar-tile range, Theil index and Gini coe¢cient). Besides those inequality treatment e¤ects, we also report results for average treatment e¤ects. Finally, we present results that compare the estimates with the population target. Those are reported by the bias, root mean squared er-ror, mean absolute error and the coverage rate of con…dence intervals.

13 _Tables

6 and 7

show some features of the distribution of bootstrapped standard errors for 1,000 Monte Carlo replications using within each replication a bootstrapping procedure with 100 repetitions.

The results indicate that the “feasible estimator” performs well according to the MSE criteria and that its variance shrinks as expected as the sample size increases. Relatively to other estimators being analyzed, the “feasible estimator” has better properties than the naive and the CD estimators, in terms of bias, variance, mean squared error, absolute error and coverage rate.

6 Empirical Application

The empirical application is on a Brazilian public-sponsored job training program, also known as P OR (Plano Nacional de Quali…cação Pro…ssional). That program, which started

in 1

6 has provided classroom training for the formation of the basic skills necessary for

certain occupations (e.g. waiters, hairdressers, administrative jobs). The program operates on a continuous basis throughout the year, with new cohorts of participants starting every month. Although funding comes from the federal government, the program was decentralized at the State level14. Each state subcontracted for classroom training with vocational propri-etary schools and local community colleges. The target population consists of disadvantaged workers, who have been de…ned as the unemployed, and individuals with low level of schooling and/or income. Enrollment of individuals in the program is voluntary, but its scale in 1 was relativley small, being around1 of the labor force in all metropolitan areas in Brazil.

The evaluation of PNFO involved the …rst attempt in the country to perform a

ran-domized study designed to measure impacts of social programs. In the years of 1 the

Brazilian Ministry of L or …nanced an experimental evaluation of the program impact on

earnings and employment.15 _{Experimental data were collected in two metropolitan areas of the}

country, namely Rio de Janeiro and Fortaleza. The process of randomization of individuals in and out of the program took place in August1 and almost all individuals that were selected

in attended the training courses in September 1 In that month, participants in both cities

1 3_{We present coverage rates of two types of}

0% con…dence intervals. In the …rst column, each Monte Carlo

replication used a bootstrap (100 repetitions) variance estimate of that replication. In the second column we used for all replications the same variance estimate, which was the one given by the sampling variation between replications.

1 4_{According to the Brazilian Ministry of}

abor, during January 01 1 and ctober 27 2007, exactly R$

4,312,42,25.55 or US$2, 1, 1,.7 (using June 2008 exchange rate) have been spent on PANFR all over

the country.

1 5_{This data set has been analyzed by Foguel (2007) and in Firpo and Foguel (2008), where more details on}

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were interviewed through the application of the same questionnaire, and retrospective questions were asked about their labor market history. A follow-up survey took place in November

and retrospective questions were asked going back to September

The total available sample size was 4, individuals, out of which 2, were from Rio de Janeiro. We selected individuals who obtained a job between the end of the treatment period and the follow-up survey. We also restricted our sample for individuals between and 50 years old. We ended up with 548 treated individuals and 408 controls in Rio de Janeiro and

7 treated and 7 7 controls in Fortaleza. Table 8 presents some summary statistics of the

sample used in this paper. A few interesting features emerge from that table, revealing that the target population in those two sites, Rio de Janeiro and Fortaleza are intrinsically di¤erent: People in our Fortaleza sample are older (average age of 27 years old) than people in our Rio de Janeiro sample, which consists basically of teenagers/young adults (average age ofyears old).

Average schooling is just above Brazilian elementary schooling of 8 years. In Rio de Janeiro, about 3 0 of sample were not working during the months before the treatment, whereas in

Fortaleza that number is around40

We can see that apparently covariates are unbalanced across treatment status. In Rio de Janeiro, a male worker who was older and unemployed during the months before treatment had a greater chance to be in the treatment group than a young female employed worker. In Fortaleza, race/ethnicity and schooling appear to be unbalanced. The main reason for those di¤erences across groups is that the randomization took place within the class level. There were many di¤erent courses being o¤ered and attracting di¤erent people. Also classes had uneven proportions of treatment and control observations. We thus checked whether at the class level, covariates are unbalanced. In other words, let C be a variable that indicates which class the workers was enrolled andY,T andX be de…ned as before. We test whetherX??TC=cas a

check for good randomization. For Fortaleza, in the original sample there wereclasses(0

with common support, that is, with at least one treated and one control). When we perform a t-test for di¤erences in means at the class level, using either the original or the restricted sample, we rarely observe any signi…cant di¤erence. 22 out of classes present at least

one unbalanced variable. For Rio de Janeiro, the proportion of unbalanced covariates at the class level is of the same order of magnitude (13 of 70). We use that as piece of evidence that the randomization process was done reasonably well.

To avoid having to control for as many as cells, as would be in the case of Fortaleza, we opted for controlling for some pre-treatment covariates. We chose those which seemed to be unbalanced when we performed an unconditional test of di¤erence in means. Thus in Rio de Janeiro we control for age and employment one month before treatment, while in Fortaleza we control for race, age and schooling. We discretized schooling and age by constructing groups.1!

Thus, with all “confounding” variables being discrete, we ended up having cells, but in a smaller number than if we had used classes. For Fortaleza we have 24 cells, whereas for Rio de Janeiro, we ended up with 8 cells. For each cell, we inspected whether we had failures of the common support assumption. All cells had treated and control units, so we did not have a failure of common support. Regarding covariates balancing between treated and control groups, we have that for Rio only 2 out of 8 cells present at least one unbalanced variable, while in Fortaleza we found 7 out of 24 unbalanced cells. That is about the same proportion of unbalancing classes when looking at the class level.

1"

For age we grouped 1#-17, 18-1$, 20-21, and 22-50, whereas for schooling we grouped 0-7 (incomplete

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The outcome variable of interest is the …rst monthly salary received in the …rst job after treatment period. For the whole treatment e¤ect analysis we dropped observations with zero earnings (2 in Rio de Janeiro and 3 in Fortaleza). In Table &we report average and inequality

treatment e¤ects estimates. We have three estimators: naive, which is a simple di¤erence between groups, reweighted and counterfactual estimators. ')* weights in the reweighted

estimators are given by the fully saturated model for the propensity-score, which is feasible as we have discrete covariates. In order to have comparability with the counterfactual estimator, we used cell dummies for the linear regression (see the previous section for details on this estimator). As expected, for average treatment e¤ects, both methods are algebraically identical and we can see that in Table&+ They however di¤er from the naive estimator, which should be inconsistent

as it does not “undo” the mixing of classes with di¤erent proportions of treated and controls. We compute bootstrapped standard errors as we do not derive analytical standard errors for the counterfactual estimator.17 _{Also, for this particular exercise, bootstrapped standard errors}

are simpler to generate for all estimators and should be valid.

We note that the program, when it a¤ects expected earnings (as in Rio de Janeiro), it has a negative impact. However, we also see that for Rio de Janeiro, although the program does seem to induce no average gains, it does reduce inequality among treated. 'ne possible interpretation

is that the program reduces signaling costs, allowing employers to set similar wages for entering workers that have program certi…cates. For Fortaleza, results are that program is ine¤ective in reducing inequality.

We correct for the fact that randomization was clustered by using the reweighted and coun-terfactual estimators. A problem that may emerge with the councoun-terfactual estimator is that it might create negative earnings, as predicted values from the linear regression are not necessar-ily bounded above zero. Having a variable with negative values creates an asymmetry between counterfactual and reweighted estimator since some inequality measures are de…ned only for positive values. We also compute estimators for a sample that drops negative predicted values from the counterfactual approach, for the inequality measures that require positive support. Re-sults in both sites are that the counterfactual estimators are very di¤erent from the reweighted and naive estimators, possibly indicating that such deletion procedure may aggravate the bias problem of the counterfactual estimator.

7 Conclusion

We proposed a method that helps applied researchers who are interested in comparing inequality measures of two or more outcome distributions. When comparing Gini coe¢cients between two groups (for example, treated and non-treated groups), it is important to acknowledge for the fact that there are many observed factors whose distributions di¤er across groups. 'ur method

allows applied researchers to identify what is the impact of the treatment that explain di¤erences in Gini coe¢cients between these two groups of workers and separate it from a composition e¤ect, induced by di¤erent distribution in covariates. ',course, we could make parametric and

functional form assumptions in order to relate covariates to the outcome distribution in each group. However, if we are not willing to impose restrictive parametric assumptions, it is not clear how to compare Gini coe¢cients …xing the distribution of observed covariates.

1 7_{Although for ATT counterfactual and reweighted should have the same variance, for other functionals this}

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-./ method consisted of the following. We …rst estimate, through a reweighing method,

the inequality measures of the potential outcomes, and then take the di¤erence between those estimates. That estimation strategy is useful for policy-making purposes when the individual decision to participate in the social program (the treatment) depends on observable character-istics. If the identi…cation restrictions hold, then the reweighing method allows identifying the distribution of potential outcomes and, therefore, many of their inequality parameters.

We showed that the proposed reweighting inequality treatment e¤ect estimators are root-N

consistent, asymptotically normal and e¢cient. Finally, we performed a series of Monte Carlo exercises and apply the method to a new data set. -2erall and in both the Monte Carlo and the

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APPENDIX

Details of the Monte Carlo Exercise

De…ne:

X= 1; X1; X2; X12; X22; X13X2

|

so a realization ofX is x= 1; x1; x2; x21; x22; x13x2

|

. De…ne also the vector of parameters:

0 = [ 00; 01; 02; 03; 04; 05]|

1 = [ 10; 11; 12; 13; 14; 15]|

s

0 = [ s00; s01; s02; s03; s04; s05]

|

s

1 = [ s10; s11; s12; s13; s14; s15]|:

Thus the under the speci…cation given in the section 5, the conditional density of the potential outcomes will be

f_Y₍₀₎_j_X1;X2(y8x1; x2) = y3

q

2 3(x

| s 0)2

1

3exp

(

1 2 3

ln (y) x| 0

x| s 0

2)

f_Y₍₁₎_j_X1;X2(y8x1; x2) = y3

q

2 3(x

| s 1)2

1

3exp

(

1 2 3

ln (y) x| 1

x| s 1

2)

thus the unconditional densities are given by:

f_Y₍₀₎(y) = 1 12 3

Z X1+

p 12 2 X1 p 12 2

Z _X2+p12 2

X2

p

12 2

f_Y₍₀₎_j_X1;X2(y8x1; x2)3dx2

!

3dx1

f_Y₍₁₎(y) = 1 12 3

Z _X1+p12 2

X1

p

12 2

Z _X2+p12 2

X2

p

12 2

f_Y₍₁₎_j_X1;X2(y8x1; x2)3dx2

!

3dx1

We compute those integrals numerically. :; <e we have the densities of the marginals we are

able to evaluate numerically the statistics that we listed in section 5, as the mean, the median and some inequality measures.

Proofs

Proof of Theorem 1:

For the …rst set of equalities, by the de…nition of observed outcomes (Y =Y (1)3T+Y (0)3

(1 T)) and Assumption 1, we have that the …rst equality holds as the conditional p.d.f. ofY

givenT = 1and X=x and evaluated atyis exactly the p.d.f. of Y (1)givenX=xevaluated aty:

fYjT;X(y; 1; x) =fY(1)jT;X(y; 1; x) =fY(1)jX(y;x)

thus

F_Y₍₁₎(a) =

Z

x2X Z a

y2R

f_Y₍₁₎_j_X(y;x)3dy3fX(x)3dx

=

Z

x2X Z a

y2R

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The second equality holds by the de…nition of weighted c.d.f.s and by the fact thatp(x) > 0

(Assumption 2):

F!=!1

Y (a) =

Z

x2X

(!1(1; p(x))=p(x)=

Z a

y2R

f_Y_j_T;X(y; 1; x)=dy

+!1(0; p(x))=(1 p(x))=

Z a

y2R

f_Y_j_T;X(y; 0; x)=dy)=fX(x)=dx

=

Z

x2X

1

p(x)=p(x)=

Z a

y2R

fYjT;X(y; 1; x)=dy=fX(x)=dx

=

Z

x2X Z a

y2R

fYjT;X(y; 1; x)=dy=fX(x)=dx

and the third is just the de…nition of a weighted expectation given in Equation 4 with the specialization to&(Y) = 1IfY >ag and !=!1:

E[!1(T; p(X))=1IfY >ag]

=

Z

x2X

(!1(1; p(x))=p(x)=

Z a

y2R

f_Y_j_T;X(y; 1; x)=dy

+!1(0; p(x))=(1 p(x))=

Z a

y2R

fYjT;X(y; 0; x)=dy)=fX(x)=dx

=F!=!1

Y (a)

The second set of equalities follows by analogy. The third is trivial as it involves the c.d.f. ofY(1)@T = 1. Finally, we consider the fourth set of equalities. We have again, by Assumption

1:

f_Y_j_T;X(y; 0; x) =f_Y₍₀₎_j_T;X(y; 0; x) =f_Y₍₀₎_j_T;X(y; 1; x)

thus

FY(0)jT=1(a) =

Z

x2X Z a

y2R

fY(0)jT;X(y; 1; x)=dy=fX

jT (x; 1)=dx

=

Z

x2X

p(x) p =

Z a

y2R

f_Y_j_T;X(y; 0; x)=dy=fX(x)=dx

and because of the Baye ABrule we have that

f_X_j_T (x; 1) = p(x)

p =fX(x):

The second equality holds once again by the de…nition of weighted c.d.f.s and by the fact that1 p(x)>0 (Assumption 2):

F!=!01

Y (a) =

Z

x2X

(!01(1; p(x))=p(x)=

Z a

y2R

fYjT;X(y; 1; x)=dy

+!01(0; p(x))=(1 p(x))=

Z a

y2R

fYjT;X(y; 0; x)=dy)=fX(x)=dx

=

Z

x2X

1 1 p(x) =

p(x)

p =(1 p(x))=

Z a

y2R

f_Y_j_T;X(y; 0; x)=dy=fX(x)=dx

=

Z

x2X

p(x) p =

Z a

y2R

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and …nally the last equality holds as

E[!01(T; p(X))C1IDY EaF]

=

Z

x2X

(!01(1; p(x))Cp(x)C

Z a

y2R

f_Y_j_T;X(y; 1; x)Cdy

+!01(0; p(x))C(1 p(x))C

Z a

y2R

f_Y_j_T;X(y; 0; x)Cdy)CfX(x)Cdx

=F!=!01

Y (a)

Q:E:D:

Proof of Corollary 1:

By de…nition of ITE parameters, we have that they are the following di¤erences in func-tionals of the distributions:

= F_Y₍₁₎ F_Y₍₀₎ = F!=!1

Y F

!=!0

Y

T=1 = FY(1)jT=1 FY(0)jT=1 = FY!=!11 F !=!01

Y

C = (FY) FY(0) = (FY) F_Y!=!0

and therefore those three parameters can be expressed as functions of the observable data (Y,T,X).

Q:E:D:

Proof of Proposition 1

In order to be able to prove Proposition 1, we need …rst to guarantee that the propensity-score estimated here as proposed by Hirano, Imbens and Ridder (2003) is uniformly consistent for the true propensity-score. They show that with an extra assumption uniform consistency is achieved. For sake of completeness we state such assumption and the desired result:

Assumption A.1 [First Step]:

(i)X is a compact subset ofR

r

G

(ii) the density ofX,f(x), satis…es0<infx2Xf(x)Esup

x2Xf(x)<H

(iii) p(x)is s-times continuously di¤erentiable, where sI7r and r is the dimension ofXG

(iv) the order ofHK(x),K, is of the formK =CCN

c_where_C_{is a constant and}_c

J

1 4(s

r 1)

;1₉

Newey KMN N5, MNN Q R has established that for orthogonal polynomials HK(x) and compact

X:

(K) = sup x2X

kHK(x)kECCK (A-1)

whereC is a generic constant. Note then that because of part (iv) of Assumption A.1, will be a function ofN sinceK is assumed to be a function of N.

Uniform consistency of the estimated propensity-score is guaranteed by the following lemma: Lemma A.1 [First Step]: Under Assumptions 2 and A.1, the following results hold: (I) sup_x_2XSp(x) pK(x)S E CC (K)CK

s=2r

E CCK

1 s=2r

E CCN

(1 s=2r)c ₌ _o(1)

G where

pK(x) =L(HK(x)0 K)and K =argmax ₂RKE p(X)Clog(L(HK(X)

0 ₎₎₊₍₁ _p(X))

Clog(1

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(II)T^K KT=Op

q

K(N)

N UCVOp

q

Nc

N UCVOp N c 1

2 =o

p(1)W

(III) sup_x_2XYp(x) bp(x)Y U C1 VN

(1 s=2r)c ₊_Op _(K)

V

q

K(N)

N U C1 VN

(1 s=2r)c ₊_C

2 V

N(3c 1)=2

VOp(1) =op(1)W

(IV) There is" >0: limN!1Pr[" <infX2Xp(X)^ Usup

X2Xp(X)^ <1 "] = 1.

Proof of Lemma A.1: See Hirano, Imbens and Ridder (2003), Z [\mas 1 and 2.

Now consider

p

NV Fb

!=b!

Y (FY!) . Under the asymptotic linearity assumption

(As-sumption 3) we have

p

NV Fb

!=!b

Y (FY!) (A-2)

= p1

N

V

N

X

i=1

b

!iV (Yi;F

!

Y) +op(1) (A-3)

= p1

N

V

N

X

i=1

(!_bi !(Ti; p(Xi)))V (Yi;F

!

Y) (A-4)

1

p

N

V

N

X

i=1

h(Ti; p(Xi))V (Yi;F

!

Y)V(pb(Xi) p(Xi)) (A-5)

+p1

N

V

N

X

i=1

h(Ti; p(Xi))V (Yi;F

!

Y)V(pb(Xi) p(Xi)) ) ]^ _ `a

1

p

N

V

N

X

i=1

E[h(T; p(X))V (Y;F

!

Y)YX=Xi]V(Ti p(Xi)) (A-7)

+p1

N

V

N

X

i=1

!(Ti; p(Xi))V (Yi;F

!

Y) (A-8)

+p1

N

V

N

X

i=1

E[h(T; p(X))V (Y;F

!

Y)YX=Xi]V(Ti p(Xi)) +op(1) ]^ _ ba