metadata: authors: 0: author: Ferreira,Francisco H. G.
1: author: Firpo,Sergio
2: author: Galvao,Antonio F.
count: Brazil,United States,World
docna: 0: docna: Estimation and inference for actual and
            counterfactual growth incidence curves
docty: Policy Research Working Paper
subtopic: Economic Theory & Research,Industrial Economics,Labor Markets,Educational Sciences,Gender and Development,Economic Growth,Inequality
teratopic: Poverty Reduction
abstracts: cdata!: Different episodes of economic growth
            display widely varying distributional characteristics, both
            across countries and over time. Growth is sometimes
            accompanied by rising and sometimes by falling inequality.
            Applied economists have come to rely on the Growth Incidence
            Curve, which gives the quantile-specific rate of income
            growth over a certain period, to describe and analyze the
            incidence of economic growth. This paper discusses the
            identification conditions, and develops estimation and
            inference procedures for both actual and counterfactual
            growth incidence curves, based on general functions of the
            quantile potential outcome process over the space of
            quantiles. The paper establishes the limiting 0 distribution
            of the test statistics of interest for those general
            functions, and proposes resampling methods to implement
            inference in practice. The proposed methods are illustrated
            by a comparison of the growth processes in the United States
            and Brazil during 1995-2007. Although growth in the average
            real wage was disappointing in both countries, the
            distribution of that growth was markedly different. In the
            United States, wage growth was mediocre for the bottom 80
            percent of the sample, but much more rapid for the top 20
            percent. In Brazil, conversely, wage growth was rapid below
            the median, and negative at the top. As a result, inequality
            rose in the United States and fell markedly in Brazil.
geo_reg_key: 517191,119228
repnme: repnme: Estimation and inference for actual and
            counterfactual growth incidence curves
display_title: Estimation and inference for actual
            and counterfactual growth incidence curves
listing_relative_url: /research/2017/01/27075287/estimation-inference-actual-counterfactual-growth-incidence-curves
topicv3: Economic Growth m1326241 471,Poverty m1326687 695,Education m1327048 470,Gender m1327621 615,Labor Earnings m1504041 2738
publishtoextweb_dt: 2017-01-05T00:00:00Z
historic_topic: Poverty Reduction
no_of_pages: 61
author_keys: 0: author_key: 125055
1: author_key: 969625
model_extractions: WPS7933 Policy Research Working Paper 7933 Estimation and Inference for Actual and Counterfactual Growth Incidence Curves _Francisco H. G. Ferreira Sergio Firpo Antonio F. Galvão_ Development Research Group Poverty and Inequality Team January 2017
Policy Research Working Paper 7933

# **Abstract**

Different episodes of economic growth display widely varying distributional characteristics, both across countries and over time. Growth is sometimes accompanied by rising and sometimes by falling inequality. Applied economists have come to rely on the Growth Incidence Curve, which gives the quantile-specific rate of income growth over a certain period, to describe and analyze the incidence of economic growth. This paper discusses the identification conditions, and develops estimation and inference procedures for both actual and counterfactual growth incidence curves, based on general functions of the quantile potential outcome process over the space of quantiles. The paper establishes the limiting null distribution of the test statistics of interest for those general functions, and proposes resampling methods to implement inference in practice. The proposed methods are illustrated by a comparison of the growth processes in the United States and Brazil during 1995–2007. Although growth in the average real wage was disappointing in both countries, the distribution of that growth was markedly different. In the United States, wage growth was mediocre for the bottom 80 percent of the sample, but much more rapid for the top 20 percent. In Brazil, conversely, wage growth was rapid below the median, and negative at the top. As a result, inequality rose in the United States and fell markedly in Brazil.

This paper is a product of the Poverty and Inequality Team, Development Research Group. It is part of a larger effort by the World Bank to provide open access to its research and make a contribution to development policy discussions around the world. Policy Research Working Papers are also posted on the Web at http://econ.worldbank.org. The authors may be contacted at fferreira@worldbank.org; firpo@insper.edu.br; and antonio-galvao@uiowa.edu.

_The Policy Research Working Paper Series disseminates the findings of work in progress to encourage the exchange of ideas about development issues. An objective of the series is to get the findings out quickly, even if the presentations are less than fully polished. The papers carry the names of the authors and should be cited accordingly. The findings, interpretations, and conclusions expressed in this paper are entirely those of the authors. They do not necessarily represent the views of the International Bank for Reconstruction and Development/World Bank and its affiliated organizations, or those of the Executive Directors of the World Bank or the governments they represent._ Produced by the Research Support Team
# Estimation and Inference for Actual and Counterfactual Growth Incidence Curves<sup>_∗_</sup>

Francisco H. G. Ferreira<sup>_†_</sup> Sergio Firpo<sup>_‡_</sup> Antonio F. Galvao<sup>_§_</sup> Keywords: Growth Incidence Curves; Potential outcomes; Inference; Quantile Process JEL Classification: C14, C21, D31, I32.

> _∗_ The authors would like to express their appreciation to Matias Cattaneo, Yu-Chin Hsu, David Kaplan, Ying-Ying Lee, Zhongjun Qu, Alexandre Poirier, Liang Wang and participants at the 2015 meeting of the Midwest Econometric Group for useful comments and discussions regarding this paper. Vitor Possebom provided excellent research assistance. Computer programs to replicate the numerical analyses are available from the authors. All the remaining errors are ours.

> _†_ Development Research Group, The World Bank, 1818 H Street, NW, Washington, DC, 20433. E-mail: `fferreira@worldbank.org`

> _‡_ Insper, Rua Quata 300, Sao Paulo, SP 04546-042. E-mail: `firpo@insper.edu.br`

> _§_ Department of Economics, University of Iowa, W284 Pappajohn Business Building, 21 E. Market Street, Iowa City, IA 52242. E-mail: `antonio-galvao@uiowa.edu`
# **1 Introduction**

Growth episodes have displayed widely different distributional characteristics across countries and over time. The same rate of growth in average incomes has been accompanied by rising inequality in some cases, and by falling inequality in others. A large literature on “pro-poor growth”and, more generally, on the incidence of economic growth processes has developed, and attracted attention among both researchers and policymakers.

Over time, this literature has come to rely heavily on the Growth Incidence Curve ( _GIC_ ), which describes the rate of income growth at each quantile _τ ∈_ (0 _,_ 1) of the (anonymous) distribution (Ravallion and Chen (2003)). It has been used to compare the distributional characteristics of growth processes both across countries and over time (see, e.g. Besley and Cord (2007)). It has also been shown to underlie changes in certain widely-used classes of poverty and inequality measures, which can be formally expressed as functionals of the _GIC_ (Ferreira (2012)).

Growth incidence curves have also featured in a long-standing literature that uses counterfactual income distributions to decompose changes (or differences) in inequality and poverty over time (or between countries), and to attribute such changes to different factors such as, for example, changes in worker characteristics or in the returns to those characteristics. The original contributions to this literature, including Juhn, Murphy, and Pierce (1993), Dinardo, Fortin, and Lemieux (1996) and Donald, Green, and Paarsch (2000), predate the Ravallion and Chen (2003) article that introduced the term _GIC_ , and hence do not use it. Yet, each of those papers sought to account for differences across entire wage or income distributions – – which can be formally expressed as _GIC_ s using counterfactual distributions. Ferreira (2012) defines counterfactual growth incidence curves as functionals of counterfactual distributions, and establishes the link to this earlier literature on distributional change.

Despite their conceptual importance and widespread practical use, however, formal conditions for identification and inference using growth incidence curves – actual or counterfactual – have not been established. In this paper, we rely on the formal analogy between distributional change and treatment heterogeneity to fill that gap. More specifically, we write both actual and counterfactual _GIC_ s in terms of vectors of potential outcomes (Rubin (1977)), and then apply suitable variants of a number of results from the literature on quantile treatment effects to formally establish the conditions for identification of the _GIC_ . Specifically, we adapt the identification results in Firpo (2007), where the relevant identification restric1
tion is the ignorability assumption.<sup>1</sup> In our context, it implies that the income distributions that we observe in two different time periods are generated by two group of factors only: observable components whose distributions may vary over time, and unobservable components whose conditional distributions given observables are fixed over time.

We then propose a simple three-step semiparametric estimator for both actual and counterfactual growth incidence curves, which relies on established sample re-weighting and quantile regression techniques. In the first step, a nonparametric estimator of the propensity score is used, and weights are computed. In our setup, the propensity score is computed by pooling the repeated cross-section data for initial and end periods and calculating the probability of being observed at the final period, given covariates. In the second step, one obtains properly weighted quantiles of the outcome from a simple weighted quantile regression. The third step is the computation of the _GIC_ as a function of the vector of quantiles of weighted outcome distributions.<sup>2</sup> When applied to counterfactual _GIC_ s, this procedure has the added advantage that it requires no assumption on the structural relationships between income and its covariates, as was the case with most of the previous literature.

We establish the asymptotic properties of these estimators, propose suitable test statistics, and discuss inference procedures in practice. For practical inference we compute critical values using resampling methods. We provide sufficient conditions and show the theoretical validity of a bootstrap approach. Moreover, we discuss in detail an algorithm for its practical implementation. We also discuss computation of critical values through a subsampling method.

The main technical contributions of the paper are as follows. The first is to develop practical statistical inference procedures for the _GIC_ . This enables researchers to conduct estimation and inference for the _GIC_ over the entire set of quantiles. Secondly, we can easily extend our results to general functionals of the vector of quantiles of potential outcomes and not only the one that yields the _GIC_ , which allows us to develop testing procedures for general hypotheses involving these functionals.<sup>3</sup> An additional by-product contribution of

> 1This condition has been employed widely in the distributional treatment effect literature. See not only Firpo (2007), but also, among others, Flores (2007), Cattaneo (2010), and Galvao and Wang (2015).

> 2A natural extension of our method – not pursued in this paper – would be to implement a fourth step, which would involve estimation and inference of real-valued functionals of the _GIC_ process, such as poverty and income inequality growth.

> 3The theoretical results derived in this paper can be applied to other functionals of the quantiles of potential outcomes processes. For instance, the quantile treatment effects in Firpo (2007), and the Makarov bounds for the quantiles of the distribution of treatment effects discussed in Fan and Park (2010), although following a more elaborate formula, are also functionals of the quantiles of the potential outcomes. In general, our final estimator can be seen as a plug-in estimator of the functional using the estimated quantiles

2
this paper is to establish the asymptotic properties of the estimator of the vector of quantiles of weighted outcome distributions for the quantile process, namely, uniform consistency and weak convergence. The provision of uniform results over the set of quantiles is a necessary condition to establish the results for the testing procedures. We also show that the estimator is uniformly efficient, as the asymptotic variance of the estimator coincides with the semiparametric efficiency bound.

These contributions are closely related to the literature on quantile treatment effects, which is a particular functional of the vector formed by the quantiles of the potential outcomes. That literature started with Doksum (1974) and Lehmann (1974) and has expanded recently (see, e.g., Abadie, Angrist, and Imbens (2002), Chernozhukov and Hansen (2005), Bitler, Gelbach, and Hoynes (2006), Firpo (2007), Cattaneo (2010), Donald and Hsu (2014), Galvao and Wang (2015), and Firpo and Pinto (2015)).<sup>4</sup> We illustrate the proposed procedure by comparing actual and counterfactual growth incidence curves (for real hourly wages) for the two largest countries in the Western Hemisphere, namely the United States and Brazil, in the twelve years prior to the onset of the last great financial crisis: 1995-2007. Although growth rates in average wages were disappointing in both countries (especially in Brazil), there were substantial differences in inequality dynamics. The _GIC_ for the US was flat until approximately the 8 _th_ decile, and sharply upward-sloping over the top quintile. In Brazil conversely, the _GIC_ peaked around the first quintile, and was downward sloping thereafter. As a result wage inequality rose sharply in the US and declined in Brazil.

We use counterfactual _GIC_ s to examine whether these changes were driven primarily by the composition of the labor force - in terms of observed worker characteristics such as gender, age, and education - or by changes in the broader structure of the economy. In both countries, we find that increases in worker age (and thus potential experience) and education contributed to income growth in a roughly equiproportional manner. Changes in inequality were driven almost entirely by changes in economic structure.

The remainder of the paper is organized as follows. Section 2 defines the _GIC_ . Section 3 presents the econometric results, describes the three-step estimator, establishes the asymptotic properties of the estimator, discusses inference for the quantile process, and its of potential outcomes.

> 4The results of this paper are also related to those on inference on the quantile process. See, e.g., Belloni, Chernozhukov, and Fernandez-Val (2011), and Qu and Yoon (2015) for the nonparametric case; Gutenbrunner and Jureckova (1992), Koenker and Machado (1999), Koenker and Xiao (2002), Chernozhukov and Fernandez-Val (2005), and Angrist, Chernozhukov, and Fernandez-Val (2006) for the parametric case.

3
practical implementation. The empirical application to the US and Brazil is presented in Section 4. Section 5 concludes. We relegate the proofs of the results to the Appendix.

# **2 Growth incidence curves: Actual and counterfactual**

In this section we formally define the growth incidence curve ( _GIC_ ), which was originally introduced by Ravallion and Chen (2003). Let _Y_ be the outcome variable of interest, say an indicator of economic welfare such as income. There are two time periods, 0 and 1. Let us say that an individual observation taken at time 1 belongs to group _A_ , ie, _G_ = _A_ . An observation taken at time 0 belongs to group _B_ , or _G_ = _B_ . Assume that income is continuously distributed over the population of interest, and denote its cumulative distribution function (CDF) at time _t_ as _FY |T_ ( _·|t_ ). The income level at the _τ_ -th quantile for groups _A_ and _B_ are given by, respectively, the inverse of the CDF, _qA_ ( _τ_ ) = _FY_<sup>_−_</sup> _|_<sup>1</sup> _T_<sup>(</sup><sup>_τ|_1)and</sup> _qB_ ( _τ_ ) = _FY_<sup>_−_</sup> _|_<sup>1</sup> _T_<sup>(</sup><sup>_τ|_0).Then,theinstantaneous</sup><sup>_GIC_atagiventime</sup><sup>_t_andquantile</sup><sup>_τ_canbe</sup> _dF_<sup>_−_1</sup> represented as _FY_<sup>_−_</sup> _<u>|T</u>_<sup>1(</sup><sup>_τ|t_)</sup><sup>_/dt_.Indiscretetime,theincomegrowthrateforagivenquantile</sup><sup>_τ_</sup> _Y |T_<sup>(</sup><sup>_τ|t_)</sup> between two time periods, 0 and 1, can then be written as

Motivated by the importance of the _GIC_ for the economic analysis of social welfare, this paper develops estimation and inference procedures for the _GIC_ ( _τ_ ), which is calculated as the difference of quantiles in time periods 1 and 0 over the quantile in time zero, for the entire set of quantiles _τ ∈_ (0 _,_ 1). We assume availability of a random sample of size _n_ from the joint distribution of ( _Y, T,_ **_X_** ), where _Y_ is the income, _T_ is a time dummy variable that equals 1 at period _T_ = 1, and **_X_** is a vector of length _d_ of covariates. We could have represented the data equivalently as ( _Y, G,_ **_X_** ).

The covariates enable us to learn how changes in their joint distribution affect growth and inequality. For an individual _i_ in our sample, if _Gi_ = _A_ we observe _Yi_ (1), otherwise _Gi_ = _B_ and we observe _Yi_ (0), where _Yi_ (1) is what individual _i_ ’s outcome would be were she observed at time _T_ = 1, and _Yi_ (0) is what individual _i_ ’s outcome would be were she observed at time _T_ = 0. Borrowing from the treatment effect literature, we call _Y_ (1) and _Y_ (0) ‘potential outcomes’; we say that individual _i_ is ‘treated’ if she is observed at period 1 or group _A,_ and ‘untreated’ if observed at period 0 or group _B_ . We may refer to _T_ as the 4
‘treatment assignment dummy’ or, more accurately, ‘time assignment dummy’. Thus, the observed outcome is _Y_ = ( _Y_ (1) _− Y_ (0)) _T_ + _Y_ (0).

Writing the problem in terms of potential outcomes is useful because it allows us to easily write both actual and counterfactual distributions. For example, the actual outcome distribution for those individuals from group _B_ , that is, those who were observed at time 0 _,_ is _FY_ (0) _|T_ ( _·|_ 0) and the actual outcome distribution for those individuals from group _A_ , that is, those who were observed at time 1 _,_ is _FY_ (1) _|T_ ( _·|_ 1). The counterfactual outcome distribution for those individuals who were observed at time 0 _,_ were they observed at time 1 _,_ is _FY_ (1) _|T_ ( _·|_ 0) and the counterfactual outcome distribution for those individuals who were observed at time 1, were they observed at time 0 _,_ is _FY_ (0) _|T_ ( _·|_ 1).

Let _τ_ be a real number in _T ⊂_ (0 _,_ 1) and _t_ = 0 _,_ 1. Let _qAt_ ( _τ_ ) be inf _q_ Pr[ _Y_ ( _t_ ) _≤ q|T_ = 1] _≥ τ_ , or the _τ_ th quantile of _FY_ ( _t_ ) _|T_ ( _·|_ 1), which is the distribution function of _Y_ ( _t_ ) for the subpopulation _A_ . For the _B_ subpopulation, let _qBt_ ( _τ_ ) be inf _q_ Pr[ _Y_ ( _t_ ) _≤ q|T_ = 0] _≥ τ_ , or the _τ_ th quantile of _FY_ ( _t_ ) _|T_ ( _·|_ 0). For both subpopulations, those distribution functions share the same support, which is _Yt ⊂_ R. Let us also define

Thus, the _GIC_ can be derived from the previous variables as the growth rate of income at the _τ_ th quantile between periods 0 and 1. We first define the observed or actual _GIC_ as The graphical depiction of _GIC_ , as proposed in Ravallion and Chen (2003), is obtained by letting _τ_ vary from zero to one and plotting the corresponding values of _GIC_ against the quantiles _τ_ . The quantiles involved in the computation of equation (1) are based on the ranking of individuals in each distribution of interest. Therefore, unless the individual _i_ keeps her ranking over time, _GIC_ will not be an appropriate tool to infer individual gains over time. This is a consequence of the veil of ignorance (anonymity) shrouding the comparison of the two distributions (see Essama-Nssah, Paul, and Bassole (2013)).

The interpretation of the graphical depiction of _GIC_ is simple. If the _GIC_ is a decreasing function for all _τ_ in its domain of definition, then all inequality measures that respect the 5
Pigou-Dalton principle of transfers and scale invariance will indicate a fall in inequality over time. If instead, the _GIC_ is an increasing function of _τ_ , then the same measures will register an increase in inequality (Ravallion and Chen (2003)). When no relative inequality measure changes over time, then the _GIC_ will present a constant growth rate over the process of quantiles _τ_ .

Using our previous notation, we can define _GIC_<sup>_∗_</sup> as the counterfactual _GIC_ . It can be expressed as which is the growth incidence curve for quantile _τ_ if the distribution of associated factors (explanatory variables, or covariates) had remained fixed from period 0 to 1. _GIC_<sup>_∗_</sup> captures only that part of distributional change associated with changes in the conditional distribution _FY_ ( _·_ ) _|T_ , which we interpret broadly as changes in the structure of the economy.

Comparing _GIC_ with _GIC_<sup>_∗_</sup> allows us to understand whether heterogeneity in economic growth is driven by changes in the joint distribution of observed covariates ( **_X_** ) that impact income, or is driven by changes in the structure of the economy. For example, if _GIC_ is decreasing in _τ_ but _GIC_<sup>_∗_</sup> is uniform (flat) over _τ_ , the decrease in inequality is driven by changes in the distribution of covariates. This interpretation can be formally obtained by decomposing the _GIC_ ( _τ_ ) into two components as following: where is the growth incidence curve that would have occurred only because of time changes in the distribution of covariates.

We will develop estimation and inference procedures for the _GIC_ ( _τ_ ) and _GIC_<sup>_∗_</sup> ( _τ_ ) and, more generally, for functionals of the quantile of potential outcomes. In that sense, our theoretical framework provides a flexible method for the practical analysis of the growth incidence curves.

6
# **3 The econometric model**

In this section we introduce the econometric model, discuss identification, estimation of the parameters of interest, and inference procedures. As previously seen, _GIC_ can be written as a function of the vector of quantiles of potential outcomes. Thus, in this section, we first obtain the results for the latter, and then, for the _GIC_ . **Notation:** Let E and E be _p p_<sup>_∗_</sup> expectation and sample average, respectively. Let ⇝, _→_ , and _→_ denote weak convergence, and convergence in probability and in outer probability, respectively. Let _|g_ ( _z_ ) _|∞_ denote sup _z |g_ ( _z_ ) _|_ for _z ∈Z_ .

## **3.1**

In order to make our setup comparable with the treatment effects literature, we maintain all definitions and notation as it is commonly used in that framework. Therefore, we have a random sample of size _n_ from the joint distribution of ( _Y, T,_ **_X_** ), where _Y_ is the outcome of interest, _T_ is a dummy variable of treatment assignment, and **_X_** is a vector of length _d_ of covariates. For completeness, in this section, we also define _qt_ ( _τ_ ) as inf _q_ Pr[ _Y_ ( _t_ ) _≤ q_ ] _≥ τ_ , for _t_ = 0 _,_ 1, which is the unconditional _τ_ th quantile of _FY_ ( _t_ ), the distribution function of _Y_ ( _t_ ) whose support is _Yt ⊂_ R.

Now we define the _p_ -score, the conditional probability of being treated (observed at time 1) given **_X_** , as _p_ ( **_X_** ), and the unconditional probability as _p_ . Let **_X_** _∈X ⊂_ R<sup>_d_</sup> .

In what follows, it is also useful to define the function _m_ as: _m_ ( _a, b_ ; _τ_ ) = _τ −_ **1** _{a < b}_ . We state assumptions on the general model for identification of the parameters of interest.

- **I.I** For each _τ ∈T_ , _t_ = 0 _,_ 1, _qt_ ( _τ_ ) uniquely solves E[ _m_ ( _Y_ ( _t_ ) _, qt_ ( _τ_ ); _τ_ )] = 0; _qAt_ ( _τ_ ) uniquely solves E[ _m_ ( _Y_ ( _t_ ) _, qAt_ ( _τ_ ); _τ_ ) _|T_ = 1] = 0; and _qBt_ ( _τ_ ) uniquely solves E[ _m_ ( _Y_ ( _t_ ) _, qBt_ ( _τ_ ); _τ_ ) _|T_ = 0] = 0 _._

- **I.II** For all _τ ∈T_ , we have ( _Y_ (1) _, Y_ (0)) _⊥ T |_ **_X_** ;

- **I.III** For some _c >_ 0, _c < p_ ( **_X_** ) _<_ 1 _− c_ , a.e. **_X_** .

– Assumptions **I.I I.III** are standard in the literature, as in Firpo (2007). Condition **I.I** is in general not a sufficient identification condition for _qt_ ( _τ_ ) because _Y_ ( _t_ ) is not always observable from the data. Therefore, the untestable condition **I.II** , the so-called ignorability assumption, is fundamental. According to condition **I.II** , the assignment to the treatment is 7
random within subpopulations characterized by **_X_** . This assumption has been used, among others, by Heckman, Ichimura, Smith, and Todd (1998), Dehejia and Wahba (1999), Hirano and Imbens (2004), Firpo (2007). Within the _GIC_ framework this assumption implies that conditional on **_X_** , there is a random mechanism that assigns individual _i_ to the exact period that she is observed (either period 0 or 1). In our model the triple ( _Y, T,_ **_X_** ) is observable, and a random sample of size _n_ can be obtained. Condition **I.III** states that for almost all values of **_X_** , both treatment assignment levels have a positive probability of occurrence.

– Under conditions **I.I I.III** the quantities _q_ 1( _τ_ ), _q_ 0( _τ_ ), _qA_ 1( _τ_ ), _qA_ 0( _τ_ ), _qB_ 1( _τ_ ) and _qB_ 0( _τ_ ) are identified from the joint distribution of ( _Y, T,_ **_X_** ). These six objects can be written as implicit functions of the observed data. For all _τ ∈T_ , where _w_ 1( _T,_ **_X_** ) = _p_ ( _<u>T</u>_ **_X_** )<sup>,</sup><sup>_w_0(</sup><sup>_T,_</sup><sup>**_X_**) =</sup> 1 _−_ <u>1</u> _p−_ ( _<u>T</u>_ **_X_** )<sup>,</sup><sup>_wA_1(</sup><sup>_T,_</sup><sup>**_X_**) =</sup><sup>_<u>T</u>_</sup> _p_<sup>,</sup><sup>_wA_0(</sup><sup>_T,_</sup><sup>**_X_**) =</sup> � <u>1</u> _−pT_ �� 1 _−p_ <u>(</u> _p_ **_X_** ( **_X_** <u>)</u> ) �, _<u>T</u>_ 1 _−p_ <u>(</u> **_X_** <u>)</u> _wB_ 1( _T,_ **_X_** ) = � 1 _−p_ �� _p_ ( **_X_** ) � and _wB_ 0( _T,_ **_X_** ) =<sup><u>1</u></sup> 1<sup>_−_</sup> _−_<sup>_<u>T</u>_</sup> _p_<sup>.Thismainidentificationresultfollows</sup> directly from Lemma 1 in Firpo (2007).

Finally, given that the elements in the vectors _Q_ ( _τ, τ_<sup>_′_</sup> ), _QA_ ( _τ, τ_<sup>_′_</sup> ) and _QB_ ( _τ, τ_<sup>_′_</sup> ) are identified, since _q_ 1( _τ_ ), _q_ 0( _τ_ ), _qA_ 1( _τ_ ), _qA_ 0( _τ_ ), _qB_ 1( _τ_ ) and _qB_ 0( _τ_ ) are identified, it follows from equations (1) and (2) that _GIC_ ( _τ_ ) and _GIC_<sup>_∗_</sup> ( _τ_ ) are also, respectively, identified from the joint distribution of ( _Y, T,_ **_X_** ).

**Remark 1.** _We note that one can also obtain other functionals of interest based on Q_ ( _τ, τ_<sup>_′_</sup> ) _, QA_ ( _τ, τ_<sup>_′_</sup> ) _and QB_ ( _τ, τ_<sup>_′_</sup> ) _, which highlights the potential relevance of the proposed methods in practice. Given the identification result, general functionals of parameters of interest are also identified, since they can be written as functions of qt_ ( _τ_ ) _, qAt_ ( _τ_ ) _, qBt_ ( _τ_ ) _, and consequently as functions of the observable variables_ ( _Y, T,_ **_X_** ) _. For example, the quantile treatment effect (QTE) will be_ ∆( _τ_ ) = _q_ 1( _τ_ ) _− q_ 0( _τ_ ) = [1 _−_ 1] _Q_ ( _τ, τ_ ) _and for quantile treatment effect on the treated (QTT) will be_ ∆ _A_ ( _τ_ ) = _qA_ 1( _τ_ ) _− qA_ 0( _τ_ ) = [1 _−_ 1] _QA_ ( _τ, τ_ ) _. Less common than the previous two treatment effect parameters, the QTU, the quantile treatment effect on the untreated, is defined as_ ∆ _B_ ( _τ_ ) = [1 _−_ 1] _QB_ ( _τ, τ_ ) = _qB_ 1( _τ_ ) _− qB_ 0( _τ_ ) _. Other functionals, such as the Makarov bounds for the CDF of Y_ (1) _−Y_ (0) _(Fan and Park (2010)) that explicitly depend on QA_ ( _τ, τ_<sup>_′_</sup> ) _and QB_ ( _τ, τ_<sup>_′_</sup> ) _at different points_ ( _τ, τ_<sup>_′_</sup> ) _, can similarly be obtained from_

8
_the quantiles of potential outcomes._

# **3.2 Estimation**

We are interested in estimation and inference for the _GIC_ ( _τ_ ) and _GIC_<sup>_∗_</sup> ( _τ_ ). Equations (1) and (2) show that _GIC_ can be written as a function of the quantiles of potential outcomes. Thus, we estimate each component of the vectors _Q_ ( _τ, τ_<sup>_′_</sup> ), _QA_ ( _τ, τ_<sup>_′_</sup> ) and _QB_ ( _τ, τ_<sup>_′_</sup> ) to construct estimators for the _GIC_ ( _τ_ ) and _GIC_<sup>_∗_</sup> ( _τ_ ).

Given identification, we are able to estimate the parameters of interest using a multi-step estimator as follows.

**Step 1** Estimate _p_ ( **_X_** ) parametrically or nonparametrically and obtain an estimator � _p_ ( **_X_** ).<sup>5</sup> The estimator of _p_ is the sample average of _T_ , i.e., _p_ � = _n_<sup>_−_1 �</sup><sup>_n_</sup> _i_ =1<sup>_Ti_.</sup>

**Step 2** For each ( _τ, τ_<sup>_′_</sup> ) _∈T × T_ , obtain

where, for _t_ = 0 _,_ 1, _q_ � _t_ ( _τ_ ), _q_ � _At_ ( _τ_ ) and _q_ � _Bt_ ( _τ_ ) satisfying the following conditions: � � � � � � where _w_ 1 _,i_ = _Ti/p_ ( **_X_** _i_ ), _w_ 0 _,i_ = (1 _− Ti_ ) _/_ (1 _− p_ ( **_X_** _i_ )), _wA_ 1 _,i_ = _Ti/p_ , � � � � � � � � _wA_ 0 _,i_ = [(1 _− Ti_ ) _/_ (1 _− p_ ( **_X_** _i_ ))] [ _p_ ( **_X_** _i_ ) _/p_ ], _wB_ 1 _,i_ = [ _Ti/p_ ( **_X_** _i_ )] [(1 _− p_ ( **_X_** _i_ )) _/_ (1 _− p_ )] � � and _wB_ 0 _,i_ = (1 _− Ti_ ) _/_ (1 _− p_ ).

In practice, estimators of _qt_ ( _τ_ ), _qAt_ ( _τ_ ) and _qBt_ ( _τ_ ) can be obtained by weighted quantile

> 5Appendix 6.3 discusses the practical estimation of _p_ ( **_X_** ).

9
regressions (QR) where _ρτ_ ( _u_ ) := _u_ ( _τ −_ **1** _{u <_ 0 _}_ ) is the check function as in Koenker and Bassett (1978).

**Step 3** Finally, we can plug-in estimates of the quantiles of the potential outcomes into the expressions to estimate _GIC_ in (1) as following

where we estimate _q_ � _A_ 1( _τ_ ) and _q_ � _B_ 0( _τ_ ) as in (7) and (8), respectively. To compute the corresponding weights, we estimate the propensity score, _p_ ( **_X_** ), by approximating its log-odds ratio by a polynomial and use the logistic link function with covariates given below in the data description.

Analogously, we can also estimate the counterfactual _GIC_<sup>_∗_</sup> in (2) as which, as described previously, is the growth incidence curve for quantile _τ_ if the distribution of explanatory variables of income had remained fixed from period 0 to 1.

There are other alternative estimators available in the literature for the quantile objects of interest defined in Step 2 above. Donald and Hsu (2014) discuss an estimator that makes use of the inverse of the cumulative distribution function (CDF) of the potential outcomes. Their approach to estimate the quantiles is a three-step procedure. In the first step one needs to compute weights; in the second step the CDF is computed for all points on its support by using an inverse probability weighted estimator; and in the third step one obtains the quantile by inverting the CDF. We show below that the estimator proposed by Donald and Hsu (2014) and our proposed method are asymptotically equivalent. Nevertheless, the 10
estimator discussed in this paper has several practical advantages. First, our estimator for the quantiles is a two-step method: the first step coincides with the one of Donald and Hsu (2014), but unlike that method, our QR estimator for the object of interest is obtained without having to invert the CDF. This is possible because of the second advantage of our method: QR has a linear program representation, which makes practical computation simple and allows using weights directly into the objective function that is solved. Finally, if one is interested in quantiles, and its transformations, using the proposed estimator is attractive due to its computational efficiency and accuracy in finite samples.<sup>6</sup>

**Remark 2.** _One can also easily use the multi-step estimator defined above to obtain estimates_ � _for other functionals of interest. For example, the estimator of QTE will be_ ∆(<sup>�</sup> _τ_ ) = _q_ 1( _τ_ ) _−_ � � � _q_ 0( _τ_ ) = [1 _−_ 1] _Q_<sup>�</sup> ( _τ, τ_ ) _and for QTT will be_ ∆<sup>�</sup> _A_ ( _τ_ ) = _qA_ 1( _τ_ ) _− qA_ 0( _τ_ ) = [1 _−_ 1] _Q_<sup>�</sup> _A_ ( _τ, τ_ ) _. Other functionals, such as the Makarov bounds for the quantiles of the distribution of treatment effects, Y_ (1) _− Y_ (0) _, are estimated using the analytical expressions of these estimated bounds as functions of Q_<sup>�</sup> _A_ ( _τ, τ_<sup>_′_</sup> ) _and Q_<sup>�</sup> _B_ ( _τ, τ_<sup>_′_</sup> ) _._

# **3.3 Asymptotic properties**

In this section, we derive the asymptotic properties of the multi-step estimator for the quantile process. We first focus on the properties of the estimator of _qt_ ( _τ_ ) and establish the uniform consistency and the weak limit of _q_ � _t_ ( _τ_ ), in _l_<sup>_∞_</sup> ( _T_ ). The extension to _q_ � _At_ ( _τ_ ) and � _qBt_ ( _τ_ ) is direct. We also establish the consistency and the weak limit of _Q_<sup>�</sup> ( _τ, τ_<sup>_′_</sup> ), _Q_<sup>�</sup> _A_ ( _τ, τ_<sup>_′_</sup> ) and _Q_<sup>�</sup> _B_ ( _τ, τ_<sup>_′_</sup> ) in _l_<sup>_∞_</sup> ( _T_ ) _× l_<sup>_∞_</sup> ( _T_ ). The asymptotic properties of the _GIC_<sup>�</sup> ( _τ_ ) and _GIC_<sup>�</sup> _∗_ ( _τ_ ) follow from these results. In addition, we derive the uniform semiparametric efficiency of the estimator. Finally, we discuss how in practice we estimate weights used to compute _q_ � _t_ ( _τ_ ). The two last results are collected in the Appendix.<sup>7</sup>

## **3.3.1 Consistency**

Consistency is a desired property for most estimators. For the consistency of process _q_ � _t_ ( _τ_ ) over _τ ∈T_ , consider the following conditions.

> 6We refer the reader to Koenker, Leorato, and Peracchi (2013) for a discussion and comparison on the statistical properties of the distribution regression and the quantile regression approaches.

> 7In Appendix 6.2, we provide results for the uniform semiparametric efficiency of the estimator. In Appendix 6.3 we discuss the practical estimation of the corresponding nuisance parameters, _wt_ ( _·_ ), _wAt_ ( _·_ ), and _wBt_ ( _·_ ).

11
- **QC.I** For _s, t ∈{_ 0 _,_ 1 _}_ , the densities _fY_ ( _s_ ) _|T_ ( _·|t_ ) are bounded above and, uniformly in _τ_ , positive. Also, for any _δ >_ 0,

**QC.II** There exists 0 _< Mw < ∞_ such that _wt_ ( _T,_ **_X_** ) _< Mw_ , a.e. ( _T,_ **_X_** ).

**QC.III** _|w_ � _t − wt|∞_ = _op_ (1).

These conditions are standard in the literature. We state **QC.I** and **QC.II** for selfcontainedness. As usual in the QR literature, **QC.I** requires the density to be bounded away from infinity. The second part of **QC.I** is a standard identification condition. It is similar to Angrist, Chernozhukov, and Fernandez-Val (2006) and Firpo (2007), and it follows – from **I.I I.III** for each _τ_ . **QC.II** imposes boundedness on the density of **_X_** . It is analogue to Assumption 1(ii) of Firpo (2007) and follows directly from **I.III** . **QC.III** requires consistent estimation of the nuisance parameter. This is a usual requirement corresponding to (1.4) of Theorem 1 of Chen, Linton, and Van Keilegom (2003).

The following result establishes consistency of the estimator over the set of quantiles.

**Theorem 1.** _Suppose that_ E[ _wt_ ( _T,_ **_X_** ) _m_ ( _Y, qt_ ( _τ_ ); _τ_ )] = 0 _, and that conditions_ **QC.I** _–_ **QC.III** _are satisfied. Then, for t_ = 0 _,_ 1 _, as n →∞_

� � The extension of Theorem 1 to _qAt_ ( _·_ ) and _qBt_ ( _·_ ), _t_ = 0 _,_ 1 is direct. The assumptions **QC.I** – **QC.III** are analogous.

# **3.3.2 Weak convergence**

Now we derive the limiting distribution of the general _q_ � _t_ ( _τ_ ) estimator. We impose the following sufficient conditions.

- **QG.I** The functions _w_ � _t_ ( _T,_ **_X_** ) _∈_ Π and _w_ � _t_ ( _T,_ **_X_** ) _→p wt_ ( _T,_ **_X_** ) uniformly in ( _T,_ **_X_** ) over compact sets, where _wt_ ( _T,_ **_X_** ) _∈_ Π, and Π is a function class of uniformly smooth functions in ( _T,_ **_X_** ) with domain _{_ 0 _,_ 1 _} × X_ .

12
**QG.II**<sup>_√_</sup> _<u>n</u>_ (E[( _w_ � _t_ ( _T,_ **_X_** ) _− wt_ ( _T,_ **_X_** ))( _τ −_ **1** _{Y < qt_ ( _τ_ ) _}_ )] + E[ _wt_ ( _T,_ **_X_** )( _τ −_ **1** _{Y < qt_ ( _τ_ ) _}_ )]) converges weakly.

Assumptions **QG.I** – **QG.III** concern the properties of the weights. They are high level conditions and will be discussed in the section of the estimation of _wt_ . Conditions **QG.I** and **QG.II** allow for estimated weights. Assumption **QG.II** is similar to Cattaneo (2010). Examples satisfying **QG.II** include smooth function classes. These assumptions allow for a wide variety of nonparametric and parametric estimators. **QG.III** strengthens **QC.III** such that the estimator of the nuisance parameter converges at a rate faster than _n_<sup>_−_1</sup><sup>_/_4</sup> . A similar assumption appears in Chen, Linton, and Van Keilegom (2003).

Now we present the weak convergence result.

� **Theorem 2.** _For t_ = 0 _,_ 1 _, suppose that |_ E[ _wt_ ( _T,_ **_X_** ) _m_ ( _Y, qt_ ( _τ_ ); _τ_ )] _|∞_ = 0 _, that |qt − qt|∞_ = _op∗_ (1) _, and that conditions_ **QC.I–QC.II** _,_ **QG.I–QG.III** _are satisfied. Then, in l_<sup>_∞_</sup> ( _T_ ) _,_

_where_ G _t is a mean zero Gaussian process with covariance function_ E[G _t_ ( _τ_ )G _t_ ( _τ_<sup>_′_</sup> )<sup>_⊤_</sup> ] = _Dt_<sup>_−_1(</sup><sup>_τ_)</sup><sup>_Stt_(</sup><sup>_τ, τ ′_)[</sup><sup>_D_</sup> _t_<sup>_−_1(</sup><sup>_τ ′_)]</sup><sup>_⊤,with,fort_= 0</sup><sup>_,_1</sup><sup>_,andl_= 0</sup><sup>_,_1</sup><sup>_,_</sup> The result in Theorem 2 shows that the limiting distribution of the estimator is a Gaussian process. Thus, if one fixes a quantile at _τ_ ̄, then the limiting distribution collapses to a simple normal distribution, as in Firpo (2007). For practical inference, below we provide inference methods over the set of quantiles that are simple to implement in applications.<sup>8</sup> As before, the extension of Theorem 2 to � _qAt_ ( _·_ ) and � _qBt_ ( _·_ ), _t_ = 0 _,_ 1 is direct. The assumptions – corresponding to **QG.I QG.III** are analogous.

> 8Firpo and Pinto (2015) present a similar result to Theorem 2. Nevertheless our proof technique is different on the treatment of both infinite dimension parameters. In addition, we do not require compactness of the support of **_X_** and impose weaker assumptions on _w_ � _t_ .

13
Given the result in Theorem 2, it is simple to establish the weak convergence to the vector _Q_<sup>�</sup> ( _τ, τ_<sup>_′_</sup> ). The results for _Q_<sup>�</sup> _A_ ( _τ, τ_<sup>_′_</sup> ) and _Q_<sup>�</sup> _B_ ( _τ, τ_<sup>_′_</sup> ) are analogous.

**Corollary 1.** _Assume the conditions of Theorem 2, as n →∞, in l_<sup>_∞_</sup> ( _T_ ) _× l_<sup>_∞_</sup> ( _T_ )

_where_ G _is the vector of Gaussian processes with covariance function_ In order to perform inference on functions of the _Q_ ( _τ, τ_<sup>_′_</sup> ) , we impose a differentiability condition on such functions and state a functional delta method result. Consider the following assumption.

- **QG.IV** (Hadamard) The functional _h_ : _l_<sup>_∞_</sup> ( _T_ ) _× l_<sup>_∞_</sup> ( _T_ ) _→ l_<sup>_∞_</sup> ( _T_ ) defined over the distribution of potential outcomes is Hadamard differentiable at _Q_ , with Hadamard derivative given by _h_ ( _·_ )<sup>_′_</sup> .

The following result is a well known application of the functional delta method, we include it for completeness.

**Lemma 1.** _Assume the conditions of Theorem 2, and_ **QG.IV** _, as n →∞,_

Donald and Hsu (2014) establish the weak convergence of a quantile estimator that makes use of the inverse of the CDF in their Theorem 3.8. Their result is similar to that in Theorem 2 above. Nevertheless, as mentioned previously, the quantile estimators are different. In addition, the assumptions required to establish the results are different. On the one hand, Donald and Hsu (2014) impose strong conditions to derive the result. For instance, their Assumption 3.1 requires that the distributions of _Y_ (0) and _Y_ (1) have convex and compact supports. Their Assumption 3.2 requires all the covariates to be continuous 14
and the support of the vector of covariates, **_X_** , to be compact. We are able to somewhat relax these assumptions. Given that we work with a standard semiparametric estimator, and a quantile regression framework in the second step, we do not require such assumptions to derive the asymptotic properties of our proposed estimator.

Now we to return to the main object of interest and analyze the growth incidence curves, _GIC_ ( _τ_ ) and _GIC_<sup>_∗_</sup> ( _τ_ ). As an application of Theorem 2 and Lemma 1, we derive the asymptotic distribution for _GIC_ ( _τ_ ). Corollary 1 implies that where G _A_ ( _τ_ ) and G _B_ ( _τ_ ) are Gaussian processes with variance-covariance functions that can be obtained as an application of Corollary 1.

> <sup>0]</sup><sup>_<u>QA</u>_</sup><sup><u>(</u></sup><sup>_τ,τ_</sup><sup><u>)</u>0]</sup><sup>_<u>QB</u>_</sup><sup><u>(</u></sup><sup>_τ,τ_</sup><sup><u>)</u></sup> Recall that _GIC_ ( _τ_ ) = [0<sup><u>[1</u></sup> 1] _QB_ ( _τ,τ_ )<sup>_−_1,and</sup><sup>_GIC∗_(</sup><sup>_τ_)=</sup><sup><u>[</u></sup> [0<sup>1</sup> 1] _QB_ ( _τ,τ_ )<sup>_−_1.Thesefunctionals</sup> are differentiable at ( _QA, QB_ ), as long as _qB_ 0( _τ_ ) = 0 with derivatives defined by

and for _GIC_<sup>_∗_</sup> we have that Therefore, from a functional delta method we have the following results.

**Corollary 2.** _Assume the conditions of Theorem 2, as n →∞, in l_<sup>_∞_</sup> ( _T_ )

# **3.4 Inference procedures**

In this section, we turn our attention to inference procedures on the _GIC_ . Important questions posed in the econometric and statistical literatures concern the nature of the 15
impact of a policy intervention or treatment on the outcome distributions of interest. The corresponding questions for the _GIC_ are, for example, whether there is significant income growth at any quantile (the null hypothesis being _GIC_ ( _τ_ ) = 0 for all _τ_ ); and whether growth is uniform or heterogeneous ( _GIC_ ( _τ_ ) equals the average growth rate, for all _τ_ ). One can also ask if growth is non-decreasing in _τ_ ( _GIC_ ( _τ_ )<sup>_′_</sup> _≥_ 0 for all _τ_ ). Since the main objective of this paper is to study the growth incidence curve, and these questions and hypotheses are formulated for the entire _GIC_ process, we develop inference procedures for the quantile process over the set of quantiles indexed by _τ_ .

# **3.4.1 Test statistics**

We seek to develop inference for _GIC_ over the index set of quantiles _T_ . We present results for functionals of quantiles of the marginal distributions of potential outcomes, and in particular, the _GIC_ ( _τ_ ) and _GIC_<sup>_∗_</sup> ( _τ_ ). Let _β_ ( _τ_ ) be a functional of _Q_ , _QA_ , and _QB_ , that<sup>0]</sup><sup>_<u>QA</u>_</sup><sup><u>(</u></sup><sup>_τ,τ_</sup><sup><u>)</u></sup> is, _β_ ( _τ_ ) = _h_ ( _Q_ ( _τ, τ_ )). In particular, we are interested in _β_ ( _τ_ ) = _GIC_ ( _τ_ ) = [0<sup><u>[1</u></sup> 1] _QB_ ( _τ,τ_ )<sup>_−_1,</sup> and the counterfactual one _β_ ( _τ_ ) = _GIC_<sup>_∗_</sup> ( _τ_ ) = [0<sup><u>[1</u></sup> 1]<sup>0]</sup> _Q_<sup>_<u>Q</u>_</sup> _B_<sup>_<u>B</u>_</sup> (<sup><u>(</u></sup> _τ,τ_<sup>_τ,τ_</sup> )<sup><u>)</u></sup><sup>_−_1.</sup> We discuss three main hypotheses of interest. First, we consider the following standard null hypothesis uniformly, where the vector _r_ ( _τ_ ) is assumed to be known, continuous in _τ_ over _T_ , and _r ∈ l_<sup>_∞_</sup> ( _T_ ). More generally, the hypothesis in (11) embeds several interesting hypotheses about the parameters of the quantile function.

**Example** (The uniformly null effect hypothesis) **.** _A basic hypothesis is that the growth incidence curve, GIC_ ( _τ_ ) _, is statistically equal to zero for all τ ∈T . The alternative is that the it differs from zero at least for some τ ∈T . In this case, r_ ( _τ_ ) = 0 _, and relative inequality remains stable._

The basic inference process to test the null hypothesis (11) is To derive the asymptotic properties of the above statistic, we need to compute the estimator _β_<sup>�</sup> ( _τ_ ), which is given by _β_<sup>�</sup> = _h_ ( _Q_<sup>�</sup> ). The _GIC_ ( _τ_ ) estimate is _β_<sup>�</sup> ( _τ_ ) = _GIC_<sup>�</sup> ( _τ_ ) = <u>[1</u> 0] _<u>Q</u>_<sup>�</sup> _<u>A</u>_ <u>(</u> _τ,τ_ <u>)</u> _∗_ <u>[1</u> 0] � _<u>QB</u>_ <u>(</u> _τ,τ_ <u>)</u> [0 1] _Q_<sup>�</sup> _B_ ( _τ,τ_ )<sup>_−_1,andtheestimatefor</sup><sup>_GIC∗_(</sup><sup>_τ_)is</sup><sup>_β_�(</sup><sup>_τ_) =</sup> _GIC_<sup>�</sup> ( _τ_ ) = [0 1] _Q_<sup>�</sup> _B_ ( _τ,τ_ )<sup>_−_1,whichfor</sup> a fixed quantile _τ_ , has an asymptotic normal distribution as given in Corollary 2.

16
General hypotheses about _β_ ( _τ_ ) can be accommodated through functions of _Wn_ ( _·_ ). We consider the Kolmogorov-Smirnov and Cram ́er-von Mises type test statistics, _Vn_ = _f_ ( _Wn_ ( _·_ )), where _f_ ( _·_ ) is a general functional of the process _Wn_ ( _·_ ). In particular, we consider different functionals that lead to different test statistics, such as There are many alternative possible statistics as: _V_ 3 _n_ :=<sup>_√_</sup> _<u>n</u>_ sup _τ ∈T Wn_ ( _τ_ )<sup>2</sup> and _V_ 4 _n_ := _√n_ � _τ ∈T_<sup>_Wn_(</sup><sup>_τ_)2</sup><sup>_, dτ_,amongothers.Inthispaperweconcentrateon</sup><sup>_V_1</sup><sup>_n_and</sup><sup>_V_2</sup><sup>_n_.</sup> These statistics and their associated limiting theory provide a natural foundation for testing the null hypothesis. Now we present the limiting distributions of the test statistics under the null hypothesis. From Corollary 1 and Lemma 1 under the null hypothesis ( _H_ 0 : _β_ = _h_ ( _Q_ ) = _r_ ), it follows that<sup>_√_</sup> _<u>n</u>_ <u>(</u> _h_ ( _Q_<sup>�</sup> ) _− h_ ( _Q_ )) ⇝ _h_ (G)<sup>_′_</sup> . Thus, the following lemma summarizes the limiting distributions.

**Lemma 2.** _Assume the conditions of Theorem 2, and_ **QG.IV** _. Under H_ 0 : _β_ ( _τ_ ) = _h_ ( _Q_ ( _τ_ )) = _r_ ( _τ_ ) _, τ ∈T , as n →∞,_

When performing tests for the _GIC_ , the limiting distributions of the test statistics under the null hypothesis follows from Theorem 2. Under the null hypothesis ( _H_ 0 : _GIC_ ( _τ_ ) = _r_ ( _τ_ )), it follows<sup>_√_</sup> _<u>n</u>_ <u>(</u> _GIC_<sup>�</sup> ( _τ_ ) _− r_ ( _τ_ )) ⇝ _GIC_ (G _A,_ G _B_ )<sup>_′_</sup> . Thus, the following corollary summarizes the limiting distributions. The result for _H_ 0 : _GIC_<sup>_∗_</sup> ( _τ_ ) = _r_ ( _τ_ ) is analogous.

**Corollary 3.** _Assume the conditions of Theorem 2. Under H_ 0 : _GIC_ ( _τ_ ) = _r_ ( _τ_ ) _, as n →∞,_

The second hypothesis of interest concerns an unknown _r_ ( _τ_ ), which needs to be estimated. In many examples of interest, the component _r_ ( _τ_ ) in the null hypothesis (11) is unknown or defined as a function of the conditional distribution and thus needs to be estimated (see, e.g., Koenker and Xiao (2002) and Chernozhukov and Fernandez-Val (2005)). _r_ ( _τ_ ) might, 17
for example, be _GIC_ ( _τ_ ) for a different country, or period. Or it might be _GIC_<sup>_∗_</sup> ( _τ_ ) _._ The natural expedient of replacing the unknown _r_ in the test statistic by estimates introduces � some fundamental difficulties. The estimate will be denoted as _r_ ( _τ_ ). Let In this framework, we follow Chernozhukov and Fernandez-Val (2005) and assume that the quantile estimates and nuisance parameter estimates satisfy the following:<sup>_√_</sup> _<u>n</u>_ <u>-consistent</u> � estimators for _β_<sup>�</sup> ( _·_ ) and � _r_ ( _·_ ), such that<sup>_√_</sup> _<u>n</u>_ <u>(</u> _β_<sup>�</sup> ( _·_ ) _−β_ ( _·_ )) ⇝ _h_ (G( _·_ ))<sup>_′_</sup> and<sup>_√_</sup> _<u>n</u>_ <u>(</u> _r_ ( _·_ ) _−r_ ( _·_ )) ⇝ G _r_ ( _·_ ) jointly in _l_<sup>_∞_</sup> ( _T_ ), where ( _h_ (G( _·_ ))<sup>_′_</sup> _,_ G _r_ ( _·_ )) is a zero mean continuous Gaussian process with a � non-degenerate covariance kernel. Thus, we have that<sup>_√_</sup> _<u>n</u>_ <u>(</u> _β_<sup>�</sup> ( _τ_ ) _− r_ ( _τ_ )) ⇝ _h_ (G( _τ_ ))<sup>_′_</sup> _−_ G _r_ ( _τ_ ). The process remains asymptotically Gaussian; however, the estimation of _r_ ( _τ_ ) introduces a new drift component that additionally complicates the covariance kernel of the process.

Under the null hypothesis _H_ 0 : _β_ ( _τ_ ) = _r_ ( _τ_ ), the test statistics become:

**Example** (The uniformly constant (but unknown) effect hypothesis) **.** _A basic hypothesis is that the growth incidence curve, GIC_ ( _τ_ ) _, is statistically equal to the mean growth rate for all τ ∈T . , i.e., growth has no distributional heterogeneity. The alternative is that GIC_ ( _τ_ ) � � _differs from the mean at least for some τ ∈T . In this case, r_ ( _τ_ ) = _γAGR, (where γAGR is the mean growth rate)._

Now we display the limiting distributions of these test statistics under the null hypothesis.

**Lemma 3.** _Assume the conditions of Theorem 2 and that_<sup>_√_</sup> _<u>n</u>_ <u>(</u> _β_<sup>�</sup> ( _·_ ) _− β_ ( _·_ )) ⇝ _h_ (G( _·_ ))<sup>_′_</sup> _and_ � _√n_ <u>(</u> _r_ ( _·_ ) _− r_ ( _·_ )) ⇝ G _r_ ( _·_ ) _jointly in l_<sup>_∞_</sup> ( _T_ ) _, where_ ( _h_ (G( _·_ ))<sup>_′_</sup> _,_ G _r_ ( _·_ )) _is a zero mean continuous Gaussian process with a non-degenerate covariance kernel.. Under H_ 0 : _β_ ( _τ_ ) = _h_ ( _Q_ ( _τ_ )) = _r_ ( _τ_ ) _, τ ∈T , as n →∞,_

This result can be applied to test for the _GIC_ . The limiting distributions of the test statistics under the null hypothesis follow from Lemma 3 . Under the null hypothesis ( _H_ 0 : 18
� _GIC_ ( _τ_ ) = _r_ ( _τ_ )), it follows<sup>_√_</sup> _<u>n</u>_ <u>(</u> _GIC_<sup>�</sup> ( _τ_ ) _− r_ ( _τ_ )) ⇝ _GIC_ (G _A,_ G _B_ )<sup>_′_</sup> , and<sup>_√_</sup> _<u>n</u>_ <u>(</u> _r_ ( _τ_ ) _− r_ ( _τ_ )) ⇝ G _r_ ( _τ_ ). The following corollary summarizes the limiting distributions. The result for _H_ 0 : _GIC_<sup>_∗_</sup> ( _τ_ ) = _r_ ( _τ_ ) is analogous.

**Corollary 4.** _Assume the conditions of Lemma 3, with β_ ( _τ_ ) = _GIC_ ( _τ_ ) _, as n →∞,_

Finally, we consider testing hypotheses concerning inequalities on both null and alternative hypotheses as The following is an example of hypotheses that may be considered.

**Example** (The first-order stochastic dominance hypothesis) **.** _An important practical hypothesis involves the composite null GIC_ ( _τ_ ) _≥ r_ ( _τ_ ) _, for all τ ∈T , versus the alternative of GIC_ ( _τ_ ) _< r_ ( _τ_ ) _, for some τ ∈T . When r_ ( _τ_ ) = 0 _and because GIC_ ( _τ_ ) =<sup>_<u>qA</u>_</sup><sup><u>1(</u></sup> _q_<sup>_τ_</sup> _B_<sup><u>)</u></sup><sup>_−_</sup> 0(<sup>_<u>q</u>_</sup> _τ_<sup>_<u>B</u>_</sup> )<sup><u>0(</u></sup><sup>_τ_</sup><sup><u>)</u></sup> _, such that for qB_ 0( _τ_ ) = 0 _, testing whether GIC_ ( _τ_ ) _≥_ 0 _is equivalent to test whether qA_ 1( _τ_ ) _≥ qB_ 0( _τ_ ) _, ie, that FY_ (1) _|T_ =1 _stochastically dominates FY_ (0) _|T_ =0 _in first-order._

Therefore, the above example describes a test which is analogous to a first order stochastic dominance as in Donald and Hsu (2014). These null hypotheses of interest can be formalized as _H_ 0 : _β_ ( _τ_ ) _≥_ 0, and the test statistic becomes: where _W_<sup> ̃</sup> _n_ ( _τ_ ) = _β_<sup>�</sup> ( _τ_ ).

We employ the test statistic _V_<sup> ̃</sup> 1 _n_ since it has been known in the literature that when the null hypothesis involves an inequality, the set of points satisfying the null hypothesis is usually not a singleton (see, e.g., Linton, Maasoumi, and Whang (2005)). The typical way to resolve this is to apply the least favorable configuration (LFC) to find a point in the null hypothesis least favorable to the alternative hypothesis. Hence, to derive the asymptotic properties of the above statistic, _V_<sup> ̃</sup> 1 _n_ , one computes the estimator _β_<sup>�</sup> ( _τ_ ) and plugs it in, and given the LFC the limiting distribution is analogous to that in Lemma 2 and Corollary 2.

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To perform practical inference we suggest the use of resampling techniques to approximate the limiting distributions and obtain critical values. To obtain the critical value for the first two criteria we use a bootstrap procedure, and for the inequality test we make use of subsampling.

# **3.4.2 Practical implementation of testing procedures**

Implementation of the proposed tests in practice is simple. First, we discuss the test _H_ 0 in (11). To implement the tests one needs to compute the statistics of test _V_ 1 _n_ or _V_ 2 _n_ . Analogously, when _r_ ( _τ_ ) is unknown, one computes _V_<sup> ̄</sup> 1 _n_ or _V_<sup> ̄</sup> 2 _n_ . We suggest the use of a recentered bootstrap procedure to calculate critical values. The steps for implementing the tests in practice are as follows.

First, the estimates of _β_ ( _τ_ ) are computed by solving the problems in equations (6)–(8) and calculating _β_<sup>�</sup> ( _τ_ ). Second, _Wn_ is calculated by centralizing _β_<sup>�</sup> ( _τ_ ) at _r_ ( _τ_ ), and _V_ 1 _n_ or _V_ 2 _n_ is computed by taking the maximum over _τ_ ( _V_ 1 _n_ ) or summing over _τ_ ( _V_ 2 _n_ ). For the general case with unknown _r_ ( _τ_ ), the tests are computed in the same fashion. The only adjustment is the use of _r_ �( _τ_ ) to compute _W_<sup> ̄</sup> _n_ . Third, after obtaining the test statistic, it is necessary to compute the critical values. We propose the following scheme. We use the test statistic _V_ 1 _n_ as an example, but the procedure is the same for the other cases. Take _B_ as a large integer. For each _b_ = 1 _, . . . , B_ : Let � _c_<sup>_B_</sup> 1 _−α_<sup>denotetheempirical(1</sup><sup>_−α_)-quantileofthesimulatedsample</sup><sup>_{V_�</sup> 1<sup>1</sup> _n_<sup>_, . . . ,_�</sup><sup>_V_</sup> 1<sup>_B_</sup> _n_<sup>_}_,</sup> where _α ∈_ (0 _,_ 1) is the nominal size. We reject the null hypothesis if _V_ 1 _n_ is larger than � _c_<sup>_B_</sup> 1 _−α_<sup>.</sup> In practice, the maximum in step (iii) is taken over a discretized subset of _T_ .

A formal justification the simulation method is stated as follows. Consider the following conditions.

**QG.IIB**<sup>_√_</sup> _<u>n</u> n_<sup><u>1</u></sup> � _ni_ =1<sup>[(</sup><sup>_τ −_</sup><sup>**1**</sup><sup>_{Yi< qt_(</sup><sup>_τ_)</sup><sup>_}_)(</sup><sup>_w_�</sup> _t_<sup>_∗_(</sup><sup>_Ti,_</sup><sup>**_X_**</sup><sup>_i_)</sup><sup>_−w_�</sup><sup>_t_(</sup><sup>_Ti,_</sup><sup>**_X_**</sup><sup>_i_))] converges weakly to a tight</sup> random element G in _l_<sup>_∞_</sup> ( _T_ ) in _P_<sup>_∗_</sup> -probability.

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**Theorem 3.** _Under_ **QC.I–QC.II, QG.IB–QG.IIB** _and_ **QG.III** _with “in probability” replaced by “almost surely”, then, for t_ = 0 _,_ 1 _, the bootstrap estimator_<sup>_√_</sup> _<u>n</u>_ <u>(</u> _q_ � _t_<sup>_∗_(</sup><sup>_τ_)</sup><sup>_−q_�</sup><sup>_t_(</sup><sup>_τ_)) ⇝</sup> G( _τ_ ) _in P_<sup>_∗_</sup> _-probability in l_<sup>_∞_</sup> ( _T_ ) _._

Theorem 3 establishes the consistency of the bootstrap procedure. It is important to highlight the connection between this result and the previous section. In fact, Theorem 3 shows that the limiting distribution of the bootstrap estimator is the same as that of Theorem 2, and hence the above resample scheme is able to mimic the asymptotic distribution of interest.

Now we move our attention to testing the _H_ 0 displayed in (12). As discussed in Linton, Maasoumi, and Whang (2005), even when the data are i.i.d. the standard bootstrap might not work well when testing the inequality under the null hypothesis. This is because one needs to impose the null, which is difficult because it is defined by a complicated system of inequalities. Thus, we follow Linton, Maasoumi, and Whang (2005) and suggest the use of a subsampling method, which is very simple to define and yet provides consistent critical values.

We first define the subsampling procedure. Let _Zi_ = _{_ ( _Yi, Ti,_ **_X_** _i_ ) : _i_ = 1 _, ..., n}_ and “ construct all possible subsets of size _b_ . The number of such subsets _Bn_ is _n_ choose _b_ .” Let _Sn_ denote our test statistic _V_<sup> ̃</sup> 1 _n_ computed over the entire sample. With some abuse of notation, the test statistic _Sn_ can be re-written as a function of the data _{Zi_ : _i_ = 1 _, ..., n}_ : where _sn_ ( _Z_ 1 _, ..., Zn_ ) is given by sup _τ ∈T_ [ _W_<sup> ̃</sup> _n_ ( _τ_ )], where _W_<sup> ̃</sup> _n_ ( _τ_ ) = _β_<sup>�</sup> ( _τ_ ). Let denote the distribution function of _Sn_ . Let _sn,b,i_ be equal to the statistic _sb_ evaluated at the subsamples _{Zi, ..., Zi_ + _b−_ 1 _}_ of size _b_ , i.e.

� This means that we have to recompute _qt_ ( _Zi, Zi_ +1 _, ..., Zi_ + _b−_ 1) using each subsample as well. We note that each subsample of size _b_ (taken without replacement from the original data) 21
is indeed a sample of size _b_ from the true sampling distribution of the original data. Hence, it is clear that one can approximate the sampling distribution of _Sn_ using the distribution of the values of _sn,b,i_ computed over _n − b_ + 1 different subsamples of size _b_ . That is, we approximate the sampling distribution _Jn_ of _Sn_ by Let _gn,b_ (1 _− α_ ) denote the (100 _− α_ )-th sample quantile of _J_<sup>�</sup> _n,b_ ( _·_ ). We call it the subsample critical value of significance level _α_ . Thus, we reject the null hypothesis at the significance level _α_ if _Sn > gn,b_ (1 _− α_ ). The computation of this critical value is not particularly onerous, although it depends on how big _b_ is.

The validity of the subsampling methods for the quantile regression process was established by Chernozhukov and Fernandez-Val (2005).

A Supplemental Appendix collects Monte Carlo simulations conducted to evaluate the finite sample properties of the proposed tests. We conduct simulations to evaluate the performance of these tests in terms of size and power. The results provide evidence that the empirical levels of the tests approximate well the nominal or theoretical levels. Moreover, the tests possess large power against selected alternatives. The results are improved when the sample size increases, nevertheless they are not very sensitive to the numbers of bootstraps.

# **4 Wage distribution dynamics in the United States and Brazil, 1995-2007**

This section illustrates the usefulness of the proposed methods with an empirical example. We compute the _GIC_ and _GIC_<sup>_∗_</sup> for the two most populous nations in the Western Hemisphere, namely the United States and Brazil, for the 1995-2007 period, and compare results. In particular, we emphasize the role of the following decomposition of the _GIC_ , introduced in Section 2 and reproduced below: The first term in this decomposition is the counterfactual _GIC_ , which keeps the joint 22
– distribution of observed covariates fixed (see equation 2). Under Assumptions **I.I I.III** , this term can be interpreted as describing the growth process that would have obtained in the absence of any changes in that joint distribution. The second term of the decomposition is correspondingly interpreted as the effect of changes in the joint distribution of covariates.

Our reweighting method allows for the direct construction of the counterfactual _GIC_ , with no need to postulate a structural relationship between wages, covariates and unobserved terms, as was required by the earlier literature that followed Juhn, Murphy, and Pierce (1993). Under that approach, economists would typically estimate OLS regressions for the two time periods separately and then construct a counterfactual wage distribution using estimated parameters and residuals from time _t_ = 1 but covariates from time _t_ = 0. This would yield a counterfactual distribution of wages at time _t_ = 1 _,_ with a distribution of covariates that was fixed at time _t_ = 0 (see, for example, Bourguignon, Ferreira, and Leite (2008)). In addition to requiring strong functional form assumptions, however, it is not clear how one would perform statistical inference on the counterfactual _GIC_ using that method.

In this section we report the estimates for _GIC_ and its counterfactual counterpart _GIC_<sup>_∗_</sup> , _GIC_ �( _τ_ ) and _GIC_<sup>�</sup> _∗_ ( _τ_ ) respectively, over _τ ∈T_ . We also report the corresponding growth rates in average wages, _γAGR_ and _γAGR∗_ , respectively, for comparison. Moreover, using the techniques developed in the previous section, we perform inference on both sets of curves. Specifically, we apply the uniform tests, Kolmogorov-Smirnov (KS) and Cram ́er-von Mises (CVM), to test the following hypotheses:

- ( _i_ ) Constant distribution: ( _H_ 0 : _GIC_ ( _τ_ ) = 0 versus _HA_ : _GIC_ ( _τ_ ) = 0);

- ( _ii_ ) Distribution-neutral growth ( _H_ 0 : _GIC_ ( _τ_ ) = _γAGR_ versus _HA_ : _GIC_ ( _τ_ ) = _γAGR_ , where _γAGR_ is the growth rate in the average wage);

- ( _iii_ ) Constant distribution, conditional on covariates, ( _H_ 0 : _GIC_<sup>_∗_</sup> ( _τ_ ) = 0 versus _HA_ : _GIC_<sup>_∗_</sup> ( _τ_ ) = 0);

- ( _iv_ ) Distribution-neutral growth, conditional on covariates ( _H_ 0 : _GIC_<sup>_∗_</sup> ( _τ_ ) = _γAGR∗_ versus _HA_ : _GIC_<sup>_∗_</sup> ( _τ_ ) = _γAGR∗_ , where _γAGR∗_ is the counterfactual growth rate in average wage).

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# **4.1 Data**

## **– 4.1.1 CPS US**

Our data for the United States come from the March supplement to the Current Population Surveys (CPS) for 1995 and 2007.<sup>9</sup> The dataset provides the distribution of labor earnings in the US in 1995 and 2007 for full-time workers of both genders. We use the following variables for our analysis. _Y_ denotes real hourly labor earnings (sum of annual pretax wages, salaries, tips, and bonuses, divided by the number of hours worked annually). The vector **_X_** consists of three covariates of _Y_ , namely: (i) the worker’s age in years; (ii) a gender dummy; and (iii) a categorical variable for the highest educational level attained (“high school”, “some college” or “college”).

We restrict the sample to individuals aged 16 to 65 that reported a positive value for real hourly earnings in the previous year. Individuals with missing values for any of the four variables in _Y_ and **_X_** were excluded from the sample. After applying these filters, we trimmed the sample by dropping the top and bottom 0.5% of the distribution of hourly wages in each year, to eliminate outliers. Hourly wages are in US dollars of March 2007. The Consumer Price Index was used to inflate 1995 incomes: nominal values in 1995 were multiplied by 1.36 to be expressed in 2007 dollars. The final sample contains a total of 165,245 observations. Summary statistics are presented in Table 1.

– Table 1: Summary Statistics US

||Mean|S.D.|Min.|Max.|Observations|
|---|---|---|---|---|---|
|CPS 1995||||||
|Hourly Work Earnings|16.730|11.402|1.047|74.162|69,494|
|Age|37.039|12.090|16|65|69,494|
|Male|0.527||||69,494|
|High School|0.333||||69,494|
|College Incomplete|0.295||||69,494|
|College|0.243||||69,494|
|CPS 2007||||||
|Hourly Work Earnings|19.626|16.748|1.202|168.280|95,751|
|Age|39.284|12.807|16|65|95,751|
|Male|0.525||||95,751|
|High School|0.305||||95,751|
|College Incomplete|0.291||||95,751|
|College|0.297||||95,751|

> 9http://www.census.gov/programs-surveys/cps.html

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# **– 4.1.2 PNAD Brazil**

The Brazilian data come from the Pesquisa Nacional por Amostra de Domic ́ılios (PNAD), an annual Brazilian household survey that samples households across (almost) the entire country.<sup>10</sup> It collects information on various household characteristics, as well as individual incomes and education levels. We use PNAD data for 1995 and 2007. For comparability, we use the same four variables as for the CPS: real hourly labor earnings, age, gender, and a categorical educational attainment variable. IBGE, the Brazilian Statistics Bureau that is responsible for running PNAD, started including the rural Northern region in the PNAD sample after 2004 but, for comparability across years, we do not use information on the rural North for 2007.

As for the US, we restrict the sample to individuals aged 16 to 65 that reported a positive labor income in the previous year. Individuals with missing values for income or any of our three covariates were excluded from the sample. The top and bottom 0.5% of the distribution of hourly wages in each year was trimmed, as in the CPS. Hourly wages are in Brazilian reais (BRL) of September 2007, which means that nominal values in 1995 were multiplied by 2.89 to be expressed in 2007 prices. The final sample contains a total of 275,749 observations. Summary statistics are presented in Table 2.

– Table 2: Summary Statistics Brazil

||Mean|S.D.|Min.|Max.|Observations|
|---|---|---|---|---|---|
|PNAD 1995||||||
|Hourly Work Earnings|5.565|7.312|0.300|61.317|119,770|
|Age|34.924|11.936|16|65|119,770|
|Male|0.631||||119,770|
|High School|0.131||||119,770|
|College Incomplete|0.046||||119,770|
|College|0.064||||119,770|
|PNAD 2007||||||
|Hourly Work Earnings|5.659|6.978|0.312|62.500|155,979|
|Age|36.367|12.025|16|65|155,979|
|Male|0.589||||155,979|
|High School|0.259||||155,979|
|College Incomplete|0.085||||155,979|
|College|0.094||||155,979|

A comparison of Tables 1 and 2 reveals considerable differences between the two labor 10In 1995, PNAD did not survey households in the rural areas of Acre, Amap ́a, Amazonas, Par ́a, Rondˆonia and Roraima — six states in the Amazon region.

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forces. US full-time workers are on average some three years older than their Brazilian counterparts, and earn much higher wages: the nominal exchange rate in September 2007 was 1.90 BRL to the USD, so average wages in 2007 in this sample were approximately 6.6 times higher in the US than in Brazil. US workers are also much more educated, and the female share of the labor force is higher there. Over the twelve years between 1995 and 2007, both labor forces became a little older, and more educated. Educational attainment rose in both countries, but more markedly in Brazil, which started from a much lower level. Completion of high school in Brazil almost doubled over the period, and the college-educated share also rose from 6.4% to 9.4%. The female share of the labor force was essentially stable at 47% in the US, but rose from 37% to 41% in Brazil, driven primarily by a higher rate of female labor force participation (Ferreira, Firpo, and Messina (2016)).

# **4.2 Results**

Before we present results for the _GIC_ , we compute standard inequality measures for hourly real wages for both countries. Table 3 summarizes some of the main changes in the wage distributions in the US and Brazil over this period. The first panel presents five common measures of relative wage inequality for the two countries in 1995 and 2007, as well as for the counterfactual wage distribution _FY_ (1) _|T_ ( _·|_ 0). The inequality measures are the Gini coefficient, the Theil-T index (that is, the Generalized Entropy measure with parameter = 1), the mean log deviation (also known as Theil-L, or _GE_ (0)), the relative mean deviation, and the standard deviation of logarithms. The second panel presents the growth rate in mean hourly wages ( _γAGR_ ) and the average of quantile-specific growth rates, across quantiles, denoted Mean _GIC_ .

– Table 3: Inequality measures hourly real wages (HRW) US and Brazil

||Fac|tual|US<br>Counter Factual|Fac|B<br>tual|ra<br>C|zil<br>ounter Factual|
|---|---|---|---|---|---|---|---|
||1995|2007||1995|2007|||
|Gini|0.355|0.383|0.381|0.539|0.490||0.473|
|Theil Entropy|0.205|0.260|0.258|0.536|0.457||0.444|
|Theil mean log deviation|0.218|0.254|0.250|0.511|0.411||0.383|
|Relative mean deviation|0.257|0.274|0.272|0.404|0.366||0.348|
|Standard deviation of logs|0.682|0.710|0.703|0.963|0.852||0.814|
|Growth of mean wage (_γAGR_)|0.1|73|0.101|0.0|17||-0.150|
|Mean _GIC_|0.1|27|0.063|0.1|38||-0.016|

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Below we discuss the findings for each country separately, including the KS and CVM tests of the four hypotheses listed earlier, before briefly comparing results across countries.

# **4.2.1 United States**

Figure 1 presents the estimates for the _GIC_ and _GIC_<sup>_∗_</sup> , and their corresponding 95% confidence intervals, for the US in the period 1995-2007. The blue line displays the _GIC_ , and the straight horizontal black line represents the corresponding average growth rate, _γAGR_ . The green line displays the counterfactual growth incidence curve, _GIC_<sup>_∗_</sup> _,_ and the dashed red line shows its corresponding mean effect, _γAGR∗_ . The number of bootstrap replications used to construct confidence intervals is 300.

The growth incidence curve for the US is essentially flat at a cumulative growth rate of approximately 10% for the first eight deciles of the distribution. From _τ_ = 0 _._ 8 onwards it begins to slope upwards, and the slope increases sharply for the uppermost decile. A growth rate of 10% over twelve years translates into an average annual wage growth rate of less than one percent over the period, supporting earlier descriptions of wage stagnation for most US workers, even during the “Goldilocks” economy that preceded the great financial crisis of 2008-09 (see, e.g. Kopczuk, Saez, and Song (2010) and Mishel, Bivens, Gould, and Shierholz (2012)). The fact that the growth in the average wage was considerably higher, at 17 _._ 3%, reflects the much better performance of the top quintile. This is also why it was higher than the average quantile-specific growth rate across quantiles, of 12 _._ 7%. The more rapid growth of wages among the top fifth of full-time workers naturally translated into rising inequality, as shown by all five inequality measures in Table 3. The commonly used Gini coefficient rose by almost three percentage points.

The basic finding that there was positive but heterogeneous wage growth in the US is found to be statistically significant by the inference results for the formal hypotheses formulated earlier, namely constant distribution and distribution-neutral growth for the _GIC_ . These results are presented in Table 4, which reports the Kolmogorov-Smirnov (KS) and Cram ́er-von Mises tests (CVM) ( _V_ 1 _n_ and _V_ 2 _n_ , respectively). First, we test the constant distribution hypothesis for the _GIC_ uniformly over quantiles ( _H_ 0 : _GIC_ ( _τ_ ) = 0), which is rejected at the 1% level of significance for both tests. Thus, we reject the hypothesis that the US wage distribution did not change at all. Second, we test whether growth was distribution-neutral over the period, i.e. whether _GIC_ ( _τ_ ) = _γAGR_ . In this test we have an estimate ( _r_ �) under the null hypothesis and apply the _V_<sup> ̄</sup> 1 _n_ and _V_<sup> ̄</sup> 2 _n_ tests. Again, we strongly 27
Figure 1: _GIC_ US 1995–2007 <!-- Start of picture text --> GIC o<br>Counterfactual GIC<br>Average<br>Counterfactual Average<br>o<br>o<br>o<br>o<br>o<br>o oo o o o<br>ooooooooooooooooooooooooooooooooooo o oooo ooo ooooooooooooooooo o ooooooooooooooooooooooooooooooo o oooo o oo o oooo oo ooooooooooo o oooooooooooooo oo oooooo ooo oooooooo o oooooooooooo o ooooo oo ooooooooo oo ooooooooooo oo ooo oo ooooooooooo oooo ooooooooooooooooo<br>0.0 0.2 0.4 0.6 0.8 1.0<br>quantiles<br>0.4<br>0.3<br>GIC 0.2<br>0.1<br>0.0<br><!-- End of picture text --> reject the null hypothesis, which is in line with the heterogeneity observed across quantiles in Figure 1.

The second interesting finding from our analysis is that the counterfactual growth incidence curve, _GIC_<sup>_∗_</sup> , lies everywhere between the no-growth line at zero, and the actual _GIC_ , and its shape is very similar to that of the latter. This implies that both changes in (broadly defined) economic structure - encompassing changes in returns to observed worker attributes, as well as changes in both the distribution of and returns to unobserved characteristics - and changes in the joint distribution of age, gender and education contributed to the modest increase in US wages during the study period. Since the _GIC_<sup>_∗_</sup> is also flat until _τ_ = 0 _._ 8 or thereabouts, and then sharply increasing, we can conclude that the rise in wage inequality is not driven by changes in the gender, age and educational make-up of the workforce. It is driven instead by changes in economic structure and by their impact on the remuneration structure of various worker attributes.

This finding is confirmed by an inspection of the wage inequality measures for the US counterfactual distribution, _FY_ (1) _|T_ ( _·|_ 0), in Table 3 above. All five measures lie strictly between the actual wage inequality levels in 1995 and 2007, but are all much closer to the higher 2007 levels. Taking the mean log deviation as an example, the decomposition indicates that changes in economic structure between 1995 and 2007 shifted the measure from 0 _._ 218 in 1995 to 0 _._ 250. Changes in the joint distribution of covariates - i.e. the age, gender and

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– Table 4: Kolmogorov-Smirnov (KS) and Cram ́er-von Mises (CVM) Tests US

|Null Hypothesis|KS|Cri|tical Val|ues|CVM|Crit|ical Va|lues|
|---|---|---|---|---|---|---|---|---|
|||1%|5%|10%||1%|5%|10%|
|No Efect: _GIC_(_τ_) = 0|0.378|0.067|0.0601|0.057|15.677|1.661|1.379|1.184|
|Mean Efect: _GIC_(_τ_)= _γAGR_|0.306|0.058|0.055|0.051|7.689|1.333|1.252|1.093|
|No Efect: _GIC_<sup>_∗_</sup>(_τ_) = 0|0.205|0.080|0.069|0.059|6.590|2.235|1.849|1.694|
|Mean Efect: _GIC_<sup>_∗_</sup>(_τ_)= _γAGR∗_|0.204|0.075|0.063|0.054|5.648|1.472|1.269|1.186|

educational make up of the full-time labor force - account only for the residual change from 0 _._ 250 to 0 _._ 254. Formal tests, also presented in Table 4, confirm that the _GIC_<sup>_∗_</sup> ( _τ_ ) was neither constant nor distribution neutral over the period. We first test whether _GIC_<sup>_∗_</sup> ( _τ_ ) = 0. The results indicate rejection of the null at 1% level for both the KS and CVM tests. And when we test distribution neutrality of growth conditional on the joint distribution of covariates, _GIC_<sup>_∗_</sup> ( _τ_ ) = _γAGR∗_ the null is again rejected at all reported levels of significance.

# **4.2.2 Brazil**

The results for the Brazilian _GIC_ and _GIC_<sup>_∗_</sup> ( _τ_ ) for 1995-2007 are displayed in Figure 2. As before, the blue line displays the actual _GIC_ , and the dashed black line denotes the growth rate of mean wages, _γAGR_ . The green line displays the counterfactual growth incidence curve, _GIC_<sup>_∗_</sup> _,_ and the dashed red line shows its corresponding mean effect, _γAGR∗_ Remarkably, there was even less growth in average wages for full-time workers in Brazil than in the US over this period. Cumulative growth in real wages was a paltry 1 _._ 7% - a tenth of the US rate.<sup>11</sup> However, the distribution of that growth was completely different from the US case. Brazil’s _GIC_ rises sharply up until the first quintile, at which point in the distribution wages grew by 40% or more over the period. The _GIC_ is then downward sloping from _τ_ = 0 _._ 2 to _τ_ = 1 _._ 0. It crosses the x-axis near the 7th decile, and is negative thereafter. This growth pattern is consistent with a substantial decline in wage inequality among full-time workers, as shown in Table 3. Whereas all five inequality indices reported rose for the US, all five declined for Brazil. The Gini coefficient fell by almost four points,

> 11It is quite likely that this dismal performance is due, at least in part, to a composition effect. Ferreira, Firpo, and Messina (2016) report that formal employment in Brazil rose by a fifth, from 48 to 58% of the labor force, between 1995 and 2012. While not strictly the same, formal employment is highly correlated with full-time status. The same authors also report that the formalization of labor contracts was more common among lower earners. Such a process is likely to lower average earnings in that sample through a composition effect.

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# Figure 2: _GIC_ Brazil 1995–2007

<!-- Start of picture text --> ooooooo oo o oo GICCounterfactual GICAverageCounterfactual Average<br>o oo oo o o o o o o o<br>oo o oooooo o oooooooooooooooooo oo o ooooooooooooooooo o ooooooooooooooo o oooo o ooooooooo o ooo<br>ooooooo o ooooo o ooooo oo ooooooooooooooo o ooo oooo ooo oo ooooooo o ooooooooo oo oooo o oo ooo oooooooooooooooo o oo o oooo oo oooooooooooo o ooooooooooooooooo oo ooo<br>0.0 0.2 0.4 0.6 0.8 1.0<br>quantiles<br>0.4<br>0.2<br>GIC<br>0.0<br>-0.2<br><!-- End of picture text --> and the mean log deviation, which is more sensitive to income gaps at the bottom of the distribution, lost almost 20% of its initial value.

This pro-poor pattern is also evident in the fact that the average growth rate across quantiles was 13 _._ 8% - higher than in the US - despite a near stagnant average wage. Unsurprisingly, then, both the constant distribution and the distribution-neutral growth hypotheses are resoundingly rejected at 1% level of significance for Brazil as well, in both the Kolmogorov-Smirnov and Cram ́er-von Mises tests. This can be seen in Table 5, which collects the results for the KS and CVM tests ( _V_ 1 _n_ and _V_ 2 _n_ , respectively).

As in the US case, the counterfactual growth incidence curve lies everywhere below the _GIC_ , and has a very similar shape. This parallelism suggests that the main drivers of distributional heterogeneity - which in this case were highly equalizing - belong to the realm of changes in economic structure, affecting remuneration patterns and unobserved worker characteristics. One plausible such candidate driver was the sustained rise in Brazil’s minimum wage over this period, which is both consistent with the shape of the _GIC_ , and with earlier findings in the literature (e.g. Ferreira, Firpo, and Messina (2016)). Changes in the joint distribution of observed attributes - gender, age and education - on the other hand, had roughly equi-proportional effects across the distribution. These effects were generally positive - i.e. wage-increasing - as one would expect from rising experience and educational levels.

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– Table 5: Kolmogorov-Smirnov (KS) and Cram ́er-von Mises (CVM) Tests Brazil

|Null Hypothesis|KS|Crit|ical Val|ues|CVM|Crit|ical Va|lues|
|---|---|---|---|---|---|---|---|---|
|||1%|5%|10%||1%|5%|10%|
|No Efect: _GIC_(_τ_) = 0|0.522|0.0648|0.058|0.055|20.187|1.267|1.151|1.027|
|Mean Efect: _GIC_(_τ_)= _γAGR_|0.504|0.066|0.062|0.057|19.395|2.024|1.686|1.532|
|No Efect: _GIC_<sup>_∗_</sup>(_τ_) = 0|0.383|0.081|0.066|0.059|20.268|1.112|1.045|1.018|
|Mean Efect: _GIC_<sup>_∗_</sup>(_τ_)= _γAGR∗_|0.533|0.090|0.067|0.055|22.666|1.423|1.264|1.178|

Once again, this finding is consistent with the inequality measures for the Brazilian counterfactual distribution, reported in the last column of Table 3. These are all lower than the actual inequality values in both 1995 and 2007, suggesting that the observed decline in inequality was due entirely to changes in economic structure. This may well reflect both the effects of a rising minimum wage and the decline in the economy-wide skill premium, as discussed earlier in the literature (see, e.g. Barros et al., 2010). The effect of changes in the observed composition of the labor force was actually to partly offset those declines, through a mildly unequalizing effect of the second term of the decomposition.<sup>12</sup> In terms of formal inference, as should be expected from Figure 2 and the above discussion, both null hypotheses (constant distribution and the distribution-neutral growth) are rejected at the 1% level of significance for both KS and CVM tests. See Table 5.

A comparison of results suggests that the 1995-2007 period saw very different distributional dynamics for real hourly wages among full-time employees across the two countries. Growth in average wages was muted in both countries; and almost zero in Brazil. But such an aggregated description misses important differences in the distribution of that growth: whereas wages were growing at less than 1% per year in the US for all but the top fifth of workers (who experienced much faster increases), Brazil saw relatively rapid wage growth for the bottom half of the distribution, while wages were actually falling for the top fourth. As a result, wage inequality rose in the US and fell markedly in Brazil.

Despite these very disparate headline stories, there were similarities too. In both cases, changes in the observed composition of the labor force - notably higher levels of education and experience - contributed to wage growth, and did so roughly equi-proportionately across

> 12The unequalizing effect of educational expansions when returns are (artificially) held constant is not a novel finding. Bourguignon, Ferreira, and Lustig (2005) refer to this as the ’paradox of progress’ and explain that it reflects the generally observed convexity of returns to schooling. As workers become more educated, mass in the schooling distribution shifts to ranges where returns are steeper, and inequality rises.

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the distribution. In other words, changes in the joint distribution of **_X_** were not responsible for the sharp movements in inequality in either country. Those changes were attributable almost entirely to changes in the distribution of wages conditional on those observables, interpreted here broadly as changes in economic structure.

# **5 Conclusion**

The recent rise in interest in inequality within the economics profession has not been accompanied by a corresponding ability to properly identify the sources of changes in income or wage distributions. The development of the growth incidence curve ( _GIC_ ) by Ravallion and Chen (2003) has spurred a wave of descriptive studies of the distributional characteristics of economic growth, across many countries and time periods. Hitherto, however, the precise requirements for identification and inference using the _GIC_ had not been formally established.

This paper fills that gap by writing the growth incidence curve as a functional of the vector formed by the quantiles of potential outcomes, where treatment assignment is formally replaced by time assignment. We establish the conditions under which both actual and counterfactual growth incidence curves are identified, and propose a simple semi-parametric procedure that allows for the estimation of the _GIC_ with no need for restrictive functional form assumptions on the relationship between income and its covariates. We establish the asymptotic properties of these estimators, and propose practical inference procedures for general functions of the quantile potential outcome. Statistical inference procedures are developed uniformly over the set of quantiles _T_ . We propose testing for general hypotheses and consider both the Kolmogorov-Smirnov and the Cram ́er-von Mises type statistics. Since the parameter of interest is infinite dimensional, for practical inference, we compute critical values using a bootstrap method. We provide sufficient conditions under which the bootstrap is valid, and discuss an algorithm for its practical implementation.

Finally, we use the proposed methods to estimate the actual and counterfactual growth incidence curves for the US and Brazil, during the 1995-2007 period. The results document important heterogeneity across the quantiles of the income distribution in both growth processes. Neither country had a constant income distribution over that period, and neither growth process was distribution-neutral. Growth in average wages was disappointing in both countries, particularly in Brazil. But these averages hide very different distributional 32
pictures: Wage stagnation was observed in the US for the bottom 80% of the distribution, while the top fifth, and particularly the top tenth, grew much more rapidly. Conversely, wages rose rapidly below the median in Brazil, and actually fell for the top 25% or so of the distribution. As a result, inequality fell substantially in Brazil and rose in the United States. In both cases, changes in economic structure, rather than in the observed make-up of the labor force, were responsible for changing inequality.

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