Estimation and Inference for Actual and Counterfactual Growth Incidence Curves

Transcription

1 Discussin Paper Series IZA DP N Estimatin and Inference fr Actual and Cunterfactual Grwth Incidence Curves Francisc H. G. Ferreira Sergi Firp Antni F. Galva January 2017

2 Discussin Paper Series IZA DP N Estimatin and Inference fr Actual and Cunterfactual Grwth Incidence Curves Francisc H. G. Ferreira The Wrld Bank and IZA Sergi Firp Insper and IZA Antni F. Galva University f Iwa January 2017 Any pinins expressed in this paper are thse f the authr(s) and nt thse f IZA. Research published in this series may include views n plicy, but IZA takes n institutinal plicy psitins. The IZA research netwrk is cmmitted t the IZA Guiding Principles f Research Integrity. The IZA Institute f Labr Ecnmics is an independent ecnmic research institute that cnducts research in labr ecnmics and ffers evidence-based plicy advice n labr market issues. Supprted by the Deutsche Pst Fundatin, IZA runs the wrld s largest netwrk f ecnmists, whse research aims t prvide answers t the glbal labr market challenges f ur time. Our key bjective is t build bridges between academic research, plicymakers and sciety. IZA Discussin Papers ften represent preliminary wrk and are circulated t encurage discussin. Citatin f such a paper shuld accunt fr its prvisinal character. A revised versin may be available directly frm the authr. Schaumburg-Lippe-Straße Bnn, Germany IZA Institute f Labr Ecnmics Phne: publicatins@iza.rg

3 IZA DP N January 2017 Abstract Estimatin and Inference fr Actual and Cunterfactual Grwth Incidence Curves * Different episdes f ecnmic grwth display widely varying distributinal characteristics, bth acrss cuntries and ver time. Grwth is smetimes accmpanied by rising and smetimes by falling inequality. Applied ecnmists have cme t rely n the Grwth Incidence Curve, which gives the quantile-specific rate f incme grwth ver a certain perid, t describe and analyze the incidence f ecnmic grwth. This paper discusses the identificatin cnditins, and develps estimatin and inference prcedures fr bth actual and cunterfactual grwth incidence curves, based n general functins f the quantile ptential utcme prcess ver the space f quantiles. The paper establishes the limiting null distributin f the test statistics f interest fr thse general functins, and prpses resampling methds t implement inference in practice. The prpsed methds are illustrated by a cmparisn f the grwth prcesses in the United States and Brazil during Althugh grwth in the average real wage was disappinting in bth cuntries, the distributin f that grwth was markedly different. In the United States, wage grwth was medicre fr the bttm 80 percent f the sample, but much mre rapid fr the tp 20 percent. In Brazil, cnversely, wage grwth was rapid belw the median, and negative at the tp. As a result, inequality rse in the United States and fell markedly in Brazil. JEL Classificatin: Keywrds: C14, C21, D31, I32 grwth incidence curves, ptential utcmes, inference, quantile prcess Crrespnding authr: Francisc H. G. Ferreira Wrld Bank 1818 H Street, NW Washingtn, DC USA fferreira@wrldbank.rg * The authrs wuld like t express their appreciatin t Matias Cattane, Yu-Chin Hsu, David Kaplan, Ying-Ying Lee, Zhngjun Qu, Alexandre Pirier, Liang Wang and participants at the 2015 meeting f the Midwest Ecnmetric Grup fr useful cmments and discussins regarding this paper. Vitr Pssebm prvided excellent research assistance. Cmputer prgrams t replicate the numerical analyses are available frm the authrs. All the remaining errrs are urs.

4 1 Intrductin Grwth episdes have displayed widely different distributinal characteristics acrss cuntries and ver time. The same rate f grwth in average incmes has been accmpanied by rising inequality in sme cases, and by falling inequality in thers. A large literature n pr-pr grwth and, mre generally, n the incidence f ecnmic grwth prcesses has develped, and attracted attentin amng bth researchers and plicymakers. Over time, this literature has cme t rely heavily n the Grwth Incidence Curve (GIC), which describes the rate f incme grwth at each quantile τ (0, 1) f the (annymus) distributin (Ravallin and Chen (2003)). It has been used t cmpare the distributinal characteristics f grwth prcesses bth acrss cuntries and ver time (see, e.g. Besley and Crd (2007)). It has als been shwn t underlie changes in certain widely-used classes f pverty and inequality measures, which can be frmally expressed as functinals f the GIC (Ferreira (2012)). Grwth incidence curves have als featured in a lng-standing literature that uses cunterfactual incme distributins t decmpse changes (r differences) in inequality and pverty ver time (r between cuntries), and t attribute such changes t different factrs such as, fr example, changes in wrker characteristics r in the returns t thse characteristics. The riginal cntributins t this literature, including Juhn, Murphy, and Pierce (1993), Dinard, Frtin, and Lemieux (1996) and Dnald, Green, and Paarsch (2000), predate the Ravallin and Chen (2003) article that intrduced the term GIC, and hence d nt use it. Yet, each f thse papers sught t accunt fr differences acrss entire wage r incme distributins which can be frmally expressed as GICs using cunterfactual distributins. Ferreira (2012) defines cunterfactual grwth incidence curves as functinals f cunterfactual distributins, and establishes the link t this earlier literature n distributinal change. Despite their cnceptual imprtance and widespread practical use, hwever, frmal cnditins fr identificatin and inference using grwth incidence curves actual r cunterfactual have nt been established. In this paper, we rely n the frmal analgy between distributinal change and treatment hetergeneity t fill that gap. Mre specifically, we write bth actual and cunterfactual GICs in terms f vectrs f ptential utcmes (Rubin (1977)), and then apply suitable variants f a number f results frm the literature n quantile treatment effects t frmally establish the cnditins fr identificatin f the GIC. Specifically, we adapt the identificatin results in Firp (2007), where the relevant identificatin restric- 1

5 tin is the ignrability assumptin. 1 In ur cntext, it implies that the incme distributins that we bserve in tw different time perids are generated by tw grup f factrs nly: bservable cmpnents whse distributins may vary ver time, and unbservable cmpnents whse cnditinal distributins given bservables are fixed ver time. We then prpse a simple three-step semiparametric estimatr fr bth actual and cunterfactual grwth incidence curves, which relies n established sample re-weighting and quantile regressin techniques. In the first step, a nnparametric estimatr f the prpensity scre is used, and weights are cmputed. In ur setup, the prpensity scre is cmputed by pling the repeated crss-sectin data fr initial and end perids and calculating the prbability f being bserved at the final perid, given cvariates. In the secnd step, ne btains prperly weighted quantiles f the utcme frm a simple weighted quantile regressin. The third step is the cmputatin f the GIC as a functin f the vectr f quantiles f weighted utcme distributins. 2 When applied t cunterfactual GICs, this prcedure has the added advantage that it requires n assumptin n the structural relatinships between incme and its cvariates, as was the case with mst f the previus literature. We establish the asympttic prperties f these estimatrs, prpse suitable test statistics, and discuss inference prcedures in practice. Fr practical inference we cmpute critical values using resampling methds. We prvide sufficient cnditins and shw the theretical validity f a btstrap apprach. Mrever, we discuss in detail an algrithm fr its practical implementatin. We als discuss cmputatin f critical values thrugh a subsampling methd. The main technical cntributins f the paper are as fllws. The first is t develp practical statistical inference prcedures fr the GIC. This enables researchers t cnduct estimatin and inference fr the GIC ver the entire set f quantiles. Secndly, we can easily extend ur results t general functinals f the vectr f quantiles f ptential utcmes and nt nly the ne that yields the GIC, which allws us t develp testing prcedures fr general hyptheses invlving these functinals. 3 An additinal by-prduct cntributin f 1 This cnditin has been emplyed widely in the distributinal treatment effect literature. See nt nly Firp (2007), but als, amng thers, Flres (2007), Cattane (2010), and Galva and Wang (2015). 2 A natural extensin f ur methd nt pursued in this paper wuld be t implement a furth step, which wuld invlve estimatin and inference f real-valued functinals f the GIC prcess, such as pverty and incme inequality grwth. 3 The theretical results derived in this paper can be applied t ther functinals f the quantiles f ptential utcmes prcesses. Fr instance, the quantile treatment effects in Firp (2007), and the Makarv bunds fr the quantiles f the distributin f treatment effects discussed in Fan and Park (2010), althugh fllwing a mre elabrate frmula, are als functinals f the quantiles f the ptential utcmes. In general, ur final estimatr can be seen as a plug-in estimatr f the functinal using the estimated quantiles 2

6 this paper is t establish the asympttic prperties f the estimatr f the vectr f quantiles f weighted utcme distributins fr the quantile prcess, namely, unifrm cnsistency and weak cnvergence. The prvisin f unifrm results ver the set f quantiles is a necessary cnditin t establish the results fr the testing prcedures. We als shw that the estimatr is unifrmly efficient, as the asympttic variance f the estimatr cincides with the semiparametric efficiency bund. These cntributins are clsely related t the literature n quantile treatment effects, which is a particular functinal f the vectr frmed by the quantiles f the ptential utcmes. That literature started with Dksum (1974) and Lehmann (1974) and has expanded recently (see, e.g., Abadie, Angrist, and Imbens (2002), Chernzhukv and Hansen (2005), Bitler, Gelbach, and Hynes (2006), Firp (2007), Cattane (2010), Dnald and Hsu (2014), Galva and Wang (2015), and Firp and Pint (2015)). 4 We illustrate the prpsed prcedure by cmparing actual and cunterfactual grwth incidence curves (fr real hurly wages) fr the tw largest cuntries in the Western Hemisphere, namely the United States and Brazil, in the twelve years prir t the nset f the last great financial crisis: Althugh grwth rates in average wages were disappinting in bth cuntries (especially in Brazil), there were substantial differences in inequality dynamics. The GIC fr the US was flat until apprximately the 8th decile, and sharply upward-slping ver the tp quintile. In Brazil cnversely, the GIC peaked arund the first quintile, and was dwnward slping thereafter. As a result wage inequality rse sharply in the US and declined in Brazil. We use cunterfactual GICs t examine whether these changes were driven primarily by the cmpsitin f the labr frce - in terms f bserved wrker characteristics such as gender, age, and educatin - r by changes in the brader structure f the ecnmy. In bth cuntries, we find that increases in wrker age (and thus ptential experience) and educatin cntributed t incme grwth in a rughly equiprprtinal manner. Changes in inequality were driven almst entirely by changes in ecnmic structure. The remainder f the paper is rganized as fllws. Sectin 2 defines the GIC. Sectin 3 presents the ecnmetric results, describes the three-step estimatr, establishes the asympttic prperties f the estimatr, discusses inference fr the quantile prcess, and its f ptential utcmes. 4 The results f this paper are als related t thse n inference n the quantile prcess. See, e.g., Bellni, Chernzhukv, and Fernandez-Val (2011), and Qu and Yn (2015) fr the nnparametric case; Gutenbrunner and Jureckva (1992), Kenker and Machad (1999), Kenker and Xia (2002), Chernzhukv and Fernandez-Val (2005), and Angrist, Chernzhukv, and Fernandez-Val (2006) fr the parametric case. 3

7 practical implementatin. The empirical applicatin t the US and Brazil is presented in Sectin 4. Sectin 5 cncludes. We relegate the prfs f the results t the Appendix. 2 Grwth incidence curves: Actual and cunterfactual In this sectin we frmally define the grwth incidence curve (GIC), which was riginally intrduced by Ravallin and Chen (2003). Let Y be the utcme variable f interest, say an indicatr f ecnmic welfare such as incme. There are tw time perids, 0 and 1. Let us say that an individual bservatin taken at time 1 belngs t grup A, ie, G = A. An bservatin taken at time 0 belngs t grup B, r G = B. Assume that incme is cntinuusly distributed ver the ppulatin f interest, and dente its cumulative distributin functin (CDF) at time t as F Y T ( t). The incme level at the τ-th quantile fr grups A and B are given by, respectively, the inverse f the CDF, q A (τ) = F 1 Y T (τ 1) and q B (τ) = F 1 Y T (τ 0). Then, the instantaneus GIC at a given time t and quantile τ can be represented as df 1 Y T (τ t)/dt F 1 Y T (τ t). In discrete time, the incme grwth rate fr a given quantile τ between tw time perids, 0 and 1, can then be written as GIC Y (τ) = q A(τ) q B (τ). q B (τ) Mtivated by the imprtance f the GIC fr the ecnmic analysis f scial welfare, this paper develps estimatin and inference prcedures fr the GIC(τ), which is calculated as the difference f quantiles in time perids 1 and 0 ver the quantile in time zer, fr the entire set f quantiles τ (0, 1). We assume availability f a randm sample f size n frm the jint distributin f (Y, T, X), where Y is the incme, T is a time dummy variable that equals 1 at perid T = 1, and X is a vectr f length d f cvariates. We culd have represented the data equivalently as (Y, G, X). The cvariates enable us t learn hw changes in their jint distributin affect grwth and inequality. Fr an individual i in ur sample, if G i = A we bserve Y i (1), therwise G i = B and we bserve Y i (0), where Y i (1) is what individual i s utcme wuld be were she bserved at time T = 1, and Y i (0) is what individual i s utcme wuld be were she bserved at time T = 0. Brrwing frm the treatment effect literature, we call Y (1) and Y (0) ptential utcmes ; we say that individual i is treated if she is bserved at perid 1 r grup A, and untreated if bserved at perid 0 r grup B. We may refer t T as the 4

8 treatment assignment dummy r, mre accurately, time assignment dummy. Thus, the bserved utcme is Y = (Y (1) Y (0))T + Y (0). Writing the prblem in terms f ptential utcmes is useful because it allws us t easily write bth actual and cunterfactual distributins. Fr example, the actual utcme distributin fr thse individuals frm grup B, that is, thse wh were bserved at time 0, is F Y (0) T ( 0) and the actual utcme distributin fr thse individuals frm grup A, that is, thse wh were bserved at time 1, is F Y (1) T ( 1). The cunterfactual utcme distributin fr thse individuals wh were bserved at time 0, were they bserved at time 1, is F Y (1) T ( 0) and the cunterfactual utcme distributin fr thse individuals wh were bserved at time 1, were they bserved at time 0, is F Y (0) T ( 1). Let τ be a real number in T (0, 1) and t = 0, 1. Let q At (τ) be inf q Pr[Y (t) q T = 1] τ, r the τth quantile f F Y (t) T ( 1), which is the distributin functin f Y (t) fr the subppulatin A. Fr the B subppulatin, let q Bt (τ) be inf q Pr[Y (t) q T = 0] τ, r the τth quantile f F Y (t) T ( 0). Fr bth subppulatins, thse distributin functins share the same supprt, which is Y t R. Let us als define Q A (τ, τ ) := [ qa1 (τ) q A0 (τ ) ], and Q B (τ, τ ) := [ qb1 (τ) q B0 (τ ) Thus, the GIC can be derived frm the previus variables as the grwth rate f incme at the τth quantile between perids 0 and 1. We first define the bserved r actual GIC as GIC(τ) := q A1(τ) q B0 (τ) q B0 (τ) = [ [ ] ] 1 0 Q A (τ, τ) ] 1. (1) 0 1 Q B (τ, τ) The graphical depictin f GIC, as prpsed in Ravallin and Chen (2003), is btained by letting τ vary frm zer t ne and pltting the crrespnding values f GIC against the quantiles τ. The quantiles invlved in the cmputatin f equatin (1) are based n the ranking f individuals in each distributin f interest. Therefre, unless the individual i keeps her ranking ver time, GIC will nt be an apprpriate tl t infer individual gains ver time. This is a cnsequence f the veil f ignrance (annymity) shruding the cmparisn f the tw distributins (see Essama-Nssah, Paul, and Bassle (2013)). The interpretatin f the graphical depictin f GIC is simple. If the GIC is a decreasing functin fr all τ in its dmain f definitin, then all inequality measures that respect the. 5

9 Pigu-Daltn principle f transfers and scale invariance will indicate a fall in inequality ver time. If instead, the GIC is an increasing functin f τ, then the same measures will register an increase in inequality (Ravallin and Chen (2003)). When n relative inequality measure changes ver time, then the GIC will present a cnstant grwth rate ver the prcess f quantiles τ. Using ur previus ntatin, we can define GIC as the cunterfactual GIC. It can be expressed as [ ] [ ] GIC (τ) := q B1 (τ) q B0 (τ) 1 1 Q B (τ, τ) 1 0 Q B (τ, τ) = [ ] = [ ] 1, (2) q B0 (τ) 0 1 Q B (τ, τ) 0 1 Q B (τ, τ) which is the grwth incidence curve fr quantile τ if the distributin f assciated factrs (explanatry variables, r cvariates) had remained fixed frm perid 0 t 1. GIC captures nly that part f distributinal change assciated with changes in the cnditinal distributin F Y ( ) T, which we interpret bradly as changes in the structure f the ecnmy. Cmparing GIC with GIC allws us t understand whether hetergeneity in ecnmic grwth is driven by changes in the jint distributin f bserved cvariates (X) that impact incme, r is driven by changes in the structure f the ecnmy. Fr example, if GIC is decreasing in τ but GIC is unifrm (flat) ver τ, the decrease in inequality is driven by changes in the distributin f cvariates. This interpretatin can be frmally btained by decmpsing the GIC(τ) int tw cmpnents as fllwing: GIC (τ) = GIC (τ) + GIC (τ) qb1 (τ) q B0 (τ), where GIC (τ) := q A1 (τ) q B1 (τ) q B1 (τ) = [ [ ] Q A (τ, τ) ] 1 Q B (τ, τ) is the grwth incidence curve that wuld have ccurred nly because f time changes in the distributin f cvariates. We will develp estimatin and inference prcedures fr the GIC(τ) and GIC (τ) and, mre generally, fr functinals f the quantile f ptential utcmes. In that sense, ur theretical framewrk prvides a flexible methd fr the practical analysis f the grwth incidence curves. 6

10 3 The ecnmetric mdel In this sectin we intrduce the ecnmetric mdel, discuss identificatin, estimatin f the parameters f interest, and inference prcedures. As previusly seen, GIC can be written as a functin f the vectr f quantiles f ptential utcmes. Thus, in this sectin, we first btain the results fr the latter, and then, fr the GIC. Ntatin: Let E and E be p expectatin and sample average, respectively. Let,, and p dente weak cnvergence, and cnvergence in prbability and in uter prbability, respectively. Let g(z) dente sup z g(z) fr z Z. 3.1 Identificatin In rder t make ur setup cmparable with the treatment effects literature, we maintain all definitins and ntatin as it is cmmnly used in that framewrk. Therefre, we have a randm sample f size n frm the jint distributin f (Y, T, X), where Y is the utcme f interest, T is a dummy variable f treatment assignment, and X is a vectr f length d f cvariates. Fr cmpleteness, in this sectin, we als define q t (τ) as inf q Pr[Y (t) q] τ, fr t = 0, 1, which is the uncnditinal τth quantile f F Y (t), the distributin functin f Y (t) whse supprt is Y t R. Nw we define the p-scre, the cnditinal prbability f being treated (bserved at time 1) given X, as p (X), and the uncnditinal prbability as p. Let X X R d. In what fllws, it is als useful t define the functin m as: m(a, b; τ) = τ 1{a < b}. We state assumptins n the general mdel fr identificatin f the parameters f interest. I.I Fr each τ T, t = 0, 1, q t (τ) uniquely slves E[m(Y (t), q t (τ); τ)] = 0; q At (τ) uniquely slves E[m(Y (t), q At (τ); τ) T = 1] = 0; and q Bt (τ) uniquely slves E[m(Y (t), q Bt (τ); τ) T = 0] = 0. I.II Fr all τ T, we have (Y (1), Y (0)) T X; I.III Fr sme c > 0, c < p(x) < 1 c, a.e. X. Assumptins I.I I.III are standard in the literature, as in Firp (2007). Cnditin I.I is in general nt a sufficient identificatin cnditin fr q t (τ) because Y (t) is nt always bservable frm the data. Therefre, the untestable cnditin I.II, the s-called ignrability assumptin, is fundamental. Accrding t cnditin I.II, the assignment t the treatment is 7

11 randm within subppulatins characterized by X. This assumptin has been used, amng thers, by Heckman, Ichimura, Smith, and Tdd (1998), Dehejia and Wahba (1999), Hiran and Imbens (2004), Firp (2007). Within the GIC framewrk this assumptin implies that cnditinal n X, there is a randm mechanism that assigns individual i t the exact perid that she is bserved (either perid 0 r 1). In ur mdel the triple (Y, T, X) is bservable, and a randm sample f size n can be btained. Cnditin I.III states that fr almst all values f X, bth treatment assignment levels have a psitive prbability f ccurrence. Under cnditins I.I I.III the quantities q 1 (τ), q 0 (τ), q A1 (τ), q A0 (τ), q B1 (τ) and q B0 (τ) are identified frm the jint distributin f (Y, T, X). These six bjects can be written as implicit functins f the bserved data. Fr all τ T, where w 1 (T, X) = ( w B1 (T, X) = E [w 1 (T, X) m(y, q 1 (τ); τ)] = E [w 0 (T, X) m(y, q 0 (τ); τ)] = E [w A1 (T, X) m(y, q A1 (τ); τ)] = E [w A0 (T, X) m(y, q A0 (τ); τ)] = E [w B1 (T, X) m(y, q B1 (τ); τ)] = E [w B0 (T, X) m(y, q B0 (τ); τ)] = 0, T, w p(x) 0(T, X) = ) ( ) T 1 p(x) 1 p p(x) directly frm Lemma 1 in Firp (2007). ( 1 T, w 1 p(x) A1(T, X) = T, w p A0(T, X) = ) ( ) 1 T p(x), p 1 p(x) and w B0 (T, X) = 1 T. This main identificatin result fllws 1 p Finally, given that the elements in the vectrs Q(τ, τ ), Q A (τ, τ ) and Q B (τ, τ ) are identified, since q 1 (τ), q 0 (τ), q A1 (τ), q A0 (τ), q B1 (τ) and q B0 (τ) are identified, it fllws frm equatins (1) and (2) that GIC(τ) and GIC (τ) are als, respectively, identified frm the jint distributin f (Y, T, X). Remark 1. We nte that ne can als btain ther functinals f interest based n Q (τ, τ ), Q A (τ, τ ) and Q B (τ, τ ), which highlights the ptential relevance f the prpsed methds in practice. Given the identificatin result, general functinals f parameters f interest are als identified, since they can be written as functins f q t (τ), q At (τ), q Bt (τ), and cnsequently as functins f the bservable variables (Y, T, X). Fr example, the quantile treatment effect (QTE) will be ( τ) = q 1 (τ) q 0 (τ) = [1 the treated (QTT) will be A (τ) = q A1 (τ) q A0 (τ) = [1 1] Q (τ, τ) and fr quantile treatment effect n 1] Q A (τ, τ). Less cmmn than the previus tw treatment effect parameters, the QTU, the quantile treatment effect n the untreated, is defined as B (τ) = [1 1] Q B (τ, τ) = q B1 (τ) q B0 (τ). Other functinals, such as the Makarv bunds fr the CDF f Y (1) Y (0) (Fan and Park (2010)) that explicitly depend n Q A (τ, τ ) and Q B (τ, τ ) at different pints (τ, τ ), can similarly be btained frm 8

12 the quantiles f ptential utcmes. 3.2 Estimatin We are interested in estimatin and inference fr the GIC(τ) and GIC (τ). Equatins (1) and (2) shw that GIC can be written as a functin f the quantiles f ptential utcmes. Thus, we estimate each cmpnent f the vectrs Q (τ, τ ), Q A (τ, τ ) and Q B (τ, τ ) t cnstruct estimatrs fr the GIC(τ) and GIC (τ). Given identificatin, we are able t estimate the parameters f interest using a multi-step estimatr as fllws. Step 1 Estimate p(x) parametrically r nnparametrically and btain an estimatr p (X). 5 The estimatr f p is the sample average f T, i.e., p = n 1 n i=1 T i. Step 2 Fr each (τ, τ ) T T, btain [ q1 (τ) ] [ qa1 (τ) ] [ qb1 (τ) ] Q (τ, τ ) = q 0 (τ ), Q A (τ, τ ) := q A0 (τ ), and Q B (τ, τ ) := q B0 (τ ), where, fr t = 0, 1, q t (τ), q At (τ) and q Bt (τ) satisfying the fllwing cnditins: E[ŵ t (τ 1{Y < q t (τ)})] = 0 (3) E[ŵ At (τ 1{Y < q At (τ)})] = 0 (4) E[ŵ Bt (τ 1{Y < q Bt (τ)})] = 0, (5) where ŵ 1,i = T i / p (X i ), ŵ 0,i = (1 T i ) / (1 p (X i )), ŵ A1,i = T i / p, ŵ A0,i = [(1 T i ) / (1 p (X i ))] [ p (X i ) / p], ŵ B1,i = [T i / p (X i )] [(1 p (X i )) / (1 p)] and ŵ B0,i = (1 T i ) / (1 p). In practice, estimatrs f q t (τ), q At (τ) and q Bt (τ) can be btained by weighted quantile 5 Appendix 6.3 discusses the practical estimatin f p(x). 9

13 regressins (QR) q t (τ) = arg min q E [ŵ t,i ρ τ (Y i q)], (6) q At (τ) = arg min q E [ŵ At,i ρ τ (Y i q)] and (7) q Bt (τ) = arg min q E [ŵ Bt,i ρ τ (Y i q)], (8) where ρ τ (u) := u(τ 1{u < 0}) is the check functin as in Kenker and Bassett (1978). Step 3 Finally, we can plug-in estimates f the quantiles f the ptential utcmes int the expressins t estimate GIC in (1) as fllwing ĜIC (τ) = q A1 (τ) q B0 (τ) q B0 (τ) = [ [ ] 1 0 QA (τ, τ) ] 1, 0 1 QB (τ, τ) where we estimate q A1 (τ) and q B0 (τ) as in (7) and (8), respectively. T cmpute the crrespnding weights, we estimate the prpensity scre, p(x), by apprximating its lg-dds rati by a plynmial and use the lgistic link functin with cvariates given belw in the data descriptin. Analgusly, we can als estimate the cunterfactual GIC in (2) as ĜIC (τ) = q B1 (τ) q B0 (τ) q B0 (τ) = [ [ ] 1 0 QB (τ, τ) ] 1, 0 1 QB (τ, τ) which, as described previusly, is the grwth incidence curve fr quantile τ if the distributin f explanatry variables f incme had remained fixed frm perid 0 t 1. There are ther alternative estimatrs available in the literature fr the quantile bjects f interest defined in Step 2 abve. Dnald and Hsu (2014) discuss an estimatr that makes use f the inverse f the cumulative distributin functin (CDF) f the ptential utcmes. Their apprach t estimate the quantiles is a three-step prcedure. In the first step ne needs t cmpute weights; in the secnd step the CDF is cmputed fr all pints n its supprt by using an inverse prbability weighted estimatr; and in the third step ne btains the quantile by inverting the CDF. We shw belw that the estimatr prpsed by Dnald and Hsu (2014) and ur prpsed methd are asympttically equivalent. Nevertheless, the 10

14 estimatr discussed in this paper has several practical advantages. First, ur estimatr fr the quantiles is a tw-step methd: the first step cincides with the ne f Dnald and Hsu (2014), but unlike that methd, ur QR estimatr fr the bject f interest is btained withut having t invert the CDF. This is pssible because f the secnd advantage f ur methd: QR has a linear prgram representatin, which makes practical cmputatin simple and allws using weights directly int the bjective functin that is slved. Finally, if ne is interested in quantiles, and its transfrmatins, using the prpsed estimatr is attractive due t its cmputatinal efficiency and accuracy in finite samples. 6 Remark 2. One can als easily use the multi-step estimatr defined abve t btain estimates fr ther functinals f interest. Fr example, the estimatr f QTE will be (τ) = q 1 (τ) q 0 (τ) = [1 1] Q (τ, τ) and fr QTT will be A (τ) = q A1 (τ) q A0 (τ) = [1 1] Q A (τ, τ). Other functinals, such as the Makarv bunds fr the quantiles f the distributin f treatment effects, Y (1) Y (0), are estimated using the analytical expressins f these estimated bunds as functins f Q A (τ, τ ) and Q B (τ, τ ). 3.3 Asympttic prperties In this sectin, we derive the asympttic prperties f the multi-step estimatr fr the quantile prcess. We first fcus n the prperties f the estimatr f q t (τ) and establish the unifrm cnsistency and the weak limit f q t (τ), in l (T ). The extensin t q At (τ) and q Bt (τ) is direct. We als establish the cnsistency and the weak limit f Q(τ, τ ), QA (τ, τ ) and Q B (τ, τ ) in l (T ) l (T ). The asympttic prperties f the ĜIC(τ) and ĜIC (τ) fllw frm these results. In additin, we derive the unifrm semiparametric efficiency f the estimatr. Finally, we discuss hw in practice we estimate weights used t cmpute q t (τ). The tw last results are cllected in the Appendix Cnsistency Cnsistency is a desired prperty fr mst estimatrs. Fr the cnsistency f prcess q t (τ) ver τ T, cnsider the fllwing cnditins. 6 We refer the reader t Kenker, Lerat, and Peracchi (2013) fr a discussin and cmparisn n the statistical prperties f the distributin regressin and the quantile regressin appraches. 7 In Appendix 6.2, we prvide results fr the unifrm semiparametric efficiency f the estimatr. In Appendix 6.3 we discuss the practical estimatin f the crrespnding nuisance parameters, w t ( ), w At ( ), and w Bt ( ). 11

15 QC.I Fr s, t {0, 1}, the densities f Y (s) T ( t) are bunded abve and, unifrmly in τ, psitive. Als, fr any δ > 0, inf q t(τ) >δ E[w t(t, X)(τ 1{Y < q t (τ)})] > ɛ δ. QC.II There exists 0 < M w < such that w t (T, X) < M w, a.e. (T, X). QC.III ŵ t w t = p (1). These cnditins are standard in the literature. We state QC.I and QC.II fr selfcntainedness. As usual in the QR literature, QC.I requires the density t be bunded away frm infinity. The secnd part f QC.I is a standard identificatin cnditin. It is similar t Angrist, Chernzhukv, and Fernandez-Val (2006) and Firp (2007), and it fllws frm I.I I.III fr each τ. QC.II impses bundedness n the density f X. It is analgue t Assumptin 1(ii) f Firp (2007) and fllws directly frm I.III. QC.III requires cnsistent estimatin f the nuisance parameter. This is a usual requirement crrespnding t (1.4) f Therem 1 f Chen, Lintn, and Van Keilegm (2003). The fllwing result establishes cnsistency f the estimatr ver the set f quantiles. Therem 1. Suppse that E[w t (T, X)m(Y, q t (τ); τ)] = 0, and that cnditins QC.III are satisfied. Then, fr t = 0, 1, as n QC.I sup q t (τ) q t (τ) = p (1). τ T The extensin f Therem 1 t q At ( ) and q Bt ( ), t = 0, 1 is direct. The assumptins QC.I QC.III are analgus Weak cnvergence Nw we derive the limiting distributin f the general q t (τ) estimatr. fllwing sufficient cnditins. We impse the QG.I The functins ŵ t (T, X) Π and ŵ t (T, X) p w t (T, X) unifrmly in (T, X) ver cmpact sets, where w t (T, X) Π, and Π is a functin class f unifrmly smth functins in (T, X) with dmain {0, 1} X. 12

16 QG.II n (E[(ŵ t (T, X) w t (T, X))(τ 1{Y < q t (τ)})] + E[w t (T, X)(τ 1{Y < q t (τ)})]) cnverges weakly. QG.III ŵ t (T, X) w t (T, X) = p (n 1/4 ). Assumptins QG.I QG.III cncern the prperties f the weights. They are high level cnditins and will be discussed in the sectin f the estimatin f w t. Cnditins QG.I and QG.II allw fr estimated weights. Assumptin QG.II is similar t Cattane (2010). Examples satisfying QG.II include smth functin classes. These assumptins allw fr a wide variety f nnparametric and parametric estimatrs. QG.III strengthens QC.III such that the estimatr f the nuisance parameter cnverges at a rate faster than n 1/4. A similar assumptin appears in Chen, Lintn, and Van Keilegm (2003). Nw we present the weak cnvergence result. Therem 2. Fr t = 0, 1, suppse that E[w t (T, X)m(Y, q t (τ); τ)] = 0, that q t q t = p (1), and that cnditins QC.I QC.II, QG.I QG.III are satisfied. Then, in l (T ), n( qt q t ) G t, where G t is a mean zer Gaussian prcess with cvariance functin E[G t (τ)g t (τ ) ] = Dt 1 (τ)s tt (τ, τ )[Dt 1 (τ )], with, fr t = 0, 1, and l = 0, 1, D t (τ) = E[w t(t, X)m(Y, q t (τ); τ)] qt(τ)=q q t (τ) t(τ) S tl (τ, τ ) = E [(w t (T, X) (m(y, q t (τ); τ) E [m(y, q t (τ); τ) X, T = t]) + E [m(y, q t (τ); τ) X, T = t]) (w l (T, X) (m(y, q l (τ ); τ ) E [m(y, q l (τ ); τ ) X, T = l]) + E [m(y, q l (τ ); τ ) X, T = l])]. The result in Therem 2 shws that the limiting distributin f the estimatr is a Gaussian prcess. Thus, if ne fixes a quantile at τ, then the limiting distributin cllapses t a simple nrmal distributin, as in Firp (2007). Fr practical inference, belw we prvide inference methds ver the set f quantiles that are simple t implement in applicatins. 8 As befre, the extensin f Therem 2 t q At ( ) and q Bt ( ), t = 0, 1 is direct. The assumptins crrespnding t QG.I QG.III are analgus. 8 Firp and Pint (2015) present a similar result t Therem 2. Nevertheless ur prf technique is different n the treatment f bth infinite dimensin parameters. In additin, we d nt require cmpactness f the supprt f X and impse weaker assumptins n ŵ t. 13

17 Given the result in Therem 2, it is simple t establish the weak cnvergence t the vectr Q(τ, τ ). The results fr Q A (τ, τ ) and Q B (τ, τ ) are analgus. Crllary 1. Assume the cnditins f Therem 2, as n, in l (T ) l (T ) n( Q Q) = n [ ( q1 q 1 ) ( q 0 q 0 ) ] G = [ G1 where G is the vectr f Gaussian prcesses with cvariance functin E[G(τ, τ )G(τ, τ ) ] = [ D 1 1 (τ)s 11 (τ, τ )[D1 1 (τ )] D1 1 (τ)s 10 (τ, τ )[D0 1 (τ )] D0 1 (τ )S 01 (τ, τ )[D1 1 (τ )] D0 1 (τ )S 00 (τ, τ )[D0 1 (τ )] G 0 ], ]. In rder t perfrm inference n functins f the Q(τ, τ ), we impse a differentiability cnditin n such functins and state a functinal delta methd result. Cnsider the fllwing assumptin. QG.IV (Hadamard) The functinal h : l (T ) l (T ) l (T ) defined ver the distributin f ptential utcmes is Hadamard differentiable at Q, with Hadamard derivative given by h( ). The fllwing result is a well knwn applicatin f the functinal delta methd, we include it fr cmpleteness. Lemma 1. Assume the cnditins f Therem 2, and QG.IV, as n, n(h( Q) h(q)) h(g). Dnald and Hsu (2014) establish the weak cnvergence f a quantile estimatr that makes use f the inverse f the CDF in their Therem 3.8. Their result is similar t that in Therem 2 abve. Nevertheless, as mentined previusly, the quantile estimatrs are different. In additin, the assumptins required t establish the results are different. On the ne hand, Dnald and Hsu (2014) impse strng cnditins t derive the result. Fr instance, their Assumptin 3.1 requires that the distributins f Y (0) and Y (1) have cnvex and cmpact supprts. Their Assumptin 3.2 requires all the cvariates t be cntinuus 14

18 and the supprt f the vectr f cvariates, X, t be cmpact. We are able t smewhat relax these assumptins. Given that we wrk with a standard semiparametric estimatr, and a quantile regressin framewrk in the secnd step, we d nt require such assumptins t derive the asympttic prperties f ur prpsed estimatr. Nw we t return t the main bject f interest and analyze the grwth incidence curves, GIC(τ) and GIC (τ). As an applicatin f Therem 2 and Lemma 1, we derive the asympttic distributin fr GIC(τ). Crllary 1 implies that n( QA Q A ) G A, and n( Q B Q B ) G B, where G A (τ) and G B (τ) are Gaussian prcesses with variance-cvariance functins that can be btained as an applicatin f Crllary 1. Recall that GIC(τ) = [1 0]Q A(τ,τ) 1, and [0 1]Q B (τ,τ) GIC (τ) = [1 0]Q B(τ,τ) [0 1]Q B 1. These functinals (τ,τ) are differentiable at (Q A, Q B ), as lng as q B0 (τ) 0 with derivatives defined by GIC(G A, G B ) = and fr GIC we have that 1 [0 1]Q B [1 0]G A [1 0]Q A ([0 1]Q B ) 2 [0 1]G B, ( GIC (G B ) 1 = [1 0] [1 0]Q ) B 2 [0 1] G B. [0 1]Q B ([0 1]Q B ) Therefre, frm a functinal delta methd we have the fllwing results. Crllary 2. Assume the cnditins f Therem 2, as n, in l (T ) n(ĝic GIC) GIC(GA, G B ) (9) n(ĝic GIC ) GIC (G B ). (10) 3.4 Inference prcedures In this sectin, we turn ur attentin t inference prcedures n the GIC. Imprtant questins psed in the ecnmetric and statistical literatures cncern the nature f the 15

19 impact f a plicy interventin r treatment n the utcme distributins f interest. The crrespnding questins fr the GIC are, fr example, whether there is significant incme grwth at any quantile (the null hypthesis being GIC (τ) = 0 fr all τ); and whether grwth is unifrm r hetergeneus (GIC (τ) equals the average grwth rate, fr all τ). One can als ask if grwth is nn-decreasing in τ (GIC (τ) 0 fr all τ). Since the main bjective f this paper is t study the grwth incidence curve, and these questins and hyptheses are frmulated fr the entire GIC prcess, we develp inference prcedures fr the quantile prcess ver the set f quantiles indexed by τ Test statistics We seek t develp inference fr GIC ver the index set f quantiles T. We present results fr functinals f quantiles f the marginal distributins f ptential utcmes, and in particular, the GIC(τ) and GIC (τ). Let β(τ) be a functinal f Q, Q A, and Q B, that is, β(τ) = h(q(τ, τ)). In particular, we are interested in β(τ) = GIC(τ) = [1 0]Q A(τ,τ) [0 1]Q B (τ,τ) 1, and the cunterfactual ne β(τ) = GIC (τ) = [1 0]Q B(τ,τ) [0 1]Q B (τ,τ) 1. We discuss three main hyptheses f interest. First, we cnsider the fllwing standard null hypthesis H 0 : β(τ) r(τ) = 0, τ T, (11) unifrmly, where the vectr r(τ) is assumed t be knwn, cntinuus in τ ver T, and r l (T ). Mre generally, the hypthesis in (11) embeds several interesting hyptheses abut the parameters f the quantile functin. Example (The unifrmly null effect hypthesis). A basic hypthesis is that the grwth incidence curve, GIC(τ), is statistically equal t zer fr all τ T. The alternative is that the it differs frm zer at least fr sme τ T. In this case, r(τ) = 0, and relative inequality remains stable. The basic inference prcess t test the null hypthesis (11) is W n (τ) := β(τ) r(τ), τ T. T derive the asympttic prperties f the abve statistic, we need t cmpute the estimatr β(τ), which is given by β = h( Q). The GIC(τ) estimate is β(τ) = ĜIC(τ) = [1 0] Q A (τ,τ) [0 1] Q 1, and the estimate fr B (τ,τ) GIC (τ) is β(τ) = ĜIC (τ) = [1 0] Q B (τ,τ) [0 1] Q 1, which fr B (τ,τ) a fixed quantile τ, has an asympttic nrmal distributin as given in Crllary 2. 16

20 General hyptheses abut β (τ) can be accmmdated thrugh functins f W n ( ). We cnsider the Klmgrv-Smirnv and Cramér-vn Mises type test statistics, V n = f(w n ( )), where f( ) is a general functinal f the prcess W n ( ). In particular, we cnsider different functinals that lead t different test statistics, such as V 1n := n sup W n (τ), V 2n := n τ T τ T W n (τ) dτ. There are many alternative pssible statistics as: V 3n := n sup τ T W n (τ) 2 and V 4n := n τ T W n(τ) 2, dτ, amng thers. In this paper we cncentrate n V 1n and V 2n. These statistics and their assciated limiting thery prvide a natural fundatin fr testing the null hypthesis. Nw we present the limiting distributins f the test statistics under the null hypthesis. Frm Crllary 1 and Lemma 1 under the null hypthesis (H 0 : β = h(q) = r), it fllws that n(h( Q) h(q)) h(g). summarizes the limiting distributins. Thus, the fllwing lemma Lemma 2. Assume the cnditins f Therem 2, and QG.IV. Under H 0 : β(τ) = h(q(τ)) = r(τ), τ T, as n, V 1n sup h(g(τ)), τ T V 2n τ T h(g(τ)) dτ. When perfrming tests fr the GIC, the limiting distributins f the test statistics under the null hypthesis fllws frm Therem 2. Under the null hypthesis (H 0 : GIC(τ) = r(τ)), it fllws n(ĝic(τ) r(τ)) GIC(G A, G B ). Thus, the fllwing crllary summarizes the limiting distributins. The result fr H 0 : GIC (τ) = r(τ) is analgus. Crllary 3. Assume the cnditins f Therem 2. Under H 0 : GIC(τ) = r(τ), as n, V 1n sup GIC(G A, G B ), τ T V 2n τ T GIC(G A, G B ) dτ. The secnd hypthesis f interest cncerns an unknwn r(τ), which needs t be estimated. In many examples f interest, the cmpnent r(τ) in the null hypthesis (11) is unknwn r defined as a functin f the cnditinal distributin and thus needs t be estimated (see, e.g., Kenker and Xia (2002) and Chernzhukv and Fernandez-Val (2005)). r(τ) might, 17

21 fr example, be GIC(τ) fr a different cuntry, r perid. Or it might be GIC (τ). The natural expedient f replacing the unknwn r in the test statistic by estimates intrduces sme fundamental difficulties. The estimate will be dented as r(τ). Let W n (t) := β(τ) r(τ), τ T. In this framewrk, we fllw Chernzhukv and Fernandez-Val (2005) and assume that the quantile estimates and nuisance parameter estimates satisfy the fllwing: n-cnsistent estimatrs fr β( ) and r( ), such that n( β( ) β( )) h(g( )) and n( r( ) r( )) G r ( ) jintly in l (T ), where (h(g( )), G r ( )) is a zer mean cntinuus Gaussian prcess with a nn-degenerate cvariance kernel. Thus, we have that n( β(τ) r(τ)) h(g(τ)) G r (τ). The prcess remains asympttically Gaussian; hwever, the estimatin f r(τ) intrduces a new drift cmpnent that additinally cmplicates the cvariance kernel f the prcess. Under the null hypthesis H 0 : β(τ) = r(τ), the test statistics becme: V 1n := n sup W n (τ), V2n := n τ T τ T W n (τ) dτ. Example (The unifrmly cnstant (but unknwn) effect hypthesis). A basic hypthesis is that the grwth incidence curve, GIC(τ), is statistically equal t the mean grwth rate fr all τ T., i.e., grwth has n distributinal hetergeneity. The alternative is that GIC(τ) differs frm the mean at least fr sme τ T. In this case, r(τ) = γ AGR, (where γ AGR is the mean grwth rate). Nw we display the limiting distributins f these test statistics under the null hypthesis. Lemma 3. Assume the cnditins f Therem 2 and that n( β( ) β( )) h(g( )) and n( r( ) r( )) Gr ( ) jintly in l (T ), where (h(g( )), G r ( )) is a zer mean cntinuus Gaussian prcess with a nn-degenerate cvariance kernel.. Under H 0 : β(τ) = h(q(τ)) = r(τ), τ T, as n, V 1n sup h(g(τ)) G r (τ), τ T V2n τ T h(g(τ)) G r (τ) dτ. This result can be applied t test fr the GIC. The limiting distributins f the test statistics under the null hypthesis fllw frm Lemma 3. Under the null hypthesis (H 0 : 18

22 GIC(τ) = r(τ)), it fllws n(ĝic(τ) r(τ)) GIC(G A, G B ), and n( r(τ) r(τ)) G r (τ). The fllwing crllary summarizes the limiting distributins. The result fr H 0 : GIC (τ) = r(τ) is analgus. Crllary 4. Assume the cnditins f Lemma 3, with β(τ) = GIC(τ), as n, V 1n sup GIC(G A, G B ) G r (τ), τ T V2n τ T GIC(G A, G B ) G r (τ) dτ. Finally, we cnsider testing hyptheses cncerning inequalities n bth null and alternative hyptheses as H 0 : β(τ) 0 vs H 1 : β(τ) < 0, τ T. (12) The fllwing is an example f hyptheses that may be cnsidered. Example (The first-rder stchastic dminance hypthesis). An imprtant practical hypthesis invlves the cmpsite null GIC(τ) r(τ), fr all τ T, versus the alternative f GIC(τ) < r(τ), fr sme τ T. When r(τ) = 0 and because GIC(τ) = q A1(τ) q B0 (τ) q B0 (τ), such that fr q B0 (τ) 0, testing whether GIC(τ) 0 is equivalent t test whether q A1 (τ) q B0 (τ), ie, that F Y (1) T =1 stchastically dminates F Y (0) T =0 in first-rder. Therefre, the abve example describes a test which is analgus t a first rder stchastic dminance as in Dnald and Hsu (2014). These null hyptheses f interest can be frmalized as H 0 : β(τ) 0, and the test statistic becmes: Ṽ 1n := n sup τ T W n (τ), where W n (τ) = β(τ). We emply the test statistic Ṽ1n since it has been knwn in the literature that when the null hypthesis invlves an inequality, the set f pints satisfying the null hypthesis is usually nt a singletn (see, e.g., Lintn, Maasumi, and Whang (2005)). The typical way t reslve this is t apply the least favrable cnfiguratin (LFC) t find a pint in the null hypthesis least favrable t the alternative hypthesis. Hence, t derive the asympttic prperties f the abve statistic, Ṽ1n, ne cmputes the estimatr β(τ) and plugs it in, and given the LFC the limiting distributin is analgus t that in Lemma 2 and Crllary 2. 19

23 T perfrm practical inference we suggest the use f resampling techniques t apprximate the limiting distributins and btain critical values. T btain the critical value fr the first tw criteria we use a btstrap prcedure, and fr the inequality test we make use f subsampling Practical implementatin f testing prcedures Implementatin f the prpsed tests in practice is simple. First, we discuss the test H 0 in (11). T implement the tests ne needs t cmpute the statistics f test V 1n r V 2n. Analgusly, when r(τ) is unknwn, ne cmputes V 1n r V 2n. We suggest the use f a recentered btstrap prcedure t calculate critical values. The steps fr implementing the tests in practice are as fllws. First, the estimates f β(τ) are cmputed by slving the prblems in equatins (6) (8) and calculating β(τ). Secnd, W n is calculated by centralizing β(τ) at r(τ), and V 1n r V 2n is cmputed by taking the maximum ver τ (V 1n ) r summing ver τ (V 2n ). Fr the general case with unknwn r(τ), the tests are cmputed in the same fashin. The nly adjustment is the use f r(τ) t cmpute W n. Third, after btaining the test statistic, it is necessary t cmpute the critical values. We prpse the fllwing scheme. We use the test statistic V 1n as an example, but the prcedure is the same fr the ther cases. Take B as a large integer. Fr each b = 1,..., B: (i) Obtain the resampled data {(Y b i, T b i, X b i ), i = 1,..., n}. (ii) Estimate β b (τ) and set W b n(τ) := ( β b (τ) β(τ)). (iii) Cmpute the test statistic f interest V b 1n = max τ T n W b n (τ). Let ĉ B 1 α dente the empirical (1 α)-quantile f the simulated sample { V 1 1n,..., V B 1n}, where α (0, 1) is the nminal size. We reject the null hypthesis if V 1n is larger than ĉ B 1 α. In practice, the maximum in step (iii) is taken ver a discretized subset f T. A frmal justificatin the simulatin methd is stated as fllws. Cnsider the fllwing cnditins. QG.IB Fr any δ n 0, sup w Π δ n 1 n n i=1 w t(t, X) E[w t (T, X)] = p (1/ n). QG.IIB n 1 n n i=1 [(τ 1{Y i < q t (τ)})(ŵ t (T i, X i ) ŵ t (T i, X i ))] cnverges weakly t a tight randm element G in l (T ) in P -prbability. 20

24 Therem 3. Under QC.I QC.II, QG.IB QG.IIB and QG.III with in prbability replaced by almst surely, then, fr t = 0, 1, the btstrap estimatr n( q t (τ) q t (τ)) G(τ) in P -prbability in l (T ). Therem 3 establishes the cnsistency f the btstrap prcedure. It is imprtant t highlight the cnnectin between this result and the previus sectin. In fact, Therem 3 shws that the limiting distributin f the btstrap estimatr is the same as that f Therem 2, and hence the abve resample scheme is able t mimic the asympttic distributin f interest. Nw we mve ur attentin t testing the H 0 displayed in (12). As discussed in Lintn, Maasumi, and Whang (2005), even when the data are i.i.d. the standard btstrap might nt wrk well when testing the inequality under the null hypthesis. This is because ne needs t impse the null, which is difficult because it is defined by a cmplicated system f inequalities. Thus, we fllw Lintn, Maasumi, and Whang (2005) and suggest the use f a subsampling methd, which is very simple t define and yet prvides cnsistent critical values. We first define the subsampling prcedure. Let Z i = {(Y i, T i, X i ) : i = 1,..., n} and cnstruct all pssible subsets f size b. The number f such subsets B n is n chse b. Let S n dente ur test statistic Ṽ1n cmputed ver the entire sample. With sme abuse f ntatin, the test statistic S n can be re-written as a functin f the data {Z i : i = 1,..., n}: S n = ns n (Z 1,..., Z n ), where s n (Z 1,..., Z n ) is given by sup τ T [ W n (τ)], where W n (τ) = β(τ). Let J n (w) = P ( ns n (Z 1,..., Z n ) z ) dente the distributin functin f S n. Let s n,b,i be equal t the statistic s b evaluated at the subsamples {Z i,..., Z i+b 1 } f size b, i.e. s n,b,i = s b (Z i, Z i+1,..., Z i+b 1 ) fr i = 1,..., n b + 1. This means that we have t recmpute q t (Z i, Z i+1,..., Z i+b 1 ) using each subsample as well. We nte that each subsample f size b (taken withut replacement frm the riginal data) 21

25 is indeed a sample f size b frm the true sampling distributin f the riginal data. Hence, it is clear that ne can apprximate the sampling distributin f S n using the distributin f the values f s n,b,i cmputed ver n b + 1 different subsamples f size b. That is, we apprximate the sampling distributin J n f S n by Ĵ n,b (w) = n b+1 1 ( ) 1 bsn,b,i w. n b + 1 i=1 Let g n,b (1 α) dente the (100 α)-th sample quantile f Ĵn,b( ). We call it the subsample critical value f significance level α. Thus, we reject the null hypthesis at the significance level α if S n > g n,b (1 α). The cmputatin f this critical value is nt particularly nerus, althugh it depends n hw big b is. The validity f the subsampling methds fr the quantile regressin prcess was established by Chernzhukv and Fernandez-Val (2005). A Supplemental Appendix cllects Mnte Carl simulatins cnducted t evaluate the finite sample prperties f the prpsed tests. We cnduct simulatins t evaluate the perfrmance f these tests in terms f size and pwer. The results prvide evidence that the empirical levels f the tests apprximate well the nminal r theretical levels. Mrever, the tests pssess large pwer against selected alternatives. The results are imprved when the sample size increases, nevertheless they are nt very sensitive t the numbers f btstraps. 4 Wage distributin dynamics in the United States and Brazil, This sectin illustrates the usefulness f the prpsed methds with an empirical example. We cmpute the GIC and GIC fr the tw mst ppulus natins in the Western Hemisphere, namely the United States and Brazil, fr the perid, and cmpare results. In particular, we emphasize the rle f the fllwing decmpsitin f the GIC, intrduced in Sectin 2 and reprduced belw: GIC (τ) = GIC (τ) + GIC (τ) qb1 (τ) q B0 (τ). The first term in this decmpsitin is the cunterfactual GIC, which keeps the jint 22

26 distributin f bserved cvariates fixed (see equatin 2). Under Assumptins I.I I.III, this term can be interpreted as describing the grwth prcess that wuld have btained in the absence f any changes in that jint distributin. The secnd term f the decmpsitin is crrespndingly interpreted as the effect f changes in the jint distributin f cvariates. Our reweighting methd allws fr the direct cnstructin f the cunterfactual GIC, with n need t pstulate a structural relatinship between wages, cvariates and unbserved terms, as was required by the earlier literature that fllwed Juhn, Murphy, and Pierce (1993). Under that apprach, ecnmists wuld typically estimate OLS regressins fr the tw time perids separately and then cnstruct a cunterfactual wage distributin using estimated parameters and residuals frm time t = 1 but cvariates frm time t = 0. This wuld yield a cunterfactual distributin f wages at time t = 1, with a distributin f cvariates that was fixed at time t = 0 (see, fr example, Burguignn, Ferreira, and Leite (2008)). In additin t requiring strng functinal frm assumptins, hwever, it is nt clear hw ne wuld perfrm statistical inference n the cunterfactual GIC using that methd. In this sectin we reprt the estimates fr GIC and its cunterfactual cunterpart GIC, ĜIC(τ) and ĜIC (τ) respectively, ver τ T. We als reprt the crrespnding grwth rates in average wages, γ AGR and γ AGR, respectively, fr cmparisn. Mrever, using the techniques develped in the previus sectin, we perfrm inference n bth sets f curves. Specifically, we apply the unifrm tests, Klmgrv-Smirnv (KS) and Cramér-vn Mises (CVM), t test the fllwing hyptheses: (i) Cnstant distributin: (H 0 : GIC(τ) = 0 versus H A : GIC(τ) 0); (ii) Distributin-neutral grwth (H 0 : GIC(τ) = γ AGR versus H A : GIC(τ) γ AGR, where γ AGR is the grwth rate in the average wage); (iii) Cnstant distributin, cnditinal n cvariates, (H 0 : GIC (τ) = 0 versus H A : GIC (τ) 0); (iv) Distributin-neutral grwth, cnditinal n cvariates (H 0 : GIC (τ) = γ AGR versus H A : GIC (τ) γ AGR, where γ AGR is the cunterfactual grwth rate in average wage). 23