Measurement Errors in Quantile Regression Models

Transcription

1 Measurement Errrs in Quantile Regressin Mdels Sergi Firp Antni F. Galva Suyng Sng June 30, 2015 Abstract This paper develps estimatin and inference fr quantile regressin mdels with measurement errrs. We prpse an easily-implementable semiparametric tw-step estimatr when we have repeated measures fr the cvariates. Building n recent thery n Z-estimatin with infinite-dimensinal parameters, cnsistency and asympttic nrmality f the prpsed estimatr are established. We als develp statistical inference prcedures and shw the validity f a btstrap apprach t implement the methds in practice. Mnte Carl simulatins assess the finite sample perfrmance f the prpsed methds. We apply ur methds t the well-knwn example f returns t educatin n earnings using a data set n female mnzygtic twins in the U.K. We dcument strng hetergeneity in returns t educatin alng the cnditinal distributin f earnings. In additin, the returns are relatively larger at the lwer part f the distributin, prviding evidence that a ptential ecnmic redistributive plicy shuld fcus n such quantiles. Key Wrds: Quantile regressin; measurement errrs, returns t educatin JEL Classificatin: C14, C23, J31 The authrs wuld like t express their appreciatin t Rger Kenker, Yuya Sasaki, Susanne Schennach, and Liang Wang fr helpful cmments and discussins. All the remaining errrs are urs. Sa Paul Schl f Ecnmics, FGV sergi.firp@fgv.br Department f Ecnmics, University f Iwa, W284 Pappajhn Business Building, 21 E. Market Street, Iwa City, IA antni-galva@uiwa.edu Department f Ecnmics, University f Iwa, W360 Pappajhn Business Building, 21 E. Market Street, Iwa City, IA suyng-sng@uiwa.edu

2 1 Intrductin Quantile regressin QR) mdels have prvided a valuable tl in ecnmics as a way f capturing hetergeneus effects that cvariates may have n the utcme f interest, expsing a wide variety f frms f cnditinal hetergeneity under weak distributinal assumptins. Under sme assumptins n the unbservable factrs, QR can als be interpreted as prviding a structural relatinship between the utcme f interest and its bservable and unbservable determinants. Als imprtantly, QR prvides a framewrk fr rbust inference when the presence f utliers is an issue. Measurement errrs ME) have imprtant implicatins fr the reliability f general standard estimatin and testing. Variables used in empirical ecnmic analysis are frequently measured with errr, particularly if infrmatin is cllected thrugh ne-time retrspective surveys, which are ntriusly susceptible t recall errrs. If the regressrs are indeed subject t classical ME, it is well knwn that the slpe cefficient f the rdinary least squares OLS) estimatr is incnsistent. In the ne regressr case r multiple uncrrelated regressrs), under standard assumptins, the OLS is biased tward zer, a prblem ften dented as attenuatin see, e.g., Carrll et. al. 2006) and references therein fr an verview f ME mdels). Recently, the tpic f ME in variables has received cnsiderable attentin in the QR literature. As in the OLS case, the standard QR estimatr has been shwn t be incnsistent in the presence f ME see, e.g., Mntes-Rjas 2011)). He and Liang 2000) cnsider the prblem f estimating QR cefficients in errrs-in-variables mdels, and prpse an estimatr in the cntext f linear and partially linear mdels. Chesher 2001) studies the impact f cvariate ME n quantile functins using a small variance apprximatin argument. Schennach 2008) discusses identificatin f a nnparametric quantile functin under varius settings when there is an instrumental variable measured n all sampling units. Identificatin and estimatin fr general quantile functins are based n Furier transfrms and previus results fr nnlinear mdels see, e.g., Schennach, 2007). Wei and Carrll 2009) prpse a methd fr a linear QR mdel that crrects bias induced by the ME by cnstructing jint estimating equatins that simultaneusly hld fr all the quantile levels. Mre recently, Trres-Saavedra 2013) and Hausman, Lu, and Palmer 2014) study ME in the dependent variable f QR mdels. We refer t Ma and Yin 2011), Wang, Stefanski, and Zhu 2012), and Wu, Ma, and Yin 2014) fr ther recent develpments in QR mdels with ME. Thus, in 1

3 the analysis f QR with mismeasured cvariates, it has been cmmn t emply estimatin methds that either impse parametric restrictins n nuisance functinals r use exgenus infrmatin as thse prvided by instrumental variables see, e.g., Wei and Carrll 2009), Schennach 2008), and Chernzhukv and Hansen 2006)). Nevertheless, methds relying n parametric assumptins are very sensitive t misspecificatin f such cnditins, which are indeed relevant fr inference as the asympttic variance typically requires estimatin f cnditinal densities. In additin, finding exgenus instrumental variables is knwn t be a nntrivial task in mst ecnmic mdels. This paper cntributes t bth the QR and ME branches f the literature by develping estimatin and inference methds fr QR mdels in the presence f ME in the cvariates. This is achieved by explring repeated measures f the true regressr. Identificatin and estimatin f cnditinal mean regressin mdels with repeated measures f the true regressr have already been studied in Li 2002), Schennach 2004) and Hu and Sasaki 2015), amng thers. Hwever, t the best f ur knwledge, there has yet been n attempt t develp estimatin and inference fr QR mdels using repeated measures f the true regressr. This paper bridges this gap. We prpse a simple, easily-implementable, and wellbehaved tw-step semiparametric estimatin prcedure that preserves the semiparametric distributin-free and heterscedastic features f the mdel. The first step emplys a general nnparametric estimatin f the density functin. The secnd step uses the estimated densities as weights in a weighted QR estimatin. We establish the asympttic prperties f the tw-step estimatr, assuming that the cnditinal densities satisfy smthness cnditins and can be estimated at an apprpriate nnparametric rate. We als develp practical statistical inference, and prpse testing prcedures fr general linear hyptheses based n the Wald statistic. T implement these tests in practice the critical values are cmputed using a btstrap methd. We prvide sufficient cnditins under which the btstrap is theretically valid, and discuss an algrithm fr its practical implementatin. Our methd leads t a simple algrithm that can be cnveniently implemented in empirical applicatins. Cmpared t the existing prcedures fr QR mdels with ME, ur apprach has several distinctive advantages. First, ur methd des nt assume glbal linearity at all quantile levels fr the estimatin f the cnditinal density functin as in Wei and Carrll 2009). Such feature makes ur prcedure applicable t any τ-quantile f interest, thus relaxing the requirement f a jint estimatin and prviding mre flexibility. Secnd, ur algrithm is cmputatinally simple and easy t implement in practice because estimatin f the weights 2

4 des nt require recursive algrithms allwing the weights fr all bservatins t be btained frm ne single step. As a result, the quantile estimate is attained by minimizing nly ne single cnvex bjective functin at the quantile f interest. Third, the methdlgy des nt rely n instrumental variables. Therefre, infrmatin frm utside the mdel is nt necessary fr identificatin. Finally, ur estimated weights exhibit a prperty f unifrm cnsistency, implying that it is feasible t establish bth the cnsistency and asympttic nrmality f the resulting estimatrs f the parameters f interest. Hence, the methd prvides standard inference and testing prcedures. Mnte Carl simulatins assess the finite sample prperties f the prpsed methds. We evaluate the estimatr in terms f empirical bias, standard deviatin, and mean squared errr, and cmpare its perfrmance with methds that are nt designed fr dealing with ME issues. The experiments suggest that the prpsed apprach perfrms relatively well in finite samples and effectively remves bias induced by ME. Our prcedure will hpefully be useful fr thse empirical settings based n QR mdels in which ME in the independent variables is a cncern because the methd des prvide intuitive and practical ways f handling the prblem. T mtivate and illustrate the applicability f the methds, we revisit and analyze the imprtant example f returns t educatin n earnings. The QR apprach is an imprtant tl in this example because it allws us t capture the hetergeneity in the returns t educatin alng the cnditinal wage distributin. At the same time, endgeneity induced by ME has been extensively discussed in the returns t educatin example, as misreprting in the number f schling years is a genuine cncern Card 1995), Card 1999), and Harmn and Osterbeek 2000)). Within that framewrk, finding valid and strng instrumental variables t slve the endgeneity prblem is nt, in general, an easy task see, e.g., Card 1999)). Thus, ur methd is a natural alternative slutin t the ME prblem when repeated measures n educatinal achievement are available. In ur empirical example we use a data set n female mnzygtic twins in the U.K. that had been previusly used in Bnjur et al. 2003) t study the prblem f returns t educatin. Bnjur et al. 2003) use the infrmatin n ne twin t btain an instrumental variable fr schling years n the ther twin. Amin 2011) pints ut that the results in Bnjur et al. 2003) are largely affected by utlier bservatins and revisited the prblem using QR. He uses the same data n twins and apply the instrumental variables methd- 3

5 lgy described in Arias, Hallck, and Ssa-Escuder 2001). We cmpare ur results with thse frm Bnjur et al. 2003) and Amin 2011). Our empirical findings exemplify and supprt the idea that the prpsed methds are a useful alternative t existing appraches in ecnmic applicatins in which ME is an imprtant cncern. We dcument strng hetergeneity in returns t educatin alng the cnditinal distributin f earnings. In additin, the returns are relatively larger at the lwer part f the distributin, prviding evidence that a ptential ecnmic redistributive plicy shuld fcus n such quantiles. The rest f the paper is rganized as fllws. Sectin 2 presents the mdel and discusses identificatin f the parameters f interest in presence f ME. Sectin 3 prpses the tw-step QR estimatr. Sectin 4 establishes the asympttic prperties f the estimatr. Inference is discussed in Sectin 5. Sectin 6 presents the Mnte Carl experiments. In Sectin 7, we illustrate empirical usefulness f the the new apprach by applying t returns t educatin. Finally, Sectin 8 cncludes the paper. 2 Mdel and identificatin 2.1 Mdel We first intrduce the mdel studied in this paper. Given a quantile τ 0, 1), we define the fllwing quantile regressin QR) mdel, Y i = Xi β 0 τ) + Zi δ 0 τ) + ε i τ), 1) where Y i is the scalar dependent variable f interest, X i is a vectr f ptentially-mismeasured cvariates, Z i is a vectr f crrectly-bserved cvariates, and ε i τ) is the innvatin term whse τ-th quantile is zer cnditinal n X i, Z i ). The structural parameters f interest are θ 0 τ) = β 0 τ), δ 0 τ)). In general, each β 0 τ) and δ 0 τ) will depend n τ, but we assume τ t be fixed thrughut the paper and suppress such a dependence fr ntatinal simplicity. Suppse Y i, X i, Z i ) are i.i.d. randm variables defined n a cmplete prbability space Ω, F, P ). Define the ppulatin bjective functin fr the τ-th cnditinal quantile as Qβ 0, δ 0 ) := E [ ψ τ Y i Xi β 0 Zi δ 0 )[X i Z i ] ] = 0, 2) where ψ τ u) := τ I{u < 0}) with the indicatr functin I{ }. When the true cvariates X, Z) are bserved, β 0 and δ 0 can be cnsistently estimated frm the standard quantile 4

6 regressin mdel with sample analg f Qβ, δ) in 2) as Q n β, δ) := 1 n n ψ τ Y i Xi β Zi δ)[x i Z i ] = 0. 3) i=1 The presence f the indicatr functin in the abve equatin implies that the slutin may nt be an exact zer. It is usual t write this estimatr as a minimizatin prblem, and then use linear prgramming t slve the ptimizatin. Thus, the abve mment cnditin is a slight abuse f ntatin, but since everything else invlving bserved data is an estimating equatin that will have a zer, we will use the estimating equatin nmenclature. Fr mre details n Z-estimatr with nn-smth bjective functins, see He and Sha 1996, 2000). 2.2 Measurement errr bias and its slutin Under the assumptin f perfectly-measured regressrs, the slutin f equatins 3) can be shwn t prduce cnsistent estimates f β 0, δ 0 ). Nevertheless, it is cmmnly bserved that researchers have t use the regressr X measured with errr. Using mismeasured X in the standard QR estimatin in 3) induces a substantial bias in the estimates f the cefficients f interest see, e.g., He and Liang, 2000). Thus, estimatin f the standard QR mdel under measurement errrs ME) leads t incnsistent estimates. T vercme this drawback we prpse a methdlgy that makes use f repeated measures. Bth variables are mismeasured bservables f the true cvariate. Suppse that true cvariate X is unbservable due t ME. Instead, a researcher bserves tw errr-laden measurements which are nisy measures f X and defined as fllws X 1i = X i + U 1i X 2i = X i + U 2i, where U 1i and U 2i are ME. Therefre, the bserved randm variables are Y i, X 1i, X 2i, Z i ), and ne seeks t estimate the parameters β 0, δ 0 ). We shw hw t use infrmatin frm the measures X 1 and X 2 t btain cnsistent estimates f the parameters f interest. Fr that purpse, it is useful t rewrite Qβ, δ) as a 5

7 functin f the density functin as well as β, δ): Qβ 0, δ 0, f 0 ) := E[ψ τ Y X β 0 Z δ 0 )[X Z]] = ψ τ y x β 0 z δ 0 )[x z] f Y XZ y, x, z)dydxdz = ψ τ y x β 0 z δ 0 )[x z] f X Y Z x y, z)f Y Z y, z)dydxdz [ ] = E ψ τ Y x β 0 Z δ 0 )[x z] f X Y Z x Y, Z)dx x = 0, 4) where f Y XZ y, x, z) and f Y Z y, z) are the jint density f Y, X, Z) and Y, Z), respectively, and where f X Y Z x y, z) f 0 is the cnditinal density f X given Y, Z). By replacing the uter expectatin with its empirical cunterpart, we write the sample analg f the ppulatin bjective functin 4) as: Q n β, δ, f) := 1 n ψ τ Y i x β Zi δ)[x Z i ] f X Y Z x Y i, Z i )dx 5) n = 0. i=1 x The integratin in 5) makes the functin cntinuus in its argument. The summand f 5) is E x [ψ τ Y i x β Z i δ)[x Z i ] Y i, Z i ], the cnditinal mean f the riginal scre functin given the bserved Y and Z. Mrever, 5) is an unbiased estimating functin, that is, has mean zer, and will be the basis fr cnstructing estimating equatins t btain cnsistent estimates f the parameters f interest. Therefre, ne wuld slve the new estimating equatin 5) t estimate the parameters f interest. In empirical applicatins, hwever, the true cnditinal density f X Y Z x y, z) is unknwn and t implement the estimatr 5) in practice ne needs t replace it with f X Y Z x y, z), a cnsistent estimate f f X Y Z x y, z). Thus, a feasible) estimatr wuld first estimate f X Y Z x y, z). The fitted density functin frm this step wuld be used t estimate the cefficients f interest in a secnd step. Finally, with a cnsistent estimate f the cnditinal density, β 0, δ 0 ) can be cnsistently estimated. Hwever, in general, the cnditinal density is nt stchastically identified due t the unbservability f the true X. In a related mdel, Wei and Carrll 2009) make use f an iterative algrithm t btain a cnsistent estimatr f the cnditinal density f X Y X1 x y, x 1 ) in the presence f ME n X. 1 1 We nte that their cnditinal density is slightly different frm urs since there is mismeasured cvariate X 1 in their cnditining set. 6

8 They fcus n mdel with ne measurement f true X here X 1 ) and with n ther bserved cvariates Z fr simplicity. Althugh their apprach can be useful in sme applicatins, it has imprtant technical challenges. First, in rder t implement the estimatr, ne needs t estimate the cnditinal density f X Y X1 x y, x 1 ) which requires pre-specified parametric frm f f X X1 x x 1 ). This suffers frm ptentially serius mdel misspecificatin. Secnd, and related t the first prblem, there is a prblem t slve the estimating equatins, since estimating the cnditinal density f X Y X1 x y, x 1 ) invlves estimatin f the entire prcess β 0 τ) ver quantiles τ. In ther wrds, the estimating equatins in Wei and Carrll 2009) need t be slved jintly fr all the τ s, which increases the dimensinality f the prblem substantially and makes implementatin cnsiderably difficult. This is reflected in the tractability f inference fr their methd. In this paper, we prpse a nvel way t nnparametrically estimate the cnditinal density withut impsing assumptins n knwn distributins f the ME. Specifically, we make use f the repeated measures, X 1 and X 2, and shw that tw mismeasured cvariates are sufficient t identify the cnditinal density in the presence f ME n the cvariate. In turn, the result guarantees cnsistent estimatin f parameters f interest. The apprach with repeated measurements has been recently studied in the ME literature. Mst f studies have fcused n i.i.d. measurement errrs e.g., Li and Vung, 1998; Delaigle, Hall, and Meister, 2008). We extend the literature by relaxing such strng cnditins. We als extend issues in smth bjective functin f mean regressin with ME e.g., Schennach, 2004) t a nn-smth bjective functin such as the QR. In the next sectin we prpse a prcedure that yields a cnsistent estimatr f β 0, δ 0 ) in 5). We develp a methd fr QR with measurement errrs, which relies n estimating the cnditinal density functin nnparametrically. The methd is a tw-step estimatr, where in the first step we estimate the density nnparametrically and then in the secnd step we emply a standard weighted QR prcedure. Befre we prceed t estimatin, we shw an identificatin result fr the density functin which is essential in the estimatin. Fr expsitinal ease, we use f X Y Z x y, z) and fx y, z) synnymusly. 2.3 Cnditinal density As described abve, fx y, z) is an imprtant element fr the identificatin f the parameters f interest in the QR with ME. This sectin describes the identificatin f the 7

9 cnditinal density functin fx y, z) which is required t cmpute the tw-step estimatr. The identificatin is based n the assumptin that repeated measures f the true regressr are bserved. We state the fllwing assumptins t btain the main identificatin result. Assumptin A.I: i) E[U 1 X, U 2 ] = 0; ii) U 2 Y, X, Z). Assumptin A.II: i) E[ X ] < ; ii) E[ U 1 ] < ; iii) E[expiζX 2 )] > 0 fr any finite ζ R. Assumptin A.III: i) sup x,y,x1,z) suppx,y,z) fx y, z) < ; ii) fx y, z) is integrable n R fr each y, z) suppy, Z). Assumptin A.I impses restrictins n the repeated measures f X. Assumptin A.I i) requires cnditinal mean zer f ME n X 1, but allws dependence f the ME and X, U 2 ). Assumptin A.I ii) requires that ME n X 2 is independent f true X as well as ther variables. Hwever, it des nt necessarily require zer mean f U 2. Thus, ur setting n the repeated measures can be useful fr an example such that there is a drift r trend in the mismeasured cvariates. Assumptin A.II impses mild restrictins n the existence f the first mments f X and U 1, and nnvanishing characteristic functin f X 2. These have been cmmnly assumed in the decnvlutin literature see, e.g., Fan, 1991b; Fan and Trung, 1993). Assumptin A.III is trivially satisfied in cmmnly-used cnditinal densities. Let φζ, y, z) E[e iζx Y = y, Z = z] be cnditinal characteristic functin f X given Y and Z. The fllwing therem presents the identificatin f fx y, z). Therem 1 Suppse Assumptins A.I A.III hld. Then, fr x, y, z) suppx, Y, Z), fx y, z) = 1 φζ, y, z) exp iζx)dζ, 6) where fr each real ζ, φζ, y, z) = E[eiζX 2 Y = y, Z = z] E[e iζx 2 ] ζ exp 0 ie[x 1 e iξx 2 ) ] dξ. E[e iξx 2 ] Prf. See Appendix. The therem implies that cnditinal density fx y, z) can be written as a functin f purely-bserved variables. Fr this, we use useful prperties f Furier transfrm. Namely, we 8

10 write fx y, z) as the inverse Furier transfrm f φζ, y, z). This simplifies identificatin since φζ, y, z) is easily identified frm Assumptins A.I A.III by remving the ME, U 1 and U 2, in the frequency dmains ζ, ξ). It is wrth nting that the identificatin result is similar t Ktlarski 1967) wh identifies density f X frm its repeated measurements by assuming mutual independence f X, U 1, and U 2. Our apprach rests n weaker assumptins than their mutual independence, which is highlighted in cnditin A.I. As a result, the prpsed methd can be applied t many interesting tpics which allw fr dependence amng variables and their ME. 2.4 Identificatin Given the result in equatin 6), we can rewrite Qβ, δ, f) as: [ Qβ, δ, f) = E ψ τ Y x β Z δ ) ] [x Z] f X Y Z x Y, Z)dx x [ = E ψ τ Y x β Z δ ) ) ] 1 [x Z] φζ, Y, Z) exp iζx)dζ dx, x which des nt depend n data n X. Thus, estimatin f β 0, δ 0 ) fllws frm slving a feasible versin f Q n β, δ, f): where Q n β, δ, f) = 1 n n i=1 x ζ ψ τ Yi x β Z i δ ) [x Z i ] f X Y Z x Y i, Z i )dx, f X Y Z x Y i, Z i ) = 1 φζ, Y i, Z i ) exp iζx)dζ, ζ and the nly feature f this sample bjective functin that had nt yet been presented is φ, the estimate f φ, which is defined in the next sectin. In practice, as we discuss next, we apprximate integrals by sums, thus actual implementatin slves a slightly different bjective functin. By apprximating the integral by a sum, we end up with a duble sum n bservatins and n grid values f X). Imprtantly n that representatin is the fact that the estimates β, δ) will be btained by a weighted QR, whse weights will be given by the estimate f X Y Z. 9

11 3 Estimatin Given the identificatin cnditin in equatin 6) f Therem 1, we are able t estimate the structural parameters f interest, β 0, δ 0 ). We prpse a semiparametric estimatr that invlves tw-step estimatin. Implementatin f the estimatr is simple in practice. In the first step, ne estimates the nuisance parameter, the cnditinal distributin, using a nnparametric methd which requires n ptimizatin. In the secnd step, by plugging-in these estimates, a general weighted quantile regressin QR) is perfrmed. 3.1 Estimatin f nuisance parameter In this subsectin we discuss the estimatin f the nuisance parameter in the first step, i.e., the cnditinal density fx y, z). It is imprtant t nte that the prpsed density estimatin is nvel in the literature and makes use f repeated measures and nice prperties f Furier transfrm. The estimatin f the nuisance parameter is very imprtant step fr implementatin f the prpsed estimatr in practice. We prpse a nnparametric methd t estimate the density cnsistently. T btain a cnsistent estimatr f fx y, z), we adapt the class f flat-tp kernels f infinite rder prpsed by Plitis and Rman 1999). Cnsider the fllwing assumptin. Assumptin A.IV: The real-valued kernel x kx) is measurable and symmetric with kx)dx = 1, and its Furier transfrm ξ κξ) is bunded, cmpactly supprted, and equal t ne fr ξ < ξ fr sme ξ > 0. Frm Assumptin A.IV, we allw fr a kernel f the frm see, e.g., Li and Vung, 1998) with its Furier transfrm such that κh x ζ) = kx) = sinx) πx, 7) 1 h x k x h x ) expiζx)dx, 8) fr a bandwidth h x. This flat-tp kernels f infinite rder has the prperty that its Furier transfrm is equal t ne ver [ 1, 1] interval and zer elsewhere, which guarantees that the bias ges t zer faster than any pwer f the bandwidth. We nte that the ill-psed 10

12 inverse prblem ccurs when ne tries t invert a cnvlutin peratin. This is true t ur prpsed estimatr because it is divided by a quantity which cnverges t zer as frequency parameter ges t infinity by Riemann-Lebesgue lemma. By estimating the numeratr using the kernel whse Furier transfrm is cmpactly supprted, ne can guarantees that the rati is under cntrl. This is because that the numeratr can decay t zer befre the denminatr cnverges t zer. This cmpact supprt f the Furier transfrm f the kernel can be easily implemented by preserving mst f the prperties f the riginal kernel. Fr instance, ne can transfrm any given kernel k int a mdified kernel k with cmpact Furier supprt by using a windw functin that is cnstant in the neighbrhd f the rigin and vanishes beynd a given frequency. The fllwing therem summarizes the result. Therem 2 Suppse Assumptins A.I A.III hld, and let k satisfy Assumptin A.IV. Fr x, y, z) suppx, Y, Z) and h x > 0, let ) 1 x x fx y, z; h x ) h k f x y, z)d x. 9) x h x Then we have fx y, z; h x ) = 1 κh x ζ)φζ, y, z) exp iζx)dζ. 10) Prf. See Appendix. Let h n h x n, h 2) n ) with h 2) n h y n, h z n) be a set f smthing parameters. Let Ê[ ] dente a sample average, i.e., 1 n n i=1 [ ]. Finally, we intrduce a cnsistent nnparametric estimatr f fx y, z) mtivated by Therem 2. Definitin 2.3 The estimatr f fx y, z) is defined as fx y, z; h n ) 1 κh x nζ) φζ, y, z, h 2) n ) exp iζx)dζ, 11) fr h n 0 as n, where φζ, y, z, h 2) n ) Ê[eiζX 2 Y = y, Z = z] ζ exp Ê[e iζx 2 ] 0 ) iê[x 1e iξx 2 ] dξ. Ê[e iξx 2 ] 11

13 The abve estimatr is useful t cmpute the structural parameters f interest. Since it has an explicit clsed frm, it requires n ptimizatin rutine unlike ther likelihd-based appraches. Estimatin f cnditinal mean, Ê[eiζX 2 any nnparametric methd. k hn ) h 1 n k /h n ) e.g., Epanechnikv kernel) defined as Y = y, Z = z], can be achieved via Fr instance, ne might use ppular kernel estimatin with Ê[e iζx 2 Y = y, Z = z] Ê[eiζX 2 k h y n Y y)k h z n Z z)]. Ê[k h y n Y y)k h z n Z z)]] 3.2 Estimatin f the structural parameters This sectin describes the general estimatr fr QR mdels with ME. The estimatr can be btained in tw steps. Given the identificatin cnditin in equatin 5) and the estimatr f the density functin described in the previus sectin, we are able t estimate the structural parameters f interest. We prpse a Z-estimatr that invlves tw-step estimatin. We estimate the parameters f interest, θ 0 = β 0, δ 0 ) fr a selected τ f interest, frm the fllwing tw steps: Step 1. Estimate fx j Y i, Z i ; h) fr each i-th bservatin and j-th grid as in equatin 5) where j J {1, 2,..., m} with m number f grids fr apprximating the numerical integral. The chice f kernels and bandwidths are prvided in Definitin 2.3 abve. The integrals in equatin 11) are perfrmed using the fast Furier transfrms FFT) algrithm. Well-behaving perfrmance f the algrithm is guaranteed by the smthness f the characteristic functin φ ) and the finiteness f the mments. Step 2. Then, t cmpute equatin 5) in practice, we have t make a numerical apprximatin t the integral ver x. We d this via translating the prblem int a weighted quantile regressin prblem. Let x = x 1, x 2,..., x m ) is a fine grid f pssible x j values, akin t a set f abscissas in Gaussian quadrature. Fr each τ, θτ) = βτ), δτ)) can be cmputed by slving n m ψ τ Y i x j β Zi δ)[ x j Z i ] f x j Y i, Z i ; h) = 0, 12) i=1 j=1 where f x j Y i, Z i ; h) is btained frm Step 1. The weighted quantile regressin f Y i n x j and Z i with crrespnding weights f x j Y i, Z i ; h) can be readily cmputed using the functin called rq in R package quantreg. 12

14 The asympttic prperties f the estimatr given in equatin 11) and als f βτ), δτ)), in equatin 12), are established in Sectin 4 belw. 4 Asympttic prperties This sectin investigates the large sample prperties f the prpsed tw-step estimatr. While these methds seem similar t the nes discussed by Wei and Carrll 2009), the nvel estimatin f the cnditinal density functin raises sme new issues fr the asympttic analysis f the estimatr. First, we establish the asympttic results fr the estimatr f the cnditinal density functin given in 11). Secnd, we establish cnsistency and asympttic nrmality f the tw-step estimatr in 12). 4.1 Asympttic prperties f the density estimatr In this subsectin we establish the asympttic prperties f the density functin estimatr in equatin 11). Let µζ) E[e iζx ], ω 1 ζ) E [ e iζx 2], and χζ, y, z) e iζx 2 f X2 Y Zx 2, y, z)dx 2. We impse the fllwing assumptins. Assumptin B.I: i) There exist cnstants C 1 > 0 and γ µ 0 such that D ζ ln µζ) = D ζ µζ) µζ) C 11 + ζ ) γµ ; ii) There exist cnstants C φ > 0, α φ 0, ν φ 0, and γ φ R such that ν φ γ φ 0 and and if α φ = 0, then γ φ < 1; sup φζ, y, z) C φ 1 + ζ ) γ φ expα φ ζ ν φ ), y,z) suppy,z) iii) There exist cnstants C ω > 0,α ω 0, ν ω ν φ 0, and γ θ R such that ν ω γ ω 0 and min{ inf χζ, y, z), ω 1ζ) } C ω 1 + ζ ) γω expα ω ζ νω ). y,z) suppy,z) Assumptin B.II: i) E[ X 1 2 ] < ; ii) E[ X 1 X 2 ] < ; i) E[ X 2 ] <. Assumptin B.III: sup y,z) suppy,z)) fy, z) fy, z) = O p ln n) 1/2 nh y h z ) 1/2 + s=y,z hs ) 2 ). These assumptins are standard fr nnparametric decnvlutin estimatrs because their rates f cnvergence will depend n the tails f the Furier transfrms see, e.g., Fan, 13

15 1991b; Fan and Trung, 1993). The literature cmmnly adpts tw types f smthness assumptins: rdinary and super smthness. Ordinary smthness admits a Furier transfrm whse tail decays t zer at a gemetric rate ζ γ, γ < 0 whereas super smthness admits a Furier transfrm whse tail decays t zer at an expnential rate exp α ζ γ ), α < 0, γ > 0. 2 Assumptin B.I simultaneusly impses rdinary and super smthness cnditins. 3 Assumptin B.II impses mild mment restrictins required fr cnsistency results. Assumptin B.III impses a standard cnditin n nnparametric estimatr f the jint density f fy, z). The next result establishes the asympttic prperties f the density functin estimatr. Therem 3 Let Assumptins A.I IV and B.I III hld. Then fr x, y, z) suppx, Y, Z) and h > 0 satisfying max{h y n) 1, h z n) 1 } = O n η ) and fr sme η > 0, we have h x n) 1 = O ln n) 1/νω η) if ν ω 0, h x n) 1 = O n 1 20η)/2γµ γω)) if ν ω = 0, sup fx y, z; h) fx y, z) x,y,z) suppx,y,z) = O h x ) 1) γ B exp α B h x ) 1) ν B )) + O p n max{ 1/2 1 + h x ) 1) δ L 1, h y h z ) }) 1 + h x ) 1) γ L exp α L h x ) 1) ν L ) ), with α B α φ ξν φ, νb ν φ, γ B γ φ + 1, α L α φ 1 {νφ =ν ω} α ω, ν L ν ω, γ L 1 + γ φ γ ω, and δ L 1 + γ µ. Prf. See Appendix. The therem abve establishes a cnsistency and unifrm cnvergence rate f the prpsed estimatr. The cnditins n the bandwidths are impsed t guarantee that asympttic behavir f the linear apprximatin f the expressin fx y, z; h) fx y, z) is 2 The typical examples f rdinarily smth functins are unifrm, gamma, symmetric gamma, Laplace r duble expnential), and their mixtures. Nrmal, Cauchy, and their mixtures are super smth functins. 3 A term exp α 1 ζ ν1 ) is mitted in Assumptin B.I i) with merely a small lss f generality since ln µ ζ) is indeed a pwer f ζ. 14

16 essentially determined by a variance term since a nnlinear remainder term is asympttically negligible. The result als shws that cnvergence rate depends n the tail behavirs f the assciated quantities. Fr instance, when χζ, y, z) and ω 1 ζ) in Assumptin B.I is rdinarily smth i.e., ν ω = 0), ne can chse small bandwidth s that resulting cnvergence rate f the estimatr is faster than when they are super smth. 4.2 Asympttic prperties f the tw-step estimatr In this subsectin, we derive the asympttic prperties f the tw-step estimatr f parameters f interest. We establish its cnsistency and asympttic nrmality Cnsistency Cnsistency is a desirable prperty fr mst estimatrs. We wish t establish cnsistency f the estimatr θ = β, δ) defined in equatin 12), where f, given in 11), is an estimatr f f 0 := fx y, z). First, ntice that frm the estimating equatin in 5) we have Q n β, δ, f) = 1 n ψy i x β Zi δ)x Z i ) fx Y i, Z i ) dx, n i=1 and its expectatin is Qβ, δ, f) = E ψy i x β Z i δ)x Z i ) fx Y i, Z i ) dx. The estimatr θ = β, δ) is btained by equating Q n β, δ, f) t zer, where f is an estimatr f f 0. Nte that Qβ, δ, f 0 ) = 0 if and nly if β, δ ) = β0, δ0 ) Θ. Nw we frmally state the fllwing sufficient cnditins fr the tw-step estimatr t be cnsistent. Assumptin C.I: Q n β, δ, f) = p 1). Assumptin C.II: X X, a cmpact set in R dx. Assumptin C.III: E[ Z ] <. 15

17 Cnditin C.I defines the estimating equatin Z-estimatr). Pakes and Pllard 1989) and Chen, Lintn, and Van Keilegm 2003) have similar assumptins. Fr a detailed discussin f this type f identificatin assumptin, see, e.g., He and Sha 1996, 2000). C.II impses cmpactness fr the true cvariate. A similar assumptin in the QR literature appears in Chernzhukv and Hansen 2006). C.III nly requires the first mment f the well-measured regressr t be finite. A unifrm law f large numbers fr the first-step estimatr fx y, z) is standard in tw-step estimatin literature; see, e.g., Newey and McFadden 1994). We nte that this is straightfrwardly satisfied by Therem 3. The fllwing therem derives cnsistency f the prpsed tw-step estimatr, θ = β, δ). Therem 4 Under assumptins C.I C.III and cnditins f Therem 3, as n θ p θ 0. Prf. See Appendix Weak cnvergence Nw we derive the limiting distributin f the tw-step estimatr in 12). We impse the fllwing assumptins fr weak cnvergence. Assumptin G.I: Q n β, δ, f) = p n 1/2 ). Assumptin G.II: The cnditinal density g Y y X = x, Z = z) is bunded and unifrmly cntinuus in y, unifrmly in x and z ver the supprt f Y, X, Z). Assumptin G.III: Let Γ 1 := Eg Y X β 0 +Z δ 0 ) X, Z)X, Z ) X, Z ) be psitive definite and V n := var[q n θ 0 )]. There exists a nnnegative definite matrix V such that V n V as n. Assumptin G.IV: f f 0 = p n 1/4 ). Assumptin G.V: Z Z is cmpact. Assumptin G.VI: Fr sme ɛ > 0, F ɛ = {f : f f 0 ɛ} is unifrmly bunded and Dnsker. 16

18 Cnditin G.I defines the estimatr. It is slightly strnger than cnditin C.I but still allws the right-hand side t be nly apprximately zer. This type f p n 1/2 ) cnditin is als assumed in Therem 3.3 f Pakes and Pllard 1989) and Therem 2 f Chen, Lintn, and Van Keilegm 2003). Cnditins G.II and G.III are standard in the QR literature; see, e.g., Kenker 2005). Cnditin G.IV impses that the estimatr f the nuisance parameter cnverges at a rate faster than n 1/4. A similar cnditin appears in cnditin 2.4) in Therem 2 f Chen, Lintn, and Van Keilegm 2003). Assumptin G.V strengthens C.III and impses cmpactness n the well-measured regressr. Finally, cnditin G.VI is similar t Chen, Lintn, and Van Keilegm 2003) and Galva and Wang 2015), and guarantees that f is asympttically well behaved. This cnditin is related t the stchastic equicntinuity f the mment functin assciated with Q n. It allws fr many nnparametric estimatrs f the cnditinal density f 0. Primitive cnditins can be btained thrugh the derivatin f asympttic nrmality f f, which requires finding a lwer bund fr the variance f the estimatr. In fact, an exact asympttic rate f cnvergence can be btained frm the assumptin that the limiting behavir f the relevant Furier transfrms has a pwer law r an expnential frm; see e.g., Fan 1991a) fr the kernel decnvlutin estimatr. We nte that Assumptin G.IV is verifiable fr particular examples thrugh Therem 3. As shwn in Therem 3, the cnvergence rate is cntrlled by the smthness f quantities such as φζ, y, z), χζ, y, z), and ω 1 ζ). Recall that φζ, y, z) is the cnditinal density f X given Y = y and Z = z i.e., fx Y = y, Z = z)), the parameter f interest in the first step; χζ, y, z) is the cnditinal characteristic functin f X 2 given Y = y and Z = z, weighted by the jint density f Y, Z) i.e., E[e iζx 2 Y = y, Z = z]fy, z)); and ω 1 ζ) is the characteristic functin f X 2. Since ω 1 ζ) = E[e iζx 2 ] = E[e iζx ]E[e iζu 2 ], the smthness f ω 1 ζ) is determined by X and U 2. Therefre, the rate f cnvergence depends n the pssible cmbinatins f the smthness f varius quantities. Fr instance, if φζ, y, z) is rdinarily smth and if χζ, y, z) and ω 1 ζ) are super smth, a cnvergence rate f the frm ln n) υ fr sme υ > 0 is achieved. This case illustrates a very slw rate f cnvergence. On the ther hand, a faster cnvergence rate, n υ fr sme υ > 0, which satisfies Assumptin G.IV, can be achieved when φζ, y, z) is als super smth. In additin, if all three quantities, φζ, y, z), χζ, y, z), and ω 1 ζ), are rdinarily smth, the slw cnvergence prblem is easily avided. Weak cnvergence f the tw-step estimatr, θ = β, δ), is established in the fllwing result. 17

19 Therem 5 Under Assumptins C.I C.III, G.I G.VI, and cnditins f Therem 3, as n n θ θ0 ) N0, Λ) fr sme psitive definite matrix Λ = Γ 1 1 V Γ 1 1. Prf. See Appendix. 5 Inference In this sectin, we turn ur attentin t inference in the quantile regressin QR) with measurement errrs ME) mdel. Imprtant questins psed in the ecnmetric and statistical literatures cncern the nature f the impact f a plicy interventin r treatment n the utcme distributins f interest; fr example, whether a plicy exerts a significant effect, a cnstant versus hetergeneus effect, r a nn-decreasing effect. It is pssible t frmulate a wide variety f tests using variants f the prpsed methd, frm simple tests n a single quantile regressin cefficient t jint tests invlving many cvariates and distinct quantiles simultaneusly. We suggest a btstrap-based inference prcedure t test general linear hyptheses. 5.1 Test statistic General hyptheses n the vectr θτ) can be accmmdated by standard tests. The prpsed statistic and the assciated limiting thery prvide a natural fundatin fr the hypthesis Rθτ) = r when r is knwn. The fllwing are examples f hyptheses that may be cnsidered in the frmer framewrk. Example 1 N effect f the mismeasured variable). Fr a given τ, if there is n dynamic effect in the mdel, then under H 0 : βτ) = 0. Thus, θτ) = βτ), δτ)), R = [1, 0] and r = 0. Example 2 Lcatin shifts). The hyptheses f lcatin shifts fr βτ) and δτ) can be accmmdated in the mdel. Fr the first case, H 0 : βτ) = β, s θτ) = βτ), δτ)), R = [1, 0] and r = β. Fr the latter case, H 0 : δτ) = δ, s that R = [0, 1] and r = δ. 18

20 Mre general hyptheses are als easily accmmdated by the linear hypthesis. Let ζ = θτ 1 ),..., θτ m ) ) and define the null hypthesis as H 0 : Rν = r. This frmulatin accmmdates a wide variety f testing situatins, frm a simple test n single QR cefficients t jint tests invlving several cvariates and distinct quantiles. Thus, fr instance, we might test fr the equality f several slpe cefficients acrss several quantiles. Example 3 Same mismeasured effect fr tw distinct quantiles). If there are the same effects fr tw given distinct quantiles in the mdel, then under H 0, βτ 1 ) = βτ 2 ). Thus, ζ = θτ 1 ),..., θτ m ) ) = βτ 1 ), δτ 1 ), βτ 2 ), δτ 2 )), R = [1, 0, 1, 0] and r = 0. Cnsider the fllwing general null hypthesis fr a given τ f interest H 0 : Rθτ) r = 0, where R is a full-rank matrix impsing q number f restrictins n the parameters, and r is assumed t be a knwn clumn vectr f q elements. Practical implementatin f testing prcedures can be carried ut based n the fllwing statistic W n τ) = R θτ) r. 13) Frm Therem 5, at given τ, and under the null hypthesis, it fllws nr θτ) r) N0, RΛR ). If we are interested in testing H 0, a Chi-square test culd be cnducted based n the statistic in equatin 13). Hwever, t carry ut practical inference prcedures, even fr a fixed quantile f interest, t cnstruct a Wald statistic ne wuld need t first estimate Λ cnsistently, and cnsequently nuisance parameters which depend n bth the unknwn θ 0 and f 0 in a cmplicated way. The estimatin f Λ is ptentially difficult because it cntains additinal terms frm the effect f θ n the bjective functin indirectly thrugh f 0. An alternative methd is t use the statistic W n directly and the btstrap t cmpute critical values and als frm cnfidence regins. Therefre, t make practical inference we suggest the use f btstrap techniques t apprximate the limiting distributin. 5.2 Implementatin f testing prcedures Practical implementatin f the prpsed tests is simple. T test H 0 with knwn r, ne needs t cmpute the test statistics W n τ) fr a given τ f interest. The steps fr implementing the tests are as fllwing: 19

21 First, the estimates f θτ) are cmputed by slving the prblem in equatin 12). Secnd, W n τ) is calculated by centralizing θτ) at r. Third, after btaining the test statistic, it is necessary t cmpute the critical values. We prpse the fllwing scheme. Take B as a large integer. Fr each b = 1,..., B: i) Obtain the resampled data {Y b i, X b 1i, X b 2i, Z b i ), i = 1,..., n}. ii) Estimate θ b τ) and set W b nτ) := R θ b τ) θτ)). iii) G back t step i) and repeat the prcedure B times. Let ĉ B 1 α dente the empirical 1 α)-quantile f the simulated sample {Wn, 1..., Wn B }, where α 0, 1) is the nminal size. We reject the null hypthesis if W n is larger than ĉ B 1 α. Cnfidence intervals fr the parameters f interest can be easily cnstructed by inverting the tests described abve. We prvide a frmal justificatin f the simulatin methd. Cnsider the fllwing cnditins. Assumptin G.IB: Fr any δ n 0, sup f f0 δ n 1 n n i=1 f ) E[f 0 )] = p 1/ n). Assumptin G.IIB: n 1 n n i=1 [τ 1{Y i < q τ0 }) f ) f ))] cnverges weakly t a tight randm element G in L in P -prbability. Lemma 1 Under Assumptins C.I C.III, G.IB G.IIB and G.VI with in prbability replaced by almst surely, the btstrap estimatr f the θ 0 is n-cnsistent and n θ θ) N0, Λ) in P -prbability. Prf. See Appendix. Lemma 1 establishes the cnsistency f the btstrap prcedure. It is imprtant t highlight the cnnectin between this result and the previus sectin. In fact, Lemma 1 shws that the limiting distributin f the btstrap estimatr is the same as that f Therem 5, and hence the abve resample scheme is able t mimic the asympttic distributin f interest. Thus, cmputatin f critical values and practical inference are feasible. 20

22 6 Mnte Carl simulatins 6.1 Mnte Carl design In this sectin, we describe the design f a small simulatin experiment that have been cnducted t assess the finite sample perfrmance f the prpsed tw-step estimatr discussed in the previus sectins. We cnsider the fllwing mdel as a data generating prcess: Y i = β 1 + β 2 X i + ε i, where ε N0, 0.25), and β 1 and β 2 are the parameters f interest. 4 We set them as β 1, β 2 ) = 0.5, 0.5). The true variable X is nt bserved by the researcher, and we use additive frms f measurement errrs ME) t generate the mismeasured X as fllws: X 1i = X i + U 1i, X 2i = X i + U 2i, where we generate X N0, 1), and we use a Laplace distributin density as L0, 0.25) t generate bth measurement errrs, U 1 and U 2. We cmpute and reprt results fr the prpsed QR estimatr. Fr cmparisn, we cmpute the density f X Y using different prcedures. First, we cnstruct ur prpsed estimatr t cntrl fr ME, using the variables Y, X 1, X 2 ), where the density is estimated by the Furier Estimatr. Secnd, we use the variables Y, X) t cnstruct an infeasible kernel estimatr f f X Y in the first step. Finally, the variables Y, X 1 ) are used fr naive kernel estimatr f f X Y which still suffers frm ME. Fr all estimatrs, we cnsider furth-rder Gaussian kernel. We apprximate the inner summatin in equatin 12) using Gauss-Hermite quadrature which is useful fr the indefinite integral. We perfrm 1000 simulatins with n = 500 and n = We scan a set f bandwidths fr X and Y in rder t find empirical ptimal bandwidths in terms f minimizing mean squared errr. 6.2 Mnte Carl results We reprt results fr the fllwing statistics f the cefficient β 2 : bias B), standard deviatin SD), and mean squared errr MSE). First f all, in rder t illustrate the prblem f ME in practice, we cnsider a mdel estimatin where the researcher ignres the ME prblem 4 Fr simplicity, the perfectly-bserved cvariate Z is absent here. 21

23 and perfrms a parametric median regressin f Y n X 1 withut crrecting fr the ME in X. This simple regressin prvides the bias f , the standard errr and the MSE f These results highlight the imprtance f crrecting fr the ME prblem. Nw we discuss and present the results fr the nnparametric estimatrs withut) crrectin f ME. Tables 1 3 reprt finite-sample perfrmance f three different tw-step estimatrs at the median: i) ur prpsed estimatr Furier estimatr); ii) infeasible kernel estimatr; iii) naive kernel estimatr. These results are fr n = 500, but the results fr n = 1000 are similar. At the bttm f each table, B, SD, and MSE frm ptimal bandwidth are reprted. In Table 4 we vary the quantiles and present results fr the different estimatrs acrss different deciles with n = Tables 1-3 Simulatin Results [ABOUT HERE] Table 1 shws that the prpsed estimatr is effective in reducing the bias when true X is measured with errrs and repeated measures f the mismeasured cvariate are available. These results are cmparable t the infeasible kernel estimatr in Table 2. On the ther hand, the results in Table 3 frm the naive kernel estimatr ignring ME in X shw much larger bias ver all selected bandwidths. Therefre, ur estimatr utperfrms the naive kernel estimatr in terms f bth bias and MSE. The minimum MSE fr ur prpsed methd is while the minimum MSE frm the naive kernel estimatr is This result cnfirms that the methds prpsed in this paper are beneficial in finite samples when repeated measures f the mismeasured regressr are available t the researcher. Table 4 reprts finite-sample perfrmance f three estimatrs ver varius quantiles with n = Fr simplicity, we use the ptimal bandwidths btained frm the simulatin results abve. The results cnfirm that ur prpsed estimatr perfrms well ver different level f quantiles. Table 4 - Simulatin Results [ABOUT HERE] 22

24 7 Empirical applicatin This sectin illustrates the usefulness f the new prpsed methds in an empirical example. One f the mst cmmnly studied tpics in labr ecnmics is the impact f educatin n earnings. The prblem f measuring returns t educatin is an imprtant research area in ecnmics with a very large literature n the subject. Fr examples f cmprehensive studies, see, e.g., Card 1995), Card 1999), and Harmn and Osterbeek 2000). The large vlume f research in this area has been explained by bth the interest in the causal effect f educatin n earnings and the inherent difficulty in measuring this effect. The difficulty arises fr several reasns. The classical ne is the fact that unbserved factrs, such as ability is prbably related t bth educatinal level and earnings. In a mean regressin framewrk, if ability is psitively crrelated with bth educatin and earnings, rdinary least squares OLS) will verestimate true causal impact f educatin n earnings. Finding strng instrumental variables IV) that are nt crrelated with unbserved ability is usually a difficult task. Nevertheless, even when available, IV estimatrs d nt necessarily prduce estimated cefficients f educatin that are significantly lwer than thse btained by OLS. A ptential reasn fr these findings in the returns t educatin literature is that IV s are used fr tw simultaneus purpses: t crrect fr bth an mitted variable bias since ability is unbservable) and measurement errrs ME) in reprted schling years. Educatin measures are frequently measured with errr, particularly if the infrmatin is cllected thrugh ne-time retrspective surveys, which are ntriusly susceptible t recall errrs, see, e.g., Ashenfelter and Krueger 1994), Kane, Ruse, and Staiger 1999), Bund, Brwn, and Mathiwetz 1999), and Black, Sanders, and Taylr 2003)). It is als knwn that ME in a simple framewrk can prvke attenuatin bias, thus OLS may nt necessarily be verestimating the true returns t educatin if ME is a quantitatively mre imprtant prblem than mitting a cvariate. Thus, it became imprtant in that literature t understand what is the islated rle f ME n the bias f estimated cefficients. We use quantile regressin QR) methds t study returns t educatin. We accmmdate pssible hetergeneity n the returns t educatin in the earnings distributin by applying QR. Indeed, this hetergeneity is nt revealed by cnventinal least squares r tw stage least squares, while the QR apprach cnstitutes a suitable way t investigate whether the returns t educatin differ alng the cnditinal wage distributin. In this paper, we primarily fcus n cntrlling fr ME in educatin, even thugh the mitted variable bias 23

25 may be an imprtant issue. T the best f ur knwledge, there is n published wrk which effectively cntrls fr bth mitted variable and ME in QR. 5 Careful research is required t cntrl fr bth surces f endgeneity f educatin in the QR framewrk. We leave this tpic fr future research. Our QR methd prpses a slutin t the ME prblem in educatin by using repeated measures f the educatin variable. The literature n the returns t educatin has used useful infrmatin n repeated measurements f educatin where ne twin is asked t reprt n bth his/her wn schling and the schling f the ther twin Ashenfelter and Krueger 1994) and Bnjur et al. 2003)). This allws ne t treat the infrmatin reprted by the ther twin as a repeated measure f the true educatin. We therefre apply ur methd t a data set n female mnzygtic twins frm the Twins Research Unit, St. Thmas Hspital frm the United Kingdm. Our data are taken frm Bnjur et al. 2003) and Amin 2011). The sample cnsists f 428 individuals cmprising 214 identical twin pairs with cmplete wage, age, and schling infrmatin. The summary statistics are described in Table 5. Table 5 - Summary Statistics [ABOUT HERE] The prpsed QR estimatr is designed t crrect fr the ME prblem while explring hetergeneus cvariate effects, and therefre prvides a flexible methd fr the practical analysis f returns t educatin. Thus, ur bjective is t estimate the fllwing cnditinal quantile functin: Q Wi τ edu i, Z i ) = βτ)edu i + Z i δτ), 14) where W i is the earnings f individual i, edu i is the true number f years f educatin which is latent, and Z i is a vectr f exgenus cvariates. The parameters f interest are βτ), δτ)). As mentined earlier, if edu i is subject t ME, and nly edu 1i and edu 2i are bserved, standard QR estimates f βτ) using edu 1i r edu 2i will be incnsistent. Fr the practical implementatin f the prcedures, the dependent variable is the lg f wage Y ). The independent variable subject t ME is educatin and the bserved repeated measures 5 Amin 2011) uses the average educatin f the twins as an additinal cvariate t prxy fr mitted ability bias and uses c-twin s estimate f educatin as an instrument t cntrl fr ME in self-reprted educatin. Hwever, this prcedure generates an issue f tw mismeasured cvariates which require tw valid instruments. Amin 2011) instruments bth mismeasured cvariates with reprted educatin variables. Hwever, there will be ME n thse instruments, which makes the IV apprach in QR invalid. 24

26 f true educatin are twin 1 s educatin X 1 ) and twin 2 s reprt f twin 1 s educatin X 2 ). These Y, X 1 and X 2 are standardized t have mean zer and standard deviatin ne, fr the purpse f bandwidth selectin. We use age and squared age as crrectly-bserved exgenus cvariates Z). Clearly, the mdel in equatin 14) is very simple: ability has a mntnically psitive r negative impact n educatin return. Hwever, as emphasized by Arias, Hallck, and Ssa-Escuder 2001), QR prvides a mre flexible apprach t distinguishing the effect f educatin n different percentiles f the cnditinal earning distributin, being cnsistent with a nn-trivial and, in fact, unknwn interactin between educatin and ability. We cmpare the estimates using ur prpsed methds with thse frm the existing literature, in particular the results presented in Amin 2011) fr QR and IV-QR. Amin 2011) presents results fr the parameter f interest using the tw-stage QR estimatr f Arias, Hallck, and Ssa-Escuder 2001) and Pwell 1983), where fitted value f educatin is estimated in the first stage and a QR f lg f wage n the fitted value f educatin fllws in the secnd stage. Hwever, fr cmparisn purpses, we reprt estimates using the standard IV-QR prpsed by Chernzhukv and Hansen 2006). Fr this, we use the variable edu 2 as an instrument fr educatin edu 1. The IV strategy is based n the assumptin that the c-twin s educatin is strngly related t the ther s reprt f the c-twin s educatin i.e., IV) but the IV is independent f unbservable factrs f earnings as well as measurement errrs e.g. Chernzhukv and Hansen 2005)). We cnjecture that the IV apprach delivers different estimates than ur prpsed ME estimatr since they rely n different set f cnditins. Our methd is particularly useful fr the data set where it is unlikely that the IV is independent f the regressin errr which cntains ME n self-reprted educatin, since the IV is als mismeasured. 6 Our results fr the estimates f the returns t educatin cefficient are reprted in Figures 1 4. The figures present results fr the cefficients and cnfidence bands, fr a range f quantiles, fr QR, IV-QR, and QRME, respectively. The shaded regin in each panel represents the 95% cnfidence interval. In additin, the estimates fr simple OLS and the IV-OLS appear in the respective figures, with dashed red lines fr cnfidence bunds. In Figure 1 we reprt standard QR and OLS estimates. The estimatin strategy fllws Kenker and 6 We nte that the independence cnditin implies independence between ME n the c-twin s educatin and the ther s reprt f the c-twin s educatin. Hwever, ur apprach requires a weaker assumptin f cnditinal mean zer as in Assumptin A.I i). 25