1 Introduction. Research Article. Tatiana Komarova, Thomas A. Severini and Elie T. Tamer Quantile Uncorrelation and Instrumental Regressions

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1 DOI / Jounal of Econometic Methods 2012; 1(1): 2 14 Reseach Aticle Tatiana Komaova Thomas A Seveini and Elie T Tame Quantile Uncoelation and Instumental Regessions Abstact: We intoduce a notion of median uncoelation that is a natual extension of mean (linea) uncoelation A scala andom vaiable Y is median uncoelated with a k-dimensional andom vecto X if and only if the sloe fom an LAD egession of Y on X is zeo Using this simle definition we chaacteize oeties of median uncoelated andom vaiables and intoduce a notion of multivaiate median uncoelation We ovide measues of median uncoelation that ae simila to the linea coelation coefficient and the coefficient of detemination We also extend this median uncoelation to othe loss functions As two stage least squaes exloits mean uncoelation between an instument vecto and the eo to deive consistent estimatos fo aametes in linea egessions with endogenous egessos the main esult of this ae shows how a median uncoelation assumtion between an instument vecto and the eo can similaly be used to deive consistent estimatos in these linea models with endogenous egessos We also show how median uncoelation can be used in linea anel models with quantile estictions and in linea models with measuement eos Keywods: quantile egession endogeneity instumental vaiables coelation Autho Notes: We thank the edito and two anonymous efeees fo oviding us with constuctive comments and suggestions We also aeciate feedback fom semina aticiants at the London School of Economics Univesity College London the Univesity of Toonto and Queen May Univesity of London and feedback fom the aticiants of the Canadian Econometic Study Gou and all UC Econometics Confeence Tatiana Komaova: London School of Economics and Political Science tkomaova@lseacuk Thomas A Seveini: Nothwesten Univesity seveini@nothwestenedu Elie T Tame: Nothwesten Univesity tame@nothwestenedu 1 Intoduction We intoduce a concet of quantile uncoelation o L 1 -uncoelation between two andom vaiables that is a natual extension of the well-known mean uncoelation o L 2 -uncoelation We tem this tye of uncoelation median uncoelation which is the counteat of the familia mean (linea) uncoelation o simly uncoelation We chaacteize the elationshi between andom vaiables that ae uncoelated in this manne We ovide a seies of oeties that imly o ae imlied by median uncoelation Natually fo examle indeendence of two andom vaiables imlies median uncoelation (o in this case L -uncoelation fo any 1) Also this uncoelation is not symmetic and is nonadditive but it etains an imotant invaiance oety We extend ou definition to median uncoelation between andom vectos which esults indiectly in a multivaiate vesion of a quantile estiction We also deive an asymmetic coelation measue based on this notion of quantile uncoelation that takes values in [ 1 1] with a value of zeo fo uncoelation In addition we ovide anothe coelation measue that is the analog of the coefficient of detemination o R 2 in linea egessions We also extend this concet to cove L -uncoelation fo 1 As two stage least squaes is based on exloiting linea uncoelation between the eo and an excluded andom vaiable (the instument) we also show that this uncoelation leads natually and unde easily inteetable conditions to instumental egessions with median uncoelation These ae analogs of Basmann and Theil s two stage least squaes o 2SLS (Theil (1953) and Basmann (1960)) as deived fom the usual mean uncoelation between two andom vaiables As in the classical 2SLS median uncoelation leads to an estimato that is deived by taking a samle analogue of the median uncoelation measue This estimato simila to one used by Chenozhukov and Hansen (2006) (o CH) is consistent ovided that this uncoelation holds (along with othe standad assumtions) Othe alications ae natual counteats of existing least squaes

2 Komaova et al: Quantile Uncoelation and Instumental Regessions 3 methods Fo examle by exloiting this uncoelation futhe we show that as instumental vaiable methods can be used in mean-based models to emedy the oblem of classical measuement eo vaiables obeying ou median uncoelation condition can be used as instuments to obtain estimates of aametes in linea models with measuement eo unde quantile estictions Futhemoe anel data quantile egession of diffeenced data delives consistent estimates of aametes of inteest without making assumtions on the individual effects unde median uncoelation estictions So this uncoelation gives suot fo unning standad quantile egession of fist diffeenced outcomes on fist diffeenced egessos unde an absolute loss function to obtain consistent estimates of the sloe aametes in linea models An imotant featue of the concet of median uncoelatedness is the fact that it is defined in tems of the linea edicto and hence is exlicitly a linea concet Basically it shaes this oety with best linea edictos in that heuistically a andom vaiable is median uncoelated with anothe if the latte is not useful as a linea edicto of the fome unde absolute loss Finally this notion of median uncoelation is geneal and is loss function based Thee is a lage liteatue in econometics on best edicto oblems Manski (1988) delineates estimatos deived fom ediction oblems fom vaious loss functions Thee best linea edictos ae deived and consistent estimatos ae ovided that ae based on the analogy incile The linea model based on quantile estictions is equally well studied stating with the wok of Koenke and Bassett (1978); see also Koenke (2005) Thee has also been a seies of aes dealing with the esence of endogenous egessos in models with quantile estictions Amemiya (1981) oosed a two-staged least absolute deviation estimato See also Powell (1983) Then based on method of moments Honoé and Hu (2004) ovide methods that can be used to do infeence on aametes defined though seaable moment models (that can be nonlinea) CH (see also Chenozhukov and Hansen (2005)) in a seies of aes shed new light on a geneal class of monotonic models with conditional quantile estictions They ovide sufficient oint identification conditions fo these models and also an estimato that they show is consistent unde those conditions CH study also the asymtotic oeties of thei estimato and chaacteize its lage samle distibution The estimato based on ou median uncoelation assumtion is the same as the one used in CH Finally Sakata (2007) and Sakata (2001) in inteesting wok ovide estimatos based on an L 1 loss function fo instumental egession models1 Both these aes use a condition that is close to conditional median indeendence but the aoach in siit is simila to ous In Section 2 we ovide fist a few elementay definitions that lead to median uncoelation Afte defining median uncoelation Section 3 chaacteizes this uncoelation concet in tems of vaious oeties of the joint distibution of andom vaiables Section 4 shows how median uncoelation leads to natual estimatos in linea models with endogenous egessos Section 5 ovides notions of median coelation among andom vaiables We ovide in Section 6 simle alications of ou median uncoelation concet to linea quantile egession with measuement eo and to anel data quantile egession Section 7 concludes 2 Definition and Poeties Let T be a scala andom vaiable and let S be a k-dimensional andom vecto such that ES< We ae inteested in the following otimization oblem since it is key in defining ou concet of median uncoelation: min ET -α- S ( α ) whee we assume2 that ET< This is done fo simlicity of notation Define M(TS) R k as the set of solutions to this otimization oblem with esect to : MT ( S) : αsuch that ( α ) = agmin ET -α - S ( α ) In geneal one can find distibutions in which M(T S) is a set Howeve unde weak conditions M(T S) is a singleton; see at 3 of Poosition 21 below Notice that fo a fixed ET -S -MedT- S = min ET -α-s whee Med(z) inf {t : P (z t) 05} Theefoe = ( ) M T S ag min E T S Med T S α 1 Fo othe aoaches to estimation in quantile egession with endogeneity see Ma and Koenke (2006) Lee (2007) and Cheshe (2003) 2 Without this assumtion we can ewite the objective function as min { ET - α - S - ET - α - 0 S 0 } fo some fixed (α ( α ) 0 0 )

3 4 Komaova et al: Quantile Uncoelation and Instumental Regessions The next oosition chaacteizes elements of the set M(T S) and also gives conditions unde which M(T S) is a singleton We collect Poofs to esults in the Aendix Poosition 21 The following hold: 1 Let R k Then M(T S) if and only if fo any α R R k E ( α+ S ) sgn( T-S Med( T S )) E α+ ( ( ) = ) S 1 T S Med T S 0 (21) as whee hee and in the est of the ae we define sgn( ) 1 x> 0 sgn( x) = 0 x= 0-1 x < 0 2 Let R k be such that P(T S Med(T S ) = 0) = 0 Then M(T S) if and only if E[S sgn(t S Med(T S ))]=0 (22) 3 Suose that any R k satisfies P(T S Med (T S )=0)=0 Then M(T S) is a singleton if and only if equation E[S sgn(t S Med(T S ))]=0 has a unique solution This solution is M(T S) We use Equation (22) as the basis fo a measue of median coelation intoduced in Section 5 The next definition intoduces the notion of median uncoelation of a andom vecto with anothe andom vecto Hee and in the emainde of the ae we take M(T S) = to mean that M(T S) contains the single value Definition 21 (Median Uncoelation) Let W denote an l-dimensional andom vecto We will say that W is median uncoelated with S if M(c W S)=0 fo all c R l (23) The definition above is loss function based So it natually caies ove to quantiles othe than the median by simly changing the absolute loss to asymmetic loss by using the check function Moeove imlicit in this definition is a fomulation fo multivaiate quantiles In aticula when defining this uncoelation oety meant fo scala quantiles to the multivaiate case we equie that median uncoelation holds fo any linea combination of the elements of the multivaiate vecto as in (23) Finally a key oety that this loss function maintains is the invaiance oety below Lemma 21 (Invaiance) Fo any constant vecto b R k and any constant scala a M(T + a + S b S)=M(T S)+b (24) This oety lays a key ole below Lineaity of T - α - S in the objective function is essential fo this invaiance oety to hold The concet of uncoelation we intoduced is intimately tied to linea models and is simila to the elationshi between uncoelation in the least squaes setu and its elationshi to linea models Median uncoelation is median linea uncoelation 3 Chaacteizations of Median Uncoelation In this section we ovide key insights that exloe futhe the meaning of median uncoelation in Definition 21 above The following chaacteization theoem collects a set of oeties that ae helful in gaining intuition about median uncoelation Theoem 31 (Poeties of Median Uncoelation) The following hold: A A sufficient condition fo an l-dimensional andom vecto W to be median uncoelated with a andom vecto S is that Med(c Ws)=Med(c W) fo all c R l B If W is median uncoelated with S it does not necessaily follow that S is median uncoelated with W C A sufficient condition fo W to be median uncoelated with S is that the conditional chaacteistic function of W given S is eal D Conside a scala andom vaiable T and any andom vecto S Assume that M(TS) is a singleton Then T can be witten as T=α 0 + S M(T S) + δ whee M(d S) = 0 and a 0 is any constant E Fo a scala andom vaiable T and andom vectos S and Z in R k assume that P(T Med(T) = 0) = 0 and M(T S + Z) is a singleton Then M(T S) = M(T Z) = 0 M(T S + Z) = 0 F Suose that fo a scala andom vaiable T and a non-degeneate binay andom vaiable S the median of TS = 1 and the median of TS = 0 ae unique The following hold: M(T S) = 0 Med(TS = 1) = Med(TS = 0);

4 Komaova et al: Quantile Uncoelation and Instumental Regessions 5 if P(T Med(T) = 0) = 0 then M(T S)=0 Med(TS =1) = Med(TS =0) Poety (A) can be diectly deived fom the definition and states median indeendence as a sufficient condition fo median uncoelation (B) means that the definition of median uncoelation is not symmetic This is in diect contast with mean uncoelation which is a symmetic oety Poety (D) is imotant and it states that any scala andom vaiable T can be decomosed into a linea combination of S s and anothe andom vaiable that is median uncoelated with S This is a diect esult of the invaiance oety in (24) above Moeove this is simila to the linea mean decomosition in best linea ediction examles See (31) below Poety (E) illustates an additivity oety of median uncoelation: If T is median uncoelated with S and Z then it is median uncoelated with thei sum S + Z Poety (F) states that unde weak estictions T is median uncoelated with a binay vaiable S if and only if T is median indeendent of S Evidently if W is median uncoelated with S then S is not useful in the L 1 ediction of linea functions of W 31 Comaison to mean uncoelation It is helful to comae the median uncoelation with the well-known mean uncoelation Conside the otimization oblem ( α ) ( -α- S ) 2 min ET whee ET 2 < ES 2 < Unde the usual ank condition on S this oblem has a unique solution Denote its solution with esect to as L(T S) This is the L 2 analogue of M(T S) It is easy to show that fo scala S fo examle L(T S) = Cov(T S)Va(S) 1 In addition W with the values in R l and S ae (mean) uncoelated if fo any c R l L(c W S) = 0 since L(c W S) = Va(S) 1 Cov(S W)c Poeties in Theoem 31 have the following L 2 vesions L 2 Poeties The following hold: A A sufficient condition fo an l-dimensional andom vecto W to be (mean) uncoelated with a k- dimensional andom vecto S is that E(c WS) = E(c W) fo all c R l This holds in aticula if W is mean indeendent of S B If W is uncoelated with S then S is uncoelated with W C A sufficient condition fo W to be uncoelated with S is that the conditional chaacteistic function of W given S is eal D Fo a scala andom vaiable T and a k-dimensional andom vecto S vaiable T can be eesented as follows: T = α 0 +S L(T S)+d (31) whee L(d S) = 0 and α 0 is any constant Clealy if W is uncoelated with S then S is not useful in the L 2 ediction of linea functions of W The main technical diffeences between median uncoelation and uncoelation ae that (1) median uncoelation is not symmetic (2) if W 1 and W 2 ae both uncoelated with S then the vecto (W 1 W 2 ) is uncoelated with S while the same is not tue fo median uncoelation (3) a condition fo W and S to be uncoelated can be given in tems of W alone (ie Cov(W S) = 0) without efeence to linea functions and (4) the additivity of L(W S) ie L(W 1 +W 2 S) = L(W 1 S) + L(W 2 S) which often geatly simlifies technical aguments This latte diffeence basically means that if W 1 is uncoelated with S and W 2 is uncoelated with S then W 1 +W 2 is uncoelated with S Two simle esults in Poosition 31 below comae the median uncoelation with the usual mean uncoelation Poosition 31 Let T be a scala andom vaiable and S be a andom vecto in R k 1 If V a scala andom vaiable is indeendent of S then cov(t +V S) = cov(t S) but in geneal M(T +V S) M(T S) 2 If V a andom vecto in R k is indeendent of T then cov(t S +V) = cov(t S) but in geneal M(T S +V) M(T S) 4 Median Uncoelation and Instumental Regession This is the main section of the ae in which we exloit the median uncoelation concet to define estimatos fo

5 6 Komaova et al: Quantile Uncoelation and Instumental Regessions aametes in linea models with endogenous vaiables The estimato (and the model) is defined via the uncoelation assumtion in the same way as some vesions of 2SLS ae defined fom the mean uncoelation Conside the following model: Y=α 0 +X 0 +ε (41) whee Y and ε ae eal-valued andom vaiables X is a k-dimensional andom vecto with a ositive definite covaiance matix α 0 is an unknown scala aamete and 0 is an unknown sloe vecto The aamete of inteest is 0 Assume that ε has median 0 but that Med(εx) 0 whee Med( ) denotes the conditional median The oblem hee is that this conditional median is allowed to deend on X Thee ae many easons fo this tye of endogeneity in economic models Classical wok on demand and suly analysis in linea (in aamete) models motivate many ealy woks in linea models with mean estictions whee instumental vaiables assumtions wee used to eliminate least squaes bias that aises fom this endogeneity See Theil (1953) Basmann (1960) and Amemiya (1985) and efeences theein Thee ae a set of aes that deal with endogeneity in linea quantile based models See fo examle Amemiya (1981) fo a 2 stage inteetation of the 2SLS and Chenozhukov and Hansen (2005) fo an aoach to infeence in quantile based models both linea and nonlinea in the esence of endogenous egessos Finally also Sakata (2007) ovides a simila aoach to ous fo estimating models based on L 1 loss which also involves instumental vaiables Recall that the 2SLS stategy is based on finding an instument vecto Z such that E[Zε] = 0 and using this uncoelation (moment) condition to deive a consistent estimato fo 0 In this section we extend this intuition to median uncoelation wheeas we assume the esence of a andom vecto Z which we call a vecto of instuments that obeys a median uncoelation assumtion (see Assumtion A1 below) This median uncoelation oety similaly to its counteat E[Ze] = 0 leads natually to a simle estimato fo 0 So the intuition fo obtaining an instument hee is simila to 2SLS in that one looks fo an excluded vaiable that is median uncoelated with the outcome ie cannot linealy exlain the outcome based on a linea median egession (hee the outcome means the outcome afte ojection on the othe egessos) Finally ou aoach is closely elated also to Sakata (2007) who ovides a novel aoach to infeence in this setu Thee the IV estimato is defined though an imlication of a conditional median indeendence assumtion Below we state the main assumtion hee Assumtion A1 Let thee be a d-dimensional andom vecto Z such that: 1 Thee exists a k d constant matix of full ank g with d k such that X = gz+δ fo some andom vecto δ 2 (δ ε) is median uncoelated with Z Fist we equie that the dimension of Z be at least equal to the dimension of X This is the necessay condition fo oint identification The key assumtion is at 2 of A1 whee we equie that not only ε be median uncoelated with Z and d be median uncoelated with Z but also that (δ ε) = (X gz ε) be jointly median uncoelated with Z (since the fact that M(ε Z) = 0 and M(δ Z) = 0 does not imly that (d ε) is median uncoelated with Z) Given Assumtion A1 we ae able to easily ove the following theoem which constitutes the main esult in this section Theoem 41 (Main Result) Conside the function y() = M(Y X Z) (42) Let assumtion A1 hold Then y() = 0 = 0 Poof: Note that by assumtion A1 we have Y = a 0 +Z g 0 +δ 0 + ε Let m M(Y X Z) = M(a 0 + Z g ( 0 )+d ( 0 ) +ε Z) By the invaiance oety in Lemma 21 thee exists m 0 M(δ ( 0 ) + ε Z) such that m=g ( 0 )+m 0 Note that δ ( 0 ) = ( 0 ) δ Hence since (δ ε) is median uncoelated with Z m 0 = 0 It follows that m = g ( - 0 ) and hence that y( ) = g ( 0 ) Since d k and g is full column ank by assumtion A1 we have y() = 0 = 0 which oves the theoem

6 Komaova et al: Quantile Uncoelation and Instumental Regessions 7 log Wage S IQ Exeience Tenue Age Least Squaes 057(74) 0041(35) 0138(311) 0054(19) 014(276) 2SLS 015(84) 017(347) 013(288) 003(121) 02(326) Quantile Reg (5) 05(475) 005(275) 008(143) 008(2) 018(24) MIR -000(-08) 024(718) 014(239) 0032(77) 019(232) Table 1: Retuns to Schooling when Contolling fo Endogenous Ability The theoem can be used as the basis fo an estimation method fo 0 Note that in case we use the least squaes function L( ) instead of M( ) we get exactly Basmann s inteetation of the 2SLS estimato of 0 Moeove note that the estimato based on the esult in Theoem 41 is the same as the one used by Chenozhukov and Hansen (2005) Let Ŷ denote an n 1 vecto of ealizations of Y let ˆX denote an n k matix of ealizations of X and let Ẑ denote an n d matix of ealizations of Z Define MYZ ˆ ˆ ˆ to be the vecto c R d that minimizes Y ˆ - - ˆ j a Zc j j when minimizing ove (a c) Then ˆ is defined as the solution in b to ( Xb Z ) M ˆ Y ˆ- ˆ ˆ = 0 ˆ can be obtained as in CH by minimizing ( Xb Z) ˆ = agmin Mˆ Yˆ-ˆ ˆ k b R A whee A is the weighted by A Euclidian nom It is inteesting to note that the sufficient condition fo identification in CH adated to the linea model is (in ou notation) that fo all Z the following has a unique solution at the tue aamete 0 : 1 P( Y< α0+ X Z) = E 1 [ Y< α0+ X ] Z = 2 while ou median uncoelation condition equies that the moment condition E[Z sgn(y X Med(Y X ))] = 0 (43) has a unique solution at 0 CH s condition above can be witten as E[sgn(Y X 0 Med(Y X 0 ))Z] = 0 which obviously imlies (43) when it is calculated at 0 Clealy it is a conditional statement as oosed to an unconditional statement But ou aoach equies an (unconditional) uncoelation assumtion on the joint distibution of (δ ε Z) We next state the asymtotic distibution without any conditions and efe the eade to Chenozhukov and Hansen (2005) who deived these esults fo details and fo ways to comute the estimato and its standad eos Unde the conditions in CH as n we have d -1-1 ( -) N ( ) n ˆ 0 C D C whee C=E[f ε (0X Z)XZ ] and D E[ ZZ ] X 0 1 = and ε=y-α Relationshi to the 2SLS Assumtions In the usual model with endogeneity we have Y=α 0 +X 0 +ε Cov(ε X) 0 Hee a andom vecto Z is an instument if Cov(X Z) and Cov(Z Z) have full ank and Cov(Z ε) = 0 o E[Zε] = 0 with a mean zeo assumtion on ε Let g = Cov(X Z)Cov(Z Z) 1 and define δ = X gz Then X = gz + δ Hee (δ ε) is uncoelated with Z because δ is uncoelated with Z by constuction and ε is uncoelated with Z by definition This is not tue in the median case whee we need to imose the joint median uncoelation condition in at 1 of A1 This is the key diffeence between the mean and the median fomulations 42 Emiical illustation We illustate ou aoach above by estimating a wage egession simila to Giliches (1976) using an extact fom the 1980 NLSY which contains data on wages schooling and many othe vaiables3 We ae inteested in the 3 Fo infomation about this samle see Blackbun and Neumak (1992)

7 8 Komaova et al: Quantile Uncoelation and Instumental Regessions elationshi between schooling and wages allowing fo Ability oxied hee with IQ to be endogenous We use the following egession as the benchmak: Ln(Wage) =α + 1 S+ 2 IQ + 2 Exeience + 4 Tenue + 5 Age + ε whee S is comleted yeas of schooling IQ is the IQ scoe and hee stands fo Ability Exeience is yeas of exeience and Tenue is yeas of tenue In this egession the vaiable IQ is endogenous and so we use KWW o knowledge of the wold test as an instument fo it Above Table 1 ovides estimates fo the aamete vecto 0 using a set of estimatos each imoses vaious assumtions on the undelying distibution of ε conditional on the egessos and the instuments We eot least squaes and two stage least squae esults a median quantile egession esults and MIR which is median instumental egession esults The Table also esents the t-stat in aentheses In least squaes all the coefficient ae significant and ae useful in edicting wages This stoy changes somehow when we conside 2sls: now it aeas that schooling becomes much less imotant (and the esult holds if we use efficient GMM) The median egession esults ae simila qualitatively to the least squaes esults So the inteesting esult to note is that the etuns to schooling when we contol fo ability is statistically and economically significant when we do not coect fo endogeneity of IQ (eithe least squaes o median egession) and is oughly aound 5% When we contol fo endogeneity of IQ using KWW as an instument schooling becomes both economically and statistically insignificant even when we use MIR o median uncoelated egessions So the MIR esults in aticula show that schooling is not useful in linealy edicting wage unde absolute loss when we include IQ (and othe egessos) and when we allow fo endogeneity as defined though the MIR model assumtions 5 Some Measues of Median Coelation In the case when two andom vaiables ae not median uncoelated we would like to be able to measue the degee of thei median coelation Two such measues ae esented below The fist genealizes the usual (mean) coelation; the second genealizes the idea of the coefficient of detemination Fist we eview the L 2 case Fo scala andom vaiables T and S intoduce the nomalized andom vaiables T-ET T = σ T S-E( S) S = σ S Coelation between T and S is measued by the coelation coefficient co(t S): co(t S)=E[T S sgn(t )sgn(s )] This definition equies T and S to have finite vaiances A second way to measue the linea elationshi between two scala andom vaiables is to conside the extent to which a linea function of one andom vaiable is useful in the ediction of the othe; when alied to data this measue is the coefficient of detemination often denoted by R 2 Thus let min ET ( -α-s α ) 2 2 R sq T S = 1 - ET ( -ET ) It is well-known that sq(t S) = co(t S) 2 Now conside the L 1 case; we begin by consideing the analogue of co Suose that ET< and4 ES< Define T and S as T Med( T) T - = E T - Med T S Med( S) S - = E S - Med S Let medco(t S) denote a measue of median coelation between T and S defined as ( ) ( ) medco T S E S sgn T sgn S Note that in geneal medco(t S) is diffeent fom M(T S) The theoem below establishes some imotant oeties of the medco measue Theoem 51 Conside the andom vaiables T and S such that ES< The following hold: 1 medco(t S) [ 1 1] 4 We can avoid assuming ET< if medco (TS) is defined in the following way: medco (T S) = E[S ~ sgn(t Med(T))sgn(S ~ )] When ET< these two definitions give the same numeical value 2

8 Komaova et al: Quantile Uncoelation and Instumental Regessions 9 2 Suose that M(T S) is a singleton and P(T M(T S) S - Med(T M(T S)S) = 0) = 0 and P(T Med(T) = 0) = 0 Then sgn(medco(t S)) = sgn(m(t S)) In addition we can show5 that medco(t S) is inceasing in M(TS) So fo examle if M(TS) > 0 we know that medco(t S) is also ositive and a highe M(T S) esults in a highe medco(t S) In the exteme case whee M(T S) = + it is easy to see that medco(t S) = 1 The L 1 analogue of sq is min E T-S-Med T-S medsq T S 1 - E T -Med T Note that ( ) E T-0S-Med T- 0S medsq T S = 1 - E T -Med( T) whee 0 is an abitay element of M(T S) This method was used in Koenke and Machado (1999) to measue the goodness of fit fo quantile egessions Koenke and Machado (1999) exlain why medsq is bounded between 0 and 1 They also show that this coelation measue takes the value of 1 whee the andom vaiable T and the andom vecto S ae linealy efectly coelated We collect some esults about medsq and about the elationshi between medco and medsq in the following theoem Theoem 52 Conside andom vaiables T and S such that ES < and ET < The following hold: 1 If M(T S) = 0 then medsq(t S) = 0; if medsq (T S) = 0 then 0 M(T S) 2 Suose that P(T Med(T) = 0) = 0 Then medsq (T S) = 0 if and only if medco(t S) = 0 Pat (1) shows that medsq takes the value of zeo when T is median uncoelated with S This is simila to the usual 5 A sketch of a oof fo this is as follows Since medco ( T S ) E ( ) ( ) Ssgn T sgn S elace T ~ with T = α + SM ( T S ) + d to get sgn ( α + ( ) + d) sgn( ) sgn ( α + ( ) + d) to MT ( S ) is equal to ( α - d S ) E S SM T S S which is in tun equal to E S SM T S The deivative of the latte with esect S S f SM T S df which is ositive R 2 in linea models Pat (2) says that this median R 2 is equal to zeo when the median coelation is zeo Also Blomqvist (1950) intoduced the following measue of median coelation between andom vaiables T and S: k(t S) = E[sgn(T Med(T))sgn(S Med(S))] o in tems of nomalized vaiables ( ) sgn( ) sgn( ) kt S= E T S if ET < ES < As we can see this measue is diffeent fom ous In aticula k(t S) is symmetic and does not satisfy the invaiance oety The value of medco(t S) measues the degee of linea elationshi between T and S while k(t S) eesents an analog of Kendall s ank coelation because k(t S) = P((T Med(T))(S Med(S)) > 0) P((T Med(T))(S Med(S)) < 0) Next we genealize the concet of L 1 -coelation to othe loss functions This will be a natual extension to the above esults 51 L -coelation fo any 1 The notion of L 1 -coelation can be genealized to the case of L -coelation fo any 1 Definition 51 Fo a andom vaiable Y and fo any 1 < define Med (Y ) as follows: Med (Y) inf{d:e[y d 1 sgn(y d)] 0} Note that Med 1 (Y ) = Med(Y ) and Med 2 (Y ) = E(Y ) Let T be a andom vaiable and S be a andom vecto with values in R k such that ET < and ES < Conside the otimization oblem min ET -α- S ( α ) We ae inteested in the solutions to this oblem with esect to Denote the set of these solutions as M (T S): M ( TS ) : α such that ( α ) = agmin ET -α - S ( α ) Notice that fo a fixed ET -S -Med T- S = min ET -α-s Theefoe α

9 10 Komaova et al: Quantile Uncoelation and Instumental Regessions = - - ( - ) M T S ag min E T S Med T S The next definition intoduces the notion of L - uncoelation of a andom vecto with anothe andom vecto Definition 52 (L P -uncoelation) Let W denote an l-dimensional andom vecto We say that W is L -uncoelated with S if M (c W S) = 0 fo all c R l To measue L -coelation of a scala andom vaiable T with a scala andom vaiable S let us nomalize these vaiable and define T and S in the following way: T Med ( T) T - = E T -Med ( T) S = 1 S-Med ( S) E S-Med ( S) 1 Define a measue of L -coelation of T with S as follows: -1 ( ) ( ) medco T S = E S T sgn T sgn S The value of medco (T S) lies in the inteval [ 1 1] and it can be shown that unde weak estictions simila to the ones in Theoem 51 sgn(medco (T S)) = sgn(m (T S)) Note that if fo some c 2 T = c 1 + c 2 S with obability 1 then medco (T S) = sgn(c 2 ) It is easy to see that medco 2 (T S) coincides with the familia coelation coefficient co (T S) The L analogue of medsq is defined as follows: medsq ( T S) and obviously min E T-S-Med T-S 1 - E T -Med T ( ) E T-0S-Med T- 0S medsq T S = 1 - E T -Med ( T) whee 0 is an abitay element of M (T S) 6 Othe Alications of Median Uncoelation We ovide two othe alications of this median uncoelation by mimicking imlications of mean uncoelation when dealing with measuement eo in linea models unde quantile estictions and in anel data models with quantile estictions 61 Quantile egession with measuement eo We aly the idea of median uncoelation to linea quantile egessions with classical measuement eo in the egessos In aticula conside the model Y = α 0 + X 0 +ε Med(ε) = 0 (61) whee we assume that M(ε X ) = 0 o that ε is median uncoelated with a k-dimensional andom vecto X We do not obseve X diectly but we obseve an eo-idden vesion of it X such that X=X +n (62) whee we assume that M(vX ) = 0 We also obseve Y To emedy the identication oblem that esults fom the measuement eo we follow the teatment of the linea model unde the mean uncoelation and use instuments Let thee exist a d-dimensional andom vecto Z and a k d constant matix g with d k such that X =gz+ψ (63) fo some andom vecto ψ and M(ψ Z) = 0 Then X=gZ+ψ+n Given the esults of the evious section we can show the following esult Theoem 61 Fo model (61) suose that we obseve (YX) such that (62) holds with M(v X ) = 0 Moeove assume that (ε n ψ) is median uncoelated with Z and that g in (63) has full ank Then M(Y X Z)=0 = 0 Note that the equiements of the above model ae that the vecto (ε n ψ) is jointly median uncoelated with Z The eal assumtion hee is that the vecto of unobsevables is equied to be median uncoelated with Z In contast in the mean uncoelation model Z is mean uncoelated with ψ by constuction So again as in the

10 Komaova et al: Quantile Uncoelation and Instumental Regessions 11 2SLS genealization it is the joint median uncoelation that is needed 62 Quantile egession with anel data We ae inteested in infeence on 0 in the following model: yit = xit 0 + αi + εit t=1 2 (64) whee α i is the individual effect that is abitaily coelated with x i = ( xi 1 xi 2) Denote y i =y i1 y i2 x i = x i1 x i2 and ε i = ε i1 ε i2 Suose that we have a data set of iid obsevations (y i x i ) fo i = 1 n whee y i =(y i1 y i2 ) If we maintain the assumtion that ε i =(ε i1 ε i2 ) is median uncoelated with x i then 0 =M( y i x i ) Indeed this follows fom E y i a x i = E ε i a x i ( 0 ) and the definition of the median uncoelation of the vecto ε i with x i We want to emhasize that we equie not only ε it be contemoaneously median uncoelated with x it t = 12 but also that the vecto ε i be jointly median uncoelated with the vecto x i of exlanatoy vaiables in both eiods On the othe hand it is ossible to elax this joint median uncoelation condition in the anel setu to equiing that the andom vaiable ε i be median uncoelated with x i 7 Conclusion The ae consides an analogue of the 2SLS estimato which is commonly used in econometics fo estimating egessions with endogenous vaiables The 2SLS estimato is based on the assumtion that even though a egesso is coelated with the eo thee exists an excluded exogenous egesso that is (linealy) uncoelated with the eo This egesso is called an instument And so 2SLS exloits imlications of this (linea) uncoelation between the instument and the eo in the main egession to obtain a consistent estimato fo the sloe This ae ties to follow the same model but uses median uncoelation instead This median uncoelation is new to ou knowledge and is exactly simila to mean uncoelation excet that it uses the absolute loss function as oosed to the squaed loss function used with the mean We chaacteize oeties of two vectos that ae linealy median uncoelated and then ovide a measue of median uncoelation which is bounded between -1 and 1 This is meant to mio the tyical coelation coefficient in linea models We also ovide counteats to R 2 the coefficient of detemination Most imotantly we show that in a linea egession model whee the egessos ae coelated with the eos a median uncoelation assumtion between a set of instuments and the eo ovides the basis fo infeence on the linea sloe aamete that is akin to what the 2SLS aoach does unde mean uncoelation We aly this uncoelation concet to othe examles like linea models with measuement eo and quantile estictions and anel data quantile models 8 Aendix Poof of Poosition 21 1 Fist suose that R k satisfies inequality (21) fo any α R R k Denote m (S) = Med(T S )+S Choose any a R b R k and denote m(s) = a +S b Then ET m (S) ET m(s) =E[(T m (S))sgn(T m (S))] ET m(s) =E[(T m(s))sgn(t m (S))]+E[(m(S) m (S))sgn (T m (S))] ET m(s) E[(T m(s))sgn(t m (S)) 1(T m (S) 0)] + E[m(S) m (S) 1(T m (S)=0)] ET m(s) (81) E[T m(s) 1(T m (S) 0)] + E[m(S) T 1(T m (S) = 0)] ET m(s) ET m(s) ET m(s) = 0 whee the fist tem in (81) is obtained using inequality (21) Thus M(T S) Now suose that M(T S) Then fo any R ET m (S)+m(S) ET m (S) 0 and theefoe ET - m( S) + ms -ET -m( S) lim inf 0 0 ET m( S) + ms ET m( S) Note that = ( = 1 E m S 1 T m S 0) + E[( T m ( S) + m( S) E T m ( S)) 1( T m ( S) 0)] When T m(s) 0

11 12 Komaova et al: Quantile Uncoelation and Instumental Regessions T m ( S) + m( S) T m ( S) 2 2( T m ( S)) m( S) + m ( S) = + + T m ( S) m( S) T m ( S) and T m ( S) + m( S) T m ( S) lim 0 2( T m ( S)) m( S) = = sgn( T m ( S)) m( S) 2 T m ( S) Taking into account that T m ( S) m( S) T m ( S) m( S) and alying Lebesgue s dominated convegence theoem we obtain ET m S ms ET m S lim inf 0 ET m S+ ms ET m S = lim 0 + = E m S T- m S= [ 1( 0)] + Esgn( T-m ( S)) m( S) 1( T-m ( S) 0)] Then E[sgn(T m (S))m(S) 1(T m (S) 0)] E[m(S) 1 (T m (S)=0)] If the same technique is alied to ET m (S) m(s) ET m (S) then E[sgn(T m (S))m(S) 1(T m (S) 0)] E[m(S) 1 (T m (S)=0)] Theefoe E[sgn(T m (S))m(S) 1(T m (S) 0)] E[m(S) 1 (T m (S)=0)] which concludes the oof of at 1 2 Use the esult of at 1 of this oosition Unde given conditions fo any α R E[α sgn(t S Med(T S ))]=0 and the ight-hand side in (21) is 0 This gives E[S sgn(t S Med(T S ))]=0 fo any R k Choosing = (10 0) we obtain that E[S 1 sgn(t S Med(T S ))]=0 In a simila way we can show that fo any i = 1 k E[S i sgn(t S Med(T S ))]=0 which means that E[S sgn(t S Med(T S ))]=0 3 This esult is obvious fom at 2 of this oosition Poof of Lemma 21 We ove this lemma in two stes In the fist ste we show that M(T S) + b M(T + a + S b S) In the second ste we establish that M(T + a + S b S) M(T S) + b Fist of all note that fo a given b and any a M( T+ a+ S b S) = agmin E T+ S'( b-q) - Med( T+ S ( b-q)) k q R Let m 1 M(T S) This imlies that fo any q R k ET + S (b q) Med(T + S (b q)) ET S m 1 Med(T S m 1 ) Obviously the inequality becomes the equality if q = m 1 + b Theefoe m 1 + b M(T + a + S b S) Now let m 2 M(T + a + S b S) This imlies that fo any R k ET S Med(T S ) ET+ S (b m 2 ) Med(T+ S (b m 2 )) The inequality becomes the equality if =m 2 b Theefoe m 2 b M(T S) and hence m 2 M(T S) + b Poof of Theoem 31 (A): Suose Med (c Ws) c Then we know that c minimizes the following oblem ove all (measuable) functions g(s): Ec W c Ec W g(s) In aticula this holds fo any linea function of S a + S with 0 (B): Conside indeendent andom vaiables S and Z such 7 that P(S = 1) = 16 P(S = 1) = and P(Z = 1) = 6 P(Z = 0) = 2 P(Z = 1) = 1 3 Define andom vaiable W as W = SZ Since Med(WS = 1) = Med(WS = -1) = 0 then fom at (A) we conclude that W is median uncoelated with S Let us now analyze whethe S is median uncoelated with W Conside the otimization oblem

12 Komaova et al: Quantile Uncoelation and Instumental Regessions 13 min E S-W-Med( S-W) Since Med(S) = -1 then the value of the objective function when = 0 is ES+1 = 7 8 Let us find the value of this objective function when = -1 Since Med(S + W) = Med(S + SZ) = 0 then ES + W-Med(S + W) = ES + SZ = E1 + Z = 5 6 which is smalle than 7 8 Thus = 0 cannot be a solution to the otimization oblem This imlies that S is not median uncoelated with W (C): This means that the conditional chaacteistic function of c W given S is eal which in at means that the conditional distibution of c W given S is symmetic aound 0 Hence Med(c Ws) = 0 = Med(c W) fo all s (D): Let δ = T a 0 S M(T S) whee a 0 is any constant Showing that M(δ S) is equal to 0 is a diect esult of the invaiance oety in (24) (E): Since by assumtion P(T Med(T) = 0)= 0 Poosition 21 and conditions M(T S) = 0 and M(T Z) = 0 imly that E[S sgn(t Med(T))] = 0 E[Z sgn(t Med(T))]=0 Then E[(S + Z) sgn(t Med(T))] = E[S sgn(t Med(T))] + E[Z sgn(t Med(T))] = 0 that is 0 M(T S + Z) Since M(T S + Z) is assumed to be a singleton M(T S + Z) = 0 (F): The fist at of the statement follows fom (A) Fo the second at of the statement note that Poosition 21 imlies E[S sgn(t Med(T))] = 0 Given that the conditional median of TS = 1 is unique we have: E[S sgn(t Med(T))] = 0 E[sgn(T Med(T))S = 1] = 0 Med(T) = Med(TS = 1) Because E[sgn(T Med(T))] = 0 E[sgn(T Med(T))S = 1] = 0 E[sgn(T Med(T))S= 0] = 0 Taking into account that that the conditional median of TS = 0 is unique we obtain that Med(T) = Med(TS = 0) Poof of Theoem 51 (1): This follows fom medco( T S) = E[ S sgn( T Med( T ))sgn( S )] ES=1 (2): Fist let us ove that M(T S) = 0 medco(t S) = 0 Taking into account the conditions of this theoem and alying Poosition 21 obtain that M(T S) = 0 E [S sgn(t Med(T))] = 0 S-Med( S) E sgn( T - Med( T )) = 0 E S-Med( S) E S sgn( T - Med( T )) = 0 medco(t S)=0 Note that medco(t S) = medco(t Med(T) S ) and M( T -Med( T ) S ) M( T S) = E S-Med( S) and hence sgn(m(t S)) = sgn(m(t Med(T) S )) Thus it is enough to show that sgn(medco(t Med(T) S )) = sgn(m(t Med(T) S )) Denote b = M(T Med(T) S ) Fo b =0 the esult is aleady oven Suose b 0 Notice that sgn(medco(t Med(T) S ))=sgn(b )sgn(e[b S sgn(t Med(T))]) and theefoe the esult will be oven if we establish that E[b S sgn(t Med(T))] >0 Denote a = Med(T Med(T) b S ) = Med(T b S ) Med(T) Accoding to Poosition 21 b satisfies E S T Med T b S a sgn( ) = 0 Then Eb S sgn(t Med(T))]=E[(b S +a )sgn(t Med(T))] =E[(b S +a )(sgn(t Med(T)) sgn(t Med(T) b S ~ a ))] =2E[(b S +a )1(T Med(T)>0)1(T Med(T) b S a <0)] -2E[(b S +a )1(T Med(T)<0)1(T Med(T) b S a >0)] Notice that both tems in the last sum ae non-negative Moeove at least one of them is stictly ositive because P(sgn(T Med(T))sgn(T Med(T) b S a )= 1)>0 o equivalently P(sgn(T Med(T))sgn(T b S Med(T b S ))= 1)>0

13 14 Komaova et al: Quantile Uncoelation and Instumental Regessions This follows fom the assumtions of the theoem and at 2 of Poosition 21 accoding to which b b = E S - Med ( S ) solves the equation E[S sgn(t bs Med(T bs))]=0 while b = 0 does not Thus E[b S sgn(t Med(T))]>0 Poof of Theoem 52 (1): If M(T S) = 0 then E T -Med( T ) medsq( T S) = 1- = 0 E T -Med( T ) If medsq(t S) = 0 then min E T -S-Med( T - S) = E T -Med( T ) so that clealy 0 M(TS) (2): If medsq(t S) = 0 then fom at (1) 0 M(T S) Fom Poosition 21 it follows that E[S sgn(t Med(T))] = 0 and hence medco(t S) = 0 If medco(t S) = 0 then E[S sgn(t Med(T))] = 0 so that by Poosition 21 0 M(T S) It follows that min E T -S-Med( T - S) = E T -Med( T ) Poof of Theoem 61 The oof of this theoem is analogous to the oof of Theoem 41 Let m M(Y X Z)=M(α 0 +Z g ( 0 )+ψ ( 0 )+ε n Z) By the invaiance oety in Lemma 21 thee exists m 0 M(ψ ( 0 )+ε n Z) such that m=g ( 0 )+m 0 Note that ψ ( 0 )=( 0 ) ψ and n = n Hence since (ε n ψ) is median uncoelated with Z m 0 = 0 It follows that m = g ( 0 ) and hence that M(Y X Z)=g ( 0 ) Since d k and g is full column ank by assumtion then M(Y X Z)=0 = 0 Refeences Amemiya T (1981): Two Stage Least Absolute Deviations Estimatos Econometica Amemiya T (1985): Advanced Econometics Havad Univesity Pess Basmann R L (1960): On the Asymtotic Distibution of Genealized Linea Estimatos Econometica 28(1) Blackbun M and D Neumak (1992): Unobseved Ability Efficiency Wages and Inteindusty Wage Diffeentials The Quately Jounal of Economics 107(4) Blomqvist N (1950): On a Measue of Deendence Between Two Random Vaiables Ann Math Statistics Chenozhukov V and C Hansen (2005): An IV Model of Quantile Teatment Effects Econometica 73(1) Chenozhukov V and C Hansen (2006): Instumental Quantile Regession Infeence fo Stuctual and Teatment Effect Models Jounal of Econometics 132(2) Cheshe A (2003): Identification in Nonseaable Models Econometica 71(5) Giliches Z (1976): Wages of Vey Young Men The Jounal of Political Economy 84(4) S69 S86 Honoé B and L Hu (2004): On the Pefomance of Some Robust Instumental Vaiables Estimatos Jounal of Business and Economic Statistics 22(1) Koenke R (2005): Quantile Regession vol 38 of Econometic Society Monogahs Cambidge Univesity Pess Cambidge Koenke R and G Bassett (1978): Regession Quantiles Econometica Koenke R and J A F Machado (1999): Goodness of Fit and Related Infeence Pocesses fo Quantile Regession Jounal of the Ameican Statistical Association 94(448) Lee S (2007): Endogeneity in Quantile Regession Models: A Contol Function Aoach Jounal of Econometics 141(2) Ma L and R Koenke (2006): Quantile Regession Methods fo Recusive Stuctual Equation Models Jounal of Econometics 134(2) Manski C (1988): Analog Estimation Methods in Econometics Chaman and Hall Powell J (1983): The Asymtotic Nomality of Two Stage Least Absolute Deviations Estimatos Econometica Sakata S (2001): Instumental Vaiable Estimation Based on Mean Absolute Deviation Estimato Univesity of Michigan Woking Pae Sakata S (2007): Instumental Vaiable Estimation Based on Conditional Median Restiction Jounal of Econometics 141(2) Theil H (1953): Estimation and Simultaneous Coelation in Comlete Equation Systems The Hague: Centaal Planbueau