Ecoomics Letters 95 (2007) 272 277 www.elsevier.com/locate/ecobase Iferece approaches for istrumetal variable quatile regressio Victor Cherozhukov a, Christia Hase b,, Michael Jasso c a Massachusetts Istitute of Techology, Uited States b Uiversity of Chicago, Graduate School of Busiess, 5807 S. Woodlaw Ave., Chicago, IL 60637, Uited States c Uiversity of Califoria-Berkeley, Uited States Received 7 February 2006; received i revised form 13 September 2006; accepted 25 October 2006 Abstract We cosider asymptotic ad fiite sample cofidece bouds i istrumetal variables quatile regressios of wages o schoolig with relatively weak istrumets. We fid practically importat differeces betwee the asymptotic ad fiite sample iterval estimates. 2006 Elsevier B.V. All rights reserved. Keywords: Quatile regressio; Istrumetal variables; Schoolig JEL classificatio: C10; J31 1. Itroductio I this ote, we outlie three approaches to obtaiig iferece statemets for quatile regressio (QR) models with edogeeity: a asymptotic approximatio provided i Cherozhukov ad Hase (2006) that exteds the results of Koeker ad Bassett (1978) for the QR model with all exogeous variables to a model with edogeeity whe istrumets are available, a alterate approach to iferece preseted i Cherozhukov ad Hase (2006) that will be asymptotically valid uder weak idetificatio, ad a approach to obtaiig fiite sample cofidece regios for parameters of a model defied by quatile restrictios developed i Cherozhukov, Hase, ad Jasso (2005). We compare Correspodig author. Tel.: +1 773 834 1702. E-mail address: chase1@chicagogsb.edu (C. Hase). 0165-1765/$ - see frot matter 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.ecolet.2006.10.016
V. Cherozhukov et al. / Ecoomics Letters 95 (2007) 272 277 273 the three approaches i a example due to Card (1995) whichusescollegeproximityasaistrumet for years of educatio i a quatile regressio of wages o educatio. We fid substatial differeces betwee the three approaches that appear to be due to the weakess of the istrumets. The fidigs suggest cautio should be used whe relyig o asymptotic approximatios i QR models with edogeeity. 2. Iferece approaches We cosider a radom coefficiet model with structural equatio give by Y ¼ DVaðUÞþXVbðUÞ; where D ad U may be statistically depedet, ð2:1þ A1. D α(u )+X β(u ) is strictly icreasig i U for almost every D ad X, A2. U U(0, 1), ad A3. U is idepedet of X ad Z variables that do ot eter the structural equatio. That is, D is a vector of potetially edogeous variables with radom coefficiets α(u ); X is a vector of exogeous variables that eter the structural equatio with radom coefficiets β(u ), ad Z is a vector of exogeous variables that are excluded from the structural equatio where it is assumed that dim (Z ) dim(d). This model icorporates the covetioal liear QR model of Koeker ad Bassett (1978) where α(u ) 0 ad Z=X. Uder coditios A1. A3., the problem of depedece betwee U ad D is overcome through the presece of istrumets, Z, that affect D but are idepedet of U. From Eq. (2.1) ad the mootoicity assumed i A1., the evet {Y D α(τ)+x β(τ)} is equivalet to the evet {U τ}. It the follows uder A2. ad A3. that P½Y V DVaðsÞþX VbðsÞjZ; X Š¼s: ð2:2þ Eq. (2.2) provides a momet restrictio that ca be used to estimate the structural parameters α(τ) ad β (τ). For example, oe may use this set of momet restrictios to form a GMM estimator; see Pakes ad Pollard (1989). Momet coditios (2.2) also suggest a differet procedure for estimatig α(τ) adβ(τ). For a give value of the structural parameter, say α, ru the ordiary QR of Y D α o X ad Z to obtai (βˆ(α,τ), γˆ (α,τ)) where γˆ (α,τ) are the estimated coefficiets o the istrumets Z. The ote that Eq. (2.2) implies that 0 is the τth coditioal quatile of Y D α(τ) X β (τ) givez ad X, so oe may estimate α(τ) by fidig a value for α that makes the coefficiet o the istrumetal variable γˆ (α,τ) as close to 0 as possible. That is, âðsþ ¼arg if ½ĝða; sþvš AðaÞ½ ˆ ĝða; sþš; aaa ð2:3þ where A is the parameter space for α, Â(α)=A(α)+o p (1) ad A(α) is positive defiite uiformly i p aaa. ffiffi It is coveiet to set A(α) equal to the iverse of the asymptotic co-variace matrix of ðĝða; sþ gða; sþþ i which case W (α) is the Wald statistic for testig γ(α,τ)=0. The parameter
274 V. Cherozhukov et al. / Ecoomics Letters 95 (2007) 272 277 estimates are the give by (αˆ (τ), βˆ (τ))=(αˆ (τ), βˆ (αˆ (τ),τ)). Cherozhukov ad Hase (2006) provide additioal details, verify that the above procedure cosistetly estimates α(τ) ad β(τ), ad fid the limitig distributio from which covetioal asymptotic iferece immediately follows. The above procedure may be modified to obtai cofidece itervals that will be asymptotically valid whe there is oly a weak relatioship betwee D ad Z. The idea is to base iferece o the Wald statistic W (α) for testig whether the coefficiets o the istrumets are zero (i.e. whether γ(α,τ)=0). The ituitio for this approach is that at the true α(τ) the structural Eq. (2.1) implies that γ(α,τ)=0. Thus, fixig a value for α(τ), γ(α,τ) should be ear zero if this value is cosistet with the structural equatio. More formally, whe a ¼ aðsþ; W ðaþ Y d v 2 ðdimðgþþ. Thus, a valid cofidece regio for α(τ) ca be based o the iversio of this Wald statistic: CR p [α(τ)]:={α:w (α)bc p } cotais α(τ) with probability approachig p, where c p is the p-percetile of a χ 2 (dim(γ)) distributio. Cherozhukov ad Hase (2006) show that this procedure provides a valid cofidece regio for α(τ) without puttig restrictios o the depedece betwee D ad Z. The fial approach to iferece we cosider is based o the observatio that P[Y D α(τ)+x β(τ) Z,X ]=τ implies that the evet {Y D α(τ)+x β(τ)} is distributed exactly as a Beroulli(τ) coditioal o X ad Z regardless of the sample size. Thus, ay test statistic that depeds oly o this evet, X, adz will have a distributio that does ot deped o ay ukow parameters i fiite samples ad so ca be used to costruct valid fiite sample iferece statemets. This basic approach exteds work o fiite sample iferece for ucoditioal quatiles; see Walsh (1960) ad MacKio (1964). Cherozhukov et al. (2005) make use of the aforemetioed distributioal fact ad cosider iferece based o the GMM objective fuctio: L ða; bþ ¼ 1 2 1 p X ffiffiffi VW 1 m i ða; bþ! p X ffiffi! m i ða; bþ ; ð2:4þ where m i (α,β)=[τ 1(Y i D i α X i β)] (Z i,x i ). I this expressio, W is a positive defiite weight matrix, which is fixed coditioal o (X 1,Z 1 ),, (X,Z ). W ¼ 1 sð1 sþ 1 X ðz i V; X i VÞVðZ i V; X i VÞ! 1 ; which equals the iverse of the variace of 1=2 P m iðaðsþ; bðsþþ coditioal o (X 1,Z 1 ),, (X,Z ), is a atural choice of W. Sice statistic L (α,β) defied by Eq. (2.4) depeds oly o X, Z, ad {Y D α+x β} it is coditioally pivotal i fiite samples whe evaluated at (α,β)=(α(τ), β(τ) ) ; that is, its distributio at the true parameter values does ot deped o ay ukow parameters ad so exact fiite sample critical values for this statistic may be obtaied. Let c ( p) deotethepth quatile of the distributio of the statistic give i Eq. (2.4), ad ote that c ( p) may be obtaied easily through simulatio by, for j =1,, J, 1.drawig(B i, j, 1 p X V ffiffiffi m 1 i; j W ffiffi p X where mi, j = i ) as iid Beroulli(τ) radom variables, 2. computig L ;j ¼ 1 2 m i; j [τ B i, j ](Z, i X ), i ad 3. obtaiig c (α) astheα-quatile of the sample (L, j, j =1,, J ) for a large umber J.
V. Cherozhukov et al. / Ecoomics Letters 95 (2007) 272 277 275 Table 1 Schoolig coefficiet Estimatio method τ=0.25 τ =0.50 τ=0.75 A. Quatile regressio (o istrumets) α(τ) 0.074 0.074 0.079 Quatile regressio (asymptotic) (0.064,0.083) (0.065,0.083) (0.071,0.087) Fiite sample (0.047,0.100) (0.050,0.101) (0.057,0.098) B. IV Quatile regressio (proximity to 2 ad 4 year college as istrumets) α(τ) 0.175 0.033 0.103 Iverse quatile regressio (asymptotic) (0.081,0.269) ( 0.077,0.142) (0.025,0.180) Weak istrumet (asymptotic) (0.018,0.500] ( 0.053,0.500] (0.068,0.483) Fiite sample [ 0.100,0.500] [ 0.100,0.500] [ 0.100,0.500] Note: 95% level cofidece iterval estimates for the schoolig coefficiet i a simple wage model. Schoolig is treated as exogeous i Pael A ad as edogeous i Pael B. The first row i each pael reports the poit estimate, ad the secod row reports the iterval estimate obtaied usig the asymptotic approximatio. I Pael A, the third row reports a iterval estimate obtaied usig the fiite sample approach. I Pael B, the third row reports a weak-istrumet robust iterval, ad the fourth row reports the fiite sample iterval. Give c (p), a valid p-level cofidece regio for (α(τ), β(τ) ) may the be obtaied as the set CR( p) {(α, β ) : L (α, β ) c ( p)}. CR( p) defies a joit cofidece set for all of the model's parameters from which oe ca defie a cofidece boud for a real-valued fuctioal ψ(α(τ), β(τ), τ) as CR(p,ψ)={ψ(α, β,τ), (α,β ) CR(p)}. It is worth otig that the joit cofidece regio CR( p) is ot coservative, but cofidece bouds for fuctioals may be. Also, formig cofidece regios by ivertig a test-statistic will geerally be computatioally demadig. Cherozhukov et al. (2005) presets various feasible algorithms for computig cofidece regios ad bouds i this cotext. 3. Case study: Card's (1995) schoolig example We illustrate the differet approaches to iferece by estimatig a wage model that allows for heterogeeity i the effect of schoolig o wages as well as other heterogeeity i the effects of other cotrol variables. Because we are cocered that the level of schoolig ad earigs may be joitly determied, we istrumet for schoolig. We employ the same data ad idetificatio strategy as Card (1995). Specifically, we suppose that the log of the wage is determied by the followig liear quatile model: Y ¼ aðuþs þ X VbðUÞ where Y is the log of the hourly wage; S is years of completed schoolig; X cosists of a costat ad 14 demographic cotrols; ad U is a uobservable ormalized to follow a uiform distributio over (0,1). We might thik of U as idexig uobserved ability, i which case α(τ) may be thought of as the retur to schoolig for a idividual with uobserved ability τ. Sice we believe that years of schoolig may be joitly determied with uobserved ability, we use a idicator for residece ear a two year college i 1966 ad a idicator for residece ear a four year college i 1966 as istrumets for schoolig. Further
276 V. Cherozhukov et al. / Ecoomics Letters 95 (2007) 272 277 details about the data sources, descriptive statistics, ad argumets for the validity of the istrumets are i Card (1995). We summarize results for the schoolig coefficiet i Table 1. For quatile regressio, we report 95%- level cofidece itervals computed via the asymptotic approximatio ad fiite sample approach. For istrumetal variable quatile regressio, we report 95%-level cofidece itervals computed usig the asymptotic approximatio, the weak-istrumet robust approach, ad the fiite sample approach. Both the fiite sample ad weak-istrumet approach require that we ivert a test-statistic for a rage of potetial values for α(τ); i all cases, we cosider values i the iterval [.1,.5] which we feel covers essetially all plausible values for the retur to a additioal year of schoolig. The results have a umber of iterestig features. Lookig first at the results i Pael A of Table 1 which treat schoolig as exogeous, we see that the poit estimates across the various quatiles are quite close, suggestig little heterogeeity i how the quatiles of wages vary with schoolig. We also see that the fiite sample ad asymptotic itervals are qualitatively similar, though the fiite sample itervals are wider tha the asymptotic itervals. I the estimates which treat schoolig as edogeous, the differeces betwee the iferece approaches are strikig. The poit estimates suggest cosiderable heterogeeity i the returs to schoolig, though eve the usual asymptotic itervals are wide eough to make fidig statistical evidece for this quite ulikely. Cosiderig ext the weak istrumet robust itervals, we see that they are substatially wider tha the usual asymptotic itervals i all cases. For each quatile we cosider, the weak istrumet robust itervals iclude essetially the etire parameter space. The weak istrumet robust itervals do exclude 0 i the two cases where the usual itervals do. While the weak istrumet robust itervals rely o less striget assumptios tha do the usual asymptotic itervals, they still rely o a asymptotic argumet. Lookig last at the fiite sample itervals which are valid uder miimal assumptios ad do ot require ay asymptotic argumet, we see that they cover the etire parameter space i every case. The likely cause for the large discrepacies is the weakess of the relatioship betwee the istrumets ad schoolig. If oe regresses educatio o the cotrol variables ad the two proximity istrumets, the F-statistic o the excluded istrumets is 8.32 which is i the rage that may would cosider weak i the usual liear IV model. The weakess of the istrumets clearly explais the differece betwee the usual asymptotic ad weak istrumet asymptotic results. The differeces betwee the weak istrumet ad fiite sample itervals is the likely drive by the fact that with a large umber of covariates ad weak istrumets the effective sample size is quite small. It would be iterestig to explore these differeces ad develop formal procedures for judgig differeces betwee asymptotic ad fiite sample itervals. Overall, the differece betwee the asymptotic ad fiite sample iferece statemets calls the validity of the asymptotics ito questio i this example. Refereces Card, D., 1995. Usig geographic variatio i college proximity to estimate the retur to schoolig. I: Christofides, L., Grat, E.K., Swidisky, R. (Eds.), Aspects of Labour Ecoomics: Essays i Hoour of Joh Vaderkamp. Uiversity of Toroto Press. Cherozhukov, V., Hase, C., 2006. Istrumetal quatile regressio iferece for structural ad treatmet effect models. J. Ecoom. 132 (2), 491 525. Cherozhukov, V., Hase, C., Jasso, M., 2005. Fiite-Sample Iferece i Ecoo-metric Models via Quatile Restrictios, mimeo.
V. Cherozhukov et al. / Ecoomics Letters 95 (2007) 272 277 277 Koeker, R., Bassett, G.S., 1978. Regressio quatiles. Ecoometrica 46, 33 50. MacKio, W.J., 1964. Table for both the sig test ad distributio-free cofidece itervals of the media for sample sizes to 1000. J. Am. Stat. Assoc. 59, 935 956. Pakes, A., Pollard, D., 1989. Simulatio ad asymptotics of optimizatio estimators. Ecoometrica 57, 1027 1057. Walsh, J.E., 1960. Noparametric tests for media by iterpolatio from sig tests. A. Ist. Stat. Math. 11, 183 188.