Local Linear GMM Estimation of Functional Coefficient IV Models with an Application to Estimating the Rate of Return to Schooling

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1 Loal Liear GMM Estimatio of Futioal Coeffiiet IV Models with a Appliatio to Estimatig the Rate of Retur to Shoolig Liagju Su, a Iria Murtazashvili, b Ama Ullah a Shool of Eoomis, Sigapore Maagemet Uiversity ljsu@smu.edu.sg b Departmet of Eoomis, Drexel Uiversity im99@drexel.edu Departmet of Eoomis, Uiversity of Califoria, Riverside ama.ullah@ur.edu November 17, 2012 Abstrat We osider the loal liear GMM estimatio of futioal oeffiiet models with a mix of disrete ad otiuous data ad i the presee of edogeous regressors. We establish the asymptoti ormality of the estimator ad derive the optimal istrumetal variable that miimizes the asymptoti variae-ovariae matrix amog the lass of all loal liear GMM estimators. Data-depedet badwidth sequees are also allowed for. We propose a oparametri test for the ostay of the futioal oeffiiets, study its asymptoti properties uder the ull hypothesis as well as a sequee of loal alteratives ad global alteratives, ad propose a bootstrap versio for it. Simulatios are oduted to evaluate both the estimator ad test. Appliatios to the 1985 Australia Logitudial Survey data idiate a lear rejetio of the ull hypothesis of the ostat rate of retur to eduatio, ad that the returs to eduatio obtaied i earlier studies ted to be overestimated for all the work experiee. JEL Classifiatios: C12, C13, C14 Key Words: Disrete variables; Edogeeity; Heterogeeity; Futioal oeffiiet; Loal liear GMM estimatio; Optimal istrumetal variable; Shoolig. 1 Itrodutio I the lassial eoometris literature, a eoometri model is ofte studied i a liear parametri regressio form with its oeffiiets derivatives or margial hages assumed to be ostat over time or aross ross setio uits. I pratie this may ot be true, e.g., it may be hard to believe that the margial propesity to save or to osume would be the same for a youger as for a older group of idividuals i a give ross setio data set, or that the elastiity of wages with respet to shoolig or the rate of retur to shoolig would be the same for idividuals with less experiee ompared to those with more experiee. I the ase of oliear parametri regressio models, the oeffiiets have bee take as ostat but derivatives do vary depedig o the speifiatio of models, e.g., the traslog produtio 1

2 futio has ostat oeffiiets, ad the elastiities derivatives based o this futio vary liearly with iputs. Realizig the fat that some or all of the oeffiiets i a regressio may be varyig, the traditioal eoometris literature has tried to osider various forms of parametri speifiatios of the varyig oeffiiets. See, e.g., the papers of Hildreth ad Houk 1968, Swamy 1970, Sigh ad Ullah 1974, ad Grager ad Teräsvirta 1999, ad the books by Swamy 1971, Raj ad Ullah 1981, ad Grager ad Teräsvirta However, it is ow well kow that the ostat or parametri varyig oeffiiet models may ofte be misspeified, ad therefore this may lead to iosistet estimatio ad testig proedures ad hee misleadig empirial aalysis ad poliy evaluatios. I view of the above issues, i reet years, the oparametri varyig/futioal oeffiiet models have bee osidered by various authors, iludig Clevelad, Grosse, ad Shyu 1992, Che ad Tsay 1993, Hastie ad Tibshirai 1993, Fa ad Zhag 1999, ad Cai, Fa, ad Yao 2000, amog others. The oeffiiets i these models are modeled as ukow futios of the observed variables whih a be estimated oparametrially. A additioal advatage of the futioal oeffiiet model is that it also osiders the ukow futioal form of the iteratig variables whih i empirial parametri models is ofte misspeified to be liear. Most of the above works o futioal oeffiiet models are foused o models with exogeous regressors. Reetly Das 2005, Cai, Das, Xiog, ad Wu 2006, CDXW hereafter, Lewbel 2007, Cai ad Li 2008, Tra ad Tsioas 2010, ad Su 2012, amog others, have osidered the semiparametri models with edogeous variables ad they suggest a oparametri/semiparametri geeralized method of momets GMM istrumetal variable IV approah to estimate them. I partiular, CDXW 2006, Cai ad Li 2008, ad Tra ad Tsioas 2010 fous o futioal oeffiiet models with edogeous regressors. CDXW 2006 propose a two-stage loal liear estimatio proedure to estimate the futioal oeffiiet models, whih ufortuately requires oe to first estimate a high-dimesio oparametri model ad the to estimate the futioal oeffiiets usig the first-stage oparametri estimates as geerated regressors. I otrast, Cai ad Li 2008 suggest a oe-step loal liear GMM estimator whih orrespods to our loal liear GMM estimator with a idetity weight matrix. Tra ad Tsioas 2010 provide a loal ostat two-step GMM estimator with a speified weightig matrix that a be hose to miimize the asymptoti variaes i the lass of GMM estimators. However, the loal ostat estimatio proedure, as is ow well kow, is less desirable tha the loal liear estimatio proedure, espeially at the boudaries. I additio, all of these papers osider varyig oeffiiets with otiuous variables oly. O the other had, Su, Che, ad Ullah 2009, SCU hereafter osider both otiuous ad ategorial variables i futioal oeffiiets ad show that the osideratio for the ategorial variables is extremely importat for empirial aalysis, ad it improves o the speifiatios of the traditioal liear parametri dummy-variable models. But they do ot osider the edogeeity issue whih prevails i eoomis. I additio, i the estimatio otext, the advatage of usig the traditioal ostat oeffiiet models rests o their validity. Nevertheless, to the best of our kowledge, there is o oparametri hypothesis testig proedure available for this whe edogeeity is preset, although there are some tests e.g., Fa, Zhag, ad Zhag 2001 ad Hog ad Lee 2009 i the absee of edogeeity. I view of the above defiieies i the existig literature we first fous o further improvemet i the estimatio area, ad the provide a osistet test for the ostay of futioal oeffiiets. If we fail to rejet the ull of ostay, the we a otiue to rely o the traditioal ostat oeffiiet models. Otherwise we may have to osider the futioal oeffiiets with ukow form. I this paper, we develop loal liear GMM estimatio of futioal oeffiiet IV models with a 2

3 geeral weight matrix. A varyig oeffiiet model is osidered i whih some or all the regressors are edogeous ad their oeffiiets are varyig with respet to exogeous otiuous ad ategorial variables. For give IVs a optimal loal liear GMM estimator is proposed where the weight matrix is obtaied by miimizig the asymptoti variae-ovariae matrix AVC of the GMM estimator. We also osider the hoie of optimal IVs to miimize the AVC amog the lass of all loal liear GMM estimators ad establish the asymptoti ormality of the loal liear GMM estimator for a datadepedet badwidth sequee. The we develop a ew test statisti for testig the hypothesis that a subvetor of the futioal oeffiiets is ostat. It is argued that the test based o the Lagragia multiplier LM priiple eeds restrited estimatio ad may suffer from the urse of dimesioality, ad similarly the test usig the likelihood ratio LR method also requires both urestrited ad ivolved restrited estimatio. For these reasos a simpler Wald type of test is proposed whih is based o the urestrited estimatio. The osistey, asymptoti ull distributio, ad asymptoti loal power of the proposed test are established. It is well kow that oparametri tests based o the ritial values of their asymptoti ormal distributios may perform poorly i fiite samples. I view of this, we also provide a bootstrap proedure to approximate the asymptoti ull distributio of our test statisti ad justify its asymptoti validity. To assess the fiite sample properties of the proposed loal liear GMM estimator ad the ew test statisti, we odut a small set of simulatios. The results show that our loal liear GMM estimator performs well i ompariso with some existig estimators i the literature ad our test has orret size ad good power properties i fiite samples. Aother importat objetive of this paper is to employ our proposed oparametri GMM estimator to study the empirial relatioship betwee earigs ad shoolig usig the 1985 Australia Logitudial Survey. Labor eoomists have log studied two major problems arisig whe estimatig the wage equatio: edogeeity of eduatio ad heterogeeity of returs to eduatio, see Card 2001 for detailed stimulatig disussios. Our oparametri estimator is able to deal with both problems i a flexible way. Speifially, i otrast to other existig estimators, our estimator allows the returs to eduatio to deped o both otiuous experiee ad disrete marital status, uio membership, et. harateristis of idividuals while otrollig for edogeeity of eduatio. Further, we use our proposed ew oparametri test to hek for ostay of futioal oeffiiets i the wage equatio. Our fidigs are uambiguous: the returs to eduatio do deped o both experiee ad the ategorial variables we use, i a o-liear maer. Additioally, we fid that the returs to eduatio ted to be overestimated for all of the observed work experiee whe the ategorial explaatory variables are ot aouted for i futioal oeffiiets as i CDXW 2006 ad Cai ad Xiog These results are also importat sie our proposed tests show the absee of the ostay of the retur to eduatio, whih is ofte assumed i most of the parametri empirial studies i labor eoomis. The paper is strutured as follows. I Setio 2 we itrodue our futioal oeffiiet IV model ad propose a loal liear GMM proedure to estimate the futioal oeffiiets ad their first order derivatives. The asymptoti properties of these estimators are studied i Setio 3. We propose a speifiatio test for our model i Setio 4. We odut a small set of Mote Carlo studies to hek the relative performae of the proposed estimator ad test i Setio 5. Setio 6 provides empirial data aalysis. Fial remarks are otaied i Setio 7. All tehial details are relegated to the Appedix. For atural umbers a ad b, we use I a to deote a a a idetity matrix, ad 0 a b a a b matrix of zeros. Let ad deote the Kroeker ad Hadamard produts, respetively. If ad d are vetors of the same dimesio, /d deotes the vetor of elemetwise divisios. For a matrix M, M meas the traspose of M, ad M = tr MM. We use 1 { } to deote the usual idiator futio whih takes 3

4 value 1 if the oditio iside the urly braket holds ad 0 otherwise, ad C to sigify a geeri ostat whose exat value may vary from ase to ase. We use d ad P to deote overgee i distributio ad probability, respetively. 2 Futioal Coeffiiet Estimatio with Mixed Data ad Estimated Covariate I this setio we first itrodue a futioal oeffiiet IV model where the oeffiiet futio may deped o both otiuous ad disrete exogeous regressors ad the edogeous regressors eter the model liearly. The we propose loal liear GMM estimates for the futioal oeffiiets. 2.1 Futioal oeffiiet represetatio We osider the followig futioal oeffiiet IV model Y i = g U i, U d i Xi + ε i = d g j U i, U d i Xi,j + ε i, E ε i Z i, U i = 0 a.s., 2.1 j=1 where Y i is a salar radom variable, g = g 1,, g d, {g j } d j=1 are the ukow strutural futios of iterest, X i,1 = 1, X i = X i,1,, X i,d is a d 1 vetor osistig of d 1 edogeous regressors, U i = U i, Ud i, U i ad Ud i deote a p 1 vetor of otiuous exogeous regressors ad a p d 1 vetor of disrete exogeous regressors, respetively, Z i is a q z 1 vetor of istrumetal variables, ad a.s. abbreviates almost surely. We assume that a radom sample {Y i, X i, Z i, U i } is observed. I the absee of U d i, 2.1 redues to the model of CDXW If oe of the variables i X i are edogeous, the model beomes that of SCU As the latter authors demostrate through the estimatio of earigs futio, it is importat to allow the variables i the futioal oeffiiets to ilude both otiuous ad disrete variables, where the disrete variables may represet rae, professio, regio, et. 2.2 Loal liear GMM estimatio The orthogoality oditio i 2.1 suggests that we a estimate the ukow futioal oeffiiets via the priiple of oparametri geeralized method of momets NPGMM, whih is similar to the GMM of Hase 1982 for parametri models. Let V i = Z i, U i. It idiates that for ay k 1 vetor futio Q V i, we have d E [Q V i ε i V i ] = E Q V i Y i g j U i, U d i Xi,j V i = Followig Cai ad Li 2008, we propose a estimatio proedure to ombie the orthogoality oditio i 2.2 with the idea of loal liear fittig i the oparametris literature to estimate the ukow futioal oeffiiets. Like Raie ad Li 2004, we use U d i,t to deote the tth ompoet of Ud i. U i,t is similarly defied. Aalogously, we let u d t ad u t deote the tth ompoet of u d ad u, respetively, i.e., u d = u d 1,, u d p d ad u = u 1,, u p. We assume that U d i,t a take t 2 differet values, i.e., j=1 4

5 Ui,t d {0, 1,, t 1} for t = 1,, p d. Let u = u, u d R p R p d. To defie the kerel weight futio, we fous o the ase for whih there is o atural orderig i U d i. Defie l { Ui,t, d u d 1 if U d t, λ t = i,t = ud t, λ t if U d i,t 2.3 ud t, where λ t is a badwidth that lies o the iterval [0, 1]. Clearly, whe λ t = 0, l U d i,t, ud t, 0 beomes a idiator futio, ad λ t = 1, l U d i,t, ud t, 1 beomes a uiform weight futio. We defie the produt kerel for the disrete radom variables by L U d i, u d, λ = L λ U d i u d = p d t=1 l U d i,t, u d t, λ t. 2.4 For the otiuous radom variables, we use w to deote a uivariate kerel futio ad defie the produt kerel futio by W h,iu = W h U i u = Π p t=1 h 1 t w Ui,t t u /ht, where h = h 1,, h p deotes the p -vetor of smoothig parameters. We the defie the kerel weight futio K hλ,iu by K hλ,iu = W h,iu L λ,iu d 2.5 where L λ,iu d = L U d i, ud, λ. To estimate the ukow futioal oeffiiets i model 2.1 via the loal liear regressio tehique, we assume that { g j u, u d, j = 1,, d } are twie otiuously differetiable with respet to u. Deote by g. j u, u d = g j u, u d / u the p 1 vetor of first order derivatives of g j with respet to u. Deote by g.. j u, u d = 2 g j u, u d / u u the p p matrix of seod order derivatives of g j with respet to u. We use g j,ss u, u d to deote the sth diagoal elemet of g.. j u, u d. For ay give u ad U i i a eighborhood of u, it follows from a first order Taylor expasio of g j U i, u d aroud u, u d that d g j U i, u d X i,j j=1 d j=1 [ g j u, u d + g. j u, u d ] U i u X i,j = α u ξ i,u 2.6 where αu = g 1 u,, g d u, ġ 1u,, ġ d u X i ad ξ i,u = X i U i are both d p + 1 u 1 vetors. Motivated by the idea of loal liear fittig, for the global istrumet Q V i we defie its assoiated loal versio as Q h,iu = Q V i Q V i U i u /h. 2.7 Clearly, the dimesio of Q h,iu is k p + 1 as Q V i is a k 1 vetor. I view of the fat that the orthogoality oditio i 2.2 otiues to hold whe we replae Q V i, V i by Q h,iu, U i, we approximate E[Q h,iu {Y i d j=1 g j U i, U d i Xi,j } U i = u] by its sample aalog 1 [ Q h,iu Yi α u ] ξ i,u Khλ,iu = 1 Q h u K hλ u [Y ξ u α] where Y = Y 1,, Y, ξ u = ξ 1,u,, ξ,u, α = α u, K hλ u =diagk hλ,1u,, K hλ,u, ad Q h u =Q h,1u,, Q h,u. To obtai estimates of g j ad g. j, we a hoose α to miimize the followig loal liear GMM riterio futio 1 [ Qh u K hλ u Y ξ u α ] Ψ u 1 [ Q h u K hλ u Y ξ u α ], 2.8 5

6 where Ψ u is a symmetri k p + 1 k p + 1 weight matrix that is positive defiite for large. Clearly, the solutio to the above miimizatio problem is give by α Ψ u; h, λ = [ ] 1 ξ u K hλ u Q h u Ψ u 1 Q h u K hλ u ξ u ξ u K hλ u Q h u Ψ u 1 Q h u K hλ u Y. 2.9 Let e j,d1+p deote the d 1 + p 1 uit vetor with 1 at the jth positio ad 0 elsewhere. Let ẽ j,p,d1+p deote the p d 1 + p seletio matrix suh that ẽ j,p,d1+p α = g. j u. The the loal liear GMM estimator of g j u ad g. j u are respetively give by ĝ j u; h, λ = e j,d1+p α Ψ u; h, λ ad ġj u; h, λ = ẽ j,p,d1+p α Ψ u; h, λ for j = 1,, d We will study the asymptoti properties of α Ψ u; h, λ i the ext setio. Remark 1 Choie of IVs The hoie of Q V i is importat i appliatios. Oe a hoose it from the uio of Z i ad U i e.g., Q V i = V i suh that a ertai idetifiatio oditio is satisfied. A eessary idetifiatio oditio is k d, whih esures that the dimesio of Q h,iu is ot smaller tha the dimesio of α u. Below we will osider the optimal hoie of Q V i where optimality is i the sese of miimizig the asymptoti variae-ovariae AVC matrix for the lass of loal liear GMM estimators give the orthogoal oditio i 2.1. We do so by extedig the work of Newey 1990, 1993, Baltagi ad Li 2002, ad Ai ad Che 2003 to our framework, but the latter authors oly osider optimal IVs for GMM estimates of fiite dimesioal parameters based o oditioal momet oditios. Remark 2 Loal liear versus loal ostat GMM estimators A alterative to the loal liear GMM estimator is the loal ostat GMM estimator; see, e.g., Lewbel 2007 ad Tra ad Tsioas I this ase, the parameter of iterest α otais oly the set of futioal oeffiiets g j, j = 1,, d, evaluated at u = u, u d, but ot their first order derivatives with respet to the otiuous argumets. As a result, oe a set Q h,iu = Q V i so that there is o distitio betwee loal ad global istrumets. I additio, our loal liear GMM estimator i 2.9 redues to that of Cai ad Li 2008 by settig Ψ u to be the idetity matrix ad hoosig k = d global istrumets. The latter oditio is eessary for the model to be loally just idetified. 3 Asymptoti Properties of the Loal Liear GMM Estimator I this setio, we first give a set of assumptios ad the study the asymptoti properties of the loal liear GMM estimator. 3.1 Assumptios To failitate the presetatio, defie Ω 1 u = E Q V i X i U i = u ad Ω 2 u = E [ Q V i Q V i σ 2 V i U i = u ] where σ 2 v E [ ε 2 i V i = v ]. Let f U u f U u, u d deote the joit desity of U i ad Ud i ad p u d be the margial probability mass of U d i at ud. We use U ad U d =Π p d t=1 {0, 1,, t 1} to deote the support of U i ad Ud i, respetively. 6

7 We ow list the assumptios that will be used to establish the asymptoti distributio of our estimator. Assumptio A1. Y i, X i, Z i, U i, i = 1,,, are idepedet ad idetially distributed IID. Assumptio A2. E ε i 2+δ < for some δ > 0. E Q V i X i 2 <. Assumptio A3. i U is ompat. ii The futios f U, ũ d, Ω 1, ũ d, ad Ω 2, ũ d are otiuously differetiable o U for all ũ d U d. 0 < f U u, u d C for some C <. iii The futios g j, ũ d, j = 1,, d, are seod order otiuously differetiable o U for all ũ d U d. Assumptio A4. i rakω 1 u = d, ad the k k matrix Ω 2 u is positive defiite. ii Ψ u = Ψ u + o P 1, where Ψ u is symmetri ad positive defiite. Assumptio A5. The kerel futio w is a probability desity futio PDF that is symmetri, bouded, ad has ompat support [ w, w ]. It satisfies the Lipshitz oditio w v 1 w v 2 C w v 1 v 2 for all v 1, v 2 [ w, w ]. Assumptio A6. As 0, the badwidth sequees h = h 1,, h p ad λ= λ 1,, λ pd satisfy i h!, ad ii h! 1/2 h 2 + λ = O 1, where h! h 1 h p. A1 requires IID observatios. Followig Cai ad Li 2008 ad SCU 2009, this assumptio a be relaxed to allow for time series observatios. A2 ad A3 impose some momet ad smoothess oditios, respetively. A4i imposes rak oditios for the idetifiatio of the futioal oeffiiets ad their first order derivatives ad A4ii is weak i that it allows the radom weight matrix Ψ to be osistetly estimated from the data. As Hall, Wolf, ad Yao 1999 remark, the requiremet i A5 that w is ompatly supported a be removed at the ost of legthier argumets used i the proofs, ad i partiular, the Gaussia kerel is allowed. A6 is stadard for oparametri regressio with mixed data; see, e.g., Li ad Raie Asymptoti theory for the loal liear estimator Let µ s,t = R vs w v t dv, s, t = 0, 1, 2. Defie Φ u = f U u Υ u = f U u Ω 1 u 0 k dp, ad kp d µ 2,1 Ω 1 u I p µ p 0,2 Ω 2u 0 k kp kp k µ 2,2 Ω 2 u I p Clearly, Φ u is a k 1 + p d 1 + p matrix ad Υ u is k 1 + p k 1 + p matrix. To desribe the leadig bias term assoiated with the disrete radom variables, we defie a idiator futio I s, by I s u d, ũ d = 1{u d ũ d s} p d 1{u d = ũ d t }. That is, I s u d, ũ d is oe if ad oly if u d ad ũ d differ oly i the sth ompoet ad is zero otherwise. Let { 1 B u; h, λ = 2 µ 2,1f U u Ω 1 u A u; h + p d ũ d U d s=1 0 kp 1 t s λ s I s u d, ũ d f U u, ũ d Ω 1 u, ũ d g u, ũ d g u, u d µ 2,1 Ω1 u, ũ d I p g u, u d, 3.3 where A u; h = p s=1 h2 sg 1,ss u,, p s=1 h2 sg d,ss u, g u = g 1 u,., g d u, ad g u =. g 1 u,, ġ d u. Now we state our first mai theorem. 7

8 Theorem 3.1 Suppose that Assumptios A1-A6 hold. The h!{h[ α Ψ u; h, λ α u] Φ Ψ 1 Φ 1 Φ Ψ 1 d B u; h, λ} N0, Φ Ψ 1 Φ 1 Φ Ψ 1 ΥΨ 1 Φ Φ Ψ 1 Φ 1, where we have suppressed the depedee of Φ, Ψ, ad Υ o u, ad H =diag1,, 1, h,, h is a d p + 1 d p + 1 diagoal matrix with both 1 ad h appearig d times. Remark 3 Optimal hoie of the weight matrix To miimize the AVC matrix of α Ψ, we a hoose Ψ u as a osistet estimate of Υu, say Υ u. The the AVC matrix of α Υ u; h, λ is give by Σ u = [Φ u Υ u 1 Φ u] 1, whih is the miimum AVC matrix oditioal o the hoie of the global istrumets Q V i. Let α u be a prelimiary estimate of α u by settig Ψ u = I kp+1. Defie the loal residual ε i u = Y i d j=1 g j u X i,j, where g j u is the jth ompoet of α u. Let Υ u = h! Q i Q i ε i u 2 Q i Q i η i u ε i u 2 Q i Q i η i u ε i u 2 Q i Q i [η i u η i u ] ε i u 2 K 2 hλ,iu where Q i Q V i ad η i u U i u /h. It is easy to show that uder Assumptios A1-A6 Υ u =Υu +o P 1. Alteratively, we a obtai the estimates α u ad thus g j u for u = U i, i = 1,,, ad the we a defie the global residual ε i = Y i d j=1 g j U i X i,j. Replaig ε i u i the defiitio of Υ u by ε i also yields a osistet estimate of Υ u, but this eeds prelimiary estimatio of the futioal oeffiiets at all data poits ad thus is muh more omputatioally expesive. By hoosig Ψ u = Υ u, we deote the resultig loal liear GMM estimator of α u as α Υ u; h, λ. We summarize the asymptoti properties of this estimator i the followig orollary, whose proof is straightforward. Corollary 3.2 Suppose that Assumptios A1-A4i ad A5-A6 hold. The h!{h[ α Υ u; h, λ α u] Φ Υ 1 Φ 1 Φ Υ 1 B u; h, λ} d N0, Φ Υ 1 Φ 1. I partiular, h!{ĝ Υ u; h, λ g u f U u 1 [Ω 1 u Ω 2 u 1 Ω 1 u] 1 Ω 1 u Ω 2 u 1 B 0 u; h, λ} d N0, µ p 0,2 f U u 1 [Ω 1 u Ω 2 u 1 Ω 1 u] 1, where ĝ Υ u; h, λ ad B 0 u; h, λ deote the first d elemets of α Υ u; h, λ ad B u; h, λ, respetively. Remark 4 Asymptoti idepedee betwee estimates of futioal oeffiiets ad their first order derivatives Theorem 3.1 idiates that, for the geeral hoie of Ψ that may ot be blok diagoal, the estimators of the futioal oeffiiets ad those of their first order derivatives may ot be asymptotially idepedet. Nevertheless, if oe hooses Ψ as a asymptotially blok diagoal matrix i.e., the limit of Ψ is blok diagoal as i Corollary 3.2, the we have asymptoti idepedee betwee the estimates of g u ad g u. If further k = d, the the formulae for the asymptoti bias ad variae of ĝ Υ u a be simplified to Ω 1 u 1 B 0 u; h, λ /f U u ad µ p 0,2 Ω 1 u 1 Ω 2 u Ω 1 u 1 /f U u, respetively. 3.3 Optimal hoie of global istrumets To derive the optimal global istrumets for the estimatio of α u based o the oditioal momet restritio give i 2.1, defie Q V i = CE X i V i /σ 2 V i 3.4 where C is ay osigular oradom d d matrix. As Q V i is a d 1 vetor, the weight matrix Ψ does ot play a role. It is easy to verify that the loal liear GMM estimator orrespodig to this 8

9 hoie of IV has the followig AVC matrix Σ u = f 1 U u µ p 0,2 Ω u 1 0 d dp 0 dp d µ2,2 /µ 2 2,1 Ω u 1 I p 1 = f 1 U u K µ p 0,2 Ω u 0 d dp 0 dp d µ 2,2 Ω K, u I p where Ω u E[E X i V i E X i V i σ 2 µ p 0,2 V i U i = u] ad K I d 0 d dp. Notig that Σ u is free of the hoie of C, hereafter we simply take C = I d ad otiue to use Q V i 0 dp d µ 2,2 /µ 2,1 I dp to deote E X i V i /σ 2 V i. We ow follow Newey 1993 ad argue that Q V i is the optimal IV i the sese of miimizig the AVC matrix of our loal liear GMM estimator of α u amog the lass of all loal liear GMM estimators. Let Q i Q V i. Defie m i,q Φ u Ψ u 1 K 1 Q i Q i η i u E m i,q m i,q Q i Q i η i u ε i K hλ,iu h! 1/2 ad m i,q ε i K hλ,iu h! 1/2. By the law of iterated expetatios ad momet alulatios, [ Q i Q i Q i Q i η i u = K 1 E Q i Q i η i u Q i Q i [η i u η i u σ 2 V i Khλ,iuh! 2 ] [ ] = K 1 Ω U i Ω U i η i u E Ω U i η i u Ω U i [η i u η i u Khλ,iuh! 2 K 1 ] = f U u K 1 µ p 0,2 Ω u 0 d dp 0 dp d µ 2,2 Ω K 1 + o 1 u I p = [Σ u] 1 + o 1. Similarly, Em i,q m i,q = Φ u Ψ u 1 Υ u Ψ u 1 Φ u+o 1 ad Em i,q m i,q = Φ u Ψ u 1 Φ u + o 1. It follows that Φ Ψ 1 Φ 1 Φ Ψ 1 ΥΨ 1 Φ Φ Ψ 1 Φ 1 Σ u = [ Em i,q m i,q ] 1 Emi,Q m i,q [ Em i,q m i,q ] 1 [ Emi,Q m i,q ] 1 + o 1 = [ { Em i,q m i,q ] 1 Em i,q m i,q Em i,q m i,q [ } Em i,q m i,q ] 1 Emi,Q m i,q [ Em i,q m i,q ] 1 + o 1 = E [R i R i] + o 1 where R i [Em i,q m i,q ] 1 {m i,q Em i,q m i,q [Em i,q m i,q ] 1 m i,q }. The positive semi-defiiteess of E [R i R i ] implies that the loal liear GMM estimator of α u based o Q V i is asymptotially optimal amog the lass of all loal liear GMM estimators of α u. I this sese, we say that Q V i is the optimal IV withi the lass. Remark 5 Compariso with the optimal IV for parametri GMM estimatio Cosider a simple parametri model i whih Y i = β X i + ε i ad E ε i V i = 0 a.s. The results i Newey 1993 imply that the optimal IV for the GMM estimatio of β is give by E X i V i /E ε 2 i V i. Suh a IV will miimize the AVC matrix of the GMM estimator of β amog the lass of all GMM estimators based o the give oditioal momet restritio. The optimal IV Q V i for our futioal oeffiiet model i ] K 1 9

10 2.1 takes the same futioal form. I additio, it is worth metioig that the squared asymptoti bias for a parametri GMM estimate is asymptotially egligible i ompariso with its asymptoti variae, whereas for our oparametri GMM estimate it is ot uless oe uses a udersmoothig badwidth sequee i the estimatio. Differet hoies of IVs yield differet asymptoti bias formulae ad it is extremely hard to ompare them. Eve if the use of the optimal IV Q V i miimizes the asymptoti variae of the estimate of eah elemet i α u, it may ot miimize the asymptoti mea squared error AMSE. We thik that this is a importat reaso why we aot fid ay appliatio of optimal IV for oparametri GMM estimatio i the literature. Aother reaso is also essetial. To apply the optimal IV, we have to estimate both E X i V i ad σ 2 V i oparametrially, ad the theoretial justifiatio is tehially hallegig ad beyod the sope of this paper. The similar results ad remarks also hold whe oe osiders the optimal IV for loal ostat GMM estimatio. I partiular, Q V i is also the optimal IV for loal ostat estimatio of g u. We will ompare loal liear ad loal ostat GMM estimates based o o-optimal ad estimated optimal IVs through Mote Carlo simulatios. 3.4 Data-depedet badwidth By Theorem 3.1 we a defie the asymptoti mea itegrated squared error AMISE of {ĝ j u, j = 1,..., d}, ad hoose h r r = 1,, p ad λ s s = 1,, p d to miimize it. By a argumet similar to Li ad Raie 2008, it is easy to obtai the optimal rates of badwidths i terms of miimizig the AMISE: h r 1/4+p ad λ s 2/4+p for r = 1,, p ad s = 1,, p d. Nevertheless, the exat formula for the optimal smoothig parameters is diffiult to obtai exept for the simplest ases e.g., p = 1 ad p d = 0 or 1. This also suggests that it is ifeasible to use the plug-i badwidth i applied settig sie the plug-i method would first require the formula for eah smoothig parameter ad the pilot estimates for some ukow futios i the formula. I pratie, we propose to use least squares ross validatio LSCV to hoose the smoothig parameters. We hoose h, λ to miimize the followig least squares ross validatio riterio futio CV h, λ = 1 Y i d j=1 ĝ i j U i ; h, λ X i,j a U i, where ĝ i j U i ; h, λ is the leave-oe-out futioal oeffiiet estimate of g j U i usig badwidth h, λ, ad a U i is a weight futio that serves to avoid divisio by zero ad perform trimmig i areas of sparse support. I pratie ad the followig umerial study we set a U i = Π p j=1 1{ U i,j U j 2s U j }, where U j ad s U j deote the sample mea ad stadard deviatio of {Ui,j, 1 i }, respetively. To implemet, oe a use grid searh for h, λ whe the dimesios of U i ad Ud j are both small. Alteratively, oe a apply the miimizatio futio built i various software; but multiple startig values are reommeded. I the followig we argue that the result i Theorem 3.1 otiues to hold whe the ostohasti badwidth h, λ is replaed by some data-depedet stohasti badwidth, say, ĥ ĥ1,..., ĥp ad λ λ 1,..., λ pd. Followig Li ad Li 2010 we assume that ĥr h 0 r/h 0 r = o P 1 ad λ s λ 0 s/λ 0 s = o P 1 for r = 1,, p ad s = 1,, p d, where h 0 h 0 1,..., h 0 p ad λ 0 λ 0 1,..., λ 0 p d deotes ostohasti badwidth sequees for U i ad Ud i, respetively. For example, ĥ, λ ould be the LSCV badwidth. If so, oe a follow Hall, Raie ad Li 2004 ad Hall, Li ad Raie 2007 ad show that the above requiremet holds for h 0, λ 0 whih is optimal i miimizig a weighted versio of the AMISE of {ĝ j u, j = 1,..., d}. The followig theorem summarizes the key result. 2 10

11 Theorem 3.3 Suppose that Assumptios A1-A5 hold. Suppose that ĥr h 0 r/h 0 r = o P 1 ad λ s λ 0 s/λ 0 s = o P 1 for r = 1,, p ad s = 1,, p d, where h 0 ad λ 0 satisfy A6. The ĥ!{ĥ[ α Ψ u; ĥ, λ α u] Φ Ψ 1 Φ 1 Φ Ψ 1 B u; h 0, λ 0 } d N0, Φ Ψ 1 Φ 1 Φ Ψ 1 ΥΨ 1 Φ Φ Ψ 1 Φ 1, where Ĥ is aalogously defied as H with h beig replaed by ĥ. 4 A Speifiatio Test I this setio, we osider testig the hypothesis that some of the futioal oeffiiets are ostat. The test a be applied to ay oempty subset of the full set of futioal oeffiiets. 4.1 Hypotheses ad test statisti We first split up the set of regressors i X i ad the set of futioal oeffiiets i g u ito two ompoets after possibly rearragig the regressors: X 1i = X i,1,, X i,d1 assoiated with g 1 u = g 1 u,, g d1 u, ad X 2i = X i,d1+1,, X i,d, assoiated with g 2 u = g d1+1 u,, g d u, where X i,1 may ot deote the ostat term i this setio. The we a rewrite the model i 2.1 as Y i = g 1 U i X 1i + g 2 U i X 2i + ε i, E ε i Z i, U i = 0 a.s. 4.1 Suppose that we wat to test for the ostay of futioal oeffiiets for a subset of the regressors X 1i ad maitai the assumptio that the futioal oeffiiets of X 2i may deped o the set of exogeous regressors U i. The the ull hypothesis is H 0 : g 1 U i = θ 1 a.s. for some parameter θ 1 R d1, 4.2 ad the alterative hypothesis H 1 deotes the egatio of H 0. Uder H 0, d 1 of the d futioal oeffiiets are ostat whereas uder H 1, at least oe of the futioal oeffiiets i g 1 is ot ostat. There are may ways to test the ull hypothesis i 4.2. Oe way is to estimate the followig restrited semiparametri futioal oeffiiet IV model Y i = θ 1X 1i + g 2 U i X 2i + ε r i 4.3 where ε r i is the restrited error term defied by 4.3 suh that Eε r i Z i, U i = 0 a.s. uder the ull. The oe a propose a Lagragia multiplier LM type of test based o the estimatio of this restrited model oly, say, by osiderig the test statisti based o the sample aalog of E[ε r i E[ε r i V i ]f V V i ] where f V is the PDF of V i = Z i, U i. The seod way is to adopt the likelihood ratio LR priiple to estimate both the urestrited ad restrited models ad ostrut various test statistis, say, by omparig the estimates of either g 1 or g = g 1, g 2 i both models through ertai distae measure e.g., Hog ad Lee, 2009, or by extedig the geeralized likelihood ratio GLR test of Fa, Zhag, ad Zhag 2001 to our framework where edogeeity is preset. Clearly tests based the LM priiple ad E[ε r i E[ε r i V i ]f V V i ] i partiular may suffer from the problem of urse of dimesioality beause the dimesio of the otiuous variables i V i is typially larger tha the dimesio p of U i. Tests based o the LR priiple requires oparametri/semiparametri estimatio uder both the ull ad alterative, ad uless d 1 = d, the estimatio of the restrited model 4.3 is more ivolved tha the estimatio of the urestrited model. For this reaso, we propose a Wald-type statisti that requires oly osistet estimatio of the urestrited model. Let ĝ Ψ u deote the first d elemet of α Ψ u α Ψ u; h, λ. It is the estimator 11

12 of g u = g 1 u, g 2 u. Split ĝ Ψ u as ĝ 1 u = ĝ 1,Ψ u ad ĝ 2 u = ĝ 2,Ψ u so that ĝ l u estimates g l u for l = 1, 2. Our proposed test statisti is T = h! 1/2 2 ĝ 1 U i ĝ 1 where ĝ 1 1 ĝ1 U i. I the ext subsetio, we show that after beig suitably ormalized, T is asymptotially distributed as N 0, 1 uder H 0 ad diverges to ifiity uder H Asymptoti distributio of the test statisti Let Φ u 1 Q h u K hλ u ξ u H 1. Defie Γ 1 u = S 1 [Φ u Ψ u 1 Φ u] 1 Φ u Ψ u 1, ad Γ 1 u = S 1 [Φ u Ψ u 1 Φ u] 1 Φ u Ψ u 1, 4.5 where S 1 = I d1, 0 d1 d 1p +d 2p +1 is a seletio matrix. We add the followig assumptios. Assumptio A7. i Ψ u = Ψ u + O P ν uiformly i u, where Ψ u is symmetri ad positive defiite for eah u ad ν 0 as. ii sup Γ1 u u < C <. Assumptio A8. As, i 1/2 h 2 + λ ν 0, ii h 2 + λ h! 1/2 log 0, ad iii h! 1/2 h 4 + λ 2 0. A7 stregthes A4ii. It is satisfied if oe hooses Ψ u as the idetity matrix I kp+1 for all u, i whih ase ν = 0. Alteratively, if oe hooses Ψ u = Υ u, the oe a verify that A7i is satisfied with ν = h 2 + λ + h!/ log 1/2. A7ii is weak give the ompat support of the otiuous regressor U i. A8i a easily be satisfied whereas A8ii requires that p 3; oe a use higher order loal polyomial estimatio if p > 3. A8iii requires that udersmoothig badwidth must be used i order to remove the effet of asymptoti bias of our oparametri estimators. Without loss of geerality, we osider the hoie of Ψ u as Υ u ad set h s 1/δ for s = 1,..., p ad λ t 2/δ for t = 1,..., p d ; i.e., h 1,..., h p pass to 0 at the same rate ad similarly for λ 1,..., λ pd. The the oditios i A8 are all satisfied by settig δ 1, 4.5 for p = 1, δ 2, 5 for p = 2, ad δ 3, 5.5 for p = 3. To proeed, we first osider the osistet estimatio of θ 1 uder H 0. We estimate it by θ 1 = ĝ 1 = 1 ĝ 1 U i. 4.6 By A.1 i the appedix, we have the followig usual bias ad variae deompositio for ĝ 1 U i : ĝ 1 U i g 1 U i = Γ 1 U i B U i + Γ 1 U i V U i, 4.7 where Γ 1 u is defied i 4.5, ad the bias term B U i ad the variae term V U i are defied i the lie after A.1. Uder H 0, θ1 θ 1 = 1/2 Γ 1 U i B U i + 1/2 Γ 1 U i V U i. 4.8 We shall show that the first term bias o the right had side of 4.8 is asymptotially egligible uder some extra oditio o the badwidth sequee, whereas the seod term otributes to the AVC of θ 1. 12

13 To haraterize the AVC matrix of θ 1, let ζ i U i, Q i, ε i, ad Q j ε j ϕζ i, ζ j Γ 1 U i Q j ε j η j U i K hλ,jui, ad ϕζ i = ϕζ, ζ i df ζ ζ, 4.9 where F ζ deotes the CDF of ζ i. Let Σ θ1 = lim E [ϕζ i ϕζ i ]. Straightforward but tedious alulatios show that p Ω 2 u 0 k pk Σ θ1 = w t wt sdtds Γ 1 u Γ 1 u f U u 2 df U u pk k 0 pk p k The followig theorem establishes the -osistey ad asymptoti ormality of θ 1 uder H 0. Theorem 4.1 Suppose Assumptios A1-A4i ad A5-A8 hold. Suppose that 1/2 h 2 + λ = o 1, ν h! 1/2 = o 1, ad h! 2 / log as. The uder H 0, θ 1 θ 1 d N 0, Σ θ1. Clearly Theorem 4.1 says that uder H 0, θ 1 a osistetly estimate θ 1 at the usual -rate. The extra oditios o the badwidth i the above theorem esures that the bias term i 4.7 vaishes asymptotially ad the replaemet of Γ 1 U i i 4.7 by Γ 1 U i has asymptotially egligible effet o the asymptoti ormality of θ 1. If all futioal oeffiiets are ostat uder H 0, the B U i = 0 a.s. so that we do ot eed the first extra oditio o the badwidth i the theorem. Let B 2 h! 1/2 j=1 ϕζ i, ζ j 2 ad σ 2 0 lim 2h!E j E l [ ϕζ, ζ j ϕζ, ζ l df ζ ζ] 2, where E j deotes the expetatio with respet to ζ j. The ext theorem studies the asymptoti distributio of T uder H 0. Theorem 4.2 Suppose Assumptios A1-A4i ad A5-A8 hold. The uder H 0, T B d N0, σ 2 0. Followig the last remark after Theorem 4.1, Assumptio A8iii is ot eeded for the above theorem if we are testig the ostay of all futioal oeffiiets. To implemet the test, we osistetly estimate B ad σ 2 0 usig where ϕ ij = Γ 1 U i B h!1/2 2 j=1 Q j ε j Q j ε j η j U i ϕ ij 2 ad σ 2 = 2h! 1 j=1 l j [ 1 ] 2 ϕ ij ϕ il, K hλ,jui, ad ε i = Y i ĝ Ψ U i X i. It is straightforward to show that B B = o P 1 ad σ 2 σ0 2 = o P 1. The we have J T B / σ 2 d N 0, 1 uder H 0. Whe is suffiietly large, we a ompare J to the oe-sided ritial value z α, the upper α peretile from the N 0, 1 distributio, ad rejet the ull at asymptoti level α if J > z α. To examie the asymptoti loal power, we osider the sequee of Pitma loal alteratives H 1 r : g 1 U i = θ 1 + r δ U i a.s. where r 0 as ad the δ s are a sequee of real otiuous vetor-valued futios suh that µ 0 lim E[ δ U i E [δ U i ] 2 ] <. The followig theorem establishes the asymptoti loal power of the J test. 13

14 Theorem 4.3 Suppose Assumptios A1-A4i ad A5-A8 hold. The uder H 1 r with r = 1/2 h! 1/4, J d N µ0 /σ 0, 1. Theorem 4.3 shows that the J test has otrivial power agaist Pitma loal alteratives that overge to zero at rate 1/2 h! 1/4. The asymptoti loal power futio is give by lim P J z H 1 r = 1 Φ z µ 0 /σ 0, where Φ is the stadard ormal CDF. The ext theorem establishes the osistey of the test. Theorem 4.4 Suppose Assumptios A1-A4i ad A5-A8 hold. The uder H 1, 1 h! 1/2 J = µ A /σ 0 + o P 1 where µ A E [g 1 U i θ 1 ] 2, so that P J > 1 uder H 1 for ay ostohasti sequee = o h! 1/ A bootstrap versio of our test It is well kow that a oparametri test based o its asymptoti ormal ull distributio may perform poorly i fiite samples. So we suggest usig a bootstrap method to obtai the bootstrap approximatio to the fiite-sample distributio of our test statisti uder the ull. We fid that it is easy to adopt the fixed-desig wild bootstrap method i the spirit of Hase 2000 i our framework; see also Su ad White 2010 ad Su ad Ullah The great advatage of this method lies i the fat that we do ot eed to mimi some importat features suh as depedee or edogeeity struture i the data geeratig proess ad a still justify its asymptoti validity. We propose to geerate the bootstrap versio of J as follows: 1. Obtai the loal liear GMM estimates ĝ 1 U i ad ĝ 2 U i by usig the weight matrix Ψ ad the badwidth h, λ, ad alulate the urestrited residuals ε i = Y i ĝ 1 U i X 1i ĝ 2 U i X 2i. 2. For i = 1,...,, geerate the wild bootstrap residuals ε i = ε ie i where e i s are IID N 0, For i = 1,...,, geerate Y i = θ 1X 1i + θ 2X 2i + ε i where θ 1 1 ĝ1 U i ad θ 2 1 ĝ2 U i are the restrited loal liear GMM estimates uder the ull hypothesis H 0s : g U i = θ a.s. for some parameter θ R d. 4. Compute the bootstrap test statisti J i the same way as J by usig {Y i, U i, X i, Z i } ad the weight matrix Ψ. 5. Repeat Steps 1-4 B times to obtai B bootstrap test statisti {Jj }B j=1. Calulate the bootstrap p-values p B 1 B j=1 1{J j J } ad rejet the ull hypothesis H 0 : g 1 U i = θ 1 a.s. if p is smaller tha the presribed omial level of sigifiae. Note that i Step 3 we impose the ull hypothesis H 0s : g U i = θ a.s., whih is stroger tha H 0 : g 1 U i = θ 1 a.s. uless d 1 = d. Ituitively speakig, i order to justify the asymptoti validity of the above bootstrap proedure, we eed to demostrate that the bootstrap test statisti J has the asymptoti distributio N 0, 1 o matter whether the origial sample is geerated uder the ull hypothesis H 0 or ot. We will show that oditioal o the origial sample J is asymptotially N 0, 1, whih implies that it is also asymptotially N 0, 1 uoditioally. Note that the origial test statisti J is asymptotially N 0, 1 uder H 0 ad our bootstrap statisti has the same asymptoti distributio. This esures the orret asymptoti size of our bootstrap test. Further, ote that the origial test statisti J diverges to ifiity at the rate h! 1/2 uder the global alterative hypothesis H 1 whereas the bootstrap test 14

15 statisti J remais asymptotially N 0, 1 i this ase. This esures the osistey of our bootstrap test. By imposig a stroger hypothesis H 0s tha the origial ull hypothesis of iterest H 0, our bootstrap test has both pros ad os. The major pros lie i two aspets. First, oe a easily odut the bootstrap test for testig the ostay of various subvetors of g i a sigle step beause we a geerate the same bootstrap depedet variable oe for all ad the omputatio burde is almost idetial to the ase of testig the ostay of a sigle subvetor of g. Seod, oe a easily justify the asymptoti validity of our bootstrap method ad there is o eed to use oversmoothig badwidth for first stage estimatio as i Härdle ad Marro Our simulatios idiate this proedure does ot result i a loss of power i ompariso with the alterative approah by geeratig Yi through θ 1X 1i + g 2 U i X 2i + ε i where oe imposes the exat ull hypothesis to be tested, θ 1 1 g 1 U i, ad g 1, g 2 is a prelimiary estimate of g 1, g 2. But the justifiatio for the validity of this latter approah would be muh harder beause oe eeds to show that the seod order derivatives of g 2 are uiformly well behaved, whih typially requires oversmoothig; see Härdle ad Marro The major os of our bootstrap proedure lie i the potetial loss of seod order effiiey. I other words, the impositio of a stroger hypothesis tha eessary i the bootstrap world is expeted to have a seod order asymptoti effet. For parametri tests, it is ofte argued that a bootstrap test based o a asymptotially pivotal statisti may yield a higher order effiiey tha a test based o the asymptoti ormal or hi-square distributios. For oparametri tests, it is extremely hallegig to demostrate higher order effiiey for a bootstrap test statisti. Therefore we thik higher order effiiey is a less importat issue tha esurig the orret asymptoti size ad osistey of a bootstrap test. Its formal study is ertaily beyod the sope of the urret paper. To show that the bootstrap statisti J a be used to approximate the asymptoti ull distributio of J, we follow Li, Hsiao ad Zi 2003 ad Su ad Ullah 2012 ad rely o the otio of overgee i distributio i probability, whih geeralizes the usual overgee i distributio to allow for oditioal radom distributio futios. The followig theorem establishes the asymptoti validity of the above bootstrap proedure. Theorem 4.5 Suppose Assumptios A1-A4i ad A5-A8 hold. Suppose that Ψ u = Ψ u + O P ν uiformly i u, where P is the probability measure idued by the wild bootstrap. Let z α be the α-level bootstrap ritial value based o B bootstrap resamples. The J overges to N0, 1 i distributio i probability, lim P J z α = α uder H 0, lim P J z α 1 Φz α µ A /σ 0 uder H 1 1/2 h p/4, ad lim P J z α = 1 uder H 1, where z α deotes the 1001 αth peretile of the stadard ormal distributio. Theorem 4.5 shows that the bootstrap provides a asymptoti valid approximatio to the limit ull distributio of J. This holds as log as we geerate the bootstrap data by imposig the ull hypothesis. If the ull hypothesis does ot hold i the observed sample, the J explodes at the rate h! 1/2 but J is still well behaved, whih ituitively explais the osistey of the bootstrap-based test J. 5 Mote Carlo Simulatios I this setio, we odut a small set of Mote Carlo experimets to illustrate the fiite sample performae of our loal liear GMM estimator of futioal oeffiiets ad that of our test for the ostay of some futioal oeffiiets. 15

16 5.1 Evaluatio of the loal liear GMM estimates To evaluate the loal liear GMM estimates, we osider two data geeratig proesses DGPs: DGP 1: Y i = Ui Ui,1 d U i,2 d + [1 + U i,1 + ϕ U i 0.5U i,1 d + 0.5U i,2 d ]X i + σ i ε i, DGP 2: Y i = 1 + e U i + [1 + 2 siu i ]X i + σ i ε i, where Ui N 0, 1 truated at ±2, Ui,1 d ad U i,2 d are both Beroulli radom variables takig value 1 with probability 0.5, X i = Z i + τε i / 1 + τ 2, Z i, ε i N 0, I 2, ad ϕ is the stadard ormal PDF. Note that there is o disrete radom variable i DGP 2. Here we use τ to otrol the degree of edogeeity; e.g., τ =0.32 ad 0.75 idiates that the orrelatios betwee X i ad ε i are 0.3 ad 0.6, respetively. We osider both oditioally homoskedasti ad heteroskedasti errors. For the homoskedasti ase, σ i = 1 i both DGPs 1 ad 2; for the heteroskedasti ase, we speify σ i as follows σ i = Z 2 i + U 2 i + 0.5U d i,1 + U d i,2 i DGP 1 ad σ i = Z 2 i + U 2 i i DGP 2. We assume that we observe { Y i, Ui, U i,1 d, U i,2 d, X } i, Z i ad {Y i, Ui, X i, Z i } i DGP 1 ad DGP 2, respetively. The defiitios of the futioal oeffiiets, g 1 u ad g 2 u, i eah DGP are self-evidet. We osider six oparametri estimates for g 1 u ad g 2 u. The first estimate is the loal liear estimate of SCU 2009 where the edogeeity of X i is egleted. The seod ad third estimates are obtaied as our loal liear GMM futioal oeffiiet estimators by hoosig the global IV respetively as Q V i = [1, Z i ] ad loal liear estimate of Q V i = [1, E X i V i ] /σ 2 V i,respetively, where V i = Z i, Ui, U i,1 d, U i,2 d ad V i = Z i, Ui i DGPs 1 ad 2, respetively. Sie the dimesio of Q V i is the same as that of [1, X i ], the weight matrix Ψ does ot affet the loal liear GMM estimate so that we a simply use the idetity weight IW matrix as the weight matrix for our seod estimate, whih also redues to the estimate of Cai ad Li Similarly, the third estimate is the optimal IV OIV estimate whih is ot iflueed by the hoie of weight matrix. The fourth ad fifth estimates are the loal ostat aalogues of the seod ad third estimates, respetively; they are also the estimates of Tra ad Tsioas 2010, TT whe the IV is hose to be Q V i ad the loal ostat estimate of Q V i, respetively. The sixth estimate is the two-stage loal liear estimate of CDXW. Below we will deote these six estimates as SCU, IW ll, OIV ll, IW l, OIV l, ad CDXW i order. For all estimators, we use the stadardized Epaehikov kerel k u = u2 1{ u 5}, ad osider two hoies of smoothig parameters [h, λ = h, λ 1, λ 2 for the oditioig variables Ui, U i,1 d, U i,2 d i the futioal oeffiiets i DGP 1 ad h = h for the oditioig variable U i i the futioal oeffiiets i DGP 2]; oe is obtaied by the LSCV method disussed i Setio 3.4, ad the other by the simple rule of thumb ROT: h = s U 1/5 i both DGPs 1 ad 2, ad λ 1 = λ 2 = 2/5 i DGP 1. Here s A deotes the sample stadard deviatio of {A i }. To estimate the optimal IVs, we eed to estimate both E X i V i ad σ 2 V i by the loal liear or loal ostat method. For both ases, we use the stadardized Epaehikov kerel ad udersmoothig ROT badwidths by speifyig h = [s Z 1/5 s U 1/5 ] i both DGPs 1 ad 2 ad λ 1 = λ 2 = 2/5 i DGP 2 whe we regress either X i or ε 2 i o V i. The use of udersmoothig badwidths helps to elimiate the effet of early stage estimates bias o the fial estimate; see Mamme, Rothe ad Shiele To obtai the CDXW estimate, we eed first to obtai the loal liear estimate of E X i V i by speifyig a similar udersmoothig badwidth. I additio, we fid that the LSCV ad ROT hoies of h, λ yield qualitatively similar results. So we fous o the ROT badwidth below for brevity. To evaluate the fiite sample performae of differet futioal oeffiiet estimates, we alulate both the mea absolute deviatio MAD ad mea squared error MSE for eah estimate evaluated at 16

17 all data poits: MAD r l = 1 ĝ r l U i g l U i ad MSE l = 1 [ ] 2 ĝ r l U i g l U i where for l = 1, 2, ĝ r l is a estimator of g l i the rth repliatio by usig ay oe of the above estimatio methods. We osider two sample sizes: = 100 ad 400. Table 1 reports the results where the MADs ad MSEs are averages over 500 repliatios for eah futioal oeffiiet. We summarize some importat fidigs from Table 1. First, i both homoskedasti ad heteroskedasti ases, the SCU estimate without takig ito aout the edogeeity issue is geerally the worst estimate amog all six estimates. Exeptios may our whe is small or o heteroskedastiity is preset. Seod, i the ase of homoskedasti errors, the loal GMM estimates obtaied by usig the estimated optimal IVs for either our loal liear method or TT s loal ostat method may or may ot outperform the oe usig simple IV with idetity weight matrix. This is similar to the fidigs i Altoji ad Segal 1996 who show that the use of optimal weights i the GMM estimatio may be domiated by the oe-step equally weighted GMM estimatio i fiite samples. Third, i the ase of heteroskedasti errors, we observe substatial gai by usig the estimated optimal IVs i the loal GMM estimatio proedure; this is true for both our loal liear GMM estimates ad TT s loal ostat estimates. Fourth, for both DGPs uder ivestigatio, the loal liear method teds to outperform the loal ostat method. We ojeture this is due to the otorious boudary bias issue assoiated with the loal ostat estimates whe the support of Ui is ompat. Fifth, the CDXW estimate geerally is outperformed i DGP 1 by both the loal liear ad loal ostat estimates with or without usig the estimated optimal IVs. For DGP 2, the CDXW may outperform the loal ostat GMM estimates but ot the loal liear GMM estimates. 5.2 Tests for the ostay of futioal oeffiiets We ow osider the fiite sample performae of our test. To this goal, we modify DGPs 1-2 as follows DGP 1 : Y i = [ 1 + δ 0.25U 2 i + 0.5U d i, U d i,2] + [1 + δu i,1 + ϕ U i 0.5U d i, U d i,2 ]X i + σ i ε i, DGP 2 : Y i = 1 + δe U i + [1 + 2δ siu i ] X i + σ i ε i, where all variables are geerated as i the above subsetio, ad we allow δ to take differet values to evaluate both the size ad power properties of our test. Whe δ = 1, DGPs 1 ad 2 redue to DGPs 1 ad 2, respetively. For both DGPs, we osider the followig three ull hypotheses: H 0,1 : g 1 U i = θ 1 a.s., H 0,2 : g 2 U i = θ 2 a.s., 5.1 H 0,12 : g 1 U i, g 2 U i = θ 1, θ 2 a.s., for some ukow parameters θ 1 ad θ 2. To ostrut the test statisti, we eed to hoose both the kerel ad the badwidth. As i the previous setio, we hoose the stadardized Epaehikov kerel ad osider the use of the bootstrap to approximate the asymptoti ull distributio of our test statistis. Assumptio A8iii suggests that we eed to hoose udersmoothig badwidth sequees. We set h = s U 1/p+3, ad λ 1 = λ 2 = 2/p+3 for DGP 1 ad h = s U 1/p+3 i DGP 2 for differet values of to hek the sesitivity of our test to the hoie of badwidth. We have tried three values for : 0.5, 1 ad 2 ad foud that our test is ot sesitive to the hoie of. To save spae, we oly fous o the ase where = 1 i the 17

18 followig aalysis. We osider two sample sizes: = 100 ad 200, four values for δ : 0, 0.2, 0.4, ad 0.6, ad two values for τ : 0.32 ad For eah seario, we osider 500 repliatios ad 200 bootstrap resamples for eah repliatio. The results for our bootstrap-based test at 5% omial level are reported i Table 2. We summarize some importat fidigs from Table 2. First, the empirial levels of our test orrespodig to δ = 0 i the table geerally behave very well for all values of τ ad all three ull hypotheses, ad uder both oditioal homoskedastiity ad heteroskedastiity. The oly exeptio ours for testig H 0,1 i DGP 2 whe = 100, i whih ase the test is moderately udersized. Seod, the power of our test is reasoably good i almost all ases the relatively low power i testig H 0,1 i DGP 2 simply reflets the diffiulty i testig expoetial alteratives of the form δe U i. As either δ or the sample size ireases, we observe a fast irease of the empirial power. Third, the degree of edogeeity has some effet o the level ad power behavior of our test but the diretio is ot obvious. 6 A Empirial Example: Estimatig the Wage Equatio Labor eoomists have bee devotig a tremedous amout of effort to ivestigatig the ausal effet of eduatio o labor market earigs. As Card 2001, p suggests, the edogeeity of eduatio i the wage equatio might partially explai the otiuig iterest i this very diffiult task of uoverig the ausal effet of eduatio i labor market outomes. The lassial framework of the huma apital earigs futio due to Mier 1974 assumes additivity of eduatio ad work experiee that are used as explaatory variables. However, reet studies have questioed the appropriateess of this assumptio. I partiular, Card 2001 approahes the matter of o-additivity of the explaatory variables by arguig that the returs to eduatio are heterogeeous sie the eoomi beefits of shoolig are idividualspeifi. Beker ad Chiswik 1966 are amog the authors who maitai that variatio i returs to eduatio a partially aout for variatio over time i aggregate iequality. Card s 2001 laim suggests that a more geeral futioal form of heterogeeity i the returs to eduatio would make the empirial relatio betwee earigs ad eduatio eve more realisti. Ideed, if, for example, work experiee is valued by employers, the oe a expet earigs to be ireasig i experiee for ay give level of eduatio. Further, the returs to eduatio may also differ substatially amog differet groups defied by some idividual-speifi harateristis, say, a perso s marital status. Therefore, we estimate the ausal effet of eduatio o earigs i the followig futioal oeffiiet model: logy = g 1 U + g 2 US + ε, 6.1 where Y is a measure of idividual earigs, S is years of eduatio, ad U is a vetor of mixed both otiuous ad disrete variables. Equatio 6.1 allows studyig ot oly the diret effets of variables i U o wage i a flexible way but also the effets of these variables o the retur to eduatio. The existig literature has already provided support for a oliear relatio betwee wage ad work experiee see, for example, Murphy ad Welh 1990 ad Ullah I additio, Card ad Lemieux 2001 emphasize that the risig retur to eduatio has bee more profoud i the youger ohorts tha i the older oes sie the 1980s. Our goal is to study the empirial relatio betwee earigs ad eduatio as preseted i 6.1 usig our proposed estimator from the previous setios. For this purpose, we use the Australia Logitudial Survey ALS oduted aually sie Speifially, we employ the 1985 wave of the ALS, ad osider youg Australia wome, who reported workig ad were aged 16 to 25 i Our sample 18