Inference in Nonparametric Instrumental Variables with Partial Identification

Transcription

1 Inference in Nonarametric Instrumental Variables wit Partial Identification Andres Santos Deartment of Economics University of California, San Diego October, 7 Abstract Tis aer develos metods for yotesis testing in a nonarametric instrumental variables IV setting witin a artial identification framework. I construct and derive te asymtotic distribution of a test statistic for te yotesis tat at least one element of te identified set satisfies a conjectured restriction. Tis rocedure can be used to test for features of te model tat may be identified even wen te true model is not. Tis framework can also be emloyed to construct confidence regions for functionals of te elements of te identified set, suc as consumer surlus and rice elasticity of demand at a oint. I aly tis rocedure to study Engel curves for gasoline and etanol in Brazil. For bot etanol and gasoline I fail to reject tat tere are log-linear Engel curves in te identified set. In addition, I derive confidence regions for te level and sloe of te Engel curves at te samle average as well as for te comensated variation associated wit a rice cange in gasoline. Keywords: Instrumental variables, nonarametric yotesis testing, artial identification. I would like to tank my advisors Peter Hansen, Arajit Maajan and Frank Wolak for teir generous and elful guidance. I am esecially grateful to Azeem Saik and Frank Wolak for teir suggestions. I am also indebted to Liran Einav, Sriniket Nagavarau, Luigi Pistaferri, Peter Reiss and seminar articiants were tis aer was resented for teir elful comments. Te financial suort of te Stanford Institute for Economic and Policy Researc in te form of a doctoral dissertation fellowsi is gratefully acknowledged.

2 Introduction Emirical work in economics is often concerned wit te estimation and analysis of econometric models tat are derived from te beavior of otimizing agents. Te underlying structural relations tyically imly some of te regressors are endogenous, so tat tese models do not fit te classical regression framework, but are instead of te form: Y = θ X + ɛ were Eɛ X. Instrumental variables IV metods ave become immensely oular in econometrics as tey allow for te estimation of te unknown function θ x. Under te arametric assumtion tat θ x is of te form θx, β for some vector β, and te aroriate identifying conditions, IV estimators are consistent and asymtotically normally distributed. Te otential missecification of arametric metods makes it desirable to extend te IV aroac to a more flexible nonarametric framework. Unfortunately, te teoretical study and emirical imlementation of nonarametric IV metods as faced two imortant callenges. First, as originally ointed out in Newey & Powell 3, nonarametric identification is ard to attain. Witout arametric assumtions limiting te ossible forms of θ x, te burden of identification falls entirely on te instrument. Nonarametric identification of θ x requires te availability of an instrument satisfying conditions far stronger tan te usual covariance restrictions needed in te arametric case. Second, few metods are available for yotesis testing in a nonarametric IV setting. Ai & Cen 3 establis te asymtotic normality of te arametric comonent in a semiarametric secification, wic allows for yotesis testing on te arametric comonent. Horowitz 7 sows te asymtotic normality of te fully nonarametric estimator roosed in Hall & Horowitz 5, and is in tis way able to build confidence intervals for te level of θ x. In te resent aer I extend te existing results in te nonarametric IV literature by constructing a family of test statistics for a wider set of yoteses tan was reviously available. In addition, since nonarametric identification is not only ard to attain but also to test, my analysis does not assume identification. Instead I aly many of te revalent ideas in te artial identification literature, see Manski 99, 3, to te nonarametric IV setting. For any deendent variable, endogenous regressor and instrument trilet Y, X, Z, I study te nonarametric IV roblem as one of artial identification. Te identified set is defined to be: Θ = {θx Θ : EY θx Z = } Tis is te relevant definition of endogeneity in tis context, in contrast to EXɛ, because Eɛ X = imlies θ x is nonarametrically identified in wile EɛX = does not.

3 were Θ is a nonarametric set of functions tat by assumtion contains θ x. Te elements of Θ are tose functions in Θ tat are consistent wit te conditional moment assumtion Eɛ Z =, wic is te instrument exclusion restriction in a nonarametric setting. For a family of restrictions on functions θx, I sow ow to construct a test statistic for te null yotesis tat tese restrictions are satisfied by at least one element of Θ. Secial cases of te restrictions we can test for include oint values of a function and its derivatives, sae restrictions suc as monotonicity, concavity, economies of scale, economies of scoe and arametric secification tests against a nonarametric alternative. In te secial case were θ x is identified, tese tests reduce to tests on θ x. Te testing framework develoed in tis aer can be used to build informative confidence intervals for functionals on θ x, even wen suc functionals are not identified. For examle, suose θ x is a demand function and we are interested in its consumer surlus CSθ. We can construct a set Ŝ suc tat if θx Θ as consumer surlus CSθ, ten P CSθ Ŝ α for α te size of te test. Tis concet is referred to as confidence regions for functions of te identifiable arameters in Romano & Saik 6a, wic elaborates on te concet of confidence regions for identifiable arameters, as originally exlored in Imbens & Manski 4 for te case of te mean. Anoter advantage of not requiring identification is tat a test for weter at least one element of Θ satisfies te yotesized restrictions may still be consistent for te null yotesis tat θ x satisfies te tested restrictions and correctly reject even wen θ x is not identified. For examle, in a arametric secification test against a nonarametric alternative, if no element of te identified set is a member of te yotesized arametric family, ten we can conclude θ x does not fit te arametric secification eiter. On te oter and, witout identification te test may not be consistent for te null yotesis tat θ x satisfies te yotesized restrictions and asymtotically fail to reject because some oter element of Θ satisfies te restrictions we are testing for. Tis is not necessarily a weakness of te rocedure, but te result of te instrument emloyed being unable to nonarametrically identify not only θ x but also te features we are testing for. I aly te test statistics derived in tis aer to study Engel curves for gasoline and etanol in Brazil wit endogenous total exenditures. For bot etanol and gasoline I fail to reject te null yotesis tat tere is at least one log-linear Engel curve in te identified set, wic is a commonly used arametric secification. I build log-linear and nonarametric confidence regions for te level and derivative of te Engel curves at te samle mean. Te nonarametric confidence regions for te level and sloe of te etanol Engel curve are considerable different from tose obtained assuming log-linearity. Tis discreancy, owever, migt be exlained by a small samle 3

4 size. In contrast, for te gasoline Engel curve I find a log-linear arametrization imlies confidence regions for te level tat are similar to te nonarametric ones. Te analogous confidence regions for te sloe of te gasoline Engel curve, owever, differ substantially. Wile te uer bounds of te confidence regions are almost identical, teir lower bounds are significantly different. I find tat tese differences do not translate into dissimilar confidence regions for te comensated variation associated wit a rice cange in gasoline. Te results illustrate tat te imact of a articular arametrization are deendent on wat te yotesis of interest is. Te literature on nonarametric IV is fairly recent. Newey & Powell 3 roose a nonarametric estimator for θ x and sow its consistency. Te autors solve te ill-osed inverse roblem by obtaining comactness troug smootness assumtions on θ x, an insigt I rely uon in tis aer. Darolles, Florens & Renault 3 and Hall & Horowitz 5 roose alternative consistent estimators and obtain rates of convergence. Ai & Cen 3 analyze semiarametric secifications and establis efficient estimators for te arametric comonent. Blundell, Cen & Kristensen 4 use a semiarametric secification and rovide te first emirical alication of semiarametric IV togeter wit low level identification conditions and new rates of convergence results. Severini & Triati 6 exlore te identification of linear functionals of θ x wen θ x is not identified, wile Severini & Triati 7 obtain an efficiency bound for te estimation of tese functionals. Santos 7 constructs N consistent estimators for suc continuous linear functionals on θ x. Horowitz 6 derives a arametric secification test against nonarametric alternatives for model. In related work, Newey, Powell & Vella 999, Ceser 3, 5, 7 and Imbens & Newey 6 exlore estimation and identification in triangular systems. Work estimating informative bounds wen instruments fail to rovide identification as also been done in te treatment effects literature. See for examle Manski & Peer, Heckman & Vytlacil, Manski 3, Saik & Vytlacil 5 and references terein. Te remainder of te aer is organized as follows. Section exands on te artial identification roblem. Section 3 develos te test statistic tat allows us to test weter at least one element of te identified set satisfies a set of yotesized restrictions. Section 4 addresses two asects of te imlementation of te test: te utilization of sieves to aroximate te functional sace and te use of subsamling if te statistic is not ivotal. In Section 5 I aly tese metods to study te Brazilian market for etanol and gasoline. Section 6 briefly concludes. All roofs are contained in a matematical aendix. 4

5 Partial Identification Tis section elaborates on te roblem of artial identification in te nonarametric IV setting. I first review te requirements for identification and ten roceed to motivate wy it may be advantageous to adot a framework tat does not assume θ x is nonarametrically identified.. Te Identified Set Identification in a nonarametric IV setting was originally discussed in Newey & Powell 3. In tis section I review teir results before discussing te identified set. For exositional uroses te discussion of nonarametric identification will roceed by drawing analogies to te familiar linear setting. Te two models are: Y = X β + ɛ Y = θ X + ɛ 3 were Y R, X R k, ɛ is unobservable and β R k is an unknown vector, wile θ x : R k R is an unknown function. Te aroriate definition of endogeneity in te linear model is EXɛ, because if EXɛ =, ten β can be consistently estimated by OLS. In contrast, te ertinent definition of endogeneity in te nonarametric model is Eɛ X, because if Eɛ X = ten θ x can be consistently estimated by a nonarametric regression. In order to simlify te discussion of te linear model I will also assume, in tis section only, tat Z and X ave te same dimension. Te strategy to obtain identification in te linear case is to multily bot sides of te first equation in 3 by Z and take exectations. For te nonarametric case we instead take conditional exectations on bot sides of te second equation in 3. In tis way we obtain: EZY = EZX β + EZɛ EY Z = Eθ X Z + Eɛ Z 4 For te linear model, obtaining covariances wit Z is a useful aroac as it yields k equations to solve for β, wic is k dimensional. For te nonarametric model, owever, tis aroac is unsatisfactory as it yields k equations to identify a wole function, wic is an infinite dimensional object. Instead, we take conditional exectations to derive a functional equation. In tis way we obtain an infinite number of equations, one for every ossible value z, to solve for θ x. Under te exogeneity assumtion on te instrument, we simlify te identifying equations from 4 to: EZY = EZX β EY Z = Eθ X Z 5 5

6 In order to identify β and θ x from te equations in 5 we need te existence of unique solutions. Uniqueness follows from te rank conditions on te instruments, wic are: EZX β = β = EθX Z = θx = 6 If te rank condition is not satisfied in te linear model, ten β will not be identified. For any β satisfying EZX β =, te model Y = X β + β + ɛ wit ɛ = ɛ X β is observationally equivalent to te true model and also satisfies EZ ɛ =. Tis rank condition will old if and only if te matrix EZX is invertible. Similarly, for te nonarametric model, if tere exists a function θx satisfying EθX Z =, ten te model Y = θ X + θx + ɛ wit ɛ = ɛ θx is observationally equivalent to te true model and also satisfies E ɛ Z =. Te requirement tat tere be no function θx satisfying EθX Z = was atly named a nonarametric rank condition by Newey & Powell 3. If Z and X satisfy te nonarametric rank condition, ten te distribution of X conditional on Z is said to be comlete in Z. Sufficient conditions for comleteness are only known for a few distributions, most notably for te exonential families. On te one and, te nonarametric rank condition is quite demanding, and one tat may not be met in many emirical alications. On te oter and, te nonarametric rank condition is recisely wat is required for nonarametric identification and ence, witout furter assumtions, doing away wit it is not ossible witout losing identification. Witout te rank condition, te model is artially identified. Te identified set is comosed of te models tat are observationally equivalent to te true model and are consistent wit te exogeneity assumtion on Z. Hence, te identified sets are given by te solutions to te equations in 5, wic can be caracterized as: β + N EZX θ x + N E Z 7 were N EZX is te null sace of EZX and N E Z is te null sace of E Z. Tat is, N E Z is te set of functions wit finite second moment suc tat EθX Z =. Te rank conditions are equivalent to assuming N EZX and N E Z are equal to {}. Witout furter assumtions te identified sets can be quite large. Te sets N EZX and N E Z are bot vector saces. Furtermore, wile te dimension of N EZX is less tan or equal to k, te dimension of N E Z can be infinite.. Limiting te Identified Set In order to derive te asymtotic distribution of te test statistic I need to assume te true model θ x is differentiable a secified number of times and tat some of its derivatives are bounded. 6

7 Newey & Powell 3 ioneered te use of tese tecnical assumtions in te nonarametric IV literature to obtain comactness of Θ and in tis manner solve te ill-osed inverse roblem. Tese assumtions also imly te statistics in our analysis beave uniformly on Θ. Besides teir tecnical advantages, tese regularity assumtions also aid in identification. Te use of known or assumed roerties of θ x to aid in identification as been reviously used in te artial identification literature. Manski 997, for examle, exlores te identifying ower of monotonicity, semi-monotonicity and semi-concavity restrictions. In a similar vein, Blundell, Cen & Kristensen 4 exloit te boundedness of Engel curves to aid in identification. Te autors require a unique bounded solution to te equation EY Z = Eθ X Z, because any unbounded solutions cannot be an Engel curve. Tis is equivalent to assuming tat tere exist no bounded functions θx suc tat EθX Z =. Tis condition is known as bounded comleteness and it is weaker tan comleteness. Since θ x is assumed to satisfy a set of regularity conditions, functions tat do not satisfy tem sould be excluded from te identified set. In some instances, tese conditions may be enoug to attain identification. Tus, before recisely defining te identified set, it is necessary to first formally state te regularity conditions I assume for te true model θ x. Let X R k and define λ to be a k dimensional vector of nonnegative integers, also known as a multi-index. In addition, define λ = k i= λ k and let D λ θx = λ θx/ x λ... x λ k k. For m, m and δ ositive integers satisfying m > k/, δ > k/ and k/m + k/δ < / define te norm: θ S = λ m+m D λ θx + x x δ dx were te function θx is assumed to be m + m times differentiable wit resect to its arguments. Weigting te integrand by + x x δ 8 imlies tat if θ S <, ten te tails of te function θx and its derivatives decay at a rate of at least o + x x δ. Tis tail condition allows us to carry out te analysis for X wit full suort. If we assume X as comact suort, ten te tail condition is unnecessary and δ can be set equal to zero. I assume te true model θ x satisfies θ S B for some known constant B. Te functional sace Θ is defined to be: Θ = {θx : θ S B} 9 Gallant & Nycka 987 sow te θ S B imlies max λ m su x D λ θx is uniformly bounded in θx Θ. Terefore, in te definition of Θ I am imlicitly assuming te true model θ x and its derivatives u to order m are bounded. Tese assumtions, wic are made for 7

8 tecnical reasons, aid te instrument Z in identifying θ x as well. By assumtion, it is only necessary to consider alternatives witin Θ, and ence te identified set is defined to be: Θ = {θx Θ : EY θx Z = } In articular, since te functions θx Θ are uniformly bounded, similarly as in Blundell, Cen & Kristensen 4, bounded comleteness is sufficient to attain identification..3 Analysis Witout Identification Even wit te restrictions imosed on Θ, it may still be a strong requirement tat θ x be identified. In addition, tere are currently not tests for te yotesis tat θ x is identified. It is terefore rudent to utilize a testing framework tat is robust to θ x not being identified. Instead of erforming yotesis tests on a function, te test statistic develoed in tis aer will allow us to test restrictions on te identified set Θ. Te kind of yotesis tests I allow for are of te form: H : Θ R H : Θ R = were R is a set of functions tat satisfy a roerty we wis to test for. Some restrictions are imosed on R, and tese will be discussed in detail in Section 3. Te null yotesis in is tat at least one element of Θ satisfies te restrictions imosed in R. For examle, R can be a arametric set of functions, in wic case te test in is a arametric secification test against a nonarametric alternative. R can also be te set of demand functions tat are inelastic at a oint x. In tis context, te null yotesis in is tat at least demand function in Θ is inelastic at x. Sae restrictions can also be included in tis framework. For examle, R can be te set of cost functions wit economics of scoe or te set of roductions functions wit economies of scale. Wen θ x is identified, te null yotesis and te alternative in simlify to H : θ x R and H : θ x / R. If nonarametric identification is attained, ten te resent testing framework reduces to tests on te true model θ x. Terefore, tere are no clear disadvantages to adoting a testing framework tat does not assume θ x is identified. Tere are, owever, tree imortant advantages. First, wen we are interested in a functional of θ x, we may be able to construct informative confidence intervals even wen te functional is not identified. Te tye of coverage requirement we adot was originally exlored in Imbens & Manski 4. I will refer to tese confidence regions as confidence regions for identifiable functionals, wic is equivalent to te concet of confidence regions for functions of te identifiable arameters in Romano & Saik 6a. 8

9 Second, even toug Z may not be able to identify θ x, it may still be able to answer interesting questions about it, suc as weter it satisfies certain sae restrictions. Tird, te inability to reac comelling conclusions using tis aroac oints out te limitations of te instrument to nonarametrically identify arameters of interest. Tis is not a weakness of te rocedure, but rater its virtue of reflecting te limits of wat can be learned witout arametric assumtions. Te following sections elaborate on te first two oints..3. Confidence Regions for Identifiable Functionals In many instances, we will not be interested in θ x but in a functional of it. Often, θ x not being identified will imly certain functionals of it will not be identified eiter. It is still ossible, owever, to use yoteses like to construct a confidence region tat asymtotically covers te value of te functional at every element of Θ wit a resecified robability tat controls te size of te test. Suose, for examle, tat te functions θx are Walrasian inverse demand functions, X is quantity, and we are interested in Walrasian consumer surlus wit endogenous quantity at some level q. Every θx Θ as a corresonding consumer surlus CSθ = q Te identified set for Walrasian consumer surlus is given by: { q } S = CSθ = θxdx θq q : θx Θ θxdx θq q. Consumer surlus will not be identified unless every θx Θ as te same consumer surlus. If we do not restrict our attention to te functional sace Θ, ten we migt not be able to learn anyting at all about consumer surlus unless θ x is identified. To see tis, recall from 7 tat te identified set witout smootness restrictions is given by θ x + N E Z. Suose θ x is not identified, and tat tere is a function θ x N E Z suc tat CSθ. For all scalars λ, te function θ x + λθ x is in te identified set as well because λθ x N E Z for all λ. Te consumer surlus of θ x + λθ x is given by CSθ + λθ = CSθ + λcsθ, and ence, by coosing λ aroriately we can find an inverse demand function in te identified set wit an arbitrary Walrasian consumer surlus. Te assumtion θ x Θ els in te artial identification roblem by limiting te ossible inverse demand functions in Θ and terefore also te ossible consumer surluses in S. As ointed out in Manski 997, owever, economic restrictions sould also be exloited to limit S. For examle, by imosing te constraint tat demand functions must be weakly ositive we can avoid S being te wole real line. If warranted, additional assumtions suc as monotonicity or convexity of te demand function can also be used to furter limit Θ and ence Θ and S. 9

10 Te set S is te limit of wat can be nonarametrically learned about consumer surlus witout furter assumtions. A large set S reflects tat te instrument Z is not able to rovide muc information about consumer surlus for te true inverse demand function θ x. Under suc circumstances, it may be advisable to adot a arametric model for θ x in order to erform inference on consumer surlus. In tis case, recise arametric estimates of consumer surlus reflect not te ability of te instrument Z to estimate it, but te imortance of te arametric assumtion in identifying it. A large identified set S uts emasis on te crucial role te arametric assumtions lay not only in estimation but also in identification. It also sows ow far from te trut te estimates migt be if te arametric model is missecified. Using a family of yoteses of te form H : Θ R, it is ossible to construct confidence regions for te identifiable functionals. Tis is a weaker coverage requirement tan building a confidence region for te wole set S, as discussed in Cernozukov, Hong & Tamer 7 and Romano & Saik 6b. We aim to construct a set Ŝ suc tat if CSθ S, ten: lim P CSθ N Ŝ α 3 were α is te cosen level of control for a Tye I error. Te construction of tis tye of confidence regions is a secial case of our framework. To see tis, consider te family of sets Rλ = {θx Θ : CSθ = λ} and te corresonding family of null yoteses H λ : Θ Rλ. Section 3 I will sow ow to construct statistics to test tis kind of yotesis wile controlling te robability of a Tye I error. Terefore, if Ŝ is te set of λ suc tat H λ is not rejected, ten Ŝ will ave te desired roerty 3 by te duality of confidence intervals and yotesis testing. Tis construction can be alied to a wide array of functionals suc as diverse kinds of elasticities and values of te function θ x and its derivatives. Te outlined rocedure can also be used to construct joint confidence regions for multile functionals. For examle, if in addition to consumer surlus CSθ we also care about rice elasticity Eθ, ten we can construct a joint confidence region for CSθ and Eθ by using te sets Rλ = {θx Θ : CSθ = λ, Eθ = λ }. In.3. Identified Features Witout Identification of θ x In some instances, deending on te distribution of Y, X and Z, it will still be ossible to infer roerties of θ x witout it being identified. I illustrate tis ossibility using a arametric secification test as an examle. Suose we wis to test weter θ x belongs to some arametric family θx, β. If θ x is

11 identified, ten we can use a yotesis test of te form H : Θ R to test weter θ x belongs to a secified arametric family by letting R = {θx : θx = θx, β for some β}. Even witout identification, owever, we may still be able to conclude tat θ x does not belong to te secified arametric family wen tis is indeed te case. Suose tat for all β: Eθ X Z EθX, β Z 4 wit ositive robability. If inequality 4 is satisfied, ten it follows tat θ x does not belong to te secified arametric family, and in addition by 5 and tat neiter do any functions in Θ. Terefore, we will asymtotically reject H : Θ R wit robability tending to one. Hence, since θ x Θ we can infer θ x does not belong to te secified arametric family eiter. On te oter and, if θ x does not belong to te secified arametric family, but some oter function θx Θ does, ten H : Θ R will be true. We will consequently fail to reject H : Θ R wit asymtotic robability α, were α is te cosen size of te test. Uon failing to reject tis null yotesis it is not ossible to reac any conclusions about θ x. Tis is not a weakness of te rocedure. On te contrary, tis reflects tat witout identification it is imossible to conclude weter te function in te identified set belonging to te arametric family is really θ x or not. In general, it will be ossible to conclude θ x / R, even witout identification, as long as Θ R =. Tis requirement on te instrument Z can still be fairly restrictive, toug it is always weaker tan assuming θ x is identified. Te derivation of low level conditions tat ensure Θ R = wen θ x / R is a callenging exercise beyond te scoe of tis aer. 3 Test Statistics In tis section I develo a test statistic for te null yotesis H : Θ R. Since Θ Θ, te null yotesis H : Θ R is equivalent to asking weter tere is a function θx Θ R satisfying EY θx Z =. Terefore, for certain sets R, H : Θ R is equivalent to: H : inf E EY θx Z fzz = 5 θx Θ R If te set R is suc tat te infimum in 5 is attained, ten te null yotesis in 5 is equivalent to H : Θ R. Tis tye of equivalence was originally used in Romano & Saik 6a to derive confidence regions for functions of te identifiable arameters and is similar in sirit to te Anderson & Rubin test 949.

12 Te advantage of using 5 over H : Θ R is tat no estimation of Θ is required. As a reliminary ste towards being able to test 5 I construct a test statistic for H : θx Θ for some arbitrary θx in Θ. Adating arguments from Hall 984 I develo a test statistic T N θ tat is asymtotically normally distributed wen evaluated at a θx Θ and diverges to infinity oterwise. Following 5, I ten use tis result to sow it is ossible to test H : Θ R by using te test statistic I N R = min Θ R T N θ. If Θ R =, ten wen comuting I N R we will minimize T N θ over values for wic it diverges to infinity and ence I N R will diverge to infinity as well. On te oter and, if Θ R ten te minimizer of T N θ over Θ R sould be close to Θ R because T N θ diverges to infinity for all oter functions θx. In Section 3.3 I formalize tis intuition and sow I N R converges in robability to min Θ R T N θ wen H : Θ R is true. Tis result in turn allows us to obtain te asymtotic distribution of I N R under te null yotesis. 3. Testing Strategy and Assumtions for H : θx Θ A function θx is in Θ if and only if it is consistent wit te exogeneity assumtion on Z. Hence, te null yotesis H : θx Θ is equivalent to H : EY θx Z =. Once θx is fixed, te second yotesis can be viewed as a secification test for weter te nonarametric regression function EY θx Z is equal to zero. In order to imlement tis secification test, I emloy te Nadaraya-Watson kernel estimator for EY θx Z. Assume Z R d and define: ˆf Z z n = zi z n K 6 N d i n were te kernel Ku is a symmetric density function and is te cosen bandwidt. Nadaraya-Watson estimator is ten given by: ÊY θx z n = zi z n K N d i n y i θx i ˆf Z z n Te 7 Hall 984 derives te asymtotic beavior of te integrated square error of te Nadaraya- Watson estimator, wic is given by ÊY θx Z = z EY θx Z = z dz. Te secification test for H : EY θx Z = can be imlemented by examining te integrated square error under te null yotesis. Te results from Hall 984 can be readily alied to derive te asymtotic beavior of ÊY θx Z = z dz and in tis way obtain a test statistic for te null yotesis H : θx Θ. In order to avoid te comutationally intensive calculation of

13 integrating over te suort of Z, I adat te arguments in Hall 984 to instead examine te asymtotic beavior of: Q N θ = N N n= ÊY θx zn ˆf Zz n 8 Using ˆf Z z n as a weigt function allows us to avoid te ossible irregular beavior of te Nadaraya- Watson estimator wen f Z z n is close to zero. In addition, using ˆf Z z n as a weigt function simlifies te derivation of te asymtotic beavior of Q N θ as it no longer is te ratio of two random variables. Te following assumtions are sufficient for establising te asymtotic beavior of te statistic Q N θ for any θx in te functional sace Θ: ASSUMPTION : Te observations {y n, x n, z n } N n= are i.i.d. wit Y R, X R k and Z R d. Tey are distributed wit density f ZXY z, x, y and generated by te model secified in wit θ x Θ and ɛ not a deterministic function of X. ASSUMPTION : Te marginal distribution f Z z is bounded and continuous almost everywere. Te marginal distribution f ZX z, x is also bounded. ASSUMPTION 3: Tere exists a δ > suc tat EY 4+δ <. ASSUMPTION 4: Te moments EY θx J Z = z are well defined for J {,, 4} and continuous and bounded in z for J {, } uniformly in θx Θ. Te moments E Y J Z are bounded for J {,, 3, 4}. ASSUMPTION 5: Te kernel Ku is a bounded, symmetric density wit full suort on R d. Te bandwidt satisfies and N d. Assumtions -5 are stronger tan wat is needed to establis te asymtotic beavior of Q N θ for any articular θx Θ. In Section 3.4, owever, I need to analyze te beavior of Q N θ as a stocastic rocess defined on te sace of bounded functionals on Θ. In tis instance te uniformity in Θ asects of Assumtions -5, wic are unnecessary for te resent analysis, will be crucial. In order to avoid te introduction of multile sets of assumtions I adot te stronger tan necessary Assumtions -5 tat are alicable trougout te aer. Te assumtion tat Ku as full suort in R d is also unnecessary. It is made to simlify te roofs and notation wen canges of variables in integration are necessary. 3

14 3. Test Statistic for H : θx Θ Under Assumtions -5, if Q N θ is roerly centered and scaled, ten it converges in distribution to a standard normal random variable wen evaluated at a function θx Θ and diverges to infinity oterwise. In order to simlify notation we define: v n θ = y n θx n 9 were te notation v n θ is meant to emasize tat te residual deends on te function θx tat imlies it. Te U-Statistic tat will rovide te roer centering for Q N θ is: ˆB N θ = N 3 d N n= i<n K zi z n v nθ + vi θ ˆB N θ is te sum of te squares tat are generated uon exanding te terms ÊY θx z n in Q N θ. It is also necessary to define te estimator for te asymtotic variance of Q N θ. We let ˆσ C θ = ˆσ CI θσ CII, were σ CII = KuKu + wdu dw and ˆσ CI is defined by: ˆσ CIθ = 4N 4! N! 4d N zn z K n= i<n j<i k<j K zi z K Te estimator ˆσ CI θ can be comutationally intensive. zj z K zk z v nθv i θv j θv kθdz For tis reason it may be desirable to instead use te also consistent estimator σ CI θ = 4 Êvnθ Z = z ˆf 4 Z zdz. Te asymtotic beavior of σ CI θ is dominated by te cross terms tat arise from exanding te fourt ower. Te sum of tese cross terms is identical to ˆσ CI θ. I use ˆσ CI θ instead of σ CI θ because it is less cumbersome to sow its uniform consistency over Θ. We now state te teorem tat will allow us to test H : θx Θ. Teorem 3.. Let T N θ = N d Q N θ ˆB N θ. If Assumtions -5 old and θx Θ, ten: d ˆσ C θ T Nθ = N ˆσ C θ Furtermore, if Assumtions -5 old and θx / Θ, ten: d ˆσ C θ T Nθ = N ˆσ C θ Q N θ ˆB L N θ N, Q N θ ˆB N θ + Since ˆB N θ is te sum of te squares tat result from exanding te terms ÊY θx z n in Q N θ, te statistic T N θ = N d Q N θ ˆB N θ is equal to a sum of cross terms of te form: d zi z n K K zj z n 4 v i θv j θ

15 Groued in trees, terms like form a symmetric kernel tat sows T N θ is a symmetric U- E Statistic of order tree. Te exectation of is equal to d E K zi z n vi θ zn, wic converges to E EY θx Z f Z Z. Hence, for sufficiently small te exectation of will be strictly ositive causing T N θ to be imroerly centered and diverge to ositive infinity. On te oter and, wen θx Θ te exectation of conditional on any air z k, v k θ for k {n, i, j} is equal to zero, wic imlies T N θ is not only roerly centered but also degenerate of order one. Due to its degeneracy, te asymtotic beavior of T N θ is governed by te second term in its Hoeffding decomosition wen θx Θ. Te scaling by N d terefore corresonds to te standard deviation of te second term in te Hoeffding decomosition of N d T N θ. Wen θx / Θ, T N θ is no longer degenerate for sufficiently small. Te asymtotic beavior of N d T N θ is ten governed by te first term in its Hoeffding decomosition, wic as a standard deviation of order ON. Consequently, T N θ diverges to ositive infinity wen θx / Θ not only because it is imroerly centered, but also because it is being scaled by a factor of N d tat is too large. Wile te imroer centering of T N θ ensures te divergence is to ositive infinity and not negative infinity, te scaling by N d tan N. causes te divergence to be at a rate faster 3.3 Testing Strategy and Assumtions for H : Θ R I construct a test statistic for te null yotesis H : Θ R by exloiting its equivalence wit H : inf θx Θ R E EY θx Z fz Z =. For tis equivalence to old we need to ensure tat te infimum is attained. Attainment will be imlied by comactness of Θ R under te norm: θ Cδ = max su λ m x D λ θx + x x δ 3 were m was secified in te definition of te norm θ S in 8 and k/ < δ < δ. Te functional E EY θx Z fz Z is continuous under θ Cδ and ence comactness of Θ R under θ Cδ will imly te infimum in 5 is attained. Gallant & Nycka 987 sow Θ is comact under θ Cδ. Terefore, since closed subsets of comact sets are comact, if R is closed under θ Cδ, ten Θ R will be comact under θ Cδ. Due to te norm θ Cδ being very strong, it is straigtforward to sow te set R will be closed for a wide array of interesting yoteses. For examle, for any M R m and transformation F : Θ R m tat is continuous under θ Cδ, te sets R = {θx : F θ M} and R = {θx : F θ = M} will be closed under θ Cδ. Since θ Cδ is a strong norm, F : Θ R m being continuous under it is a weak requirement. Examles 5

16 of functionals tat are continuous under θ Cδ include levels of θx and its derivatives, consumer surlus and rice elasticities. Relacing H : Θ R by 5 and assuming R is closed under θ Cδ as two limitations. Te first limitation is tat requiring R to be closed is not just a tecnical condition but actually restricts te economic content of te yoteses tat can be tested. As an examle consider testing weter an Engel curve is weakly decreasing. By letting F θ = inf x θ x and R = {θx : F θ }, we see tat R is closed under θ Cδ because F θ is continuous under θ Cδ. Terefore, a test for weak monotonicity fits te resent framework. Suose, owever, tat we wis to test for strict monotonicity instead, wic corresonds to R = {θx : F θ < }. Te set R is now oen, wic imlies H : Θ R and H : inf θx Θ R E EY θx Z f Z Z = may no longer be equivalent. Assume, for simlicity, tat θ x is identified and in addition tat it is not strictly monotonic but only weakly so. In tis case H : Θ R is false. On te oter and, te yotesis H : inf θx Θ R E EY θx Z f Z Z = is actually true. By evaluating E EY θx Z f Z Z at a sequence of strictly monotonic Engel curves aroacing te true model θ x we can make E EY θx Z f Z Z arbitrarily close to zero. If we utilize te resent framework regardless, ten we are likely to incorrectly infer θ x is strictly monotonic wen in reality it is only weakly so. A second limitation of te resent testing framework is tat yoteses are restricted to te functional sace Θ. Te smootness assumtions on Θ ariori limit te set of yoteses tat can be tested. For examle, for many arametric families θx, β, te set R = {θx Θ : θx, β for some β} will be closed under θ Cδ. Terefore, te resent framework includes a arametric secification test as a secial case. Tere is an imlicit assumtion, owever, tat te arametric family θx, β itself is sufficiently smoot and ence included in Θ. It is not ossible, for examle, to utilize te resent framework to do a secification test for a arametric family θx, β tat is not differentiable. Te equivalence of H : Θ R and H : inf θx Θ R E EY θx Z f Z Z = for a large class of sets R allows us to base our test on te latter null yotesis. Te advantage of using te second null yotesis is tat it does not require estimation of te identified set Θ. To test tis yotesis we relace E EY θx Z f Z Z wit te statistic T N θ from Teorem 3. and define: I N R = inf Θ R T Nθ 4 From Teorem 3. we know T N θ diverges to infinity for all θx / Θ. Terefore, I N R diverges to infinity wen Θ R =, because in tis case te infimum in 4 is taken over functions tat 6

17 are not in Θ. On te oter and, wen Θ R tere are functions θx Θ R tat are also in Θ. Since T N θ converges to a normal distribution for suc functions and it diverges to infinity for te rest, T N θ sould be minimized on functions tat are close to te identified set. We terefore exect I N R to beave like inf Θ R T N θ wen Θ R. In rincile, tis rationale can be used to construct similar test statistics tat utilize alternatives to T N θ as a first ste. An excellent reference to te large number of admissible tests for te null yotesis H : EY θx Z = can be found in Hart 997. Wile tere is an extensive literature examining te ower of tese tests for te null yotesis H : EY θx Z, it is unclear ow te coice of a articular test statistic affects te ower in te second ste for te null yotesis H : Θ R. Tis is a comlex question beyond te scoe of tis aer. I need te following assumtions to formalize te intuition develoed in tis section: ASSUMPTION 6: Te set R is closed under te norm θ Cδ. ASSUMPTION 7: For some l, te bandwidt satisfies N l and N d+l m +δ k m δ. ASSUMPTION 8: Te density f Z z is differentiable, wit bounded gradient, and for small enoug tere exists a function gu, z, y, x satisfying Ku f ZY X u+z, y, x f ZY X z, y, x y θx i gu, z, y, x for all θ, i {, } and gu, z, y, xdudzdydx <. Te role of Assumtion 6 as already been extensively discussed. Assumtion 7 is necessary to ensure tat certain biases converge to zero sufficiently fast. Te assumtion tat k/m +k/δ < /, wic is necessary to control te covering numbers of Θ, also guarantees Assumtion 7 is comatible wit Assumtion 5, wic requires N d. Alternatively, if iger order kernels are used for Ku to reduce tese biases, ten it is ossible to allow for a iger range of rates for te bandwidt. Assumtion 8 is necessary to allow us to excange te order of integration and differentiation. It also imlies certain error terms in Taylor exansions will beave uniformly in Θ. 3.4 Test Statistic for H : Θ R Before stating te asymtotic distribution of I N R it will be elful to rovide a basic overview of convergence in distribution in te sace of bounded functionals. For a detailed discussion lease refer to Cater.5 in van der Vaart & Wellner 998. Te set of functionals on Θ, F : Θ R satisfying su θx Θ F θ < form a metric sace denominated L Θ. In tis sace te distance between two functionals F and F is measured by su θx Θ F θ F θ. It is elful to tink of T N θ as an element of L Θ. Te statistic T N θ mas every θx Θ to R. In 7

18 addition, because θx Θ are uniformly bounded, we ave su θx Θ T N θ <, wic imlies T N θ is indeed an element of L Θ. Te value T N θ assigns to any articular θx, owever, is random as it deends on te realization of te data. As suc, T N θ is a random variable defined on L Θ because every realization of te data generates a different maing T N θ from Θ to R. Just as for random variables defined on te real line, it is also ossible to tink of T N θ converging in distribution to a random variable on L Θ. In fact, T N θ converges in distribution to a Gaussian rocess Gθ in L Θ. A Gaussian rocess in L Θ is a random maing from Θ to R wit te roerty tat Gθ,..., Gθ M is jointly normally distributed for any finite vector of functions θ x,..., θ M x in Θ. We now roceed to te statement of te main teorem: Teorem 3.. If Assumtions -8 old and Θ R, ten: I N R = inf T Nθ = inf Θ R Θ R Nd Q N θ ˆB L N θ + o inf Gθ Θ R were Gθ is a Gaussian rocess in L Θ. If Assumtions -8 old and Θ R =, ten: I N R = inf T Nθ + Θ R Teorem 3. imlies tat if Θ R is a singleton wit unique element θ U x, ten te test statistic I N R will converge in distribution to Gθ U, wic is a normal random variable. examle, if θ x is identified, ten Θ R will be eiter a singleton, wen θ x R, or emty, wen θ x / R. Corollary 3. establises tat it is ossible to estimate te asymtotic variance of Gθ U and construct a ivotal statistic tat will converge in distribution to a standard normal random variable wen Θ R = {θ U x}. Corollary 3.. Suose Assumtions -8 old and θ x arg min Θ R T N θ. R = {θ U x} is a singleton, ten ˆσ C θ I N R ˆσ C θ I N R +. For If te set Θ L N,. Furtermore, if Θ R = ten Tere are a number of interesting tecnical callenges resent in te roof of Teorem 3.. Te statistic T N θ does not converge to an asymtotically tigt random variable in L Θ and ence te asymtotic beavior of I N R cannot be derived troug a direct use of te continuous maing teorem. Even toug for every samle of size N te U-Statistic T N θ as continuous samle ats in all of Θ, it is only asymtotically uniformly equicontinuous wit resect to θ L X on Θ, were θ L X = Eθ X. Te lack of asymtotic uniform equicontinuity follows immediately from Teorem 3., as T N θ converges to a normal random variable for every θx Θ but diverges to infinity for every θx / Θ. Te identified set Θ, owever, is small relative to Θ; in fact its 8

19 interior relative to Θ is emty. Terefore, T N θ actually fails to be asymtotically uniformly equicontinuous in almost all of Θ, wic imlies it does not converge in distribution in L Θ. I address tis callenge by examining te beavior of te Hoeffding decomosition of T N θ witin a srinking neigborood Θ ɛ N of te identified set Θ, and outside tis neigborood. Since T N θ is a U-Statistic of order tree, its Hoeffding decomosition generates four fully degenerate U-Statistics P N θ, P N θ, P N θ and P 3 N θ, were P i N θ is of order i and P N θ = ET N θ. For θx Θ, T N θ is degenerate and terefore P N θ =. In addition, since P N θ and Θ Θ ɛ N, te following inequality follows: 3 i= inf θ Θ ɛ N R P i Nθ inf T N θ θ Θ ɛ N R 3 inf P θ Θ Nθ i 5 R Te triangular array nature of te roblem and T N θ not being linear in θx revent te direct use of te results in Arcones & Gine 993. Instead, I aly te maximal inequalities from Arcones & Gine 993 to canging classes of functions tat are Liscitz in Θ. Wit tese maximal inequalities it is ossible to sow P 3 N θ converges in robability to zero uniformly in Θ and P N θ is uniformly asymtotically equicontinuous wit resect to θ L X on all of Θ. From Teorem 3. we know te marginals of PN θ on Θ are normally distributed and terefore PN θ converges in distribution to a Gaussian rocess in L Θ, toug robably to Gaussian caos in L Θ. For θx / Θ, te standard deviation of N d P N θ is of order ON, but its level decreases to zero as θx gets closer to Θ. If ɛ N fast enoug, ten te variance of PN θ converges uniformly to zero witin Θ ɛ N. Tis result follows by letting te rate at wic te level of te variance of N d P N θ decreases because of θx being close to Θ comensate for T N θ being scaled by te factor N d i= instead of N. Terefore, since PN θ and P N 3 θ converge in robability to zero uniformly witin Θ ɛ N and PN θ is uniformly asymtotically equicontinuous wit resect to θ L X in Θ, inequality 5 imlies inf ɛ Θ N R T Nθ = inf ɛ Θ N R P N θ + o L inf Θ R Gθ. If ɛ N slowly enoug, ten ET N θ + PN θ, and ence T Nθ, will diverge to infinity uniformly in Θ ɛ N c because of te imroer centering. Terefore, te infimum of T N θ over Θ R will wit ig robability be attained on Θ ɛ N R, wic imlies I N R = inf Θ ɛ N R T Nθ + o and ence establises te teorem. 4 Imlementation Tere are two callenges tat need to be addressed before being able to imlement te test secified in Teorem 3.. First, te test statistic I N R is te solution to an otimization roblem over te 9

20 ossibly nonarametric set of functions Θ R, wic is comutationally roblematic. Second, te aroriate critical values are known only wen Θ R is a singleton, in wic case Corollary 3. can be used to construct a ivotal test statistic. In order to address te first roblem, I sow tat it is ossible to comute I N R by otimizing over saces tat aroximate Θ R witout losing te asymtotic results. For te second roblem, I discuss wen we migt exect Corollary 3. to aly and sow tat subsamling can be used to estimate te aroriate critical values wen tis is not te case. 4. Aroximations to Θ R For certain yoteses, suc as a arametric secification test, te set Θ R is arametric and te comutation of I N R is straigtforward. For most yoteses, owever, te comutation of I N R will require solving a minimization roblem over a nonarametric set of functions. I address tis callenge by sowing in Teorem 4. tat it is ossible to solve te minimization roblem over an aroximating sieve witout losing te asymtotic results of Teorem 3. and Corollary 3.. Teorem 4.. Suose Assumtions -8 old, let {Θ J } Θ be a sequence of closed sets under te norm θ L X = Eθ X and define θj x arg min Θ J R T N θ.. If Θ J R Θ R and Θ R =, ten min ΘJ R T N θ + and ˆσ C θ J T NθJ +.. If Θ R, su θ Θ R inf θj Θ J R θ θ J L X = o l for l wit N l and N d+l m +δ k m δ, ten min ΘJ R T N θ = min Θ R T N θ + o. If in addition te set Θ R is a singleton, ten ˆσ C θ J T NθJ L N,. Teorem 4. requires tat {Θ J R} be able to aroximate Θ R uniformly well, wic is different from te tyical requirement tat {Θ J } be able to aroximate Θ. If we wis to test weter demand is inelastic at a oint x using slines, for examle, ten te slines must satisfy tis constraint and still be able to aroximate tose demand functions in Θ tat are inelastic at x. For ointwise restrictions tis is a straigtforward requirement. For global restrictions, suc as non-negativity monotonicity or concavity, we will need to coose sae-reserving slines; see Cen 6 for examles. Te sieve {Θ J R} is required to aroximate Θ R under θ L X = Eθ X, wic is weaker tan te norm θ Cδ under wic R is required to be closed. Te aroximation error must decrease to zero at a rate o l, wic is governed by te bandwidt. Desite deending only on

21 te bandwidt, tis rate cannot be made arbitrarily slow because Assumtion 7 rovides a lower bound on ow slow can decrease to zero. Te use of iger order kernels would increase te rate at wic biases vanis, allowing for a slower rate requirement on te bandwidt and terefore also a slower rate requirement on te aroximation error from using {Θ J R} instead of Θ R. 4. Coice of Critical Values In tis section I discuss ow to coose te aroriate critical values for te test statistic. Since for most yoteses te set Θ R will be nonarametric, we will need to comute our test statistic over an aroximating sace {Θ J R}. Terefore, I will resent te results in tis section in terms of te test statistics from Teorem 4.: Ĩ N R = inf Θ J R T Nθ ˆσ C θ JT N θ J 6 were θ J x arg min Θ J R T N θ. As imlied by Teorem 4., te results in tis section will also aly to te statistics I N R and ˆσ C θ T N θ, were θ x arg min Θ R T N θ. 4.. Critical Values From te Normal Aroximation Corollary 3. and Teorem 4. imly tat if te set Θ R is a singleton, ten ˆσ C θ J T NθJ L N,. Terefore, if we exect Θ R to be a singleton wenever Θ R, ten we can use te quantiles of te standard normal distribution as critical values. Tis will be te case, for examle, if θ x is identified. Nonarametric identification of θ x, owever, is not a necessary condition for ˆσ C θ J T NθJ to be ivotal. As an illustrative examle suose tat X, Z R and we wis to test weter tere is a linear function θx = α + xβ in Θ. Te constraint set R l corresonding to tis yotesis is: R l = {θx = α + xβ : α, β R} 7 Suose tere exist two linear functions, θ x = α x + β x and θ x = α x + β x in Θ. It follows tat EY Z = α i + EX Zβ i for bot i = and i =, and terefore α α = EX Zβ β. Tis equality can old for α, β α, β if and only if EX Z = EX. Terefore, if X is not mean indeendent of Z and te set Θ R l is not emty, ten Θ R l will be a singleton. Hence, ˆσ C θ J T Nθ J will be ivotal as long as EX Z EX, even if θ x is not nonarametrically identified. In general, assuming Θ R is a singleton wen it is not emty is equivalent to assuming te instrument can identify a unique model witin R tat is consistent wit te exogeneity assumtion

22 on Z. In te linear secification test examle, EX Z EX is exactly wat is needed to identify te linear model wen Eɛ Z =. For certain yoteses, te set R will rovide enoug structure for te instrument to identify a unique model witin R. In tese cases, Θ R will be a singleton if it is not emty and te statistic ˆσ C θ J T NθJ will be ivotal. Tis discussion, owever, is secondary to weter θ x is nonarametrically identified or not. For examle, even toug in a linear secification test we exect ˆσ C θ J T Nθ J to be ivotal, if θ x is not nonarametrically identified, ten uon failing to reject H : Θ R l we cannot conclude tat θ x is indeed linear. Furtermore, for certain yoteses, suc as sae restrictions or levels of a function at a oint, te requirements for Θ R to be a singleton may be eiter ard to satisfy or interret, in wic case it migt be advisable to use te statistic ĨNR instead of ˆσ C θ J T Nθ J. If te set Θ R is a singleton wen it is not emty, ten we can use te quantiles of te standard normal distribution as critical values. We can construct a test wit asymtotic size α for H : Θ R by rejecting wenever: ˆσ C θ JT N θ J > q α 8 were q α is te α quantile of te standard normal distribution. Wen te null yotesis is true and Θ R is a singleton, Corollary 3. and Teorem 4. imly P ˆσ C θ J T NθJ > q α α, wic controls te robability of a Tye I error. Te test is also consistent, since wen te null is not true P ˆσ C θ J T NθJ > q α because ˆσ C θ J T NθJ diverges to ositive infinity. 4.. Critical Values From Subsamling If we do not exect ˆσ C θ J T Nθ J to be ivotal, ten we need to instead use ĨNR as our test statistic. Teorem 3. and Teorem 4. imly tat under te null yotesis ĨNR were Gθ is a Gaussian rocess defined on L Θ. L min Θ R Gθ, Since te distribution of min Θ R Gθ deends on te infinite dimensional set Θ, it is unclear ow to obtain critical values troug simulation. I ave also not yet been able to sow te consistency, or lack tereof, of te bootstra in te resent context. Since te bootstra can sometimes be inconsistent, see Andrews for examles, it seems imrudent to utilize te bootstra witout first establising its validity. For tis reason I resort to using subsamling, wic is consistent under very general conditions. Te subsamling construction I use is from Politis, Romano & Wolf 999. Let b N be a sequence of ositive integers satisfying b N and b N /N. Define B N to be te set of all ossible subsets of size b N of a dataset wit N observations and ˆB N a random subset of B N satisfying