Semiparametric Efficiency in GMM Models with Nonclassical Measurement Error

Transcription

1 Semiarametric Efficiency in GMM Models with Nonclassical Measurement Error Xiaohong Chen New York University Han Hong Duke University Alessandro Tarozzi Duke University August 2005 Abstract We study semiarametric efficiency bounds and efficient estimation of arameters defined through general nonlinear, ossibly non-smooth and over-identified moment restrictions in the resence of nonclassical measurement error in the variables of interest. Identification is achieved by assuming the existence of an auxiliary database that contains information about the conditional distribution of the variables of interest given the mismeasured and ossibly other roxy variables, and assuming that this conditional distribution is the same in both the rimary and auxiliary data sets. We rovide semiarametric efficiency bounds for both the verify-out-of-samle case, where the two samles are indeendent, and the verify-in-samle case, where auxiliary information is available for a subset of the rimary samle. We derive searate bounds for the cases where the roensity score here interreted as the robability of not observing the true value of the variables of interest given the roxy variables is unknown, or known, or it belongs to a correctly secified arametric family. We find that the roensity score is ancillary for arameter estimation in the verify-in-samle case, but not in the verify-out-ofsamle case. We show that sieve conditional exectation rojection based GMM estimators achieve the semiarametric efficiency bounds for all the above mentioned cases, and establish their asymtotic efficiency under mild regularity conditions. Although inverse robability weighting based GMM estimators are also shown to be semiarametrically efficient, they need stronger regularity conditions and clever combinations of nonarametric and arametric estimates of the roensity score to achieve the efficiency bounds for various cases. JEL: C, C3 Key words: Semiarametric Efficiency Bounds, GMM, Measurement Error, Missing Data, Auxiliary data, Sieve Estimation. We thank John Ham, Guido Imbens, Oliver Linton, Whitney Newey, Bernard Salanié and seminar articiants at several institutions for insightful comments and suggestions. We also thank the National Science Foundation and the Sloan Foundation for generous research suorts. We are solely resonsible for any remaining errors and omissions. Xiaohong Chen, Det of Economics, New York University, 269 Mercer Street, New York, NY 0003, xiaohong.chen@nyu.edu; Han Hong, Det of Economics, Duke University, Social Sciences Building, PO Box 90097, Durham, NC 27708, hanhong@econ.duke.edu; Alessandro Tarozzi, Det of Economics, Duke University, Social Sciences Building, PO Box 90097, Durham, NC 27708, taroz@econ.duke.edu.

2 Introduction It is well known that many emirical studies are comlicated by the resence of measurement error in the variables under examination. The interretation of the results is esecially comlicated in the resence of non-linear models, and when the measurement error is correlated with the true unobserved variables. In such circumstances, identifying assumtions become necessary to overcome the lack of identification that results from the missing information. One solution to this identification roblem is based on the assumtion that information on the true value of the variables in the data set of interest the rimary data set) can be recovered using auxiliary data sources under a conditional indeendence assumtion. The key element of the identification strategy is that the auxiliary data set must rovide information about the conditional distribution of the true variables of interest given a set of roxy variables, where the roxy variables are observed in both the rimary samle and the auxiliary samle. In other words, conditional on the roxy variables, the distributions of the variables of interest are assumed to be indeendent of whether they belong to the rimary samle or the auxiliary samle. In this aer, we study semiarametric efficiency bounds and efficient estimation of arameters defined through general nonlinear, ossibly non-smooth and over-identified moment conditions under a conditional indeendence assumtion. We rovide semiarametric efficiency bounds for the cases when the roensity score is either unknown, or known, or it belongs to a correctly secified arametric family. In our context, the roensity score is defined as the robability that one observation belongs to the subsamle where only the roxy variables are observed. We calculate efficiency bounds both for the verify-out-of-samle case, where the auxiliary samle and the rimary samle are indeendent, and for the verify-in-samle case, where the auxiliary samle is a subset of the rimary samle. These efficiency variance bounds indicate that the roensity score is ancillary for the verify-in-samle case but not for the verify-out-of-samle case. That is, more information on the roensity score will not affect the asymtotic efficiency variance bounds for arameters defined in the verify-in-samle case, but will imrove the asymtotic efficiency for arameters defined in the verify-out-of-samle case. Semiarametric efficiency bounds are imortant for understanding the limits of semiarametric estimation methods, as they reresent the equivalent of the Cramer-Rao lower bound in semiarametric models. Standard references for this literature are given by Newey 990) and Bickel, Klaassen, Ritov, and Wellner 993) among others. Recent works on nonlinear models with classical measurement error include Hausman, Ichimura, Newey, and Powell 99), Hausman, Newey, and Powell 995), Newey 200), Hsiao and Wang 995), Li 2002) and Schennach 2004). However, the resence of non-classical measurement errors in economic data is well documented by Bound and Krueger 99a), Bound, Brown, Duncan, and Rodgers 994), Bound, Brown, and Mathiowetz 200) and Bollinger 998). Inference in nonlinear models in resence of nonclassical measurement error has been studied in Carroll and Wand 99), Seanski and Carroll 993), Carroll, Ruert, and Stefanski 995), Lee and Seanski 995) and Chen, Hong, and Tamer 2005). Our efficiency bound calculations build on existing results in the literature on missing data and on rogram evaluation, results that we generalize and comlement in several ways. Efficiency bounds in regression models when variables are missing at random in the sense of Rubin 976)) are studied in Robins, Rotnitzky, and Zhao 994), who develo a unified framework for the case of missing regressors,

3 and in Robins and Rotnitzky 995) and Rotnitzky and Robins 995) for the case of missing resonse variables. While the framework of these contributions is equivalent to our validate-in-samle case, we also calculate efficiency bounds for the verify-out-of-samle case, which arises when auxiliary information is rovided through an indeendent validation data set. These new results are interesting, since they show that more information on the roensity score leads to smaller efficiency variance bounds. This latter finding is analogous to an imortant result due to Hahn 998), who ioneered the analysis of semiarametric efficiency bounds in the rogram evaluation literature, under the assumtion that the latent outcomes are indeendent on actual treatment conditional on observable covariates. In this framework, Hahn 998) shows that the roensity score interreted as the robability of treatment conditional on observed covariates is ancillary for estimation of the average treatment effect, but not of the average treatment on the treated. The structure of the derivations in Hahn 998) reresents the oint of dearture for our calculations of semiarametric efficiency variance bounds for arameters imlicitly defined through general nonlinear, ossibly non-smooth and over-identified moment restrictions in resence of non-classical measurement error. We derive results under the assumtions of unknown, known or arametric roensity score, for both the verify-in-samle case and the verify-out-of-samle case. For each one of these cases, we also develo two classes of sieve-based GMM estimators that achieve the efficiency bounds. Each estimator relies only on one nonarametric estimate; a conditional exectation rojection based GMM hereafter CEP-GMM) estimators only requires the nonarametric estimation of a conditional exectation, while an inverse robability weighting based GMM hereafter IPW-GMM) estimator only needs a nonarametric estimate of the roensity score. We establish asymtotic normality and efficiency roerties of both estimators under weaker regularity conditions than the existing ones in the literature. In articular, we allow for non-smooth moment conditions, and for unbounded suort of conditioning or roxy) variables, which is very imortant for measurement error alications. Overall, we find that the CEP-GMM estimator resents some advantages over the IPW-GMM estimator. First, its root-n asymtotic normality and efficiency can be derived without the strong assumtion that the unknown roensity score is uniformly bounded away from zero and one. Second, the CEP-GMM estimator is characterized by a simle common format that achieves the relevant efficiency bound for all the cases we consider, and the contributions to the variance deriving from each of the two stages of the estimation rocedure are orthogonal to each other. Instead, the IPW-GMM estimator will be generally inefficient when the roensity score is known and, interestingly, even if it is arametrically estimated using a correctly secified arametric model; in such instances, clever combinations of nonarametric and arametric estimates of the roensity score are needed to achieve the bounds. In a recent aer, Chen, Hong, and Tamer 2005) roose the CEP-GMM estimator in the context of non-classical measurement error models, but do not examine its efficiency roerties. Robins, Mark, and Newey 992a) study the semiarametric efficiency roerties of a conditional exectation rojection estimator in the context of a causal regression model, when the roensity score takes a correctly secified arametric form. In the context of the treatment effect literature, a CEP-based estimator is also used in Imbens, Newey, and Ridder 2005), who rovide a criterion for choosing the number of terms in the Semiarametric efficiency bounds of average treatment effects with or without knowledge of the roensity score are also studied in Heckman, Ichimura, and Todd 998), Hirano, Imbens, and Ridder 2003), Firo 2004) and Firo 2005). 2

4 nonarametric first ste series estimator of the rojection. They demonstrate the asymtotic otimality of the criterion they roosed in terms of exected mean-squared-error. The IPW-GMM estimator extends the roensity score based estimators for treatment effects in Hahn 998), Heckman, Ichimura, and Todd 998) and Hirano, Imbens, and Ridder 2003) to a broad class of arameters defined by general nonlinear, ossibly non-smooth and over-identified moment restrictions. For missing data models that corresond to the verify-in-samle case, Robins, Rotnitzky, and Zhao 994), Robins and Rotnitzky 995), Rotnitzky and Robins 995) and Robins, Rotnitzky, and Zhao 995) have roosed various semiarametric estimators that require arametric estimation of the roensity score, and Wooldridge 2002) and Wooldridge 2003) showed that the use of an estimated rather than known arametric roensity score leads to efficiency gains within the framework of M-estimators. Yet the issue of how to use a correctly secified arametric roensity score to obtain asymtotically efficient estimators for arameters defined by a general moment restriction under the verify-out-of-samle case has not been addressed before. In section 2 we describe the model and give motivating examles in the context of the literature on nonlinear models with nonclassical measurement error. Section 3 resents the semiarametric efficiency bounds. Section 4 shows that the otimally weighted CEP-GMM estimators achieve the bounds when the roensity score is unknown, or known, or it belongs to a arametric family. Section 5 shows that the otimally weighted IPW-GMM estimators achieve the bounds when the roensity score is unknown, while aroriate modifications are necessary to achieve efficiency when the roensity score is known or correctly arameterized. In Section 6 we illustrate emirically the erformance of the different estimators in the estimation of earnings quantiles and cumulative distribution functions, when non-classical measurement errors are resent. Section 7 concludes. All roofs are given in the aendixes. 2 The Model We assume that the researcher has access to two data sets: the rimary data set is a random samle from the oulation of interest, while an auxiliary samle will serve the urose of ensuring the identification of arameters that would not be identified by the rimary data set alone. Let D denote a binary variable which is observed by the researcher and is equal to zero for observations that belong to the auxiliary samle, and equal to one otherwise. We distinguish two cases. We refer to the first case as to the verify-out-of-samle framework, which is relevant when the rimary and the auxiliary data are two different and indeendent data sets. In this case D = for observations that belong to the rimary data set. In the second framework, the verify-insamle case, the auxiliary data set is a subset of the rimary data set. In this case the rimary data set includes both observations for which D = and others for which D = 0. 2 We are interested in the estimation of arameters β R d β defined imlicitly in terms of general 2 In the rogram evaluation literature, the analogue of the validate-out-of-samle case is reresented by the estimation of the treatment effect for the treated, while the average treatment effect for the whole oulation is the analogue of our validate-in-samle framework. 3

5 nonlinear moment conditions. In the verify-out-of-samle case such conditions are described by E m Z; β) D = = 0 if and only if β = β 0 ) while in the verify-in-samle case the condition is E m Z; β) = 0 if and only if β = β 0 2) where Z = Y, X) and m ; β) is a set of functions with dimension d m d β. Notice that in both case ) and 2) the moment conditions are assumed to hold in the rimary samle. The main challenge is that at least some of the variables in Y are only observed with error for observations for which D =, that is, in the whole rimary samle in case ) and in a subset of it in case 2). Identification is ossible if the researcher has access to an auxiliary data set D = 0) which contains both Y, the true value of the variables of interest, and X, a set of roxy variables that are also otentially of interest, and if the following fundamental conditional indeendence assumtion holds: Assumtion Y D X. If assumtion holds, identification follows by noting that, under case ) E m Z; β) D = = E m Z; β) x, D = 0 f x D = ) dx, while under case 2), E m Z; β) = E m Z; β) x, D = 0 f x) dx. Therefore, E m Z; β) x, D = 0 can be recovered using observations where D = 0, and it can be integrated against either fx D = ) or fx) to recover the arameters of interest. Conditional indeendence assumtions have been roosed in several different settings in the econometrics and statistics literature to recover identification lost due to the resence of missing information. Such assumtions have been used to study inference in models with attrition or nonresonse e.g. Robins and Rotnitzky 995), Rotnitzky and Robins 995), Wooldridge 2002), Wooldridge 2003)), in the rogram evaluation literature see e.g. the references surveyed in Heckman, LaLonde, and Smith 999)), in estimation of overty and inequality in small areas merging information from censuses and household surveys Elbers, Lanjouw, and Lanjouw 2003)), in recovering comarability over time of statistics calculated using data collected with different methodology e.g. Clogg et al., Schenker 2003), Tarozzi 2004a)). In the measurement error setu, the moment condition tyically deends only on Y, which contains the unobserved true variables e.g. income, exenditure, or union status), while X contains reorted variables of interest. In general, X does not necessarily have to be the reorted values of Y, but can also reresent some roxy variables that contain information about Y. Assumtion in these models is stated as f Y X, D = ) = f Y X, D = 0). For examle, assumtion is satisfied in the stratified samling design where a nonrandom resonse based subsamle of the rimary data is validated. The stratified samling rocedure can be illustrated as 4

6 follows. Let U be i.i.d U0, ) random variables indeendent of both X and Y, and let X) 0, ) be a measurable function of the rimary data. A stratified samle is obtained by validating every observation for which U < X). In other words, X) secifies the robability of validating an observation after X is observed. This samling scheme corresonds to case 2), and it is commonly used to oversamle a suboulation of the rimary data set where more severe measurement error is susected to be resent. Assumtion is valid as long as the samling rocedure adoted to create the auxiliary data set is based only on information rovided by the distribution of the rimary data set. If a simle random subset of the rimary data is validated, X) is a constant and the auxiliary data set Y, X is characterized by the same distribution of Y, X as the rimary data set. In this case assumtion is easily seen satisfied. In this case, which is common in the statistics literature, the auxiliary data set is usually called a validation data set. 3 Semiarametric Efficiency Bounds In this section we calculate the efficiency bound for the estimation of β defined by either moment conditions ) or 2). The derivation is very closely related to Hahn 998), but we use a different factorization of the likelihood function for case ). The semiarametric efficiency bounds are derived by calculating the efficient influence function associated with the athwise derivatives of the arameters β. These efficient influence functions are the rojection of the moment conditions onto the tangent sace of all regular arametric submodels satisfying the moment restrictions see Aendix A for more details). To state the efficiency bounds we introduce some notations. observations on Z i = Y i, X i ), D i, where Y i is only observed when D i = 0. Let n denote the size of a samle of Let = P rd = ) and X) = P rd = X). In this aer we use β to mean an arbitrary value in the arameter sace, but to save notation β is sometimes used as the true arameter value β 0 in this section. Define E X; β) = E m Z; β) X to be the conditional exectation of the moment conditions given X, and define V m Z; β) X) = E m Z; β) m Z; β) X E X; β) E X; β) to be the conditional variance of the moment conditions given X. In addition, also define Jβ = β E m Z; β) D = and J β 2 = E m Z; β). β Assumtion 2 i) Both J β and J 2 β have full column rank equal to d β; ii) The data X i, Y i, D i comes from an i.i.d. samle; iii) = P rd = ) 0, ). Theorem Under assumtion and assumtion 2, the asymtotic variance lower bound for ) n ˆβ β for any regular estimator ˆβ is given by J β Ω β J β). When the moment condition case ) holds, J β Jβ and Ω β = Ω β where Ω β = E X) 2 X) 2 V m Z; β) X) + X)) 2 E X; β) E X; β). 5

7 When the moment condition case 2) holds, J β J 2 β and Ω β = Ω 2 β where Ω 2 β = E V m Z; β) X + E X; β) E X; β). X) Remark : The results resented above, as well as the ones in the remaining of the aer, also encomass maximum likelihood models as a secial case. In such models, the researcher is interested in the estimation of arameters of a likelihood model in the rimary samle, where the log-likelihood function is given by log gy ; β). All the results will hold as long as one secializes the results to the case of exact identification, and uses the score β log g Y ; β) as the relevant moment function mz; β). 3. Information content of the roensity score It is interesting to analyze whether knowing X) decreases the semiarametric efficiency bounds for the arameters β. Hahn 998) showed that it does for estimation of the average effect of treatment on the treated, while the roensity score is ancillary for the average treatment effect. A similar result holds for the GMM model discussed here. Theorem 2 Under assumtions and 2, if X) is known, then the asymtotic variance bound for estimating β is J β Ω β β) J. When the moment condition case ) holds, J β J β and Ω β = Ω β where Ω β = E X) 2 X)2 2 V m Z; β) X) + X)) 2 E X; β) E X; β). When the moment condition 2) holds, J β J 2 β and Ω β = Ω 2 β given in Theorem. In other words, knowledge of X) reduces the semiarametric efficiency bound for β with out of samle validation, but it does not with in samle validation. The following argument rovides an intuition for this result. When 2) holds, β is defined through the relation m y, x; β) f y x) dyf x) dx = 0. The roensity score X) does not enter the definition of β, therefore its knowledge should not affect the variance bound for β. However, the relation that identifies β when ) holds clearly deends on X): m y, x; β) x) f y x) dyf x) dx = 0. The results for in samle validation in theorems and 2 can also be derived from roosition 8.2 of Robins, Rotnitzky, and Zhao 994) when there is a single hierarchy in the case of monotone missing data atterns and when the instrument functions are given in the conditional mean model. Another interesting question is what is the efficiency bound for the estimation of β defined by moment condition ) if the roensity score is unknown but is assumed to belong to a correctly secified arametric 6

8 family and the following theorem rovides an answer. Assume that the roensity score takes a arametric form X; γ) and let γ X) = X; γ)/ γ, and define the score function for γ as S γ = S γ D, X) = D X; γ) X; γ) X; γ)) γx). Theorem 3 Under assumtions and 2, if X) = X; γ) belongs to a correctly secified arametric family indexed by γ and ES γ D, X)S γ D, X) is ositive definite, then the efficient variance bound for estimating β defined by moment condition ) is given by J β Ω β β) J where Jβ = Jβ and Ω β = Ω β + E E X; β) γ X) ) ESγ S γ E E X; β) ) γ X). This variance bound is clearly larger than Ω β stated in Theorem 2, but it is smaller than the bound in Theorem. This latter result can be verified noting first that while the bound in Theorem 3 corresonds to the variance of the following influence function: ) D) X) EX; β) mz; β) EX; β)) + Proj D X)) X)) S γ D, X) + X)EX; β), where we use Proj Z Z 2 ) to denote the oulation least squares rojection of a random variable Z onto the linear sace sanned by Z 2. The conclusion follows noting that the variance bound stated in Theorem for moment condition ) is instead the variance of the following influence function D)X) DEX; β) + mz; β) EX; β), X)) and noting that the corresonding variance is larger Validation samles A secial case of assumtion is when the auxiliary samle is randomly drawn from the same oulation of the rimary samle. In this case the auxiliary samle is called a validation samle e.g. Carroll and Wand 99), Seanski and Carroll 993), Lee and Seanski 995)), and the following assumtion holds. Assumtion 3 Y, X D. It is easy to see that assumtion 3 is equivalent to adding to assumtion the statement that X D, or that X) = an unknown constant). Within this framework, the moment conditions ) and 2) coincide with each other, so that the semiarametric variance bound stated in the next theorem alies to both conditions. 3 In a revious version of the aer we also derive the information bound when the stratifying scheme and fx D = 0) is known. Interestingly, we found that knowledge of the stratifying samling scheme does not reduce the variance bound if the auxiliary data set is disjoint from the rimary data set, but does reduce the variance bound if the auxiliary data set is a subset of the rimary data set. The following theorem formalizes the revious discussion. 7

9 Theorem 4 Under assumtions 2 and 3, the semiarametric variance bound for the arameter β defined ), through the moment condition of ) or 2) is given by J β where Jβ = Jβ = J β 2 and J β Ω v Ω v = E V m Z; β) X + E X; β) E X; β). The semiarametric variance bound Ω v in theorem 4 is identical to Ω 2 β given in theorem. This is because X) is ancillary to β defined through the moment condition 2) as shown in Theorem. However, knowledge of assumtion 3 does decrease the semiarametric variance bound for β defined through moment condition ). Intuitively, assumtion 3 imlies that f X D = 0) rovides useful information about β in addition to those rovided by f Y X, D = 0) in the auxiliary samle. On the one hand, when moment condition 2) holds, the entire samle is used in the estimation, so that knowledge of assumtion 3 in addition to assumtion does not hel increase estimation efficiency. On the other hand, when moment condition ) holds, if only assumtion is being used when in fact assumtion 3 also holds, then only the information contained in f X D = ) and f Y X, D = 0) is being used in the estimation of β, and the information contained in f X D = 0) is not being utilized. Therefore in this case assumtion 3 rovides additional information for estimation efficiency. 4 CEP-GMM Estimation There are two alternative aroaches for semiarametric estimation of the arameter β when assumtion holds and if the true value Y is only observed when D = 0. The first one is based on a conditional exectation rojection method. The second one is based on the inverse robability weighting IPW) or roensity score weighting method. In this section we show that the otimally weighted GMM estimator of β using a sieve conditional exectation rojection aroach is semiarametrically efficient. We discuss IPW in the Section Efficient estimation with unknown roensity score 4.. The estimator and heuristics for asymtotic variances Under assumtion, E X; β) = E mz; β) X, D = 0 for all β, and moment condition ) is equivalent to E E X i ; β 0 ) D i = = 0, while moment condition 2) is simly E E X i ; β 0 ) = 0. The rojection method first estimates E X; β) nonarametrically from the auxiliary samle, and then averages this nonarametric estimator over the rimary samle. In the following, we use subscrits and a to refer to observations belonging to the rimary samle and to the auxiliary samle resectively. Let n be the size of the rimary samle and n a be the size of the auxiliary samle. Observations in the rimary samle are indexed by i =,..., n. Observations in the auxiliary samle are indexed by 8

10 j =,..., n a. Under moment condition ) verify-out-of-samle case), n = n + n a. Under moment condition 2) verify-in-samle case), n = n. Let Ê X; β) denote a nonarametric estimate of E X; β) using the auxiliary samle. Chen, Hong, and Tamer 2005) hereafter CHT) used a sieve based method for this nonarametric estimation. Let q l X), l =, 2,...} denote a sequence of known basis functions that can aroximate any square-measurable function of X arbitrarily well. Also let q kna) X) = q X),..., q kna) X) ) and Q a = q kna) X a ),..., q kna) X ana ) for some integer kn a ), with kn a ) and kn a )/n 0 when n. Then for each given β, the first ste nonarametric estimation can be defined as, n a Ê X; β) = m Z aj ; β) q kna) X aj ) Q ) aq a q kn a) X). j= A generalized method of moment estimator for β 0 can then be defined as n n ) ˆβ = arg min Ê X i ; β)) Ŵ Ê X i ; β). 3) β B n i= This rojection based GMM method CEP-GMM) is closely related to the imutation method of Hahn 998). The difference is that Hahn 998) s imutation method lugs in Y whenever it is observed while the rojection method only uses Ê X; β). The -consistency and asymtotic normality of this CEP-GMM estimator have been established in CHT. Following the roof of their claim A.2), we have the following asymtotic reresentation: n i= ) n Ê X i ; β 0 ) = n i= n E X i ; β 0 ) + n i= n a n a j= f X X aj ) fx aj D = 0) mz aj; β 0 ) EX aj ; β 0 ) + o ), where we use f X X) to denote the density of X in the rimary data set, and o ) reresents a term that converges to 0 in robability. When moment condition ) holds, n = n + n a, f X X) = f X D = ) and f X X) ) X) = fx D = 0) X)). In this case we can also write the influence function for i= n n i= Ê X i; β 0 ) as } D X i ) iex i ; β 0 ) + D i ) mz i ; β 0 ) EX i ; β 0 ) + o ). 4) X i )) The roof of Theorem shows that the two terms in the influence function corresond to the two comonents of the efficient influence functions that contain information about f X D = ) and f Y X) resectively. These two terms are orthogonal to each other, so that n n ) Avar Ê X i ; β 0 ) = Ω β, n i= 9

11 where Ω β is given in Theorem. When moment condition 2) holds, f X X) = f X), n = n and The influence function for i= n f X X) ) = fx D = 0) X)). n i= Ê X i; β 0 ) can then be written as } E X i ; β 0 ) + D i ) mz i ; β 0 ) EX i ; β 0 ) + o ). 5) X i ) The two terms in the influence function corresond to the two comonents of the rojected efficiency influence function that contain information about f X) and f Y X) resectively in the roof of Theorem. The orthogonality between these two terms imlies that n n ) Avar Ê X i ; β 0 ) = Ω 2 β, n i= where Ω 2 β is given in Theorem. The semiarametric efficiency bounds given in Theorem are then achieved by an otimally weighted GMM estimator ˆβ for β 0 that uses a weighting matrix Ŵ = Ω β +o ) Asymtotic roerties Before we formally resent the semiarametric efficiency roerty of the CEP-GMM estimator, we need to introduce some notations and assumtions. Let the suort of X be X = R dx. We could use more comlicated notations and let X = X c X dc, with X c being the suort of the continuous variables and X dc the suort of the finitely many discrete variables. Further we could decomose X c = X c X c2 with X c = R d x, and X c2 being a comact and connected subset of R d x,2. Then, under simle and usual modification of the assumtions, the large samle results stated below remain valid. To avoid tedious notation yet to allow for some unbounded suort elements of X, we assume X = X c = R dx in this aer. For any d x vector a = a,..., a dx ) of non-negative integers, we write a = d x k= a k, and for any x = x,..., x dx ) X, we denote the a -th derivative of a function h : X R as: a hx) = a x a... xa dx d x For some γ > 0, let γ be the largest integer smaller than γ, and let Λ γ X ) denote a Hölder sace with smoothness γ, i.e., a sace of functions h : X R which have u to γ continuous derivatives, and the highest γ -th) derivatives are Hölder continuous with the Hölder exonent γ γ 0,. The Hölder hx). sace becomes a Banach sace when endowed with the Hölder norm: h Λ γ = su x hx) + max a =γ su x x a hx) a hx) x x) γ γ <. x x) Let Λ γ X, ω ) denote a weighted Hölder sace of functions h : X R such that h ) + 2 ω /2 is in Λ γ X ). We call Λ γ c X, ω ) h Λ γ X, ω ) : h ) + 2 ω /2 Λ γ with radius c). 0 c < } a weighted Hölder ball

12 The sieve estimator ÊX; β) needs to converge to EX; β) in some metric. We allow suorts of the roxy variables to be unbounded, and use a weighted su-norm metric defined as g,ω su gx, β) + x 2 ω/2 x X,β B for some ω > 0. Also we let Π n g denote the rojection of g onto the closed linear san of q kna) x) = q x),..., q kna)x)) under the norm,ω. Let f Xa x) = f X D=0 x) and f X x) = f X D= x). The following assumtion is sufficient to ensure that Ê ; β) converges to E ; β) under the suremum norm,ω. Assumtion 4 Let Ŵ W = o ) for a ositive semidefinite matrix W, and the following hold:. for all β B, E ; β) belongs to a weighted Hölder ball Λ γ c X, ω ) for some γ > 0 and ω 0; 2. + x 2 ) ω f X x)dx <, + x 2 ) ω f Xa x)dx < for some ω > ω 0; 3. For each fixed x, Ex; β) is continuous at β for all β B; 4. V armz i ; β) X i = x, D i = 0 is bounded uniformly over x and β. 5. For any E ; β) Λ γ c X, ω ), there is a sequence Π n E in the sieve sace G n = g ; β) Λ γ c X, ω ) : gx; β) = q kna) x) πβ)} such that E ; β) Π n E ; β),ω = o). Also E a q kna) X)q kna) X) is non-singular. Theorem 5 Let β be the CEP-GMM estimator given in 3). Under assumtions, 2 and 4, if kn a ), kn a) n a 0, then ˆβ β 0 = o ). Additional regularity conditions are required for stating the asymtotic normality results. Let E ) = E D = ) and E a ) = E D = 0). Denote h 2 2,a = hx) 2 f Xa x)dx = E a hx) 2 } and Π 2n h be the rojection of h onto the closed linear san of q kna) x) = q x),..., q kna)x)) under the norm 2,a. Assumtion 5 Let β 0 intb), E EX; β 0 )EX; β 0 ) be ositive definite, and the following hold:. assumtion 4. is satisfied with γ > d x /2 and assumtion 4.2 is satisfied with ω > ω + γ; 2. For each fixed x, and for some δ > 0 Ex;β) β is continuous in β B with β β 0 δ, E su EX ; β) β < ; β: β β 0 δ 3. There exist a constant ɛ 0,, a δ > 0 and a measurable function b ) with E bx ) < such that Ẽx;β) β with Ẽ E,ω δ. fx ) X) 2 4. E a f Xa X) < ; Ex;β) β bx) Ẽ E,ω ɛ for all β B with β β 0 δ and all Ẽ Λ γ c X, ω ) 5. kn a ) = O n a ) dx 2γ+dx ), n a ) γ f Xa ) Π 2n 2γ+dx f X ) f X ) f Xa ) = on /2 ). 2,a

13 Theorem 6 Let β be the CEP-GMM estimator given in 3). Under Assumtions, 2, 4 and 5, we have β β0 ) N 0, V ), with V = J β W J β) J β W Ω βw J β J β W J β), where Ω β is given in Theorem. Furthermore, if W = Ω β ), then β β 0 ) N 0, V 0 ), with V 0 = J β Ω β J β), where J β = J β and Ω β = Ω β under moment condition ), and J β = J 2 β and Ω β = Ω 2 β condition 2). under moment Remark 2: i) Assumtions 4 and 5 allow for m Z; β) to be non-smooth such as in quantile based moment functions. ii) The weights ω and ω are needed since the suort of the conditioning variable X is allowed to include the entire Euclidean sace. When X has bounded suort and f X is bounded above and below over its suort, we can simly set ω = 0 = ω in Assumtions 4 and 5, and relace assumtion 5. with the assumtion that 4. holds with γ > d x /2. iii) Since f X X), assumtion 5.4 is f Xa X) = X) ) X)) automatically satisfied under the condition 0 < x) < imosed in Hirano, Imbens, and Ridder 2003) and Firo 2004). Assumtion 5.5 will be satisfied under mild smoothness conditions imosed on X) X) that are weaker than those imosed in Hirano, Imbens, and Ridder 2003) and Firo 2004). In ) dx 2γ+dx articular, if we let kn a ) = O na, the growth order which leads to the otimal convergence rate ) of Ê ; β 0) E ; β 0 ) 2,a = O n γ 2γ+dx a, then assumtion 5.5 is satisfied with f X ) f Xa ) Π f X ) 2n f Xa ) = ) 2,a o n dx ) 22γ+dx) = o kn a ) 2. For examle, both assumtions 5.4 and 5.5 will be satisfied as long as a ) ) Λγ X, ω ) with γ > d x /2. The roofs of Theorems 5 and 6 follow directly from those in CHT, who also rovide simle consistent estimators of V and V 0 : V = Ĵ W Ĵ ) Ĵ W ΩW Ĵ Ĵ W Ĵ ) and V0 = Ĵ Ω Ĵ ), where for moment condition ), and Ĵ = n n i= ÊX i; β) β, Ω = n a ) ) n n a υ n aj Û aj υ aj Û aj + ÊXi a n 2 ; β)êx i; β) ) j= i= n Û aj = my aj, X aj ; β) ÊX aj; β), Q ) υ aj = q kna) X i ) a Q a q kna) X aj). n i= n a 2

14 4.2 Efficient estimation with known roensity score Suose now that the roensity score X) is known. Theorems 2 and 6 show that the otimally weighted CEP-GMM estimator given in 3) still achieves the semiarametric efficiency bound for β defined by moment condition 2). Even if the estimator is no longer efficient for β defined through moment condition ), it is ossible to construct an efficient estimator for case ) using the sieve estimate Ê X; β) and the known X). Notice that under assumtion, the moment condition ) is equivalent to E E X i ; β 0 ) X i) = 0. Hence the otimally weighted GMM using the following samle moment condition will give an efficient estimator for β 0 defined through ): n i= Ê X i ; β) X i), ˆ where ˆ = n n. From the roof in CHT, n n i= Ê X i; β 0 ) X i) ˆ i= X i ) is asymtotically equivalent to E X i ; β 0 ) + D } i X i ) mz i; β 0 ) EX i ; β 0 ) + o ), whose asymtotic variance is equal to Ω β, so that the estimator achieves the bound stated in Theorem Efficient estimation with a validation samle Suose now that assumtion 3 holds, that is, the auxiliary data set is actually a validation data set, so that X) =. Theorems 4 and 6 show that the otimally weighted CEP-GMM estimator defined in 3) still achieves the semiarametric efficiency bound for β defined by moment condition 2), but not for β defined through moment condition ), as under assumtion 3 Ω β > Ω2 β in Theorem. Intuitively, this is because the estimator using 3) for case ) does not exloit the fact that n a n a j= m Z j ; β 0 ) 6) should be close to zero. This set of moment conditions is not orthogonal to n n i= Ê X i; β), which are used in 3) but it does contain additional information about f X D = 0) that is useful for estimating β 0 defined by ). Under assumtion 3, one might be temted to estimate β 0 using the validation data set alone: ˇβ = arg min n a m Z j ; β) Ŵ n a m Z j ; β). β n a n a j= When Ŵ is otimally chosen, ) ˇβ β0 has asymtotic variance J β ˇΩ β β) J where j= ˇΩ β = V ar m Z j; β 0 )) > Ω v, 3

15 where Ω v = E V m Z; β 0) X + E X; β 0 ) E X; β 0 ) = V m Z; β 0)) V E X; β 0 )) is the efficient bound stated in theorem 4 under assumtion 3. Therefore the estimator ˇβ that uses the validation samle alone is not efficient. Efficiency is achieved combining otimally the two moment conditions 3) and 6). However, a simler alternative efficient estimator is the otimally weighted GMM using the following samle moment n Ê X i ; β), i= which is the same moment condition as 3) but taking n to be the entire samle instead of just the subsamle where D =. This is because under assumtion 3, moment condition ) becomes the same as case 2): E E X i ; β 0 ) = 0, and therefore one should use all the observations in the samle. Again from the roofs in CHT and under assumtion 3, n n i= Ê X i; β 0 ) has the asymtotic reresentation i= E X i ; β 0 ) + D } i mz i ; β 0 ) EX i ; β 0 ) + o ), which has asymtotic variance Ω v, and hence allows to achieve the bound stated in Theorem Efficient estimation with arametric roensity score Suose now that the roensity score X) is correctly arameterized as X; γ) u to a finite-dimensional unknown arameter γ. Theorems 2 and 6 show that the otimally weighted CEP-GMM estimator defined in 3) still achieves the semiarametric efficiency bound for β defined by moment condition 2). However, according to Theorems 3 and 6, such an estimator is no longer efficient for β defined through moment condition ). Using the equivalent exression of moment condition ): E E X i ; β 0 ) X i) = 0, we can again construct an efficient estimator for β 0 based on the sieve estimate Ê X; β) and the correctly secified arametric form X; γ). In articular, the otimally weighted GMM estimator using the following samle moment condition will achieve the efficient bound in Theorem 3 for β defined through ): n i= Ê X i ; β) X i; ˆγ), 7) ˆ where ˆ = n n and ˆγ is the arametric MLE estimator that solves the score equation for γ: n Sˆγ D i, X i ) = n i= i= D i X i ; ˆγ) X i ; ˆγ) X i ; ˆγ)) ˆγX i ) = 0. 4 In a recent communication, Whitney Newey has suggested another simle efficient GMM estimator of β 0 using an increasing set of moments n n i= DimZi; β) and n i= Di ˆ)AXi), where AXi) is a growing set of measurable function of Xi. 4

16 Theorem 7 Let X; γ) be the arametric roensity score function known u to the arameters γ and let ES γ0 D, X) S γ0 D, X) be ositive definite. Let β 0 satisfy the moment condition ) and β be its CEP- GMM estimator using the samle moment 7). Under assumtions, 2, 4 and 5, we have β β 0 ) N 0, V ), with V = Jβ W J β ) Jβ W Ωβ W Jβ J β W J β ), where Ω β is given in Theorem 3. Further, if W = Ω β, then β β 0 ) N 0, V 0 ), where V 0 = ) Jβ Ω β J β is the efficiency variance bound given in Theorem 3. The roof of this theorem is very similar to the revious one and hence omitted. It suffices to oint out that n n i= Ê X i; β 0 ) X i;ˆγ) ˆ can be shown to be asymtotically equivalent to i= X i ) E X i ; β 0 ) + D } i X i ) mz i; β 0 ) EX i ; β 0 ) + E E X i ; β 0 ) γx i ) nˆγ γ0 ), where ˆγ γ0 ) = ES γ0 D, X) S γ0 D, X) n S γ0 D i, X i ) + o ). i= We remark that even when a arametric assumtion is being made about the roensity score X; γ) in fact even if in addition f Y ) is assumed to be a arametric likelihood), the inference about β is still semiarametric. This is because the marginal density fx) is still nonarametric and contains semiarametric information about β. This exlains why nonarametric estimation is still needed to achieve the efficient variance bound for β. We also remark that if we are willing to assume that a arametric assumtion f γ Y X) is correctly secified, then we can relace Ê X; β) by a arametric estimate: E X; β, ˆγ) = m y, x; β) fˆγ y X) dy, where ˆγ is a maximum likelihood estimate for γ using the subsamle where D = 0: D i ) Sˆγ Y i X i ) = i= i= D i ) log f Y i X i ; ˆγ) γ = 0. It is easy to show that this achieves the semiarametric efficiency bound when f γ Y X) is correctly arameterized. For illustration, consider only the case when X i ) is unknown. In this case, the efficient influence functions for the moment conditions given a correctly secified arametric model of f γ Y X) are ) D X) Proj m Z; β) E X; β) X) D) S E X; β) γ Y X) + D under moment condition ) and ) D Proj m Z; β) E X; β) X) D) S γ Y X) + E X; β) 5

17 under moment condition 2). n i= Next, we note that the arametric estimated moment condition has the following influence function: n n E X i ; β 0, ˆγ) = E X i ; β 0, γ 0 ) + γ E n E X i ; β 0, γ 0 ) ˆγ γ) + o ), n i= where E n denotes exectation taken with resect to the rimary samle. Under moment condition ), it can be calculated that γ E n E X i ; β 0, γ 0 ) = γ E E X i; β 0, γ 0 ) D = and hence u to o ), i= n n X) = E Di E X i; β 0 ) + Proj D X) m Z; β 0) E X; β 0 ) D) S γ Y X) i= E X i; β 0 ; ˆγ) is ) D X) m Z; β 0) E X; β 0 ) D i) S γ Y i X i ), which is identical to the desired efficient influence function. Similarly, under moment condition 2), it can be calculated that γ E n E X i ; β 0, γ 0 ) = γ E E X i; β 0, γ 0 ) ) D = E X) m Z; β 0) E X; β 0 ) D) S γ Y X), and that u to o ), i= n n E X i ; β 0 ) + Proj i= E X i; β 0, ˆγ) is ) D X) m Z; β 0) E X; β 0 ) D i) S γ Y i X i ). This also achieves the efficient influence function. Similarly, using the arametric E X; β, ˆγ) also achieves the semiarametric efficiency bound in cases when the roensity score is known, or is correctly arametrically secified. ), 5 IPW-GMM Estimation An alternative estimation method for β is the inverse robability weighting based GMM IPW-GMM). Several authors have considered inverse robability weighting aired with a conditional indeendence assumtion for estimation in resence of missing information. Recent examles include arametric IPW as in Wooldridge 2002), Wooldridge 2003), Robins, Mark, and Newey 992b) and Tarozzi 2004b), for missing data models, and nonarametric inverse robability weighting as in Hirano, Imbens, and Ridder 2003) for the case of mean treatment effect analysis. In this section, we extend existing results and first show that the otimally weighted IPW-GMM estimator of β is semiarametrically efficient when the roensity score is unknown. The same estimator, however, will be generally inefficient when the roensity score is known or it belongs to a correctly secified arametric family; clever combinations of nonarametric and known or arametric estimated roensity scores are needed to achieve the semiarametric efficiency bounds for these cases. 6

18 5. Efficient estimation with unknown roensity score 5.. the estimator and heuristics for asymtotic variances The IPW-GMM method uses the fact that under assumtion, moment condition ) is equivalent to: X) ) E mz; β 0 ) X)) D = 0 = 0; while moment condition 2) is equivalent to: E mz; β 0 ) X) D = 0 = 0. Let ˆ X) be a consistent estimate of the true roensity score. Then we can estimate β 0 defined by case ) using GMM with the following samle moment: n a n a j= m Z j ; β) ˆ X j ) ˆ ˆ X j ) ˆ, 8) and estimate β 0 defined by case 2) using GMM with the following samle moment: n a n a j= m Z j ; β) ˆ ˆ X j ). 9) The inverse robability weighting aroach is considered semiarametric when ˆ X) is estimated nonarametrically. In this case, it can be shown that the samle moment 8) evaluated at β 0 is asymtotically equivalent to n i= X i ) D i ) m Z i ; β 0 ) X i ) + E X i; β 0 ) D i X i ) + o ). X i ) The two comonents of this influence function are negatively correlated. Because of this, the asymtotic variance might be smaller than that of the estimator of β 0 based on moment condition 8) with the known X). This is ointed out by Hirano, Imbens, and Ridder 2003) in the treatment effect literature. The influence function can be rewritten in terms of two orthogonal terms as D i ) m Z i ; β 0 ) E X i ; β 0 ) n i= j= X i ) X i ) + D ie X i ; β 0 ) which is identical to the influence function in 4). Therefore, Avar n a ˆ X j ) ˆ m Z j ; β 0 ) = Ω β n a ˆ X j ) ˆ, } + o ), where Ω β is given in Theorem. An otimally weighted GMM estimator for β 0 defined by case ) using samle moment 8) should then achieve the semiarametric efficiency bound stated in Theorem. The influence function reresentation for samle moment 9) can be calculated as D i ) m Z i ; β 0 ) + o ), i= X i ) + E X i; β 0 ) D i X i ) X i ) 7

19 whose two comonents are again negatively correlated. As in the revious case, the influence function can be written in terms of two orthogonal comonents as i= } D i ) m Z i ; β 0 ) E X i ; β 0 ) X i ) + E X i; β 0 ) + o ), which identical to the influence function in 5). Hence, an otimally weighted GMM estimator for β 0 defined by case 2) using this samle moment 9) achieves the semiarametric efficiency bound for case ) stated in Theorem Asymtotic roerties In this subsection to emhasize that the true roensity score function is unknown and has to be estimated nonarametrically, we use o x) ED X = x) to indicate the true roensity score and x) to denote any candidate function. 5 Let ) be a sieve estimator of o x) that uses the combined samle D i, X i ) : i =,..., n = n a + n }. Let Z ai = Y ai, X ai ) : i =,..., n a } be the auxiliary i.e. D = 0) data set. We define the IPW-GMM estimator β for moment condition ) as ) n a ) X ai ) n a X ai ) β = arg min mz ai ; β) Ŵ mz ai ; β) β B X ai ) X ai ) n a i= and the IPW-GMM estimator β for moment condition 2) as ) n a β = arg min mz ai ; β) Ŵ β B X ai ) n a i= There are two oular sieve nonarametric estimators of o ): n a n a i= n a i) a sieve LS estimator ls x) as in Hahn 998), Das, Newey, and Vella 2003): ls = arg min ) H n n i= D i X i )) 2 /2. i= 0) ) mz ai ; β). ) X ai ) In the aendix we establish the consistency and convergence rate of ls x) under the assumtion that the variables in X have unbounded suort. ii) a sieve ML estimator mle x) as in Hirano, Imbens, and Ridder 2003): mle = arg max D i logx i ) + D i ) log X i )}, ) H n n i= H n = h Λ γ c X ) : hx) = A kn x) π 2} } or h Λ γ c X ) : hx) = exa kn x) π). Recall that E a ) = )f X D=0 x)dx. We define a weighted su-norm as h,ω su x X hx) + x 2 ω/2 for some ω > 0. Assumtion 6 Let Ŵ W = o ) for a ositive semidefinite matrix W, and the following hold: 5 Note that to save notations in the rest of the main text x) denotes true roensity score function. 8

20 . o ) belongs to a Hölder ball H = ) Λ γ c X ) : 0 < x) < } for some γ > 0; 2. + x 2 ) ω f X x)dx < for some ω > 0; 3. there is a non-increasing function b ) such that bδ) 0 as δ 0 and E a su mz i ; β) mz i, β) 2 bδ) β β <δ for all small ositive value δ; 4. E a suβ B mz i ; β) 2 < ; 5. for any h H, there is a sequence Π n h H n such that h Π n h,ω = o). Theorem 8 Let β be the IPW-GMM estimator given in 0) or ). Under assumtions, 2 and 6, if k nn 0, k n, then: β β 0 = o ). Let E ) = )f X x)dx, h 2 = hx) 2 f X x)dx, and Π 2n h be the rojection of h onto the closed linear san of q kn x) = q x),..., q kn x)) under the norm 2. We need the following additional assumtions to obtain asymtotic normality. ox) Assumtion 7 : Let β 0 intb), E EX; β ox) 0)EX; β 0 ) be ositive definite, and the following hold:. assumtions 6. and 6.2 are satisfied with γ > d x /2 and ω > γ; 2. There exist a constant ɛ 0, and a small δ 0 > 0 such that E a su mz i ; β) mz i, β) 2 const.δ ɛ β β <δ for any small ositive value δ δ 0 ; 3. E a su β B: β β0 δ 0 mz i ; β) 2 + X i 2 ) ω < for some small δ 0 > 0; EX;β 4. E 0 ) β + X 2 ) ω 2 <, and for all x X, Ex;β) β is continuous around β 0 ; 5. k n = O n dx 2γ+dx ), n γ 2γ+dx E ;βo) Π o ) 2n E ;βo) o ) = on /2 ) either one of the following is satisfied: 6a. su β B: β β0 δ 0 su x X Ex, β) const. < for some small δ 0 > 0; 6b. E a su β B: β β0 δ 0 EX, β) 4 const. < for some small δ 0 > 0, and f X D=0 ) Λ γ c X ) with γ > 3d x /4; 6c. E a su β B: β β0 δ 0 EX, β) 2 const. < for some small δ 0 > 0, and f X D=0 ) Λ γ c X ) with γ > d x. Theorem 9 Let β be the IPW-GMM estimator given in 0) or ). Under Assumtions, 2, 6 and 7, we have β β 0 ) N 0, V ), with V the same as that in Theorem 6. 9

21 Remark 3: i) The weighting ω is needed since the suort of the conditioning variable X is assumed to be the entire Euclidean sace. When X has bounded suort and f X D=0 is bounded above and below over its suort, we can simly set ω = 0 in Assumtions 6 and 7 and relace 7. with the assumtion that 6. holds with γ > d x /2. Note that assumtion 7.6a is easily satisfied when X has comact suort. When X = R dx, assumtion 7.6a rules out Ex, β) being linear in x; assumtions 7.6b or 7.6c allow for linear Ex, β) but need smoother roensity score x) and density f X D=0. ii) Assumtions 6 and 7 again allow for non-smooth moment conditions. iii) Since f X D=0X) f X X) = ox), the assumtion 0 < o x) < imlies that f X D=0X) f X X), hence E) and E a) in assumtions 6 and 7 are effectively equivalent. iv) Although assumtion 6. imoses the same strong condition 0 < o x) < as that in Hirano, Imbens, and Ridder 2003) and Firo 2004), we relax their other conditions by allowing for unbounded ) n dx 2γ+dx, suort of X and assume weaker smoothness on o x) and E ; β o ). In articular, if we let k n = O the growth order which leads to the otimal convergence rate of ) o ) 2 = O n ) ) assumtion 7.5 is satisfied with E ;βo) Π o ) 2n E ;βo) o ) = o n dx 22γ+dx) = o k 2 n Efficient estimation with known roensity score γ 2γ+dx ), then Suose now that the roensity score X) is known. According to Theorems, 2 and 9, the otimally weighted IPW-GMM estimator using the samle moment 9) with a nonarametric estimate ˆX) will still be efficient for β 0 defined by case 2). However, the otimally weighted IPW-GMM estimator using the samle moment 8) with nonarametric estimate ˆX) will not be efficient for β defined by case ), as knowledge of the roensity score leads to a reduction in the bound in such case. In section 4, we already ointed out that the CEP-GMM estimator that uses the samle moment condition n n i= Ê X i; β) X i) ˆ is semiarametrically efficient for β defined by case ). Another moment condition that leads to efficiency is the one that uses both the sieve estimate ˆ X) and the known X): 6 n a n a j= m Z j ; β) X j ) ˆ ˆ X j ) ˆ. Efficiency can be verified noting that this moment condition has an asymtotic linear reresentation identical to the one in 4). 5.3 Estimation with arametric roensity score In section 4.4 we have shown that one can construct sieve based CEP-GMM estimators that achieve the efficiency bound even if the roensity score has a known arametric form. This raises the interesting question of whether the arametric roensity score versions of 8) and 9) take advantage of the efficiency imrovement of the arametric assumtions. The answer turns out to be no Parametric Proensity Score with In Samle Validation For the sake of simlicity, we consider the case of moment condition 2) first. As theorem and theorem 2 show that the semiarametric efficiency bound is not affected by knowledge of the roensity score 6 An analogous result is obtained in Hirano, Imbens, and Ridder 2003) in the context of the rogram evaluation literature. 20