The Asymptotic Variance of Semi-parametric Estimators with Generated Regressors

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1 The Asymptotic Variance of Semi-parametric stimators with Generated Regressors Jinyong Hahn Department of conomics, UCLA Geert Ridder Department of conomics, USC October 7, 00 Abstract We study the asymptotic distribution of three-step estimators of a nite dimensional parameter vector where the second step consists of one or more non-parametric regressions on a regressor that is estimated in the rst step. The rst-step estimator is either parametric or non-parametric. Using Newey s (994) path-derivative method we derive the contribution of the rst-step estimator to the in uence function. In this derivation it is important to account for the dual role that the rst-step estimator plays in the second-step non-parametric regression, i.e., that of conditioning variable and that of argument. We consider three examples in more detail: the partial linear regression model estimator with a generated regressor, the Heckman, Ichimura and Todd (998) estimator of the Average Treatment ect and a semi-parametric control variable estimator. JL Classi cation: C0, C4. Keywords: Semi-parametric estimation, generated regressors, asymptotic variance. Financial support for this research was generously provided through NSF SS 0896 and We thank Guido Imbens and seminar participants at UC Riverside, the Tinbergen Institute, Yale, FGV-Rio de Janeiro, FGV-São Paulo, Harvard/MIT, UCLA, Mannheim, Sciences Po and CMFI for comments. Geert Ridder thanks the Department of conomics, PUC, Rio de Janeiro for their hospitality. Addresses: Jinyong Hahn, Department of conomics, Bunche Hall, UCLA, Los Angeles, CA 90095, hahnecon.ucla.edu; Geert Ridder, Department of conomics, Kaprilian Hall, USC, Los Angeles, CA 90089, ridderusc.edu.

2 Introduction In a seminal contribution Pagan (984) derived the asymptotic variance of regression coe cient estimators in linear regression models, if (some of) the regressors are themselves estimated in a preliminary step. Pagan called such regressors generated regressors and he characterized the contribution of the estimation error in the generated regressors to the total asymptotic variance of the regression coe cient estimators. xamples of generated regressors are linear predictors or residuals from an estimated equation as in Barro (977) or Shefrin (979). The estimators considered by Pagan are special cases of standard two-step estimators, and such estimators can be conveniently analyzed as single-step GMM estimators, as in Newey (984) or Murphy and Topel (985). These methods of adjusting the asymptotic variance for the rst-stage estimation error are now so well-understood that they can be found in textbooks such as Wooldridge (00, Chapter.4). Pagan (984) considered parametric linear regression models with parametrically estimated generated regressors. However, econometrics has evolved since then, and the rst-step estimators these days can be non-parametric estimators obtained by kernel or sieve methods. Newey (994) discusses a general method of characterizing the asymptotic variance of two-step GMM estimators of a nite dimensional parameter vector, if the moment condition depends on a conditional expectation or a density that is estimated non-parametrically. A special instance of his method deals with the case of a linear regression model with a non-parametrically estimated generated regressor. Newey uses path derivatives to obtain the in uence function for semiparametric GMM estimators. The asymptotically linear representation of the estimator gives the asymptotic variance of the estimator. After this derivation it still has to be shown that the di erence between the semi-parametric GMM estimator and its asymptotically linear representation converges to 0 at a rate that is faster than the parametric rate. Su cient conditions for this in general depend on the non-parametric estimator and the smoothness of the conditional expectation or density that is estimated. Given the complexity of the multi-step estimators it is useful to have the in uence function before one considers the asymptotic properties of remainder terms. The asymptotic properties of non-parametric two-step estimators where both the generated regressor and the second-stage regression are estimated non-parametrically have been studied by Sperlich (009) and Song (008). Non-parametric multi-step estimators are not considered in this paper. As in Newey (994) we will only consider semi-parametric estimators for nite dimensional parameters. The di erence with Newey is that we consider three-step estimators where the second step is a non-parametric regression on a generated regressor. As we discuss in this paper the e ect of the rst-stage estimation error on the asymptotic variance of the estimator of the nite dimensional parameter is qualitatively di erent for the two- and threestep semi-parametric estimators. Also the results for two-steon-parametric estimators cannot be used directly to obtain the in uence function for semi-parametric three-step estimators. The purpose of this note is to use Newey s path-derivative method to derive the asymptotic variance of three- or even multi-step estimators of a nite dimensional parameter in which one of the steps is a non-parametric regression with a generated regressor. The generated regressor that is estimated in the rst step can be estimated parametrically or non-parametrically. Since Newey (994), a number of estimators have been suggested that have this structure with one

3 of the steps a non-parametric regression on a generated regressor. We consider three examples: (i) the partially linear regression model with a generated regressor in Wooldridge and Lee (00) and Newey (009), (ii) the Average Treatment ect (AT) estimator for the case of unconfounded treatment assignment suggested by Heckman, Ichimura, and Todd (998) that involves two non-parametric regressions on the estimated propensity score, (iii) a parametric control variate estimator that depends on a non-parametric regression on a residual estimated in a rst stage. These examples illustrate the method that can also be used to derive the asymptotic variance of other estimators with the same structure not covered here, for instance the production function estimators of Pakes and Olley (995) and Olley and Pakes (996). The key issue in the application of Newey s path-derivative method is to account for the contribution of the rst-stage estimation error of the generated regressor to the sampling variation of the second-stage non-parametric regression. This contribution consists of two parts. First, there is the e ect of the rst-step estimation error on the estimate of the generated regressor. However, there is a second contribution to the sampling variation of the conditional expectation, because we condition on an estimated instead of a population value of the regressor. It is the latter contribution that is easily forgotten. One can wonder whether the reformulation of the two-step estimator of Pagan (984) as a one-step GMM estimator as in Newey (984) or Murphy and Topel (985) can be generalized to the three or more step estimator considered here. In particular, Ai and Chen (007) recently considered a variety of conditional moment restriction estimators, some with a more complicated structure than in this paper, where the conditioning variables are not estimated. Therefore our results are not a special case of, but rather complementary to the results in Ai and Chen. Whether our asymptotic variance can be derived from a one step GMM problem as in Ai and Chen (007) is the subject of ongoing research. This paper has the following structure. In Section, we present a parametric example that provides the basic intuition underlying our results. Our main result is in Section. In Section 4 and 5 we discuss two applications of the main result. In Section 6 we consider estimators that involve partial means with an application to regression on the estimated propensity score in Section 7. A Parametric xample To gain intuition for the results later on we consider a fully parametric example. Consider the following scenario. We have a random sample s i (y i ; x i ; z i ) ; i ; : : : ; n from a joint distribution. The scalar parameter is estimated by a three-step estimator. In the rst step, we estimate the scalar parameter by b such that (b ) (x i ; z i ) + o p () with [ (x i ; z i )] 0 and the population value of the parameter. In the second step, we estimate the coe cients ( ; ; ) of the linear projection of y on ; x; v with v ' (x; z; ), i.e., the solution to min ; ; (y x v ). Because we do not know, we use the estimated bv i ' (x i ; z i ; b), so that the estimator b of is the OLS estimator of y

4 on ; x; bv. The estimator of is obtained in the third step b P n n (b + b x i + b ' (x i ; z i ; b)), so that [ + x + '(x; z; )]. Our interest is to characterize the rst order asymptotic properties of this estimator. A standard argument suggests that it su ces to consider the expansion of the form b ( + x i + ' (x i ; z i ; ) ) + [x] [' (x; z; )] (b ) ' (x; z; ) pn + (b ) + o p () : Let us now focus on the adjustments to the in uence function that account for the estimation error in the rst and second step, i.e., the sum of the second and third terms on the right, which we will call. A routine calculation (presented in Appendix A) reveals that where [x] [' (x; z; )] G G 4 4 i x i i ' (x i ; z i ; ) i x ' (x; z; ) x x x' (x; z; ) ' (x; z; ) x' (x; z; ) ' (x; z; ) 5 : 5 + o p () ; () The expansion () can be given an intuitive interpretation by considering an infeasible estimator. Assume that is known to the econometrician, and v i ' (x i ; z i ; ) is used in the regression. Let e denote the resulting OLS estimator of. The rst order asymptotic properties of e P n n (e + e x i + e ' (x i ; z i ; )) can be analyzed using the expansion e ( + x i + ' (x i ; z i ; ) ) + [x] [' (x; z; )] (e ) + o p () A routine calculation (presented in Appendix A) also establishes that [x] [' (x; z; )] (e ) () [x] [' (x; z; )] G 4 i x i i ' (x i ; z i ; ) i 5 + o p () Comparing the correction terms () and () leads us to an interesting conclusion: The in uence function for b is equal to that of the unfeasible estimator e that ignores the estimation error in the rst step, i.e., that in b! In order to understand this apparent puzzle, it is convenient to de ne b () (b () ; b () ; b ()) as the OLS estimator with y as the dependent and x and v '(x; z; ) as the independent variables. Note that b b (b) and e b ( ). Also () is the vector of coe cients of the linear projection of y on ; x; '(x; z; ). A naïve derivation of the in uence function of b would use the following decomposition 4

5 . Main term that re ects the uncertainty left if we know and : ( + x i + ' (x i ; z i ; ) ). A term that accounts for the sampling variation in b ( ) if we know : [x] [' (x; z; )] G 4. A term that accounts for the sampling variation in b: ' (x; z; ) pn (b ) This naïve decomposition is missing one additional term, i.e., i x i i ' (x i ; z i ; ) i 5 [x] [' (x; z; )] G G pn (x i ; z i ) () where 6 G 4 '(x;z; ) x i '(x;z; ) ' (x; z; ) '(x;z;) P As shown in Appendix A, G G n pn (x i; z i ) is the e ect of the sampling variation in b on the sampling distribution of b. De ning () [ () + () x + () ' (x; z; )], we show in Appendix B that the missing term is asymptotically equivalent to ( (b) ( )). The expression () + () x + () ' (x; z; ) that appears in the de nition of () can be given an interesting interpretation. It is the linear projection of y on ; x; ' (x; z; ) when after projection we substitute ' (x; z; ) for ' (x; z; ). Note that the linear projection of y on ; x; ' (x; z; ) has coe cients (). This speci es a function of x; ' (x; z; ) that can be evaluated at any value of these arguments and here we choose the values x; '(x; z; ). Hence, plays two roles. First, it determines the functional form of the projection, here only the coe cients (), because the projection is restricted to be linear. Second, enters in the variables at which the (linear) projection is evaluated, here x; ' (x; z; ). If we substitute the estimator b then the two correction terms that account for the estimation error in b correspond to these two roles of and in this example these two correction terms are opposites so that their sum is 0. The naïve derivation of the in uence function ignores the e ect of on the coe cients of the linear projection. In this paper we propose a method that accounts for the full contribution of b to the in uence function, i.e., we improve on step above. The full (accounting for the two distinct See Appendix A

6 roles of ) contribution of the sampling variation of b, i.e., with the projection coe cients equal to (b); (b); (b), is ( (b) + (b)x i + (b)' (x i ; z i ; b) x i ' (x i ; z i ; )) n ( p ( ) + ()x i + ()' (x i ; z i ; )) n(b ) + o p () [ p () + ()x i + ()' (x i ; z i ; )] n(b ) + o p () Now the projection of y on ; x; '(x; z; ) implies that for all constants p ; p ; p and for all 0 [(y () () x () ' (x; z; )) (p + p x + p ' (x; z; ))] Taking p, p 0, and p 0, and di erentiating the rst equation with respect to and evaluating the derivative at, we obtain [ ( ) + ( ) x + ( ) ' (x; z; )] 0 Therefore we conclude that the contribution of the sampling variation in b to the sampling variation of b is 0. This derivation is simpler than that in Appendix A and can be generalized to the case of general projections that are not restricted to be linear. In general the rst step estimate plays these two distinct roles. The example in this section was relatively simple because the linear functional relation can be summarized by a nite dimensional vector (). The challenge to the econometrician is that when the projection is non-parametric, as is the case when the generated regressor is used in a non-parametric regression, such simplicity disappears. By separately considering the two roles that sampling variation in the rst step plays when we evaluate its e ect on the second-stage projection, we can properly adjust the in uence function. In general the two corresponding correction terms are not opposite as in the simple example considered here. The In uence Function of Semi-parametric Three-Step stimators We now present our two main results on semi-parametric three-step estimators. In the rst step we estimate a regressor. In the second step we estimate a non-parametric regression with the generated regressor as one of the independent variables. In the third step we estimate a nite dimensional parameter (without loss of generality we consider the scalar case) that satis es a moment condition that also depends on the non-parametric regression estimated in the second step. We distinguish between two cases: (i) the rst step is parametric, (ii) the rst step is non-parametric. Moreover, we rst consider estimators that can be expressed as a sample average of a function of the second-stage non-parametric regression only. Next, we allow that function to depend on other variables besides the second-stage non-parametric regression. In 6

7 both cases the estimators are, in Newey s (994) terminology, full means, because they average over all arguments of the second-stage non-parametric regression. In Section 6 we consider estimators that average over most but not all independent variables in the second-stage nonparametric regression, i.e. the estimator is a partial mean. This makes the in uence function more complicated, which is the reason that we start with the full mean case. As was emphasized in the introduction, in all cases our characterization is based on Newey s (994) path-derivative method.. Parametric First Step, Non-parametric Second Step We assume that we observe i.i.d. observations s i (y i ; x i ; z i ; ) ; i ; : : : ; n. The rst step is identical to that in Section, i.e., we have an estimator b such that p P n (b ) p n n (x i; z i ) + o p () with [ (x i ; z i )] 0. The parameter vector indexes a relation between a dependent variable that is a component of x (and that we later denote by u) and independent variables that are some or all of the other variables in x and those in z. ither the predicted value or the residual of this relationship is an independent variable in the second-step non-parametric regression. The notation '(x; z; ) covers both cases. The second step is di erent from the parametric example, because our goal is to estimate (x; v ) [y j x; v ] where v ' (x; z; ), i.e., we no longer restrict the projection to be linear. Because we do not observe, we use bv i ' (x i ; z i ; b) in the non-parametric regression. The non-parametric regression estimator of y on x; bv is denoted by b. Our goal is to characterize the rst order asymptotic properties of b h (b (x i ; ' (x i ; z i ; b))) n We can consider b as the solution of a sample moment equation that is derived from a population moment equation that depends on and (x; '(x; z; )). As will be seen below it matters whether h is linear (as in Section ) or not. Using Newey s (994) path-derivative approach, we express the in uence function of b as a sum of three terms: (i) the main term (h ( (x i ; ' (x i ; z i ; ))) ) (ii) a term that adjusts for the estimation of b, i.e., p (h (b (x i ; ' (x i ; z i ; ))) h ( (x i ; ' (x i ; z i ; )))) n and (iii) an adjustment related to the estimation of b, i.e., p (h ( (x i ; ' (x i ; z i ; b))) h ( (x i ; ' (x i ; z i ; )))) : n 7

8 The decomposition here is based on the fact that Newey s approach can be used term-by-term. Therefore, we may without loss of generality assume that is a scalar. The second component in the decomposition can be analyzed as in Newey (994, pp. 60 6). It is equal to h((xi ; v i )) x i; v i (y i (x i ; v i )) + o p () (4) h((x i ; v i )) (y i (x i ; v i )) + o p () As in Section we therefore focus on the analysis of the third component We de ne (h ( (x i ; ' (x i ; z i ; b))) h ( (x i ; ' (x i ; z i ; )))) (x; v ; ) [y j x; ' (x; z; ) v ] g (s; ; ) h ( (x; ' (x; z; ) ; )) Note that the two roles that plays are made explicit in g (s; ; ) that is obtained by substituting v '(x; z; ) in (x; v ; ). Note also that (x; v ) (x; v ; ). The notation ; is just an expositional device, since. With these de nitions, we can now write n h ( (x i ; ' (x i ; z i ; b) ; b)) n g (s i ; b ; b ) where b b b, but we keep them separate to emphasize the two roles of b. This forces us to deal with the two roles that b plays in the linearization that involves partial derivatives: (h ( (x i ; ' (x i ; z i ; b) ; b)) h ( (x i ; ' (x i ; z i ; ) ; ))) p (g (s i ; b ; b ) g (s i ; ; )) n g (s; ; ) g (s; ; ) pn(b + ) + o p () The fact that Newey s approach can be used term-by-term is illustrated in an earlier version of the paper, which is available upon request. There, we consider the case where the moment function includes multiple nonparametric objects, all of which are obtained by non-parametric regressions with possibly di erent independent variables. 8

9 h i h Therefore we must compute g(s;;) and g(s;;) i. The computation of the rst expectation is easy. Because (x; ' (x; z; ) ; ) (x; ' (x; z; )), we have g (s; ; ) h ( (x; ' (x; z; ))) (x; ' (x; z; )) v ' (x; z; ) The headache is to compute the second expectation. By the chain rule g (s; ; ) h ( (x; ' (x; z; ))) (x; ' (x; z; ) ; ) (5) Unfortunately, it is not obvious how to di erentiate (x; ' (x; z; ) ; ) with respect to. After all, (x; ' (x; z; ) ; ) has the functional form of [y j x; ' (x; z; ) v ] that depends on. The next lemma gives the solution in a generic case. Lemma t(x; v ) (x; v ) v [t(x; '(x; z; ))(x; '(x; z; ); )] (6) '(x; z; ) + ((x; z) (x; v )) t(x; v ) '(x; z; ) v Proof. Because (x; '(x; z; ); ) is the solution to we have that for all min p (y p(x; '(x; z; ))) [(y (x; '(x; z; ); )) t (x; ' (x; z; ))] 0 Di erentiating with respect to and evaluating the result at we nd after rearranging (6). This key lemma is used repeatedly in the sequel, beginning with the proof of the following theorem. Theorem (Contribution parametric rst-stage estimator) The adjustment to the in- uence function that accounts for the rst-stage estimation error is g (s; ; ) g (s; ; ) pn(b + ) (7) ( (x; z) (x; v )) h ( (x; v )) (x; v ) '(x; z; ) pn(b ) v with v '(x; z; ). 9

10 Proof. We compute the right hand side of (5) that by Lemma is equal to h ( (x; ' (x; z; ))) (x; ' (x; z; ) ; ) h(x; (x; v )) (x; v ) '(x; z; ) v + ((x; z) (x; v )) h(x; (x; v )) (x; v ) '(x; z; ) v h i Adding g(s;;) that is equal to the opposite of the rst term on the right-hand side, we nd the desired result... Linear h and Index Restrictions There are two cases in which the adjustment term is 0, so that there is no contribution of the rst-stage estimate to the in uence function. The rst case is that h is linear in. This was illustrated for the fully parametric case in Section. The second case is that the index restriction (yjx; z) (yjx; '(x; z; )) holds. A su cient condition is that for (almost) all x the index '(x; z; ) is a one-to-one function of z, so that regressing on the index and x does not lead to dimension reduction. If (yjx; v ) (yjv ) then there is no contribution of the rst stage if (yjx; z) (yj'(x; z; )). The fact that index restrictions simplify in uence functions was noted by Newey (994). He considered a two-step estimator based on the moment condition [m(x; ; )] 0 with (x; z) (yjx; z) ('(x; z; )) (yj'(x; z; )), i.e. depends on through the index '(x; z; ). Note that in this case we do not need a rst-stage estimator. De ne (x; z; ) [yj'(x; z; )] so that (x; z; ) (x; z) ('(x; z; )). For the in uence function we need to compute [m(x; ; )] Now appears in three places: (i) directly in m (this the linearization step that is skipped above), (ii) as an argument in, (iii) in the conditioning variable that de nes. To nd the contribution of (ii) and (iii) to the in uence function we need to compute m(x; 0 ; (x; z)) (x; z; ) Newey obtained an expression for (x; z; ) 0

11 under the index restriction. Because (x; z; ) [yj'(x; z; )] [[yj'(x; z; ); '(x; z; )]j'(x; z; )] [('(x; z; ))j'(x; z; )] we have that (x; z; ; ) [('(x; z; ) '(x; z; ) + '(x; z; ))j'(x; z; )] is equal to (x; z; ) if. We take derivatives with respect to and (x; z; ; ) v ('(x; z; )) ' (x; z; )j'(x; z; ) (x; z; ; ) [('(x; z; ))j'(x; z; )] The nal conditional expectation is just ('(x; z; )) so that taking the derivative with respect to the in the conditioning variable is trivial. Compare this to the general case where the argument and conditioning variable are not the same. Adding the derivatives we obtain Newey s result (x; z; ) ' v ('(x; z; )) (x; z; ) ' (x; z; )j'(x; z; ) Newey uses this result to simplify the moment condition. If the moment condition depends on x; z through the index only then this result implies that the estimation of has no contribution to the asymptotic variance. This is in line with Theorem. This result is used by Klein, Shen and Vella (00)... Multidimensional Second Step Suppose now that the is multidimensional, i.e., y is a J-dimensional random vector. The estimator is now b h (b (x i ; ' (x i ; z i ; b)) ; : : : ; b J (x i ; ' (x i ; z i ; b))) n The product rule of calculus suggests that we can tackle this problem by adding the derivatives. This is formalized in the next theorem. Theorem (Contribution parametric rst-stage estimators) The adjustment to the in- uence function that accounts for the rst-stage estimation error is X h ( (x; ' (x; z; ))) (y j j (x; ' (x; z; ))) j (x; ' (x; z; )) ' (x; z; ) pn (b ) : v j j Proof See Appendix C.

12 . Non-parametric First Step, Non-parametric Second Step We now assume that the rst step is non-parametric. Again we have a random sample s i (y i ; x i ; z i ; d i ) ; i ; : : : ; n. The rst-step projection of one of the components of x, that we denote by u, on some or all of the other components of x and z is denoted by v ' (x; z) [u j x; z]. The rst step is to estimate this projection by non-parametric regression. In the second step we estimate (x; v ) [y j x; v ] by non-parametric regression of y on x; bv b'(x; z). Our interest is to characterize the rst-order asymptotic properties of b n h (b (x i ; b' (x i ; z i ))) where b (x i ; bv i ) is the non-parametric regression estimate. We de ne (x; v ; v ) [y j x; ' (x; z) v ] g (w; v ; v ) h ( (x; v ; v )) with v '(x; z) and conditioning on '(x; z) v. Note that v and v play the roles of and. With these de nitions, we can now write n h ( (x i ; bv ; bv )) n g (s i ; bv ; bv ) where bv bv bv. We keep them separate to emphasize their di erent roles. Our objective is to approximate g (s i ; bv ; bv ) g (s i ; v ; v ) n n To nd the contribution of the sampling variation in ^v we have taken as known. As in Newey (994) we consider a path v indexed by R such that v v. First, using the calculation in the previous section we obtain that for [h ( (x; v ; v ))] [D (s; v )] D (s; v ) h ( (x; v )) (y (x; v )) (x; v ) v : v which is linear in v. Second, we have that for any v '(x; z), [D (s; v)] [ (x; z) '(x; z)] with h ( (x; ' (x; z))) (x; z) (y (x; ' (x; z))) (x; ' (x; z)) v x; z (8)

13 where it is understood that we condition on the variables that are in ' so that we average over all x that are not in the generated regressor. By Newey (994) Proposition 4 these two facts imply that the adjustment to the in uence function is equal to (x i ; z i ) (u i [u j x i ; z i ]) (x i ; z i ) (u i ' (x i ; z i )) with u the component of x that is projected on x; z. We summarize the result in a theorem: Theorem (Contribution non-parametric rst-stage estimator) The adjustment to the in uence function that accounts for the rst-stage estimation error is p (x i ; z i ) (u i ' (x i ; z i )) n with ' (x; z) [ujx; z] and as in (8). The adjustment for the estimation of is as in (4) in Section.. The generalization to multidimensional is obvious.. Additional Variables in the Third Step So far, we have assumed that the parameter of interest is [h((x; v ))] where h depends only on (x; v ). We now consider the extension to [h(w; (x; v ))] where w is a vector of other variables that may have x; z as subvectors. We consider both the case that ' is parametric and the case that this function is non-parametric. Because as before the main term and the contribution of the estimation of [yjx; v ] do not raise new issues, the next two theorems only give the contribution of the rst-stage estimator. In these theorems we use the function h (w; (x; v )) (x; v) x; ' (x; z; ) v with ' (x; z) substituted in the non-parametric case. Theorem 4 (Contribution parametric rst-stage estimator) The adjustment to the in- uence function that accounts for the rst-stage estimation error is h (w; (x; ' (x; z; ))) (x; ' (x; z; )) ' (x; z; ) (x; ' (x; z; )) + v (x; ' (x; z; )) ((x; z) (x; ' (x; z; ))) ' (x; z; ) pn (b ) v with v '(x; z; ).

14 Proof See Appendix C Now, we consider the case where the rst step is non-parametric. The discussion preceding Theorem, which summarizes Newey s argument, implies that Theorem 5 (Contribution non-parametric rst-stage estimator) The adjustment to the in uence function that accounts for the rst-stage estimation error is (x i ; z i ) (u i ' (x i ; z i )) with ' (x; z) [ujx; z] and h (w; (x; ' (x; z))) (x; z) (x; ' (x; z)) + ((x; z) v (x; ' (x; z)) (x; ' (x; z)) v x; z (x; ' (x; z))) x; z Theorems 4 and 5 are easily generalized to the case of multidimensional. Suppose that (x; ' (x; z)) 0 in Theorem 4. The adjustment is then equal to the derivative with respect to, i.e., the naive derivative (see equation () in the proof of Theorem 4). Therefore, it may be useful to check whether (x; ' (x; z)) 0 in speci c models. If it is the case, we need not worry about the e ect of rst-step estimation on the second-stage non-parametric regression. Section 4 has an example of this. The theorems can be applied to general semi-parametric GMM estimators. If we consider the moment condition [m(w; (x; v ); )] 0 and we linearize the corresponding sample moment condition we obtain p n( b m(w; (x; v ); ) ) pn m (w 0 i ; b(x i ; b'(x i ; z i )); ) + o p () Therefore, the contribution of the rst-stage estimate to the asymptotic distribution of b can be found by applying Theorem 5 to P n m (w i; b(x i ; b'(x i ; z i )); )..4 Two-step and Three-Step stimators The e ect of the rst-stage estimation error is qualitatively di erent for three-stage and twostage semi-parametric estimators. To show this we contrast our results with two results available in the literature. First, consider the standard two-stage estimator (with a non-parametric rst stage) of the form b h (x i ; b' (x i ; z i )) n where b' is an estimator of ' (x; z) [uj x; z]. As discussed in Newey (994), among others, the contribution of the estimation of ' to the in uence function is h(x;'(x;z)) v (u ' (x; z)). 4

15 This involves the rst derivative of h, so that this contribution is nonzero if h is linear. This in contrast to the three-stage estimator in the full mean case, where the contribution is zero with h linear. Second, we can compare our results with those on the asymptotic distribution of the non-parametric regression estimator b (x; ' (x; z; b)) following a rst-step parametric estimation. Because the b typically converges at the parametric rate, the asymptotic distribution of b (x; ' (x; z; b)) for all x; z is una ected by the rst-step estimation error. If we would take this result to the third-step estimation of by b n h (b (x i ; ' (x i ; z i ; b))) we would incorrectly conclude that the rst-step estimation of b does not a ect the third-step estimator whether h is linear or not. This example makes it clear that our results cannot be derived from the results in, e.g., Song (008) or Sperlich (009) for the non-parametric regression on generated regressors estimated in the rst step. 4 Application: The Partial Linear Model with a Generated Regressor In this section, we apply the results in the previous section to a semi-parametric model, the partial linear regression model, y i x i + m ('(x i ; z i ; )) + i ; where m is non-parametric. The error term i satis es [ i j x i ; v i ] 0. The parameter of interest is. We initially consider the case that the generated regressor is estimated parametrically, but we also give the contribution to the in uence function for the case that it is estimated non-parametrically. The model is estimated by regressing an estimate of y i [y j v i ] on an estimate of x i [x j v i ], i.e., b P n (x i b (' (x i ; z i ; b))) (y i b (' (x i ; z i ; b))) P n (x i b (' (x i ; z i ; b))) We denote (v ) [xj v ], (v ) [yj v ], and the non-parametric estimators of these functions are b and b. By Newey (994), Proposition the estimation of the conditional expectations (v ); (v ) has no contribution to the in uence function of, b because these estimates are obtained by minimizing P n (y i x i m('(x i ; z i ; b))) over m, i.e., they satisfy the rst-order condition for this minimization problem. Hence, with the notation (v ; ) [xj'(x; z; ) v ], (v ; ) [yj'(x; z; ) v ] we have P n b (x i (' (x i ; z i b) ; b)) (y i (' (x i ; z i ; b) ; b)) P n (x i (' (x i ; z i ; b) ; b)) + o p n 5

16 Because (bv; b) (bv; b)+ (bv; b)+ 4 (bv; b) with (v ; ) [m(v )j'(x; z; ) v ] and 4 (v ; ) [j'(x; z; ) v ] we obtain after substitution b n (x i (bv i ; b)) ((m(v i ) (bv i ; b)) + ( i 4 (bv i ; b))) + o p n with [(x (v )) ]. If we de ne h(x; v ; ; ; ; 4 ) (x ) ((m(v ) )) + ( 4 )) we can use the multi-dimensional version of Theorem 4 to nd the contribution of the rst stage to the in uence function. First, note that h(x; v ; ; (v ); (v ); 4 (v )) j (v ) j v 0 for j ; ; 4 and for all v. This implies that the e ect of the rst-stage estimator on the conditional expectation is 0, so that we only need to deal with bv as an argument. We have because (v ) m(v ) and by assumption (jv ) 4 (v ) 0 h(w; (x; v )) (x; v ) v Further with x (v ) h(w; (x; v )) and (x; v ) v h(w; (x; v )) 4 (x; v ) 4 v ' (x; z; ) ' (x; z; ) ' (x; z; ) (x; v ) v (x; v ) v 4(x; v ) v ' (x; z; ) ' (x; z; ) ' (x; z; ) Because [jv v] 4 (v) 0 for all v the nal expression is 0. To conclude, the adjustment in the in uence function of b corresponding to the estimation error in b is ('(x; z; )) ' (x; z; ) + m ('(x; z; )) ' (x; z; ) pn (b ) h i Note that the 0 if we assume that [j x; z] 0, and in that case ('(x;z; )) '(x;z; ) our result is the same as in Newey (009) or Li and Wooldridge (00). Combining this result with Newey (994) we nd that the contribution in the case that '(x; z) is estimated by non-parametric regression of u on x; z is equal to [jx; z] (' (x; z)) + [jx; z] m (' (x; z)) (u ' (x; z)) 6

17 5 Application: A Semi-parametric Control Variable stimator Hahn, Hu and Ridder (008) consider a model that is nonlinear in a mismeasured independent variable. The details of their model are not important here. For our purpose it su ces to note that their estimator uses a control variable and the asymptotic analysis requires dealing with a generated regressor in a V-statistic. Because of the V-statistic structure, Theorem 4 cannot be used directly, but we will use the same approach as in that theorem. A minor di erence is that the second-stage non-parametric regression is on a residual not on a predicted value of a rst-stage regression. Suppose that we have a random sample (y i ; x i ; z i ) ; i ; : : : ; n. The estimator of a parameter has the following three steps:. stimate a nite dimensional parameter b by nonlinear least squares of x on (z; ) and obtain the residual bv x (z; b) '(x; z; b).. stimate (x; v ) [y j x; v ] non-parametrically using the sample (y i ; x i ; bv i ) ; i ; : : : ; n. Call the estimator b(x; bv). Note that the non-parametric regression is on a residual not on a predicted value. Let L (x) v [(x; v )] and b L (x) n P n j b(x; bv j).. The identi cation result for control variable estimators implies that L (x) R (x; ) for a known function R. De ne b as the solution of the minimization problem min n C (x i ) bl (xi ) R (x i ; ) for some set C. In the sequel we will ignore the indicator function C for simplicity. Let b denote the solution to the preceding minimization problem that satis es the moment condition 0 bl (xi ) R x i ; n b R x i ; b : Characterization of asymptotic distribution of b requires characterization of the in uence function of bl(x i )r (x i ) b(x n n i ; bv j )r(x i ) where r (x i ) R (x i ; )/. The contribution of b and can be derived as in Newey (994). The double sum requires an application of the V-statistic projection theorem (see Hahn, Hu, and Ridder (008) for details). Therefore the contribution of b can be derived from n Z j (x i ; '(~x; ~z; b); b)r(x i )df xz (~x; ~z) 7

18 We have to compute the derivative with respect to of Z x (x; '(~x; ~z; ); )r(x)df xz (~x; ~z) which can be written as the sum of Z x (x; '(~x; ~z; ); )r(x)df xz (~x; ~z) xv (x; v ; )r(x) f(x)f(v ) f(x; v ) (9) and x Z (x; '(~x; ~z; ))r(x)df xz (~x; ~z) (0) We compute the derivative as the sum of the derivatives of (9) and (0). For (0), we nd Z x (x; '(~x; ~z; ))r(x)df xz (~x; ~z) x Z (x; ' (~x; ~z; )) v r (x) (~z; ) df xz (~x; ~z) : (This method of computing the derivative of (0) easier than doing so by putting (0) in the form of Theorem 4, which involves a transformation that depends on and this complicates the computation of the derivative.) For (9), which is as in Theorem 4, we have by Lemma (x; v ) v xv (x; v ; )r(x) f(x)f(v ) f(x; v ) ln f(v ) ((x; z) (x; v )) v ln f(x; v ) v The contribution of the rst step estimation to the in uence function is then ( + ) (b ) r (x) f (x) f (v ) f (x; v ) (z; ) : 6 The In uence Function of Semi-parametric Three-step stimators: Partial Means In Section the estimator b averaged over all arguments of the second-steon-parametric regression function. In Newey s (994) terminology the estimator is a full mean. In this section we consider the case that the discrete independent variable d is xed, i.e., b does not average over this variable. Hence the estimator is a partial mean. Because the variable that is xed in the partial mean is discrete, the parametric rate of convergence applies. (If that variable were continuous we would have a slower rate.) Let the discrete variable d take the values d ; : : : ; d K. In the second step we estimate (x; v ; d) [y j x; v ; d] 8

19 As Section we use bv i in the non-parametric regression. De ne b k (x i ; bv i ) b(x i ; bv i ; d k ), i.e., b k is the non-parametric regression function if we set x and bv to the observed values for i, but x d at value d k which may not be its value for i. The K vector b(x i ; bv i ) stacks the b k (x i ; bv i ). We also de ne k (x; v ) [y j x; v ; d k ] and (x; v ) as the K vector with components k (x; v ). Our goal is to characterize the rst order asymptotic properties of b n h (w i ; b (x i ; bv i )) with b the K vector of non-parametric regression functions where the discrete variable d is xed at its K distinct values. As in Section. we allow for a vector of additional variables w in h. 6. Partial Means: Parametric First Step, Non-parametric Second Step We assume that the rst step is parametric, and bv i ' (x i ; z i ; ; b). As in Section, we can use Newey s (994) path-derivative approach, and express the in uence function of b as a sum of three terms: (i) the main term, (ii) a term that adjusts for the estimation of b, and (iii) an adjustment related to the estimation of b. De ne k (x; '(x; z; )) Pr(d d k jx; '(x; z; )) and k (x; z) Pr(d d k jx; z) h (w; (x; v )) k (x; v) k x; ' (x; z; ) v The second component in the decomposition can be analyzed as in Newey (994, pp. 60 6) and is equal to KX k (d i d k )(y i k (x i ; v i )) k(x i ; v i ) k (x i ; v i ) + o p () () As in Section we therefore focus on the analysis of the third component (h (w i ; (x i ; ' (x i ; z i ; b) ; b)) h (w i ; (x i ; ' (x i ; z i ; )) ; )) (g (s i ; b ; b ) g (s i ; ; )) 9

20 with k (x; v ; ) [y j x; ' (x; z; ) v ; d d k ] (x; v ; ) ( (x; v ; ) K (x; v ; )) 0 g (w; ; ) h (w; (x; ' (x; z; ) ; )) Lemma can be generalized to the partial means case: Lemma For partial means we have for all k ; : : : ; K [t(x; '(x; z; )) k (x; '(x; z; ); )] () k (x; z) k (x; z) + k (x; v ) ( k(x; z) k (x; v )) with v '(x; z; ). k (x; v ) t(x; v ) k(x; v ) v k (x; v ) t(x; v ) v Proof. Because k (x; '(x; z; ); ) is the solution to '(x; z; ) min p (d d k )(y p(x; '(x; z; ))) t(x; v ) k(x; v ) v we have that for all t(x; '(x; z; )) (d d k ) (y k (x; '(x; z; ); )) 0 k (x; '(x; z; ) '(x; z; ) Di erentiating with respect to and evaluating the result at we nd after rearranging (). The next theorem generalizes Theorem 4: Theorem 6 (Contribution parametric rst-stage estimator) The adjustment to the in- uence function that accounts for the rst-stage estimation error is KX! h(w; (x; v )) k (x; z) k k (x; v ) k (x; v ) ' (x; z; ) k(x; v ) v k! KX k (x; z) + k (x; v ) ( k(x; z) k (x; v )) k(x; v ) ' (x; z; ) v k!! KX k (x; z) k (x; v ) ( k(x; z) k (x; v )) k (x; v ) k(v; v ) ' (x; z; ) pn (b ) v k with v '(x; z; ). 0

21 Proof See Appendix D. is + If h only depends on, then k (x; v ) h((x;v)) k KX k KX! k (x; z) k (x; v ) h ( (x; v )) k (x; v ) k (x; v ) k v k k 0 KX k so that the contribution of the rst stage '(x; z; )! KX k (x; z) k (x; v ) ( k (x; z) k (x; v )) h ( (x; v )) k 0 (x; v ) '(x; z; ) k k 0 v!! k (x; z) k (x; v ) ( k (x; z) k (x; v )) h ( (x; v )) k (x; v ) '(x; z; ) pn(b ) k v 6. Partial Means: Nonparametric First Step, Non-parametric Second Step Now assume that the rst step consists of non-parametric estimation of v ' (x; z) [u j x; z]. We can obtain the adjustment by replicating the arguments in Section. leading to Theorem 5. Letting X K h(w; (x; v )) k (x; z) (x; z) k k (x; v ) k (x; v ) k(x; v ) v x; z + k K X k (x; z) k (x; v ) ( k(x; z) k (x; v )) k(x; v ) v x; z (4) k K X k (x; z) k (x; v ) ( k(x; z) k (x; v )) k (x; v ) k(v; v ) v x; z k we obtain an analog of Theorem : Theorem 7 (Contribution non-parametric rst-stage estimator) The adjustment to the in uence function that accounts for the rst-stage estimation error is (x i ; z i ) (u i ' (x i ; z i )) with ' (x; z) [ujx; z] and (x; z) given in (4). ()

22 If h depends on only we replace above by X K k (x; z) k (x; ' (x; z)) h ( (x; ' (x; z))) k (x; ' (x; z)) (x; z) k (x; v ) k v + KX k KX k k 0 KX k k (x; z) k (x; ' (x; z)) ( k (x; z) k (x; z) k (x; ' (x; z)) ( k (x; z) k (x; ' (x; z))) h ( (x; ' (x; z))) k 0 (x; ' (x; z)) k k 0 v k (x; ' (x; z))) h ( (x; ' (x; z))) k k (x; ' (x; z)) v 7 Application: Regression on the stimated Propensity Score We consider an intervention with potential outcomes y 0 ; y that are the control and treated outcome, respectively. The treatment indicator is d and y dy + ( d)y 0 is the observed outcome. The vector x contains covariates that are not a ected by the intervention. As shown by Rosenbaum and Rubin (98) unconfounded assignment, i.e., the assumption that y ; y 0? djx, implies y ; y 0? dj' (x) with ' (x) Pr(d jx) the probability of selection or propensity score. This observation has led to a large number of estimators that can be classi ed into three groups. Most of these estimators rely on the propensity score, but some do not. The asymptotic variance of the estimators can be compared to the semi-parametric e ciency bound for the AT derived by Hahn (998). The most popular estimators are the matching estimators that estimate the AT given x or given ' (x) by averaging outcomes over units with a similar value of x or ' (x) (and that subsequently average over the distribution of x or ' (x) to estimate the AT). Abadie and Imbens (009a), (009b) are recent contributions. They show that matching estimators that have an asymptotic distribution that is notoriously di cult to analyze, are not asymptotically e cient. The second class of estimators do not estimate the AT given x or ' (x) but use the propensity scores as weights. Hahn s (998) estimator and the estimator of Hirano, Imbens and Ridder (00) are examples of such estimators. These estimators are asymptotically e cient, which suggests that the propensity score is needed to achieve e ciency. The third class of estimators use non-parametric regression to estimate [yjd ; x], [yjd 0; x] or [yjd ; ' (x)], [yjd 0; ' (x)]. Of these estimators the estimator based on [yjd ; x], [yjd 0; x], the imputation estimator, is known to be asymptotically e cient, which suggests that there is no role for the propensity score. The missing result is that for the estimator that uses the non-parametric regression on the propensity score that is estimated in a preliminary step. This estimator that was suggested and analyzed by Heckman, Ichimura, and Todd (HIT) (998) ts into our framework and is analyzed here. Our conclusion is that the HIT estimator has the same asymptotic variance as the imputation estimator, so that there is no e ciency gain in using the propensity score. This should settle the issue whether there is a role for the propensity score in achieving semi-parametric e ciency. Heckman, Ichimura, and Todd actually consider an estimator of the Average Treatment ect on the Treated (ATT) that we also analyze. x; z

23 That does not mean that there is no role for the propensity score in assessing the identi cation or in improved small sample performance of AT estimators. 7. Parametric First Step, Non-parametric Second Step We have a random sample s i (y i ; x i ; d i ) ; i ; : : : ; n. The propensity score Pr(d jx) '(x; ) is parametric and its parameters are estimated in the rst step, by e.g. Maximum Likelihood or OLS (Linear Probability model) or any other method, such that (b ) (d i ; x i ) + o p () with [ (d i ; x i )] 0. In the second step, we estimate ('(x; )) ( [y j '(x; ); d ] ; [y j '(x; ); d 0]) 0 ; Because we do not observe, we use ' (x i ; b) in the non-parametric regression. Our interest is to characterize the rst order asymptotic properties of b n (b (' (x i ; b)) b (' (x i ; b))) This estimator can be handled by applying Theorem 6 for the special case that h only depends on. The vector ('(x; )) is a vector of partial means and d is either 0 or. Further '; k depend on x only and (x) '(x; ) and ('(x; )) Pr(d j'(x; )) '(x; ). Also h() so that the second derivatives are 0. Upon substitution the rst two terms on the right-hand side of () are 0. Because h k k for k ; we nd that the contribution v of the rst-stage estimator to the in uence function is g (s; ; ) g (s; ; ) + pn(^ ) [yj x; d ] (' (x; )) + [yj x; d 0] (' (x; )) ' (x; ) pn(^ ) ' (x; ) ' (x; ) The contribution of b can be derived using Newey (994), and is given in the next section. We also consider the HIT estimator of the Average Treatment ect on the Treated (ATT) ^ n d i p (b (b'(x i )) b (b'(x i ))) with p Pr(d ). This estimator is a special case of that considered in Theorem 6 with h(w; ; ) d( p ). Because h(w; ; ) d p h(w; ; ) d p

24 and (v) v p (v) v p we have for the expressions in Theorem 6 X! h(w; (x; v )) k (x; z) k k (x; v ) k (x; v ) k(x; v ) v k ('(x; )) v ('(x; )) d v p '(x; ) p by the law of iterated expectations (condition on x), and! X k (x; z) k (x; v ) ( k(x; z) k (x; v )) k(x; v ) v and k ' (x; z; ) '(x; ) 0 ' (x; z; ) p (( (x) ('(x; ))) ( (x) ('(x; )))) ' (x; z; ) X k! k (x; z) k (x; v ) ( k(x; z) k (x; v )) k (x; v ) k(v; v ) v p ( (x) ('(x; ))) '(x; ) '(x; ) ( (x) ('(x; ))) By taking the di erence of the last two terms we nd that the contribution is g (s; ; ) g (s; ; ) + pn(^ ) [yjx; d 0] (' (x; )) '(x; ) pn(^ ) p ( ' (x; )) ' (x; z; ) ' (x; z; ) 7. Non-parametric First Step, Non-parametric Second Step The analysis in the previous section combined with the results in Newey (994) shows that in the case that the rst stage is non-parametric the contribution of the rst-stage estimation to the in uence function of the AT estimator is [yj x; d ] (' (x)) ' (x) which can be alternatively written as + [yj x; d 0] (' (x)) ' (x) (d ' (x)) [yj x; d ] (' (x)) d + ( [yj x; d ] (' (x))) ' (x) + [yj x; d 0] (' (x)) ( d) ( [yj x; d 0] (' (x))) (5) ' (x) 4

25 To obtain the complete in uence function of b we need the contribution of the estimation error in b. This contribution is derived in Appendix and is equal to d ' (x) (y ( (' (x)) (' (x)) ) + (6) (' (x))) d ' (x) (y (' (x))) Adding (5) and (6), we obtain the in uence function of the estimator based on regressions on the estimated propensity score: ( [yj x; d ] [yj x; d 0] )+ d ' (x) (y [yj x; d ]) d (y [yj x; d 0]) ' (x) which is the in uence function of the e cient estimator and also that of the imputation estimator b I ( n b (x i ) b (x i )) with (x) [yjx; d ]; (x) [yjx; d 0]. The imputation estimator involves nonparametric regressions on x and not on the estimated propensity score. However these two estimators have the same in uence function which shows that regressing on the non-parametrically estimated propensity score does not result in an e ciency gain. The infeasible estimator that depends on non-parametric regressions on the population propensity score is less e cient than the estimator that uses the estimated propensity score. For the estimator of the ATT the contribution of the rst stage is [yjx; d 0] (' (x)) (d ' (x)) p ( ' (x)) The main term and the contribution of the estimation of the (infeasible) non-parametric regressions is d p (y (' (x))) ( d)' (x) p ( ' (x)) (y (' (x))) + d p ( (' (x)) (' (x)) ) which can be derived using an argument virtually identical to Appendix. expressions we obtain the full in uence function Adding these d p (y [yjx; d ]) ( d)' (x) p( ' (x)) (y [yjx; d 0]) + d p ([yjx; d ] [yjx; d 0] ) As in the case of the AT the in uence function is the same as that for the estimator that involves non-parametric regressions on x and not on the estimated propensity score, so that again there is no rst-order asymptotic e ciency gain if we use the estimated propensity score in the non-parametric regressions. It should be noted that the in uence functions derived in this section are di erent from those found in the literature. Recently, Mammen, Rothe, and Schienle (00) derived the in uence function for the AT estimator considered in this section. They concluded that it is 5

26 identical to that of the infeasible estimator that regresses on the population propensity score. This is because they imposed the index assumption [yj d; x] [yj d; '(x)], which is not made in the standard program evaluation literature, because it restricts the distribution of the potential outcomes. For instance, in a linear selection (on observables) model the index restriction implies that the regression coe cients in the outcome equations are proportional to those in the selection equation. HIT derived the in uence function for the ATT estimator that is also di erent from ours. In this case, the derivation fails to account for the e ect of the rst-stage estimation on the conditional expectation in the second stage. Only the variability of the rst-stage estimator as an argument is considered. 7. Approximating the In uence Function for the Non-parametric First Step with a Parametric First Step We assume that for the population propensity score ' (x) '(x; ) p (x) 0 where p (x) is a nite, possibly high-dimensional vector of functions of x. We can think of this expression as a series approximation of the propensity score with basis functions in the vector p (x). The in uence function for the least squares estimator of is p (x) p (x) 0 p (x) (d ' (x)) (7) Using the result in subsection 7., the adjustment to the in uence function for the rst step estimation is [yj x; d ] (' (x; )) + [yj x; d 0] (' (x; )) ' (x; ) pn (b ) ' (x; ) ' (x; ) 0 where (x) p (x) 0 (b ) (8) (x) [yj x; d ] (' (x; )) ' (x; ) + [yj x; d 0] (' (x; )) ' (x; ) for simplicity. Combining (7) and (8), we conclude that the adjustment to the in uence function can be written as (x) p (x) 0 p (x) p (x) 0 p (x) (d ' (x)) (9) Now p (x) p (x) 0 [p (x) (x)] are the coe cients of the linear projection of (x) on p (x). In other words, we can write p (x) 0 p (x) p (x) 0 [p (x) (x)] ( (x)j p (x)) where (j p (x)) denotes the projection on the linear space spanned by p (x). If the dimension of p (x) is su ciently large, then approximately ( (x)j p (x)) [ (x)j x] (x). It 6