The Asymptotic Variance of Semi-parametric Estimators with Generated Regressors

Transcription

1 The Asymptotic Variace of Semi-parametric stimators with Geerated Regressors Jiyog Hah Departmet of coomics, UCLA Geert Ridder Departmet of coomics, USC Jue 30, 20 Abstract We study the asymptotic distributio of three-step estimators of a ite dimesioal parameter vector where the secod step cosists of oe or more o-parametric regressios o a regressor that is estimated i the rst step. The rst-step estimator is either parametric or o-parametric. Usig Newey s (994) path-derivative method we derive the cotributio of the rst-step estimator to the i uece fuctio. I this derivatio it is importat to accout for the dual role that the rst-step estimator plays i the secod-step o-parametric regressio, i.e., that of coditioig variable ad that of argumet. JL Classi catio: C0, C4. Keywords: Semi-parametric estimatio, geerated regressors, asymptotic variace. Fiacial support for this research was geerously provided through NSF SS ad We thak Guido Imbes ad semiar participats at UC Riverside, the Tiberge Istitute, Yale, FGV-Rio de Jaeiro, FGV-São Paulo, Harvard/MIT, UCLA, Maheim, Scieces Po ad CMFI for commets. Geert Ridder thaks the Departmet of coomics, PUC, Rio de Jaeiro for their hospitality. Addresses: Jiyog Hah, Departmet of coomics, Buche Hall, UCLA, Los Ageles, CA 90095, haheco.ucla.edu; Geert Ridder, Departmet of coomics, Kaprilia Hall, USC, Los Ageles, CA 90089, ridderusc.edu.

2 Itroductio We study the asymptotic distributio of estimators of a ite dimesioal parameter i a semiparametric three (or more) step estimatio problem. This topic has become quite importat especially due to the recet developmets i the ecoometric aalysis of treatmet e ects ad i the ideti catio ad estimatio of o-liear models with edogeous covariates usig cotrol variables. We udertake a theoretical ivestigatio of such estimators, ad illustrate the usefuless of our result by examiig the asymptotic variace of the estimator of the Average Treatmet ect (AT) proposed by Heckma, Ichimura ad Todd (998) that is based o o-parametric regressios o the estimated propesity score. The estimators uder cosideratio are all characterized by three steps. I the rst step we estimate a regressor. I the secod step we estimate a o-parametric regressio with the geerated regressor as oe of the idepedet variables. I the third step we estimate a ite dimesioal parameter (without loss of geerality we cosider the scalar case) that satis es a momet coditio that depeds o the o-parametric regressio estimated i the secod step. Paga (984), who cosidered regressio estimators ivolvig geerated regressors i the parametric cotext, is a itellectual predecessor. We heavily use Newey s (994) path-derivative based characterizatio of the asymptotic variace of semi-parametric estimators. We exted his result to three-step estimators, where the secod step is a o-parametric regressio o a variable estimated i the rst step. This paper has the followig structure. Our mai result is i Sectio 2. I Sectio 3 we cosider estimators that ivolve partial meas with a applicatio to regressio o the estimated propesity score i Sectio 4. 2 The I uece Fuctio of Semi-parametric Three-Step stimators We ow preset our two mai results o semi-parametric three-step estimators. We distiguish betwee two cases: (i) the rst step is parametric, (ii) the rst step is o-parametric. Moreover, we rst cosider estimators that ca be expressed as a sample average of a fuctio of the secodstep o-parametric regressio oly. Next, we allow the third-step estimator to deped o other variables besides the secod-step o-parametric regressio. I both cases the estimators are, i Newey s (994) termiology, full meas, because they average over all argumets of the secodstep o-parametric regressio. I Sectio 3 we cosider estimators that average over most but ot all idepedet variables i the secod-step o-parametric regressio, i.e. the estimator is a partial mea. This makes the i uece fuctio more complicated, which is the reaso that we start with the full mea case. 2. Parametric First Step, No-parametric Secod Step We assume that we observe i.i.d. observatios s i = (y i ; x i ; z i ; ) ; i = ; : : : ;. I the rst step, we compute a estimator b such that p P (b ) = p (x i; z i ) + o p () with [ (x i ; z i )] = 0. The parameter vector idexes a relatio betwee a depedet variable that is a compoet of 2

3 x (ad that we later deote by u) ad idepedet variables that are some or all of the other variables i x ad those i z. ither the predicted value or the residual of this relatioship is a idepedet variable i the secod-step o-parametric regressio. The otatio '(x; z; ) for the geerated regressor covers both cases. I the secod step, our goal is to estimate (x; v ) = [y j x; v ] where v = ' (x; z; ). Because we do ot observe, we use bv i = ' (x i ; z i ; b) i the oparametric regressio. The o-parametric regressio estimator of y o x; v = ' (x; z; ) is deoted by b. (The b is distict from the oparametric regressio b of y o x; v = ' (x; z; ).) Our goal is to characterize the rst order asymptotic properties of b = h (b (x i ; ' (x i ; z i ; b))) We ca cosider b as the solutio of a sample momet equatio that is derived from a populatio momet equatio that depeds o ad (x; '(x; z; )). Usig Newey s (994) path-derivative approach, it ca be show that we have the approximatio p b = p (h ( (x i ; ' (x i ; z i ; ))) ) (i) + p + p (h (b (x i ; ' (x i ; z i ; ))) h ( (x i ; ' (x i ; z i ; )))) (ii) (h ( (x i ; ' (x i ; z i ; b))) h ( (x i ; ' (x i ; z i ; )))) + o p () (iii) The rst term (i) is the mai term, the secod term (ii) is the adjustmet for the estimatio of b, ad the third term (iii) is the adjustmet related to the estimatio of b. The decompositio here is based o the fact that Newey s approach ca be used term-by-term. Therefore, we may without loss of geerality assume that is a scalar. The secod compoet (ii) i the decompositio ca be aalyzed as i Newey (994, pp ), ad we therefore focus o the aalysis of the third compoet We de e p (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; ; 2 ) = h ( (x; ' (x; z; ) ; 2 )) Note that the two roles that plays are made explicit i g (s; ; 2 ) that is obtaied by substitutig v = '(x; z; ) i (x; v ; 2 ). Note also that (x; v ) = (x; v ; ), where (x; v ) = [y j x; ' (x; z; ) = v ]. The otatio ; 2 is just a expositioal device, sice = 2 =. 3

4 With these de itios, we ca ow write h ( (x i ; ' (x i ; z i ; b) ; b)) = g (s i ; b ; b 2 ) where b = b 2 = 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 i the liearizatio that ivolves partial derivatives: p (h ( (x i ; ' (x i ; z i ; b) ; b)) h ( (x i ; ' (x i ; z i ; ) ; ))) = p (g (s i ; b ; b 2 ) g (s i ; ; )) g (s; ; ) g (s; ; ) p(b = + ) + o p () 2 h i h Therefore we must compute g(s;;) ad g(s;;) 2 i. The computatio of the rst expectatio is easy; it is straightforward to show that g (s; ; ) h ( (x; ' (x; z; ))) = (x; ' (x; z; )) v ' (x; z; ) The headache is to compute the secod expectatio. By the chai rule g (s; ; ) h ( (x; ' (x; z; ))) (x; ' (x; z; ) ; ) = 2 2 () Ufortuately, it is ot obvious how to di eretiate (x; ' (x; z; ) ; ) with respect to. After all, (x; ' (x; z; ) ; ) has the fuctioal form of [y j x; ' (x; z; ) = v ] that depeds o. The ext lemma gives the solutio i a geeric case. Lemma Let t (x; '(x; z; )) deote a arbitrary mea square itegrable fuctio that is cotiuously di eretiable i the secod argumet. The, [t(x; '(x; z; ))(x; '(x; z; ); )] = = t(x; v ) (x; v ) '(x; z; ) + ((x; z) (x; v )) t(x; v ) '(x; z; ) (2) v v with v = '(x; z; ) ad (x; z) = (yjx; z). Our aalysis is predicated o the assumptio that the derivative exists o the left of () exists. Aalysis of the case whe the derivative does ot exist is beyod the scope of this paper. A ote that aalyzes ad gives su ciet coditios for the existece of the derivative is available o request. 4

5 Proof. Because (x; '(x; z; ); ) is the solutio to we have that for all mi p (y p(x; '(x; z; ))) 2 [(y (x; '(x; z; ); )) t (x; ' (x; z; ))] = 0 Di eretiatig with respect to ad evaluatig the result at =, we d after rearragig (2). This key lemma is used repeatedly i the sequel, begiig with the proof of the followig theorem. Theorem (Cotributio parametric rst-step estimator) The adjustmet to the i uece fuctio that accouts for the rst-stage estimatio error is g (s; ; ) g (s; ; ) p + (b ) 2 = ( (x; z) (x; v )) 2 h ( (x; v )) (x; v ) '(x; z; ) p(b ) (3) 2 v with v = '(x; z; ). Proof. We compute the right had side of () 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 )) 2 h(x; (x; v )) (x; v ) '(x; z; ) 2 v h i Addig g(s;;) that is equal to the opposite of the rst term o the right-had side, we d the desired result. Remark From Theorem, it ca be easily deduced that the rst-stage estimate has o cotributio to the i uece fuctio if h is liear i. Although straightforward ex post, this simpli catio does ot seem to have bee recogized i the past. The literature focused istead o the simpli - catio that occurs if the idex restrictio [yj x; z] = [yj x; '(x; z; )] holds. See, e.g., Newey (994) ad Klei, She ad Vella (200). Suppose ow that the is multidimesioal, i.e., y is a J-dimesioal radom vector. The estimator is ow b = h (b (x i ; ' (x i ; z i ; b)) ; : : : ; b J (x i ; ' (x i ; z i ; b))) The product rule of calculus suggests that we ca tackle this problem by addig the derivatives. This is formalized i the ext theorem. 5

6 Theorem 2 (Cotributio parametric rst-step estimators) The adjustmet to the i uece fuctio that accouts for the rst-stage estimatio error is X 2 h ( (x; ' (x; z; ))) (y j j (x; ' (x; z; ))) j (x; ' (x; z; )) ' (x; z; ) p (b ) : v j 2 j Proof. See Appedix. 2.2 No-parametric First Step, No-parametric Secod Step We ow assume that the rst step is o-parametric. Agai we have a radom sample s i = (y i ; x i ; z i ) ; i = ; : : : ;. The rst-step projectio of oe of the compoets of x, that we deote by u, o some or all of the other compoets of x ad z is deoted by v = ' (x; z) = [u j x; z]. The rst step is to estimate this projectio by o-parametric regressio. I the secod step we estimate (x; v ) = [y j x; v ] by o-parametric regressio of y o x; bv = b'(x; z). Our iterest is to characterize the rst-order asymptotic properties of b = h (b (x i ; b' (x i ; z i ))) where b (x i ; bv i ) is the o-parametric regressio estimate. We de e (x; v ; v 2 ) = [y j x; ' (x; z) = v ] g (w; v ; v 2 ) = h ( (x; v ; v 2 )) with v 2 '(x; z) ad coditioig o '(x; z) = v. Note that v ad v 2 play the roles of ad 2. With these de itios, we ca ow write h ( (x i ; bv ; bv 2 )) = g (s i ; bv ; bv 2 ) where bv = bv 2 = bv. We keep them separate to emphasize their di eret roles. Our objective is to approximate g (s i ; bv ; bv 2 ) g (s i ; v ; v 2 ) To d the cotributio of the samplig variatio i ^v we take as kow, i.e., the samplig variatio i the secod-step o-parametric regressio is accouted for i a separate term that has a well-kow form, sice it follows directly from Newey (994). As i Newey (994) we cosider a path v idexed by 2 R such that v = v. First, usig the calculatio i the previous sectio we obtai that [h ( (x; v ; v ))] = [D (s; v )] = 6

7 for D (s; v ) = 2 h ( (x; v )) (y (x; v 2 )) (x; v ) v : v which is liear i v. Secod, we have that for ay v = '(x; z), with [D (s; v)] = [ (x; z) '(x; z)] 2 h ( (x; ' (x; z))) (x; z) = (y (x; ' 2 (x; z))) (x; ' (x; z)) v x; z where it is uderstood that we coditio o the variables that are i ' so that we average over all x that are ot i the geerated regressor. By Newey (994, Propositio 4), these two facts imply that the adjustmet to the i uece fuctio 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 compoet of x that is projected o x; z. We summarize the result i a theorem: Theorem 3 (Cotributio o-parametric rst-step estimator) The adjustmet to the i- uece fuctio that accouts for the rst-stage estimatio error is p (x i ; z i ) (u i ' (x i ; z i )) with ' (x; z) = [ujx; z] ad as i (4). 2.3 Additioal Variables i the Third Step So far, we have assumed that the parameter of iterest is = [h((x; v ))] where h depeds oly o (x; v ). We ow cosider the extesio to = [h(w; (x; v ))] where w is a vector of other variables that may have x; z as subvectors. We cosider both the case that ' is parametric ad the case that this fuctio is o-parametric. Because as before the mai term ad the cotributio of the estimatio of [yjx; v ] do ot raise ew issues, the ext two theorems oly give the cotributio of the rst-stage estimator. I these theorems we use the fuctio h (w; (x; v )) (x; v) = x; ' (x; z; ) = v with ' (x; z) substituted i the o-parametric case. 7 (4)

8 Theorem 4 (Cotributio parametric rst-step estimator) The adjustmet to the i uece fuctio that accouts for the rst-stage estimatio 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; ) p (b ) v with v = '(x; z; ). Proof. See Appedix. Now, we cosider the case where the rst step is o-parametric. The discussio precedig Theorem 3 implies that Theorem 5 (Cotributio o-parametric rst-step estimator) The adjustmet to the i- uece fuctio that accouts for the rst-stage estimatio error is p 2 (x i ; z i ) (u i ' (x i ; z i )) with ' (x; z) = [ujx; z] ad h (w; (x; ' (x; z))) (x; ' (x; z)) 2 (x; z) = (x; ' (x; z)) v x; z (x; ' (x; z)) + ((x; z) (x; ' (x; z))) v x; z Remark 2 Theorems 4 ad 5 are easily geeralized to the case of multidimesioal. Remark 3 Suppose that (x; ' (x; z; )) = 0 i Theorem 4. The adjustmet is the equal to the derivative with respect to, i.e., the aive derivative that oly accouts for as a argumet (see equatio (5) i the proof of Theorem 4). Therefore, it may be useful to check whether (x; ' (x; z; )) = 0 i speci c models. If it is the case, we eed ot worry about the e ect of rst-step estimatio o the secod-stage o-parametric regressio. Such a characterizatio turs out to be useful for the partially liear regressio model y i = x i + m ('(x i ; z i ; )) + i ; where m is o-parametric ad [ i j x i ; v i ] = 0. 2 Remark 4 The theorems ca be applied to geeral semi-parametric GMM estimators. If we cosider the momet coditio [m(w; (x; v ); )] = 0 ad we liearize the correspodig sample momet coditio to obtai p ( b m(w; (x; v ); ) ) = p m (w 0 i ; b(x i ; b'(x i ; z i )); ) + o p () Therefore, the cotributio of the rst-stage P estimate to the asymptotic distributio of b ca be foud by applyig Theorem 5 to p m (w i; b(x i ; b'(x i ; z i )); ). 2 A detailed discussio ca be foud i a previous versio of the paper, which is available o request. 8

9 3 The I uece Fuctio of Semi-parametric Three-step stimators: Partial Meas I Sectio 2, the estimator b averaged over all argumets of the secod-step o-parametric regressio fuctio. I Newey s (994) termiology the estimator is a full mea. I this sectio we cosider the case that the discrete idepedet variable d is xed, i.e., b does ot average over this variable. Hece the estimator is a partial mea. Because the variable that is xed i the partial mea is discrete, the parametric rate of covergece applies. (If that variable were cotiuous we would have a slower rate.) Let the discrete variable d take the values d () ; : : : ; d (K). I the secod step we estimate (x; v ; d) = [y j x; v ; d] As i Sectio 2 we use bv i i the o-parametric regressio. De e b k (x i ; bv i ) = b(x i ; bv i ; d (k) ), i.e., b k is the o-parametric regressio fuctio if we set x ad bv to the observed values for i, but x d at value d (k) which may ot be its value for i. The K vector b(x i ; bv i ) stacks the b k (x i ; bv i ). We also de e k (x; v ) = y j x; v ; d (k) ad (x; v ) as the K vector with compoets k (x; v ). Our goal is to characterize the rst order asymptotic properties of b = h (w i ; b (x i ; bv i )) with b the K vector of o-parametric regressio fuctios where the discrete variable d is xed at its K distict values. As i Sectio 2.3, we allow for a vector of additioal variables w i h. 3. Partial Meas: Parametric First Step, No-parametric Secod Step We assume that the rst step is parametric, ad bv i = ' (x i ; z i ; b). As i Sectio 2, we ca use Newey s (994) approach, ad express the i uece fuctio of b as a sum of three terms: (i) the mai term, (ii) a term that adjusts for the estimatio of b, ad (iii) a adjustmet related to the estimatio of b. De e ad k (x; '(x; z; )) = Pr(d = d (k) jx; '(x; z; )) k (x; z) = Pr(d = d (k) jx; z) h (w; (x; v )) k (x; v) = k x; ' (x; z; ) = v The secod compoet i the decompositio ca be aalyzed as i Newey (994, pp ) ad is equal to p KX (d i = d (k) )(y i k (x i ; v i )) k(x i ; v i ) k (x i ; v i ) + o p () (5) 9

10 As i Sectio 2 we therefore focus o the aalysis of the third compoet p (h (w i ; (x i ; ' (x i ; z i ; b) ; b)) h (w i ; (x i ; ' (x i ; z i ; )) ; )) = p (g (s i ; b ; b 2 ) g (s i ; ; )) with k (x; v ; ) = y j x; ' (x; z; ) = v ; d = d (k) (x; v ; ) = ( (x; v ; ) K (x; v ; )) 0 g (w; ; 2 ) = h (w; (x; ' (x; z; ) ; 2 )) Lemma ca be geeralized to the partial meas case: Lemma 2 For partial meas we have for all k = ; : : : ; K [t(x; '(x; z; )) k (x; '(x; z; ); )] = k (x; z) = k (x; v ) t(x; v ) k(x; v ) '(x; z; ) v k (x; z) + k (x; v ) ( k(x; z) 2 k (x; v )) k (x; v ) t(x; v ) v with v = '(x; z; ). Proof. Because k (x; '(x; z; ); ) is the solutio to mi (d = d (k) )(y p(x; '(x; z; ))) 2 p 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 eretiatig with respect to ad evaluatig the result at =, we d after rearragig (6). The ext theorem geeralizes Theorem 4: Theorem 6 (Cotributio parametric rst-step estimator) The adjustmet to the i uece fuctio that accouts for the rst-stage estimatio error is KX! h(w; (x; v )) k (x; z) k k (x; v ) k (x; v ) ' (x; z; ) k(x; v ) v! KX k (x; z) + k (x; v ) ( k(x; z) k (x; v )) k(x; v ) ' (x; z; ) v!! KX k (x; z) k (x; v ) ( k(x; z) 2 k (x; v )) k (x; v ) k(v; v ) ' (x; z; ) p (b ) v 0 (6)

11 Proof. See Appedix. If h oly depeds o, the k (x; v ) = h((x;v)) k so that the cotributio of the rst stage is! KX k (x; z) k (x; v ) h ( (x; v )) k (x; v ) '(x; z; ) k (x; v ) k v! KX KX k (x; z) + k 0 k (x; v ) ( k (x; z) k (x; v )) 2 h ( (x; v )) k 0 (x; v ) '(x; z; ) k k 0 v =!! KX k (x; z) k (x; v ) 2 ( k (x; z) k (x; v )) h ( (x; v )) k (x; v ) '(x; z; ) p(b ) k v 3.2 Partial Meas: Noparametric First Step, No-parametric Secod Step Now assume that the rst step cosists of a o-parametric regressio estimate of v = ' (x; z) = [u j x; z]. We ca obtai the adjustmet by replicatig the argumets i Sectio 2.3 leadig to Theorem 5. Lettig X K h(w; (x; v )) k (x; z) 3 (x; z) = k k (x; v ) k (x; v ) k(x; v ) v x; z + X K k (x; z) k (x; v ) ( k(x; z) k (x; v )) k(x; v ) v x; z (8) X K k (x; z) k (x; v ) ( k(x; z) 2 k (x; v )) k (x; v ) k(v; v ) v x; z we obtai a aalog of Theorem 3: Theorem 7 (Cotributio o-parametric rst-step estimator) The adjustmet to the i- uece fuctio that accouts for the rst-stage estimatio error is p 3 (x i ; z i ) (u i ' (x i ; z i )) with 3 (x; z) give i (8). If h depeds o oly we replace 3 above by X K k (x; z) k (x; ' (x; z)) h ( (x; ' (x; z))) k (x; ' (x; z)) 3 (x; z) = k (x; v ) k v + KX KX k 0 = KX k (x; z) k (x; ' (x; z)) ( k (x; z) k (x; z) k (x; ' (x; z)) 2 ( k (x; z) (7) k (x; ' (x; z))) 2 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 x; z

12 4 Applicatio: Regressio o the stimated Propesity Score We cosider a itervetio with potetial outcomes y 0 ; y that are the cotrol ad treated outcome, respectively. The treatmet idicator is d ad y = dy + ( d)y 0 is the observed outcome. The vector x cotais covariates that are ot a ected by the itervetio. As show by Rosebaum ad Rubi (983) ucofouded assigmet, i.e., the assumptio that y ; y 0? djx, implies y ; y 0? dj' (x) with ' (x) = Pr(d = jx) the probability of selectio or propesity score. This observatio has led to a large umber of estimators. The asymptotic variace of the estimators ca be compared to the semi-parametric e ciecy boud for the AT derived by Hah (998). The most popular estimators are the matchig estimators that estimate the AT give x or give ' (x) by averagig outcomes over uits with a similar value of x or ' (x). Abadie ad Imbes (2009a, 2009b) are recet cotributios. They show that matchig estimators that have a asymptotic distributio that is otoriously di cult to aalyze, are ot asymptotically e ciet. The secod class of estimators do ot estimate the AT give x or ' (x) but use the propesity scores as weights. Hah s (998) estimator ad the estimator of Hirao, Imbes ad Ridder (2003) are examples of such estimators. These estimators are asymptotically e ciet. The third class of estimators use o-parametric regressio to estimate [yjd = ; x], [yjd = 0; x] or [yjd = ; ' (x)], [yjd = 0; ' (x)]. Of these estimators the estimator based o [yjd = ; x], [yjd = 0; x], the imputatio estimator, is kow to be asymptotically e ciet, which suggests that there is o role for the propesity score. The missig result is that for the estimator that uses the o-parametric regressio o the propesity score that is estimated i a prelimiary step. This estimator that was suggested ad aalyzed by Heckma, Ichimura, ad Todd (HIT) (998) ts ito our framework ad is aalyzed here. 3 Our coclusio is that the HIT estimator has the same asymptotic variace as the imputatio estimator, so that there is o e ciecy gai i usig the propesity score. This should settle the issue whether there is a role for the propesity score i achievig semi-parametric e ciecy Parametric First Step, No-parametric Secod Step We have a radom sample s i = (y i ; x i ; d i ) ; i = ; : : : ;. The propesity score Pr(d = jx) = '(x; ) is parametric ad its parameters are estimated i the rst step, by e.g. Maximum Likelihood or OLS (Liear Probability model) or ay other method, such that p (b ) = p (d i ; x i ) + o p () with [ (d i ; x i )] = 0. I the secod step, we estimate ('(x; )) = ( [y j '(x; ); d = ] ; [y j '(x; ); d = 0]) 0 ; 3 Heckma, Ichimura, ad Todd actually cosider a estimator of the Average Treatmet ect o the Treated (ATT) that we also aalyze. 4 That does ot mea that there is o role for the propesity score i assessig the ideti catio or i improved small sample performace of AT estimators. 2

13 Because we do ot observe, we use ' (x i ; b) i the o-parametric regressio. Our iterest is to characterize the rst order asymptotic properties of b = (b (' (x i ; b)) b 2 (' (x i ; b))) This estimator ca be hadled by applyig Theorem 6 for the special case that h oly depeds o. The vector ('(x; )) is a 2-vector of partial meas ad d is either d () = or d (2) = 0. Further '; k deped o x oly ad (x) = '(x; ) ad ('(x; )) = Pr(d = j'(x; )) = '(x; ). Also h() = 2 so that the secod derivatives are 0. Upo substitutio the rst two terms o the right-had side of (7) are 0. Because h = for k = ; 2 we d that the cotributio k k v of the rst-stage estimator to the i uece fuctio is g (s; ; ) g (s; ; ) + p(^ ) = 2 [yj x; d = ] (' (x; )) + [yj x; d = 0] 2 (' (x; )) ' (x; ) p(^ ) ' (x; ) ' (x; ) The cotributio of b ca be derived usig Newey (994), ad is give i (0) below. We also cosider the HIT estimator of the Average Treatmet ect o the Treated (ATT) ^ = d i p (b ('(x i ; b)) b 2 ('(x i ; b))) with p = Pr(d = ). This estimator is a special case of that cosidered i Theorem 6 with h(w; ; 2 ) = d( p 2 ), so that h ot oly depeds o ; 2, so that the simpli catio i (7) does ot apply. Substitutio of h(w; ; 2 ) = d h(w; ; 2 ) = d p 2 p ad ( ; 2 are fuctios of v oly) (v) = v p 2 (v) = v p give that the rst term i Theorem 6 is 0, ad the secod term is p (( (x) ('(x; ))) ( 2 (x) 2 ('(x; )))) ' (x; ) ad the third p ( (x) ('(x; ))) '(x; ) ' (x; '(x; ) ( ) 2(x) 2 ('(x; ))) so that by takig the di erece of the last two terms we d that the cotributio is g (s; ; ) g (s; ; ) + p(^ ) = 2 [yjx; d = 0] 2 (' (x; )) '(x; ) p(^ ) p ( ' (x; )) 3

14 4.2 No-parametric First Step, No-parametric Secod Step The aalysis i the previous sectio combied with the results i Newey (994) shows that i the case that the rst stage is o-parametric the cotributio of the rst-stage estimatio to the i uece fuctio of the AT estimator is [yj x; d = ] (' (x)) + [yj x; d = 0] 2 (' (x)) (d ' (x)) ' (x) ' (x) which ca be alteratively writte as [yj x; d = ] (' (x)) d + ( [yj x; d = ] (' (x))) ' (x) + [yj x; d = 0] 2 (' (x)) ( d) ( [yj x; d = 0] 2 (' (x))) (9) ' (x) To obtai the complete i uece fuctio of b we eed the cotributio of the estimatio error i b. This cotributio is equal to 5 ( (' (x)) 2 (' (x)) ) + d ' (x) (y d (' (x))) ' (x) (y 2 (' (x))) (0) Addig (9) ad (0), we obtai the i uece fuctio of the estimator based o regressios o the estimated propesity score: ( [yj x; d = ] [yj x; d = 0] )+ d ' (x) (y [yj x; d = ]) d (y [yj x; d = 0]) ; ' (x) which is the i uece fuctio of the e ciet estimator ad also that of the imputatio estimator b I = ( b (x i ) 2 b (x i )) with (x) = [yjx; d = ]; 2 (x) = [yjx; d = 0]. The imputatio estimator ivolves oparametric regressios o x ad ot o the estimated propesity score. However these two estimators have the same i uece fuctio which shows that regressig o the o-parametrically estimated propesity score does ot result i a e ciecy gai. The ifeasible estimator that depeds o o-parametric regressios o the populatio propesity score is less e ciet tha the estimator that uses the estimated propesity score. For the estimator of the ATT the cotributio of the rst stage is [yjx; d = 0] 2 (' (x)) (d ' (x)) p ( ' (x)) 5 Derivatio is straightforward, ad is cotaied i a previous versio of the paper, which is available o request. 4

15 The mai term ad the cotributio of the estimatio of the (ifeasible) o-parametric regressios is 6 d p (y (' (x))) ( d)' (x) p ( ' (x)) (y 2 (' (x))) + d p ( (' (x)) 2 (' (x)) ) : Addig these expressios we obtai the full i uece fuctio d p (y [yjx; d = ]) ( d)' (x) p( ' (x)) (y [yjx; d = 0]) + d p ([yjx; d = ] [yjx; d = 0] ) As i the case of the AT the i uece fuctio is the same as that for the estimator that ivolves o-parametric regressios o x ad ot o the estimated propesity score, so that agai there is o rst-order asymptotic e ciecy gai if we use the estimated propesity score i the o-parametric regressios. It should be oted that the i uece fuctios derived i this sectio are di eret from those foud i the literature. Recetly, Mamme, Rothe, ad Schiele (200) derived the i uece fuctio for the AT estimator cosidered i this sectio. They cocluded that it is idetical to that of the ifeasible estimator that regresses o the populatio propesity score. This is because they imposed the idex assumptio [yj d; x] = [yj d; '(x)], which is ot made i the stadard program evaluatio literature, because it restricts the distributio of the potetial outcomes. For istace, i a liear selectio (o observables) model the idex restrictio implies that the regressio coe ciets i the outcome equatios are proportioal to those i the selectio equatio. HIT derived the i uece fuctio for the ATT estimator that is also di eret from ours. I this case, the di erece appears to be due to a error i their derivatio, which fails to accout for the e ect of the rst-stage estimatio o the coditioal expectatio i the secod stage. 5 Summary We studied the asymptotic distributio of three-step estimators of a ite dimesioal parameter vector where the secod step cosists of oe or more o-parametric regressios o a regressor that is estimated i the rst step. The rst step estimator is either parametric or o-parametric. We showed that Newey s (994) method ca be used to determie the cotributio of the rst-step estimatio error o the i uece fuctio. I doig so it is essetial to recogize that the rststage estimate has two e ects o the samplig distributio of the ite-dimesioal parameter vector. First, the rst-stage estimate eters the argumet at which the coditioal expectatio is evaluated, secod, the rst-stage estimate chages the coditioal expectatio itself. I the literature the secod cotributio of the rst-stage estimate to the i uece fuctio is sometimes forgotte. Our cotributio is that we show how to derive this cotributio so that we obtai the correct i uece fuctio for three- or more stage estimators. 6 This ca be derived usig a argumet that leads to (0). 5

16 Appedix Proof of Theorem 2 We write p (h ( (x i ; ' (x i ; z i ; b) ; b)) h ( (x i ; ' (x i ; z i ; ) ; ))) = p (g (s i ; b ; b 2 ) g (s i ; ; )) g (s; ; ) g (s; ; ) p(b = + ) + o p () 2 h i h Therefore we must compute g(s;;) ad g(s;;) 2 i. The computatio of the rst expectatio is easy. Because (x; ' (x; z; ) ; ) = (x; ' (x; z; )), we have g (s; ; ) = JX j= h ( (x; ' (x; z; ))) j (x; ' (x; z; )) j v ' (x; z; ) where j deotes the j-th compoet of, etc. We ow tackle the secod expectatio. By the chai rule g (s; ; ) JX h ( (x; ' (x; z; ))) j (x; ' (x; z; ) ; ) = () 2 j 2 j= We compute the right had side of () usig Lemma. g (s; ; ) = X h ( (x; ' (x; z; ))) j (x; '(x; z; ); ) = 2 j j 2 JX j= Addig JX j= h ( (x; ' (x; z; ))) j (x; '(x; z; )) j v (y j j (x; ' (x; z; ))) 2 h ( (x; ' (x; z; ))) 2 j h i g(s;;) we d the desired result. '(x; z; ) + j (x; ' (x; z; )) v '(x; z; ) Proof of Theorem 4 ad The cotributio of b is the sum of h (w; (x; ' (x; z; ) ; )) p (b ) (2) h (w; (x; ' (x; z; ) ; )) p (b ) (3) 2 6

17 Note that h (w; (x; ' (x; z; ) ; )) h (w; (x; v )) = = Because (x;'(x;z;);) is a fuctio of (x; ' (x; z; )), we have (x; v ) v ' (x; z; ) h (w; (x; ' (x; z; ) ; )) 2 h (w; (x; ' (x; z; ))) (x; ' (x; z; ) ; ) = 2 h (w; (x; ' (x; z; ))) = (x; ' (x; z; x; ' (x; z; ) ; ) ) 2 = (x; ' (x; z; )) (x; ' (x; z; ) ; ) 2 (4) (5) By Lemma (x; ' (x; z; ) ; ) (x; ' (x; z; )) 2 = ((x; z) (x; v )) (x; v ) ' (x; z; ) v (x; v ) (x; v ) ' (x; z; ) v (6) Combiig (4) - (6), we coclude that h (w; (x; ' (x; z; ) ; )) h (w; (x; ' (x; z; ) ; )) + 2 h (w; (x; v )) (x; v ) ' (x; z; ) = (x; v ) v (x; v ) + ((x; z) (x; v )) ' (x; z; ) v which gives us the desired result. Proof of Theorem 6 Note that h (w; (x; ' (x; z; ) ; )) = = KX! h (w; (x; v )) k (x; v ) k v ' (x; z; ) (7) 7

18 Because k(x;v ; ) is a fuctio of (x; v ), we have h (w; (x; ' (x; z; ) ; )) X K h (w; (x; v )) = 2 k x; v k (x; v ; ) K X = k (x; v ) k(x; v ; ) By Lemma 2 K X k (x; v ) k (x; v ; ) X k (x; z) = k (x; v ) k (x; v ) k(x; v ) '(x; z; ) v k X K k (x; z) + k (x; v ) ( k (x; z) k (x; v )) k(x; v ) v X K k (x; z) k (x; v ) ( 2 k (x; z) k (x; v )) k (x; v ) k (x; v ) v '(x; z; ) '(x; z; ) (8) (9) Combiig (7) - (9), we obtai the desired coclusio. 8

19 Refereces [] Abadie, A. ad G. Imbes (2009a), A Martigale Represetatio for Matchig stimators, upublished workig paper. [2] Abadie, A. ad G. Imbes (2009b), Matchig o the stimated Propesity Score, upublished workig paper. [3] Hah, J. (998): O the Role of the Propesity Score i ciet Semiparametric stimatio of Average Treatmet ects, coometrica, 66,pp [4] Heckma, J.J., H. Ichimura, ad P. Todd (998): Matchig as a coometric valuatio stimator, Review of coomic Studies 65, pp [5] Hirao, K., G. Imbes, ad G. Ridder (2003): ciet stimatio of the Average Treatmet ect Usig the stimated Propesity Score, coometrica, 7, pp [6] Klei, R., C. She, ad F. Vella (200): Triagular Semiparametric Models Featurig Two Depedet dogeous Biary Outcomes, upublished workig paper. [7] Mamme,., C. Rothe, ad M. Schiele (200): Noparametric Regressio with Noparametrically Geerated Covariates, upublished workig paper. [8] Murphy, K. M. ad R. H. Topel (985): stimatio ad Iferece i Two-Step coometric Models, Joural of Busiess ad coomic Statistics 3, pp [9] Newey, W.K. (994): The Asymptotic Variace of Semiparametric stimators, coometrica 62, pp [0] Paga, A. (984): coometric Issues i the Aalysis of Regressios with Geerated Regressors, Iteratioal coomic Review 25, pp [] Rosebaum, P., ad D. Rubi (983): The Cetral Role of the Propesity Score i Observatioal Studies for Causal e ects, Biometrika 70, pp