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2 Sulement to Identi cation and stimation of Distributional Imacts of Interventions Using Changes in Inequality Measures, Part I: Proofs, derivations and some additional technical material Sergio Firo (sergio. Cristine Pinto December 204 Introduction In the accomanying aer to this sulement we resented several theoretical results. We did not, however, rovide formal roofs for their validity in that aer. In this sulement we establish formally and in a very detailed manner the validity of those results. We also resent for the sake of comleteness some seci c formulas of in uence functions used in that aer. In articular, we resent the formulas for the Coe cient of Variation, the Interquartile Range, the Theil Index and the Gini Coe cient. 2 Lemma Proof of Lemma. A roof of this Lemma has two arts. The rst concerns the identi cation of the marginal (and conditional on T ) CDFs of otential outcomes and is omitted because it follows trivially from Firo (2007), Lemma. The second art follows by the de nition of IT arameters. They are di erences in the functionals of these marginal (conditional on T ) identi ed distributions and therefore they can be exressed as functions of the observable data (; ; T ). 3 Lemma 2 Proof of Lemma 2. The roof of this second lemma is based on a similar decomosition to that resented in the aendix of Hirano, Imbens and Ridder (2003, henceforth HIR) and therefore we use the same notation to what hadbeen used in that aer. Thus, our estimator of the roensity score is b () b () H () > b. In fact, we set notation as in HIR to simlify the comarison of the decomosition resented below with results resented in their aendix. We break down the di erence

3 bf b (y) F (y) into six comonents: bf b (y) F (y) () ( (T i; b ( i)) I f i yg (T i; ( i)) I f i yg (T i; ( i)) I f i yg (b ( i) ( i)) ( i) (Ti; ( i)) I f i yg (b ( i) ( ( (Ti; (x)) I f i (x) x (b (x) (x)) df (x) (T; (x) e ( i) I f yg x (b (x) (x)) df (x) T i ( i) ( i) ( ( i)) (3) (4) e T i ( i) ( i) ( i) ( i) ( ( i)) T i ( i) T i ( i) ( i) 0 ( i) ( i) ( ( i)) (i) ( ( i)) ( (T i; ( i)) I f i yg F T i ( i) (y)) + 0 ( i) (i) ( ( i)) (5) (6) (7) with e (x) (x) (Ti; (Ti; (z) I f i yg z 0 H (z) > e H (z) > df (z) e r H 0 (x) > H (x) (8) I f i yg z 0 H (z) > H (z) > df (z) r H 0 (x) > H (Ti; (x)) 0 ( (x) (9) I f i yg x (i) ( ( i)) (0) h 0 H () > H () H () >i e 0 H ( i) > e H ( i) H ( i) > ; where e lies between b and. ow we show that each term can be bounded uniformly in y. 2

4 By Taylor exansion, (T i; b ( i)) (T i; ( i)) (Ti; ( i) (b ( i) ( i)) + O kb ( i) ( i)k 2 : We can rewrite (2) (T i; ( i)) (b ( ( i) O kb ( i) ( i)k 2 ; ( (Ti; ( i)) I f i yg (b ( ( i) ( i)) where by triangle inequality kb ( i) ( i)k 2 kb ( i) ( i)k 2 + k ( i) ( i)k 2 : () By the mean value theorem, b (x) (x) 0 H (x) > e H (x) > (b ) : Using this result, the rst comonent of () is therefore kb ( i) ( i)k 2 C ( ()) 2 kb k 2 where () su x kh (x)k. ext, using Lemma in the aendix of HIR, the rst term of the sum in () is bounded by kb ( i) ( i)k 2 O C ( ()) 2 r s : Using again Lemma 2 in the aendix of HIR, we can bound the last term in the exansion, k ( i) ( ()) ( i)k 2 O : At the end, O kb ( i) ( i)k 2 ( ()) O + O ( ()) 2 r s : 3

5 ow, we nd a bound on (3). First, we rewrite this term (Ti; ( i)) I f i yg (b ( i) ( i)) ( (T; (x)) I f (x) x (b (x) (x)) df (Ti; ( i)) I f i yg ( ( i) ( i)) ( (T; (x)) I f (x) x ( (x) (x)) df (x) + and denote (3) by V. ote that [V ] 0 and (T; ()) Var [V ] Var I f () ( () ()) (T; ()) +Var I f () ( () ()) # # (T; ()) I f () ( () ()) 2 C ( ()) 2 s r and and [jv j] Var [V ] C ( ()) su jv j O ( ()) y2 2r s : s 2r ow consider (2), by the mean value theorem this term can be rewritten (Ti; ( ( (T; (x)) I f (x) x I f i yg 0 H ( i) > e H ( i) > 0 H (x) > e H (x) > df (x) (b ) : By a second alication of the mean value theorem, we write the rst term in the exression above as W () + W 2() W 3() with W (Ti; ( ( (T; (x)) I f (x) x I f i yg 0 H ( i) > H ( i) > 0 H (x) > H (x) > df (x) ; 4

6 and W 2() 2 W 3() (T i; ( i)) I f i yg 00 H ( i) > e H ( i) H ( i) > (e ( (T; (x)) I f (x) x 00 H (x) > e H (x) H (x) > df (x) (e ) where e lies between e and. First, we calculate the variance of W () ; Var W () and (T; ()) Var I f () 0 2 H () > H () H () (T; ()) +Var I f () 0 H () > H () > # (T; ()) I f yg () 0 2 H () > H () H () > C hh () H () >i W() qtr Var W () C r tr hh () H () >i ow, working with W 2() ; we rst notice that for (; ; T ) and all (T; () C ( ()) : (4) I f yg 00 H () > e H () H () > (e (T; ()) I f yg 00 H () > e H () H () > () k and note (T; ()) I f yg 00 H () > e H () H () > () k C ( ()) 3 2 and, By analogy we work with W 3() ; 2 C ( ()) 3 2 : W2() C ( ()) 3 2 (T; (x)) I f (x) x 00 H (x) 0 e H (x) H (x) > df (x) (e ) 5

7 Using the triangle and Cauchy-Schwartz inequality, h > i W() + W 2() W 3() (b ) C ( ())2 : Combining the bounds above, we (Ti; ( i)) su I f i yg (b ( i) ( i)) ( (T; (x)) I f (x) x (b (x) (x)) df (x) O ( ()) 2r s ( ()) 2 + O : ow, we work with (4). ote that + Using the mean value exansion we obtain, and de (T; (x)) I f (x) x (b (x) (x)) df (T; (x)) I f (x) x (b (x) (x)) df (T; (x)) I f (x) x ( (x) (x)) df (x) (T; (x)) I f (x) x (b (x) (x)) df (T; (x)) I f (x) x 0 H (x) > e H (x) > df (x) (b ) e Using this de nition, we can rewrite (8) (T; (x)) I f (x) x 0 H (x) > e H (x) > df (x) : (T; (z)) (x) I f (z) z 0 H (z) > e H (z) > df (z) r H 0 (x) > H (x) e e r H 0 (x) > H (x) : The rst order condition of the seudo-maximum likelihood aroach to calculate b is H ( i) T i H ( i) > b 0 6

8 and by the mean value theorem, (b ) e H ( i) H ( i) > 0 H ( i) > e H ( i) (T i ( i)) H ( i) (T i ( i)) (T; (x)) I f (x) x (b (x) (x)) df (T; (x)) I f yg x 0 H (x) > e H (x) > df (x) (b (x) e e e ( i) e ( i) r H 0 ( i) > (T i ( i)) H ( i) r H 0 ( i) > (T i ( i)) r 0 H ( i) > (T i ( i)) ( i) ( ( i)) and hence @ (T; (x)) I f (x) x (b (x) (x)) df (T; (x)) I f (x) x ( (x) (x)) df (x) : i x is bounded away on we have that su (T; (x) I f yg ( (x) (x)) df (x) C ( ()) s 2r O ( ()) s 2r : e (T i ( i)) ( i) ( i) ( ( i)) ow we work with (5). Let us de (T; (x)) I f (x) x 0 H (x) > H (x) > df (x) : 7

9 Using this de nition, we can rewrite (9) (T; (z)) (x) I f (z) z 0 H (z) > H (z) > df (z) r H 0 (x) > H (x) r H 0 (x) > H (x) : Therefore, (5) can be rewritten as: e ( i) ( i) T i ( i) ( i) ( ( i)) e e B () with B () H ( i) (T i ( i)) : ote that e e B () e e B () + e B () and working with the rst term we obtain e e B () B () e min e where min (A) is the smallest eigenvalue of a matrix A. By the mean value theorem and using the fact that (x) is bounded away from zero, C C (T; (z) I f yg z 0 H (z) > H (z) > e kh (z)k 2 df (z) k e k kh (z)k 2 df (z) k e k C ( ()) 2 k e k 0 H (z) > e H (z) > df (z) and e e B () C ( ()) 2 k e k B () : ow, we work with the second term e B () e e B () B() e : min( ) e 8

10 De ne b P 0 H ( i) > H ( i) H ( i) > and W ; e W e b W + b W H (i)> (e ) 00 H ( i) > e H ( i) H ( i) > W + C ( ()) 3 k e k kw k + ote that 0 H ( i) > H ( i) H ( i) > 0 H ( i) > H ( i) H ( i) > kw k C k k C ( ()) h 0 H () > H () H () >i W h 0 H () > H () H () >i W : and that Var h i 0 H () > H () H () > W W > Var h 0 H () > H () H () >i W W > 0 H () > H () H () > 2 W CW > H () H () > 2 W C ( ()) 4 : ow, we look at B () B () H ( i) (T i ( i)) + Since (x) is bounded away from zero and one, and Var B () h () ( H ( i) ( ( i) ( i)) : ()) H () H () >i + h( () ()) 2 H () H () >i C ( ()) 2 + C 2 ( ()) 4 s r By Cauchy-Schwartz inequality we have B () C ( ()) 2 : h i e e B () C ( ()) 7 2 9

11 and h e B () i C ( ()) 2 + C 2 ( ()) 3 : At the end, e ( i) ( i) T i ( i) ( i) ( ( i)) O ( ()) 2 : ow we work with (6). Since (6) is a sum of iid random variables with mean di erent than zero, we bound this term by deriving the order of its second moment, () T () () ( ()) 0 () 2 # T () () ( ()) 2 # T () T () () () () ( ()) () ( ()) 2 # T () T () + () 0 () () ( ()) () ( ()) T i () T () +2 () () () ( ()) () ( ()) # T () T () () 0 () () ( ()) () ( ()) + +2 () 2 T () () ( ()) 2 # ( () 0 ()) 2 T () () ( ()) () ( () 0 ()) 2 # T () () ( ()) T () () ( ()) T () () ( ()) # T () : () ( ()) We can aroximate (x) as the least squares rojection of 0 (x) on H (x) (x) ( (x)). If we assume that 0 (x) is t times continuously di erentiable, and using Lemma of HIR we have, su j 0 (x) (x)j < C r t x2 and hence 2 # ( () 0 ()) 2 T () () ( ()) ( () 0 ()) 2 C 2t r : 0

12 ow, consider the rst term in the exansion, () 2 T () () ( ()) () 2 ( () ()) 2 + () ( ()) 2 # T () () ( ()) () ( ()) () 2 () ( ()) 2 # : Using the aroximation of (x), (x) 2 0 (x) 2 + j 0 (x)j C t r and therefore () 2 ( () ()) 2 () ( ()) 0 () 2 ( () ()) 2 + C sr ( () ()) 2 j 0 ()j () ( ()) () ( ()) (T; ()) 2 I f () () ( ()) ( () ())2 () ( ()) ## + C r (T; ()) I f yg () ( () ( () ()) 2 : () ( ()) Since (x) is bounded from 0 and on and using Lemma in the aendix of HIR, (Ti; (x)) I f i yg (x) ( ( i)) ( ( i) ( i)) 2 ( i) ( ( i)) ( ( i) ( i)) 2 C C ( ()) 2 r s ( i) ( ( i)) and () 2 ( () ()) 2 () ( ()) C ( ()) 2 s r + C2 ( ()) 2 s r t r C ( ()) 2 s r : By analogy, () ( ()) () 2 () ( ()) 2 # C ( ()) 2 s r and since (x) is bounded and (x) and (x) are bounded away from 0 and, T () T () () ( () 0 ()) () ( ()) () ( ()) C [j () 0 ()j] C r t : # T () () ( ())

13 Finally, su y2 O max ( i) t 2r ; ( ()) T i ( i) ( i) ( ( i)) s 2r : 0 ( i) T i ( i) (i) ( ( i)) Combining the bounds, we have su bf b (y) F (y) y2 ( ()) 3 O + O ( ()) 2 r s +O ( ()) s 2r +O ( ()) ( ( ()) 2 + O s 2r O ( ()) 2 r s + O ( ()) 5 s 2 2r ( (T i; ( i)) I f i yg F (y)) 0 ( i) ( ()) 2 + O + O max 2r t ; ( ()) ( ()) 2 + O : s 2r ) T i ( i) (i) ( ( i)) And under the assumtions on ( ()) and s, this sum is o () : 4 Proof of Theorem This roof is divided into two arts. First we demonstrate that H is P-Donsker, where H is the collection of measurable functions from ( f0; g) R, H f (; ; T; y)j y 2 g that are right continuous and whose limits from the left exist everywhere in ( f0; g). The sace D[ f0; g] is equied with the uniform norm. This rst art of the result be shown in an intermediate lemma. Then in the second art we aly Donsker s Theorem. Lemma S. (Donsker)Under assumtions 2, 3 and 5, for j A; B, H j j (; ; T; y) y 2 is Donsker with a nite enveloe function. Proof of Lemma S.. For notational simlicity we x and omit subscrit j. The measurable collection of functions W f I f ygj y 2 g is Donsker since the bracketing number [] ( ; W; L 2 (P )) 2 2 are of the olynomial order. The bracketing integral is nite since it converges at a slower rate than 2 : ote that the collection of indicators functions W has a nite enveloe function such that ji f ygj (y) < : ow, we work the measurable functions f F ( yj )j y 2 g. Using the roof of Lemma A.2 in Donald and Hsu (204), we can claim that is Donsker. In this roof, Donald and Hsu (204) show that [] (; ; L 2 (P )) + 2, and the bracketing integral is nite since it converges at a slower rate than 2. Donald and Hsu (204) argue that we can nd a collection of 0 y 0 < y < ::: < y k such that F y ( y jj ) F y ( y j j ) 2 for all j k: In this case, the aroximating functions satisfy the inequality jf y ( y j j )j F, and the collection has a nite enveloe function F. Since (x) is bounded away from zero and one in, d (T; ) (T; ()) is a uniformly bounded measurable function, and d (T; ) W is Donske with a nite enveloe function. Similarly, de ne d 2 (T; i (T ()), d 2 (T; ) is a uniformly bounded measurable function, and d 2 (T; ) is 2

14 Donsker with a nite enveloe function. Hence, H f d (T; ) W+d 2 (T; ) j y 2 g is Donsker with a nite enveloe function.we can nd a nite enveloe function for the class H j such that j (; ; T; y) (y) < for every y and j. Proof of Theorem. In the second art of the roof, we let P be the emirical measure of the samle (; ; T ) : Using Lemma, H is P-Donsker. ow we aly Donsker s Theorem (Theorem 9.3 at age 266 of van der Vaart, 998) to bf b A bf b B F A F B de ned by G A;B, with variance-covariance matrix, which shall converge to a zero mean Gaussian rocess, [G j (s) G k (t)] j (; ; T; s) F j (s) ( k (; ; T; t) F k (t)) ; for (s; t) 2, where j and k A; B. 5 Proof of Theorem 2 The roof is divided into two arts. In the rst art, resented as an intermediate lemma, we x j A; B and (i) derive the asymtotic distribution of v bf bw ; (ii) demonstrate that for a xed y, F b bw is e cient for F ; and (iii) therefore v bf bw will be e cient because v is Hadamard. Finally, in the second art, we derive the asymtotic distribution of the di erence v bf bw A v bf bw B and demonstrate that it is e cient. In what follows, let denote convergence in law. Lemma S. 2 (Semiarametric ciency) For j A; B, if assumtions -6 hold then, bf b j G j ; F j F j j (T i; ( i)) i; F (T; ( i)) ( i) ; F j i (T i ( i)) + o () where is the functional s Hadamard derivative, ( i; ) ( i ; ) and i observation i. Moreover, bf b j is asymtotically e cient. is the Dirac measure at Proof of Lemma S.2. Again, we x j and therefore dro the subscrit. Using results in Theorem, bf b F G + o () ) G where G is a Gaussian rocess with variance-covariance matrix given by [G G ] (s; t) [( (; ; T; s) F (s)) ( (; ; T; t) F (t))] for (s; t) 2. Because the ma : F R is Hadamard di erentiable at F der Vaart s (998) Theorem 20.8: bf b (F ) (G ; F ) + o () : 2 F we can aly van 3

15 And since is Hadamard di erentiable, its functional derivative (; F ) is linear, imlying that (G ; F ) (T i; ( i)) ( i; F (T; (i)) + ( ; F ( i) i (T i ( i)) : We now establish e ciency for bf b as an estimator for (F ). Consider a (regular) arametric submodel of the joint distribution of (; ; T ) with cdf F (y; x; t; ). The log-likelihood is ln f (y; x; tj) t [ln f (yjx; ) + ln (xj)] + ( t) [ln f 0 (yjx; ) + ln ( (xj))] + ln f (xj) t ln f (yjx; ) + ( t) ln f 0 (yjx; ) + t ln (xj) + ( t) ln ( (xj)) + ln f (xj) where for j 0; we use the ignorability assumtion to write f (yjx; T j; ) as f j (yjx; ), which is the conditional density of (j) given x for arameter value. Following results in Hahn (998), we have that the corresonding score function: S (y; x; tj) ts (yjx; ) + ( t) s 0 (yjx; ) + d (xj) d (t (xj)) (xj) ( (xj)) + s (xj) where for j 0;, s j (yjx; ) d ln f j (yjx; ) d and s (xj) d ln f (xj) d. The tangent sace for this model is: L fs (y; x; t) : S (y; x; t) ts (yjx) + ( t) s 0 (yjx) + a (x) (t (x)) + s (x)g where a (x) is a square-integrable function of x, s j (yjx) f j (yjx; 0) 0, 8x, j 0; ; s (xj) f (xj 0) 0; and for notational simlicity, (j 0) (). A arameter () is athwise di erentiable if there is a di erentiable zero-mean function F (; ; T ) such that F (; ; T ) 2 L and for all regular arametric [F (; ; T ) S (; ; T j 0)] 0 We secialize () to F (y; ), the weighted distribution of at a given y 2, which can be written as F (y; ) Ifz ygf (zjx; ) dz (; (xj)) (xj) f (xj) dx + Ifz ygf 0 (zjx; ) dz (0; (xj)) ( (xj)) f (xj) dx 4

16 and calculate its derivative with resect to, which can be written (y; Ifz ygs (zjx; ) f (zjx; ) dz (; (xj)) (xj) f (xj) dx + Ifz ygs 0 (zjx; ) f 0 (zjx; ) dz (0; (xj)) ( (xj)) f (xj) dx + Ifz ygf (zjx; ) dz (; (xj)) Ifz ygf 0 (zjx; ) dz (0; (xj)) d (xj) f (xj) dx d d (xj) f (xj) dx d + + Ifz ygf (zjx; ) dz (; (xj)) (xj) s (xj) f (xj) dx Ifz ygf 0 (zjx; ) dz (0; (xj)) ( (xj)) s (xj) f (xj) dx + (; (xj)) Ifz ygf (zjx; ) (0; (xj)) Ifz ygf 0 (zjx; ) (xj) ow, evaluating that derivative at 0 we have and after some tedious algebra one can show that d (xj) (xj) f (xj) dx d d (xj) d ( (xj)) f (xj) (y; [ (T; ()) If ygs (; ; T j 0)] 0 (T; ()) + If yg d () d 0 (T; ()) If yg d () d (T; ()) If () (T ()) S (; ; T j 0) and (y; [F (; ; T ) S (; ; T j 0)] 0 5

17 where, for () F (y; ), we have that F (; ; T ) (T; ()) If yg F (y; (T; ()) + If () (T ()) (; ; T; y) F (y; 0) : We now show that F (; ; T ) 2 L. We rewrite (; ; T; y) as (; ; T; y) T (; ()) If yg F j;t (yj; ; 0) + ( T ) (0; ()) If yg F j;t (yj; 0; (T; ()) +( If () + (; ()) F j;t (yj; ; 0) (0; ()) F j;t (yj; 0; 0) ) (T ()) + () (; ()) F j;t (yj; ; 0) + ( ()) (0; ()) F j;t (yj; 0; 0) and we can easily check that for j 0; : and (j; ()) If yg F j;t (yj; j; 0) j; T j 0 F (y; 0) F j;t (yj; ; 0) (; ()) () + F j;t (yj; 0; 0) (0; ()) ( ()) : According to Bickel, laassen, Ritov and Wellner (993), because the estimator for F, F b b, is asymtotically linear with in uence function F (; ; T ) 2 L, F b b is also asymtotically e cient at P, the joint distribution of (; ; T ). Therefore, since is Hadamard di erentiable, by theorem of van der Vaart (998) bf b is asymtotically e cient at P for estimating (F ). Proof of Theorem 2. This result follows mechanically from revious lemma. By de nition, b bf b A G A ; F A bf b B (G B ; F B ) ( (F A ) (F B )) ow, for the e ciency art, let us de ne the functional vector : F v R 2. Thus, bf b A > bf b B ( (F A ) (F B )) # bf b A F A bf b B F B : Result from revious lemma allows us to write F (y; ) F ( (y; ) A (; ; T; y) F A (y; 0)) 0 ( B (; ; T; y) F B (y; 0)) S (; ; T j 0) 6

18 and therefore we have that v is e cient for > v bf b A bf b B F A F B #. F A is e cient for v F B. Moreover, > v bf b A bf b B 6 The In uence Functions of Inequality Measures For comletion, we secify the format of the function ( ; F ) for each one of four inequality measures considered in this aer: the Gini coe cient, the Theil Index, the Coe cient of Variation and the Interquartile Range. 6. Gini Coe cient The in uence function of the Gini coe cient was derived by Hoe ding (948) as an examle of the results on the asymtotic distribution of U-Statistics. We follow a more recent literature on the statistical roerties of inequality measures (Cowell, 2000 and Schluter and Trede, 2003), and we write the in uence function of the Gini coe cient as: Gini (y; F ) A G (F ) + B G (F ) + C G (y; F ) where A G (F ) 2R G (F ) (F ) # B G (F ) A G (F ) (F ) C G (y; F ) 2 (F ) [y [ F (y)] + GL (F (y) ; F )] with (F ) y df (y) F (y) [ (T; ()) If yg] R (F ) 0 GL (; F ) d and (F GL (; F ) () ) y df (y) F (F ) () : 6.2 Theil Index Alying the Delta Method, we obtain de in uence function of the Theil index, T heil (y; F ) (y log (y) v ) v + ( ) 2 (y ) where (F ) v y log (y) df (y) 7

19 6.3 Coe cient of Variation In addition, by an alication of the Delta Method, we also obtain the in uence function of the Coe cient of Variation, where CV (y; F ) (y ) Interquartile Range y 2 df (y) ( ) 2 2 ( ) 2 (y ) Following Ferguson (996) and Van de Vaart (998), we de ne the in uence function of the -th quantile of the weighted distribution of as q (y; F ) fy q g f (q ) where f (q ) df (q ) is the density evaluated at the quantile q, where F (q ) 2 [0; ]. The in uence function of the interquartile range is the di erence of the two quantile in uence functions, IQR (y; F ) 0:75 fy q 0:75g 0:25 fy q 0:25g f (q 0:75 ) f (q 0:25 ) 7 Proof of Theorem 3 We divide the roof of this theorem into two arts. We rst show in an intermediate lemma that for gj j A; B; su x2 (x) H j (x) > b jl o (). Then, we sow that because all of the nonarametric comonents of the variance estimator converge uniformly in robability to their oulation counterarts, the variance estimator is consistent for the asymtotic variance. Lemma S. 3 (Uniform Consistency of Regression Comonent) For j A; B and under assumtions -7 Proof of Lemma S.3. e arg min Using triangle inequality, su g (x) x2 su g j (x) x2 H j (x) > b j o () Again, we x j and omit the subscrit. Let us de ne e, (T i; () ()(i ) i; b F (T; ()) arg min ; () b (T; ()) arg min ( ; F () H (x) > b su g (x) x2 H (x) > H ( i) > # 2 H () > # 2 H () > : 2 + () (k k + k e k + ke b k) 8

20 where () su x kh (x)k. First, under the assumtion that the function g () is s times continuously di erentiable we have that for a xed : su g (x) H (x) > C r s : x2 We then work with di erences in the coe cients: k k H () C (T; () h ; F b b ( ; F ) ; b F b ( ; F )i : ote that because () is bounded away from zero and is less than one, for l 0; su (l; (x) C l su l C l C and as a result of Theorem 2 ; F b b ( ; ( ; z) C 2 zf # su F b b (y) y2 F (y) + bf b F 2 thus ow, the di erence k k C () 2 : k e k C 2 V H ( i) H () C V (T; ()) H (Ti; (i)) ( b F b H () (T; ()) ; () b 2 b ( ; F ) + ; F b (T; ()) ; () b b 2 ( ; F ) 9

21 and working with the (T; ()) V H () V H () +V +2 Cov H (T; ()) ( ; F (Ti; () ( ; F ) + ; F b b ; F b b ( ; F (T; ()) H () ( ; F ) ; H () C 2 () h( ( ; F )) 2i +C 2 () ; F b 2 b ( ; F ) h +C 2 () Cov ( ; F ) ; ; F b b i ( ; F ) ( ; F (T; () ; F b b ( ; F ) C 2 () : Therefore we have that k e k C 2 2 () 2 C () : Finally, ke b (Ti; b ( (T i; ( i)) H ( i) () b F (Ti; C () b ( (T i; ( i)) () b F b C () (t; 2 (x) (b ( i) ( i)) i; F b b t2f0;g;x2 and since the second derivative of with resect to (x) is bounded,we have that ke b k C () (b ( i) C () i; F b b ( i)) i; b F b su x2 0 H (x) > e H (x) > k(b )k + su j (x) x2 (x)j : ow let us work rst with su 0 H (x) > e H (x) > k(b x2 ( ) 2 + s2r 2 )k + su j (x) x2 (x)j 20

22 and then with i; F b b ( i; F ) + i; F b b O () + O 2 : ( i; F ) We reach that Therefore, we have that ke b k C () () su g (x) x2 O 2 H (x) > b 2 + s2r s r () + O 2 () 2 +O 2 ( ()) + +O 2 ( ()) ( ()) 2 2 +O 2 ( ()) ( ()) s2r o () : : Proof of Theorem 3. We have that su x jb (x) (x)j o (), for j A; B, su x jbg j (x) g j (x)j b o () and su b y2 F j (y) F j (y) o (). We can rewrite V b AB as bv AB h T i; i; b ( i) ; bg A ( i) ; bg B ( i) ; b F b A ; F b b B where h is a continuously di erentiable function with resect to W [ () ; g A () ; g B () ; F A ; F B ]>. For convenience, de ne W c h b () ; bg A () ; bg B () ; F b b A ; F b i > b B : Thus, a simle linearization of VAB b yields b V AB (Ti; i; Wi) su jb (x) (x)j su jbg A (x) g A (x)j su jbg B (x) g B (x)j x2 x2 x2 su F b b A (y) F A (y) su F b b B (y) F B (y) + o () o () : y2 y2 8 Proof of Theorem 4 We divide this roof into two arts. Consider the bootstra scheme in the main text. First, we show that given the samle, bb b converges conditionally in distribution to the same limit as b. Based on that, we show that we can use the ercentile bootstra to construct con dence intervals for : 2

23 Proof of Theorem 4. First art. Fix j A; B and omit that subscrit. In the roof of Theorem, we show that the class of measurable functions H f (; ; T; y)j y 2 g is Donsker with a nite enveloe function. De ne ` (H) as the sace of bounded functions on H with suremum norm. From theorems and 2, G is a sequence of maings with values onto the normed sace, ` (H), converging in distribution to the Gaussian Process G. Following van der Vaart (998), section 23.2., G is a sequence of maings with values onto the normed sace, ` (H), converging (conditionally on ) in distribution to G. Putting it more formally, we use van der Vaart s theorem 23.7 to write su j [h (G )] [h (G )]j 0 h2b L (`(H)) where denotes the exectation conditionally on f( i; i; T i) : i ; :::; g. Because () : F R is Hadamard di erentiable tangentially to the subset F ; and F ` (H), by the Delta-method for bootstra in robability (Theorem 23.9 in van der Vaart, 998), conditionally on, the sequence bf b b bf b should converge in distribution to the same limit as bf b (F ). Consequently and given, bb b converges conditionally in distribution to the same limit as b. Second art. Fix with 0 < <. De ne the following quantities F b (r) and F b (r) for some r 2 R as: and their inverses evaluated at as 2 h n F b (r) b F b (r) b b n bb d F b () d F () : b oi r b o r ote that for an aroriate choice of h (), we could write d [h (G )] and d [h (G )]. Therefore, from the rst art of the theorem, d d 0 uniformly. ow, we show that the bootstra con dence interval for, CI (; ( ) %), can be rewritten using d 2 and d 2: CI (; ( ) %) 2 b b [( 2)b] ; 2 b b [(2)b] b d 2 ; b d 2 : The set BL (` (H)) consists of all functions h : ` (H) [ ; ] that are uniformly Lischitz. See van der Vaart (998), age If the cdfs admit at regions, we then de ne their inverses (the quantiles) as being: d inf b d2r (d) d inf b d2r (d) : 22

24 That haens because b b b F b b n bb b o [b] n bb b[b] b b[b] b F b (d ) : b o Then, CI (; ( ) %) b d 2 ; b d 2 and therefore Pr [ 2 CI (; ( ) %)] Pr b d 2 b d 2 h Pr d 2 i b d 2 F b d 2 F b d F b d 2 F b d 2 ; 2 + F b d 2 F b d 2 which holds since F b () is continuous, imlying that F b (d ) F b (d ) 0 uniformly. RFRCS Bickel, P, laassen, C, Ritov,, Wellner J cient and Adative Inference in Semiarametric Models. Johns Hokins University Press, Baltimore. Cowell, F Measurement of Inequality, in Handbook of Income Distribution. orth Holland, Amsterdam. Donald, S, HSU, stimation and Inference for Distribution Functions and Quantile Functions in Treatment ect Models. Journal of conometrics 78: Ferguson, T A Course in Large Samle Theory. Chaman & Hall Texts in Statistical Science Series, Taylor & Francis. Firo, S cient Semiarametric stimation of Quantile Treatment ects. conometrica 75: Hahn, J On the Role of the Proensity Score in cient Semiarametric stimation of Average Treatment ects. conometrica 66: Hirano,, Imbens, G, Ridder G cient stimation of Average Treatment ects Using the stimated Proensity Score. conometrica 7: Hoe ding, W A Class of Statistics with Asymtotically ormal Distribution. The Annals of Mathematical Statistics 9:

25 Schluter, C, Trede, M Local Versus Global Assessment of Mobility. International conomic Review 44: Tarozzi, A Calculating Comarable Statistics from Incomarable Surveys, with an Alication to Poverty in India. Journal of Business and conomic Statistics 25(3): van der Vaart, A Asymtotic Statistics. Cambridge, Cambridge University Press. 24