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1 Ecoometrica Supplemetary Material SUPPLEMENT TO NONPARAMETRIC INSTRUMENTAL VARIABLES ESTIMATION OF A QUANTILE REGRESSION MODEL : MATHEMATICAL APPENDIX (Ecoometrica, Vol. 75, No. 4, July 27, ) BY JOEL L. HOROWITZ AND SOKBAE LEE THIS APPENDIX provides proofs of Theorems 1 3. Theorems 4 ad 5 ca be proved by followig the same steps after coditioig o Z. A.1. PROOF OF THEOREM 1 The proof is a modificatio of the proof of Theorem 2 i Bissatz, Hohage, ad Muk (24). By (2.5), (A1) ˆT (ĝ) q ˆ f W 2 + a ĝ 2 ˆT (g) q ˆ f W 2 + a g 2 I additio, E ˆT (g) T (g) 2 = O(δ ) ad E ˆ f W f W 2 = O(δ ). Therefore, by Assumptio 3, E ˆT (g) q ˆ f W 2 2E ˆT (g) T (g) 2 + 2qE ˆ f W f W 2 = O(δ ) Combiig this result with (A1) gives E ˆT (ĝ) q ˆ f W 2 + a E ĝ 2 Cδ + a g 2 for some costat C< ad all sufficietly large. Therefore, by Assumptio 3, lim sup E ĝ 2 g 2 Note, i additio, that E ˆT (ĝ) qf ˆ W 2 as. Moreover, Assumptios 1(b) ad 2 imply that T is weakly closed. Cosistecy ow follows from argumets idetical to those used to prove Theorem 2 of Bissatz, Hohage, ad Muk (24, p. 1777). Q.E.D. A.2. PROOF OF THEOREM 2 Assumptios 1, 2, ad 4 7 hold throughout this sectio. Let deote the ier product i L 2 [ 1].Defieω g = (T g T g ) 1 T g g ad (A2) g = g a (T g T g + a I) 1 T g ω g 1

2 2 J. L. HOROWITZ AND S. LEE where I is the idetity operator. Observe that by (3.4), (A3) L ω g < 1 Let ˆr ad r be Taylor series remaider terms with the properties that (A4) T (ĝ) = qf W + T g (ĝ g) + ˆr ad (A5) T ( g) = qf W + T g ( g g) + r where qf W = T (g). By (3.3), ˆr (L/2) ĝ g 2 ad r (L/2) g g 2. LEMMA A.1: For ay g G, (1 L ω g ) ĝ g 2 PROOF: By (A5), g g 2 + a 1 ˆT ( g) T ( g) + r + qf W qf ˆ W 2 + 2T g ( g g) + a ω g ω g +a 1 T g( g g) qf W qf ˆ W ω g +2a 1 r + qf W qf ˆ W T g ( g g) + 2 ˆT (ĝ) T (ĝ) ω g +2a 1 ˆT ( g) T ( g) T g ( g g) (A6) ˆT ( g) q ˆ f W 2 = ˆT ( g) T ( g) + r + qf W q ˆ f W 2 + T g ( g g) r + qf W q ˆ f W T g ( g g) +2 ˆT ( g) T ( g) T g ( g g) Also, (A7) ˆT (ĝ) q ˆ f W 2 = ˆT (ĝ) q ˆ f W + a ω g 2 + a 2 ω g 2 2a ˆT (ĝ) T (ĝ) ω g 2a T (ĝ) qf W ω g 2a qf W qf ˆ W ω g 2a 2 ω g 2 Moreover, (A8) g g g = g g T g ω g = T g ( g g) ω g By (A4), (A9) ĝ g g = ĝ g T g ω g

3 NONPARAMETRIC ESTIMATION OF REGRESSION MODEL 3 = T g (ĝ g) ω g = T (ĝ) qf W ω g ˆr ω g By (2.5), (A1) ˆT (ĝ) q ˆ f W 2 + a ĝ 2 ˆT ( g) q ˆ f W 2 + a g 2 Rearragig ad expadig terms i (A1) gives (A11) ĝ g 2 [ a 1 ˆT ( g) qf ˆ W 2 ˆT (ĝ) qf ˆ W 2] + g g g g g 2 ĝ g g Combiig (A11) with (A6) (A9) gives (A12) ĝ g 2 2 ˆr ω g a 1 ˆT (ĝ) qf ˆ W + a ω g 2 + g g 2 + a 1 ˆT ( g) T ( g) + r + qf W qf ˆ W 2 + 2T g ( g g) + a ω g ω g +a 1 T g( g g) qf W qf ˆ W ω g +2a 1 r + qf W qf ˆ W T g ( g g) + 2 ˆT (ĝ) T (ĝ) ω g +2a 1 ˆT ( g) T ( g) T g ( g g) The lemma follows by otig that the secod term o the right-had side of (A12) is opositive ad that 2 ˆr ω g L ω g ĝ g 2. Q.E.D. LEMMA A.2: For ay g G, (A13) (A14) (A15) a 1 ˆT ( g) T ( g) + r + qf W qf ˆ W 2 4a 1 ( ˆT ( g) T ( g) 2 + L2 4 g g 4 + qf W q ˆ f W 2 ) 2T g ( g g) + a ω g ω g +a 1 T g( g g) 2 = a 3 (T gt + a g I) 1 ω g 2 2 qf W qf ˆ W ω g +2a 1 r + qf W qf ˆ W T g ( g g) L ω g g g 2 + 2a qf W q ˆ f W (T g T g + a I) 1 ω g + La g g 2 (T g T g + a I) 1 ω g

4 4 J. L. HOROWITZ AND S. LEE ad (A16) 2 ˆT (ĝ) T (ĝ) ω g +2a 1 ˆT ( g) T ( g) T g ( g g) 2a ˆT ( g) T ( g) (T g T + a g I) 1 ω g + 2 ω g [ ˆT (ĝ) T (ĝ)] [ˆT (g) T (g)] + 2 ω g [ ˆT ( g) T ( g)] [ˆT (g) T (g)] PROOF: Iequality (A13) follows from (A5) ad the relatio A + B + C 2 4( A 2 + C 2 + C 2 ) for ay fuctios A, B,adC. To show (A14), ote that (A17) a (T g T g + a I) 1 = I T g (T g T g + a I) 1 T g By (A2), (A18) T( g g) = a T g (T g T g + a I) 1 T g ω g It follows from (A17) ad (A18) that (A19) a ω g + T( g g) = a 2 (T gt g + a I) 1 ω g Takig the squares of the orms of both sides of (A19) ad expadig the term o the left-had side yields (A2) a 2 ω g 2 + T g ( g g) a ω g T g ( g g) = a 4 (T gt g + a I) 1 ω g 2 The (A14) follows by dividig both sides of (A2) by a. We ow tur to (A15). First ote that (A21) r + qf W q ˆ f W T g ( g g) = r + qf W q ˆ f W T g ( g g) + a ω g r + qf W q ˆ f W a ω g It follows from (A19) ad (A21) that (A22) 2 qf W qf ˆ W ω g +2a 1 r + qf W qf ˆ W T g ( g g) + a ω g = 2a r + qf W q ˆ f W (T g T g + a I) 1 ω g 2 r ω g

5 NONPARAMETRIC ESTIMATION OF REGRESSION MODEL 5 The (A15) follows by applyig the Cauchy Schwarz ad triagle iequalities to (A22). Now we prove (A16). Observe that by (A19) ad algebra like that yieldig (A21), 2 ˆT (ĝ) T (ĝ) ω g +2a 1 ˆT ( g) T ( g) T g ( g g) = 2a ˆT ( g) T ( g) (T g T + a g I) 1 ω g + 2 [ ˆT (ĝ) T (ĝ)] [ˆT ( g) T ( g)] ω g which yields (A16) by the Cauchy Schwarz ad triagle iequalities. Q.E.D. LEMMA A.3: The followig relatios hold uiformly over H H. (a) g g 2 = O[ (2β 1)/(2β+α) ]; (b) a 1 g g 4 = O[ (2β 1)/(2β+α) ]; (c) a 3 (T gt + a g I) 1 ω g 2 = O[ (2β 1)/(2β+α) ]; (d) a g g 2 (T g T + a g I) 1 ω g =O[ (2β 1)/(2β+α) ]; (e) a qf W qf ˆ W (T g T + a g I) 1 ω g =O p [ (2β 1)/(2β+α) ]; (f) a ˆT ( g) T ( g) (T g T + a g I) 1 ω g =O p [ (2β 1)/(2β+α) ]; (g) there are radom variables = O p [ (β 1/2)/(2β+α) ] ad Ɣ = o p (1) such that [ ˆT (ĝ) T (ĝ)] [ˆT (g) T (g)] ĝ g +Ɣ ĝ g 2 ; (h) [ ˆT ( g) T ( g)] [ˆT (g) T (g)] = O p [ (2β 1)/(2β+α) ]. PROOF: To prove (a), ote that by (A2) ad T g ω g = g, Therefore, g(x) g(x) = a g g 2 = a 2 = a 1 λ j + a φ j (x) φ j T g ω g b j λ j + a φ j (x) b 2 j (λ j + a ) 2 = O[ (2β 1)/(2β+α) ] where the secod lie follows from argumets idetical to those used to prove equatio (6.4) of Hall ad Horowitz (25). This proves (a). It follows from (a) that a 1 g g 4 = O[ (2β 1)/(2β+α) ]

6 6 J. L. HOROWITZ AND S. LEE wheever α < 2β 1, thereby provig (b). We ow tur to (c). Defie ψ j = T g φ j / T g φ j.useω g = (T g T g ) 1 T g g ad the sigular value decompositio T ψ g j = λ 1/2 j φ j to obtai (T g T + a g I) 1 ω g = = = = 1 λ j (λ j + a ) ψ j ψ j T g g 1 λ j (λ j + a ) ψ j T g ψ j g 1 λ j (λ j + a ) ψ j λ 1/2 j φ j g b j λ 1/2 j (λ j + a ) ψ j Therefore, (A23) (T g T g + a I)ω g 2 = b 2 j λ j (λ j + a ) 2 = O[a (2β 3α 1)/α ] by argumets like those used to prove equatio (6.4) of Hall ad Horowitz (25). Therefore, (c) follows from a = C a α/(2β+α). To prove (d), ote that by (A23), (A24) a (T g T g + a I) 1 ω g =O[a (2β α 1)/(2α) ] = O[ (2β α 1)/(4β+2α) ] Therefore, (d) follows from (A24) ad (a), because α<2β 1. Now by Assumptios 2 ad 5(b), ˆ f W f W 2 = O p [ (2β 1+α)/(2β+α) ] Moreover, because T is a desity oly with respect to its w argumet ad ca be estimated with the oe-dimesioal oparametric rate of covergece, we have ˆT T 2 = O p [ (2β 1+α)/(2β+α) ] uiformly over H. Therefore, (e) ad (f) follow from (A24).

7 NONPARAMETRIC ESTIMATION OF REGRESSION MODEL 7 We ow tur to (g). By the mea value theorem, { [ ˆT (ĝ) T (ĝ)] [ˆT (g) T (g)] } (w) = { ˆ fyxw [ḡ(x) x w] f YXW [ḡ(x) x w] } [ĝ(x) g(x)] dx where ḡ is betwee ĝ ad g. The by the Cauchy Schwarz iequality, [ ˆT (ĝ) T (ĝ)] [ˆT (g) T (g)] 2 = ( { ˆ fyxw [ḡ(x) x w] f YXW [ḡ(x) x w] } [ĝ(x) g(x)] dx) 2 dw But = ( { fyxw ˆ [ḡ(x) x w] f YXW [ḡ(x) x w] }2 dx ) [ĝ(x) g(x)] 2 dx dw { ˆ fyxw [ḡ(x) x w] f YXW [ḡ(x) x w] }2 dxdw ĝ g 2 { fyxw ˆ [ḡ(x) x w] f YXW [ḡ(x) x w] }2 dxdw = O p [ (2β 1)/(2β+α) ]+ ĝ g 2 o p (1) by Assumptios 2 ad 5(b), thereby yieldig (g). Fially, (h) ca be proved by combiig (a) with argumets similar to those used to prove (g). The lemma is ow proved because the foregoig argumets hold uiformly over H H. Q.E.D. PROOF OF THEOREM 2: The theorem follows by combiig the results of Lemmas A.1 A.3 with L ω g < 1. Q.E.D. A.3. PROOF OF THEOREM 3 It suffices to fid a sequece of fiite-dimesioal models {g } H for which lim if P H[ g g 2 >D (2β 1)/(2β+α) ] >

8 8 J. L. HOROWITZ AND S. LEE To this ed, let m deote the iteger part of 1/(2β+α) ad let f XW deote the desity of (X W ).Letr w = (2β + α 1)/2. Assume that f XW (x w) C for all (x w) [ 1] 2 ad some costat C<.Let Y = g (X) + U where U is idepedet of (X W ) ad P(U ) = q. LetF U ad f U,respectively, deote the distributio fuctio ad desity of U. Assume that f U () > ad that F U is twice cotiuously differetiable everywhere with F (u) <M for all u ad some M<. Defie the operator Q o L U 2[ 1] by (Πg)(x) = for ay g L 2 [ 1],where π(x z)g(z)dz π(x z) = f U () 2 f XW (x w)f XW (z w)dw Let {λ j φ j : j = 1 2 } deote the orthoormal eigevalues ad eigevectors of Π ordered so that λ 1 λ 2 >. Assume that j α λ j is bouded away from ad1forallj.set g (x) = θ j β φ j (x) j=m for some fiite, costat θ>. The for ay h L 2 [ 1], (T h)(w) = ad the Fréchet derivative of T at g is (T g h)(w) = f U () F U [h(x) g (x)]f XW (x w)dx f XW (x w)[h(x) g (x)] dx Assumptio 6 is satisfied with L = MC wheever θ> is sufficietly small. Now let ˆθ be a estimator of θ. The ĝ(x) ˆθ j β φ j (x) j=m is a estimator of g (x). Moreover, (A25) ĝ g 2 = ( ˆθ θ) 2 R

9 NONPARAMETRIC ESTIMATION OF REGRESSION MODEL 9 where R = j=m j 2β. Note that (2β 1)/(2β+α) R is bouded away from ad 1 as. I additio, f W is estimated by ˆ f w (w) = q 1 (T ĝ)(w) Defie ψ j = T g φ j / T g φ j. The a Taylor series approximatio ad sigular value expasio give Now ˆ f W (w) f W (w) = q 1 ( ˆθ θ) = q 1 ( ˆθ θ) j β (T g φ j )(w) + ( ˆθ θ) 2 O[ (2β 1)/(2β+α) ] j=m j=m (2β+α 1)/(2β+α) j=m j β λ 1/2 j ψ j (w) + ( ˆθ θ) 2 O[ (2β 1)/(2β+α) ] j β λ 1/2 j ψ j 2 is bouded away from ad as. Therefore, there is a fiite costat C θ > such that (A26) ( ˆθ θ) 2 C θ (2β+α 1)/(2β+α) ˆ f W f W 2 Combiig (A25) ad (A26) shows that there is a fiite costat C g > such that (A27) (2β 1)/(2β+α) ĝ g 2 C g (2β+α 1)/(2β+α) ˆ f W f W 2 The theorem ow follows from (A27) ad the observatio that with r w = (2β + α 1)/2 O p [ (2β+α 1)/(2β+α) ] is the fastest possible miimax rate of covergece of f ˆ W f W 2. Q.E.D. Rewrite (3.4) as A.4. PROOF OF (3.5) (A28) ( L< b 2 j λ j ) 1

10 1 J. L. HOROWITZ AND S. LEE By a Taylor series expasio, T (g 1 ) T (g 2 ) T g2 (g 1 g 2 ) = 5 f YXW [ḡ(x) x w] [g 1 (x) g 2 (x)] 2 dx y where ḡ is betwee g 1 ad g 2. Therefore, it follows from (3.3) that (A29) L<sup f YXW (y x w) y y x w It follows from (A28) ad (A29) that (3.5) is a sufficiet coditio for (3.4). Q.E.D.

11 NONPARAMETRIC ESTIMATION OF REGRESSION MODEL 11 A.5. AN ADDITIONAL FIGURE FOR MONTE CARLO EXPERIMENTS FIGURE A1. Mote Carlo results for = 8.

12 12 J. L. HOROWITZ AND S. LEE Dept. of Ecoomics, Northwester Uiversity, 21 Sherida Road, Evasto, IL 628, U.S.A.; ad Dept. of Ecoomics, Uiversity College Lodo, Gower Street, Lodo WC1E 6BT, U.K.;