Econometrica Supplementary Material

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1 Econometrca Supplementary Materal SUPPLEMENT TO IDENTIFICATION AND ESTIMATION OF TRIANGULAR SIMULTANEOUS EQUATIONS MODELS WITHOUT ADDITIVITY Econometrca, Vol. 77, No. 5, September 009, 48 5) BY GUIDO W. IMBENS AND WHITNEY K. NEWEY PROOFS OF LEMMAS 0 AND AND THEOREMS AND 3 Throughout ths supplementary materal, C wll denote a generc postve constant that may be dfferent n dfferent uses. Also, we wll abbrevate the phrases wth probablty approachng as w.p.a., postve semdefnte as p.s.d., and postve defnte as p.d.; λ mn A) λ max A), anda / wll denote the mnmum egenvalues, the maxmum egenvalues, and the square root, respectvely, of a symmetrc matrx A. Let denote n. Also, let CS, M, and T refer to the Cauchy Schwarz, Markov, and trangle nequaltes, respectvely. Also, let CM refer to the followng well known result: If E[ Y n Z n ]=O p r n ), then Y n = O p r n ). PROOF OF LEMMA 0: The jont PDF of x η) s f Z x η)f η η), where f Z ) s the PDF of Z and f η ) s the PDF of η. By a change of varable v = F η η), the PDF of x v) s f Z x F v)) η where F η ) s the CDF of η 0. Consder α =ᾱ + δ> R )/R = σ /σ. η Z Then for η = F η v) and 0 <v<, f Z x F v)) η = C exp v α v) α x η σ Z ) ) ) α η Φ Φ η ) α σ η σ η It s well known that φu)/φu) s monotoncally decreasng, so there s C>0 such that Φu) Cφu), u 0, and smlarly Φu) Cφu) u 0. Then by Φu) forallu, Φu) Φ u) Cφu) Therefore, for η = σ η Φ v), f Z x F v)) η C exp v α v) α x η σ Z ) } exp ) αη ση x = C exp + xη + η ασ Z σz σz σz ση )} 009 The Econometrc Socety DOI: 0.398/ECTA708

2 G. W. IMBENS AND W. K. NEWEY The expresson followng the equalty s bounded away from zero for x B and all η R by α>σ /σ. η Z The upper bound follows by a smlar argument, usng the fact that there s a C wth φu)/φu) u +C for all u. Before provng Lemma, we prove some prelmnary results. Let q = q L Z ) and ω j = X j X ) F X ZX Z j ). LEMMA S.: For Z = Z Z n ) and L vectors of functons b Z) = n), f n b Z) ˆQb Z)/n = O p r n ), then b Z) q j ω j / } / n n = O p r n ) PROOF: Note that ω j. Consder j k and suppose wthout loss of generalty that j otherwse reverse the role of j and k becausewecannothave = j and = k). By ndependence of the observatons, E [ ω j ω k Z ] = E [ E[ω j ω k ZX X k ] Z ] = E [ ω k E[ω j ZX X k ] Z ] = E [ ω k E[ω j Z j Z X ] Z ] = E [ ω k E[Xj X ) Z j Z X ] F X ZX Z j ) } Z ] = 0 Therefore, t follows that [ E b Z) q j ω j / } ] / n n Z = b Z) q j E[ω j ω k Z]q k }b /n Z)/n jk= b Z) q j E[ω j Z]q j }b /n Z)/n b Z) ˆQb Z)/n so the concluson follows by CM. LEMMA S. Lorentz 986, p. 90, Theorem 8): If Assumpton 3 s satsfed, then there exsts C such that for each x there s γx) wth sup z Z F X Zx z) p K z) γx) CK d /r.

3 TRIANGULAR SIMULTANEOUS EQUATIONS MODELS 3 LEMMA S.3: If Assumpton 4 s satsfed, then for each K there exsts a nonsngular constant matrx B such that p K w) = Bp K w) satsfes E[ p K w ) p K w ) ] = I K, sup w W p K w) CK α K V, sup w W p K w)/ V CK α+ V K, and sup t [0] p K V t) CK +α V. PROOF: Foru [0], letp α j u) be the jth orthonormal polynomal wth respect to the weght u α u) α.denotex = r [x x l= l l]. By the fact that the order of the power seres s ncreasng and that all terms of a gven order are ncluded before a term of hgher order, for each k and λk l) wth p k w) =, there exsts b kj j k) such that s l= wλkl) l k r b kj p j w) = P 0 [x λkl) l x l ]/[ x l x l ])P α t) λks) l= Let B k denote a K vector B k = b k b kk 0 ), b kk 0, where 0 s a K k)-dmensonal zero vector, and let B be the K K matrx wth kth row B.Then k B s a lower trangular matrx wth nonzero dagonal elements and so s nonsngular. As shown n Andrews 99), there s C such that P αu) j Cjα+/ + ) Cj α+/ and dp α j u)/du Cjα+5/ for all u [0] and j }. Then for p K w) = Bp K w), t follows that p k w) Cλk s) α+/ s λk l= l)/, so that p K w) CK α K V, and sup w W p K w)/ t CK α+ V K. Then by Assumpton 4, t follows that K = E[ p K w ) p K w ) ] CI K.Let B = / K and defne p K w) = B p K w). Then p K w) = p K w) p K w) p K w) p K w) C p K w) and an analogous nequalty holds for p K w)/ t, gvng the concluson. Henceforth defne ζ = CK αk V and ζ = CK α+ V K. Also, snce the estmator s nvarant to nonsngular lnear transformatons of p K w), we can assume that the concluson of Lemma S.3 s satsfed wth p K w) replacng p K w). PROOF OF LEMMA : Let δ j = F X ZX Z j ) q j γk X ),wth δ j K d /r by Lemma S.. Then for V = ã K Z X X ) ), where V V = Δ I + ΔII Δ I = q ˆQ + Δ III q j ω j /n Δ II = q ˆQ q j δ j /n Δ III = δ Note that Δ III CK d /r by Lemma S.. Also, by ˆQ p.s.d. and symmetrc, there exsts a dagonal matrx of egenvalues Λ and an orthonormal matrx B

4 4 G. W. IMBENS AND W. K. NEWEY such that ˆQ = BΛB.LetΛ denote the dagonal matrx of nverse of nonzero egenvalues and zeros, and let ˆQ = BΛ B.Then ˆQ q q = tr ˆQ ˆQ) CL. By CS and Assumpton 3, Δ II ) /n q ˆQ q ) / δ /n j n C = CK d /r tr ˆQ ˆQ) CK d /r Note that for b Z) = q ˆQ / n we have b Z) ˆQb Z)/n = tr ˆQ ˆQ ˆQ ˆQ )/n = tr ˆQ ˆQ )/n CK /n = O p K /n) q ˆQ q )L d /n so t follows by Lemma S. that n ΔI ) /n = O p L/n). The concluson then follows by T and by τ V) τv ) V V, whchgves ˆ V V ) /n V V ) /n. Before provng other results, we gve some useful lemmas. For these results let p = p K w ), ˆp = p K ŵ ), p =[p p n ], ˆp =[ˆp ˆp n ], ˆP = ˆp ˆp/n, and P = p p/n, P = E[p p ]. Also, as n Newey 997), t can be shown that wthout loss of generalty we can set P = I K. LEMMA S.4: If the hypotheses of Theorem are satsfed, then E[Y XZ]= mx V ). PROOF: By the proof of Theorem, V = F X ZX Z) s a functon of X and Z that s nvertble n X wth nverse X = hz V ), where hz v) s the nverse of F X Zx z) n ts frst argument. Therefore, V Z) s a one-to-one functon of X Z). By ndependence of Z and ε η), ε s ndependent of Z condtonal on V, so that by equaton 4), E[Y XZ]=E[Y VZ]=E [ g hz V ) ε) VZ ] = g hz V ) e)f ε ZV de ZV ) = g hz V ) e)f ε V de V)= mx V ) Let u = Y mx V ) and let u = u u n ).

5 TRIANGULAR SIMULTANEOUS EQUATIONS MODELS 5 LEMMA S.5: If ˆ V V /n = O p Δ n ) and Assumptons 3 6 are satsfed, the followng equaltes hold: ) P P =O p ζ K /n) ) p u/n =O p K /n) ) ˆp p /n = O p ζ Δ) n v) ˆP P =O p ζ Δ + n K ζ Δ n ) v) ˆp p) u/n =O p ζ Δ n / n) PROOF: The frst two results follow as n equaton A.) and page 6 of Newey 997). For ), a mean value expanson gves ˆp = p +[ p K w )/ V ] Vˆ V ) where w = x V ) and V les n between Vˆ and V.SnceVˆ and V le n [0 ], t follows that V [0], so that by Lemma S.3, p K w )/ V Cζ.ThenbyCS, ˆp p Cζ Vˆ V. Summng up gves S.) ˆp p /n = ˆp p /n = O p ζ Δ) n For v), by Lemma S.3, n p /n = O p E[ p ]) = tri K ) = K.Thenby T, CS, and M, ˆP P ˆp ˆp p p /n ˆp p /n ) / + ˆp p /n p /n ) / = O p ζ Δ n + K ζ Δ n ) Fnally, for v), for Z = Z Z n ) and X = X X n ), t follows from Lemma S.4, Assumpton 6, and ndependence of the observatons that E[uu X Z ] CI n, so that by p and ˆp dependng only on Z and X, E [ ˆp p) u/n X Z ] = tr ˆp p) E[uu X Z ] ˆp p)/n } C ˆp p /n = O p ζ Δ n /n) LEMMA S.6: If Assumptons 3 6 are satsfed and K ζ Δ n 0, then w.p.a., λ mn ˆP) C, λ mn P) C. PROOF: By Lemma S.3 and ζ K /n CK ζ K /n, wehave ˆP P p 0and P P p 0, so the concluson follows as on page 6 of Newey 997).

6 6 G. W. IMBENS AND W. K. NEWEY Let m = mw ) mw n )),and ˆm = mŵ )mŵ n )). LEMMA S.7: If ˆ V V /n = O p Δ n ), Assumptons 3 6 are satsfed, K ζ Δ n 0, and K ζ /n 0, then for α = ˆP ˆp ˆm/n and ᾱ = ˆP ˆp m/n, the followng equaltes hold: ) ˆα ᾱ =O p K /n) ) α ᾱ =O p Δ n ) ) α α K =Op K d /r ) PROOF: For ), E [ ˆP / ˆα ᾱ) X Z ] = E[u ˆp ˆP ˆp u/n X Z ] = tr ˆP / ˆp E[uu X Z ] ˆp ˆP /} /n C tr ˆp ˆP ˆp }/n C tri K )/n = CK /n Snce by Lemma S.6, λ mn ˆP) C w.p.a., ths mples that E[ ˆα ᾱ X Z ] CK /n. Smlarly, for ), ˆP / α ᾱ) C ˆm m) ˆp ˆP ˆp ˆm m)/n C ˆm m /n = O p Δ n ) whch follows from mw) beng Lpschtz n V, so that also α ᾱ = O p Δ ). n Fnally for ), ˆP / α α K ) = α ˆP ˆp ˆpα K /n C ˆm ˆp α K ) ˆp ˆP ˆp ˆm ˆp α K )/n ˆm ˆpα K /n C sup m 0 w) p K w) α K w W d = O p K /r ) so that ˆP / α α K ) = O p K d /r ). PROOF OF THEOREM : Note that by Lemma, for Δ = K n /n+k d /r we have ˆ V V /n = O p Δ ),sobyk n ζ /n CK ζ K /n, the hypotheses of Lemma S.7 are satsfed. Also by Lemma S.7 and T, ˆα α K =,

7 TRIANGULAR SIMULTANEOUS EQUATIONS MODELS 7 O p K /n + K d /r + Δ n ).Then [ ˆmw) mw)] F w dw) = [p K w) ˆα α K ) + p K w) α K mw) ] F w dw) C ˆα α K + CK d /r = O p K /n + K d /r + Δ n) For the second part of Theorem, sup ˆmw) mw) w W = sup p K w) ˆα α K ) + p K w) α K βw) w W = O p ζ K /n + K d /r + Δ n ) / ) d + Op K /r ) = O p ζ K /n + K d /r + Δ n) / ) PROOF OF THEOREM 3: Let p = 0 pk V t) dt and note that by Lemma S.3, p p CK +α V.Also, S.) px) def = 0 p K w) dt = p Kx x) p As above, E[uu X Z ] CI n, so that by Fubn s theorem, [ E px) ˆα ᾱ)} F X dx) X ] Z = px) ˆP ˆp E[uu X Z ] ˆp ˆP px) } F X dx)/n C px) ˆP px)f X dx)/n CE[ px) px)]/n = C E[p Kx X) p Kx X)] p p) } /n = K x K +α V /n It then follows by CM that px) ˆα ᾱ)} F X dx) = O p K x K +α V /n).also, px) px) F X dx) = I Kx p p CI K p p CI K K +a V

8 8 G. W. IMBENS AND W. K. NEWEY so that by Lemma S.7 and T, px) ᾱ α K )} F X dx) ᾱ α K ) px) px) F X dx)ᾱ α K ) CK +a V ᾱ α K = O p K +a d V K /s + Δn)) Also, by CS, px) α K μx)} F X dx) p K w) α βw)} dv F X dx) = O ) K d /s 0 Then the concluson follows by T and [ˆμx) μx)] F 0 dx) = px) ˆα α K ) + px) α K μx)} F X dx) = O p K +α V Kx /n + K d /r + Δn)) REFERENCES ANDREWS, D. W. K. 99): Asymptotc Normalty of Seres Estmators for Nonparametrc and Semparametrc Regresson Models, Econometrca, 59, LORENTZ, G. 986): Approxmaton of Functons. New York: Chelsea Publshng Company. NEWEY, W. K. 997): Convergence Rates and Asymptotc Normalty for Seres Estmators, Journal of Econometrcs, 79, Dept. of Economcs, Lttauer Center, Harvard Unversty, 805 Cambrdge Street, Cambrdge, MA 038, U.S.A.; mbens@fas.harvard.edu and Dept. of Economcs, Massachusetts Insttute of Technology, Cambrdge, MA , U.S.A.; wnewey@mt.edu. Manuscrpt receved Aprl, 007; fnal revson receved January, 009.