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1 Econometrica Sulementary Material SUPPLEMENT TO PARTIAL IDENTIFICATION IN TRIANGULAR SYSTEMS OF EQUATIONS WITH BINARY DEPENDENT VARIABLES : APPENDIX Econometrica, Vol. 79, No. 3, May 2011, ) BY AZEEM M. SHAIKH AND EDWARD J. VYTLACIL PROOF OF LEMMA 2.1: First recall the simlifications following from Assumtions 2.1 and 2.2 noted at the beginning of Section 2. Next, note from equation 1) and Assumtion 2.1 that and Thus, for >, is equal to Pr{D 1 Y 1 X x P Pr{ε 2 ε 1 ν 1 1 x ) Pr{D 1 Y 1 X x P Pr{ε 2 ε 1 ν 1 1 x ) Pr{D 1 Y 1 X x P Pr{D 1 Y 1 X x P Pr{ <ε 2 ε 1 ν 1 1 x ) It follows similarly that and Therefore, is equal to Pr{D 0 Y 1 X x P Pr{ε 2 > ε 1 ν 1 0 x) Pr{D 0 Y 1 X x P Pr{ε 2 > ε 1 ν 1 0 x) Pr{D 0 Y 1 X x P Pr{D 0 Y 1 X x P Pr{ <ε 2 ε 1 ν 1 0 x) Hence, Pr{ <ε 2 ν 1 0 x)<ε 1 ν 1 1 x ) if ν 1 1 x )>ν 1 0 x) hx x 0 ) if ν 1 1 x ) ν 1 0 x) Pr{ <ε 2 ν 1 1 x )<ε 1 ν 1 0 x) if ν 1 1 x )<ν 1 0 x) 2011 The Econometric Society DOI: /ECTA9082

2 2 A. M. SHAIKH AND E. J. VYTLACIL The desired conclusion now follows immediately from Assumtion 2.2. Q.E.D. PROOF OF THEOREM 2.1: Consider art i) of the theorem. We derive bounds on G 1 0 x) Pr{Y 0 1 X x; the bounds on G 1 1 x) and on G 1 x) follow from arallel arguments. Note that Pr{Y 0 1 X x P Pr{D 0 Y 0 1 X x P Pr{D 1 Y 0 1 X x P By Lemma 2.1, equation 1), and Assumtion 2.1, Pr{D 1 Y 0 1 X x P Pr{D 1 Y 1 X x P for all x X 0 x) and Pr{D 1 Y 0 1 X x P Pr{D 1 Y 1 X x P for all x X 0 x). Thus,Pr{Y 0 1 X x P is bounded from below by Pr{D 0 Y 1 X x P su x X 0 x) Pr{D 1 Y 1 X x P and from above by Pr{D 0 Y 1 X x P inf x X 0 x) Pr{Y 1 D 1 X x P where all suremums and infimums are only taken over regions where all conditional robabilities are well defined, and with the convention that the suremum over the emty set is 0 and the infimum over the emty set is 1. The stated result now follows by noting that equation 1) and Assumtion 2.1 imly that Pr{Y 0 1 X xpr{y 0 1 X x P. Consider art ii) of the theorem. We rove the result for the term L 0 x); the result for the other terms follows from arallel arguments. Suose sup) is not a singleton, for otherwise there is nothing to rove. Since sux P) sux) sup), hx x ) is well defined for some < with ) sup) 2 and any x x ) sux) 2. Hence, by Lemma 2.1, we have that 4) X 0 x) {x : ν 1 1 x ) ν 1 0 x)

3 TRIANGULAR SYSTEMS OF EQUATIONS 3 It follows from Assumtions 2.3 and 2.4 that X 0 x) is comact. Hence, by Assumtion 2.4, there exists x l 0 x) X 0 x) such that ν 1 1 x l 0 x)) su x X 0 x) ν 1 1 x ) From equation 1), we therefore have for any sup) that su Pr{D 1 Y 1 X x P x X 0 x) Pr{D 1 Y 1 X x l 0 x) P from which it follows that { L 0 x) su Pr{D 0 Y 1 X x P Pr{D 1 Y 1 X x l 0 x) P To comlete the argument, note for any > that Pr{D 0 Y 1 X x P Pr{D 1 Y 1 X x l 0 x) P ) Pr{D 0 Y 1 X x P Pr{D 1 Y 1 X x l 0 x) P ) Pr { ε 1 ν 1 1 x l 0 x)) <ε 2 0 Pr{ε 1 ν 1 0 x) <ε 2 where the final inequality follows from the fact that x l x) X 0 0 x) and 4). Finally, consider art iii) of the theorem. Before roceeding, we introduce some notation. Let ε 1 ε ) denote a random vector with 2 ε 1 ε ) 2 X Z) and with ε 1 ε ) having density f with resect to Lebesgue measure on R 2.Letf denote the corresonding marginal density of 2 ε and let f denote the corresonding density of ε conditional on 1 ε.letf 2 1 2, f 1 2,andf 2 denote the corresonding density functions for ε 1 ε 2 ).Wewillalsomakeuse of F 1 2, the cumulative distribution function c.d.f.) for ε 1 ε 2 ),andf 1 2, the c.d.f. for ε 1 ε 2 ). To show that our bounds on G 1 0 x), G 1 1 x),andg 1 1 x) G 1 0 x) are shar, it suffices to show that for any x sux) and s 0 s 1 ) [L 0 x) U 0 x)] [L 1 x) U 1 x)], there exists a density function f 1 2 such that the following claims hold:

4 4 A. M. SHAIKH AND E. J. VYTLACIL A) f is strictly ositive on 1 2 R2. B) the roosed model is consistent with the observed data, that is, i) Pr{D 1 X x P Pr{ε, 2 ii) Pr{Y 1 D 1 X x P Pr{ε ν 1 11 x) ε, 2 iii) Pr{Y 1 D 0 X x P Pr{ε ν 1 10 x) ε > 2 for all x ) sux P). C) The roosed model is consistent with the secified values of G 1 0 x) and G 1 1 x), that is, i) Pr{ε ν 1 10 x)s 0, ii) Pr{ε ν 1 11 x)s 1. Let x sux) and s 0 s 1 ) [L 0 x) U 0 x)] [L 1 x) U 1 x)] be given. We rove the result for the case where X d x), X d x), andx d x) X d x) for d {0 1; the result in the other cases follows from analogous arguments. Note that by arguing as in Remark 2.2, this imlies in articular that L d x) < U d x) for d {0 1. For brevity, we also only consider s 0 s 1 ) L 0 x) U 0 x)) L 1 x) U 1 x)); the case where s d equals L d x) or U d x) for some d {0 1 follows from a straightforward modification of the argument below. Recall that hx x ) is well defined for some < with ) sup) 2 and any x x ) sux) 2 because sux P) sux) sup). Arguing as in the roof of art ii) of the theorem, we have that 5) L 0 x) Pr{D 0 Y 1 X x P Pr{D 1 Y 1 X x l 0 x) P U 0 x) Pr{D 0 Y 1 X x P Pr{D 1 Y 1 X x u 0 x) P L 1 x) Pr{D 1 Y 1 X x P Pr{D 0 Y 1 X x l 1 x) P U 1 x) Pr{D 1 Y 1 X x P Pr{D 0 Y 1 X x u 1 x) P where x l x) and d xu d x) for d {0 1 denote evaluation oints such that Pr{D 1 Y 1 X x l 0 x) P su x X 0 x) Pr{D 1 Y 1 X x P Pr{D 1 Y 1 X x u 0 x) P inf Pr{D 1 Y 1 X x P x X 0 x)

5 TRIANGULAR SYSTEMS OF EQUATIONS 5 Pr{D 0 Y 1 X x l 1 x) P su x X 1 x) Pr{D 0 Y 1 X x P Pr{D 0 Y 1 X x u 1 x) P inf x X 1 x) Pr{D 0 Y 1 X x P Let s 0 s 0 Pr{D 0 Y 1 X x P s 1 s 1 Pr{D 1 Y 1 X x P Using equation 5) and the fact that s d L d x) U d x)) for d {0 1,wehave that 6) s 0 F 1 2 ν1 1 x l 0 x)) ) F 1 2 ν1 1 x u 0 x)) )) s 1 F 1 2 ν1 0 x l 1 x)) ) F 1 2 ν1 0 x u 1 x)) )) These intervals are nonemty because L d x) < U d x) for d {0 1. It follows by Lemma 2.1 that 7) ν 1 d x l 1 d x)) < ν 11 d x) < ν 1 d x u 1 d x)) for d {0 1, where the strict inequalities follow from our assumtion that X d x) X d x) for d {0 1. Furthermore, by the construction of x l x) d and x u d x) for d {0 1, it must be the case for d {0 1 and x sux) that 8) ν 1 d x) / ν 1 d x l 1 d x)) ν 1d x u 1 d x))) We now construct the roosed density f 1 2 as follows. Let f 1 2 t 1 t 2 ) f 1 2 t 1 t 2 )f 2 t 2),wheref 2 t 2) f 2 t 2 ) I{0 t 2 1 and f t t 2 ) at 2 )f 1 2 t 1 t 2 ) if ν 1 1 x l 0 x)) < t 1 <ν 1 0 x)and t 2 < bt 2 )f 1 2 t 1 t 2 ) if ν 1 0 x) t 1 <ν 1 1 x u 0 x)) and t 2 < ct 2 )f 1 2 t 1 t 2 ) if ν 1 0 x l 1 x)) t 1 <ν 1 1 x)and t 2 > dt 2 )f 1 2 t 1 t 2 ) if ν 1 1 x) t 1 <ν 1 0 x u 1 x)) and t 2 > f 1 2 t 1 t 2 ) otherwise

6 6 A. M. SHAIKH AND E. J. VYTLACIL with at 2 ) Pr{ν 11 x l 0 x)) < ε 1 <ν 1 1 x u 0 x)) ε 2 t 2 Pr{ν 1 1 x l 0 x)) < ε 1 <ν 1 0 x) ε 2 t 2 s F 0 1 2ν 1 1 x l 0 x)) ) F 1 2 ν 1 1 x ux)) ) F 0 1 2ν 1 1 x l 0 x)) ) bt 2 ) Pr{ν 1 1 x l 0 x)) < ε 1 <ν 1 1 x u 0 x)) ε 2 t 2 at 2 ) Pr{ν 1 1 x l 0 x)) < ε 1 <ν 1 0 x) ε 2 t 2 ) / Pr{ν 1 0 x)<ε 1 <ν 1 1 x u 0 x)) ε 2 t 2 ct 2 ) Pr{ν 10 x l 1 x)) < ε 1 <ν 1 0 x u 1 x)) ε 2 t 2 Pr{ν 1 0 x l 1 x)) < ε 1 <ν 1 1 x) ε 2 t 2 s 1 F 1 2ν 1 0 x l 1 x)) ) F 1 2 ν 1 0 x ux)) ) F 1 1 2ν 1 0 x l 1 x)) ) dt 2 ) Pr{ν 1 0 x l x)) < ε 1 1 <ν 1 0 x u x)) ε 1 2 t 2 ct 2 ) Pr{ν 1 0 x l x)) < ε 1 1 <ν 1 1 x) ε 2 t 2 ) / Pr{ν 1 1 x)<ε 1 <ν 1 0 x u 0 x)) ε 2 t 2 These quantities are well defined because of the fact that the intervals in 6) are nonemty, because of 7), and Assumtion 2.2. We now argue that f 1 2 satisfies claim A), that is, that it is a strictly ositive density on R 2. For this urose, it suffices to show that f 1 2 integrates to 1 and is strictly ositive on R. First consider whether f integrates to 1. For t [ ], f t 1 2 2) f 1 2 t 2 ) and so the result follows immediately. For t 2 <, f 1 2 t 1 t 2 )dt 1 ν1 1 x l 0 x)) ν1 0 x) f 1 2 t 1 t 2 )dt 1 at 2 ) f 1 2 t 1 t 2 )dt 1 ν 1 1 x l 0 x)) ν1 1 x u 0 x)) bt 2 ) ν 1 0 x) f 1 2 t 1 t 2 )dt 1 f 1 2 t 1 t 2 )dt 1 ν 1 1 x u 0 x)) Pr { ε 1 ν 1 1 x l x)) ε 0 2 t 2 1 Pr { ν 1 1 x l 0 x)) < ε 1 <ν 1 1 x u 0 x)) ε 2 t 2 Pr{ε 1 ν 1 1 x u 0 x)) ε 2 t 2

7 TRIANGULAR SYSTEMS OF EQUATIONS 7 A similar argument shows that f t t 2 )dt 1 1fort 2 >. Since f 1 2 is strictly ositive on R, to establish that f 1 2 is strictly ositive on R, it suffices to show that at 2 ) bt 2 ) ct 2 ),anddt 2 ) are all strictly ositive. Consider at 2 ) and bt 2 ); the roof for ct 2 ) and dt 2 ) follows from similar arguments. From 6), we have that s >F 0 1 2ν 1 1 x l 0 x)) ), which together with 7) and Assumtion 2.2 imlies that at 2 )>0. Similarly, from 6), we have that s <F 0 1 2ν 1 1 x u 0 x)) ), which imlies that s F 0 1 2ν 1 1 x l 0 x)) ) F 1 2 ν 1 1 x ux)) ) F < ν 1 1 x l 0 x)) ) It therefore follows from 7) and Assumtion 2.2 that Pr { ν 1 1 x l 0 x)) < ε 1 <ν 1 1 x u 0 x)) ε 2 t 2 at 2 ) Pr { ν 1 1 x l 0 x)) < ε 1 <ν 1 0 x) ε 2 t 2 Pr { ν 1 1 x l x)) < ε 0 1 <ν 1 1 x u x)) ε 0 2 t 2 s 0 1 F 1 2ν 1 1 x l ) 0 x)) ) F 1 2 ν 1 1 x ux)) ) F 0 1 2ν 1 1 x l 0 x)) ) > 0 so bt 2 )>0. We now argue that f satisfies claim B). Since f f , we have immediately that Pr{ε 2 Pr{D 1 X x P for all x ) sux P). Consider Pr{ε ν 1 11 x) ε. From8), we have that ν 2 11 x) ν 1 1 x l x)) 0 or ν 1 1 x) ν 1 1 x u 0 x)) for any x sux). For x ) sux P) such that ν 1 1 x) ν 1 1 x l 0 x)), wehave Pr{ε 1 ν 11 x) ε 2 1 ν1 1 x) 0 ν1 1 x) f 1 2 t 1 t 2 )dt 1 dt 2 1 f 1 2 t 1 t 2 )dt 1 dt 2 0 Pr{ε 1 ν 1 1 x) ε 2 Pr{Y 1 D 1 X x P For x ) sux P) such that ν 1 1 x) ν 1 1 x u 0 x)), wehave Pr{ε 1 ν 11 x) ε 2 ν1 1 x) 1 0 f 1 2 t 1 t 2 )dt 1 dt 2

8 8 A. M. SHAIKH AND E. J. VYTLACIL { ν1 1 x) 1 f 1 2 t 1 t 2 )dt 1 dt 2 [ ν1 1 x l 0 x)) 0 bt 2 ) ν1 1 x u 0 x)) ν 1 0 x) ν1 0 x) f 1 2 t 1 t 2 )dt 1 at 2 ) f 1 2 t 1 t 2 )dt 1 ν 1 1 x l 0 x)) ν1 1 x) f 1 2 t 1 t 2 )dt 1 f 1 2 t 1 t 2 )dt 1 ν 1 1 x u 0 x)) ] dt 2 1 { Pr{ε1 ν 1 1 x) <ε 2 Pr{ε 1 ν 1 1 x) ε 2 Pr{ε 1 ν 1 1 x) ε 2 Pr{Y 1 D 1 X x P The roof that Pr{ε ν 1 10 x) ε 2 >Pr{Y 1 D 0 X x P for all x ) sux P) follows from an analogous argument. Finally, we argue that f satisfies claim C). Consider 1 2 Pr{ε ν 1 10 x). From 8), we have that ν 1 1 x) ν 1 1 x l x)) or ν 0 11 x) ν 1 1 x u 0 x)). In the former case, we have that Pr{ε 1 ν 10 x) ν1 0 x) 0 { { ν1 1 x l 0 x)) 0 ν1 0 x) ν1 1 x l 0 x)) 0 ν1 0 x) f 1 2 t 1 t 2 )dt 1 dt 2 ν1 0 x) ) f t t 2 )dt 1 f t ν 1 1 x l t 2 )dt 1 0 x)) dt 2 f t t 2 )dt 1 dt 2 ν1 0 x) ) f 1 2 t 1 t 2 )dt 1 at 2 ) f 1 2 t 1 t 2 )dt 1 ν 1 1 x l 0 x)) dt 2 f 1 2 t 1 t 2 )dt 1 dt 2 s 0 Pr{D 0 Y 1 X x P s 0 In the latter case, it suffices to show that ν1 0 x) f 1 2 t 1 t 2 )dt 1 dt 2 ν1 0 x) f 1 2 t 1 t 2 )dt 1 dt 2

9 TRIANGULAR SYSTEMS OF EQUATIONS 9 For this urose, it suffices to show that ν1 0 x u 1 x)) ν 1 0 x l 1 x)) ν 1 0 x l 1 x)) f 1 2 t 1 t 2 )dt 1 dt 2 ν1 0 x u 1 x)) since outside of this region of integration f 1 2 f 1 2. Note that ν1 0 x u 1 x)) f 1 2 t 1 t 2 )dt 1 dt 2 ν 1 0 x l 1 x)) f 1 2 t 1 t 2 )dt 1 dt 2 ct 2 ) ν1 0 x) dt 2 ) ν 1 0 x l 1 x)) f 1 2 t 1 t 2 )dt 1 dt 2 ν1 0 x u 1 x)) ν 1 0 x) f 1 2 t 1 t 2 )dt 1 dt 2 ct 2 ) Pr { ν 1 0 x l 1 x)) < ε 1 <ν 1 0 x) t 2 dt2 dt 2 ) Pr { ν 1 0 x)<ε 1 <ν 1 0 x u 1 x)) t 2 dt2 Pr { ν 1 0 x l 1 x)) < ε 1 <ν 1 1 x u 1 x)) t 2 dt2 ν1 0 x u 1 x)) ν 1 0 x l 1 x)) f 1 2 t 1 t 2 )dt 1 dt 2 as desired. The roof that Pr{ε ν 1 11 x)s 1 follows from an analogous argument. Q.E.D. Det. of Economics, University of Chicago, 1126 East 59th Street, Chicago, IL 60637, U.S.A.; amshaikh@uchicago.edu and Det. of Economics, Yale University, New Haven, CT , U.S.A.; edward.vytlacil@yale.edu. Manuscrit received February, 2010; final revision received October, 2010.