Online Supplement for "Threshold Regression with Endogeneity" by Ping Yu and Peter C. B. Phillips

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1 Ole Sulemet fo "Tesold Regesso wt Edogeety" y Pg Yu ad Pete C. B. Plls. D cultes Alyg te DKE We tee ae o ote covaates esdes q, te DKE s a oula ocedue fo estmatg : Pote ad Yu () ovde some dscusso ad efeeces to te elated lteatue. I ts smle case, we ave te model y g(q) ( q) (q ) e wt E[ejq]. Te DKE s de ed as te extemum estmato DKE ag max () () wee () E[yjq ] E[yjq ] wt E[yjq ] P j w j()y j d j () ad E[yjq ] P j w j()y j ( d j ()) eg estmatos of E[yjq ] ad E[yjq ], ad d j () (q j ). I te de to of E[yjq. ], te wegt fucto w j () (q j ) (q l ), wee () l () s a escaled eel desty, ad s te adwdt. Due to te wegted aveage atue of eel smootes, () would e ea zeo f tee wee o jum at. Otewse, te d eece would e ea te magtude of te jum ( ) wc s assumed to e ozeo. Ts d eece esues tat te estmato DKE s cosstet. Pote ad Yu () ave ecetly sow tat DKE coveges at ate ad te asymtotc dstuto s elated to a comoud Posso ocess. Ts lmt teoy s exlaed y teetg as a mddle ouday ot of q (see Yu, ). Fo ouday ot estmato, t s wellow tat oly data a O egoood s fomatve, so te egoood te costucto of te DKE s tycally lage eoug to esue te -cosstecy of DKE. Gve DKE, te lteatue as also cosdeed te estmato of te jum magtude : But o estmato of s esetly avalale. We tee ae addtoal covaates, Delgado ad Hdalgo () suggested tat te DKE cotue to e used to estmate. I ts case, te ocedue ca e emloyed y xg some ot (say x o ) te suot of x ad ede g () as E[yjx o q ] E[yjxo q ], wee E[yjx o q ] s a estmato of te codtoal mea of y gve x x o ad q. Te ojectve fucto coveges to zeo we 6, ad to o (E[yjx o q ] E[yjx o q ]) we, so DKE s cosstet f o 6. Tee ae seveal d cultes alyg te DKE ts way. Fst, te selecto of x o ases d cultes, as sow te followg examle. Suose y (x q) (q > ) ", wee, te suots of x ad q ae ot [ ], ad edogeety taes te fom E ["jx q] x q. Fgue sows E[yjx q] ad E[yjx q ]. To detfy successfully we eed to select x o so tat o s lage, wc meas tat x o sould e o te ouday of x s suot. O te ote ad, we also eed f xjq (x o j ) to e lage so tat tee s su cet data to detfy. We te desty of f xjq (xj ) taes o a ell sae, as a tycal case, x o sould deally e te mddle of x s suot. Hece, te selecto of x o oses a dlemma ad a otetal tadeo tat s esetly uesolved fom ot teoy ad actcal esectves. Secod, cosstecy of DKE eques tat o 6, ut o ca e as sow te examle of Fgue. Delgado ad Hdalgo () aly te DKE to estmate, assumg tat x q ad 6 so tat o 6 does ot deed o te coce of x o. Moeove, te eel fucto uses data te egoood of q e cetly, so tat te covegece ate of DKE s qute slow, as dscussed fute Secto.. Futemoe, gve DKE, te duced estmato of uses oly data te egoood of (x o DKE ), so te covegece ate of DKE s also vey slow, esecally we te dmeso of x s lage. Oe mgt cosde eglectg te data of x, ad usg oly te data of q ad y to estmate. Ts wll geeate te DKE of Pote ad Yu (). Now, te jum sze E[x jq ] s a aveage of te jums at all x values, so may e zeo o small, wc esults det cato falue o wea det cato. Eve f E[x jq ] s lage, ts DKE mgt e less e cet ta te IDKE ecause te jum fomato at s ot fully exloed see footote 6 fo fute aalyss.

2 4 3 Fgue : E[yjx q] ad E[yjx q ] Wt Edogeety: Te Blue Les Reeset te Case Wtout Edogeety. D cultes Alyg Two Alteatve Estmatos It s ow (Secto 4.. of Pote ad Yu, ) tat te DKE s asymtotcally equvalet to te LSE ad te PLE we q s a sgle covaate. I wat follows, we de e te LSE ad te PLE we ote covaates ae eset ad dscuss te d cultes tat ase devg te asymtotc dstutos. De e te oaametc LSE of te geeal case as follows, wee f N LSE ag m jj6 y f cm f (x q )(q ) cm f (x q )(q > ) K j wt K j K x j (q j q ) s te eel estmato of f f(x q ), cm f (x q ) cm f (x q ) jj6 jj6 y j K x j (q j q q ) y j K x j (q j q q ) wt ( (u t) u (u t) ( u u t f t f t f t u t f t :

3 I te costucto of N LSE, we elmate te adom deomato as. We ext de e te PLE as wee P LE ag m g f (x q ) y f x (q ) f g f (x q ) () jj6 y j x j(q j ) K j : Ts desty-wegted ojectve fucto of te PLE was suggested L (996) wtout cosdeg te tesold e ects. I ot te LSE ad te PLE, s estmated y dg te est t etwee y ad a estmato of E[y jx q ] te d eece les tat d eet estmatos of E[y jx q ] ae used. Te ojectve fucto of te IDKE s sueo to tat of te LSE ad te PLE two esects. Fst, accodg to Yu (8, ), oly te fomato aoud te tesold ot s fomatve fo, so () te ojectve fucto of te IDKE s costucted usg oly data te egoood of. I cotast, te ojectve fuctos of te LSE ad te PLE use fomato ote aeas, ad te esultg ases eed to e adled caefully. Te ojectve fucto of te IDKE teefoe taes advatage of ts local costucto, weeas te gloal ojectve fuctos of te LSE ad te PLE ae ueced y te e ects of fomato tougout te dstuto. Secod, sce () te ojectve fucto of te IDKE s qj qj lea ad, t s easy to localze te egoood of, wc s ey to devg te covegece ate ad te asymtotc dstuto of. Howeve, te ojectve fuctos of te LSE ad PLE ae comlcated olea fuctos of, wc maes localzato extemely ad. I addto, te ojectve fucto of te IDKE does ot ely o te assumto tat (x q) x, weeas tat of te PLE does. 3. Poofs fo Teoem, 3, 5 ad Coollay Poof of Teoem. (998), we ave We st deve te fomula fo. Followg Aedx A. of Hecma et al. a (x ) (x ) (I d ) W W Y (I d ) H H W H H W Y (I d ) H W W Y (I d ) z jzj w A j (I d ) H M jj6 z jw j y j A 3

4 wee (x x q ) S. (x x q ) S (x x q ) S C. A (x x q ) S P (d )(d ) Y H dag I d I (d )(d ) H z j (x j x q j ) S H W dag K x (x x x ) (q ) K x (x x x ) (q ) dag w w ad a (x ) (x ) ae smlaly de ed wt y. y y W dag K x (x x x ) (q ) K x (x x x ) (q ). y C A elacg W. Next xq a (x ) a (x ) (x ) (x ), ad P (x ) (x ) (x ) xq P A : Te st ste devg te asymtotc dstuto of s to sow tat ca e elaced y. I ote wods, xq xq wee estmatos wt suesct deotes te ogal estmatos ut wt elaced y. Of couse, we eed oly sow A sce s just a lea comato xq xq xq xq xq xq A. Poosto 5 followg ts oof gves ts esult. A xq xq a (x ) a (x ) a (x ) a (x ) (x ) (x ) xq, a (x ) a (x ) a (x ) a (x ), (x ) (x ) (x ) (x ) 4

5 wee a (x ) (x ) s de ed y a (x ) m (x ) lm m(x ) ad y (x ) m (x ) lm (@m(x )@), wt a smla de to fo a (x ) (x ), ad a (x ) a (x ), : Note tat, ude ou sec cato, (x ) (x ) xq ad a (x ) a (x ) (x )( (x ) (x )) fo ay x. Also, (x ) (x ) (x ) We st deve te asymtotc dstuto (x ) (x ) xq Gve assumtos E, F, G, ad H, we ca aly te agumets Teoem 3 of Hecma et al. (998) to xq xq A 8 >< (I d ) >: M M e xq m M e A ad te cosde M m R R. 9, > > : wee M s te squae matx of sze P ( d )(d ) wt te l-t ow, t-t colum loc eg, fo l t, m (u x u q ) S(l) (u x u q ) S(t) K x (u x x ) (u q )f(x u x u q )du x du q s a P ( d )(d ) y vecto wt te t-t loc eg te tasose of (u x u q ) S(t) (u x u q ) S() m () (x ) K x (u x x ) (u q )f(x )du x du q ad m () (x) eg a ( d)( )(d ) vecto of te atal devatves of m(x q) at q, e jj6 z jw j e j wt z j ad w j eg z j ad w j ut avg elaced y, (I d ) R o () 5

6 ad te ojects wt suesct ae smlaly de ed. It tus out tat te tems assocated wt m wll cotute to te as ad te tems assocated wt e, wc s a U-statstc, wll cotute to te P vaace. Gve tat fq ( ), we eed oly cocetate o te umeato. Fst, aalyze te as. " E M (I d ) (I d ) M o B (I d ) m M m M m M m f(x jq )dx (q ) f(q )dq M o B E[g () (x ) q ]f q ( ) wee Mo ad B ae de ed te ma text, ad m () (x ) m () (x ) g () (x ) ude Assumto G. Note ee tat te eel K x s elaced y K ecause te data te egoood of te ouday of ca e eglected asymtotcally. Also, we ca calculate tat te vaace of ts tem s O o(), so t coveges oalty to ts exectato. Secod, aalyze te vaace. Tag te lt elemet A, we cosde xq xq # e l M e M e wc s a secod-ode U-statstc. Fom Lemma 8.4 of Newey ad McFadde (994), ts U-statstc s asymtotcally equvalet to P m (x j q j e j ) E e j m (x q e ), wee e l e l M z j w j e j M z j w j M z j w j e j x j q j e j M z j w j f(x q )dx dq We aly te Lauov cetal lmt teoem to deve te asymtotc dstuto. It s stadad to cec tat te Lauov codto s sats ed, so we cocetate o calculatg te asymtotc vaace as follows. " E e j "e E j "e E j R E e j 6 4 R e l T T T 3: e l M z j w j # e l M z j w j f(x q )dx dq # e l M z j w j f(x q )dx dq 3 M z j w j f(x q )dx dq 7 M z j w j f(x 5 q )dx dq e l # M z j w j f(x q )dx dq 6

7 We aalyze T, T ad T 3 tu. T E 4e j qj (u q ) e l M o " u x q # S 3 j K (u x ) du x du q 5 (x j ) (v q ) (u q ) e l M o (u x v q ) S K (ux ) du x du q f(x j )dx j dv q E (v q ) (x j )C l (v q) dv q jq j f q ( ): Smlaly, qj ad T 3 sce m (x q e ) T E (v q ) (x j )C l (v q ) dv q jq j f q ( ) qj. I summay, d N E (v q ) (x)c l (v q) (v q ) (x)c l (v q ) dv q q f q ( ) ad te asymtotc dstuto of xq xq follows as te teoem. We ext deve te asymtotc dstuto of. Gve tat O () ude Assumto H, te tem ca e eglected, ad as te same asymtotc dstuto as (x ) (x ) (x ) Fo te as, ote tat " E (x ) ( I d ) E (x ) ( I d ) (x ) ( I d ) M m M o B, (x ) (x ) M m Fo te vaace, te coesodg U-ojecto m (x q e ) s e j (x ) ( I d ) M z j w j M m # : M m f(x jq )dx (q ) f(q )dq M o B g () (x ) q f q ( ): M z j w j f(x q )dx dq : We ca oceed a smla faso to te aove devg te asymtotc vaace. Fo examle, te 7

8 coesodg fom to T s T E 4e j (q j ) (u q ) (x j ) ( I d ) M o (x j ) (v q ) (u q ) (x j ) ( I d ) M o E (v q ) (x j )C (x j v q ) dv q jq j f q ( ): " u x q # S 3 j K (u x ) du x du q 5 (u x v q ) S K (ux ) du x du q f(x j )dx j dv q Poosto 5: Ude te assumtos of Poof. We eed oly to sow (a (x ) a (x )) (x ) (x ) xq xq A : a (x ) a (x ) (x ) (x ) ad : (3) It s easy to see tat te st esult s mled y (a (x ) a (x )) a (x ) a (x ) ufomly x, (x ) (x ) (x ) (x ) ufomly x. Sce O ( ), falls to C C fo some ostve C wt ay lage oalty we s lage eoug. So we ca just ove tese esults y elacg y C C. Te coesodg a (x ) ad (x ) ae deoted as a C (x ) ad C (x ). Sce te esults fo a (x ) ad (x ) ae smlaly oved, we eed oly ove tat a C (x ) a (x ) ufomly x, (4) C (x ) (x ) ufomly x. Wtout loss of geealty, suose C >. Lemma 6 sows (3), ad Lemma 7 sows (4). Poof of Coollay. Te asymtotc dstuto of s moe volved sce t cludes vaatos fom two comoets as : 8

9 Fst ote tat N f q ( ) f q ( ) f q ( ) N f q ( ) fq ( ) f q ( ) f q ( ) N N N f q ( ) fq ( ) f q ( ) f q ( ) wee N ad N ae te umeatos of ad, ad f q ( ) P. Fom te eale aalyss te oof of Teoem, N N sats es N N e M o B M o B E[g () (x ) q ]f q ( ) e M e M e e j e M z j w j M z j w j f(x q )dx dq ad also j N E N fq ( ) E fq ( ) a (x ) a (x ) E a (x ) E : a (x ) It s ot ad to see tat tese two uece fuctos ae ucoelated, so te vaace of s te sum of te vaaces of tese two ats. Te vaace of te st at s deved te oof of Teoem. As to te secod at, ote tat " E a (x ) a (x ) # a (x ) a (x ) f(x q )dx dq (v q ) a (x ) a (x ) f(x )dx dv q (v q ) dv q E[ a (x ) a (x ) j ]f q ( ): 9

10 Smlaly, " E a (x ) a (x ) # " E # (v q ) dv q f q ( ) (v q ) dv q f q ( ) so te vaace of te secod at s aoxmately (v q ) dv q E[ a (x ) a (x ) j ]f q ( ) (v q ) dv q f q ( ) (v q ) dv q E[ a (x ) a (x ) j ] f q ( ): (v q ) dv q f q ( ) Fo te as of te secod at, ote tat E N f q ( ) (q ) a (x ) a (x ) f(x q )dx dq f q ( ) (v q ) a (x ) a (x ) f(x v q )dx dv q f q ( ) (v q ) a (x ) a (x ) (x ) (v q ) l dx dv q l l l (v q ) v q l dv q l l f (l) a (x ) a (x ) f (l) (x )dx wee f (l) (x ) s te lt ode atal devatve of f(x ) wt esect to evaluated at, ad E fq ( ) f q ( ) (q ) f(q )dq f q ( ) (v q ) f( v q )dv q f q ( ) (v q ) l l f (l) ( ) (v q ) l dv q l (v q ) v l q dv q f (l) ( l l ) wee f (l) ( ) s te lt ode devatve of f q () wt esect to evaluated at. I sum, te asymtotc dstuto of s as stated te teoem. Poof of Teoem 3. Fst deve te fomula fo e. x Fom (??), e x ( x ) ( x ) (q ) ( x ) (q ) (a (x ) a (x )) : By smla aalyss to te oof of Teoem, e x ca e elaced y wtout a ectg ts asymtotc dstuto. Also, a (x ) a (x ) ca e elaced y ts lea aoxmato wt o

11 asymtotc mact. I summay, e x q x ( x ) ( x ) (q ) e M m M m M ( x ) e M e wee M, m ad e ae de ed te oof of Teoem. By stadad metods, te deomato coveges oalty to M f q ( ), wee M s de ed te ma text, so we cocetate o te umeato. Fst, cosde te as tem. Fom te oof of Teoem, ( x ) e M m M m E ( x ) e M o B M o B g () (x ) q f q ( ): Next cosde te vaace. We eed to calculate te covaace etwee te lt ad tt elemet of te umeato, l t d. Tag te (l )t elemet of te umeato, l :d, we cosde x l e M e M e wc s a secod-ode U-statstc. Fom Lemma 8.4 of Newey ad McFadde (994), ts U-statstc s asymtotcally equvalet to P m l (x j q j e j ) E x l e j x l It s ot ad to sow tat m l (x q e )m t (x q e ) m l (x q e ), wee e e M z j w j e j M z j w j M z j w j e j x j q j e j M z j w j f(x q )dx dq : E x l x t (v q ) (x)c (v q) (v q ) (x)c (v q ) dv q q f q ( ): Te, alyg te Lauov cetal lmt teoem, te asymtotc dstuto of q ad exl xl, l d, follows as te teoem. We, e q O ( )O ( ) o (), so e ave te same asymtotc dstuto as. We 6, te covegece ate of e s. It s ovous tat q

12 q O ( )O ( ) o (). Also, q q q o () q q : q So e as te same asymtotc dstuto as q q. Poof of Teoem 5. Note tat GMM GMM G G G z (q ) z (q > ) (" x ( < q )) : By te cosstecy of ad Glveo-Catell, G G. Followg te oof of Teoem 3 of Cae ad Hase (4), we ca sow tat ude te momet estctos o x q, " ad z. We stll eed to sow tat z (q ) z (q x ( < q ) > P z x A ( < q ) ad z ( < q ) z ( < q ) " : Fo tese two esults, cosstecy of s ot eoug we eed ( ). But ts case, P z x P ( < q ) o z x o (), ad te secod esult olds smlaly. Gve tese two esults, stadad agumets yeld te asymtotc dstuto of te GMM estmato. 4. Poofs fo te Lemmas Lemma su Q () Q () : Poof. Notg tat Gve tat s comact we ave fom Lemma B. of Newey (994) tat su x su E () O (), x Q () () () E E [ q ()] O l d : E () () E [ ()] () O ql d

13 ufomly. By a Glveo-Catell teoem, su E () Note tat E E () Q (), te esult of teest follows. E E () : Lemma P l 4P T l o () ufomly v: Poof. We tae T 4 to llustate ad ave T 4 jj6 ( ) jj6 e j (q j ) K v j K j ( ) ( ) jj6 jj6 e j (q j ) K v j K j e j (q j ) Kx j e j ( q j ) qj v Kj x A ( ) qj jj6 P ( j ) A ufomly v, wee te secod to last equalty s fom te Lsctz cotuty of (). By te U-statstc ojecto, see, e.g., Lemma 8.4 of Newey ad McFadde (994), ( ) jj6 P ( j ) E [P ( j ) j j ] O P E ( j ) : j ( I ou case, E [P ( j ) j j ] e q j ) R j K x j f(x )dx O(e j ( q j )), ad E P ( j ) R (x d )f(x )dx O, so d E [P ( j ) j j ] e j ( q j ) A o () j j E P ( j ) O o(): d Lemma 3 P P (T 5 T 6 ) f (x ) () [( x ) e ] ( < q v )f(x ) f (x ) o (): 3

14 Poof. P T 5 f (x ) s a U-statstc ad we wte T 5 f (x ) jj6 e j ( < q j v )K j f (x ) E [P ( j ) j j ] O P E ( j ) j ( ) jj6 P ( j ) wee P ( j ) e j ( < q j v )K j f (x ) wt (x q e ), ad te last equalty s fom Lemma 8.4 of Newey ad McFadde (994). Te E [P ( j ) j j ] e j ( < q j v ) () d K x (x x j x ) f(x ) f (x )dx e j ( < q j v ) () K x (u x x j u x ) f(x j u x ) f (x j u x )du x e j ( < q j v ) ()f(x j ) f (x j ) ad so tat E P ( j ) O d d O d E [P ( j ) j j ] j E P ( j ) e ( < q v ) ()f(x ) f (x ) d d o(): Smlaly, T 6 f (x ) jj6 x j e j ( < q j v )K v j f (x ) [( x ) e ] ( < q v ) ()f(x ) f (x ) o (): Te esult follows y otg tat () (). Lemma 4 P 4 T l l P 4 T l o () ufomly v: l Poof. Tae T 4 as a examle. T 4 d O j d j e j (q j ) K v j K d e j (q j )K x xj j x o x o d e j ( q j )K x xj x o qj v x o A qj 4

15 ufomly v, wee te last equalty s fom te Lsctz cotuty of (). Sce E[T 4 ] O d o() T 4 o (). Lemma 5 T 5 T 6 T 5 T 6 d Do (v): Poof. Tae T 5 T 5 as a examle. We use te caactestc fucto to d ts wea lmt. De e T 5 P j T 5j P j t 5j (v < q j ) ad T 5 P j T 5j P j t 5j ( < q j v ), wee t 5j d e j K j, t 5j d e j K j, v < ad v >. Note tat Hece ex s T 5j ( v < q j ) ex s t 5j ex s T 5j ( < q j v ) ex s t 5j : E ex s T 5j s T 5j E ex s T5j E ex s T 5j E ( v < q j )E ex s t 5j qj E ( < q j v )E ex s t 5j qj R d R R o qj ex s v ej K(u x ) f(e j x o jq j )de j du x f(q j )dq j R d v R R o ex s qj e j K(u x ) f(e j x o jq j )de j du x f(q j )dq j v f q( ) R R ex s e j K(u x ) () f(e j x o jq j )de j du x v f q( ) R R ex s e j K(u x ) () f(e j x o jq j )de j du x v f q( ) R R d ex s e j K(u x ) () (K(u x)>) Vol(K(u f(e x)>) j x o jq j )de j du x v f q( ) R R d ex s e j K(u x ) () (K(u x)>) Vol(K(u f(e x)>) j x o jq j )de j du x v d f q ( )f xjq (x o j )E ex s e j K(U j ) () x j x o q j v d f q ( )f xjq (x o j )E ex s e j K(U j ) () x j x o q j v d f(x o )E ex s e j K(U j ) () xj x o q j v d f(x o )E ex s e j K(U j ) () xj x o q j wee Vol(K(u x ) > ) d s te volume of te aea of u x suc tat K(u x ) >, ad U j ad U j ae deedet of (e j x j q j) ad follow a ufom dstuto o te suot of K(). It follows tat 8 < E : Y E ex j T 5j s j T 5j j s T 5j s T 5j 93 A 5 ex v d f(x o )E ex s ek(u ) () x x o q v d f(x o )E ex s ek(u ) () x x o q : 5

16 Ts s te caactestc fucto of a comoud Posso ocess D 5 () evaluated at v 8 >< D 5 (v) >: N (jvj) P N (v) P ez, f v ez, f v > ad v, wee s a cadlag ocess wt D 5 (), ez e K(U ) (), ez e K(U ) (), ad e e U U, N () ad N () ae de ed Coollay. Geealzg ts agumet, we get te esult of teest. Lemma 6 P C P : Poof. C C C C ( q C ) O o () wee te equalty s fom te Lsctz cotuty of (). Lemma 7 Ufomly x, a C (x ) a (x ) C (x ) (x ) : Poof. Tae te st esult as a examle. We ave a C (x ) e e M C M e M e M a (x ) C e M C M C M C M M e M M C M M C M M M C C M M C wee M C ad C ae smlaly de ed as M ad ut wt elaced y C, ad te decomosto te last equalty s fom Lemma of Yu (). Sce M, M ad ae O (), we eed oly to sow tat M C M ufomly, C ufomly. 6

17 Tae te secod esult as a examle. C jj6 jj6 (x j x q j C ) S K x qj C (x j x x ) y j (x j x q j C ) S K x qj (x j x x ) y j : Tae te followg tem of C as a examle sce t s te adest to aalyze. jj6 jj6 jj6 jj6 q j C K x (x j x x ) qj C y j (q j ) K x qj (x j x x ) y j K x (x j x x ) C qj C y j K x (x j x x ) (q j ) ( q j C ) C y j : Fom Lemma B. of Newey (994), ot tems o te gt sde covege to te exectatos ufomly, ut t s easy to see tat tese exectatos ae O o(). Te esults of teest follow. Addtoal Refeeces L, Q., 996, O te Root-N-Cosstet Semaametc Estmato of Patally Lea Models, Ecoomcs Lettes, 5,