ESTIMATING A CLASS OF TRIANGULAR SIMULTANEOUS EQUATIONS MODELS WITHOUT EXCLUSION RESTRICTIONS

Transcription

1 ESTIMATING A CLASS OF TRIANGULAR SIMULTANEOUS EQUATIONS MODELS WITHOUT EXCLUSION RESTRICTIONS Roger Klen Francs Vella THE INSTITUTE FOR FISCAL STUDIES DEPARTMENT OF ECONOMICS, UCL cemmap workng paper CWP08/05

2 Estmatng a Class of Trangular Smultaneous Equatons Models Wtout Excluson Restrctons Roger Klen Rutgers Unversty Francs Vella European Unversty Insttute July, 005 Abstract Ts paper provdes a control functon estmator to adjust for endogenety n te trangular smultaneous equatons model were tere are no avalable excluson restrctons to generate sutable nstruments. Our approac s to explot te dependence of te errors on exogenous varables (e.g. eteroscedastcty) to adjust te conventonal control functon estmator. Te form of te error dependence on te exogenous varables s subject to restrctons, but s not parametrcally spec ed. In addton to provdng te estmator and dervng ts largesample propertes, we present smulaton evdence wc ndcates te estmator works well. 1 Introducton Instrumental varables (IV) s a metod commonly employed n emprcal applcatons for estmatng models wt endogenous regressors. However, We are grateful to partcpants at numerous semnars over te past four years for varous comments wc ave resulted n mprovements to te paper. We would also lke to tank Etel Fonseca for elpful comments. We are partcularly grateful to Wtney Newey for detaled comments wc led to te current formulaton of te problem. Any remanng errors are te sole responsblty of te autors. 1

3 wle tere s general agreement tat IV s approprate for a large class of models wt endogenety, tere s frequently lttle agreement about te excluson restrctons tat ts metod typcally requres n spec c emprcal applcatons. In fact, te d culty n obtanng nstruments as generated a rapdly growng and mportant lterature related to nference n te presence of weak nstruments (see, for example, Stager and Stock 1999). Wen te prmary equaton of nterest contans an endogenous regressor, t s well known tat IV s equvalent to an OLS regresson tat ncludes an addtonal regressor to control for endogenety. Commonly, ts addtonal varable or control s te reduced form resdual for te endogenous regressor. In te lnear case, as te control s a lnear combnaton of te endogenous regressor and exogenous varables, te model s only dent ed n te presence of at least one excluson restrcton. 1 In te above case te mpact of te control s a constant tat s estmated along wt te parameters of nterest. As a result, wtout furter nformaton, dent caton requres an excluson restrcton. However, wen te error dstrbuton depends on te exogenous varables, we sow tat t s possble and n some sense natural to develop a control wose mpact s not constant. Wtout provdng parametrc functonal form assumptons, we provde assumptons on te manner n wc errors depend on exogenous varables. In partcular, as elaborated on below, we assume a generalzed form of eteroscedastcty for bot errors. We ten develop a "feasble" control wose mpact s not constant and sow tat te model s dent ed wtout excluson restrctons. As dscussed n secton 3, oter papers ave explored dent caton va second moments (e.g. Vella and Verbeek 1997, Rummery et al 1999, Sentana and Forentn 001, Rgobon 003 and Lewbel 004). For te model tat we consder, dent caton depends on tere beng eteroscedastcty n one or bot equatons of nterest and tat t "d ers" across equatons n a manner made precse below. Te estmator s ten based on estmatng a generalzed form of eteroscedastcty n eac equaton. For te structural equaton of nterest, suc eteroscedastcty must be estmated smultaneously wt te model s parameters as consstent resduals are unavalable. We do ts n a settng were te condtonal varance of eac error s an unknown functon of an ndex wc needs to be estmated. Wle ts semparametrc treatment of te unknown functons complcates te analyss, t ensures tat we 1 Ts control functon approac s equvalent to two-stage-least-squares.

4 can consder a general form of eteroscedastcty wtout avng to rely on parametrc assumptons for dent caton. In te followng secton we outlne te model. In secton 3 we dscuss te estmaton metod and ow to mplement t. Formal results are stated n secton 4. Ts secton also outlnes te proof strategy for obtanng tese results. Secton 5 provdes smulaton evdence and secton 6 concludes. Te Appendx contans detaled proofs of all teorems and ntermedate lemmas. Model and Ident caton Sources Wt o and o as vectors of true parameter values, consder te followng lnear trangular model: Y 1 = X 1o + Y o + u W o + u (1) Y = X o + v ; () were Y 1 and Y are contnuous endogenous varables; X s a vector of varables tat are mean-ndependent of te error components u and v : We furter assume tat tese errors are correlated. Te man objectve of estmaton s to conduct nference on o ; te vector of true parameter values n te prmary equaton. We use te terms prmary and secondary to refer to te rst and second equatons respectvely. Notce tat te model allows te same X 0 s n bot equatons wtout mposng any restrctons on te parameter values. Wen te errors do not depend on X, te (lnear) relaton between errors s captured by te followng uncondtonal populaton regresson: a o = arg mne [u a av] ) a o = cov (u; v) =V ar(v): By constructon, " u a o v s uncorrelated wt v, wc provdes te bass for te controlled regresson: Y 1 = W o + a o v + " : Provded tat te matrx [W; v ] as full column rank, te OLS estmator for ts regresson s consstent and would be mplemented n practce by replacng v by te correspondng resdual. However, n te absence of an excluson restrcton ts full rank condton s not sats ed. 3

5 Wen te dstrbuton of te errors depends on X, we would capture te (lnear) condtonal relaton between te errors by te followng condtonal populaton regresson: A o (X) = arg mne [u Av j X] ) A A o (X) = cov (u; v j X) =V ar(v j X): In ts case, " u A o (X) v s uncorrelated wt v condtoned on X, wc provdes te bass for te controlled regresson: Y 1 = W o + A o (X )v + " : Provded tat A o depends on X, wc would be reasonable wen te error dstrbutons depend on X, te matrx [W A o (X)v] wll ave full column rank. Accordngly, wen A o (X) s known, te above model s dent ed wtout excluson restrctons. As A(X) s unknown, t must be estmated and restrctons must be mposed to obtan dent caton. Here, we explore te restrctons mpled by a generalzed form of eteroscedastcty. To ts end, assume: were u S u u ; v S v v ; S u V ar (ujx) S v V ar (vjx) E (ujx) = E(vjX) = 0: Furter, tere s a constant relaton between unscaled error components: 3 o E (u v jx) = E (u v ) : Subject to te above restrctons, te error components can arbtrarly depend on X. Wt te correlaton o constant, te control s gven as: A o (X)v = o [S uo =S vo ] v: We would lke to tank Wtney Newey for ts nterpretaton of te control. 3 Note tat Bollerslev (1990) also employs a constant correlaton assumpton n a tmeseres context. 4

6 Before dscussng ow to mplement te above control, note tat f te scalng functons are known or can be consstently estmated, ten dent - caton olds f tese scalng functons "d er" n tat te matrx [S uo S vo ] as full column rank. As a specalzed nterpretaton, vew u and v as unobserved varables wt non-constant mpacts on te endogenous varables. Tese mpacts are functons of X and are gven by te functons S uo and S vo respectvely. Oter papers explot second moment nformaton as a source of dent- caton. Vella and Verbeek (1997) and Rummery et al (1999) develop an estmaton procedure based on te rank order of an ndvdual s poston n te reduced form resdual dstrbuton for subsets of te data. Te varable determnng te selecton of subsets s also assumed to be responsble for te eteroscedastcty. In te context of normal factor models, Sentana and Forentn (001) examne eteroscedastcty as a source of dent caton. Rgobon (003) formulates a model n wc tere are two known regmes. Te parameters of nterest and te covarance between te equatons errors do not depend on te regme ndcator. However, te error varances do depend on te known regme ndcator. Employng an error covarance restrcton smlar to tat n Rgobon, Lewbel (004) examnes a model of eteroscedastcty wt second moment nformaton dependng on a known vector of varables Z. As Z may concde wt X, for comparatve purposes we focus on ts case and wtout loss of generalty take E(X) = 0. He ten consders a model n wc: E (X u v ) = E [X E (u v jx )] = 0; E X v 6= 0. Te model consdered ere d ers n several respects from tose above. Frst, for te model outlned earler, E (u v jx ) depends on X. Consequently, wle te rst restrcton above may old n specal cases, t wll not old n general for te model consdered ere. Second, wt te condtonal covarance and varance functons dependng on X ; ere te condtonal varance of eac error s modeled as an unknown functon of an ndex. In a d erent model, Klen and Vella (004) we also explot eteroscedastcty to estmate a trangular treatment model were te endogenous regressor s bnary. To exbly model bot te sape and condtonal varance for te error dstrbuton n te bnary model, a double ndex formulaton s employed. In so dong, wt te estmated bnary response probablty as an nstrument, te model s "well-dent ed" wtout excluson restrctons. 5

7 Wle te treatment paper s related to ts paper te dent caton and estmaton strateges are fundamentally d erent from tose employed ere. Fnally note tat te use of nstruments n te absence of excluson restrctons s not lmted to cases of eteroscedastcty. Dagenas and Dagenas (1997) and Lewbel (1999) also dscuss estmaton of models were tere are endogenous regressors and no excluson restrctons. Tey sow tat wen tere s measurement error of a spec c form one s able to use nstruments based on te ger powers of te ncluded varables. Te model and estmator presented below bot d er from te approac n tese papers. 3 Te Estmator: Implementng Strateges Before presentng te man results, ts secton outlnes and motvates te estmaton strategy. From te above dscusson, we wll requre resduals and te condtonal varance functon for te secondary equaton. Accordngly, rst obtan consstent estmates of te secondary equaton s condtonal mean parameter values by regressng Y on X to get b: We ten estmate te resduals as: 4 bv = Y Xb: To estmate S v ; we mpose a sngle ndex structure: S v E v j X = E v j I v ( o ) ; were I v ( o ) X 1 + X o. Next, estmate o usng semparametrc least squares wt bv as te dependent varable (see Icmura, 1993). Namely: X ^ = arg mn ^ ^v ^E ^v j I v () ; Em- were ^ s a trmmng functons tat restrcts X to a compact set. 5 ployng te estmated ndex: ^S v = E b bv j I v ^ were b E s a non-parametrc estmator for te ndcated condtonal expectaton. Employng te above ntal estmator ^S v, we ten repeat te above 4 Tese resduals could be obtaned n a more general non-parametrc or semparametrc regresson. 5 Here, te set wll depend on sample quantles for te X0s. 6

8 process n a GLS step. 6 For notatonal convenence below, denote te vector of parameter estmates as: ^ ^ 0 0 ^0. As our focus wll be on te prmary equaton, we wll refer to tese parameters as nusance parameters. For te prmary equaton of nterest, we employ an estmator for S u tat s tself a functon of unknown parameters. Let I u (b o ) X 1 + X b o and assume te sngle ndex assumpton: E u j X = E u j I u (b o ) : As te above functon can not be estmated drectly, de ne te functon: ^S u (; b) E (Y 1 W ) j I u (b) were te estmated expectaton s obtaned from a kernel-based nonparametrc regresson of (Y 1 W ) on I u (b). Wt (; ; b) and = 1; :::; N observatons, de ne: ^A (; ^) ^Su (; b) = ^S v ^M (; ^) W + ^A () ^v ^S () 1 X ^ Y 1 ^M (; ^). N Te estmator for te prmary equaton s now de ned as: ^ arg mn ^S () : In te next secton, we provde assumptons and de ntons under wc tese estmators are consstent and asymptotcally dstrbuted as normal. Wt detaled proofs relegated to an appendx, ts secton also summarzes te man results and outlnes te proof strategy for obtanng tem. 4 Assumptons, De ntons, and Results We employ te followng assumptons to establs te asymptotc results proved n te Appendx: 6 Wle t s possble to avod a GLS step, we ave found tat te estmator for S v based on ^ GLS s mproved and tat tere s a correspondng mprovement n te estmates of te prmary equaton of nterest. 7

9 A1 Te vector (Y 1 ; Y ; X ; u ; v ) s..d dstrbuted over, wt E (u 4 jx ) and E (v 4 jx ) bounded. A Te parameter vector: (; ; ; b; ) s n a compact parameter space, ; were o s n te nteror of : A3 Wrte te error components as: u S u u ; S u V ar (ujx) v S v v ; S v V ar(vjx): Assume: E (u jx) = E (v jx) = 0 o E (u v jx) = E (u v ) : A4 Wt I u (b o ) X 1 + X b o and I v ( o ) X 1 + X o : S u E u jx = E u j I u (b o ) > 0 S v E v jx = E v j I v ( o ) > 0: For X n a compact set, tese functons and ter rst four dervatves are unformly bounded. Furter, eac ndex depends on a contnuous varable. A5 Te condtonal densty g(x 1 jx ) s bounded away from zero on te nteror of ts support and as bounded dervatves up to te fourt order. A6 For estmatng expectatons and denstes, assume tat te kernel functon, K, s a symmetrc densty wt up to 4 bounded dervatves. 7 A7 Te matrx [X; Y ; (S u =S v ) v] : N x (K + ) as full column rank. Te above assumptons are somewat standard, wt te last assumpton requred for dent caton (see Teorem of ts secton). In addton to tese assumptons, we need to de ne te estmators and a bas reducton devce used to establs asymptotc normalty. Accordngly, we adopt te followng de ntons: 7 In te smulatons, K s a normal kernel. 8

10 D1 Indcator Trmmng. Let c k and c k be lower and upper populaton quantles for X k ; k = 1; :::; K: Let q o be te vector of tese quantles. Wt x: 1xK, de ne P fx : c k < x k < c k ; k = 1; :::; Kg : Wt X [X 1 ; :::X K ], de ne te trmmng ndcator: o (q o ) f X P g : Wt ^q as a vector of sample quantles, te estmated trmmng functon s gven as: ^ (^q) : To nsure tat varous estmated denomnators are bounded away from zero n large samples, te above trmmng functon restrcts te components of X to a compact set dependng on estmated sample quantles. As a result, te trmmng functon sould be vewed as beng estmated rater tan taken as xed. 8 Employng ts trmmng functon, (D-4) provde te estmators for te parameter values of te secondary and prmary equatons. D Y -Model: Let ^ be te GLS estmator from te regresson of Y on X. 9 De ne te resdual: ^v Y X ^: Te estmated ndex parameters of te condtonal error varance are ten gven by: ^ = arg mn X N ^R () ; ^R () ^ ^v =1 ^E ^v j I v () =N; were ^E s a nonparametrc estmated expectaton de ned below. Wt ^ ^; ^ estmatng o ( o ; o ), we refer to ^ as te (nusance) vector of estmates for te secondary equaton. 8 As dscussed n Lemma G1 of te Appendx, we wll be able to take estmated trmmng as known under a result due to Pakes and Pollard (1989). 9 Frst, obtan OLS resduals ^v. Second, obtan I v ^ ; from te SLS estmator of o. Next, de ne estmate ^S v : ^S v = ^E ^v j I v ^ : Rewegtng observatons n te Y model provdes te GLS estmator of o : All of te results n ts paper old usng te OLS estmator of o : We ave found, owever, tat te nte sample propertes of te estmator for te Y 1 model are mproved n employng te GLS estmator for te secondary equaton. 9

11 D3 Estmated Condtonal Varance. Wt ^ gven n (D): ^S v ^E ^v ji v ^ : D4 Y 1 -Model. Wt te Y 1 model gven as: Y 1 = [X Y ] o + u W o + u; de ne u () Y 1 (; b; ) and W and ^S u (; b) ^E[u () j I u (b)]. Ten, wt ^M () W + ^Su (; b) = ^S v ^v, for te prmary equaton: 10 ^ = 1 X arg mn ^S () ; ^S () ^ Y 1 ^M () =N: Bot secondary and prmary equatons depend on a nonparametrc estmator of a condtonal expectaton. De ntons (D5-9) provde ts estmator and te underlyng wndows upon wc t depends. D5 Estmated Condtonal Expectatons. Let Z X 1 + X and de ne: ^E [Y j Z = z ; ] ^f =^g ; were wt te kernel functon satsfyng te assumptons n (A6): ^f ^f z ; ^L; X Y j ^Lj (N 1) K z z j ^L j j6= ^g ^g z ; ^; X ^ j (N 1) K z z j ^ j : j6= Here, ^Lj and ^ j are estmated local smootng parameters tat are de ned below and are employed for bas control. Note tat te local smootng parameters depend on a plot wndow, p ; and tat te above estmators also depend on a global wndow,. Restrctons on te wndows and p are gven n (D10) after local smootng parameters ave been de ned. 10 Te factor 1/ avods avng to account for a factor of n a number of gradent expressons examned n te Appendx. 10

12 Wt te excepton of te estmated local smootng parameters, (D5) s a standard nonparametrc estmator for a condtonal expectaton. For reasons detaled n Lemmas -3 of te Appendx, suc local smootng serves as a devce to reduce te bas n te components of ts estmator. De ntons (D6-9) provde te components needed to de ne estmated local smootng parameters. D6 Smoot Trmmng. Local smootng parameters wll be smootly trmmed away from zero usng te smooted trmmng functon: N (l; a) [1 + exp [ ( Ln(N)Ln(N) ) (l a) ] ] 1 : D7 Plot Estmators. Te estmated smootng parameters are based plot estmators. De ne te followng plot estmators for te densty g(z) and te condtonal expectaton E(Y jz = z): ^g p (z j ) X 1 zj z k K (N 1) p p k 6= j ^E p (Y j Z = z j ) X Y j zj z k K =^g p (z j ) (N 1) p p k 6= j ^f p (z j ) ^E p (Y jz = z j )^g p (z j ) ; were p s termed a plot wndow and z j s an ndex. D8 Trmmed Plot Estmators. Refer to te trmmng functon n (D1) and select populaton quantles: ck < c k; c k > c k De ne P fx : c k < x k < c k ; k = 1; :::; Kg so as to contan P n (D1) as a proper subset. Ten, de ne: f nf ^fp (z j ) ; ḡ nf ^g P P p (z j ) : Referrng to (D7), de ne te trmmed estmators ^f 1 N ^fp (z ) ; f f + N ^fp (z ) ; ^fp (z ) f ^g 1 N ^g p (z ) ; ḡ ḡ + N ^g p (z ) ; ḡ ^g p (z j ) : 11

13 D9 Estmated Local Smootng Parameters. Lettng m 1 and m be te geometrc means of ^f j and ^g (z j ) respectvely; te local smootng parameters n (D5) are gven by: ^L j ^f j j =m 1 1= ; ^j ^g j =m 1= : If te local smootng parameters were known and bounded away from zero, ten from Lemma -3 n te Appendx tey would serve as an approprate bas reducng devce. Trmmng s requred to nsure tat tese parameters are bounded away from zero. In ts context, smoot rater tan ndcator trmmng s requred for tecncal reasons. Wt a smoot trmmng functon approxmatng an ndcator, te dervatve of ts functon must ncrease wtout bound as te sample sze ncreases. Te Ln(N) factor of te trmmng functon nsures tat ts dervatve slowly ncreases. D10 Wndows. Set plot and second stage global wndows suc tat: ^ = O p N 1=. 11 Assume: p ^N rp ; ^N r ; 1=8 < r < 1=6; r > r p ; r + r p > 1=: Te man asymptotc results of ts paper follow from te above assumptons and de ntons and are proved n te Man secton of te Appendx. In te remander of ts secton, we summarze te asymptotc results and provde a bref outlne of te proof strategy. Detaled arguments are gven n te Appendx. Begnnng wt te secondary equaton (Y Model); Teorem 1 provdes te large sample results for te estmators of te nusance parameters. Teorem 1 (Te Y -Model): Under te above assumptons and de - ntons, estmates of regresson and ndex parameters satsfy te caracterzatons: 1) p N[^ o ] = p NX N " =N ) p N ^ o = p N =1 NX " =N; 11 Te estmated component ^ need not be p N convergent, and may d er for plot and second stage wndows. 1 =1

14 were " and " eac are ::d. wt 0 expectaton and nte varance. Te rst result above s mmedate and te second follows from a standard Taylor seres argument and Icmura (1993). Ts result also follows from te same type of U-statstc arguments used to establs asymptotc normalty for estmates of te prmary equaton (Y 1 Model): For te Y 1 Model, Teorem below provdes te consstency/dent caton result. Teorem (Consstency: te Y 1 Model): Wt o ( o ; o ; b o ) and ^ ^; ^; ^b, under te above assumptons and de ntons: ^ p! o : To outlne te consstency argument, wc s provded n detal n te Appendx, recall from D4 tat: ^ = arg mn ^S (): Equvalently, employng a well-known devce tat avods avng to establs convergence for ^S ( o ): ^ = arg mn ^Q () ; ^Q () ^S () ^S (o ) Referrng to (D4), obtan M () from ^M () by replacng all estmated functons wt ter unform probablty lmts. Ten, de ne Q () by replacng ^M () n ^Q () wt M (). It can be sown tat ^Q () Q () and ^Q () E [Q ()] eac converge n probablty, unformly n to zero. Accordngly, consstency wll follow f E [Q ()] s unquely mnmzed at o : Wt M o M ( o ) ; (; o ) (M M o ) ; and wt (; o ) as te correspondng vector of d erences: E [Q ()] = E [ (; o )] 0 [ (; o )] =N: As (; o ) = 0 at = o ; o s a mnmzer and te ssue s one of unqueness. Under a constant correlaton assumpton, n te appendx we 13

15 establs dent caton (unqueness) wen te matrx [X; Y ; (S u =S v )v] as full column rank. Teorem 3 below, wc s proved n te appendx, provdes te normalty result. Teorem 3 (Normalty: te Y 1 Model): Under te assumptons and de ntons above: p N [^ d o ]! Z; Z~N (0; ) : To outlne te argument, note tat under a standard Taylor seres argument for te gradent to te objectve functon and a unform convergence argument for te Hessan, normalty wll follow f te normalzed gradent s asymptotcally dstrbuted as normal. For expostonal purposes, neglect rst-stage estmaton uncertanty as t matters, 1 but poses no tecncal df- cultes. Ten, evaluatng all functons at true parameter values and lettng ^w r ^Mo te gradent s gven as: p p X N ^G = N ^ [Y 1 M o ] ^w =N + p N X ^ ^Mo M o ^w =N p N ^G 1 ( o ) + p N ^G ( o ) : Wt te estmated trmmng functon dependng on sample quantles, Lemma G1 employs results n Pakes and Pollard (1989) to sow tat te trmmng functon may be taken as known. A mean-square convergence argument s ten used to sow tat te estmated wegt functon, ^w ; may be taken as known. Accordngly: wt " 1 [Y 1 M o ] w and wt " 1 as te correspondng sample mean: p N ^G1 = p N" 1 + o p (1): For te second gradent component, Lemma 5 sows tat te estmated trmmng and wegt functons may be taken as known: p N ^G = p N X o ^Mo M o w =N + o p (1): 1 See Newey and McFadden (1994). 14

16 Wt ^M o dependng on estmated expectatons, te bas propertes of tese expectatons are mportant to te argument. Recall tat an estmated expectaton, wc s calculated under local smootng, as te form: ^E ^f =^g : Many of te ntermedate lemmas n te Appendx are concerned wt sowng tat te bas n numerator and denomnator s su cently small. Once t s establsed tat local smootng delvers a bas of order o(n 1= ), Lemmas 7 and G of te Appendx establs tat p N ^G s close n probablty to a lnear combnaton of U-statstcs. From a standard projecton argument, t s ten possble to caracterze ts gradent component n te same form as te rst component. 13 Namely n te Appendx we de ne a vector " wc s..d. wt expectaton zero and nte varance components. Ten, wt " as te correspondng sample mean, we sow tat: p N ^G = p N" + o p (1) : Wt te estmaton uncertanty component avng a smlar..d. caracterzaton, asymptotc normalty follows. mean 5 Smulaton Evdence To analyze te performance of te estmator we examne te followng settng. We smulated te followng model were te same exogenous varables appear n te condtonal means and te condtonal varances of bot endogenous varables. Te two ndces underlyng te eteroscedastcty are also gly correlated. Moreover, we use te same functonal form for te eteroscedastcty n eac equaton. Te model as te form: Y 1 = 1 + x 1 + x + Y + u Y = 1 + x 1 + x + v u = 1 + exp(: x 1 + :6 x ) u v = 1 + exp(:6 x 1 + : x ) v u = :33 v + N(0; 1) and v s N(0; 1): 13 See Ser ng (1980) and Powell, Stock, and Stoker (1989). 15

17 We examne two dstrbutons for te x 0 s. In te rst desgn we generate bot x 1 and x as standard normal random varables. In te second desgn we retan x 1 but transform x nto a c-squared varable wt 1 degree of freedom. We ten estmated te model by OLS and te control functon procedure developed ere, wc we denote CF n te tables. Te smulaton results for n = 1000 and 100 replcatons are reported n Table 1. An examnaton of Table 1 reveals a number of nterestng features of te smulatons. Frst, turn to te left and sde of Table 1 wc reports results for te normal desgn. Te entres n te Table represent te mean value from te 100 replcatons wt te standard devaton of te replcatons reported n parenteses under te estmate. Te OLS estmates for te man equaton s parameters n ts spec caton are badly based wt respect to ter true values of 1 ndcatng tat tere s a large degree of endogenety n ts model. Te estmates for eac of te x 0 s are approxmately.7, ndcatng a bas of around 30 percent, wle te bas on te coe cent for te endogenous regressor s approxmately 9 percent. Te lower panel of columns 1 sows results for te control functon procedure under te normal desgn. For te -parameter underlyng te ndex generatng eteroscedastcty n te secondary equaton, te mean of te estmates s.351. Ts value s reasonably close to te true value of.33, but tere s a relatvely large standard devaton. Te parameter b s te coe - cent n te ndex generatng eteroscedastcty n te man equaton. Te average pont estmate of.61 s reasonable relatve to te true value of.33, but note tat tere s a very large standard devaton assocated wt ts estmate. Recall tat ts estmate s obtaned smultaneously wt te slope coe cents and ts mprecson re ects tat t s d cult to estmate ts parameter accurately wle smply mnmzng te squared resduals for ts model. If ts parameter s of drect nterest, ten as descrbed below t s possble to explot oter sources of nformaton to ncrease ts precson. Te parameter corresponds to te coe cent on te control functon. Gven te desgn te true value s.33 and tus te average estmate of.304 s good. Most mportantly, owever, s tat te average value of te coe cents for te x 0 s and Y are all close to 1 ndcatng tat te ncluson of te control functon s accountng for te endogenety bas. Moreover, wle tere s more varablty n te estmates, n comparson to te OLS estmates, te estmates are generally qute accurate as ndcated by ter standard devatons. In te remander of te Table we report te correspondng estmates for 16

18 te c-square desgn. Te OLS estmates ndcate tat wt te c-squared x te problem of endogenety s reduced. Te remanng estmates are generally consstent wt te normal desgn. Toug not reported ere, we also consdered several varants on te estmator presented. Here, we ave explored te case n wc te condtonal varance of te errors s caracterzed by a sngle ndex. However, suppose tat te entre dstrbuton of eac error s caracterzed by a sngle ndex (as s te case n te smulatons). Under ts more restrctve ndex assumpton, t s possble to develop a mod ed verson of te estmator presented ere wt sgn cantly better ntes sample performance (especally for ndex parameters). 14 We ave also examned several "GLS" varants of te CF metod presented ere. Wle tese resulted n a notceable mprovement n te estmates, we judged te mprovement not su cent to warrant any furter (albet mnor) lengtenng of te Appendx Wen te entre dstrbuton of te errors depends on a sngle ndex, any functon of te squared resdual wll satsfy a sngle ndex assumpton. Accordngly, n an SLS regresson, tere wll be many ways of estmatng te ndex parameters. For te multplcatve eteroscedastcty employed n te monte-carlo, te (trmmed) log transform of te squared resdual would appear to a natural transformaton to employ, and we found tat t resulted n a notceable nte sample mprovement estmates of bot secondary and prmary equatons. 15 We examned estmators based on consstent resduals from te prmary equaton to estmate te condtonal varance parameter n a manner smlar to tat employed for te secondary equaton. Not surprsngly, as te nformaton on te condtonal varance s largely contaned n te resduals, te resultng estmator for te ndex parameter ad a muc smaller varance tat obtaned from te control metod reported ere. Indeed, t would be nterestng to combne ts moment nformaton wt tat n te rst-order condtons to te mnmzaton problem de ned ere. We ave not explored ts possblty n te present paper. 17

19 Table 1: Smulaton Results Normal OLS C-sq OLS constant (.058) (.1) x (.049) (.10) x (.060) (.11) Y (.039) (.108) CF CF constant (.78) (.304) x (.76) (.30) x (.81) (.98) Y (.73) (.305) (.184) (.00) b (.717) (.848) (.158) (.4) 18

20 6 Concluson In summary, we ave examned a generalzed form of eteroscedastcty wt condtonal covarance and varance functons dependng on exogenous varables. For ts case, we ave sown tat te model s dent ed and ave formulated a metod for estmatng t. We ave establsed tat te estmator s consstent and asymptotcally dstrbuted as normal. In a montecarlo study, across several desgns, te estmator for te parameters of nterest n te prmary equaton performed qute well n nte samples. As ndcated prevously, tere s scope for furter mprovements n te estmator for te ndex parameters. Suc mprovements would come from fully explorng ndex structure n te monte-carlo (see footnote 11 ) or from makng use of all avalable moment nformaton (see footnote 1 ). We ave focused on te lnear structure n part because t s most often used n practce. More mportantly, n te absence of oter nformaton, t s ts structure for wc dent caton fals wtout excluson restrctons. Neverteless, t would seem relatvely stragt-forward to extend te model to allow nonlnear functons of te exogenous varables to enter bot prmary and secondary equatons. Wt a mod ed control based on E(ujv; X), t would also seem possble to allow for nonlneartes n te endogenous varables. 19

21 7 Appendx 7.1 Intermedate Lemmas Te Appendx s organzed nto a secton on ntermedate lemmas and a man secton provdng consstency and asymptotc normalty for te proposed estmator. In ts secton of te Appendx, we provde a caracterzaton for te gradent from wc asymptotc normalty wll follow. To prevew te argument, we requre te followng notaton. Denote ^ as te vector of estmated parameters from te Y -equaton (ncludng estmated parameters of te condtonal error varance). Wrte o ( 0 o; o ; b 0 o) 0 for te column vector of true parameter values from te Y 1 -model (ncludng correlaton and condtonal varance parameter values). Refer to te averaged objectve functon for te prmary equaton (Y 1 model) sown n (D4). Taken wt respect to let ^G ( o ; ^) and ^H ( o ; ^) be te correspondng gradent and Hessan. Employng te above notaton, from a Taylor seres expanson: p N [^ o ] = ^H + ; ^ 1 p N ^G ( o ; ^) ; (1A) + [ o :^]. Obtan H (; ) from ^H by replacng all estmated nonparametrc expectatons by ter probablty lmts n ^H. Unformly n te parameters, t can be sown tat ^H (; ) H (; ) and jh (; ) EH (; )j eac converge n probablty to zero. Terefore, once consstency s establsed, ^H ( + ; ^) wll converge n probablty to H o EH ( o ; o ). Asymptotc normalty wll ten follow f te gradent component s asymptotcally normal. To analyze te gradent, wt ^M ( o ; ^) W o + ^A ( o ; ^) ^v, de ne: ^M o ^M ( o ; o ) W o + ^A ( o ; o ) v M o W o + A ( o ; o ) v : Denote ^w r ^M ( o ; o ) and ^G 3 r ^G ( o ; b o ; + ) [^ o ]. Ten, from (D4), te gradent wt respect to at o s gven as: ^G ( o ; ^) = ^G ( o ; o ) + ^G 3 (A) 0

22 = X ^ [Y 1 M o ] ^w =N + X ^ ^Mo M o ^w =N + ^G 3 ^G 1 + ^G + ^G 3 : Lemmas G1 and G below provde approprate caracterzatons for p N ^G 1 and p N ^G : Te requred caracterzaton for te trd component wll mmedately follow from Teorem 1 n te next secton wc caracterzes te rst-stage estmator, ^. Te oter ntermedate lemmas n ts secton provde results requred for tese caracterzatons. For a number of terms, we wll need to deal wt te fact tat te trmmng functon depends on estmated sample quantles. For one suc term, we provde a complete argument n Lemma G1 to sow tat te trmmng functon may be taken as known. As a smlar argument apples to oter terms, subsequently we wll take te trmmng functon as known. In wat follows, we wll make use of several standard convergence results. Frst, recall from (D5) tat wt Z X 1 + X, an estmated condtonal expectaton as te form: ^E ^E [Y j Z = z; ] ^f=^g: De nng te dervatve operator on a functon f as: t can be sown tat sup z; r f; r0 f f, r d ^E r d E! p 0; d = 0; 1; : Wt E f=g; estmated local smootng functons depend on ^f p and ^g p ; plot estmates of f and g respectvely (D8-9). Obtan ^f from ^f by evaluatng local smootng functons at te expectaton of plot estmators. Ten, for d = 0; 1; and wt all functons evaluated at o : X r d ^f r d ^f =N = Op (r 1 ) ; r 1 = 1 X r d ^f X E r d ^f N d p E r d ^f =N = Op (r ) ; r = 1 N d r d f =N = O (r3 ) ; r 3 = 4 : 1

23 Te rst rate above follows from a Taylor seres expanson, Lemma 1 below, and te rate at wc te varance of a plot estmator tends to zero. Te oter rates follow from te rates at wc te expectaton of eac sum tends to zero. It follows tat: 16 X r d ^f r d f =N = Op (max (r )) ; = 1; ; 3: Smlar results old for te denomnator for te estmated densty ^g : Lemma 1 below provdes a basc result used n analyzng gradent components based on te above rates. Lemma 1. Assume: S a X ^a =N = O p N s ; S b X ^b =N = O p N t ; were s + t > 1: Ten, p X N ^a^b = o p (1): Proof of Lemma 1. Te result follows from Caucy s nequalty: p N X ^a ^b =N p NS 1= a S 1= b : Lemma G1 (Frst Gradent Component). Wt W (X ; Y ) and Y 1 M o = u o (S u =S v ) v : W + o r ^Su v = ^S 3 v ^w ^Su = ^S v v : o r b ^Su v = ^S 5 v : 16 Wt r d ^f r d ^f ; let: 1 r d ^f r d ^f ; r d ^f r d f : Snce 1 + > 1 ; 1 +. Te rate follows by wrtng as a sum of approprate terms.

24 De ne w by replacng ^S u wt S u and ^S v wt S v n ^w : Wt o (q o ), let: NX " 1 o [Y 1 M o ] w ; " 1 " 1 =N: Ten, p N ^G 1 N P 1= ^ [Y 1 M o ] ^w : p N ^G1 N X 1= ^ [Y 1 M o ] ^w = p N" 1 + o p (1) : =1 Proof of Lemma G1. Wt ^ (^q), p N ^G 1 s te sum of te followng tree terms: A N 1= X [Y 1 M o ] [^ o ] w B N 1= X [Y 1 M o ] [^ o ] [ ^w w ] C N 1= X [Y 1 M o ] o [ ^w w ] : Te proof wll follow f eac of tese terms s o p (1) : Employng te same strategy as n Klen (1993), denote q o as a vector of populaton quantles (see (D1), Secton 4) and let N " q : jq q o j < " ; " = o(1): Ten, A = o p (1) f A supn X 1= [Y 1 M o ] [ (q) (q o )] w = o p (1) N " for all " = o(1): 17 From Pakes and Pollard [1989, Lemma.17, p. 1037], A = o p (1): For te term B; t su ces to sow tat: B supn X 1= [Y 1 M o ] [ (q) (q o )] [ ^w w ] = o p (1): N " Lettng (q) = 1 : (q) = 1 and/or (q o ) = 1 0 : Oterwse; ; 17 If unformty olds for N " for all " = o(1); ten unformty olds over o p (1) negboroods of q o : 3

25 t follows from te de nton of (q) and Lemma 1 tat: B N 1= B 1B ; X B 1 = sup [Y1 M o ] [ (q) (q o )] =N N " X 1= B = sup (q) [ ^w w ] =N : N " From Klen (1993), wt ndcators approxmated by smoot functons, t can be sown tat for any xed arbtrarly small : B 1 = o p N 1=+ : It can be sown tat B = o p N ; wc completes te argument for B. Turnng to C; te analyss s smlar to tat n Klen and Spady (1993). For ts term, t can be sown tat ^w w may be replaced by a lnear combnaton of estmated functons. Te result ten follows from a mean-square convergence argument. To llustrate te argument, wt " [Y 1 M o ] v, one of te components of C s gven as: D 1 N 1= X o " ^Su S u = ^S v : From a Taylor seres expanson and Lemma 1: 1= D 1 = N 1= X o " ^f =^g f =g =a + o p (1) ; a S u S v : It can now be sown tat D 1 = D 1 + o p (1); D 1 N 1= X o " ^f =^g f =g [^g =g ] =a : Fnally, t can be sown tat E (D 1)! 0: A smlar argument apples to te oter terms of C, wc completes te argument. Abramson (198) and Slverman (1986) provde a unform bas result densty estmaton under known local smootng. Hall (1998) extends ts argument n several drectons and provdes a metod for makng explct bas calculatons n a varety of contexts. Te Lemma below examnes te extenson for ntegratng wt respect to a functon tat need not be a densty. 4

26 Lemma. Let p(z) be a non-negatve functon wt unformly bounded dervatves to order 4 and let K be a symmetrc kernel wt a bounded dervatves to order 4. For t P ; assume tat p(t) > 0. Denote nf p (t), t P p. Employng te smoot trmmng functon n (D6), de ne: p (z) p (z) ; p p (z) + [1 ( p (z) ; p)] p. Ten, for P a proper subset of P, unformly n t P: Z! p B (^p (t) ; p (z) 1= ) K (t z) p (z) 1= p (z) dz p(t) = O 4 : Proof of Lemma. Make a cange of varable wt " (z t)=) and ten take a Taylor seres expanson about = 0 to obtan: 3X B p(t) = C (t) + O 4 : =1 From symmetry of te kernel, terms n odd powers of vans. Snce 1 and ts dervatves vans exponentally on P, te second order term, C (t) ; s arbtrarly close to C (t) ; te second order term n te analogous expanson of B [^g (t) ; g]. For te case n wc p s a densty, Abramson and Slverman sow tat C (t) = 0. As sown by Hall, ts result also olds wen p s not a densty. Te estmator for E(Y j Z = t) as te form ^f (t)/^g(t), wt Lemma provdng te bas for ^g(t): For ^f (t) ; under known local smootng wt L j fp (z j ) 1= (see D8-9): ^f (t; L) 1 N 1 X j6= Lj t y j K zj L j : Lemma 3 below, wc s due to Hall (1990), examnes te bas n ^f (t; L) wt known local smootng. Lemma 3. De ne ^f (t; L) and f(t) as above as above. Assume tat f (t) as unformly bounded dervatves up to order 4, and tat te kernel 5

27 functon, K, also as unformly bounded fourt dervatve. Assume tat f (t) > 0 for t P : Let P be a proper subset of P. Ten, unformly n t P : E ^f (t; L) = E(Y jz = t)g v (t) + O 4. Proof of Lemma 3. Hall as proved ts result n a muc more general context. For ts partcular functon and kernel employed ere, te result follows from Lemma wt p (z) [EY (jz = z) g (z)] : Recall from (D7-9) tat local smootng functons depend on te plot estmators ^f p and ^g p. To deal wt estmated local smootng parameters we pursue te followng strategy. Frst, we sow n Lemma 4 tat wt local smootng functons evaluated at ter condtonal expectatons, ten te low order bas results of Lemmas -3 stll old. Subsequently, we wll sow n Lemmas 5-6 tat all gradent terms based on estmated local smootng parameters are asymptotcally close to tose n wc local smootng functons are evaluated at te expectaton of te plot estmators. De ne " xed" local smootng parameters: L j f j 1= ; j g j 1= : Here, wt f j E ^fj jy ; X and g j E [^g j jy ; X ] ; f j and g j denote smootly trmmed versons of tese denstes as gven n (D8). An nfeasble estmator for E (Y j z r ) s gven by E r ^f z r ; L =^g z r ; : ^f z r ; L = ^g z r ; = NX Y j K j6= NX K j6= zr zr z j z j L j L j (N 1) ; j j (N 1) : Below, Lemma 4 provdes an approprately low order of te bas for te nfeasble estmators f r and g r : 6

28 Lemma 4. Let te plot wndow for estmatng local smootng parameters be gven as: p = O(N rp ) and te wndow for estmatng second stage denstes as: = O (N r ) ; r p + r > 1=: Ten for P and P as gven above: sup 1 (t) sup E ^f t; L E [Y jz = t] g (t) = o N 1= t P sup (t) sup E^g t; g (t) = o N 1= : t P Proof of Lemma 4. expanson n about = 0: Referrng to Lemma 4, from a Taylor seres (t) = 4X ^C (t) : =1 From symmetry of te kernel, ^C 1 = ^C 3 = 0: For te second order term: ^C (t) = C (t) + ^C (t) C (t) ; were C s te probablty lmt of ^C : Wt ^C contanng estmated densty dervatves up to order and wt C contanng te correspondng expected dervatves, t can be sown tat sup t P ^C (t) C (t) = p = o N 1= : Te results now follows for e (t) : Te argument for 1 (t) s smlar. To analyze te second gradent component, Lemma 5 below sows tat te estmated trmmng and wegt functons n ts component may be taken as known. 7

29 Lemma 5. Wt w and ^w de ned as n Lemma G1, for te second gradent component de ned above: N 1= ^G p N X ^ ^Mo M o ^w =N = N 1= ^G + o p (1) ; ^G = X o ^Mo M o w =N: Proof of Lemma 5. From te de ntons of ^G and ^G : N 1= ^G ^G = N X 1= ^Mo M o [^ ^w o w ] =N: Te result follows from repeated applcaton of Lemma 1 and a convergence rate for ndcators n Klen (1993). Employng Lemma 5 and earler results, t s now possble to sow tat estmated local smootng parameters may be taken as xed at values for wc te requred degree of bas reducton olds. Let ^E 1 ^S u and ^E ^S v: Ten, usng notaton ntroduced earler, wrte: ^E k ^f k =^g k : Recall tat ^f k and ^g k eac depend on a local smootng functons evaluated at plot estmators. Furter denote f k and g k as te correspondng quanttes wt local smootng functons evaluated at te expectaton of te plot estmators upon wc tey depend. Ten, de ne: S u E 1 f 1 =g 1 ; S v E f =g : Employng ts notaton, Lemma 6 below sows tat te gradent based on ^S u and ^S v s approprately close to tat based on S u and S v: Lemma 6 Wt ^M o de ned as above, obtan M o by replacng ^S u and ^S v wt S u and S v respectvely. In ^G ; replace ^M o wt M o to obtan G. Ten: p N ^G G = o p (1). 8

30 Proof of Lemma 6. components (1A-A): From Lemma 5 and te de nton of gradent p p X N N ^G G = o N ^Su = ^S v =1 S u = S v v w =N + o p (1). Te above d erence wll depend on te estmaton error n condtonal varance components for u and v errors. Denotng tese terms by o u and o v respectvely, eac s o p (1). For te former (te analyss for te latter s dentcal): u N 1= N X =1 v w ^Su Su = ^S v. From a Taylor seres expanson, Lemma 1, and wt v w = (S u S v ) : u = N 1= N X =1 ^S u S u X N = N 1= ^f1 =^g 1 =1 + o p (1) f 1 =g 1 + o p (1) : Ts term nvolves d erentals n bot numerator and denomnator. As te arguments for bot terms are very smlar, ere we sow tat te former term s o p (1). Ts term s gven as: f = N 1= X N ^f1 =1 f1 ( =g 1 ) + o p (1). In more compact notaton, wt Y j u j and z j I u (b o ) ; let: 1= ^f (z j ) ; Lj f (z j ) 1= ^L j K j ^Lj Y j K z z j ^L j ; Kj Lj Y j K z z j L j. Ten: f = N 1= N X =1 Kj Kj Yj ( =g 1 ) + o p (1). 9

31 De ne: t k z l t K z t l + K b j 1 f 1= j Y j ; a =g 1 " X j6= 1 (N 1) k z z z j 0 t l z L j a b j ^fj l fj : From a Taylor seres expanson n ^f j about f j and from Lemma 1: For te leadng term : f = N 1= N X =1 " =N + r; r = o p (1) : E ( f r) = 1 N P P P r 6= s E ("r " s ) + 1 N N =1 E (" ) : C S For te cross-product (C) terms: C = O(N)E [" r " s ] = C 1 + C ; C 1 = O(N)E X 1 (N 1) k j6=r;s C = O(N)E X X 6 4 k l6=k zr 1 k zr z k (N 1) z j 1 k zs z l (N 1) L j k zs z j L k bk a r ^fk L l bl a s ^fl L j b ja r a s ^fj fj fk fl : Wrte: ^f j fj = ^f j [r; s] N 3 f N 1 j + O [N p ] 1 c rs + O N 1 fj ; were ^f j [r; s] s obtaned by removng te two terms from ^f j tat depend on observatons r and s and were c rs depends on observatons r and s and s bounded. Ten, t can be sown tat C 1! 0: 30

32 Turnng to C, wt r 6= s 6= k 6= l : C = O(N)E 6 4 a r k zr z k L as k N 3 ^fk [r; s] f N 1 k + O [N p ] N 3 ^fl [r; s] f N 1 l + O [N p ] k zs z l L l bk b l Te analyss of ts term s based on te d erental ^f property of te functon k. For ts functon, note tat: Z Z y t z k l f (y; z) dydz = O. 1 c rs + O (N 1 ) f j 1 c rs + O (N 1 ) f j To establs ts rate for ; wt " [t z] =: Z Z = y k ["l (t ")] f (t " j y) d" g y (y)dy: f : and on a Take a Taylor seres expanson n about = 0. Snce k s an even functon, odd moments wll vans. Te result ten follows because te nner ntegral of te followng expresson s zero: Z Z yf (t j y) k ["l (t) t] d" g y (y)dy = 0: Employng te above result for te functon k and an terated expectatons argument, t can be sown tat C = O(N)O( 4 )O( 4 p) vanses under te wndow condtons. A smlar and somewat smpler argument sows tat te S terms also vans n probablty, wc completes te proof. Te gradent component Ḡ s composed of functons of estmated condtonal varances. To analyze ts component, Lemma 9 below sows tat suc functons can be wrtten as U-statstcs and analyzed by standard projecton arguments. To state and prove ts result, we wll requre some notaton, some of wc as already been ntroduced. Recall tat ^S u ^f 1 =^g 1 and tat wt k 1 [; j] and k1 [; j] as te kernel functons underlyng ^m 1 and ^g 1 respectvely: " # " # ^S u 1 NX u 1 NX N 1 jk 1 [; j] = k1 [; j] : (3A) N 1 j6= 31 j6=

33 A smlar expresson de nes ^S v: For notatonal convenence, let I 1 I u and I I v : Ten, wt S1 E (u ji 1 ) and S E (v ji ), wrte: S k [E k g k ] =g k f k =g k ; k = 1; : (4A) It wll also be convenent to wrte te vector valued wegt functon, sown n Lemma G1, compactly as: w r (X ) + s (X ) v ; were r and s are vector-valued functons of X : (5A) Lemma 7 (U-Statstc Projecton). Denote r d kf (x 1 ; x ) as te d t dervatve of te functon f wt respect to ts k t argument. For (x 1 ; x ) bounded and c > 0 nte, assume tat te functon f as bounded dervatves: r d kf (x 1 ; x ) c for k = 1; and d = 1; de ne: ^f f ^S u; ^S v ; f f Su; Sv ; dk r 1 kf Su; Sv Wt o as te trmmng functon evaluated at populaton quantles and wt te wegt functon caracterzed as n 5A, let: Ten: k [s (X ) o d k =g k ] v k E [ k ji k ] ; k = 1; : N 1= X ^ ^f f v w = p N" u + p N" v + o p (1) ; " u X u Su 1 =N " v X v Sv =N: Proof of Lemma 7. Employng te same arguments as n Lemma G1, we may take te trmmng functon as known. Ten, from (4A) and wt ^S u ^f 1 =^g 1, t follows from Lemma 1 and a Taylor seres expanson tat: = o p (1) ; k p N X o v w d k ^fk =^g k f k =g k =N: 3

34 Smplfyng 1 : 1 = N 1= X o v w d 1 [ ( ^m 1 =^g 1 ) m 1 =g 1 ] [ ^g 1 =g 1 ] + N 1= X o v w d 1 [ ( ^m 1 =^g 1 ) m 1 =g 1 ] [(^g g ) =g 1 ] : From Lemma 1, te second term s o p (1): Terefore, from (3A) and wt 1 o v w d 1 =g 1 : 1 = N 1= X 1 [ ^m 1 m 1^g =g 1 ] + o p (1) : = N 1= N (N 1) = N 1= U N ; U N X X j; j 1 u j k 1 [; j] (m 1 =g 1 ) k1 [; j] + o p (1) j6= N 1 X X j ; j = j + j =: Snce U N s a U-statstc and E ( 1jI 1 ) = 1, from standard projecton arguments: 18 1 = p N" u + o p (1) : pemployng te same arguments as above, s smlarly caracterzed as N" v and te lemma follows. j> Employng te above lemma, t s now possble to caracterze te second gradent component. 18 Wt Z (Y 1 ; Y ; X ) ; under local smootng (see Lemmas -3), for C(Z ) bounded and " > 0 : E jjz = N 1= " C (Z ) ) E j = o N 1= : Terefore: Te nner expectaton as te form: were E (" u ) = 0: Furter: p NUN ^UN = o p (1); ^UN 1 X E N jjz : E jjz = " u + r ; r = O (1) ; E j = o(n 1= ) ) E (r ) = o N 1= : pn Wt r r =N, t can be sown tat Var r! 0: Te result follows. 33

35 Lemma G. Te second gradent component as te caracterzaton: p N G = p N" ; were te " are..d. wt expectaton 0 and nte varance. Proof of Lemma G. From Lemmas 5-8, we may evaluate p N ^G as te true wegts and wt expected denstes replacng estmated denstes n te local smootng functons. Employng te same arguments as n Lemma G1, we wll also be able to take te trmmng functon as known. Terefore: p p X N N ^G = o N o ^Su = ^S v S u =S v v w =1 p X N o N o f =1 ^S u; ^S v ; v w f S u; S v; v w : Te proof now mmedately follows from Lemma 9 wt " " u + " v: 7. Man Results Recall tat te trd gradent component for te second stage estmator depends on ^; te estmator for te nusance parameter vector from te Y 1 - model. To analyze suc rst-stage estmaton uncertanty, Teorem 1 below caracterzes te components of ^. Teorem 1: ne: were De ne ^R (; ) 1 N Frst Stage Consstency and Caracterzaton. De- v () (Y X ) ; E v ( o ) j I v ( o ) = E [Y () j X ] : NX ^ ^r (; ) ; ^r (; ) v () =1 ^ () = arg mn ^R (; ) ; ^S v () E (Y X ) ji v ^ () 1= : 34 ^E v () j I v ()

36 Wt ^ ols as te OLS estmator for o ; let: ^X X = ^S v (^ ols ) ; X X =S v ( o ) : Ten, wt p lm X 0 X =N and wt " X 0 v ; te GLS estmator of, ^, sats es: De ne R (; ) 1 N 1) p N [^ o ] = 1p N X " =N + o p (1) : NX o r (; ) ; r (; ) v () =1 E v () j I v w r ( o ; o ) ; w w E [w ji v ( R 11 p 0 R ( o; o ) ; R 1 p 0 R ( o; o ) : Te estmator for te ndex parameters, ^ (^), sats es: 19 ) p p X N ^ o = R11 1 N r ( o ; o ) w =N + R 1 [^ o ] + o p (1) : Proof of Teorem 1. For (1), te proof s mmedate. For (), accountng for estmaton uncertanty n ^; te proof follows an adaptaton of Icmura (1993) under local smootng or from an applcaton of te arguments used n Teorem 3 below. 0 Recall from (1A-A) at te begnnng of te Appendx tat te second stage estmator as tree gradent components, wt te rst two beng caracterzed n Lemmas G1- above. Lemma G3 caracterzes te trd gradent component. 19 Ts caracterzaton olds under a more general semparametrc formulaton of te Y -model: Here, to empasze dent caton ssues, we ave focused on te case were te Y model s lnear wt an unknown condtonal varance functon. 0 Wt te wegt rede ned for te second-stage estmator, rst and second-stage gradents ave a smlar structure. Te ntermedate lemmas used to prove Teorem 3 could also ten be employed to prove Teorem 1 under estmated local smootng. 35

37 Lemma G3. Referrng to (1A-A), let Q 1 p lm (rg ( o ; b o ; o )) : Employng te notaton Teorem 1 above, de ne: " 3 Q 1 " 1 R11 1 [r ( o ; o ) w + R 1 1 " ] : Ten: p N ^G3 = p N" 3 + o p (1); " 3 NX " 3 =N: =1 Proof of Lemma G3. Te proof follows from (1A-A), Teorem 1, and a standard unform convergence result. Teorem : Second Stage Consstency and Ident caton. Let Z be te matrx wt t row: [W ( S u ( o ; b o ) =S v ) v ] : Ten, te model s dent ed under te constant correlaton assumpton f Z as full column rank. Proof of Teorem. Recall from (D5) tat: wt (; ; b) : W [X Y ] ; S u (; b) E (Y 1 W ) ji u (b) 1= M () W + S u (; b) M; M ( o ; o ; b o ) M o ^ arg ^S () = arg mn ^Q () ; ^Q () ^S () ^S (o ) : Replace all estmated functons ^Q () wt ter unform probablty lmts to obtan Q (), It can be sown tat ^Q () Q () s, unformly n, o p (1): Furter, te functon Q () converges unformly n te parameters to ts expectaton gven as: E[ 1 + ], 1 W ( o ) [S u (; b) o S u ( o ; b o )] v =S v : Wt ( 1 + ) mnmzed at te true parameter values, consstency wll follow f ts mnmum s unque. If te mnmum s not unque, t must be 36

38 te case tat 1 + = 0 at all potental mnmzng parameter values. Ten: W ( o ) + [S u (; b) o S u ( o ; b o )] v =S v = 0; (A) from wc t follows tat: Su (; b) v =Sv = o Su ( o ; b o ) v =Sv o S u ( o ; b o ) (v =S v ) W ( o ) Takng an expectaton condtoned on X : + ( o ) 0 W 0 W ( o ) : S u (; b) = os u ( o ; b o ) o S u ( o ; b o ) S v ( o ) (B) From te de nton of S u (; b) : + ( o ) 0 E [ W 0 W j X ] ( o ) : S u (; b) = E (Y W ) j X = E ( u W ( o )) j X = S u ( o ; b o ) E(u v jx ) ( o ) + ( o ) 0 W 0 W ( o ) = S u ( o ; b o ) o S u ( o ; b o ) S v ( o ) + ( o ) 0 W 0 W ( o ) : Wt [(1 o) = (1 )] 1= ; d erencng te expressons n (B) and (C): S u (; b) S u (; b) = os u ( o ; b o ) S u ( o ; b o ) (D) Substtutng (D) nto (A):, S u (; b) = S u ( o ; b o ) : W ( o ) + ( o ) S u ( o ; b o ) v =S v = 0, o [W, ( S u ( o ; b o ) =S v ) v ] = 0: Under a full rank assumpton, = o and = o : Snce = o ; from (C), S u ( o ; b) = S u ( o ; b o ) : Consequently, from (D), = 1: Wt = o ; t follows tat = o : As sown above, Su ( o ; b) = Su ( o ; b o ) : It can now also readly be sown tat b = b o wen S u s subject to a sngle ndex assumpton (Icmura 1993). o (C) 37