Bias Corrections in Testing and Estimating Semiparametric, Single Index Models

Transcription

1 Bas Correctons n Testng and Estmatng Semarametrc, Sngle Index Models Roger Klen and Chan Shen July, 2009 Abstract Semarametrc methods are wdely emloyed n aled work where the ablty to conduct nferences s mortant. To establsh asymtotc normalty for makng nferences, bas control mechansms are often used n mlementng semarametrc estmators. The rst contrbuton of ths aer s to roose a mechansm that enables us to establsh asymtotc normalty wth regular kernels. In so dong, we argue below that the resultng estmator erforms very well n nte samles. Semarametrc models are commonly estmated under a sngle ndex assumton. Snce the consstency of the estmator crtcally deends on ths assumton beng correct, our second objectve s to develo a test for t. To ensure that the test statstc has good sze and ower roertes n nte samles, we emloy a bas control mechansm smlar to that underlyng the estmator. Furthermore, we structure the test so that ts form adats to the model under the alternatve hyothess. Monte Carlo results con rm that the bas control and the adatve feature sgn cantly mrove the erformance of the test statstc n nte samles. 1 Introducton Semarametrc models are commonly estmated under a sngle ndex assumton (e.g., Ahn (1997), Clmov, Delecrox and Smar (2002), Fraga and Martns (2001), Ger n (1996), Gorgens (2000), Gorgens and Horowtz (1999), Ichmura (1993), Klen and Sherman (2002), Klen and Sady (1993)). In ths aer, we focus on develong bas controls for estmatng and testng semarametrc models n such a settng. Before dscussng bas controls, we begn wth a bref descrton of these sngle ndex models Ths aer s based on a orton of Chan Shen s Ph.D. dssertaton, Economcs Deartment, Rutgers Unversty. We would lke to thank the edtor and anonymous referees for helful comments. Any errors are the sole resonsblty of the authors. 1

2 wth whch ths aer s concerned. To ths end, let Y be a deendent varable of nterest, X a vector of exlanatory varable, and V (X; o ) a arametrc functon of X where o s a vector of arameter values. Wth V termed an ndex, these sngle ndex models are characterzed by the assumton: E(Y jx) = G(V (X; o )) = E(Y jv (X; o )) The lnear ndex, where V s lnear n observed varables, s the most commonly used. Such a structure can cover a wde class of models because t ermts V to be a lnear combnaton of hgher order and nteracton terms. To cover an even wder class of models, t may be necessary to consder multle ndex models where the condtonal exectaton s an unknown functon of two or more searate ndces. As above, let o be a arameter vector of nterest and denote ^ as ts estmator. To ensure that ths estmator has desrable roertes, n large and nte samles, t s crtcal to have a good estmator for the condtonal exectaton, E(Y jv (X; o )). In estmatng ths exectaton, hgher order kernels 1 are often used n the lterature as a bas control mechansm to obtan asymtotc normalty for ^: Ths bas control method can fal to satsfy restrctons on the true condtonal exectaton of nterest, and hence does not erform well n nte samles. For examle, n bnary resonse models, where exectatons are robabltes, such a method can delver estmated robabltes that fall outsde of the nterval [0,1]. One objectve of ths aer s to rovde an estmator wth desrable large samle roertes and that also erforms well n nte samles. To obtan these roertes, we roose a bas control mechansm wth two controls and regular kernels. The bas controls wll ensure normalty, whle the use of regular kernels allows us to mose known restrctons on the estmated condtonal exectaton. 2 We nd that the resultng 1 Wth V..d. dstrbuted as g, a kernel densty estmator for g(t) s gven as: ^g (t) = X 1 h K [(t V ) =h] : When K s a densty that s symmetrc about zero (e.g., a standard normal), we refer to K as a regular kernel. In ths case, t can be shown that the bas n ^g s O(h 2 ); where h tends to zero at a rate gven below. When K s a functon that s symmetrc about zero, ntegrates to one, and Z z 2 K(z)dz = 0; = 1; 2; :::; then K s termed a hgher order kernel. It can be shown that the bas n a densty estmator based on ths kernel s O h 2(1+) : otce that unlke regular kernels, hgher order kernels must take on negatve values. 2 There are other alternatve methods that control for the bas under regular kernels. For examle, Powell and Honore (2005) emloy the followng jackknfe aroach. Let ^(h) be an estmator based on the wndow arameter h. Then, under ths aroach the nal estmator s a lnear combnaton of such estmators usng d erent wndows. In contrast, here we emloy a two-stage aroach that exlots a result due to Whtney ewey (see Theorem 0 n secton 2) to ensure asymtotc normalty 2

3 estmator erforms well n nte samles, an mortant feature for ts use n alcatons. It should be remarked that smlar bas control mechansms can be used for lkelhood-based estmators n double ndex models (see Klen, Shen, and Vella, 2009). Such models are mortant and naturally arse when one or more of the exlanatory varables s endogenous. For an alcaton n the context of healthcare decsons, see Shen (2009). When, as n the examles just mentoned, the model s characterzed by more than one ndex, the moston of the sngle ndex assumton wll result n an nconsstent estmator for the condtonal exectaton of nterest. 3 Gven the senstvty of the estmator to a sngle ndex assumton and gven ts wde use, a second objectve of ths aer s to formulate a test for ths assumton. There have been aers n the lterature on testng arametrc aganst semarametrc models (e.g., Härdle, Mammen, and Müller (1998), Härdle, Sokony, and Serlch (1997), Horowtz and Härdle(1994)). Related tests of arametrc models are gven by ewey (1985), Berens (1990), and Härdle and Mammen (1993). Ths aer d ers from those above n that t formulates a test for a man assumton n semarametrc models. We note that n a lkelhood context wth a arametrc null hyothess, ewey develos condtonal moment tests that have otmal local ower roertes. It may be ossble to extend these results to the resent context, but ths extenson s beyond the scoe of the current aer. Some aers focus on testng sngle ndex restrctons. Escancano and Song (2007) rovde a test focusng on average margnal e ects and show that t has a mnmax roerty; Andrews (1993) rovdes hgh level condtons for testng moment restrctons. Our aer d ers from these n that we rovde rmtve condtons for a condtonal moment test and for the estmator on whch t s based. Trathy and Ktamura (2003) emloy an emrcal lkelhood aroach for testng moment condtons of the form: E [G(z; )jx] = 0: Wth G a known functon and X contnuous, they establsh an otmalty roerty for ther test. The test roosed here s also an orthogonalty test n that we test whether a functon G s correlated wth functons M(X). However, unlke the above test, here the functon G wll be unknown as we set G = Y E(Y jv ), where V s an ndex and the condtonal exectaton functon E(Y jv ) s unknown. Further, the M-functons wll be unknown and we wll requre nonarametrc estmates of them. Ths feature s needed to ensure that the form of test statstc adats to the model under the alternatve hyothess. Most mortantly, the roosed statstc d ers from those n the lterature n the bas control mechansm that t emloys. Ths mechansm s smlar to that underlyng the estmator, and results under regular kernels. In addton, we also mlement a smoothng adjustment to the nal estmator. We nd that the resultng estmator erforms qute well n nte samles. It s an oen queston as to whether or not a further mrovement would be obtaned f we jackknfed our estmator. 3 As an alternatve examle of a double ndex model, return to the bnary resonse model dscussed earler, and let the error term have a condtonal varance that deends on another ndex. amely, let u = s(x o )", where " s ndeendent of X. 3

4 n a test statstc that has good sze and ower roertes n nte samles. In organzng ths aer, we begn by dscussng the moment condtons that characterze the estmator and the test statstc n Secton 2. These condtons ncororate methods for controllng ther bas usng regular kernels. Secton 3 contans assumtons and asymtotc results. Here, we wll also outlne the basc roof strategy, wth the Aendx contanng all formal and comlete roofs. In secton 5, we carry out Monte Carlo studes, where we evaluate the erformance of the estmators and test statstcs n nte samles. To revew the results, we nd that a bas corrected estmator based on regular kernels erforms the best and the bas-corrected form of the test statstc has good sze and ower roertes. 2 Moment Condtons and Bas Control 2.1 The Estmator In descrbng these condtons and the nature of the bas controls, t wll be useful to have sml ed notaton for samle averages of the quanttes of nterest. For ths urose, de ne: hab X [A B ] =; ha=b =1 X [A =B ] = =1 Further, we use the " " symbol above a quantty of nterest to ndcate an estmator for t. Then, lettng V ( 0 ) V (X; 0 ) be a sngle ndex deendng on exlanatory varables, X, and on a vector of true arameter values, 0 ; assume that we are nterested n an extremum estmator for 0 : Let be a trmmng functon that controls for small denomnators n a manner that we wll make exlct below. In ths secton, for exostonal smlcty, we take ths trmmng functon as known. In the Aendx, we let ths functon deend on an estmated argument and show that t may be taken as known. Emloyng ths trmmng functon, consder estmators whose gradents have the followng structural form: Dh ^G ( 0 ) Y ^E(Y jv (0 ) ^W E ; where the weghtng functon, ^W ; has the form: ^W ^ (V ( 0 )) r ^E For SLS estmators (Ichmura (1993)), ^ (V ( 0 )) = 1: For a QMLE estmator for bnary resonse models (Klen and Sady (1993)), ^ (V ( 0 )) = 1= ^E h 1 ^E : For a QMLE estmator of ordered models (Klen and Sherman (2002)), the gradent conssts of a number of comonents, all of whch have the structure above. The weghts d er 4

5 above, but all consst of a functon of the ndex and the dervatve of a nonarametrc exectaton estmator. If the gradent, when normalzed by s asymtotcally dstrbuted as normal, then t s not d cult to show that the underlyng estmator of nterest has an asymtotc normal dstrbuton. Accordngly, n what follows, we focus on these gradent exressons. For such estmators characterzed by the gradent structure above, wrte the gradent as: ^G ( 0 ) = h ^A ( 0 ) ^B ( 0 ) ^A ( 0 ) ^B ( 0 ) D[Y E (Y jv 0 )] ^W E h ^E (Y jv0 ) E (Y jv 0 ) ^W For the rst term, wth the argument gven n the Aendx, t can be shown that: h! ^A ( 0 ) A ( 0 ) 0; A ( 0 ) h[y E (Y jv 0 )] W The second or B-comonent above contrbutes a bas to the estmator that we need to control n order to show that the gradent has an asymtotc normal dstrbuton. Below we wll de ne ^E (Y jv 0 ) as a rato of estmated functons: ^f=^g, each of whch converges to ts true lmtng value. We wll be able to show that: 4 h ^B ( 0 ) ^B S! 0; ^B S = = Dh ^f=^g E (Y jv0 ) ^W E (^g=g) Dh E ^f ^ge (Y jv0 ) W =g + o (1): Snce the above quantty s lnear n the estmated comonents ^f and ^g; t s ossble to control for the bas n ^B S by controllng for the bas n these estmated functons. Hgher order kernels are commonly emloyed for ths urose. In ths case a standard U- statstc rojecton argument, whch we rovde n the Aendx, mmedately rovdes the result: h ^B S B S! 0; B S = h[y E (Y jv 0 )] E [W jv 0 ] 4 ote that: Dh ^f=^g E (Y jv0 ) ^W! [(^g=g) 1]E 0 wth the rst and thrd term each convergng to zero at a rate somewhat below 1=2. We wll show n the Aendx that the overall or combned rate of the roduct s su cent to rovde the desred result. 5

6 Asymtotc normalty for the normalzed gradent now follows from a standard central lmt theorem. In usng hgher order kernels to control for the bas and delver ths result, t should be noted that such kernels can result n negatve densty estmates and (as s the case here) often do not erform as well as methods based on regular kernels that do not delver the desred large samle roertes. Here, we seek alternatve bas controls that delver the desred large samle results wth regular kernels. Recallng that the weght functon contans the dervatve of the exectaton functon, we exlot a roerty of ths dervatve due to Whtney ewey n the followng theorem: 5 Theorem 0: Wth V ( 0 ) V (X; 0 ) as a sngle ndex, assume the followng sngle ndex restrcton holds: E (Y jx) = E (Y jv ( 0 )) F ( V ( 0 )) Then: E [ r E (Y jv ()) j V ( 0 )] =0 = 0. Proof: Let () V ( 0 ) V () and observe that ( 0 ) = 0 and that r () = r V () : Then, emloyng the ndex restrcton and usng terated exectatons: E (Y jv ()) = E X [E (Y jv ( 0 )) j V ()] E X [F [V ( 0 )] j V ()] = E X [F [V () + ()] j V ()] G (V () ; ()) Let G k be the artal dervatve of G taken w.r.t. n the k th argument of G, k = 1,2. From the chan rule: r G (V () ; ()) j =0 = G 1 (V () ; 0) j =0 + G 2 (V ( 0 ) ; ()) j =0 = r F ( V ()) j =0 E [r F ( V ()) j V ( 0 )] =0 The roof now follows. From above, r E [Y jv ()] =0 behaves as an error comonent wth condtonal exectaton 0. As ths comonent enters multlcatvely nto the gradent, we exlot ts resdual-lke roertes as a bas control. To utlze ewey s result, return to the gradent dscussed above and let h H(V ) E ^f ^ge (Y jv0 ) j X 5 Ths result and ts roof were rovded to one of the authors n a rvate communcaton. The roof, whch s very short and can be found n Klen and Sherman (2002), s also rovded here. 6

7 Then, take an terated exectaton to obtan: h D h E E ^B S = E X E ^f ^ge (Y jv0 ) W =g j X + o(1) = he X (H(V )W =g) = he V f[h(v )=g]e [(W jv )]g If the trmmng functon, ; deends on X, t s not ossble to emloy ewey s result and obtan 0 for ths exectaton. If the trmmng deends on V, then ths exectaton would be zero by constructon. Based on the above observaton, we consder a mult-stage estmaton method. In the rst stage, we trm on X and obtan consstent estmates for the ndex arameters. Usng these arameter estmates, we construct an estmated ndex uon whch to base trmmng. In the second stage, we then trm on the bass of the (estmated) ndex rather than X. In so dong, the exected value of the gradent would be zero. However, such trmmng usets the consstency argument because t rovdes no rotecton for small denomnators outsde of a small neghborhood of the truth. To resolve ths roblem, we adjust exectatons as follows. Recallng that ^E = ^f=^g, de ne an adjusted exectaton as: ^f ^E a = ^g + Below, we wll de ne such that t vanshes radly n regons where g s bounded away from zero. In regons where g tends to zero, tends to zero very slowly. In ths manner, we are able to reserve the consstency argument and establsh asymtotc normalty for the gradent. 6 It s ossble to further mrove the erformance of the estmator n nte samles under a smoothng adjustment, but we defer dscusson of ths ssue untl Secton Test Statstcs In what follows, we wll rst consder a general test of moment condtons and then secalze t to the test for the sngle ndex restrcton. Consder test statstcs based on "resduals"of the form: G kt ( 0 ) h[y E(Y jv ( 0 )] W T k, k = 1; ::; K where W T k = W T k (X k ) s a vector of observatons on a functon of the k th exogenous varable, X k. De ne G T as a column vector wth G kt as the k th element. Under a null hyothess of nterest, H 0 ; we assume that the followng orthogonalty condton holds: E [G T ( 0 )] = 0; 6 A smlar strategy s emloyed n Klen and Sady (1993) so as to let trmmng deend on an estmated densty. That aer, however, reles on hgher order kernels or local smoothng to obtan large samle results. 7

8 In testng whether or not these condtons hold, we allow the condtonal exectaton E(Y jv ) and the weght W k to be unknown functons that can be estmated nonarametrcally. Accordngly, wrte the estmated k th moment as: Dh E ^G k ^ Y ^E(Y jv ^ ^WT k : Wth a test statstc based on these estmated moments, we wll need to show that ^GkT ^ has an asymtotc normal dstrbuton under the null hyothess of nterest. Emloyng a standard Taylor exanson and wth ^ as a consstent estmator that has an asymtotc lnear reresentaton 7, we wll be able to wrte ^ ^GkT = ^G kt ( 0 ) + re [G kt ( 0 )] h^ 0 + o (1) : As the second or arameter-uncertanty comonent oses no d culty, here we focus on the rst comonent and dscuss the nature of the bas control that we emloy. Wth V o V ( 0 ), we wll be able to decomose these moment condtons n the same form as the gradent for the estmator above and wrte ^GkT ( 0 ) = h A T ( 0 ) ^BT ( 0 ) + o (1) A T ( 0 ) [Y E (Y jv 0 )] W Dh T k E ^B T ( 0 ) ^E (Y jv0 ) E (Y jv 0 ) ^WT k As for the estmator, here the second or B-comonent contrbutes a bas that we seek to control. As above, we wll be able to show: h ^BT ( 0 ) ^BS! 0; ^B S = = Dh ^f=^g E (Y jv0 ) ^WT k (^g=g)e Dh E ^f ^ge (Y jv0 ) W T k =g + o (1) Usng hgher order kernels to control for the bas n ^f and ^g, n a standard U-statstc argument, whch s rovded n the Aendx, we can show: h! ^BS B 0; B = h[y E (Y jv 0 )] E [W k jv 0 ] 7 The estmators we consder are all of the form: h^ 0 = Ho 1 hg ; where H 0 s the Hessan matrx, and hg s asymtotcally dstrbuted as (0; ): 8

9 The moment condton n large samles now has a form to whch a central lmt theorem would aly under the null hyothess. amely: ^GkT ( 0 ) = h[y E (Y jv 0 )] [W k E [W k jv 0 ]] + o (1) Takng estmaton uncertanty nto account, the "full" gradent has the form: ^ ^GkT = ^G kt ( 0 ) + re [G kt ( 0 )] h^ 0 Lettng G ^ be the vector wth k th element ^G kt ^, the test statstc s then gven by a standard quadratc form: T ^ 0 G ^ 1 G ^ ; where ^ s a consstent estmator for the covarance matrx of G ^ : Varous alternatve estmators for ths covarance matrx wll be rovded below and examned n the Monte Carlo secton. As n the case for the estmator, we nd that the test statstc based on hgher order kernels can be domnated by one based on an alternatve bas control and regular kernels. Unfortunately, the weght need not and wll not here have the resdual roerty of the dervatve weght enterng the gradent for the estmator. Therefore, we roose to recenter the weght so that t has the same resdual-lke roerty as n the estmator case. amely, wth ^V ^ V de ne ^G (^) as a vector wth the k th element beng: ^G kt ^ Dh Y ^W k ^W k E h ^W k j ^V E ^E(Y jv ^ ^W k T ^G ^ 0 ^ 1 ^G ^ For the test statstc roosed below, we wll show that such recenterng rovdes a bas control that makes t ossble to emloy regular kernels and stll obtan the same large samle result obtaned under hgher order kernels. amely, we wll show that T s close n robablty to T, wth T havng a 2 dstrbuton. We nd below that T ; whch s based on ths alternatve bas control, has much better nte samle roertes than T. To secalze the above moment condtons and develo a corresondng test statstc (a quadratc form n the moments) for the sngle ndex assumton, we need to secfy the weght functon. A natural choce for ths functon would not be a functon of any artcular exogenous varable, but rather the full condtonal exectaton: 9

10 E (Y jx). In ths case, the exected moment condton becomes: E ( [Y E (Y jv )] E (Y jx) ) = E ( [E (Y jx) E (Y jv )] E (Y jx) ) = E [E (Y jx) E (Y jv )] 2 otce that ths exected moment condton s zero E (Y jx) = E (Y jv ) : The above weght would seem natural as the exected moment condton reduces to the dstance between nonarametrc and ndex exectatons. However, t s d cult to obtan reasonable estmates of the full condtonal exectaton W = E (Y jx) when the dmenson of X s large. We are therefore motvated to seek low dmensonal weghts that are close to ths full exectaton. Wth "close" de ned n a mean-squared error sense, low dmensonal weghts are gven by: W k = arg mn! E(W!) 2 jx k = E (Y jxk ) otce that ths weght deends on the actual form of the deendence of Y on X. In other words, t s adatve to the alternatve model. Ths roerty s desrable comared to xed weghts, because ntutvely t yelds better test ower by beng able to exbly cature d erent volatons of the null hyothess. Our Monte Carlo study comarng one common xed weght and our adatve weght con rms the above observaton. The xed weght we use s the quadratc weght. Detaled dscussons are n the Monte Carlo secton. 3 Assumtons, De ntons, and Results To obtan the above results, we requre standard assumtons on the data generatng rocess, smoothness condtons on unknown denstes, and gven sets over whch denstes are ostve. Assume: (A1) Observatons. Wth (Y ; X ) as the th observaton on the deendent and exlanatory varables, assume that (Y ; X ) s..d. Wth X as the xk matrx of observatons on the exlanatory varables (ncludng a column vector of ones), assume that X has full column rank wth robablty 1. (A2) Model. Under the null hyothess E(Y jx ) = E(Y jv ); V X 1 + X 2 0 ; where X 1 s contnuous and 0 s n the nteror of a comact arameter sace, : Furthermore, to smlfy arguments we assume that X s bounded. 8 In addton, V ar(y jx ) s bounded. 8 The assumton on X beng bounded s not necessary, but sml es several of the arguments. 10

11 H o os- (A3) Estmator Characterzaton. Under the null hyothess, wth tve de nte and G beng..d., the estmator for 0 sats es: (^ 0 ) = H 1 o 1=2 nx G + o (1); =1 E (G ) = 0; V ar(g ) = O(1): (A4) Contnuous Varable Densty. Wth X k as any of the contnuous X varables, denote g k (jy) as ts densty condtoned on Y = y. Denote r d g k (tjy) as the d th artal dervatve wth resect to t, wth r o g k (tjy) g k (tjy). Wth g k suorted on [a k ; b k ]: g k > 0 on (a k ; b k ); a k < a k < b k < b k jr d g k j = O(1) on [a k ; b k ]; d = 0; 1; 2; 3. (A5) Index Densty. Wth V X 1 + X 2 0, let g(x 1 jy; x 2 ) be the ndcated condtonal densty suorted on [a; b], Assume g > 0 on (a; b) jr d gj = O(1) on [a; b]; d = 0; 1; 2; 3. (A6) Tal Condton. Wth g y as the densty for the deendent varable, Y, assume that there exsts T such that for t > T and df > 4: g y (t) < 1=[ 1 + t 2 (df+1)=2 ]. The above assumtons are somewhat standard n the lterature. amely, the model must nclude a contnuous varable (A2) and denstes for contnuous varables and the ndex must be su cently smooth, as mled by (A4-5). otce that (A4-5) also sec es when densty denomnators become zero, whch facltate the trmmng strategy. To establsh unform convergence results for estmated exectatons, we requre a tal condton on the densty for the deendent varable, Y. Whle ths assumton can be made n terms of the number of nte moments for Y, here we drectly assume n (A6) that the densty has tals that are no thnner than those for a t-dstrbuton wth 11

12 d > 4. Addtonal wndow condtons wll be requred and are stated drectly n the Theorems for whch they are needed. To de ne the estmators and test statstcs, we wll also requre the de ntons below. (D1) Trmmng. Wth Z k as the th observaton on a contnuous varable, Z k ; k = 1; :::K, let ^ k 1 : ^ak < Z k < ^b k 0 : otherwse, ^ k^ k where ^a k and ^b k are resectvely lower and uer samle quantles for Z k : Wth X k as an exogenous varable, when Z k = X k ; we refer to ^ as X-trmmng and wrte ^ x = ^ ; wth ^V as the estmated ndex, when Z k = ^V ; k = 1; we refer to ^ as ndex-trmmng and wrte ^ v = ^ : In the case where a smooth trmmng functon s requred, de ne: (z; ) [1 + ex ( Ln()Ln() [z ])] 1 as a smoothed aroxmaton to an ndcator on z > : A smoothed ndcator on z [a; b] s then de ned as (z; a) (b; z) : (D2) Kernels. The kernel functon K(z) s termed regular f K(z) 0, R K(z)dz = 1, and K(z) = K( z). The functon K(z) wll be termed a (normal) twcng kernel f K(z) = 2 (z) z= 2 = 2: (D3) Exectatons. Wth h = O ( r ) and K j K [(z z j ) =h], the estmated condtonal exectaton wth wndow arameter r s denoted as ^E ^E (Y jz = z ) and s gven by: " # " # 1 X 1 X ^E Y j^ j K j = ^ + ^ j K j ( 1) h ( 1) h ^f =^g j6= The exectaton s referred to as beng: a) regular ( ^E) f ^ j = 1, ^ = 0, and K s a regular kernel. b) twcng f ^ j = 1, ^ = 0, and K s a (normal) twcng kernel (ewey, Hseh, and Robns (2004)). c) adjusted ( ^E a ) f ^ j = 1, K s regular, and wth ^q as a lower samle quantle, (e.g., 0.01) of ^g (z ) ; = 1; :::; : ^ h ^q h1 ^ ^a; ^b, 0 < < 1 12 j6=

13 (D4) Frst and Second Stage Estmators. 9 ^1 = arg max ^2 = arg max ^Q 1 ; ^Q1 1 2n ^Q 2 ; ^Q2 1 2n nx ^ x [Y E ~ (Y jv(x ; ))] 2 ; =1 nx ^ v [Y ^Ea (Y jv(x ; ))] 2 (D5) Smoothng Adjustment. Lettng ^H() be the Hessan w.r.t. ^Q2 ; and ^E be a regular exectaton wth wndow arameter r = 1=5, de ne: =1 ^B ^2 ^B ^2 = = nx ^ v ( ^E ^2 =1 nx ^ v ( ^E ^2 =1 E ^2 )r ^E ^2 E ^2 )r ^E ^2 Then, de ne an adjusted estmator as: 1 h ^ = ^2 ^H ^2 ^B ^2 ^B ^2 (D6) Test Statstcs. The test statstcs, T and T ; are de ned as above. As dscussed earler, we emloy a two-stage estmator (D4) so as to utlze ewey s result as a bas control. The rst stage of ths estmator requres X-trmmng (D1) and regular exectatons (D3), whle the second stage requres ndex-trmmng (D1) and adjusted exectatons (D3). We wll comare results under regular and hgher order kernels (D2). otce that the twcng kernel n (D2) s a hgher order kernel n that: Z = 2 2 Z = 2 2 z h2 2 (z) Z z= 2 = 2 dz [z= 2] 2 z= 2 = 2dz w 2 (w)dw = 0; w z= 2 In examnng the second stage estmator and the test statstc for varous desgns, we had one desgn where the nte samle bas for the estmator was sgn cantly 9 As dscussed earler, there are many d erent estmators to whch ths aer ales. We focus on varants of the SLS estmator so as to emloy the same estmator over desgns where the deendent varable s contnuous or dscrete. 13

14 larger than that for the other desgns. As a result, we found that the test statstc had oor sze roertes n ths case. The smoothng adjustment (D5) mroved the sze roertes of our test statstc sgn cantly n ths case by reducng the bas n the estmator. To exlan why ths adjustment "works", recall the de nton of ^A(0 ) ^B( 0 ) n the revous secton. Then, a standard Taylor exanson yelds: ^2 0 = ^H 1 ( + )( ^A( 0 ) ^B(0 )); + h^2 ; 0 : De nng an estmator wth an nfeasble adjustment as: ^I = ^ 2 ^H 1 + h ^B (0 ) ^B ( 0 ) ; then t mmedately follows that ^I 0 = ^H 1 + h ^A (0 ) ^B ( 0 ) : Ths nfeasble estmator s the same as ^ 2 ; excet the B-comonent now deends on an otmal exectaton estmator. As a result, we would exect t to erform better n nte samles. Below, we show that ths nfeasble estmator can be aroxmated by the feasble estmator based on the adjustment n (D5) n that: h^i ^ 2! 0: Begnnng wth the estmator, Theorem 1 below establshes consstency at both stages. Theorem 1: (Estmator Consstency). Wth df = 4 gven n (A6), set df= (1 df) : Denote ^ 1 and ^ 2 as the rst and second stage estmators resectvely and assume (A1-6). Base the rst-stage estmator on a regular exectaton (D3) wth wndow r 1 : 1=8 < r 1 < 1=6; 0 < r 1 < [1=2 ] = [ + "] Base the second-stage estmator on an adjusted exectaton (D3) wth adjustment arameter 0 < < 1 and wndow r 2 : 1=8 < r 2 < 1=6; and 0 < r 2 < [1=2 ] = [ (1 + a) + "] ; Then: ^1 o = o (1) ; ^ 2 o = o (1). The normalty arguments are very smlar for the estmator and the test statstc as they are both based on smlar moment condtons. After rovdng these results n Theorems 2-3 below, we wll outlne the common structure of the argument. 14

15 Theorem 2: (Estmator: Asymtotc Lnearty and ormalty). Assume (A1-6) and base the second stage estmator, ^ 2 ; on an adjusted exectaton (D3) wth adjustment and wndow arameters as gven n Theorem 1. Lettng G ( 0 ) r 0Q 2 ( 0 ), H 0 r 0Q 2 ( 0 ), and H 1 0 E a) : b) : ^ 1 o = o 1=4 ^2 0 h G0 G 0 0 H 1 0 : = H0 1 G (0 ) + o (1) c) : ^ 2 ^2 = o (1) d) : ^ 2 0 d! Z ~ ( 0; ) In the secal case when V ar(y jx ) = 2 o s constant, = 2 oh 1 0 : Theorem 3. (Test Statstc: Asymtotc ull-dstrbuton): Let ^M ^E Y j ^V ; ^Mk ^E (Y jx k ) ; ^MT ^E T Y j ^V ; where the rst two exectatons are regular wth wndow arameter r : 1=6 < r < 1=4 and the thrd s twcng wth wndow arameter r T : 1=8 < r T < 1=6: De ne: ^w k ^ k ^Mk ; ^w k ^w k ^E ^w k j ^V ; where the above exectaton s a regular wth wndow arameter r : 1=6 < r < r < 1=4. Denote ^T and ^T as the unscented and centered moments wth resectve k th elements: E D ^T k ^ = D^ v Y ^MT ^w k ; ^T k ^ = Y E ^M ^w k Then, wth " (Y M) ; under the null hyothess of a sngle ndex: ^ a) : ^T k = S k + o (1) ; S k = h"wk hr wk H 1 h ^ b) : ^T k ^T k ^ = o (1) c) : T 0 1 T! d X ~ 2 (K) ; T = ^T ^, ^T ^, where wth S k as the k th element of S: E[SS 0 ]: 0 G o To outlne the roof for Theorems 2b and 3a (other arts follow drectly), note that the moment condtons underlyng the estmator and the test statstc have the structure: E D^(Y ^M)^! = h^(y M)^! 15 D ^( ^M E M)^! :

16 Here ^! s an estmated weght vector whose form s gven above, deendng on whether the above moment condtons descrbe the estmator or the test statstc. Denote! as the lmtng value of the estmated weght. Then, for the rst comonent above, a mean-squared convergence argument s emloyed n the Aendx together wth a result from Pakes and Pollard (1989) to show that : h^(y M)^! = h(y M)! + o (1) : Wth o (1) of ^M = ^f=^g, n the Aendx we show that the second comonent s wthn D ( ^M E M)!^g=g = D ( ^f E ^gm)!=g : As a U-statstc, ths last term can be analyzed by conventonal rojecton arguments. Provded that ts exectaton tends to zero, ths term vanshes for the estmator and the centered test statstc. For the uncentered test statstc, t contrbutes recsely the term that makes t asymtotcally close to the centered form. As dscussed n secton 2, the above exresson wll have exectaton tendng to zero f arorate hgher kernels are emloyed or when the trmmng functon, ; deends only on the ndex: For both the estmator and the centered test statstc, regular kernels can be emloyed as ths last condton holds. 4 Monte Carlo Desgns and Results In ths secton Monte Carlo exerments are used to nvestgate d erent estmators and test varants. Frst, we use a Monte Carlo exerment to evaluate d erent estmators that are de ned n Secton 3. Our Monte Carlo study shows that the two-stage normal kernel estmator wth bas correcton has the smallest root mean-square error (RMSE). Therefore, we choose ths estmator to study the emrcal sze and ower of d erent varants of the test statstc. Second, n evaluatng the test statstcs T and T, we wll examne both bascorrected and regular forms of the test statstcs as de ned n Secton 3. Both test statstcs deend on an estmated covarance matrx, and we rovde results for two d erent estmates. Wth S de ned as n Theorem 3, the covarance matrx s gven by = E(SS 0 ); whch may be estmated by a samle analogue. Alternatvely, wth " Y E(Y jv ); note that the k th element of S has the form: S k = X =1 w k" hr w k H 1 0 X r E (Y jv ) " =1 Accordngly, elements of S wll deend on " 2 and condtonng on X, wrte: terms. Takng an terated exectaton = E [E(SS 0 jx)] 16

17 Gven the form of S k ; t can be shown that the nner exectaton deends on the varance of Y condtoned on the ndex. If ths condtonal varance s known to be constant, as t s n all but one of the desgns below, then t can be factored out of the above exectaton and drectly estmated as an average of squared resduals. We wll use the terms KCV (known constant condtonal varance) and UCV (unknown condtonal varance) to refer to these two covarance matrx estmates. Test statstcs wll be comuted and comared under these two covarance matrx estmators. Thrd, we comare the erformance of our adatve weght verson of the test statstc and the xed weght verson. Recall that the test statstc deends on a weght that s a functon of X k ; the k th exogenous varable enterng the model. The adatve or redctve weght s gven by w(x k ) = E(Y jx k ); whch s the otmal redctor of Y under quadratc loss. otce that ths weght has an unknown functonal form that s model deendent. In contrast, a xed weght has a known functonal form that does not deend on the alternatve. The xed weght we use n our Monte Carlo study s the common quadratc weght w(x k ) = Xk 2. The Monte Carlo exerment con rms that the adatve weght verson of the test s robust. amely, n some desgns the two versons erform smlarly, however, n other desgns the adatve weght strongly domnates the xed one. 4.1 Desgns All of the desgns have sngle ndex structures under the null hyothess. For each desgn, the alternatve does not satsfy a sngle ndex assumton. Under the alternatve, the rst desgn s a double ndex model wth contnuous deendent varables; the second desgn s a bnary resonse model wth ndex heteroscedastcty; the thrd desgn s a double ndex model wth dscrete deendent varables; the last two desgns are general lnear models wth no ndex structure. In all the desgns, we normalze such that E(Y jx ) has standard devaton 2 under the null and alternatve models. In the rst (basc) desgn, we use the followng data generatng method for the null hyothess: Y = M 0 + " ; M 0 _ (X 1 + X 2 ) 2 ; where the X0s ~ 2 (1) and "~(0; 1): Under the alternatve: Y = M 1 + ", M 1 _ [(X 1 + X 2 ) 2 + (X 1 X 2 ) 3 ]: The second desgn s a bnary resonse desgn. Wth " beng..d. (0; 1), under the null of a sngle ndex model: 1 : M0 > " Y = 0 : otherwse, M o _ X 1 + X 2 0:5 Unlke the revous two desgns, here the two X0s are correlated. In artcular, the X0s are lnear combnatons of the same 2 varable and d erent normal shocks. 17

18 The alternatve model ntroduces heteroscedastcty, wth M 1 " relacng " above and wth M 1 _ 1 + (X 1 X 2 ) 2. We normalze M 0 and M 0 =M 1 so that they have exectaton zero and standard devaton 2. Ths desgn s the only one that does not have a constant condtonal varance. Snce dscrete ndeendent varables are very common n ractce, the thrd desgn has a dscrete regressor. The structure of the null and alternatve are the same as n the basc desgn, however, here X 2 s a bnary varable. In the fourth (general lnear model) desgn, under the null hyothess we generate data by: Y = M 0 + ", M 0 _ (X 1 + X 2 + 2X 3 ) 2 ; where the X0s ~ 2 (1) and "~(0; 1): Under the alternatve, whch has no ndex structure: Y = M 1 + ", M 1 _ [3X X X ]: A fth desgn s constructed to comare the roertes of our adatve weght verson of the test statstc and the xed weght verson. Ths desgn s d erent from the other four n a way that wll be exlaned below. Under the null hyothess we generate data by: Y = M 0 + ", M 0 _ (X 1 + X 2 + X 3 ) 2 ; where the X0s ~ (0; 1) and "~(0; 1): Under the alternatve, whch has no ndex structure: Y = M 1 + ", M 1 _ [X X X ]: For all the desgns, the samle sze we use s n=1000, and the number of Monte Carlo relcatons s We rovde results for theoretcal szes of 0.05 and For the estmator, there are a number of wndow and trmmng choces that need to be sec ed. Wth wndows havng the form h = O( r ); for the stage1 and stage2 estmators, we set r at 1/6.1. Wthn the range of ermssble values, the value gves the fastest ont-wse convergence rate of the estmated exectaton to the truth. For the smoothng adjustment, we select an otmal ontwse rate of 1/5. Fnally, for the twcng kernel, we set ths wndow at 1/7. In the case of trmmng, all trmmng s based on the.99 quantle for the relevant varables. Recall that n the second stage estmator, we adjust the denomnator of estmated exectatons. Here, we smoothly kee the ndex between the.005 and the.995 ndex quantles. Recall also that ths adjustment deends on a lower densty quantle, and we select the.01 quantle for ths urose. Fnally the adjustment deends multlcatvely on a wndow rased to the ower of ; 0 < < 1: In ths case, we set to be 1/2. For the test statstcs, wth one exceton gven below, we set the wndow arameter r to be 1/5 for the exectatons E(Y jv ) and M k = E(Y jx k ): The wndow arameter for E(M k jv ) s 1/7. The ndex-trmmng s set at.95, whle the X-trmmng s set at

19 4.2 Monte Carlo Results The rst ste s to use a Monte Carlo study to evaluate d erent estmators. The estmators studed are SLS estmators usng twcng kernels (SLS-TW), usng X-trmmng n the rst stage (S1SLS), and usng ndex-trmmng n the second stage (S2SLS). For each SLS varant there are two versons: havng smoothng correcton or not; the corrected ones have an extra "C" n front (e.g., CSLS-TW). Among the unadjusted estmators, n the general lnear model and basc desgns, S2SLS has an RMSE about 30% lower than that of the other estmators. The reducton n RMSE s smaller (8%) n the bnary resonse case and close to zero n the dscrete regressor desgn reorted here. Ths last ndng s desgn deendent and does not hold for other dscrete desgns. 10 In terms of RMSE, bas adjusted estmators are qute close to uncorrected ones n all the desgns excet n the dscrete regressor case, where t reduced RMSE by about 16% by cuttng the bas n half. 11 Essentally, the bas correcton makes lttle d erence when the bas n the uncorrected estmator s very small, but can have a large mact when ths bas s large. Hence our concluson would be that the bas corrected two-stage normal kernel estmator wth bas reducng structure s the best choce. Detaled results can be found n Table 1 Estmaton Results. ote that wth exceton of the dscrete regressor desgn, n all the other desgns the twcng kernel desgn are not reorted because there are severe outlers resultng n msleadng bas and varance values. After the CS2SLS estmator s chosen, we comare all the varants of test statstcs we mentoned n our Monte Carlo study, nvolvng known or unknown condtonal varance (KCV or UCV) and d erent bas reducng mechansms. The bas reducng mechansms we emloy are Twcng Kernel (TW), Regular Kernel usng a wndow r > 1 (BRR); and Recenterng. We nvestgate the emrcal sze, ower, and adjusted 4 ower, whch s the emrcal ower usng bootstra crtcal value adjustng the emrcal sze to be equal to the theoretcal sze. For reasons dscussed below, our Monte Carlo results recommend the centered BRR as the best among all those varants. In addton, n all cases where the condtonal varance s constant, t s better to mose ths nformaton. In comarng d erent varants of the test statstc, note rst that the recentered test statstcs erform much better than uncentered ones n that the emrcal szes are much closer to theoretcal value and ower s also better. The uncentered test statstcs n all the desgns have hghly n ated emrcal szes. As a result, the adjusted ower 10 Sec cally, we nterchanged quadratc and cubc comonents so that the condtonal mean functon was cubc under the null. For ths case, we found that the gan s substantal as s shown n the detaled table below: An Alternatve Dscrete Regressor Desgn SLS-TW CSLS-TW S1SLS S2SLS CS2SLS Bas Dscrete Regressor (Fled) Rvar Rmse The resultng bas reducton substantally mroved the erformance of the test statstc. 19

20 s substantally d erent from the unadjusted ower. For examle, turn to the general lnear model desgn. The recentered KCV BRR gves emrcal szes of and for 5% and 10% theoretcal szes; whle the uncentered verson gves and resectvely. The recentered test also has better ower roertes. Smlar results occur for the other desgns. Second, we comare results that deend on whether or not a known constant condtonal varance (KCV) assumton s correctly mosed n estmatng the covarance matrx. In all three desgns where ths assumton holds, the erformance of the test statstc s mroved. The szes are reasonable and smlar, but the ower of KCV s hgher than UCV. For examle, n the basc desgn, the ower of the UCV verson gves adjusted ower of only 0.7 and for 5% and 10% theoretcal szes; whle the KCV verson gves owers of and resectvely. ot surrsngly, a better test statstc results from mosng correct (constant condtonal varance) nformaton when estmatng the covarance matrx. As for kernel selecton, the results are qute close to one another. However, BRR s the most stable over desgns. For the dscrete regressor desgn, the recentered KCV wth BRR gves emrcal szes of and for 5% and 10% theoretcal szes; whle the corresondng ones for smle exectaton are 0.22 and 0.353; twcng kernel yelds and The ower s also slghtly better than the other two. The d erence among them n other desgns s often small. For examle n the general lnear model desgn our recentered KCV under BRR gves sze ower combnatons of (0.049, 0.817) and (0.097,0.872) for 5% and 10% theoretcal szes; whle corresondng exectaton by ndex gves (0.045,0.85) and (0.089, 0.889); twcng rovdes (0.045, 0.806) and (0.088, 0.863). As a concluson, the recentered test statstc usng BRR stands out among all the varatons we tred. It erforms well under all the desgns. Furthermore, when t s known that the condtonal varance s constant, such nformaton should be mosed. To comare xed wth adatve weghts, recall that the xed weghts are the squares of the exogenous varables that aear n the model, whle the adatve or redctve weghts are the otmal (MSE) redctors of Y: Wth the exceton of the fth desgn, all of the other desgn have mortant quadratc elements. As a result, the xed and adatve weghts exlan a comarable roortons of the varaton n the deendent varable n those desgns. ot surrsngly, n these cases we nd, but do not reort, that xed and adatve weghts erform smlarly. In the fth desgn, whch s gven above, quadratc elements are not mortant n the alternatve model. Even collectvely, such elements only exlan 3.5% of the varaton n Y. In contrast, collectvely the adatve weghts exlan 78.1%. Table 3 rovdes Monte Carlo results for the comarson between our adatve weght verson of the test and the xed weght verson. It s shown that our adatve weght test statstc domnates the xed weght verson n ths desgn by havng much better ower results. For examle, at the 5% theoretcal crtcal value anel, we nd the adatve weght verson of the recentered BRR test wth KCV has an emrcal ower of 0.962; whle the number for the xed weght 20

21 verson s much lower at Conclusons In summary, we have rst develoed an estmator that has desrable large samle roertes (consstency and asymtotc normalty), and that also erforms well n nte samles. We have obtaned these roertes by emloyng bas controls that make t ossble to base the estmator on regular kernels. These nte and large samle roertes are mortant n aled work and are also central to the erformance of test statstcs that deend on the estmator. Second, we have formulated a test statstc for testng the frequently made sngle ndex assumton n semarametrc models. We establsh the large samle dstrbuton of the test statstc under the null hyothess and show that t erforms well across a varety of desgns n Monte Carlo exerments. Ths erformance s obtaned by an embedded bas control mechansm, the adatve nature of the test statstc, and also the estmator uon whch t s based. 21

22 Table 1. Estmaton Results Basc Desgn S1SLS S2SLS CS2SLS Bas Rvar Rmse Bnary Resonse Desgn S1SLS S2SLS CS2SLS Bas Rvar Rmse Dscrete Regressor Desgn SLS-TW CSLS-TW S1SLS S2SLS CS2SLS Bas Rvar Rmse General Lnear Model Desgn S1SLS S2SLS CS2SLS Bas Rvar Rmse

23 Table 2. Test Results Basc Desgn 5% theoretcal crtcal value 10% theoretcal crtcal value Uncentered Recentered Uncentered Recentered UCV KCV UCV KCV UCV KCV UCV KCV sze R ower adjusted ower sze TW ower adjusted ower sze BRR ower adjusted ower Bnary Resonse 5% theoretcal crtcal value 10% theoretcal crtcal value Uncentered Recentered Uncentered Recentered UCV KCV UCV KCV UCV KCV UCV KCV sze R ower adjusted ower sze TW ower adjusted ower sze BRR ower adjusted ower

24 Table 2. Test Results Contnued Dscrete Regressor Desgn 5% theoretcal crtcal value 10% theoretcal crtcal value Uncentered Recentered Uncentered Recentered UCV KCV UCV KCV UCV KCV UCV KCV sze R ower adjusted ower sze TW ower adjusted ower sze BRR ower adjusted ower General Lnear Model Desgn 5% theoretcal crtcal value 10% theoretcal crtcal value Uncentered Recentered Uncentered Recentered UCV KCV UCV KCV UCV KCV UCV KCV sze R ower adjusted ower sze TW ower adjusted ower sze BRR ower adjusted ower

25 Table 3. Comarson of Fxed Weght and Adatve Tests Quadratc Weght Test 5% theoretcal crtcal value 10% theoretcal crtcal value Uncentered Recentered Uncentered Recentered UCV KCV UCV KCV UCV KCV UCV KCV sze R ower adjusted ower sze TW ower adjusted ower sze BRR ower adjusted ower Adatve Test 5% theoretcal crtcal value 10% theoretcal crtcal value Uncentered Recentered Uncentered Recentered UCV KCV UCV KCV UCV KCV UCV KCV sze R ower adjusted ower sze TW ower adjusted ower sze BRR ower adjusted ower

26 References Ahn, H. (1997), Semarametrc Estmaton of a Sngle-Index Model wth onarametrcally Generated Regressors, Econometrc Theory 13, Bhattacharaya, P.K. (1967), Estmaton of a Probablty Densty Functon and ts Dervatves. Indan Journal of Statstcs Seres A, Berens, Herman J., (1990), A consstent condtonal moment test of functonal form. Econometrca 58, Clmov, D., M. Delecrox & L. Smar (2002), Semarametrc estmaton n sngle ndex Posson regresson: a ractcal aroach, Journal of Aled Statstcs 29, Delgado, M A. & J. Mora (1995), onarametrc and semarametrc nference wth dscrete regressors. Econometrca 63, Delgado, M A. & T. Stengos (1994), Semarametrc sec caton testng of nonnested econometrc models. Revew of Economc Studes 61, Fraga, M. & O. Martns (2001), Parametrc and semarametrc estmaton of samle selecton models: an emrcal alcaton to the female labour force n Portugal, Journal of Aled Econometrcs 16, Ger n, M. (1996), Parametrc and Sem-arametrc Estmaton of the Bnary Resonse Model of Labor Market Partcaton, Journal of Aled Econometrc 11, Gorgens, T. (2000), Semarametrc Estmaton of Sngle-Index Transton Intenstes, Econometrc Socety World Congress 2000 Contrbuted Paers 0596, Econometrc Socety Gorgens,T. & J. L. Horowtz (1999), Semarametrc Estmaton of a Censored Regresson Model wth an Unknown Transformaton of the Deendent Varable, Journal of Econometrcs 90, Härdle, W. & E. Mammen (1993), Comarng nonarametrc versus arametrc regresson ts. Annals of Statstcs 21(4), Härdle, W., E. Mammen & M. Müller (1998), Testng arametrc versus semarametrc modellng n generalzed lnear models. Journal of Amercan Statstcal Assocaton 93, Härdle, W., V. Sonkony & S. Serlch (1997), Semarametrc sngle ndex versus xed lnk functon modellng. Annals of Statstcs 25, Hoe dng, H. (1963), Probablty Inequaltes for Sums of Bounded Random Varables, Journal of the Amercan Statstcal Assocaton 48, Honore, B. E. & J. L. Powell, Parwse D erence Estmaton of onlnear Models, n D. W. K. Andrews and J. H. Stock, eds., Ident caton and Inference n Econometrc Models. Essays n Honor of Thomas Rothenberg (Cambrdge: Cambrdge Unversty Press, 2005), Horowtz, J. L. & V. G. Sonkony (2001), An adatve, rate-otmal test of a arametrc mean-regresson model aganst a nonarametrc alternatve, Econometrca 69,

27 Horowtz, J. L. & W. Härdle (1994), Testng a arametrc model aganst a semarametrc alternatve. Econometrc Theory 10, Ichmura, H. (1993), Semarametrc least squares(sls) and weghted SLS estmaton of sngle-ndex models, Journal of Econometrcs 58, Klen, R. W. (1993), Sec caton tests for bnary choce models based on ndex quantles, Journal of Econometrcs 59, Klen, R. W. & R. H. Sady (1993), An E cent Semarametrc Estmator for Bnary Resonse Models, Econometrca 61, Klen, R. W. & F. Vella (2007), Estmatng a class of trangular smultaneous equatons models wthout excluson restrctons, manuscrt. Klen, R. W., C. Shen & F. Vella (2009), Jont Bnary Selecton and Treatment Models, manuscrt. ewey, W. K. (1985), Maxmum lkelhood sec caton testng and condtonal moment tests, Econometrca 53, ewey, W. K., F. Hseh & J. Robns (2004), Twcng Kernels and a Small Bas Proerty of Semarametrc Estmators, Econometrca 72, Pakes, A. & D. Pollard, (1989), Smulaton and the asymtotcs of otmzaton estmators, Econometrca 57, Powell, J. L., J. H. Stock & T. M. Stoker (1989), Semarametrc Estmaton of Weghted Average Dervatves, Econometrca 57, Ser ng, R. J. (1980), Aroxmaton Theorems of Mathematcal Statstcs, ew York: John Wley & Sons. Shen, C. (2009), Determnants of Healthcare Decsons: Insurance, Utlzaton and Exendtures, manuscrt. Trathy, G. & Y. Ktamura (2003), Testng Condtonal Moment Restrctons, Annals of Statstc 31, Aendx 6.1 Man Results In the roofs of Theorems 1-2 below we rovde roofs for the large samle roertes of the second stage estmator, wth the argument for the rst-stage estmator beng smlar but shorter as t s based on a regular exectaton. In so dong, we smlfy notaton by not subscrtng objectve functons, gradents, and hessan exressons. Proof of Theorem 1. (Consstency: ^ 2 ). De ne: h 2 ^Q () ^ v Y ^f=^g ; Q () v [Y f=g] 2 Then, recallng (D3), lettng ^ v ^f =^g f =g ; and " Y f =g : ^Q () Q () C + S + T 27