Which Moments to Match? Durham NC USA. Phone: September Last Revised September 1995

Transcription

1 Which Mmets t Match? A. Rald Gallat Departmet f Ecmics Uiversity f Nrth Carlia Chapel Hill NC USA Phe: Gerge Tauche Departmet f Ecmics Duke Uiversity Durham NC USA Phe: September 1992 Last Revised September 1995 Supprted by the Natial Sciece Fudati. We thak Laura Baldwi, Ravi Basal, Jh Clema, ad Athy Smith fr helpful discussis ad tw referees fr very useful cmmets. 1

2 Ruig head: Which Mmets t Match? Crrespdig authr: A. Rald Gallat Phe: r Departmet f Ecmics FAX: Uiversity f Nrth Carlia r gallat@uc.edu CB# 3305, 6F Garder Hall ftp: ftp.ec.duke.edu, ( ), Chapel Hill NC USA user:aymus, directry: pub/arg 2

3 Abstract We describe a ituitive, simple, ad systematic apprach t geeratig mmet cditis fr GMM estimati f the parameters f a structural mdel. The idea is t use the scre f a desity that has a aalytic expressi t dee the GMM criteri. The auxiliary mdel that geerates the scre shuld clsely apprximate the distributi f the bserved data but is t required t est it. If the auxiliary mdel ests the structural mdel the the estimatr is as eciet as maximum likelihd. The estimatr is advatageus whe expectatis uder a structural mdel ca be cmputed by simulati, by quadrature, r by aalytic expressis but the likelihd cat be cmputed easily. 3

4 1 Itrducti We preset a systematic apprach t geeratig mmet cditis fr the geeralized methd f mmets (GMM) estimatr [27] f the parameters f a structural mdel. The apprach is a alterative t the cmm practice f selectig a few lw rder mmets a ad hc basis ad the prceedig with GMM. The idea is simple: Use the expectati uder the structural mdel f the scre frm a auxiliary mdel as the vectr f mmet cditis. This scre is the derivative f the lg desity f the auxiliary mdel with respect t the parameters f the auxiliary mdel. Thus, the mmet cditis deped up bth the parameters f the auxiliary mdel ad the parameters f the structural mdel. The parameters f the auxiliary mdel are replaced by their quasi-maximum likelihd estimates, which are cmputed by maximizig the pseud-likelihd f the auxiliary mdel. The estimates f the structural parameters are cmputed by miimizig a GMM criteri fucti. As see belw, the ptimal weightig matrix fr frmig the GMM criteri frm the mmet cditis depeds ly up the auxiliary mdel ad is easily cmputed. We call the auxiliary mdel the scre geeratr. The scre geeratr eed t ecmpass (est) the structural mdel. If it des, the the estimatr is as eciet as the maximum likelihd estimatr. Hece ur apprach esures eciecy agaist a give parametric mdel. If the scre geeratr clsely apprximates the actual distributi f the data, eve thugh it des t ecmpass it, the the estimatr is early fully eciet. The estimati ctext that we have i mid is e where a structural mdel dees a data geerati prcess fr the data. The key feature f this data geerati prcess is that it is relatively easy t cmpute the expectati f a liear fucti give values fr the structural parameters. A expectati may be cmputed by simulati, by umerical quadrature, r by aalytic expressis, whichever is the mst cveiet. Examples f this estimati ctext are the pael data mdels mtivatig the simulated methd f mmets apprach fpakes ad Pllard [40] ad McFadde [35]. Ather is the asset pricig mdel that mtivates the dyamic methd f mmets estimatr f Due ad Siglet [10]. I Secti 4 belw we preset three such situatis draw frm macre- 4

5 cmics, ace, ad empirical aucti mdelig. I these examples, the likelihd is dicult t cmpute, s maximum likelihd is ifeasible. Simulati ad mmet matchig thus aturally arise. As idicated, there is presumpti that the scre geeratr ecmpasses the structural mdel, althugh a rder cditi fr ideticati requires a miimal level f cmplexity f the scre geeratr. Uder weak regularity cditis, ur estimatr is rt- csistet ad asympttically rmal with a asympttic distributi that depeds up bth the structural mdel ad the scre geeratr. If there exists a lcal, smth mappig f the structural parameters it the parameters f the scre geeratr the the estimatr has the same asympttic distributi as the maximum likelihd estimatr uder the structural mdel. The asympttic thery f the estimatr subsumes situatis with strictly exgeus variables, where e cditis particular values f the explaatry variables. It als subsumes situatis with predetermied but t strictly exgeus variables, as is typical f statiary Markv data geerati prcesses. The mst geeral versi allws fr prcesses with time-depedet laws f mti ad depedece extedig it the ideite past. Secti 2 immediately belw presets the asympttic justicati f the estimatr. Secti 3 presets sme cadidate specicatis fr the scre geeratr. Secti 4 presets three prpsed applicatis, each f which is a substative empirical prject. 2 Thery Fr a stchastic prcess described by a sequece f desities p 1 (x 1 j); fp t (y t jx t ;)g 1 fr which expectatis f liear fuctis are easily cmputed by simulati, by quadrature, r by aalytical expressis, we derive a cmputatially cveiet GMM estimatr fr that uses the scres (@=@)lf t (y t jx t ;) frm ather sequece f desities f 1 (x 1 j); ff t (y t jx t ;)g 1 t geerate the mmet cditis. As a example, p 1 (x 1 j); fp t (y t jx t ;)g 1 might be a mdel that describes asset prices yt i terms f exgeus variables x t ad structural parameters ad f 1 (x 1 j); ff t (y t jx t ;)g 1 might be the sequece f desities f a GARCH prcess. The estimatr is csistet ad asympttically r- 5

6 mally distributed i geeral. It is fully eciet if the mdel p 1 (x 1 j); fp t (y t jx t ;)g 1 is smthly embedded (Deiti 1) withi the mdel f 1 (x 1 j); ff t (y t jx t ;)g 1 : The estimatr is attractive whe the desity f t (y t jx t ;) has a cveiet aalytical expressi whereas the desity p t (y t jx t ;) des t. Thrughut, the bserved data f~y t ; ~x t g are assumed t have bee geerated frm the sequece f desities p 1 (x 1 j ); fp t (y t jx t ; )g 2R (1) which istsay that detes the true value f the parameter i the mdel p 1 (x 1 j); fp t (y t jx t ;)g 1 2R where R detes the parameter space. The mdel f 1 (x 1 j); ff t (y t jx t ;)g 1 (2) (3) is called the scre geeratr. The variables y t ad x t ca be uivariate, multivariate, r have a dimesi that depeds t: The fuctial frm f the scre geeratr may act t exclude elemets f the vectr y t ; that is, the scre geeratr may dee a stchastic prcess fr ly sme elemets f y t : Whe we say that a prcess is time ivariat we mea that the desities f the prcess d t deped t; i which case the t subscripts the desities may be suppressed ad the dimesis f y t ad x t are xed. Writig p X N(0;V ) meas p (V 1=2 ),1 X L! N(0;I) where V =(V 1=2 )(V 1=2 ) 0 : Smthly embedded is deed as fllws. is said t be smthly embed- DEFINITION 1 The mdel p 1 (x 1 j); fp t (y t jx t ;)g 1 ded withi the scre geeratr f 1 (x 1 j); ff t (y t jx t ;)g 1 2R if fr sme pe eighbrhd R f there is a twice ctiuusly dieretiable mappig g : R! such that p t (y t jx t ;)=f t [y t jx t ;g()] t ;2;::: (4) fr every R ad p 1 (x 1 j) =f 1 [x 1 jg()] fr every R. We csider three cases. 6

7 CASE 1 All desities are time ivariat, the aalysis is cditial the bserved sequece f~x t g ; ad the data (~y t ; ~x t ) are a sample frm Q p(y tj~x t ; ). A example fr which these assumptis are apprpriate is the liear regressi mdel crss sectial data with idepedetly ad idetically distributed errrs. Fr this case, usig simulati t cmpute the GMM estimatr prpsed here requires N simulated sequeces f^y t g frm the desity Q p(y tj~x t ;). We impse Assumptis 1 thrugh 6 f [17, Chapter 3] bth p(yjx; ) ad f(yjx; ). CASE 2 (~yt ; ~x t ) All desities are time ivariat, the aalysis is ucditial, ad the data are a sample frm Q p(y tjx t ; )p(x 1 j ). A example fr which these assumptis are apprpriate is a autregressive prcess where x t is cmprised f L lagged values f y t. We impse Assumptis 1 thrugh 6 f [17, Chapter 7] bth p(yjx; ) ad f(yjx; ). I additi, (y t ;x t ) 1 t=,1 N is assumed t be statiary with jit desity p(y; xj); margial desity p(xj) = R p(y; xj) dy; ad cditial desity p(yjx; ) =p(y; xj)=p(xj): Similarly fr f(yjx; ): Fr this case, usig simulati t cmpute the GMM estimatr requires ly a sigle simulated sequece (^y ; ^x ) N frm the desity Q N p(y jx ;)p(x 1 j); geerated as fllws: Start at a arbitrary ^x 1 =(^y 0 ;;^y,l+1 ); simulate ^y 1 frm p(y 1 j^x 1 ;); put ^x 2 =(^y 1 ;:::;^y,l+2 ); simulate ^y 2 frm p(y 2 j^x 2 ;); ad s. S that ^x is plausibly a sample frm p(xj); eugh iitial simulatis are discarded fr trasiets t die ut ad the ext N simulatis are retaied as the sequece (^y ; ^x ) N CASE 3 (~yt ; ~x t ) Desities are t time ivariat, the aalysis is ucditial, ad the data are a sample frm Q p t(y t jx t ; )p 1 (x 1 j ). This framewrk des permit cditiig the iitial bservati x 1 ad cditiig exgeus variables. Cditiig x 1 is accmplished by lettig p 1 (x 1 j) put its mass a sigle pit. Cditiig exgeus variables w t is accmplished thrugh the depedece f p t (y t jx t ;)tby puttig p t (y t jx t ;)=p(y t jx t ;w t ;). A example fr which these assumptis are apprpriate is a liear regressi with xed regressrs w t ad lagged depedet variables x t : Fr Case 3, usig simulati t cmpute the GMM estimatr may require N simulated sequeces N f^yt ; ^x t g frm the desity Q p t(y t jx t ;)p 1 (x 1 j): Hwever, i the cmm case where the structural mdel is Case 2 ad the scre geeratr describes a asympttically strictly statiary prcess, a sigle simulated sequece as i Case 2 suces. We impse : 7

8 Assumptis 1 thrugh 6 f [17, Chapter 7] bth p t (y t jx t ;) ad f t (y t jx t ;). Our idea is t use the scres (@=@)lf t (y t jx t ;) (5) evaluated at the quasi maximum likelihd estimate ~ = argmax 2 1 X l f t (~y t j~x t ;) (6) t geerate GMM mmet cditis. The GMM mmet equatis are Case 1 m (; ~ )= 1 Case 2 m (; ~ )= Case 3 m (; ~ )= 1 ZZ X Z (@=@)lf(yj~x t ; ~ )p(yj~x t ;)dy (7) (@=@)lf(yjx; ~ )p(yjx; ) dy p(xj) dx (8) Z Z Y (@=@)lf t (y t jx t ; ~ ) p (y jx ;)dy p 1 (x 1 j)dx 1 (9) X These are the mmet cditis that dee the estimatr. I mst applicatis, aalytical expressis fr the itegrals will t be available ad simulati r quadrature will be required t cmpute them. If the itegrals are cmputed by simulati the the frmulas used i practice are Case 1 m (; ~ ) = 1 Case 2 m (; ~ ) = 1 N Case 3 m (; ~ ) = 1 X NX X 1 N NX (@=@)lf(^y t j~x t ; ~ ) (10) (@=@)lf(^y j^x ; ~ ) (11) 1 N NX (@=@)lf t (^y t j^x t ; ~ ) (12) We assume that N is large eugh that the Mte-Carl itegral apprximates the aalytical itegral t withi a egligible errr f the same srt as is made i cmputig ay mathematical expressi a digital cmputer. There are istaces where the itegral ca be cmputed t a give accuracy at less cst by quadrature. Quadrature rules have the geeric frm Case 1 m (; ~ ) = 1 Case 2 m (; ~ ) = 1 N X NX 1 N NX (@=@)lf(^y t j~x t ; ~ )W (^y t j~x t ;) (13) (@=@)lf(^y j^x ; ~ )W(^y ; ^x j) (14) 8

9 where W (^y ; ^x j) ad (^y ; ^x ) are the weights ad the abscissae implied by the quadrature rule. Of curse N is dramatically smaller fr quadrature rules tha fr Mte Carl itegrati. Quadrature fr Case 3 will be at t high a dimesi t be practical i mst applicatis. As t statistical thery, Cases 1 ad 2 are special cases f Case 3 s that thrughut the rest f the discussi we ca discuss Case 3 ad specialize t Cases 1 ad 2 as required. The radmess i m (; ~ ) is slely due t the radm uctuati f the maximum likelihd estimatr ~. Uder the regularity cditis impsed abve, there is a sequece fg such that m ( ;)=0;lim!1 ( ~, ) = 0 almst surely, ad p ( ~, ) N[0; (J ),1 (I)(J ),1 ]; (15) where J =(@=@ 0 )m ( ; ) ad I =Var " # 1 X p t (~y t j~x t ;) (@=@)lf (16) [17, Chapter 7, Therem 6]. Nte that I ad J are t radm quatities because we have assumed that either quadrature has bee emplyed t cmpute m (; ) r that N is as large as ecessary t make the average essetially the same as the expected value. Usig Taylr's therem p m ( ; ~ ) = p m ( ;)+[J + s (1)] p ( ~, ) (17) = (J )p ( ~, )+ p(1): This implies that p m ( ; ~ ) N(0; I ): (18) Thus, give a estimatr ~ I f I that is csistet i the sese that lim!1 ( ~ I,I )=0 almst surely, the GMM estimatr with a eciet weightig matrix is The cmputatis ecessary t estimate I ^ = argmi 2R m0 (; ~ )(~I ),1 m (; ~ ): (19) deped up hw well e thiks that the scre geeratr apprximates the true data geeratig prcess. If e is cdet that the 9

10 scre geeratr is a gd statistical apprximati t the data geeratig prcess the the estimatr ~I = 1 X h (@=@)lft (~y t j~x t ; ~ ih ) (@=@)lf t (~y t j~x t ; ~ i 0 ) : (20) ca be used. This estimatr ca als be used with Gaussia QMLE scres if the cditial mea ad variace fuctis are crrectly specied [6]. A suciet (but t ecessary) cditi is Assumpti 2. Aweaker assumpti that facilitates estimati f I is the fllwig. ASSUMPTION 1 There isa such that Z Z Y (@=@)lf t (y t jx t ; ) p (y jx ; )dy p 1 (x 1 j ) dx 1 =0 (21) fr every t : Case 2 will always satisfy Assumpti 1 because f statiarity ad time ivariace. Thus, it is a assumpti that ly aects Cases 1 ad 3. Fr Case 1, the estimatr abve, ~I = 1 X h (@=@)lft (~y t j~x t ; ~ ih ) (@=@)lf t (~y t j~x t ; ~ i 0 ) ; (22) retais its csistecy uder the weaker Assumpti 1. where ad Fr Cases 2 ad 3, the fllwig estimatr is csistet uder Assumpti 1. ~S = 8 >< >: w(x) = ~I = 8 >< >: [ 1=5 ] X =,[ 1=5 ] w [ 1=5 ]! ~S (23) 1,6jxj 2 +6jxj 3 if 0 x 1 2 (24) 2(1,jxj) 3 if 1 x P h + (@=@)lft (~y t j~x t ; ~ ih ) (@=@)lf t, (~y t, j~x t, ; ~ i 0 ) if 0 ( ~ S;, ) 0 if <0 [17, Chapter 7, Therem 5]. See Adrews [1] fr alterative suggestis as t apprpriate weights ad rates e might use istead f w(x) ad 1=5. The Parze weights suggested 10 (25)

11 abve guaratee the psitive deiteess f I ~ which is essetial. Weights that d t guaratee psitive deiteess cat be used. If e is uwillig t accept Assumpti 1 the the estimatr ~I is mdied as fllws. First cmpute the iitial estimatr # = argmi 2R m0 (; ~ )m (; ~ ): (26) Cmpute ~ t = Z Z (@=@)lf t (y t jx t ; ~ ) Y p (y jx ; # )dy p 1 (x 1 j # ) dx 1 (27) usig the itegrati methds described abve. Fr Case 1 use the estimatr ~I = 1 X h (@=@)lft (~y t j~x t ; ~ ih ), ~ t (@=@)lft (~y t j~x t ; ~ i 0: ), ~ t (28) Fr Case 3 use the frmula ~I = [ 1=5 ] X =,[ 1=5 ] w! ~S [ 1=5 (29) ] with ~ S abve mdied t read X h (@=@)lft (~y t j~x t ; ~ ), ~ t ih (@=@)lft, (~y t, j~x t, ; ~ ), ~ t, i 0 (30) ~S = 1 + fr 0. It is ulikely that this geerality will be ecessary i practice because the use f this frmula meas that e thiks that the scre geeratr is a pr statistical apprximati t the data geeratig prcess which is ulikely t be true fr the fllwig reass. The scre geeratr is cceptually a reduced frm mdel, t a structural mdel. Thus it is rdiarily easy t mdify it by addig a few parameters s that it ts the data well. The situati where e thiks the scre geeratr is a pr apprximati might arise i hypthesis testig, but eve the the ull hypthesis will usually imply either Assumpti 1 r Assumpti 2 ad the geerality is, agai, uecessary. Therem 1 gives the asympttic distributi f ^ : THEOREM 1 Fr Case 1, let Assumptis 8 thrugh 11 f [17, Chapter 3] hld. Fr Cases 2 ad 3, let Assumptis 8 thrugh 11 f [17, Chapter 7] 11

12 hld. The lim ^ = a.s. (31)!1 p (^, ) N 0;[(M ) 0 (I ),1 (M )],1 (32) lim ( ^M, M!1 )=0 a.s. (33) where ^M = M (^ ; ~ );M =M ( ; ); ad M (; ) =(@=@ 0 )m (; ): Prf Apply Therems 7 ad 9 f [17, Chapter 3] fr Case 1 ad Therems 8 ad 10 f [17, Chapter 7] fr Cases 2 ad 3. Make these assciatis: = ; = ;m ()=m (; ~ ); m () =m (; ); ad S =Var[ p m ( )] = I : 2 The ideticati cditi m (; )=0 ) = fr all larger tha sme ; (34) is amg the regularity cditis f Therem 1. The situati is aalgus t vericati t the rder ad rak cditis f simultaeus equatis mdels. The rder cditi is that the dimesi f must exceed the dimesi f. Hwever, due t liearity, aalytic vericati f the aalg f the rak cditi, which is that the equatis m (; )=0 d t have multiple slutis fr 2 R; is dicult. See [16] fr discussi ad examples. It is usually adequate t rely the ptimizati prgram used t cmpute argmi 2R m 0 (; ~ )( I ~ ),1 m (; ~ ) t idicate the at spts the surface m 0 (; ~ )( I ~ ),1 m (; ~ ) that suggest ideticati failure. Fr example, the parameters f the mixig prcess f a stchastic vlatility mdel (see Subsecti 4.2), require third ad furth rder mmet ifrmati fr ideticati. Usig the scre f a Gaussia vectr autregressi will t prvide this ifrmati. We have actually de this iadvertetly by settig sme tuig parameters erreusly i a cmputati ad leared f the errr by the behavir f the ptimizer with respect t the parameters f the mixig prcess. Direct use Therem 1 fr settig cdece itervals the elemets f ^ r testig hyptheses with the Wald test requires cmputati f M (; ): This is prbably easiest t d by savig the trial values f ad m(; ~ ) geerated ver the curse f the ptimizati that cmputes argmi 2R m 0 (; ~ )( I ~ ),1 m (; ~ ), ttig the lcal quadratic regressis m i = b i + b 0 i(, ^ )+(,^ ) 0 B i (,^ ) fr i = 1;2;:::;dim() t the elemets f 12

13 m (; ~ ) at pits ear ^ ; ad takig ^M t be the matrix with rws b 0 i : Cmputati f M (; ) ca be avided by testig hyptheses usig the criteri dierece test statistic [17, Chapter 7, Therem 15] ad settig cdece itervals byivertig it. Uder Assumpti 1 the cditi HV H 0 = HJ,1 H 0 f [17, Chapter 7, Therem 15] will be satised. It is imprtat t te that we have t, as yet, made use f a assumpti that the scre geeratr f 1 (x 1 j); ff t (y t jx t ;)g 1 yet impsed the fllwig assumpti. ctais the true mdel. That is, we have t ASSUMPTION 2 There isa such that p t (y t jx t ; )=f t (y t jx t ; ) fr,2,... ad p 1 (x 1 j )=f 1 (x 1 j ). Because Assumpti 2 implies that the scre geeratr is a crrectly specied mdel, it implies Assumpti 1 ad the fllwig stadard results frm the thery f maximum likelihd estimati: Z (@=@)lf t (y t jx t ; )p t (y t jx t ; )dy t =0 (35) fr t ;2;::: Z (@=@)lf 1 (x 1 j )p 1 (x 1 j )dx 1 =0 (36) ad Z Z h ih i (@=@)lfs (y s jx s ; ) (@=@)lf t (y t jx t ; 0 ) whe t 6= s: These results allw use f the estimatr ~I = 1 Y p (y jx ; )dy p 1 (x 1 j ) dx 1 =0 X h (@=@)lft (~y t j~x t ; ~ ih ) (@=@)lf t (~y t j~x t ; ~ i 0 ) i Cases 1, 2, ad 3. Mrever, Z fr,2,..., s that Z h (@=@)lft (y t jx t ; ) ih (@=@)lf t (y t jx t ; ) i 0 Z Z =, (@ 2 =@@ 0 )lf t (y t jx t ; ) Y Y (37) (38) p (y jx ; )dy p 1 (x 1 j ) dx 1 p (y jx ; )dy p 1 (x 1 j ) dx 1 (39) p ( ~, ) N[0; (I ),1 ] (40) 13

14 Nw let us examie the csequeces f the smthly embedded assumpti (see Deiti 1 abve). ASSUMPTION 3 The mdel p 1 (x 1 j); fp t (y t jx t ;)g 1 the scre geeratr f 1 (x 1 j); ff t (y t jx t ;)g 1 : 2R is smthly embedded withi Assumpti 3 implies Assumpti 2. Mrever, the csistecy f ^ implies that ^ is tail equivalet [17, p. 187] t a GMM estimatr btaied by ptimizig ver the clsure f R istead f ver R: Therefre, withut lss f geerality, wemay assume that the twice ctiuusly dieretiable fucti g give by Deiti 1 is deed ver R: Let G() =(@=@ 0 )g(); G =G( );ad ^G = G(^). A csequece f Assumpti 3 is that the miimum chi-square estimatr ^ mcs = argmi 2R h ~, g() i 0 ( ~ I ) h ~, g() i (41) is as eciet as the maximum likelihd estimatr fr p 1 (x 1 j); fp t (y t jx t ;)g 1 this, rst te that : T see p (^mcs, ) N 0; [(G ) 0 (I)(G )],1 : (42) Nw, if ^ mle detes the maximum likelihd estimatr fr f 1 [x 1 jg()]; ff t [y t jx t ;g()]g 1 the [17, Chapter 7, Secti 4]. Because p (^mle, ) N 0; [(G ) 0 (I )(G )],1 (43) f 1 [x 1 jg()]; ff t [y t jx t ;g()]g 1 = p 1 (x 1 j); fp t (y t jx t ;)g 1 ; (44) ^ mle is als the maximum likelihd estimatr fr the prcess p 1 (x 1 j); fp t (y t jx t ;)g 1 If g cat be cmputed, the miimum chi-square estimatr is t practical. : Hwever, E( ~ j) ca be cmputed by simulati ad the remarks abve suggest that miimum chi-square with E( ~ j) replacig g() wuld be a practical, fully eciet estimatr. See Gurierux, Mfrt, ad Reault [25] ad Smith [41] fr examples. The diculty with this apprach is that the simulated miimum chi-square estimatr is cmputatially ieciet relative t the GMM estimatr prpsed here because at each f the NMte Carl repetitis i the expressi E( ~ j) =(1=N) P N ~ a ptimizati t cmpute ~ is required. 14

15 The GMM estimatr requires ly the e ptimizati t cmpute ~ ad avids the N extra ptimizatis required t cmpute E( ~ j). Mrever, e wuld actually have t ivke Assumpti 2 r estimate J t fllw this apprach. See [25] fr additial remarks the relatiships amg varius appraches. We cclude this secti by shwig that ^ has the same asympttic distributi as ^ mle. THEOREM 2 Assumpti 3 implies Prf Frm the rst rder cditis p (^, ) N 0;[(G ) 0 (I )(G )],1 : (45) 0 = (@=@) h m 0 (^ ; ~ )(~I ),1 m (^ ; ~ ) i (46) = 2 h (@=@)m 0 (^ ; ~ ) i ( ~ I ),1 m (^ ; ~ ) wehave, after a Taylr's expasi f m (^ ; ~ ); h ( ^M ) 0 (~I ),1 ( M ) i p (^, )=, h (^M ) 0 (~I ),1 (@=@)m i p( ~, ) (47) where the verbars idicate that the rws f M (; ) =(@=@ 0 )m (; ) ad (@=@ 0 )m (; ) have bee evaluated at pits the lie segmet jiig (^ ; ~ )t( ; ). Recall that ^M ad M idicate evaluati f M (; ) at (^ ; ~ ) ad ( ; ) respectively. Nw lim!1 ( M, M ) = 0; lim!1( ^M, M ) = 0; lim!1[,(@=@)m,i ] = 0; ad lim!1 ( ~ I,I ) = 0 a.s. Further, p ( ~, ) N[0; (I ),1 ]: (48) Therefre, the equati abve ca be rewritte as h ( ^M ) 0 ( ~ I ),1 ( M ) i p (^, )=(M ) 0p ( ~, )+ p (1) (49) which implies that p (^, ) N 0; h (M )0 (I ),1 (M )i,1: (50) 15

16 We cmplete the prf by shwig that M = I G : M = (@=@ 0 ) 1 X Z Z (@=@)lf t (y t jx t ; ) X Z Z Y Y p (y jx ;)dy p 1 (x 1 j) dx 1 j = = (@=@ 0 ) 1 (@=@)lf t (y t jx t ; ) f [y jx ;g()] dy p 1 (x 1 j) dx 1 j = = 1 X Z Z X (@=@)lf t (y t jx t ; ) (@=@ 0 )f s [y s jx s ;g()] = = 1 = 1 = 1 = X Y 6=s X Z X s + 1 X Y 6=s X Z s X Y X 6=s s X = I G X Y s f (y jx ; )dy dy s p 1 (x 1 j ) dx 1 Z Z Z (@=@)lf t (y t jx t ; ) Y f (y jx ; )dy (@=@ 0 )p 1 (x 1 j)j = dx 1 (@=@)lf t (y t jx t ; )[(@=@ 0 )f s (y s jx s ;)j =g() G()] = f (y jx ; )dy dy s p 1 (x 1 j ) dx 1 0(@=@ 0 )p 1 (x 1 j)j = dx 1 Z Z [(@=@)lf t (y t jx t ; )][(@=@)lf s (y s jx s ; )] 0 f s (y s jx s ; )G f (y jx ; )dy dy s p 1 (x 1 j ) dx 1 Z Z [(@=@)lf t (y t jx t ; )][(@=@)lf s (y s jx s ; )] 0 f (y jx ; )dy p 1 (x 1 j ) dx 1 G Z Z [(@=@)lf t (y t jx t ; )][(@=@)lf t (y t jx t ; )] 0 Y f (y jx ; )dy p 1 (x 1 j ) dx 1 G (51) 2 3 Geeral Purpse Scre Geeratrs As pited ut i Secti 2, if a mdel f 1 (x 1 j); ff t (y t jx t ;)g 1 2 is kw t accurately describe the distributi f the data f~y t g the that mdel shuld be the scre geeratr 16

17 that dees m (; ~ ) ad ^. If t, we ca suggest tw geeral purpse scre geeratrs. The rst is the SNP scre which ca be expected t clsely apprximate ay liear Markvia prcess. A example f its use i cecti with the estimatr ^ prpsed here is Basal, Gallat, Hussey, ad Tauche [3] wh t a geeral equilibrium, tw-cutry, metary mdel usig high frequecy acial market data. The secd is the eural et scre which ca be expected t clsely apprximate ay crss-sectial liear regressi r ay dyamic liear autregressi, icludig determiistic chas. A example f its use i cecti with ^ is Eller, Gallat, ad Theiler [11] wh use data widely believed t exhibit chatic dyamics t calibrate the parameters f the SEIR mdel, which is a mdel f epidemics fte used i health ecmics. The cited applicatis ctai descriptis f the SNP ad eural et scres, respectively. I terms f cveiece, what e wuld like is fr Z (@=@)lf t (y t jx t ; )p t(y t jx t ; )dy t (52) t be small eugh that ~ t ca be put t zer with little eect up the accuracy f the cmputati f ~ S ad small eugh that ~S 6= 0 (53) ca be put t zer with little eect up the accuracy f the cmputati f ~ I : The estimatr f I wuld the assume its simplest frm X h (@=@)lft (~y t j~x t ; ~ ih ) (@=@)lf t (~y t j~x t ; ~ i 0 ) : (54) ~I = 1 Bth SNP ad eural ets are series expasis that have the prperty that (52) ca be made arbitrarily small by usig eugh terms i the expasi [20, 23]. Hece, ~ t ad (53) ca be made arbitrarily small by usig eugh terms. The apprpriate umber f terms relative t the sample size are suggested by the results f Fet ad Gallat [15] ad McCarey ad Gallat [34]. Hwever, there is as yet geeral thery givig the rate at which terms ca be added s as t retai p -asympttic rmality s e must guard agaist takig t may terms ad the claimig that stadard asympttics apply. 17

18 4 Applicatis We discuss three classes f applicatis f the estimatr develped i the previus sectis. I the setup fr each applicati, it is relatively simple t geerate simulated realizatis frm the structural mdel while cmputati f the likelihd is ifeasible. Hece simulati ad mmet matchig are apprpriate estimati strategies. 4.1 Csumpti ad Asset Returs i a Prducti Ecmy Csider the fllwig versi f the Brck-Mirma e-sectr setup. The represetative aget's prblem is subject t 1 max E t [ 1, 1X i=0 i c 1, t+i v 2;t+i] (55) c t + k t+1, k t Ak t v 1t (56) where c t is csumpti at time t; k t the capital stck at the begiig f perid t (that is, iherited frm perid t, 1); v 1t ad v 2t are strictly psitive shcks t techlgy ad prefereces; E t () is shrt-had fr the cditial ifrmati give all variables i the mdel dated time t ad earlier; the parameters satisfy 0 <<1;0;A>0;ad 0 <1. The aget's chice variables at time time are c t ad k t+1. The stchastic prcess v t =(v 1t ;v 2t ) 0 is strictly statiary ad Markvia f rder r; with cditial desity (v t+1 jv t ;);where v t =(v0 t ;:::;v0 t,r )0 ad is a parameter vectr. The Euler equati fr this prblem is The sluti f the ptimizati prblem is where c ad k are the plicy fuctis. c, t = E t (c, t+1ak,1 t+1 ) (57) k t+1 = k (k t ;v t) (58) c t = c (k t ;v t) (59) There is kw clsed frm sluti fr the plicy fuctis, thugh the plicy fuctis ca be well-apprximated usig e f the ewly develped methds fr slvig liear ratial expectatis mdels. The 1990 sympsium i the Jural f Busiess ad 18

19 Ecmic Statistics [42] surveys may f the extat methds. Fr this mdel, ad the prpsed applicati, the methd f Clema [8], which uses quadrature fr umerical itegrati ad cmputes the plicy fucti ver a extremely e grid, is prbably the mst accurate ad umerically eciet. Usig Clema's methd t evaluate the plicy fuctis, e ca the easily simulate frm this mdel. Give a iitial value k fr the capital stck, ad a simulated realizati f^v g geerated frm (vjv;); e geerates simulated f^k ; ^c g by recursively feedig the ^v ad ^k thrugh the plicy fucti fr capital. Gd practice is t allw the iteratis t ru fr a lg while i rder t let the eects f the trasiets wear. A simulated realizati f legth N; f^k ; ^v g N ; wuld be the last N values f the iteratis. Strategies t implemet empirically the crrespdig cmpetitive equilibrium f this mdel dier depedig whichvariables are used t cfrt the mdel t data, that is, which variables eter the scre geeratr. Fr example, with gd data bth csumpti ad capital, the researcher culd use (c t ;k t ) 0.Hwever, if capital is prly measured but utput well measured, the it wuld be better t use (c t ;q t ) 0 ; where q t is ttal utput, which ithe simulati wuld be cmputed as ^q = A^k ^v 1. Neither f these strategies, thugh, makes use f price data. A strategy that icrprates price ifrmati is t use c t alg with the returs a pure discut risk-free bd, r bt ; ad a stck, r st. Asset returs are determied via asset pricig calculatis, carried ut as fllws. (It turs ut t be a bit easier t thik f the equatis deig returs betwee t ad t +1:) The bd retur, r b;t+1 ; is the sluti t c, t = E t (c, t+1)(1 + r b;t+1 ); (60) ad r b;t+1 is kw t agets at time t. Fr the stck retur, the divided prcess is d st = Ak t v 1t, r b;t k t ; ad the stck price prcess fp st g is the sluti t the expectatial equati p st c, t = E t [c, t+1(p s;t+1 + d s;t+1 )] (61) The stck retur betwee t ad t +1isr s;t+1 =(p s;t+1 + d s;t+1 )=p st. Slvig fr the asset returs etails additial cmputati that culd ptetially be as umerically itesive as apprximatig the plicy fuctis. 19

20 This frmulati presumes that, i the cmpetitive equilibrium, the rm uses 100 percet debt acig t ret frm a husehld the capital stck k t+1 fr e perid at iterest rate r b;t+1. (Bth r b;t+1 ad k t+1 are determied ad kw ad time t:) The rm distributes t the husehld as the divided d s;t+1 = Ak v t+1 1;t+1,r b;t+1 k t+1 ; which is the rm's cash w i perid t + 1;that is, the prceeds after payig the bdhlder. Other cceptualizatis are pssible, ad, i particular, the stck price ad returs prcess culd be dieret if the rm retais earigs r uses dieret frms f debt acig. Oe typically des t bserve a risk-free real bd retur. Cmm practice i empirical asset pricig is t use the csumpti series alg with either the real ex-pst retur the stck (deated usig a price idex) r the excess f the stck retur ver the bd retur, r et = r st, r bt ; frm which iati cacels ut. This practice presumes that the bserved data cme frm a metary ecmy with exactly the same real side as abve ad a mial side characterized by a bidig cash-i-advace cstrait, which implies uitary metary velcity. We shw hw t implemet the estimatr data csistig f csumpti ad the excess stck retur. This is de fr illustrative purpses. The prpsal ers a alterative t the stadard SMM strategy f selectig ut a set f lw-rder mmets, as i Gette ad Marsh [24], fr estimati f a asset pricig mdel. I actual practice, e wuld wat t emply mre sphisticated versis f the mdel with time separabilities i csumpti ad prducti ad als iclude additial latet taste ad/r techlgy shcks whe additial asset returs are bserved. Cmm practice i stchastic mdelig is t iclude suciet shcks r measuremet errrs t preclude the predicted distributi f the data frm beig ccetrated a lwer dimesial maifld, which is rmally cuterfactual, ad the mdel beig dismissed ut f had immediately. Put y t =(r et ;c t ) 0. Let =(; A; ; 0 ) 0 dete the vectr f structural parameters. The umerical sluti f the mdel prvides a meas t simulate data give a value f : Experiece with acial data suggests a reasable chice fr the scre geeratr is the sequece f desities deed by aarch [12] r GARCH prcess [13]. Fr ease f 20

21 expsiti, we shw the ARCH case. Csider the multivariate ARCH mdel y t = b 0 + vech( t ) = c 0 + XL 1 j XL 2 j B j y t,j + u t (62) C j vech(u t,j u t,j ) (63) where u t N(0; t ): Let pdf(y t jx t ; ) dete the implied cditial desity ffy t guder the ARCH mdel, where x t =(y 0 t,l ;:::;y0 t,1 )0 ;L=L 1 +L 2 ;ad =(b 0 ; vec([b 0 1 B 2 :: B L1 ]) 0 ; c 0 ;vec([c 0 1 C 2 :: C L2 ]) 0 ) 0 (64) Cmm practice i ARCH mdelig is t impse a priri restrictis, such as diagality r factr restrictis, s as t cstrai = () t deped a lwer dimesial parameter,. Let f(y t jx t ;)=pdf(y t jx t ; []) dete the ARCH cditial desity uder the restrictis, which we take as the scre geeratr. Give the bserved data set f~y t g ; the rst step i the estimati is t apply quasimaximum likelihd t the ARCH mdel The secd step is t estimate by ~ = argmax 2 1 X l f t (~y t j~x t ;) (65) ^ = arg mi 2R m0 (; ~ )(~I ),1 m (; ~ ) (66) where m (; ~ )=(1=N) P N (@=@)lf t(^y j^x ; ~ ); f^y g is a simulated realizati give frm the mdel, ad ^x =(^y 0 t,l ;:::;^y0 t,1) 0. The relevat asympttic distributi thery is that f Case 2 i Secti 2 abve. The rder cditi fr ideticati is that the legth f be at least as lg as the legth f. The aalgue f the rak cditi is give i the discussi fllwig Therem 1. It is exceedigly dicult t determie aalytically whether the ARCH scres suce t idetify the asset pricig mdel, which, as ted earlier, is typical f liear statistics. I practice, ear at spts i the sample bjective fucti wuld be a strg idicatr f failure f ideticati. I such a case, further expasi f the scre geeratr such as relaxig cditial rmality r usig a -Markv (GARCH) mdel culd brig i additial 21

22 scre cmpets t achieve ideticati. The mechaics f implemetig a GARCH-type scre geeratr are similar t that just described fr ARCH, thugh the tati is mre cumbersme. I either case, use f this estimatr prvides a meas t brig t bear the task f selectig mmets the kwledge that ARCH-GARCH mdels t returs data well. Ather pssible scre geeratr wuld be t use the SNP mdel f Gallat ad Tauche [21, 22]; this strategy is emplyed by Basal, Gallat, Hussey, ad Tauche [3] fr estimati f a mdel f weekly currecy market data. Use f a SNP mdel wuld give the exercise a mre parametric slat, as the chice f dimesi f the scre geeratr mdel wuld be data determied. Either chice wuld esure eciecy agaist a class f mdels kw t capture much f the rst ad secd mmet dyamics f asset prices ad ther macr aggregates. 4.2 Stchastic Vlatility Csider the stchastic vlatility mdel y t, y = c(y t,1, y ) + exp(w t )r y z t w t = aw t,1 + r w ~z t The rst equati is the mea equati with parameters y ;c;ad r y ; the secd is the vlatility equati with parameters a ad r w. fy t g is a bserved acial returs prcess ad fw t g is a ubserved vlatility prcess. I the basic specicati, z t ; ad ~z t are mutually idepedet iid N(0; 1) shcks. The mdel ca be geeralized i a bvius way t accmmdate lger lag legths i either equati. Versis f this mdel have bee examied by Clark [7], Meli ad Turbull [36]; Harvey, Ruiz, ad Shephard [30], Jacquier, Pls ad Rssi [32], ad may thers. The appeal f the mdel is that it prvides a simple specicati fr speculative price mvemets that accuts, i qualitative terms, fr brad geeral features f data frm acial markets such as leptkurtsis ad persistet vlatility. The cmplicatig factr fr estimati is that the likelihd fucti is t readily available i clsed frm, which mtivates csiderati f ther appraches. Gallat, Hsieh, ad Tauche [18] emply the estimatr f this paper t estimate the stchastic vlatility mdel a lg time series cmprised f 16,127 daily bservatis 22

23 f~y t g 16;127 adjusted mvemets i the Stadard ad Pr's Cmpsite Idex, The scre geeratr is a SNP mdel as described i Secti 3 abve. The specicati search fr apprpriate auxiliary mdels fr f~y t g 16;127 leads t tw scres: a \Nparametric ARCH Scre," whe errrs are cstraied t be hmgeeus, ad a \Nliear Nparametric Scre," whe errrs are allwed t be cditially hetergeeus. The Nparametric ARCH Scre ctais idicatrs fr bth deviatis frm cditial rmality ad ARCH. Tgether, these scres suce t idetify the stchastic vlatility mdel; ideed, the stchastic vlatility mdel places veridetifyig restrictis acrss these scres. The Nliear Nparametric ARCH Scre ctais additial idicatrs fr cditial hetergeeity, mst imprtatly, the leverage type eect f Nels [38] which is a frm f dyamic asymmetry. These additial idicatrs idetify dyamic asymmetries like thse suggested by Harvey ad Shephard [31], which the Nparametric ARCH Scre des t idetify. Whe tted t either f these tw scres, the stadard stchastic vlatility mdel fails t apprximate the distributi f the data adequately; it is verwhelmigly rejected the chi-square gdess f t tests. After alterig the distributi f z t t accmmdate thickess i bth tails alg with left skewess ad geeralizig the vlatility equati t iclude lg memry Harvey [29], the stchastic vlatility mdel ca match the mmets deed the simpler Nparametric ARCH Scre, but t thse deed by the Nliear Nparametric Scre. Itrducig crss-crrelati betwee z t,1 ad ~z t as i Harvey ad Shephard [31] imprves the t t the Nliear Nparametric Scre substatially, but still the stchastic vlatility mdel cat t that scre. Overall, Gallat, Hsieh, ad Tauche [18] d the estimati prvides a cmputatially tractable meas t assess the relative plausibility f a wide class f alterative specicatis f the stchastic vlatility mdel. They shw hw t use the scre vectr f a rejected mdel t elucidate useful diagstic ifrmati. There are ther -gig applicatis f the estimatr i the ctext f stchastic vlatility. Egle [14] emplys it t estimate a ctiuus time stchastic vlatility mdel, with the scre geeratr beig a GARCH mdel tted t the discrete time data. Ghysels ad Jasiak [26] use it t estimate a ctiuus time mdel f stck returs ad vlume subject t time defrmati like that f Clark [7] ad Tauche ad Pitts [43]. Their scre geeratr is a a SNP mdel very similar t that f Gallat, Rssi, ad Tauche [19] tted t the discrete 23

24 time returs ad vlume data. 4.3 Empirical Mdelig f Aucti Data Auctis are cmmly used t sell assets. Game theretic mdels f auctis prvide a detailed thery f the mappig frm the disparate values that bidders place the asset t the al utcme (the wier ad the sales price). The predictis f this thery deped strgly the assumptis regardig the characteristics f the aucti ad the bidders. Geerally, the specic rules f the aucti alg with the ifrmati structure, the attitudes f the bidders twards risk, ad the bidders' strategic behavir, all matter a great deal i determiig the al utcme [37]. Empirical implemetati f game theretic mdels f auctis lags well behid the thery. The extreme liearities ad umerical cmplexity f aucti mdels presets substatial bstacles t direct implemetati. Tw recet papers, Paarsch [39] ad Lat, Ossard, ad Vug [33] make substatial prgress, hwever. I bth papers, the task is t estimate the parameters f the distributi f values acrss bidders. Paarsch develps a framewrk based stadard maximum likelihd. His apprach ca hadle a variety ifrmatial evirmets, but is restricted t a relatively arrw set f parametric mdels fr the valuati distributi essetially the Paret ad Weibull. Lat, Ossard, ad Vug use a simulati apprach, ad they ca thereby hadle a much brader class f valuati distributis. Hwever, their apprach impses ly the predictis f the thery regardig rst mmets ad igres higher rder structure, which ca cause prblems f ieciecy ad ideticati. The methd set frth i Secti 2 impses all restrictis ad geerates a eciet estimate f the valuati distributi. I what fllws, we illustrate hw e wuld implemet the methd fr sme f the simpler mdels f auctis. A full empirical study wuld g much further ad, i particular, wuld relax ur strg assumptis ad csider ther evirmets kw t be theretically imprtat. We rst prvide a shrt verview f sme f the simplest aucti mdels ad the prceed t the ecmetrics. 24

25 Tw Aucti Mdels Uder Idepedet Private Valuatis A item, such as a tract f lad r stad f timber, is t be sld at aucti. The item will be sld s lg as a sellig price at least as large as a reservati price r 0 > 0 is realized; therwise, it is left usld. There are tw cmmly used aucti desigs. I a Oral Ascedig Aucti, the sellig value f the item starts at r 0 ad the icreases. Bidders drp ut as the sellig value rises util e bidder remais, wh pays the sellig value at which the last f the ther bidders drpped ut. I a Sealed Bid First Price aucti, all bids are cllected simultaeusly. The bject is sld t the highest bidder wh pays his bid s lg as it exceeds the reservati price. The idepedet private value paradigm is a set f assumptis regardig the characteristics f bidders; the paradigm is applied t either type f aucti. I this paradigm, each f B bidders is assumed t have a private valuati, v i ;i;2;:::;b; fr the item t be sld. Each bidder kws his w private valuati but des t kw the valuati f ther bidders. The bidders act as if the B valuatis are iid drawigs frm a cmm valuati distributi H(vjq; ); with desity h(vjq; ); where q is a vectr f cvariates deig characteristics f the item t be sld ad is a parameter vectr. Each bidder kws q; ; the fuctial frm f H(vjq; ); ad the reservati price, r 0. Als, each bidder is assumed t be risk eutral ad the equilibrium ccept is the symmetric Bayesia Nash equilibrium. Fr the ral ascedig aucti, the wiig bid, y; is y = max[v (B,1:B) ;r 0 ]I(v (B:B) r 0 ) if B 2 (67) y = r 0 I(v 1 r 0 ) if B (68) where v (1:B) v (B:B) are the rder statistics f v 1 ;:::;v B ; ad I() is the zer-e idicatr fucti. O the evet v (B:B) <r 0 ;the wiig bid is deed as zer ad the item is usld. Let p a (yjr 0 ;B;q;); r simply p a (yjx; ) with x =(r 0 ;B;q); dete the cditial prbability desity f wiig bid. Belw, we write either p a (yjr 0 ;B;q;)rp a (yjx; ); depedig up whether we wish t emphasize depedece each f the dieret cmpets f x r t. I geeral, p a (yjx; ) is a rdiary desity the regi y > r 0 ; 25

26 s lg as h(vjq; ) is smth, while p a (yjx; ) has atms at y = 0 ad y = r 0. I certai circumstaces, fr example, h(vjq; ) isparet r Weibull as i Paarsch [39], p a (yjx; ) has a maageable clsed-frm expressi. I ther circumstaces, fr example, h(vjq; ) is lgrmal as i Lat, Ossard, ad Vug [33], ad p a (yjx; ) admits tractable expressi. Hwever, s lg as it is easy t simulate frm h(vjq; ) the it is easy t simulate frm p a (yjx; ): Fr the sealed bid rst price aucti, the wiig bid is y = Efmax(v (B,1:B) ;r 0 )jv (B:B) gi(v (B:B) r 0 ) if B 2 (69) y = r 0 I(v 1 r 0 ) if B (70) Thus whe there are tw r mre bidders ad v (B:B) r 0 ; the the wiig bid fllws the distributi f the cditial expectati f max(v (B,1:B) ;r 0 ) give v (B:B). Let p sp (yjr 0 ;B;q;);r p sp (yjx; ); dete the implied cditial desity f wiig bid i the sealed bid case. Geerally, p sp (yjx; ) is less maageable i practice tha is p a (yjx; ). Geerati f simulated draw frm p sp (yjx; ) etails either umerical itegrati f the cumulative distributi f the valuati distributi r a duble ested set f simulatis. A New Estimati Strategy fr Aucti Mdels Suppse a ecmetricia bserves f~y t ; ~x t g ; where ~y t is the wiig bid ad ~x t = (~r 0t ; B ~ t ; ~q t ) ctais the reservati price, the umber f bidders, ad cvariates fr each f auctis. I what fllws, we take the auctis t be ral ascedig auctis ad pit ut, where apprpriate, hw thigs dier fr sealed bid auctis. The ecmetricia assumes that the same valuati desity, h(vjq; ); describes the bidder valuatis fr each aucti. The aalysis is cditial (the x 0 s are strictly exgeus); the ecmetricia assumes that ~y t ad ~y s are statistically idepedet fr t 6= s; cditial the sequece f~x 1 ; ~x 2 ;:::;~x g. The task is t estimate the true uderlyig parameter vectr, : Oe estimati strategy is straight maximum likelihd. Uder special distributial assumptis the valuati desity such asweibull r Paret, the cditial desity f the wiig bid p a (yjr 0 ;B;q;) has a maageable clsed frm. Cvetial maximum 26

27 likelihd estimati ca the be udertake. This is the strategy f Paarsch [39]. Lat, Ossard, ad Vug [33] develp a Simulated Nliear Least Squares (SNLLS) estimatr which ca hadle a brader class f paret desities fr the valuati distributi. Their apprach is t apply liear least squares: X ~ = arg mi [~y t, a (~r 0t ; B ~ t ; ~q t ;)] 2 (71) 2R where a (~r 0t ; ~ B t ; ~q t ;)= R yp a (yj~r 0t ; ~ B t ; ~q t ;)dy. I practice, a (~r 0t ; ~ B t ; ~q t ;) is apprximated via Mte Carl itegrati: where v ; ~ Bt,1: ~ Bt a (~r 0t ; ~ B t ; ~q t ;) 1 N NX max(v ;( ~ B t,1: ~ B t) ; ~r 0t) (72) is the secd highest rder statistic f the th idepedet simulated realizati f (v 1 ;:::;v ~ Bt) iid frm h(vj~q t ;). I their mtivatig examples ad empirical applicatis, v is cditially lgrmal with a mea that depeds up ~q t ; ad ctais the parameters f this cditial lgrmal distributi. The SNLLS estimatr is liear least squares with a heterskedasticity-rbust estimate f the asympttic variace f ~ that accuts fr cditial heterskedasticity f t =~y t, a (~r 0t ; ~ Bt ; ~q t ; ) (73) Lat, Ossard, ad Vug te that reveue equivalece implies the same frmulati f the cditial mea fucti applies fr a sealed bid aucti. Reveue equivalece implies a (r 0 ;B;q;) = = Z Z yp a (yjr 0 ;B;q;)dy yp sb (yjr 0 ;B;q;)dy = sb (r 0 ;B;q;) (74) fr all r 0 ;B;q, ad. Hece e ca evaluate the cditial mea fucti at the data, i.e, cmpute sb (~r 0t ; B ~ t ; ~q t ;); by simulatig ad averagig exactly as e des uder ralascedig rules. The result ca be a sigicat reducti i cmputatial demads. The SNLLS apprach wrks f the cditial rst mmet implicatis ale, thugh, ad aucti mdels place additial structure the data. A aucti mdel has secd mmet implicatis as well as rst mmet implicatis. I fact, it actually dictates 27

28 the fuctial frm f the cditial heterskedasticity i the liear regressi equati, which suggests additial mmet cditis. There are practical csequeces frm t icrpratig additial restrictis beyd rst mmet ifrmati. Lat, Ossard, ad Vug [33] ad Baldwi [2] d it dicult t estimate the variace f the uderlyig paret lgrmal usig SNLLS. Brigig i secd mmet estimati ca be expected t alleviate this diculty. I geeral, there are further implicatis beyd rst ad secd mmets as well; impsiti f all implicatis f the mdel ca be expected t sharpe eve further the estimates f the parameter. Ideally, e wats t d this by dig maximum likelihd usig either p a (yjr 0 ;B;q;)r p sb (yjr 0 ;B;q;) as apprpriate t dee the likelihd. The diculty is that bth desities are itractable, except i the special circumstaces assumed by Paarsch. The apprach utlied Secti 2 ca cme clse t the maximum likelihd ideal. Our aalysis pertais t the just-described situati where the likelihd is smth, but itractable; it des t cver cases where the likelihd is dieretiable i parameters. The csistecy f the estimatr ^ is t aected by dieretiability but asympttic rmality may be. See [28, Appedix C] fr a discussi f dieretiability csideratis with respect t GMM estimatrs. The apprach wuld be applied t the aucti data as fllws. ~ is btaied as 1 ~ = arg max X lg(f[~y t j~x t ;] (75) where f(yjx; ) is a scre geeratr that gives a gd apprximati t the cditial distributi f y give the exgeus variables. Oe chice fr f(yjx; ) is a trucated Hermite expasi, r SNP mdel f Gallat ad Tauche [22]; which has bee fud i practice t be sucietly exible t apprximate well a wide class f desities. Nte that f(yjx; ) des t have t smthly embed p a (yjx; ); thugh if it des, the the estimatr is equally eciet as maximum likelihd. Our estimatr is GMM usig the scre fucti f the estimati t dee the mmet cditis: ~ = arg mi 2R m0 (; ~ )(~I ),1 m (; ~ ) (76) 28

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