Computatio Of Asymptotic Distributio For Semiparametric GMM Estimators Hideiko Icimura Departmet of Ecoomics Uiversity College Lodo Cemmap UCL ad IFS April 9, 2004
Abstract A set of su ciet coditios for computig te asymptotic distributio of estimators wic are de ed via momet coditios wit i ite dimesioal parameters are preseted. We te coditios old, te mai teorem reduces te computatio of te asymptotic distributio to computig limits of a few momets.
Itroductio Adrews (994), Newey (994), Serma (994) ad Ai ad Ce (2003) ave exteded te ite dimesioal asymptotic aalysis to iclude i ite dimesioal parameters ad clari ed te structure of te computatio of asymptotic aalysis greatly. However, we give a estimator, eiter teir framework is limited or te coditios put forward are ot ecessarily easy to verify. Tis paper presets a set of su ciet coditios for computig te asymptotic distributio of estimators wic are de ed via momet coditios wit i ite dimesioal parameters. Te coditios are oped to be easy to verify i may applicatios. We te coditios old, te mai teorem reduces te computatio of te asymptotic distributio to computig limits of a few momets. 2 Model Let R p, Z R k, X R d, ad for eac, (; ) be a fuctio from X ito R`. We cosider a mappig g (z; ; (; )) from Z (; (; )) 2 ito R m. Geerally we cosider a Baac space of fuctios o R d wit some properties suc as give degree of di eretiability ad assume tat g is well de ed over Z. Te orm o is deoted by kk. We te domai of suc fuctios are restricted to X, we deote it by (X ). We assume for eac 2, (; ) 2 (X ). For brevity we sometimes write istead of (; ) ad, we z is evaluated at z i, write g i (; ) istead of g (z i ; ; (; )). Ofte g i (; ) = g (z i ; ; ( 2 (z i ; ) ; )) for a ukow fuctio ito R` ad a kow fuctio 2 so tat g ca be regarded as a fuctio from Z R` ito R m. Te added geerality is useful to adle applicatios were is a coditioal expectatio of a ukow variable wic eeds to be estimated, for example. Te geerality is also useful i applicatios were idividuals decisios deped o te etire distributio of a variable, wic i tur is estimated. Tis is te case for idividual decisios i auctio models, for example, or more geerally ay decisio uder explicitly stated expectatio wic is to be estimated. Let G (; ) = g i (; ) :
Tis paper cosiders a set of su ciet coditios wic imply asymptotic ormality of a ite dimesioal compoet at te rate te square root of te sample size i a class of te geeralized metod of momet (GMM) estimator wic is de ed as a solutio to te followig problem: if G (; ^ ) T ^AG (; ^ ) 2 were ^ is a estimator of ad ^A is a m m matrix wic coverges i probability to a positive de ite matrix A. 3 Asymptotic Distributio Our approac is a direct applicatio of te stadard aalysis of te GMM estimators. Like te stadard aalysis, te basic result appeals to te Taylor s series expasio. Let B be a Baac space equipped wit orm kk B ad let kk be a orm o R K. First we state a Taylor s series expasio teorem for a geeral mappig F from a ope subset of space B ito R K. To state tis teorem, we rst eed to de e te cocept of Frécet di eretiability of a mappig from a ope subset O of a ormed space X ito aoter ormed space Y. Let kk X ad kk Y be te orms of X ad Y, respectively. De itio (Frécet Di eretiability) A mappig f : O! Y is Frécet di eretiable if ad oly if at x 2 O tere is a cotiuous liear operator L x suc tat for ay " > 0 tere exists " > 0 suc tat for ay kk X < " te followig iequality olds: kf (x + ) f (x) L x k Y " kk X : We write tis as f (x + ) f (x) L x = o () : Next we discuss secod order di eretiability of mappig f or di eretiability of L x. Note tat L x ca be regarded as a mappig from X ito L (X; Y ), a space of liear operators from X ito Y. Because X is a ormed space ad L (X; Y ) is a ormed space, we ca discuss Frécet di eretiability of tis mappig L x we f is Frécet di eretiable over O. 2 We L x See for example Kolmogorov ad Fomi (957), Fourt editio Capter 0. 2 For ay L 2 L (X; Y ) te orm of L (X; Y ) is de ed by sup klk Y : kk X 2
is Frécet di eretiable, we ave L x+ L x Q x = o () for some liear operator Q x. We regard tis liear operator as te secod derivative of f. Note tat Q x is a elemet of L (X; L (X; Y )). Because L (X; L (X; Y )) ca be ideti ed wit a space of biliear operators B X 2 ; Y via B (x ; x 2 ) = A (x ) x 2 ; were A 2 L (X; L (X; Y )) ad B 2 B X 2 ; Y, we will regard Q x as a elemet of B X 2 ; Y. Aalogously oe ca de e te t order derivative of mappig f ad will regard tem as a elemet of te space of te rt order liear operators R (X r ; Y ). From ow o, we will deote te derivative of f (x) by f 0 (x), te secod derivative by f 00 (x), te rt derivative by f (r) (x). As discussed above, for eac x, f 0 (x) is a liear operator, f 00 (x) is a biliear operator, ad i geeral f (r) (x) is te rt order liear operator ito Y. Tus for ay elemet 2 X, f 0 (x) (), f 00 (x) (; ), ad i geeral f (r) (x) (; :::; ) are all well de ed ad take values i Y. Usig tese otatios, we ca state te Taylor s series expasio teorem: See Kolmogorov ad Fomi (976). 3 Teorem 2 (Taylor s Series Expasio) Let F be a mappig from B ito R K ad let F be de ed over a ope subset O of B. If F (r) (x) exists for ay x 2 O ad is uiformly cotiuous, te F (x + ) = F (x)+f 0 (x) ()+ 2! F 00 (x) (; )+ + r! F (r) (x) (; :::; )+! (x; ) were k! (x; )k B = o (kk r B ). If te rt derivative satis es te Lipscitz coditio wit expoet > 0, te k! (x; )k B = O kk r+ B : We prove asymptotic distributio of te semiparametric GMM estimator uder te followig assumptios. Coditio 3 fz i g are idepedet ad idetically distributed. Coditio 4 ( 0 ; 0 (; ) is a iterior poit of f(; )g 2. 3 Capter 0, Teorem 2. 3
Coditio 5 g (z; ; ) is Frécet di eretiable wit respect to (; ) i ad te Frécet derivatives satis es te Lipscitz cotiuity coditios: for C j (z) > 0 E fc j (z)g < (j = ; 2; 3; 4) @g (z; ; ) =@ @g z; 0 ; 0 =@ R mp C (z) 0 R p + C 2 (z) 0 @g (z; ; ) =@ @g z; 0 ; 0 =@ L( ;L( ;R m )) C 3 (z) 0 R p + C 4 (z) 0 We sall deote te Frécet derivative wit respect to iclusive of te e ect of o by rg (z; ; (; )). Note tat rg (z; ; (; )) = @g (z; ; (; )) =@+@g (z; ; (; )) =@ @ (; ) =@: Coditio 6 sup 2 k@g (z; ; (; )) =@k L( ;L( ;R m )) +sup 2 k@g (z; ; (; )) =@k R mp C 0 (z) ad E fc 0 (z)g <. Coditio 7 E frg (z; 0 ; 0 (; )g rg is ite ad as full rak. Coditio 8 plim! ^A = A were A is symmetric ad positive de ite. Coditio 9 7! 0 (; ) as a mappig from ito 0. is cotiuous at We de e te cocept of asymptotic liearity. Let deote a sample size ad fr g be a determiistic sequece wic coverges to 0. We cosider a geeral estimator ^ of a elemet 0 i a Baac space B wit orm kk B. De itio 0 A statistic ^ i B is asymptotically liear for 0 i B wit te residual rate r if tere exist a stocastic sequece f i g wit i 2 B ad E ( i ) = 0 ad a determiistic sequece fb g wit b 2 B suc tat ^ 0 i b = o p (r ) : B I our applicatio, for eac i, typically i is a fuctio of some argumets. Coditio sup 2 k^ (; ) 0 (; )k = o p () ad tat ^ (; ) is asymptotically liear for 0 (; ) i wit rate =2. We impose te followig coditio as well: 4
Coditio 2 plim! 3=2 P @g i ( 0 ; =@ 0 i = 0. Coditio 3 plim! =2 P @g i ( 0 ; =@ 0 b = g b. Typically we will d coditios uder wic g b = 0. Uder te coditios, te term ca be bouded: =2 @g i ( 0 ; =@ 0 b R C 0 (z i ) p kb k : m Tus if p kb k = o (), te g b = 0. We also assume tat te oparametric estimator is smoot ad te derivative beaves as expected. Coditio 4 ^ (; ) is cotiuously di eretiable ad sup 2 k@^ (; ) =@ @ 0 (; ) @k = o p (). For asymptotic ormality, te two coditios above are eeded oly i te eigborood of 0. For cosistecy, owever, tat is ot eoug, but peraps ot as strog as te coditio above. Let i = i E f i g. Coditio 5 E f i g 2 o =4. ad tat P i + ke f i gk = Let H = (rg) T A (rg) ad deote te expectatio coditioal o z i by E fjz i g. Let = V ar g (z ; 0 ; + E @g (z ; 0 ; =@ 0 2jz + E @g (z 2 ; 0 ; =@ 0 jz ad for ay c 2 R p, c = lim! c 0 H (rg) T A ArGH c. Teorem 6 Suppose ^ is cosistet to 0. Uder te coditios above, for ay c 2 R p for wic c is positive ad ite, p c ^ 0 0 coverges i distributio to a ormal radom variable wit mea 0 ad variace c. Te proof makes use of te followig two lemmas. rg ( 0 ; 0 (; ) = o p () i te eigbor- Lemma 7 rg (; (; )) ood 0 ad 0 (;. 5
Proof. To see tis, just ote tat krg (; (; )) rg ( 0 ; 0 (; )k R mp krg (; (; )) rg ( 0 ; (; ))k R mp + krg ( 0 ; (; )) rg ( 0 ; 0 (; )k R mp [C (z i ) k 0 k R p + C 3 (z i ) k@ (; ) =@k k 0 k R p] + C 2 (z i ) [k (; ) 0 (; )k + k 0 (; ) 0 (; k ] + C 4 (z i ) [k (; ) 0 (; )k + k 0 (; ) 0 (; k ] k@ (; ) =@k + C 0 (z i ) [k@ (; ) =@ @ 0 (; ) @k + k@ 0 (; ) =@ @ 0 (; @k ] Tis implies te result. Lemma 8 p G ( 0 ; ^ (; ) is asymptotically equivalet to @g (zi ; p 0 ; g (z i ; 0 ; ( ) + E @g (zj ; 0 ; @ 0 j jz i + E @ 0 i jz i Proof. By te Taylor s series expasio teorem for some R G ( 0 ; ^ (; ) = G ( 0 ; 0 (; ) + @g i ( 0 ; 0 (; ) @ 0 (^ (; 0 (; ) + R : We te Frécet derivative satis es te Lipscitz coditio wit expoet, te last term ca be bouded: jr j C 4 (z i ) k^ (; (; k 2 : Tus p jr j coverges to zero uder te coditios. Usig te asymptotic liearity of te oparametric estimator ^, te rst term of te rigt-ad side equals 0 @g i ( 0 ; ) @ 0 @ j= j + b A 6 +g b
wic, by exploitig te liearity of te Frécet derivative equals 2 + 2 + j= j6=i @g i ( 0 ; @ 0 j @g i ( 0 ; @ 0 i @g i ( 0 ; @ 0 b : Te secod term coverges to zero after multiplied by p uder te coditio. Te tird term multiplied by p coverges to g b uder te coditio. By te U-statistics cetral limit teorem, te rst term coverges wit te rate te square root of te sample size. To obtai te asymptotic variace formula, we compute te projectio: First by symmetrizatio we ave ( ) = times 2 ( ) j>i Te projectio is (for j 6= i) 2 = @g (zi ; 0 ; @ 0 j + @g (z j; 0 ; @ 0 i =2: @g (zi ; 0 ; E E @ 0 j + @g (z j; 0 ; @ 0 i @g (zi ; 0 ; @ 0 j jz i + E =2jz i @g (zj ; 0 ; @ 0 i jz i : Tus combiig wit te rst term, we obtai te result. Tus te asymptotic distributio is drive by @g (z ; 0 ; = V ar g (z ; 0 ; 0 (; ) + E @g (z2 ; 0 ; @ 0 2 jz + E @ 0 jz : Now we tur to a proof of te mai teorem. Proof. Let rg (; ^ ) = @G (; ^ ) =@ + @G (; ^ ) =@ @^ () =@ were @G (; ) =@ deotes te Frécet derivative of G (; ) wit respect usig te orm kk. Te te rst order coditio solves 0 = rg ^; ^ ; ^ i T ^AG ^; ^ ; ^ : 7
We cosider te expasio of G ^; ^ ; ^ at = 0 : By te stadard Taylor s series expasio teorem, G ^; ^ ; ^ = G ( 0 ; ^ (; ) + rg ( 0 ; ^ (; ) ^ 0 + rg ; ^ ; rg ( 0 ; ^ (; ) ^ 0 for some wic lies o a lie coectig ^ ad 0. After substitutio tis expressio ito te rst order coditio ad rearragig, we ave [rg ( 0 ; (; )] T ^A [rg ( 0 ; ^ (; )] ^ 0 = [rg ( 0 ; (; )] T ^AG ( 0 ; ^ (; ) + T + T2 + T3 were T = T2 = T3 = rg ^; ^ ; ^ T rg ( 0 ; 0 (; )i ^AG ( 0 ; ^ (; ) ; rg ^; ^ ; ^ T rg ( 0 ; 0 (; )i ArG ( 0 ; ^ (; ) ^ rg ^; ^ ; ^ i T A rg ; ^ ; rg ( 0 ; ^ (; ) ^ 0 ; ad 0 : Uder te coditio, clearly rg ( 0 ; 0 (; ) coverges to a full rak matrix rg. Te limit is @gi ( 0 ; rg = E 0 (; ) + @g i ( 0 ; 0 (; ) @ @ 0 @ 0 (; : @ Note tat equals ^ 0 o = rg ( 0 ; (; ) T ^ArG ( 0 ; ^ (; ) i rg ( 0 ; (; ) T ^AG ( 0 ; ^ (; ) + T + T2 + T3 Earlier lemma implies rg ^; ^ ; ^ rg ( 0 ; 0 (; ) = o p () ; rg ; ^ ; rg ( 0 ; ^ (; ) = o p () ; ad G ( 0 ; ^ (; ) rg ( 0 ; 0 (; ) = o p () : Tese also imply tat rg ( 0 ; (; ) T ^ArG ( 0 ; ^ (; ) coverges i probability to a ivertible matrix. 8
Multiply bot sides by p = + p R ^ 0 ad take te orm of p te left-ad side. Suppose p R ^ 0 diverges wit positive probability. Te wit te positive probability, te left-ad side coverges to. p Te rigt-ad side owever coverges to zero from te earlier lemma. Tus p R ^ 0 = O p (). p Next multiply bot sides wit p c 2 R p ad applyig te coditio implies te result. Tese calculatios clarify wat we eed to kow to compute te asymptotic distributio of a semiparametric GMM estimator. Tey are rg ad. 4 Applicatios To carry out tese computatios, we eed to d out te relevat Frécet derivatives ad kow wat te asymptotic liear expressios are for te oparametric estimators used i te estimatio. For te kerel desity estimators te followig are te expressios: i = d K zi z E d K zi z ad b = E d K zi z f (z) : To cotrol te bias, so tat te asymptotic liearity coditio olds wit rate =2, a certai type of kerel fuctio eeds to be used. Te followig iger order kerel by Bartlett (963) is a stadard device i te literature. Let j0 = if j = 0 ad 0 for ay oter iteger value j. De itio 9 K`, ` is te class of symmetric fuctios k : R! R aroud zero suc tat R tj k (t) dt = j0 for j = 0; ; :::; ` ad for some " > 0 lim k (t) = + jtj`++" < : jtj! Dimesio d kerel fuctio K of order ` is costructed by K (t ; :::; t d ) = k (t ) k (t d ) for k 2 K`. I order to improve te order of bias by te iger order kerel, te uderlyig desity is required to be smoot accordigly. Te followig otio of smootess is used by Robiso (988). Let [] deote te largest iteger ot equal or larger ta. 9
De itio 20 G, > 0, > 0, is te class of fuctios g : R d! R satisfyig: g is []-times partially di eretiable for all z 2 R d ; for some > 0, sup y2fky zkr d <g jg (y) g (z) Q (y; z)j = ky zk R d (z) for all z; Q = 0 we [] = 0; Q is a []-t degree omogeeous polyomial i (y z) wit coe ciets te partial derivatives of g at z of orders troug [] we [] ; ad g (z), its partial derivatives of order [] ad less, ad (z) ave ite t momets. Bouded fuctios are deoted by G. Let K be a iger order kerel costructed as above. Robiso (988) as sow te followig result: E Lemma 2 (Robiso) E d K ((z 2 we f 2 G for some > 0 ad k 2 K []+. z ) =) jz f (z ) 2 o = O 2 Lemma 22 (Robiso) E (g (z2 ) g (z )) d K ((z 2 z ) =) = O mi(;+;+) we f 2 G, g 2 G, ad k 2 K []+[]+. Tese results are useful to examie estimators we rg ad g are liear i. Usig tese results we will examie various examples. Te followig estimator ^ of E ffg is examied by Amad (976): Example 23 0 = i ^f (zi ) : I tis applicatio g (z; ; ) = (z). Tree aspects of tis applicatio makes it particularly easy to directly verify te coclusios of te lemmas: tat does ot deped o, rg (z; ; ) =, ad tat g (z; ; ) is liear i. Te rst two imply tat te coclusio of te rst lemma olds witout ay furter assumptios. Te tird implies tat tere is o approximatio error to be cocered, so we just eed to compute @g (z ; 0 ; V ar g (z ; 0 ; 0 (; ) + E @g (z2 ; 0 ; @ 0 2 jz + E @ 0 jz : Frécet derivative wit respect to ca be directly computed as mius te liear mappig from ito R wic evaluates a give fuctio at a poit g 0
is evaluated so tat @g (z ; 0 ; @ 0 2 = @g (z 2 ; 0 ; @ 0 = Tus @g (z ; 0 ; E @ 0 2 jz @g (z2 ; 0 ; E @ 0 jz d K d K = 0 = E z2 z z z 2 d K + E + E z z 2 Terefore @g (z ; 0 ; V ar g (z ; 0 ; 0 (; ) + E @ 0 2 jz = V ar 0 0 (z ) E d K z z 2 jz + E! 4E [ 0 0 (z )] 2o : d K z2 z jz d K z z 2 jz 2 : jz + E d K z z 2 @g (z2 ; 0 ; + E z z 2 d K Also, Robiso s result allows us to d coditios uder wic g b = 0. Aoter example is te partial liear regressio model of Cosslett (984), Sciller (984) ad Waba (984). Example 24 For x 2 R K, y 2 R, w 2 R d te model is y = x T 0 + (w) + " were E ("jw; x) = 0. equatios: 0 = Cosider a estimator wic solves te followig y i x 0 i^ ^E (yjwi ) + ^E x 0 jw i ^i ^Ii x i were ^I i = I ^f (wi ) > b ad I is te idicator fuctio. Te followig lemma is useful. Let I i = I (f (w i ) > b). Lemma 25 Pr at least oe of ^I i I i 6= 0! 0 we f 2 G, for some > 0, k 2 K []+, jk (0)j <, b is positive ad bouded, d b 2 = log!, b=!, ad we tere is o positive probability tat f (w i ) = b. @ 0 jz
Note tat b is ot ecessarily required to coverge to zero. Tis result allows us to cosider 0 = y i x 0 i^ ^E (yjwi ) + ^E x 0 i jw i ^ I i x i istead of te feasible GMM. Proof. Te probability is bouded by P o ^Ii Pr I i 6= 0. Note tat o Pr ^I I 6= 0 oo = E Pr ^I I 6= 0jw = E Pr ^f (w ) > bjw o ( I ) o o o + E Pr ^f (w ) < bjw I : Let ~ b = b d K (0) [( ) =] E d K ((w 2 w ) =) jw i : Te by Berstei s iequality for some positive umbers C ad C 2, o Pr ^f (w ) > bjw ( I ) ( = Pr d X ) wi w wi w K E K jw > ~ b jw ( I ) i=2 ( ) d ~ b 2 exp I (f (w C + C 2 ~ ) < b) b ad tat o Pr ^f (w ) < bjw I ( = Pr d X ) wi w wi w K + E K jw > ~ b jw I i=2 ( ) d ~ b 2 exp I (f (w C 2 ~ ) > b) : b C Te a applicatio of Lebesgue domiatig covergece teorem implies tat P o ^Ii Pr I i 6= 0 coverges to zero we all te rates coditios old. 2
For te kerel regressio estimators of g (x) = E (Y jx = x), deotig " = Y g (X), te liear approximatio of (^g g) I ^f > b takes te followig form: i = " i d K ((x i x) =) I (f (x) > b) ad f (x) b = E I (f (x) > b) (g (x i) g (x)) xi x K Let z = (w; x; y). I tis example, g (z; ; ) = I x so tat d rg (z; ; ) = I x [x 2 (w)] T : =f (x) : y x T (w) 2 (w) T Sice rg (z; ; ) is liear i ad does ot deped o, te direct veri catio of te lemma is easier. Oe ca verify o o rg = E I x [x E (xjw)] T! E x [x E (xjw)] T we b! 0 x we E [x E (xjw)] T K2o <. To examie te asymptotic distributio ote tat g (z; ; ) is liear i so tat direct calculatio is simpler. Te Frécet derivative of g wit respect to is @g=@ () = (w) 2 (w) T 0 x so tat writig u = y E (yjw) ad v = x E (xjw) ad " = y x T 0 (w) @g (z ; 0 ; @ 0 2 = @g (z 2 ; 0 ; @ 0 = u 2 v2 T 0 d K ((w 2 w ) =) I (f (w ) > b) x f (w ) u v T 0 d K ((w w 2 ) =) I (f (w 2 ) > b) x 2 : f (w 2 ) Tus otig tat u = y E (xjw) T 0 (w) = " + v T 0 @g (z ; 0 ; E @ 0 2 jz = 0 @g (z2 ; 0 ; E d K ((w w 2 ) =) @ 0 jz = " E I (f (w 2 ) > b) x 2 jw f (w 2 ) so tat @g (z ; 0 ; g (z ; 0 ; 0 (; ) I + E V ar @ 0 2 jz = V ar " d K ((w w 2 ) =) x I E I (f (w 2 ) > b) x 2 jw f (w 2 )! V ar [" [x E (x jw )]] as b! 0: 3 i @g (z2 ; 0 ; + E @ 0 jz
We ext cosider trimmig fuctio suitable to adle idex models. I tis part of te paper let ^I i = I if ^f w2br(wi ) (w) > b ad I i = I if w2br(wi ) f (w) > b Te followig lemma is useful. Lemma 26 Pr at least oe of ^I i I i 6= 0! 0 we f 2 G, for some > 0, k 2 K []+, jk (0)j <, b is positive ad bouded, d b 2 = log!, b=!, ad we tere is o positive probability tat f (w i ) = b. Proof. Te probability is bouded by P o ^Ii Pr I i 6= 0. Note tat o Pr ^I I 6= 0 oo = E Pr ^I I 6= 0jw = E Pr if ^f (w) > bjw ( I ) + E Pr if ^f (w) < bjw I : w2b r(w ) w2b r(w ) Let ~ b = b d K (0) E d K ((w 2 w ) =) jw. Te by Berstei s iequality for some positive umbers C ad C 2, o Pr ^f (w ) > bjw ( I ) ( = Pr d X ) wi w wi w K E K jw > ~ b jw ( I ) i=2 ( ) d ~ b 2 exp I (f (w C + C 2 ~ ) < b) b ad tat o Pr ^f (w ) < bjw I ( = Pr d X ) wi w wi w K + E K jw > ~ b jw I i=2 ( ) d ~ b 2 exp I (f (w C 2 ~ ) > b) : b C Te a applicatio of Lebesgue domiatig covergece teorem implies tat P o ^Ii Pr I i 6= 0 coverges to zero we all te rates coditios old. 4
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