On the Asymptotic Optimality of the LIML Estimator with Possibly Many Instruments

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1 O the Asymptotic Optimality of the LIML Estimator with Possibly May Istrumets T. W. Aderso Naoto Kuitomo ad Yukitoshi Matsushita November 23, 2009 (Fial Versio) Abstract We cosider the estimatio of the coefficiets of a liear structural equatio i a simultaeous equatio system whe there are may istrumetal variables. We derive some asymptotic properties of the limited iformatio maximum likelihood (LIML) estimator whe the umber of istrumets is large; some of these results are ew as well as old, ad we relate them to results i some recet studies. We have foud that the variace of the limitig distributio of the LIML estimator ad its modificatios ofte attai the asymptotic lower boud whe the umber of istrumets is large ad the disturbace terms are ot ecessarily ormally distributed, that is, for the micro-ecoometric models of some cases recetly called may istrumets ad may weak istrumets. Key Words Structural Equatio, Simultaeous Equatios System, May Istrumets, May Weak Istrumets, Limited Iformatio Maximum Likelihood, Asymptotic Optimality; JEL Code: C3, C30 AKM This is a revised first part of Discussio Paper CIRJE-F-32 uder the title A New Light from Old Wisdoms : Alterative Estimatio Methods of Simultaeous Equatios ad Microecoometric Models (Graduate School of Ecoomics, Uiversity of Tokyo, February 2005) which was preseted at the Ecoometric Society World Cogress 2005 at Lodo (August 2005). We thak Yoichi Arai, the co-editor ad the referees of this joural for some commets to the earlier versios. Departmet of Statistics ad Departmet of Ecoomics, Staford Uiversity Graduate School of Ecoomics, Uiversity of Tokyo, Bukyo-ku, Hogo 7-3- Tokyo, JAPAN, kuitomo@e.u-tokyo.ac.jp Graduate School of Ecoomics, Uiversity of Tokyo

2 . Itroductio Over the past three decades there has bee icreasig iterest ad research o the estimatio of a sigle structural equatio i a system of simultaeous equatios whe the umber of istrumets (the umber of exogeous variables excluded from the structural equatio), say K 2, is large relative to the sample size, say. The relevace of such models is due to collectio of large data sets ad the developmet of computatioal equipmet capable of aalysis of such data sets. Oe empirical example of this kid ofte cited i the ecoometric literature is Agrist ad Krueger (99) ; there has bee some discussio by Boud, Jaeger ad Baker (995) sice the. Asymptotic distributios of estimators ad test criteria are developed o the basis that both K 2 ad. These asymptotic distributios are used as approximatios to the distributios of the estimators ad criteria whe K 2 ad are large. Bekker (994) has writte To my kowledge a first metio of such a parameter sequece was made, with respect to the liear fuctioal relatioship model, i Aderso (976 p.34). This work was exteded to simultaeous equatios by Kuitomo (980, 982) ad Morimue (983), who gave asymptotic expasios for the case of a sigle explaatory edogeous variable. Followig Bekker there have bee may studies of the behavior of estimators of the coefficiets of a sigle equatio whe K 2 ad are large. The mai purpose of the preset paper is to show that oe estimator, the Limited Iformatio Maximum Likelihood (LIML) estimator, has some optimum properties whe K 2 ad are large. As backgroud we state ad derive some asymptotic distributios of the LIML ad Two-Stage Least Squares (TSLS) estimators as K 2 ad. Some of these results are improvemets o Kuitomo (98, 982), Morimue (983) ad Bekker (994), Chao ad Swaso (2005), Va Hasselt (2006), Hase, Hausma, ad Newey (2008), ad they are preseted i a uified otatio. I additio to the LIML ad TSLS estimators there are other istrumetal variables (IV) methods. See Aderso, Kuitomo, ad Sawa (982) o the earlier studies of the fiite sample properties, for istace. Several semiparametric estimatio methods have bee developed icludig the geeralized method of momets (GMM) estimatio ad the empirical likelihood (EL) method. (See Hayashi (2000) 2

3 for istace.) However, it has bee recetly recogized that the classical methods have some advatages i microecoometric situatios with may istrumets. I this paper we shall give the results o the asymptotic properties of the LIML estimator whe the umber of istrumets is large ad we develop the large-k 2 asymptotic theory or the may istrumets asymptotics icludig so-called the case of may weak istrumets. The TSLS ad the GMM estimators are badly biased ad they lose eve cosistecy i some of these situatios. Our results o the asymptotic properties ad optimality of the LIML estimator ad its variats give ew iterpretatios of the umerical iformatio of the fiite sample properties ad some guidace o the use of alterative estimatio methods i simultaeous equatios ad micro-ecoometric models with may weak istrumets. There is a growig literature o the problem of may istrumets i ecoometric models. We shall try to relate our results to some recet studies, icludig Doald ad Newey (200), Hah (2002), Stock ad Yogo (2005), Chao ad Swaso (2005, 2006), va Hasselt (2006), va der Ploeg ad Bekker (995), Bekker ad va der Ploeg (2005), Chioda ad Jasso (2007), Hase et al. (2008), ad Aderso, Kuitomo ad Matsushita (2008). I Sectio 2 we state the formulatio of a liear structural model ad the alterative estimatio methods of ukow parameters with possibly may istrumets. I Sectio 3 we develop the large-k 2 asymptotics (or may istrumets asymptotics) ad give some results o the asymptotic ormality of the LIML estimator whe ad K 2 are large. The we shall preset some results o the asymptotic optimality of the LIML estimator i the sese that it attais the lower boud of the asymptotic variace i a class of cosistet estimators with may istrumets uder reasoable assumptios. Also we discuss a more geeral formulatio of the models ad the related problems. I Sectio 4 we show that the asymptotic results i Sectio 3 agree with the fiite sample properties of estimators. The brief cocludig remarks will be give i Sectio 5. The proof of our theorems will be give i Sectio Alterative Estimatio Methods i Structural Equatio Models with Possibly May Istrumets We first cosider the estimatio problem of a structural equatio i the classical 3

4 liear simultaeous equatios framework. Let a sigle liear structural equatio i a ecoometric model be (2.) y i = β 2y 2i + γ z i + u i (i =,,), where y i ad y 2i are a scalar ad a vector of G 2 edogeous variables, z i is a vector of K (icluded) exogeous variables i (2.), γ ad β 2 are K ad G 2 vectors of ukow parameters, ad u,,u are idepedet disturbace terms with E(u i ) = 0 ad E(u 2 i )=σ2 (i =,,). We assume that (2.) is oe equatio i a system of + G 2 equatios i + G 2 edogeous variables y i =(y i, y 2i ). The reduced form of the model is (2.2) Y = ZΠ + V, where Y =(y i ) is the ( + G 2) matrix of edogeous variables, Z =(Z, Z 2 )= (z () i ) is the K matrix of K + K 2 istrumetal vectors z () i V =(v i ) is the ( + G 2) matrix of disturbaces, Π = π Π 2 π () 2 Π () 22 is the (K + K 2 ) ( + G 2 ) matrix of coefficiets, ad E(v i v i )=Ω = ω ω 2 ω 2 Ω 22 =(z i, z() 2i ), is a positive defiite matrix. The vector of K (= K + K 2, > 2) istrumetal variables z () i satisfies the orthogoality coditio Eu i z () i ]=0(i =,,). The relatio betwee (2.) ad (2.2) gives π Π 2 (2.3) π () 2 Π () 22 β 2 = γ 0 u i =(, β 2 )v i = β v i ad σ 2 = β Ωβ with β =(, β 2 ). Let Π 2 =(π () 2, Π () 22 )beak 2 ( + G 2 ) matrix of coefficiets. Defie the ( + G 2 ) ( + G 2 ) matrices by (2.4), G = Y Z 2. A 22. Z 2. Y = P 2 A 22.P 2, We itetioally cosider the stadard classic situatio ad state our results maily because they are clear. Noetheless a geeralizatio of the formulatio ad the correspodig results will be discussed i Sectio

5 ad (2.5) H = Y ( I Z(Z Z) Z ) Y, where A 22. = Z 2. Z 2., Z 2. = Z 2 Z A A 2, P 2 = A 22.Z 2. Y, (2.6) ad (2.7) A = Z = Z Z 2 z. z, Z 2 = (Z, Z 2 )= z () 2. z () 2, A A 2 A 2 A 22 is a osigular matrix (a.s.). The the LIML estimator ˆβ LI of β =(, β 2 ) is the solutio of (2.8) ( G λ H)ˆβ q LI = 0, where q = K (>2) ad λ (>2) is the smallest root of (2.9) G l q H =0. The solutio to (2.8) miimizes the variace ratio L = i= z () i (y i γ z i β 2 y 2i)] i= z () i z () i ] i= z () i (y i γ z i β 2 y 2i)] i= (y i γ z i β 2 y. 2i) 2 (2.0) The TSLS estimator ˆβ TS (= (, ˆβ 2.T S ) )ofβ =(, β 2 ) is give by (2.) Y 2 Z 2.A 22. Z 2. Y = 0, ˆβ 2.T S where Y 2 =(y i )isa G 2 matrix. The TSLS estimator miimizes the umerator of the variace ratio (2.0). The LIML ad the TSLS estimators of γ are (2.2) ˆγ =(Z Z ) Z Y ˆβ, where ˆβ is ˆβ LI or ˆβ TS, respectively. I this paper we shall discuss the asymptotic properties of ˆβ 2 because of its simplicity although it is straightforward to exted to treat ˆβ 2 β 2, ˆγ γ ] with some additioal otatios. The LIML ad TSLS estimators ad their properties i the geeral case were origially developed by Aderso ad Rubi (949, 950). See also Aderso (2005). 5

6 3 O Asymptotic Optimality of the LIML Estimator 3. Asymptotic Normality of the LIML Estimator We state the limitig distributio of the LIML estimator uder a set of alterative assumptios whe K 2 ad Π 2 ca deped o ad. We first cosider the case whe (I) (II) K 2 c (0 c<), Π() 22 A 22. Π () 22 p Φ 22., where Φ 22. is a osigular costat matrix. Coditio (I) implies that the umber of coefficiet parameters is proportioal to the umber of observatios. Because we wat to estimate the covariace matrix of v () i (i =,,), we wat c <. The (I) implies q as. Coditio (II) cotrols the ocetrality (or cocetratio) parameter to be proportioal to the sample size. Sice K 2 grows, it is ofte called the case of may istrumets. These coditios defie the rates of growth of the umber of icidetal parameters. Coditio (II) shall be weakeed to Coditio (II) where K 2 icreases with but at a smaller rate as stated i Sectio 3.3, which is sometimes called the case of may weak istrumets 2. We first summarize the basic results o the asymptotic distributios as Theorems. Although the preset formulatio ad Theorem are similar to the correspodig results reported i va Hasselt (2006) ad Hase et al. (2008), we shall give the proofs i Sectio 6 because the method of our proof gives some isights o the uderlyig assumptios, its extesios ad the asymptotic optimality. To state our results coveietly we trasform v i to w 2i = (0, I G2 ) I +G2 ] (3.) σ 2 Ωββ v i = (0, I G2 ) v i ] σ Cov(v,u)u 2 i ad u i = β v i. The E(w 2i u i )=0 ad (3.2) E(w 2i w 2i) = σ 2 Ωσ 2 Ωββ Ω ] 22, 2 A referee has poited out that Coditio (II) is the case of may strog istrumets. 6

7 where ] 22 is the G 2 G 2 lower right-had corer of the matrix. Theorem : Let z () i,i=, 2,,,be a set of K vectors (K = K +K 2,> 2). Let v i,i =, 2,,, be a set of ( + G 2 ) idepedet radom vectors idepedet of z (),, z () such that E(v i )=0 ad E(v i v i)=ω (a.s.). Suppose that (I) ad (II) hold. I additio assume (III) max i Π() 22 z i 2 p 0, where z i is the i-th row vector of Z 2. = Z 2 Z (Z Z ) Z Z 2. (i) For c =0, suppose that E v i 2 ] are bouded. The (3.3) (ˆβ2.LI β 2 ) d N(0, Ψ ), where Ψ = σ 2 Φ 22. ad σ 2 = β Ωβ. (ii) For 0 < c <, defie E(u 2 w 2 w 2) σ 2 E(w 2 w 2) = Γ 44.2 ad assume that E v i 4+ɛ ] < for some ɛ>0 (ad E Π () 22 z i 2+δ ] < for some δ>0 whe z () i are stochastic) 3. Suppose also that there exist limits where p () ii (IV) Ξ 3.2 = c (V) η = c =(Z 2. A 22.Z 2.) ii. The ] plim ] 2 plim Π() 22 z i i= i= p () ii c ] 2, p () ii c ] E(u 2 w 2 ), (3.4) (ˆβ2.LI β 2 ) d N(0, Ψ ), where (3.5) Ψ = σ 2 Φ Φ 22. { c Ωσ 2 Ωββ Ω ] 22 + (Ξ Ξ 3.2 )+ηγ 44.2 ]} Φ 22. ad c = c/( c). If G 2 =, the Ωσ 2 Ωββ Ω] 22 = ω ω 22 ω2 2 = Ω. Corollary : Whe v i (= (v ji )) (i =,,; j =,,G 2 + ) has a elliptically cotoured (EC) distributio i Theorem, the fourth order momets E(v ji v ki v li v mi ) = ( + κ/3)(ω jk ω lm + ω jl ω km + ω jm ω kl ) ad E(β v i ) 2 v i v i ] = ( + 3 We thak a referee for suggestig the possible improvemets o the momet coditios i the previous versio. 7

8 κ/3)(σ 2 Ω +2Ωββ Ω), where Ω =(ω jk ), E(v ji v ki )=ω jk ad κ is the kurtosis of EC(Ω) 4. The Γ 44.2 =(κ/3) Ωσ 2 Ωββ Ω ] 22 ad (3.5) is give by (3.6) Ψ = σ 2 Φ 22. +(c + 3 ηκ)φ 22. Ωσ 2 Ωββ Ω ] 22 Φ 22.. Istead of makig a assumptio o the distributio of disturbace terms except the existece of their momets, alteratively we assume (VI) plim i= p () ii c ] 2 =0. A simple example for Coditio (VI) is the case whe we ormalize (/)A 22. I K2 ad z i N(0, I K 2 ). Coditio (VI) is the same as η = 0 i Coditio (V), which i tur implies Ξ 3.2 = O i Coditio (IV) by the Cauchy-Schwarz iequality. These cosequeces of Coditio (VI) imply the followig theorem : Theorem 2 : For 0 c< assume Coditios (I), (II), (III), (VI) ad assume that E v i 2+ɛ ] < for some ɛ>0 (ad E Π () 22 z i 2+δ ] < for some δ>0whe are stochastic). The z () i (3.7) (ˆβ2.LI β 2 ) d N(0, Ψ ), where (3.8) Ψ = σ 2 Φ c Φ 22. Ωσ 2 Ωββ Ω ] 22 Φ 22. ad c = c/( c). The asymptotic properties of the LIML estimator hold whe K 2 icreases as ad K 2 / 0. I this case the limitig distributio of the LIML estimator ca be differet from that of the TSLS estimator. (The proof of Theorem 3 will be give i Sectio 6.) Theorem 3 : Let {v i, z () i ; i =,,} be a set of idepedet radom vectors. Assume that (2.) ad (2.2) hold with E(v i z i )=0 (a.s.) ad E(v i v i z () i )= Ω () i (a.s.) is a fuctio of z () i, say, Ω i, z () ]. The further assumptios o (v i, z () (v i =(v ji )) are that E(vji 4 z() i ) are bouded, there exists a costat matrix Ω such i i ) 4 The precise defiitio of elliptically cotoured (EC) distributio has bee give by Sectio 2.7 of Aderso (2003). 8

9 that Ω () i Ω is bouded ad σ 2 = β Ωβ > 0. Suppose (I ) (II) (III) K 2 η Π() c (0 η<, 0 <c< ), 22 A 22. Π () p 22 Φ 22., max i Π() 22 z i 2 p 0, where Φ 22. is a osigular costat matrix. (i) The for the LIML estimator whe 0 η<, (3.9) (ˆβ2.LI β 2 ) d N(0,σ 2 Φ 22. ), where σ 2 = β Ωβ. (ii) For the TSLS estimator whe /2 <η<, (3.0) η (ˆβ 2.T S β 2 ) p Φ 22. c(ω 2, Ω 22 )β, whe η =/2, (3.) (ˆβ2.T S β 2 ) d N cφ 22. (ω 2, Ω 22 )β,σ 2 Φ 22.], where (ω 2, Ω 22 ) is the G 2 ( + G 2 ) lower submatrix of Ω. Whe 0 η</2, (3.2) (ˆβ2.T S β 2 ) d N(0,σ 2 Φ 22.). It is possible to iterpret the stadard large sample theory as a special case of Theorem 3. The asymptotic properties of the LIML ad TSLS estimators for γ ca be derived from Theorem. Doald ad Newey (200) (i their Lemma A.6) has ivestigated the asymptotic properties of the LIML estimator whe K 2 / 0. Also Stock ad Yogo (2005) have discussed the asymptotic properties of the GMM estimators i some cases of the large-k 2 theory whe 0 <η</2. I this case, the asymptotic lower boud of the covariace matrix is the same as i the case of the large sample asymptotic theory although there are icidetal parameters (the umber of parameters is growig with the sample size). Chao ad Swaso (2005) have cosidered the cosistecy of some estimators whe K 2 is depedet o ad the disturbaces are ot ecessarily ormally distributed. Hase et al. (2008) have obtaied the limitig distributio of the LIML estimator for a more geeral model with o-ormal disturbaces ad differet assumptios. 9

10 3.2 Asymptotic Optimality with May Istrumets For the estimatio of the vector of structural parameters β, it seems atural to cosider procedures based o the two ( + G 2 ) ( + G 2 ) matrices G ad H, which are sufficiet statistics i the classical stadard situatio. We shall cosider a class of estimators which are fuctios of these matrices. The typical examples of this class are the OLS estimator, the TSLS estimator, ad the LIML estimator. It cotais the modified versios ad the combied versios of these estimators icludig the oe proposed by Fuller (977). (It also icludes other estimators which are asymptotically equivalet to these estimators.) The we have a basic result o the asymptotic optimality of the LIML estimator ad its (asymptotically equivalet) modificatios, which attais the lower boud of the asymptotic covariace uder alterative assumptios i most cases. The proof will be give i Sectio 6. Theorem 4 : Assume that (2.) ad (2.2) hold. Defie a class of cosistet estimators for β 2 by (3.3) ˆβ 2 = φ(g, H), where φ is cotiuously differetiable ad its derivatives are bouded at the probability limits of (/)G ad (/q )H as K 2 ad ad 0 c<. The uder the assumptios of the case (i) of Theorem, Corollary, Theorem 2 or Theorem 3, (3.4) AE (ˆβ 2 β 2 )(ˆβ 2 β 2 ) ] Ψ (or Ψ ), ad Ψ (or Ψ ) is give by (3.3), (3.6), (3.8) or (3.9), where the right-had side of (3.4) is the covariace matrix of the limitig distributio of the ormalized estimator (ˆβ 2 β 2 ) for the class of (3.3). This is the basic result o the asymptotic optimality of the LIML estimator whe there are may istrumets. Whe the distributio of V is ormal N(0, Ω) ad Z is exogeous, P =(Z Z) Z Y ad H = Y I Z(Z Z) Z ]Y are a sufficiet set of statistics for Π ad Ω, the parameters of a model. Whe K is fixed, it is kow that of all cosistet estimators of β 2 the LIML estimator suitably ormalized has the miimum asymptotic variace ad the optimality of ˆβ 2.LI exteds to the class of all cosistet estimators ot oly the form of (3.3). Whe K is depedet o, however, there is a further problem with (may) icidetal parameters. 0

11 The above theorems are the geeralized versios of the eariler results give by Kuitomo (98, 982, 987) because they assumed that the disturbaces are ormally distributed. Furthermore, Kuitomo (987) has ivestigated the higher order efficiecy property of the LIML estimator whe G 2 =, 0 c<. I the large-k 2 asymptotic theory with 0 <c<, the LIML estimator is asymptotically efficiet ad attais the lower boud of the variace-covariace matrix, which is strictly larger tha the iformatio matrix ad the asymptotic Cramér-Rao lower boud uder a set of assumptios, while both the TSLS ad the GMM estimators are icosistet. This is a o-regular situatio because the umber of icidetal parameters icreases as K 2 icreases i the simultaeous equatio models 5. There are further commets o recet studies. Bekker (994) derived the approximate distributios of the TSLS ad LIML estimators uder the ormal disturbaces. va der Ploeg ad Bekker (995) ad Bekker ad va der Ploeg (2005) cosidered the asymptotic distributios ad the asymptotic boud whe the data have a group structure with a fixed umber m groups ad the disturbaces are ormally distributed. Chioda ad Jasso (2007) obtaied a asymptotic optimality of the LIML estimator uder the ormal disturbaces whe G 2 =. Furthermore, Hah (2002) has derived a asymptotic lower boud whe G 2 =,K = 0 ad the disturbaces are ormally distributed. It is the same as the asymptotic Cramér-Rao lower boud, which ca be smaller tha Ψ uder some coditios o the icidetal parameters. The trivial case which satisfies such coditio is the case whe a K 2 G matrix Π () 22 has o-zero fiite compoets. Aother example is Coditio of Hah (2002), which correspods to the case of 0 η</2i our Theorem 3. These cases could be reduced to the case whe c =0 i Theorem 4 ad there is o cotradictio with our results. The mai differece i two approaches come from the fact that we have pursued the asymptotic boud ad optimality over the icidetal parameters uiformly i some sese 6 ad we do 5 As a o-trivial example, we take the bias-adjusted TSLS estimator by settig λ = K 2 / i (2.8) ad deote ˆβ 2.BT S. The the asymptotic variace of ˆβ 2.BT S is greater tha Φ i Theorem ] 3 if 0 <c< ad Ωββ Ω If we assume the ormal disturbaces with Ω = ω 2 I, the dummy istrumets ad the j-th row of Π (j =,,K )asπ j N +G (0, Ω π ), for istace, we have (so-called) a structural relatioship model (see Aderso (984) for istace). The the asymptotic boud of estimators is similar to ours. A referee also poited out the problem of quality of istrumets, which is certaily related to the problem of choosig istrumets. Sice the related discussio o the icidetal

12 ot have ay particular coditio o the icidetal parameters except (I) ad (II). Sice we ofte do ot have a prior iformatio o the reduced form coefficiets i applicatios, it may be atural to derive the asymptotic boud with some uiformity coditio over the space of reduced form coefficiets. 3.3 Geeral formulatios of the asymptotic optimality We ca geeralize the asymptotic optimality of LIML i several directios. We cosider (2.) ad a oliear replacemet 7 for the last G 2 colums of the reduced form (2.2). We treat (2.) ad (3.5) Y 2 = Π () 2Z + V 2, where Π () 2Z =(π 2i(z () i )) is a G 2 matrix, the i-th row of which π 2i(z () i ) depeds o the K vector z () i (i =,,), V 2 is a G 2 matrix, v = u + V 2 β 2, ad V =(v, V 2 ). Whe the reduced form equatios (3.5) are liear, (2.) ad (3.5) has a represetatio (2.2). I this formulatio, Coditio (II) is replaced by (II ) d 2 Π () 2Z Z 2. A 22.Z 2.Π () p 2Z Φ 22., where d 2 = tr(π() 22.Z 2. Π() 2Z ), Φ 22. is a positive (costat) defiite matrix p ad d as. We replace (III) by 2Z Z 2.A (III) d 2 max π 2i(z () i ) 2 p 0. i A possible additioal coditio (due to oliearilty i (3.5)) is (VII) Π () 2Z I Z(Z Z) Z ] Π () p 2Z O. q Coditio (VII) is automatically satisfied i the liear case. It is possible to weake this coditio such that the probability limit of (VII) is Φ 3, say. The we eed some additioal otatios to re-express Theorem -Theorem 4 without chagig their essetial results. (See Sectio 3 of Kuitomo (2008).) Three cases ca be cosidered. We have already ivestigated the first case of d = O p ( /2 ) ad K 2 = O(). The asymptotic covariace of the LIML estimator parameters problem ad the related issues, which are importat, is far beyod the scope of the preset paper, we omit the details. 7 This model is very similar to the oe studied by Hase et al. (2008), which has geeralized some results of Aderso et al. (2005). 2

13 is give by (3.5) i Theorem or (3.8) i Theorem 2 uder alterative assumptios with (II) istead of (II). The secod case is the stadard large sample asymptotics, which correspods to the cases of d = O p ( /2+δ )(δ>0), or d = O p ( /2 ) ad K 2 / = o(). I this case (3.6) d (ˆβ 2.LI β 2 ) d N(0,σ 2 Φ 22. ). Theorem 3 is oe result i this case, which ca be exteded directly to the oliear model of (2.) ad (3.5). The third case occurs whe d = o p ( /2 ) ad /d 2 0, which may correspod to oe case i Hase et al. (2008) with slightly differet ormalizatio ad assumptios. The ext result is a extesio of Theorem 4 ad it could be iterepreted as a asymptotic optimality of the LIML estimator for may weak istrumets. The variace (3.9) below is simpler tha (3.5) ad (3.8) because the effects of domiate the first, the third ad the fourth terms of (3.5) i Theorem. Theorem 5 : Assume Coditios (II), (III) ad (VII) with d = o p ( /2 ) ad /d 2 0iTheorem 2 ad Theorem 4 istead of Coditios (II) ad (III). The (3.7) ] d 2 d (ˆβ 2.LI β 2 ) N(0, Ψ ) ad (3.8) ) AE ( d 2 2 (ˆβ 2 β 2 )(ˆβ 2 β 2 ) Ψ, where the right-had side of (3.8) is the covariace matrix of the limitig distributio of the ormalized estimator d 2 / ](ˆβ 2 β 2 ) for the class of (3.3) ad (3.9) { Ψ = Φ 22. c Ωσ 2 Ωββ Ω ] } Φ is the covariace matrix of the limitig distributio of the LIML estimator. 3.4 Heteroscedasticity ad the asymptotic properties Recetly, there has bee some iterest o the role of heteroscedasticity with may istrumets. Let Ω i = E(v i v i z() i ) be the coditioal covariace matrix ad we assume (VIII) 3 Ω i i= p Ω,

14 where Ω is a positive defiite (costat) matrix. The i the case whe both Coditios (VI) ad (VIII) hold, the LIML estimator has still some desirable asymptotic properties. I the more geeral cases, the distributio of the LIML estimator could be sigificatly affected by the presece of heteroscedasticity of disturbace terms with may istrumets. O this issue, however, there are alterative ways to improve the LIML estimatio. Chao ad Swaso (2004) has ivestigated the JIVE estimatio method, ad Hausma, J., W. Newey, T. Wouterse, J. Chao ad N. Swaso (2007) have proposed the HLIM estimatio method. Kuitomo (2008) has also cosidered the class of MLIML estimators ad ivestigated the problem of asymptotic optimality uder heteroscedasticity coditios 8. The details of these issues shall be discussed i a aother occasio. 4 Discussios o Asymptotic Properties ad Fiite Sample Properties It is importat to ivestigate the fiite sample properties of estimators partly because they are ot ecessarily similar to their asymptotic properties. Oe simple example would be the fact that the exact momets of some estimators do ot ecessarily exist. (I that case it is meaigless to compare the exact MSEs of alterative estimators ad their Mote Carlo aalogues.) Although we discuss the asymptotic properties of the LIML estimator, we eed to ivestigate their relevace for practical applicatios. There is a otable differece betwee the results i Theorem ad Theorem 2, that is, the asymptotic variace depeds o the 3rd ad 4th order momets of the disturbace terms i the former. The fiite sample properties of the LIML estimator have bee ivestigated by Aderso, Kuitomo ad Matsushita (2005, 2008) i a systematic way. As typical examples we preset oly six figures (Figures A-6A) i Appedix whe α =0.5,.0 ad G 2 =. We have used the umerical estimatio of the cumulative distributio fuctio (cdf) of the LIML estimator based o the simulatio ad we have eough umerical accuracy i most cases. 8 It icludes a extesitio of Sectios 3.2 ad 3.3 of this paper with the defiitios of weak heteroscedasticity ad persistet heteroscedasticity. 4

15 The key parameters i figures ad tables are K 2 (or K 2 ), K (or K ), α =ω 22 / Ω /2 ](β 2 ω 2 /ω 22 )(Ω =(ω ij )) ad δ 2 = Π () 22 A 22. Π () 22 /ω 22. I additio to the LIML estimator, we have added the distributio fuctios of the TSLS ad OLS(ordiary least squares) estimators for comparisos i some cases. (See Aderso et al. (2008) for the details.) The figures (Figures A-6A) show the estimated cdf of alterative estimators i the stadard form, that is, (4.) σ Φ/2 22. ( ˆβ2 β 2 ). By usig (3.3) the limitig distributios of the LIML ad TSLS estimators are N(0, ) i the large sample asymptotics ad they are deoted by o. By usig (3.8) the correspodig limitig distributios of the LIML estimators i the large K 2 asymptotics are N(0,a)(a = Ψ Ψ,a ), which are deoted by large- K-ormal i Figures A-6A, ad they are traced by the dashed curves. I some figures we also have the approximatios based o the variace formula (3.5) with the third ad fourth order momets of disturbace terms, which are deoted by large-k-oormal ad traced by x. From these figures we have foud that the effects of may istrumets o the cdfs of the estimators are sigificat ad the approximatios based o the large sample asymptotics are ofte iferior. At the same time we also have foud that the effects of o-ormality of disturbace terms o the cdf of the LIML estimator are ofte very small. (The dashed curves ad x are almost idetical.) The distributios of the TSLS ad OLS estimators have sigificat bias with may istrumets. Oe importat applicatio of the asymptotic variace is to costruct a t-ratio for testig a hypothesis o the coefficiets. We ca use the asymptotic variace of the LIML estimator give by (3.5) or (3.8) replaced by its estimator. (We have used P 2 for Π 2, (/q )H for Ω ad the sample momets from residuals for σ 2 ad E(u 2 w 2 ), for istace.) We have ivestigated this problem ad as a typical example we give Table A o the cdf of t-ratio (4.2) t( ˆβ 2.LI )= ( ˆβ2.LI β 2 ) s( ˆβ 2.LI ), which is costructed by the LIML estimatio, where s 2 ( ˆβ 2.LI ) is the estimator of the variace. The formulas (3.3), (3.4), (3.5), (3.6) ad (3.8) are used. (Matsushita (2006) has ivestigated the fiite sample properties of t-ratios ad derived their 5

16 asymptotic expasios of their distributio fuctios i a systematic way.) We ca see that the effects of may istrumets o the cdf of the ull distributios of t-ratios are ofte sigificat while the approximatios based o the large sample asymptotics are ofte iferior. At the same time we also have foud that the effects of o-ormality of disturbace terms o the ull-distributios of the t-ratios are ofte small, that is, the differeces amog the effects of (3.5) i Theorem are ot substaitial for practical purposes. Bekker (994) derived the asymptotic variace formula (3.8) for the LIML estimator uder the coditio that the disturbace terms are ormally distributed. It is idetical to the asymptotic covariace matrix of the LIML estimator i the large-k 2 asymptotics reported by Kuitomo (98, 982). From our ivestigatios it may be advisable to use (3.8) for statistical ifereces o the structural coefficiets eve uder the cases whe the disturbaces are ot ormally distributed for practical purposes. These observatios agree with the recet studies reported by Kuitomo ad Matsushita (2008), Hase et al. (2008) ad Aderso et al. (2005, 2008). 5. Cocludig Remarks I this paper, we have discussed the asymptotic optimality whe the umber of istrumets is large i a structural equatio of the simultaeous equatios system. Although the limited iformatio maximum likelihood (LIML) estimator ad the two stage-least squares (TSLS) estimator are asymptotically equivalet i the stadard large sample theory, they are asymptotically quite differet i the large-k 2 asymptotics with may istrumets or may weak istrumets. I some recet microecoometric models ad models o pael data, it is ofte a commo feature that K 2 is fairly large ad this asymptotic theory has some practical relevace. (See Hsiao ad Tahmiscioglu (2008), for istace.) We have show that the LIML estimator ad its variats have ofte the asymptotic optimality with may istrumets. I practical applicatios it is ofte ot easy to idetify whether may istrumets are weak or strog such as Coditios (II) or Coditio (II). The TSLS estimator (ad hece the GMM method) crucially depeds o whether there are may istrumets or ot i ay forms, while the LIML estimator does ot. The oly additioal cost is that we eed to use the asymptotic covariace, which ca be slightly larger tha the stadard oe. I this sese the LIML estimator has some 6

17 asymptotic robustess. The asymptotic optimality results i this paper give some further reasos why we have the fiite sample properties of the alterative estimatio methods icludig the classical LIML ad TSLS estimators. The LIML estimator is also quite attractive over the semi-parametric estimatio methods of the geeralized method of momets (GMM) ad the empirical likelihood (EL) estimators i the situatios with may istrumets or may weak istrumets. 6 Proof of Theorems I this sectio we give the proofs of Theorems ad the mathematical derivatio i Sectio 3. Proof of Theorem : Substitutio of (2.2) ito (2.4) yields The G = (Π Z + V )Z 2. A 22. Z 2. (ZΠ + V) = Π 2 A 22.Π 2 + V Z 2. A 22. Z 2. V + Π 2 Z 2. V + V Z 2. Π 2. (6.) G Π 2A 22. Π 2 + K 2 Ω] = Π 2Z 2.V + V Z 2. Π 2 + V Z 2. A 22.Z 2.V K 2 Ω ]. Coditio (II) implies that as (6.2) ad (6.3) The as, (6.4) G Π 2Z 2.V p O, V Z 2. A 22.Z 2.V K 2 Ω ] p O. p G 0 = β 2 I G2 Φ 22. (β 2, I G2 )+c Ω ad (6.5) H p Ω. q The ˆβ p p LI β ad λ c as. 7

18 Defie G, H,λ, ad b by G = ( G G 0), H = q ( q H Ω), λ = (λ c), b = (ˆβ LI β). From (2.8), G 0 c Ω]β + G λ Ω]β + G 0 c Ω]b + q ch ]β = o p ( ). Sice (G 0 c Ω)β = 0 ad ˆβ LI =(, ˆβ 2.LI ), (2.8) gives (6.6) β 2 I G2 Φ 22. (ˆβ2.LI β 2 )=(G λ Ω cc H )β + o p (). Multiplicatio of (6.6) o the left by β =(, β 2 ) yields (6.7) λ = β (G cc H )β β Ωβ + o p (). Also multiplicatio of (6.6) o the left by (0, I G2 ) ad substitutio for λ from (6.7) yields (6.8) (ˆβ2.LI β 2 ) = Φ 22. (0, I G 2 )(G λ Ω cc H )β + o p () By usig the relatio Vβ = u, we obtai = Φ 22. (0, I G 2 )I G2 + Ωββ β Ωβ ](G cc H )β + o p (). (6.9) (G cc H )β = Π 2Z 2.u + c V Z 2. A K2 22.Z 2.u K 2 Ωβ ] cc q V (I Z(Z Z) Z )u q Ωβ ], where K + q =. Sice we have the coditioal expectatio give Z as E Π 2 Z 2. Vββ V Z 2. Π 2 Z ] = σ 2 Π 2 A 22.Π 2, we apply the cetral limit theorem with Lideberg coditio to the first term of (6.9). (See Theorem of Aderso ad Kuitomo (992) for istace). Coditios (II) ad (III) imply that (/ )Π () 22 Z 2.u has a limitig ormal distributio with covariace matrix σ 2 Φ 22.. This proves (i) of Theorem. Next we shall cosider (ii) of Theorem. We eed to prove that the limitig distributio of T = T + ct 2 cc T 3 is ormal by applyig a cetral limit 8

19 theorem, where T = a (/ )Π () 22 Z 2. u,t 2 = a (/ K 2 )W 2 Z 2.A 22.Z 2. u, T 3 = a (/ q )W 2 (I Z(Z Z) Z )u for ay costat vector a ad W 2 =(0, I G 2 )I G2 + Ωββ. β Ωβ ]V For the secod ad third terms o the right-had side of (6.9), we otice that each row vector of W 2 (w 2i =(0, I G2 )(v i u i Cov(v () i u i ))/σ 2 )) ad u i (i =,,) are ucorrelated ad Ew 2i w 2i ]=(/σ2 )σ 2 Ω Ωββ Ω] 22. Thus (6.0) ad (6.) = = (0, I G2 )I G2 + Ωββ β Ωβ ] V Z 2. A 22. Z 2. u K 2Ωβ ] K2 w 2i u j p () ij K2 i,j= (0, I G2 )I G2 + Ωββ β Ωβ ] q V (I Z(Z Z) Z )u q Ωβ ] q i,j= w 2i u j δ j i q () ij ], where p () k= ] ij = z i z k z k z j,q () ij = z () i,δ j i =0(i j). The the variaces of T 2 ad T 3 are k= z () ] () k z() k z j ad δi i = = E{a ( w 2i u i p () ii K 2 K 2 i= i= + i j Eu 2 i a w 2i w 2i ap()2 ii ]+ w 2i u j p () ij )] 2 Z} K 2 i j E(u 2 j )E(a w 2i w 2i a)p()2 ij, ad E{a ( q i= = q i= w 2i u i ( q () ii ) i j w 2i u j q () ij ] 2 Z} Eu 2 i a w 2i w 2ia]( 2q() ii + q ()2 ii )+ By usig the relatios i,j= p ()2 ij q ()2 ii )+ i j q ()2 ij (6.2) a K 2 ad (6.3) q i j = K 2, i,j= q ()2 ij E(u 2 j )E(a w 2j w 2i a)q()2 ij. = K ad i= ( 2q () ii + = q, the limitig variaces of T 2 ad T 3 are the limits of K 2 σ 2 E(w 2i w 2i )+ i= a q σ 2 E(w 2i w 2i q )+( 2K + 9 ] p ()2 ii Γ 44.2 a i= ] q ()2 ii )Γ 44.2 a.

20 I order to evaluate the covariaces of three terms of T, we first otice E{ Π () 22 Z 2. u] ( (6.4) W 2 Z2. A 22. Z 2. c (I Z(Z Z) Z ) ) u] Z} Secod, = i= = ( Π () 22 z () i p ii c ( q () ii ) ) E(u 2 w 2 )=Ξ() 3.2 (say). E{ W 2 Z 2.A 22. Z 2. u] W 2 (I Z(Z Z) Z )u] Z} i= = K 2 E(u 2 i w 2iw 2i )p() ii i= p () ii q () ii ]Γ 44.2 q () ii ]+ i j σ 2 E(w 2i w 2i )p() ij δ j i q () ij ] by usig the relatios that i,j= p () ij δ j i = K 2 ad i,j= p () ij q () ji = K 2. Hece we have evaluated each term E(T 2 ) = E(T 2 )+ce(t 2 2 )+cc E(T 2 3 ) +2 ce(t T 2 ) 2 cc E(T T 3 ) 2c c E(T 2 T 3 ). The we use the relatio c(+c )=c for the coefficiets of two terms of E(u 2 i w 2iw 2i ). Also by usig the relatio cc ( c ) 2cc = c 2 for the coefficiets of Γ 44.2, we fid that lim c p ()2 ii + cc ( 2K + q ()2 ii ) K 2 i= q i= ] 2c c K 2 q (K 2 p () ii q () ii ) i= = lim η, where η =(/) i= p () ii + c q () ii ] 2 c 2. By usig (6.4), the limitig covariace matrix of (ˆβ 2.LI β 2 ) is (3.5). Fially, by usig the cetral limit theorem (CLT) i Lemma 3 below for every costat vector a, we have the asymptotic ormality of (3.4) with the asymptotic covariace matrix Ψ ad it proves (ii) of Theorem. Q.E.D The ext two lemmas are the results of straightforward evaluatios o projectio matrices, ad we have omitted their derivatios. Lemma : Assume Coditio (VI) ad c = lim K 2 /. The (6.5) plim p () ii i= + c q () ii ] 2 c 2 = plim 20 p () ii i= c ( q () ii )] 2 =0,

21 where c = c/( c), p () ij =(Z 2. A 22.Z 2. ) ij ad q () ij =(ZA Z ) ij. Lemma 2 : Let a matrix P =(p ij ) satisfyig P 2 = P = P ad rak(p) = r. The (6.6) p ii p jj p ij r. i,j= Let also (6.7) B =(b ij )=Z 2. (Z 2.Z 2. ) Z 2. c I Z(Z Z) Z ]. The (6.8) i,j= b ii b jj b ij = O(). Lemma 3 : Let t () i costat vector a. As, = a Π () 22 z i ad t() 2i = a w 2i (i =,,) for ay (o-zero) (6.9) T = i= d N(0, Δ), t () i u i + i,j= t () () 2i u j p ij c (δ j i q () ij ) ] where Δ = a Φ 22. Ψ Φ 22. a or Δ = a Φ 22. Ψ Φ 22. a i Theorems ad 2. Proof of Lemma 3 : We first cosider the case whe z () i o-stochastic variables. s () 4i Set s () i =(/ )t () 2i = (/ )t () i u i,s () 2i i j= u jb ij ad b ij = p () ij = (/ )t () 2i u i b ii,s () 3i are the sequece of = (/ )u i i j= t () 2j b ji, c (δ j i q () ij )(i, j =,,). Let F,i be the σ field geerated by the radom variables u j, v j (j i, i ) ad F,0 be the iitial σ field. The T = i= X i ca be decomposed X i = s () i +s () 2i +s () 3i +s () 4i ad EX i F,i ] = 0 (i =,,). Sice each term X i (i =,,) are martigale differece sequeces, by direct calculatios we fid E X 2 i F ],i = t() i ] 2 σ 2 + b2 ii Et() 2i u i ] 2 + i σ2 b ij t 2j ] 2 + i E(t2 2i ) b ij u j ] t() i b ii Et () 2i u 2 i ]+ 2 σ2 t () i i j= j= b ij t 2j + 2 E(u2 i t i 2i)b ii b ij t 2j + 2 E(u it 2 2i )b i ii b ij u j. j= j= j= 2

22 The we apply a martigale cetral limit theorem to T = i= X i. The most importat step is to show i ( b ij t 2j ) 2 i (6.20) E( b ij t 2j ) 2 p 0, i=2 j= j= i E(t 2 2i )( b ij u j ) 2 E(t 2 2i )E( i (6.2) b ij u j ) 2 p 0, i=2 j= j= () t i b ii E(t () 2i u 2 i ) a Ξ 3.2 a ] p (6.22) 0, i= ad (6.23) (6.24) (6.25) i=2 i=2 i=2 σ 2 t () i i j= b ij t 2j p 0, E(u 2 i p i t 2i )b ii b ij t 2j 0, j= i E(u i t 2 2i )b p ii b ij u j 0. j= Uder the assumptios i Theorem, it is straightforward but tedious to show these relatios by usig Lemma 2. We oly give a illustratio. We evaluate E 2 t i ( b ij t 2i ) ( i=2 j=i )2 E t i t i b ij b i j E(t 2 2i) i,i =2 j= = ( )2 E t i t i b ii E(t 2 2i) i,i =2 because we ca utilize Z 2. (I Z(Z Z) Z)=O. The we have (6.23). Fially, we apply the martigale CLT as Theorem 3.5 of Hall ad Heyde (980). We set V = i= E Xi 2 F,i ]. The by utilizig that for ay ξ>0ad ν>0, i= E(X i ) 2 I( X i ξ)] (/ξ) ν i= EXi 2+ν ], we ca show their (3.33), (3.34) ad (3.36) uder Assumptios of Theorem. Thus we have the result by usig the momet coditio E v i 4+ɛ ] <. Whe Coditio (VI) holds, we oly eed E v i 2+ɛ ] < because i= s () p 2i 0. Whe z () i are stochastic, we utilize the additioal momet coditio to show the coditios of Theorem 3.5 of Hall ad Heyde (980). Q.E.D. Proof of Theorem 3 : (i) We make use of the fact that Z(Z Z) Z ad Z 2. (Z 2. Z 2.) Z 2. are idempotet of rak K ad K 2, respectively, ad that the 22

23 boudedess of Evji 4 z() i ] implies a Lideberg coditio sup i E v iv i I(v iv i >a) z (),, z () ] p 0 (a ). Let (6.26) G = G Π 2A 22. Π 2 ] = Π 2 Z 2. V + V Z 2. Π 2 + V Z 2. A 22. Z 2. V. Sice the matrix V Z 2. A 22.Z 2. there is a (costat) Ω such that V is positive defiite ad Ev() i v () i z () i ] is bouded, (6.27) E V Z 2. A 22. Z 2. V] = E i= K 2 Ω O Ω () i p () ii ] whe 0 <η</2. The (6.28) G β Π 2 Z 2. Vβ = V Z 2. A 22. Z 2. Vβ p 0. For the LIML estimator (2.8) implies (6.29) (0, I G2 ) Π 2A 22. Π 2 + ] G λ H q ˆβ 2.LI p = 0. By usig the facts that (/ )G p O, λ 0 ad /q ]H p Ω, we have Φ 22. (β 2, I G2 )plim = 0, ˆβ 2.LI which implies plim ˆβ2.LI (2.8) implies = β 2 because Φ 22. is positive defiite. The agai (6.30) Π 2 A 22.Π 2 + ] G λ β H +(ˆβLI β) ] = 0. q Lemma 4 : Let λ (>2) be the smallest root of (2.9). (i) For 0 <ν< η ad 0 η<, ν p (6.3) λ 0 as. (ii) For 0 η<, λ K ] 2 p (6.32) 0 23

24 as. Proof of Lemma 4 : Write (6.33) λ = b mi b b q Hb q β Gβ β Hβ = q β V Z 2. A 22.Z 2. Vβ β V (I Z(Z Z) Z )Vβ. By usig the boudedess of the fourth order momets of v i, we have (6.34) v i v p i Ω. i= Also ( ν) V Z 2. A 22.Z 2. V p O by usig the similar argumets as (6.28). The (6.35) ] ν q ( ν) β V Z 2. A λ 22.Z 2. Vβ p 0 β V (I Z(Z Z) Z )Vβ as. The result (ii) follows from (6.3), (6.34) ad σ 2 = β Ωβ > 0. Q.E.D. Due to Lemma 4, p λ 0 whe 0 η</2 (ad the asymptotic distributios of the LIML ad TSLS estimators are equivalet). The (6.36) (0, I G2 ) Π 2 A 22.Π () 22 (ˆβ2.LI β 2 ) (0, I G2 )G β p 0. We otice that = Ω () i i= (Ω () i i= + i= Π () 22 z iz iπ () 22 Ω Φ 22. Ω) Π () 22 z i z i Π() 22 Ω ] Π () 22 z i z i Π() 22 Φ 22. p O because Coditio (II ) the coditios imposed o Ω () i (i =,,). The by applyig CLT to (/ )Π () 22 Z 2.Vβ, we obtai the limitig ormal distributio N(0,σ 2 Φ 22. ). This proves (i) of Theorem 3 for 0 η</2. (ii) We cosider the asymptotic distributio of the LIML estimator whe /2 p η <. By usig the argumet of (6.29) ad the fact that λ 0, we have p ˆβ 2.LI β 2 0. By multiplyig β from the left to (6.30), we have { K β 2 λ ]Ω + V Z 2. Π 2 + V Z 2. A 22. Z 2. V K 2Ω] λ q H } β +(ˆβ LI β) ] = 0. 24

25 Multiply (6.30) o the left by (0, I G2 ) to obtai (0, I G2 ) { Π 2A 22. Π 2 + K ] 2 Ω + Π 2 Z 2. V + V Z 2. Π 2 + (V Z 2. A 22. Z 2. V K 2)Ω) } λ H β +(ˆβ q LI β) ] = 0. We cosider the asymptotic behavior of the quadratic term V Z 2. A 22. Z 2. V K 2Ω] = i,j= p () ij ( vi v j δj i Ω () ) i ]+ i= p () ii ] ( Ω () i Ω ) ], where δ j i is the idicator fuctio (δi i = ad δj i =0(i j)). For ay costat vectors a ad b, there exists a positive costat M such that E = E i,j= i= p () ij p ()2 ii 2 a (v () i v () j δ j i Ω () i )b a (v i v i Ω() i )b] 2 + i j p ()2 ij a v i v j b] 2 + i j p ()2 ij a v i v j ba v j v i b] M K 2 0 because the coditioal momets of vji 4 are bouded, i= p () ii = K 2 ad i= p ()2 ii K 2. The we fid (6.37) V Z 2. A 22. Z 2. V K 2Ω ] p O whe 0 η<. We ca use (6.30) ad the fact that Π 2 A 22.Π 2 + K ] 2 Ω λ H β = o p ( ). q By multiplyig the precedig equatio out to separate the terms with factor β ad with the factor (ˆβ LI β), we have (6.38) (0, I G2 ) Π 2A 22. Π 2 (ˆβLI β)+ ] Π 2Z p 2.Vβ 0, which is equivalet to Φ 22. (ˆβ2.LI β 2 ) Π () 22 Z 2. Vβ p 0. By applyig the CLT to the secod term, we complete the proof of (i) of Theorem 3 for the LIML estimator of β whe /2 η<. (iii) Next, we shall ivestigate the asymptotic property of the TSLS estimator. If 25

26 we substitute λ for 0 i (2.8), we have the TSLS estimator. The we fid that the limitig distributio of the TSLS estimator is the same as the LIML estimator whe 0 η</2. Whe η =/2, however, we have G β cωβ + ] Π 2Z p (6.39) 2.Vβ O. We set ˆβ TS =(, ˆβ 2.T S), which is the solutio of (2.). By evaluatig each term of (0, I G2 ) Π 2 A 22.Π 2 + β G ] +(ˆβTS β) ] = 0, we have ] (6.40) Π() 22 A 22. Π 2 (ˆβTS β) (0, I G2 )G β = o p(). The the limitig distributio of (ˆβ 2.T S β 2 ) is the same as that of Φ 22.(0, I G2 )G β. By usig (/ )V Z 2. A 22.Z 2.Vβ p cωβ ad applyig the CLT as (i), we have the result for the TSLS estimator of β whe η =/2. Whe /2 <η<, we otice (6.4) η G Π 2 A 22.Π 2 ] β = K 2 Ωβ + η η Π 2 Z 2. Vβ + V Z η 2. A 22. Z 2. V K 2Ω ] β. Because the last two terms of the right-had side of (6.4) except the first term are of the order o p ( η ), we have η G ] (6.42) Π 2 A 22.Π 2 β p cωβ as. Hece by usig the similar argumets as (i), (6.43) (0, I G2 ) Π 2 A 22.Π () 22 η (ˆβ 2.T S β 2 ) (0, I G2 )cωβ p 0 ad we complete the proof of (ii) of Theorem 3 for the TSLS estimator whe /2 η<. Q.E.D. Proof of Theorem 4 : We set the vector of true parameters β =(, β 2 )= (, β 2,, β +G2 ). We write (6.44) ˆβ k = φ k ( G, q H) (k =2,, +G 2 ). For the estimator to be cosistet we eed the coditios (6.45) β k = φ k 2 Φ 22. (β 2, I G2 )+cω, Ω I G2 (k =2,, +G 2 ) 26

27 as idetities i β 2, Φ 22., ad Ω. Let a ( + G 2 ) ( + G 2 ) matrix ( ) T (k) φk (6.46) = =(τ (k) ij ) (k =2,, +G 2 ; i, j =,, +G 2 ) g ij evaluated at the probability limits of (6.45). We write a ( + G 2 ) ( + G 2 ) matrix Θ (= (θ ij )) Θ = β 2 I G2 Φ 22. (β 2, I G2 )= β 2 Φ 22.β 2 β 2 Φ 22. Φ 22. β 2 Φ 22. where Φ 22. =(ρ m,l )(m, l =2,, +G 2 ), (Φ 22. β 2 ) l = +G 2 j=2 β j ρ lj (l =2,, + G 2 ), (β 2Φ 22. ) m = +G 2 i=2 β i ρ im (m =2,, +G 2 ), ad β 2Φ 22. β 2 = +G 2 i,j=2 ρ ij β i β j. By differetiatig each compoets of Θ with respect to β j (j =,,G 2 ), we have Θ =( θ lm (6.47) ), β j β j where θ β j =2 +G 2 i=2 ρ ji β i (j =2,, +G 2 ), θ m β j = ρ jm (m =2,, +G 2 ), θ l β j = ρ lj (l =2,, +G 2 ), ad θ lm β j =0(l, m =2,, +G 2 ). Hece ( ) (k) Θ tr T =2τ (k) +G 2 +G 2 (6.48) ρ ji β i +2 ρ ji τ (k) ji = δj k β, j i=2 i=2 where we defie δk k = ad δj k =0(k j). Defie a ( + G 2 ) ( + G 2 ) partitioed matrix T (k) = τ (k) τ (k) 2 (6.49) τ (k) 2 T (k). 22 The (6.48) is represeted as, (6.50) 2τ (k) Φ 22. β +2Φ 22. τ (k) 2 = ɛ k, where ɛ k =(0,, 0,, 0,, 0) with i the k-th place ad zeros i other elemets. Sice Φ 22. is positive defiite, we solve (6.50) as (6.5) τ (k) 2 = 2 Φ 22. ɛ k τ (k) β 2. Further by differetiatig Θ with respect to ρ ij, we have Θ =( θ lm (6.52) ), ρ ii ρ ii = β i (m = i), 0(m i), θ l ρ ii where θ ρ ii = βi 2, θ m ρ ii =(l = m = i), 0 (otherwise). For i j θ lm ρ ii (6.53) Θ ρ ij =( θ lm ρ ij ), 27 = β i (l = i), 0(l i) ad

28 where θ ρ ij =2β i β j, θ m ρ ij i),β i (l = j), 0(l i, j), ad θ lm ρ ij for (2 l, m +G 2 ). The we have the represetatio (6.54) ( ) βi (k) Θ 2τ (k) tr T = ρ ij = β j (m = i),β i (m = j), 0(m i, j), θ l ρ ij = β j (l = =(l = i, m = jorl= j, m = i), 0 (otherwise) +2τ (k) i 2β i β j τ (k) +2τ (k) β i + τ (k) ii (i = j) j β i +2τ (k) i I the matrix form we have a simple relatio as β j +2τ (k) ij (i j). (6.55) τ (k) β 2 β 2 + τ (k) 2 β 2 + β 2 τ (k) 2 + T (k) 22 = O. The we have the represetatio T (k) 22 = τ (k) β 2 β 2 τ (k) 2 β 2 β 2τ (k) = τ (k) β 2 β Φ 22. ɛ kβ 2 + β 2ɛ k Φ 22.]. Next we cosider the role of the secod matrix i (6.44). By differetiatig (6.45) with respect to ω ij (i, j =,, +G 2 ), we have the coditio c φ k = φ k (k =2,, +G 2 ; i, j =,, +G 2 ) g ij h ij evaluated at the probability limits. Let (6.56) S = G cc H = s s 2 s 2 S 22. Sice φ( ) is differetiable ad its first derivatives are bouded at the true parameters by assumptio, the liearized estimator of β k i the class of our cocer ca be represeted as +G 2 g,h= τ (k) gh s gh = τ (k) s +2τ (k) 2 s 2 + tr T (k) 22 S 22 ] = τ (k) s + ( ɛ kφ 22. 2τ (k) = τ (k) β 2 ) s2 + tr ( τ (k) ) ] β 2 β 2 Φ 22.ɛ k β 2 S22 s 2β 2s 2 + β 2S 22 β 2 ] + ɛ kφ 22.(s 2 S 22 β 2 ) = τ (k) β Sβ + ɛ kφ 22.(s 2, S 22 )β. Let (6.57) τ = τ (2). τ (+G 2) 28

29 ad we cosider the asymptotic behavior of the ormalized estimator (ˆβ 2 β 2 ) as (6.58) ê = τ β +(0, Φ 22. )] Sβ. Sice the asymptotic variace-covariace matrix of Sβ has bee obtaied by the proof of Theorem, Theorem 2 ad Lemma 5 below, we have = E ê ê ] (τ + (0, Φ σ2 22. )Ωβ)β +(0, Φ 22. )(I G 2 + Ωββ β Ωβ ) ESββ S] (τ + (0, Φ σ2 22.)Ωβ)β +(0, Φ 22.)(I G2 + Ωββ β Ωβ ) = Ψ + E (β Sβ) 2] τ +(0, Φ 22. ) ] σ Ωβ τ 2 + σ 2 β Ω ] 0 Φ 22. ] + o(), where Ψ has bee give by Theorem or Theorem 2. This covariace matrix is the sum of a positive semi-defiite matrix of rak ad a positive defiite matrix. It has a miimum if (6.59) τ = (0, Φ σ2 22. )Ωβ. Hece we have completed the proof of Theorem 4. : Q.E.D. Lemma 5 : Uder the assumptios of Theorem 2, (6.60) (0, I G2 )I G2 + Ωββ β Ωβ ]E Sββ Sβ Z ] = o p (). Proof of Lemma 5 : We eed to evaluate each term of E { u Z 2. A 22. Z 2. u c u (I Z(Z Z) Z )u ] Π () 22 Z 2. u + W 2 Z 2.A 22. Z 2. u c W 2 (I Z(Z Z) Z )u ] Z }, where W 2 = V 2 (0, I G 2 )Ωβu /σ 2. By usig the similar calculatios as (6.2)-(6.4) o the third ad fourth order momets, it is equivalet to i= ( Π () 22 z () i p ii c ( q () ii ) ) E(u 3 i )+ i= ( p () ii c ( q () ii ) ) 2 E(u 3 i w 2i ). The by usig Lemma, we have the desired result. Q.E.D 29

30 Proof of Theorem 5 We use the argumets i a parallel way to the proof of Theorem. I the oliear case we set G = Π (z) 2 Z 2. A 22. Z 2. Π(z) 2 +V Z 2. A 22. Z 2. V+Π(z) 2 Z 2. A 22. Z 2. V+V Z 2. A 22. Z 2. Π(z) 2 ad (6.6) H = Π (z) 2 I ZA Z ]Π (z) 2 + V I ZA Z ]V +Π (z) 2 I ZA Z ]V + V I ZA Z ]Π (z) 2, where Π (z) 2 = Π () 2Z β, I G 2 ] ad Π () 2Z is give by (3.5). Because of Coditio (VII), (/q )H (/q )V I ZA Z ]V = o p (), the the essetial argumets of the proof of Theorem hold. I the third case, however, we otice that the ocetrality term (i.e. the first term) of (/)G is of smaller order tha the secod term (/)V Z 2. A 22.Z 2.V. Hece i this case because (/)G p cω ad (/q )H p Ω, we fid (6.62) cω plimλ Ω =0 ad hece plimλ = c. The by usig (2.8) we cosider (6.63) d 2 ( G cω) (λ c)ω c( q H Ω) By evaluatig each terms as i the proof of Theorem, (6.64) β 2 I G2 Φ 22. (β 2, I G2 ) plimˆβ LI = o p () ] plimˆβ LI = o p (). ad thus ˆβ p LI β as because Φ22. is osigular. For the asymptotic ormality of the LIML estimator, we use the similar argumets as (6.6)-(6.8) i the proof of Theorem. I the preset case, the equatio correspodig to (6.8) becomes (0, I G2 )I G2 + σ 2 Ωββ ](G d 2 (6.65) cc H )β = Φ 22. (ˆβ 2.LI β 2 )+o p (), where G ad H are defied i a similar way as the proof of Theorem. Because d 2 / 0, the first term of (6.9) coverges to zero vector ad Ξ 3.2 = O as. The we have the result. The proof of optimality is similar to Theorem 4, which is omitted. Q.E.D. 30

31 Refereces ] Aderso, T.W. (976), Estimatio of Liear Fuctioal Relatioships : Approximate Distributios ad Coectios to Simultaeous Equatios i Ecoometrics, Joural of the Royal Statistical Society, B, Vol. 38, ] Aderso, T.W. (984), Estimatig Liear Statistical Relatioships, Aals of Statistics, Vol. 2, ] Aderso, T.W. (2003), A Itroductio to Multivariate Statistical Aalysis, Joh-Wiley, 3rd Editio. 4] Aderso, T.W. (2005), Origis of the Limited Iformatio Maximum Likelihood ad Two-Stage Least Squares Estimators, Joural of Ecoometrics, Vol. 27, -6. 5] Aderso, T.W. ad N. Kuitomo (992), Asymptotic Distributios of Regressio ad Autoregressio Coefficiets with Martigale Differece Disturbaces, Joural of Multivariate Aalysis, Vol. 40, ] Aderso, T.W., N. Kuitomo, ad Y. Matsushita (2005), A New Light from Old Wisdoms : Alterative Estimatio Methods of Simultaeous Equatios with Possibly May Istrumets, Discussio Paper CIRJE-F- 32, Graduate School of Ecoomics, Uiversity of Tokyo ( 7] Aderso, T.W., N. Kuitomo, ad Y. Matsushita (2008), O Fiite Sample Properties of Alterative Estimators of Coefficiets i a Structural Equatio with May Istrumets, Discussio Paper CIRJE-F-576, Graduate School of Ecoomics, Uiversity of Tokyo, forthcomig i the Joural of Ecoometrics. 8] Aderso, T.W., N. Kuitomo, ad T. Sawa (982), Evaluatio of the Distributio Fuctio of the Limited Iformatio Maximum Likelihood Estimator, Ecoometrica, Vol. 50, ] Aderso, T.W. ad H. Rubi (949), Estimatio of the Parameters of a Sigle Equatio i a Complete System of Stochastic Equatios, Aals of Mathematical Statistics, Vol. 20,

32 0] Aderso, T.W. ad H. Rubi (950), The Asymptotic Properties of Estimates of the Parameters of a Sigle Equatio i a Complete System of Stochastic Equatio, Aals of Mathematical Statistics, Vol. 2, ] Agrist, J. D. ad A. Krueger (99), Does Compulsory School Attedace Affect Schoolig ad Earigs, Quarterly Joural of Ecoomics, Vol. 06, ] Bekker, P.A. (994), Alterative Approximatios to the Distributios of Istrumetal Variables Estimators, Ecoometrica, Vol. 63, ] Bekker, P.A. ad J. va der Ploeg (2005), Istrumetal Variables Estimatio Based o Group Data, Statistica Neerladica, Vol. 59, 3, ] Boud, J., D.A. Jaeger, ad R.M. Baker (995), Problems with Istrumetal Variables Estimatio whe the Correlatio betwee the Istrumets ad the Edogeous Explaatory Variables is Weak, Joural of the America Statistical Associatio, Vol. 90, ] Chao, J. ad N. Swaso (2004), Asymptotic Distributios of JIVE i a Heteroscedastic IV Regressio with may Istrumets, workig paper. 6] Chao, J. ad N. Swaso (2005), Cosistet Estimatio with a Large Number of Weak Istrumets, Ecoometrica, Vol. 73, ] Chao, J. ad N. Swaso (2006), Asymptotic Normality of Sigle-Equatio Estimators for the Case with a Large Number of Istrumets, i Ecoometric Theory ad Practice, ed. by D. Corbae, S.Durlauf, ad B. Hase, Cambridge Uiversity Press. 8] Chioda, J. ad Jasso (2007), Optimal Ivariat Iferece whe the Number of Istrumets is Large, Upublished Mauscript. 9] Doald, S. ad W. Newey (200), Choosig The Number of Istrumets, Ecoometrica, Vol. 69-5, ] Fuller, W. (977), Some Properties of a Modificatio of the Limited Iformatio Estimator, Ecoometrica, Vol. 45,