Improving the Rank-Adjusted Anderson-Rubin Test with Many Instruments and Persistent Heteroscedasticity

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1 CIRJE-F-588 Improvig the Rak-Adjusted Aderso-Rubi Test with May Istrumets ad Persistet Heteroscedasticity Naoto Kuitomo Uiversity of Tokyo Yukitoshi Matsushita Graduate School of Ecoomics, Uiversity of Tokyo September 2008 CIRJE Discussio Papers ca be dowloaded without charge from: Discussio Papers are a series of mauscripts i their draft form. They are ot iteded for circulatio or distributio except as idicated by the author. For that reaso Discussio Papers may ot be reproduced or distributed without the writte coset of the author.

2 Improvig the Rak-Adjusted Aderso-Rubi Test with May Istrumets ad Persistet Heteroscedasticity Naoto Kuitomo ad Yukitoshi Matsushita August 27, 2008 Abstract Aderso ad Kuitomo (2007) have developed the likelihood ratio criterio, which is called the Rak-Adjusted Aderso-Rubi (RAAR) test, for testig the coefficiets of a structural equatio i a system of simultaeous equatios i ecoometrics agaist the alterative hypothesis that the equatio of iterest is idetified. It is related to the statistic origially proposed by Aderso ad Rubi (949, 950), ad also to the test procedures by Kleiberge (2002) ad Moreira (2003). We propose a modified procedure of RAAR test, which is suitable for the cases whe there are may istrumets ad the disturbaces have persistet heteroscedasticities. Key Words Structural Equatio, Likelihood Ratio Criterio, RAAR test, Modified RAAR test May Istrumets, Persistet Heteroscedasticity. KM 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

3 . Itroductio Aderso ad Kuitomo (2007) have developed a likelihood ratio test for a hypothesis about the coefficiets of oe structural equatio i a set of simultaeous equatios, which is called the Rak-Adjusted Aderso-Rubi (RAAR) test. The ull hypothesis is that the vector of coefficiets is a specified vector; the alterative hypothesis is that the structural equatio is idetified. They have derived the limitig distributio of the RAAR test uder the stadard model ad a model of may istrumets situatio. The limitig distributio of 2 times the logarithm of the likelihood ratio criterio is ofte chi-square with degrees of freedom equal to oe less tha the umber of coefficiets specified i the ull hypothesis whe the disturbace terms are homoscedastic. The problem of testig a ull hypothesis o the coefficiets of the structural equatio has bee studied by may ecoometricias sice Aderso ad Rubi (949). See Moreira (2003) ad Adrews, Moreira ad Stock (2006) for a recet review of these studies. Whe there are may istrumets ad/or the disturbace terms are heteroscedastic, the distributio of the RAAR test may ot be a chi-square. The there is a importat questio how to exted the existig testig procedures i such cases, which may be useful for practical applicatios. The mai purpose of this paper is to propose a ew way of improvig the RAAR test, which may be called the MRAAR test. We show that the MRAAR test statistic improves the asymptotic properties of the RAAR test ad may other testig procedures eve whe there are may istrumets icludig the cases of may weak istrumets ad the disturbaces have persistet heteroscedasticity. The particular type heteroscedasticity with may istrumets has bee recetly discussed by Hausma, J., W. Newey, T. Wouterse, J. Chao ad N. Swaso (2007), ad Kuitomo (2008) called it the Persistet Heteroscedasticity. Also we expect that the resultig procedure based o the MRAAR test should have good asymptotic properties because the MRAAR test is essetially the same as the likelihood ratio test uder the stadard situatio. I Sectio 2 we state the structural equatio model ad the alterative testig 2

4 procedures of ukow parameters i simultaeous equatio models with possibly may istrumets. The i Sectio 3 we develop a ew way of improvig the RAAR test procedure ad discuss its asymptotic properties. We also relate our test statistic to the testig procedures developed by Kleiberge (2002) ad Moreira (2003). I Sectio 4 we shall discuss possible extesios ad i Sectio 5 we shall report the fiite sample properties of the ull distributios of the MRAAR test ad other test statistics based o a set of Mote Carlo experimets. The some brief cocludig remarks will be give i Sectio 6. The mathematical derivatios will be give i Sectio Tests i Structural Equatio Models with Possibly May Istrumets Let a sigle liear structural equatio 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, respectively, ad 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 i are mutually idepedet disturbace terms with E(u i ) = 0 ad E(u 2 i z () i ) = σ 2 i (i =,, ). We assume that (2.) is oe equatio i a system of + G 2 edogeous variables y i = (y i, y 2i) ad Y = (y (), Y () 2 ) is a ( + G 2 ) vector observatios of edogeous variables. We cosider (2.2) Y () 2 = Π (z) 2 + V 2, where Π (z) 2 = (π 2i(z () i )) is a G 2 matrix, each row π 2i(z () i ) depeds o a K vector z () i, V () 2 is a G 2 matrix, v () = u+v () 2 β 2, ad V = (v (), V () 2 ). The we ca write (2.3) where Π (z) Y = Π (z) + V, = (Z γ +Π (z) 2 β 2, Π (z) 2 ) is a (+G 2 ) matrix, Z = (Z, Z 2 ) = (z () i ) is a K matrix of K + K 2 istrumets (z () i 3 = (z i, z () 2i ) ), V = (v i) is a

5 ( + G 2 ) matrix of disturbaces with E(v i z () i ) = 0 ad (2.4) E(v i v i z () i ) = Ω i = The vector of K (= K + K 2 ) istrumets z () i ω.i ω 2.i ω 2.i Ω 22.i. satisfies the orthogoal coditio Eu i z () i ] = 0 (i =,, ). The relatio betwee (2.) ad (2.2) gives u i = (, β 2)v i ad (2.5) σi 2 = (, β 2)Ω i = β Ω i β, β 2 where β = (, β 2). Sice our mai iterest is the applicatio to micro-ecoometric data, we impose the coditio (2.6) Ω i i= p Ω ad Ω is a positive defiite (costat) matrix. The (2.7) σi 2 i= p σ 2 (= β Ωβ > 0). Defie the ( + G 2 ) ( + G 2 ) matrices by (2.8) ad (2.9) G = Y Z 2. A 22.Z 2.Y, H = Y ( I Z(Z Z) Z ) Y, where Z 2. = Z 2 Z A A 2, A 22. = Z 2.Z 2. ad (2.0) A = Z Z 2 is a osigular matrix (a.s.). (Z, Z 2 ) = A A 2 A 2 A 22 The RAAR test developed by Aderso ad Kuitomo (2007) is that the ull hypothesis that H 0 : β = β 0 (i.e. β = (, β (0) 2 ) is rejected if (2.) + ˆβ Gˆβ ˆβ Hˆβ + β 0Gβ 0 β 0Hβ 0 < c(k 2, q ), 4

6 where ˆβ is the solutio of (2.2) G ] λ H ˆβ q LIML = 0 ad λ is the smallest root of (/)G l(/q )H = 0 (q = K ad c(k 2, q ) is a costat to be chose). As a ifluetial study, Aderso ad Rubi (949) proposed the Aderso-Rubi (AR) test, which is to reject H 0 if (2.3) β 0Gβ 0 β 0Hβ 0 > K 2 q F K2,q (ϵ), where F K2,q (ϵ) deotes the ϵ sigificace poit of the F-distributio with K 2 ad q degrees of freedom. We call the left-had side of (2.2) the Rak-Adjusted Aderso-Rubi (RAAR) criterio. Moreira (2003) arrived at a statistic which is similar by a somewhat differet route ad proposed a simulatio based test procedure, which is the coditioal likelihood statistic. 3 The modified RAAR tests 3. Modifyig the RAAR statistic for May Istrumets with Persistet Heteroscedasticity We cosider the situatio that there are may istrumets. Aderso, Kuitomo ad Masushita (2007) have discussed the estimatio problem of the structural equatio of iterest with may istrumets uder a set of assumptios. The basic coditios for may istrumets which Aderso et al. (2007) have used are K 2 (3.) c (0 c < ), Π (z) d 2 2 P 2. Π (z) p (3.2) 2 Φ 22. p as d ( ), where P2. = (p (2.) ij ) = Z 2. A 22.Z 2. ad Φ 22. is a osigular costat matrix. I the followig aalysis we maily cosider the stadard case whe d 2 =. 5

7 Whe c = 0 ad the disturbaces are homoscedastic, Aderso ad Kuitomo (2007) have show that the limitig ull distributio of the RAAR test is the χ 2 distributio with G 2 degrees of freedom uder a set of stadard coditios. Whe 0 < c < ad/or the disturbaces are heteroscedastic, however, it is ot ecessarily χ 2 distributio. The mai reaso why the RAAR test does ot ecessarily have stadard properties whe the disturbaces are heteroscedastic with may istrumets is the presece of icidetal parameters with may istrumets ad the effects of possible correlatio betwee the coditioal covariace Ω i ad p (2.) ii (i =,, ). This prevets from satisfyig the Weak Heteroscedasticity coditio (3.3) (WH) plim i= ] p (2.) ii Ω i c Ω = O. If this coditio is ot satisfied, we say that the disturbace terms have the Persistet Heteroscedasticity coditio as (PH). Let P Z = (p () ij ) = Z(Z Z) Z, Q Z = (q () ij ) = I P Z ad P Z = Z (Z Z ) Z be projectio matrices. The we ca utilize the relatios P 2. = (I P Z )P Z (I P Z ) ad Q Z = (I P Z )(I P Z )(I P Z ). We costruct P M = (p (m) ij ) ad Q M = (q (m) ij 0 (i, j =,, ) ad (3.4) ) = I P M such that p (m) ij plim i= = p () ij p (m) ii c ] 2 = 0. The we defie two ( + G 2 ) ( + G 2 ) matrices by (i j), p (m) ii K 2 / p (3.5) G M = Y P MY ad (3.6) H M = Y Q MY, where P M = (p ij) = (I P Z )P M (I P Z ) ad Q M = (q ij) = (I P Z )Q M (I P Z ). By usig G M ad H M, the MRAAR test procedure is defied by the ull hypoithesis 6

8 H 0 : β = β 0 is rejected if (3.7) MRAAR = ( )( K ) log + ˆβ MG M ˆβM ˆβ MH M ˆβM + β 0G M β 0 β 0H M β 0 > c (K 2, q ), where c (K 2, q ) is a costat, ˆβ M is the modified LIML (MLIML) estimator defied by the solutio of (3.8) G M ] λ H M ˆβ q M = 0 ad λ is the smallest root of (/)G M l(/q )H M = 0. The choice of c (K 2, q ) will be discussed i Sectio 3.3. The testig procedure ad the MRAAR test statistic ca be ofte very close to the RAAR testig procedure; they are exactly the same i the case whe there are (orthogoal) or dummy istrumetal variables such that (/)A 22. = I K2 ad p () ii = K /. 3.2 Asymptotic Properties of the MRAAR test We shall ivestigate the asymptotic properties of the MRAAR test statistic whe there are may istrumets. Oe of the attractive features of the MRAAR statistic is that it satisfies (3.4), which leads to (WH) with p () ij (i =,, ) uder (2.6) withi the LIML estimatio. (See Sectio 4 of Aderso et al. (2007) ad Kuitomo (2008).) This coditio plays a importat role for the asymptotic variace of the LIML estimator, which is free from the form of higher order momets of disturbaces. It also makes the limitig distributio of the MRAAR statistic to have a simple form. The we have the represetatio of the limitig distributio of the MRAAR statistic as Theorem ad the proof will be give i Sectio 7. 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 ) mutually idepedet radom vectors such that E(v i z () i ) = 0, E(v i v i z () i ) = Ω i (a.s.) is a fuctio of z (), say, i 7

9 Ω i, z () i ] ad E( v i 4 ) are bouded. For (2.) ad (2.2), suppose (2.6), (3.), (3.9) max π 2i(z () i ) 2 p 0 i ad (3.0) Π(z) 2 (P M c Q M)Π (z) p 2 Φ M is a positive defiite matrix as (q ), where (/)Π (z) 2 P MΠ (z) p 2 Φ M, (/q )Π (z) 2 Q MΠ (z) p 2 Φ 2M, Π (z) 2 = (π 2i (z () )) ad c = c/( c). (i) The (3.) ad (3.2) MRAAR = σ0 2 MRAAR MRAAR i p 0, u (P M c Q M) Π (z) ] (z) 2 + W 2 Π 2 (P M c Q M)Π (z) ] 2 Π (z) 2 + W 2] (P c Q )u, where σ 2 0 = β 0Ωβ 0 ad a G 2 matrix W 2 = (w 2i) is defied by w 2i = v 2i u i (0, I G2 )Ωβ 0 /σ 2 0 (i =,, ). (ii) Furthermore (3.2) ca be decomposed as (3.3) MRAAR = Λ + Λ 2 + 2Λ 3, where Λ = u (P σ0 2 M c Q M)Π (z) (z) 2 Π 2 (P M c Q M)Π (z) ] (z) 2 Π 2 (P M c Q M)u, Λ 2 = u (P σ0 2 M c Q (z) M)W 2 Π 2 (P M c Q M)Π (z) ] 2 W 2(P M c Q M)u, Λ 3 = u (P σ0 2 M c Q (z) M)W 2 Π 2 (P M c Q M)Π (z) ] (z) 2 Π 2 (P M c Q M)u. (iii) If c = 0, the Λ 2 = o p () ad Λ 3 = o p (). I the stadard case whe K is a fixed umber ad the disturbaces are homoscedastic, it has bee kow that the limitig distributio of the RAAR statistic 8

10 is χ 2 with G 2 degrees of freedom uder H 0. More geerally, i that case uder the local alterative (3.4) β = β 0 + ζ, ζ = 0 ζ 2, the limitig distributio of the likelihood ratio statistic is the ocetral χ 2 with G 2 degrees of freedom ad the ocetrality (3.5) ξ = plim ζ 2 ] Π(z) 2 P 2. Π (z) 2 ζ 2. Aderso ad Kuitomo (2007) have exteded this result slightly whe there are icidetal parameters ad c = 0. I this case the power fuctio of the RAAR test attais the asymptotic boud as the stadard situatio. Corollary : Assume that max i σ 2 i σ 2 0 p 0 as. The the limitig distributio of MRAAR uder H 0 is χ 2 with G 2 degrees of freedom if c = 0. I the more geeral situatio three cases ca be cosidered. We have already ivestigated the first case of d = O p ( /2 ) ad K 2 = O(). Aderso et al. (2007) have give the asymptotic covariace of the LIML estimator uder alterative assumptios, which is useful for ivestigatig the limitig behavior of the MRAAR statistic. 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 we have the stadard χ 2 as the limitig distributio uder the ull hypothesis. The third case occurs whe d = o p ( /2 ) ad /d 2 0, which may correspod to the case of may weak istrumets. I this case (3.6) d 2 MRAAR d Λ 2 (c), where d 2 /]Λ 2 (c) d Λ 2 (c) ad Λ 2 (c) follows a weighted χ 2 distributio depedig o c. The secod ad third terms of (3.3) have ofte less impacts o the limitig distributio of the MRAAR statistic because the secod ad the third terms are domiated by the first term of MRAR due to the effects of ad the ocetrality. 9

11 I the case of may weak istrumets, however, the approximatio of the stadard χ 2 distributio could be poor eve if the disturbaces are homoscedastic whe the ocetrality is relatively small. 3.3 Modifyig the RAAR testig procedure There are alterative ways to use Theorem to costruct the testig procedure for H 0 : β = β 0. Oe possible method is to use the simulated distributio of a approximate radom variable of the MRAAR statistic itroduced. For this purpose we utilize (3.7) (P M c Q M)(Π (z) 2 + W 2 ) = (P M c Q M)(Π (z) + V) I G2 + β 0β ] 0Ω 0 β 0Ωβ 0 I G2. Whe u i (i =,, ) are idepedetly distributed, the radom variable i (3.3) ca be further approximated by (3.8) where MRAAR 2 = X Ξ X, X = Ψ M] /2 Π (z) 2 + W 2] (P c Q )u, ad Ξ = ] ] σ0 2 Ψ /2 M Π(z) 2 (P M c Q M)Π (z) ] 2 σ 2 0 Ψ /2 M Ψ M = Π(z) 2 P Σ P Π (z) 2 + i,j= u 2 i w 2j w 2j + w 2i u i w 2ju j ] p ij c q ij which is give i Lemma 2 of Sectio 6, where Σ = (diagu 2 i ). Sice the limitig distributio of X is N G2 (0, I G2 ) ad the rak of Ξ is G 2, the limitig distributio of MRAAR 2 uder H 0 is χ 2 whe the disturbaces are homoscedastic ad c = 0. I the more geeral case uder H 0 whe u i (i =,, ) are idepedetly distributed it ca be re-expressed as (3.9) G 2 MRAAR 2 = a i Xi 2, 0 i= ] 2,

12 where a i (i =,, G 2 ) are the o-zero characteristic roots of Ξ ad Xi (i =,, G 2 ) are idepedetly distributed as N(0, ). Whe u i (i =,, ) are idepedetly distributed but they are possibly heteroscedastic, we eed a further modificatio because each elemets of u ad W 2 are ot ecessarily (asymptotically) idepedet eve i the asymptotic sese. Oe simple way of modificatio is to use (3.20) ˆΛ = σ0 2 (z) ˆΠ 2 P ˆΠ (z) 2 ] ˆΨ M, where P = P M c Q M, ˆΨ M = (3.2) ˆΠ (z) 2 (z) ˆΠ 2 P ˆΣ P (z) ˆΠ 2 + i,j= ad ˆΣ = (diag û 2 i ) are the estimates of Π (z) 2 û2 i ŵ 2j ŵ 2j + ŵ 2i û i ŵ 2jû j ] p ij c q ij ] 2, ad Σ = (diag σ 2 i ), respectively, ad ŵ 2i ad û i (i =,, ) are the residuals uder the ull hypothesis (or the residuaals of the MLIML estimatio). As a simple method, we ca take ˆΠ (z) 2 P (z) ˆΠ 2 = Y 2P Y 2 ad (3.22) ˆΠ (z) 2 P ˆΣ P Whe u i ad w 2i are heteroscedastic, Ew 2i u i ] = (0, I G2 ) (z) ˆΠ 2 = Y 2P ˆΣ P Y 2. Ω i σ2 i Ω σ0 2 are ot ecessarily zero vectors ad the we eed the secod term i order to estimate (3.27). The we ca evaluate umerically the distributio fuctio of ] β (3.23) G 2 MRAAR 3 = λ i Xi 2, i= where λ i are the characteristic roots of ˆΛ ad X i (i =,, G 2 ) are idepedetly distributed as N(0, ). Whe c = 0 ad the disturbaces are homoscedastic, we take P = P M = P 2. ad the (3.2) ca be writte as (3.24) MRAAR 4 = ] β σ 0Y P Y 2 Y 2P 2. Y 2 Y 2P 2. Yβ 0, data. We may use the Mote Carlo experimets to obtai the fiite sample distributio give the

13 which is quite similar to the statistic used by Kleiberge (2002). Thus (3.2) could be regarded as its extesio (or a modificatio) to the case of may istrumets ad the heteroscedastic disturbaces. Because of the form (3.24), the resultig testig procedure based o the fiite sample (coditioal) distributio of (3.8) could be iterpreted as a extesio of the coditioal likelihood ratio (CRL) approach proposed by Moreira (2003) to the may weak istrumet situatio. Whe u i N(0, σ0) 2 (i =,, ) ad c = 0, (3.24) follows the χ 2 distributio with G 2 degrees of freedom exactly. Thus the above testig procedure could be also regarded as a extesio of the stadard likelihood ratio ad the RAAR test procedures. 4. A Extesio The RAAR test ad MARRA test discussed i Sectio 3 ca be exteded to some directios. Let (4.) ad (4.2) G M = H M = Z Y Z Y P M (Z, Y) Q M (Z, Y) be (K + + G 2 ) (K + + G 2 ) matrices ad P M = (p (m) ij defied i Sectio 3.. ) ad Q M = (q (m) ) are We set the true vector θ 0 = ( γ (0),, β (0) 2 ). For the ull hypothesis H 0 : γ = γ (0) (a specified vector), β 2 = β (0) 2 (a specified vector) is rejected if (4.3) MRAAR = ( )( K ) log + ˆθ G M ˆθ ˆθ H M ˆθ + θ 0G M θ 0 θ 0H M θ 0 > c (K 2, q ), ij where ˆθ = ( ˆγ,, ˆβ 2 ) (4.4) is the the solutio of G M ] λ H M ˆθ M = 0 q 2

14 ad λ is the smallest root of (/)G M l(/q )H M = 0. The the ull distributio of the MRAAR statistic ca be developed as Theorem. The proof is similar to Theorem ad it is omitted. Theorem 2 : Suppose the assumptios of Theorem hold. Uder the ull hypothesis H 0, as (q ), (4.5) MRAAR MRAAR 5 p 0, ad (4.6) MRAAR 5 = σ0 2 u (P M c Q M)(Π (z) + W) ] ] Π (z) (P M c Q M)Π (z) (Π (z) + W) (P c P ) ] u, where Π (z) = (Z, Π (z) 2 ) ad W = (O, W 2 ). The ull distributio of the MRAAR statistic i this case ca be geerated as H 0. Whe c = 0, P = P M = P 2. ad Σ = σ 2 I, the limitig distributio is χ 2 with K + G 2 degrees of freedom because u i ad w 2i (i =,, ) are asymptotically ucorrelated i this situatio. More geerally, the extesio of the MRAAR test to ay hypothesis for the subset of parameter vector θ = ( γ,, β 2) ca be costructed i the same way. 5. O Fiite Sample Null Distributios of the MRAAR test ad other statistics We have ivestigated the fiite sample ull distributio of the RAAR statistic ad the MRAAR statistic based o a set of Mote Carlo experimets i a systematic way. We have used the umerical estimatio procedure for the cumulative distributio fuctio (cdf) based o the simulatio ad we have eough umerical accuracy i most cases. See Aderso et al. (2008) for the details of the umerical computatio method. The key parameters i figures ad tables are 3

15 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. (See Aderso et al. (2008) for the detail of these otatios.) I Tables -6 we have give the ull-distributios or the ull-sizes of the MRAAR test statistic. For compariso we also preset the ull-distributios or the ull-size of each test statistics icludig t-type test, RAAR, Kleiberge s test, Aderso-Rubi test, Moreira s Coditioal Likelihood test. Because Hausma et al. (2007) have ivestigated a particular case of persistet heteroscedasticity, we have also tried to reproduce their Mote Carlo experimets. Hece i some tables we have give the ull-distributios or the ull-size of the t-type test give by Hausma et al. (2007) ad the MRAAR test we have developed i their settig. Because they used the otatio slightly differet from ours, we deote the ocetral parameter µ 2 H (= δ 2 ) ad the correlatio coefficiet ρ (= α/ + α 2 ). I additio to these test statistics, we have give the ull-distributios of each test statistics icludig t-type test, RAAR, Kleiberge s test, Aderso-Rubi test, Moreira s Coditioal Likelihood test whe the disturbaces have the persistet heteroscedasticity. Whe the ocetrality is large or moderate as i Tables ad 2, the differeces of sigificace levels are ot large amog the test statistics we have discussed. Whe the ocetrality is small as i Tables 3-6, however, the differeces become sigificat with or without heteroscedasticities. From these tables we have foud that the ull-distributio or the ull-sizes of the MRAAR test is quite robust agaist the cases of may istrumets, may weak istrumets as well as the persistet heteroscedasticities of disturbaces. There is a atural questio o the power compriso of the MRAAR test ad the t-type test 2 whe there are may istrumets ad the disturbaces have persistet heteroscedasticity. We have foud that the empirical sizes of two statistics are ofte similar, but also we have foud that the RAAR test has a better power property agaist the t-type statistic i some situatio whe we eed a two-sided 2 See Matsushita (2006) ad Aderso et al. (2008) for some details of t-statistic i the may istrumets situatio. 4

16 t-test i particular. I this respect, we oly illustrate this situatio by showig the power fuctios of two test statistics as Figure i Appedix. We are curretly ivestigatig the coditios whe we have this pheomeo. 6. Cocludig Remarks I this paper, we propose a particular modificatio of the RAAR test procedure for may weak istrumets ad the heteroscedastic disturbaces. Whe there are may istrumets ad the disturbaces have persistet heteroscedasticity, it might be argued that the RAAR test does ot ecessarily have desirable asymptotic properties as the likelihood ratio test i the stadard asymptotic theory. However, as we have show that a simple modificatio of the RAAR test procedure based o the MRAAR statistic gives a reasoable way of testig hypothesis o coefficiets of structural equatio whe there are may istrumets ad the disturbaces are heteroscedastic. We have show that our test procedure could be iterpreted as the extesios of test statistics developed by Kleiberge (2002) ad Moreira (2002) to the cases of may istrumets ad the persistet heteroscedasticity. As a prelimiary study the MRAAR test has the reasoable power property. The fiite sample properties of the MRAAR test ad other statistics icludig the t-type statistic are curretly uder a further ivestigatio. 7 Mathematical Derivatios I this sectio we give the proof of Theorem. The method of proof is basically a modificatio of the argumets i Sectio 6 of Kuitomo (2008). Some of the details are omitted. Proof of Theorem : We set the true coefficiet vector as β 0. The MRAAR statistic i (3.7) ca be approximately the same as (7.) LR = ( ) ˆβ G ˆβ M ˆβ q H M ˆβ β 0 G Mβ 0 β 0 q H M β 0 5

17 = ( { D ) 2(ˆβ β 0 ) G Mβ 0 ( ) ] N0 H M β q 0 D 0 +(ˆβ β 0 ) G M ( ) ] N0 H M (ˆβ β q D 0 )} 0 + o p (), where N 0 = (/)β 0G M β 0, D 0 = (/q )β 0H M β 0, D = (/q )ˆβ H M ˆβ ad ˆβ is the modified LIML estimator of β by defiig the LIML estimatio with G M ad H M. We use the fact that D D 0 = o p () uder H 0 ad G Mβ 0 ( ) ] N0 (7.2) H M β q 0 = D 0 I G2 + Ωβ 0β ] 0 ( β 0Ωβ 0 G M G (0) M )β 0 cc q ( ] H M H (0) M )β q 0 where G (0) M = plim G M ad H (0) M = plim q H M. We first prepare the followig lemma. Lemma : As ad q, (7.3) ad (7.4) Proof : V P MV p c Ω V Q q MV p Ω. Because {v i } are mutually idepedet ad we have (2.6), we eed to ivestigate the diagoal parts of P M ad Q M. We use the relatio tr(p M) = tr(i P Z )P M (I P Z ) = tr(i P Z )P Z (I P Z ) + tr(i P Z )(P M P Z )(I P Z ) = tr(p Z P Z ) + tr(p M P Z )(I P Z ). Sice tr(p Z P M ) = o p (), p ii (i =,, ), K / < ad tr (P Z P M )P Z ] = tr (diag(p ii K ] ))P Z 2K, we fid that (/)tr(p M) c as ad we have (7.3). Also we use the relatio q tr(i P Z )Q M (I P Z ) 6,

18 = q tr I P Z ] + q tr(i P Z )(P Z P M )(I P Z ) = + q tr(p Z P M )(I P Z ) = + q tr(p Z P M ) q tr(p Z P M )P Z. The by usig a similar argumet we have (7.4). Q.E.D. The ext argumet is a modificatio of Kuitomo (2008) ad thus the developmet should be oly sketchy to avoid some duplicatio. By substitutig the relatio Y = Π (z) + V ito G M ad H M, we have G M = β(0) 2 Π (z) 2 + V (7.5) P M I G2 ( β (0) 2, I G2 ) Π (z) 2 + V ] p G 0 = B Φ MB + c Ω, ad (7.6) q H M = β(0) 2 q I G2 Π (z) 2 + V Q M ( β (0) 2, I G2 ) Π (z) 2 + V ] p H 0 = B Φ 2MB + Ω, where B = (β (0) 2, I G2 ) ad ad β 0 = (, β (0) 2 ) p. The (3.8) implies λ c ad p N 0 /D 0 c because of the rak codito o Π (z) ad (3.0). The ˆβ p M β0 ad G M N 0 p H M B plim ] (7.7) D 0 Π(z) 2 (P M c Q M)Π (z) 2 B. Defie G, H, λ, ad b by G = ( G M G 0 ), H = q ( q H M H 0 ), λ = (λ c) ad b = (ˆβ M β 0 ). From (3.8), we have G 0 c H 0 ]β 0 + G λ H 0 ]β 0 + G 0 c H 0 ]b + q ch ]β 0 = o p ( ) ad the for ˆβ M = (, ˆβ 2M) ad β = (, β (0) 2 ), (7.8) B Φ M ˆβ2.LI β 2 ] = (G λ H 0 cc H ) β 0 + o p (). 7

19 By multiplyig (7.5) from the left by β 0 = (, β 2), we fid λ = β 0(G cc H )β 0 + o β p (). Ωβ 0 Also by mutilyig (7.6) o the left by (0, I G2 ) ad substitutig λ, we have ˆβ2.LI β (0) ] 2 = Φ (7.9) M (0, I G2 ) I +G2 Ωβ 0β 0 β 0Ωβ 0 By usig the relatio Vβ 0 = u, we have the represetatio ] (G cc H ) β 0 + o p (). (G cc H ) β 0 = Π (z) 2 (P M c Q M) u + V (P M c Q M) ] u. The (7.9) is rewritte as (7.0) where ˆβ2.LI β 2 ] = Φ M = Φ M Π (z) (P M c Q M) u +Φ M 2 0, I G2 ] I +G2 Ωβ 0β ] 0 V (P β M c Q M) u + o p () 0Ωβ 0 Π (z) 2 (P M c Q M)u + Φ c W K2 2(P M c Q M)u, W 2 = (0, I G2 ) M I +G2 Ωβ 0β 0 β 0Ωβ 0 Thus we have obtaied the asymptotic distributio of ˆβ 2M, which ca be summarized as the followig Lemma 2. By usig (7.), (7.2) ad (7.9), LR is asymptotically equivalet to (7.) LR 2 = ( ) (ˆβ β0 ) ] σ0 2 G M H M q ( N0 D 0 ] V. ) ] (ˆβ β0 ) ] + o p (). The by substitutig (7.0) ito LR 2 ad (ˆβ β 0 ) = 0, (ˆβ 2 β (0) 2 ], fially we obtai (3.2) by usig Lemma 2 below. Q.E.D. I our derivatio of this sectio we eed the asymptotic distributio of the MLIML estimator with may istrumets whe there are persistet heteroscedasticity, which 8

20 is summarized i the ext lemma. The proof is silimar to the oe i Kuitomo (2008) ad it is omitted. Lemma 2 : 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 idepedet radom vectors such that E(v i z () i ) = 0, E(v i v i z () i ) = Ω i (a.s.) is a fuctio of z () i ad E( v i 4 ) are bouded. Suppose (2.7), (3.), (3.9) ad (7.2) Π(z) 2 (P M c Q M)Π (z) p 2 Φ M is a positive defiite matrix as ad q. The (7.3) ] d ˆβ2.MLI β 2 N(0, Φ M Ψ MΦ M ) where (7.4) Ψ M = Ψ M + Ψ 2M, Ψ M = plim Ψ 2M = plim i,j,k= i,j= π 2i (z () i )p ij c qij]σ j 2 p jk c qjk]π 2k (z () k ), σ 2 i E(w 2j w 2j z () j ) + E(w 2i u i z () i )E(w 2ju j z () j ) ] p ij c qij] 2, provided that Ψ M ad Ψ 2M coverge i probability as, ad w 2i = v 2i u i (0, I G2 )Ωβ 0 /σ 2 (i =,, ). Refereces ] Aderso, T.W. ad N. Kuitomo (2007), O Likelihood Ratio Tests of Structural Coefficiets : Aderso-Rubi (949) revisited, Discussio Paper CIRJE- F-499, Graduate School of Ecoomics, Uiversity of Tokyo ( 2] 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, 9

21 Graduate School of Ecoomics, Uiversity of Tokyo ( forthcomig i the Joural of Ecoometrics. 3] Aderso, T.W., N. Kuitomo ad Y. Matsushita (2007), O Asymptotic Optimality of the LIML Estimator with Possibly May Istrumets, Discussio Paper CIRJE-F-542, Graduate School of Ecoomics, Uiversity of Tokyo. 4] 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, ] 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, ] Adrews, Doald W.H., Marcelo J. Moreira, ad James H. Stock (2006), Optimal Two-Sided Ivariat Similar Tests for Istrumetal Variables Regressio, Ecoometrica, Vol.74, ] Hausma, J., W. Newey, T. Wouterse, J. Chao ad N. Swaso (2007), Istrumetal Variables Estimatio with Heteroscedasticity ad May Istrumets, Upublished Mauscript. 8] Kleiberge, F. (2002), Pivotal Statistics for Testig Structural Parameters i Istrumetal Variables Regressio, Ecoometrica, Vol.70, ] Kuitomo, N. (2008), Improvig the LIML estimatio with may istrumets ad persistet heteroscedasticity, Discussio Paper CIRJE-F-576, Graduate School of Ecoomics, Uiversity of Tokyo. 0] Matsushita, Y. (2006), t-tests i a Structural Equatio with May Istrumets, Discussio Paper CIRJE-F-399, Graduate School of Ecoomics, Uiversity of Tokyo. 20

22 ] Moreira, M. (2003), A Coditioal Likelihood Ratio Test for Structural Models, Ecoometrica, Vol. 7, APPENDIX : Tables I Tables -6 the ull-distributios or the ull-sizes of alterative test statistics are give. They iclude the t (t LIML ) test, the RAAR test, Aderso-Rubi (AR) test, Kleiberge (K) test, Moreira (CLR) test, Hausma s t test (t HLIML ) ad the modified RAAR (mraar) test. I Figure we give the empirical power fuctios of the RAAR test ad the t-type test i a particular case. 2

23 Table : Empirical sizes of statistics that test H 0 : β = β 0 with K = 00, K 2 = 3 ad homoscedastic errors t LIML RAAR AR K CLR t HLIML mraar δ 2 = 00 α = α = δ 2 = 30 α = α = mraar t HLIM Figure : Power of tests: K = 00, K 2 = 0, α =, δ 2 = 0 22

24 Table 2: Empirical sizes of statistics that test H 0 : β = β 0 with K = 00, K 2 = 0 ad homoscedastic errors t LIML RAAR AR K CLR t HLIML mraar δ 2 = 00 α = α = δ 2 = 50 α = α = Table 3: Empirical sizes of statistics that test H 0 : β = β 0 with = 500, K = 5 ad homoscedastic errors µ 2 H µ 2 H t LIML RAAR AR K CLR t HLIML mraar = 0 ρ = ρ = = 3 ρ = ρ =

25 Table 4: Empirical sizes of statistics that test H 0 : β = β 0 with = 500, K = 5 ad heteroscedastic errors µ 2 H µ 2 H t LIML RAAR AR K CLR t HLIML mraar = 0 ρ = ρ = = 3 ρ = ρ = Table 5: Empirical sizes of statistics that test H 0 : β = β 0 with = 500, K = 20 ad homoscedastic errors µ 2 H µ 2 H t LIML RAAR AR K CLR t HLIML mraar = 20 ρ = ρ = = 0 ρ = ρ =

26 Table 6: Empirical sizes of statistics that test H 0 : β = β 0 with = 500, K = 20 ad heteroscedastic errors µ 2 H µ 2 H t LIML RAAR AR K CLR t HLIML mraar = 20 ρ = ρ = = 0 ρ = ρ =