t-tests in a Structural Equation with Many Instruments

Transcription

1 CIRJE-F-467 t-tests i a Structural Equatio with May Istrumets Yukitoshi Matsushita CIRJE, Faculty of Ecoomics, Uiversity of Tokyo February 2007 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 t-tests i a Structural Equatio with May Istrumets Yukitoshi Matsushita December 2006 Abstract This paper studies the properties of t-ratios associated with the limited iformatio maximum likelihood (LIML) estimators i a structural form estimatio whe the umber of istrumetal variables is large. Asymptotic expasios are made of the distributios of a large t-ratio statistic uder large- asymptotics. A modified t-ratio statistic is proposed from the asymptotic expasio. The power of the large t-ratio test domiates the AR test, the -test by leiberge (2002), ad the coditioal LR test by Moreira (2003); ad the differece ca be substatial whe the istrumets are weak. ey Words May istrumets, Asymptotic expasios, t-ratio, Limited Iformatio Maximum Likelihood(LIML), Liear Simultaeous Equatios System I am grateful to Yoichi Arai ad Naoto uitomo for their support ad suggestios, ad to T.W. Aderso for givig useful commets to the previous versio.

3 . Itroductio I recet microecoometric applicatios, some ecoometricias have used may istrumetal variables i estimatig a importat structural equatio. Oe empirical example of this kid, ofte cited i ecoometric literatures, is Agrist ad rueger (99), who used 78 istrumets i oe of their specificatios. However, i such cases, it has bee foud that approximate distributios of estimators ad statistics based o the covetioal large sample asymptotic theory ca be iaccurate. See, for istace, Aderso, Sawa, ad uitomo (982); Boud, Jaeger, ad Baker (995); ad Aderso, uitomo, ad Matsushita (2005). I order to overcome this problem, several ew test statistics have recetly bee proposed. leiberge (2002) ad Moreira (200) proposed a score-type statistic, while Moreira (2003) proposed a coditioal likelihood ratio (CLR) test, both of which are show to be robust to the weak istrumets. Several papers exted these tests to a more geeral framework icludig heteroscedasticity. See, for istace, leiberge (2005) ad Adrews, Moreira ad Stock (2006). There has bee aother approach to provide better approximatio usig large- asymptotics, where the umber of istrumets () is allowed to icrease with the umber of observatios (). uitomo (980, 982) ad Morimue (983) were the earlier developers of the large- asymptotics, ad they derived asymptotic expasios of the distributios of the k-class estimators icludig the two stage least squares (TSLS) ad the limited iformatio maximum likelihood (LIML) estimators i the case of two edogeous variables. Multivariate first order approximatios to the distributios were derived by Bekker (994) ad Aderso et al (2005). Bekker (994) foud that the large- asymptotics provides better approximatios tha the oe where is fixed eve whe the umber of istrumets is ot large. Hase, Hausma ad Newey (2006) cosider the same model ad show that Bekker (994) stadard error corrects the size problem. This paper focuses o the secod approach. The mai purpose of this paper is to 2

4 explore the fiite sample properties of t-ratio statistics uder the large- asymptotic theory. Sice the t-test is oe of the most commoly used procedures to test hypotheses o a coefficiet i a structural equatio, there have bee several literatures ivestigatig the fiite sample properties of the t-ratio. See Richardso ad Rohr (97), Morimue (989), Hase et al (2006), for istace. Morimue (989) derived asymptotic expasios of the distributios of (stadard) t-ratio statistics associated with the k-class estimators uder the stadard large sample asymptotic theory i the case of ormal disturbaces. This paper exteds his work ito the case with may istrumets. We derive a asymptotic expasio of the ull distributio of (large ) t-ratio statistic based o the LIML estimator uder the large- asymptotics: both i the case of ormal disturbaces ad o-ormal disturbaces. A asymptotic expasio of the distributio of the LIML estimator is also derived, which is ew i the may edogeous variables case. We fid that the absolute values of the secod terms of the asymptotic expasio of the (stadardized) LIML estimator ad large t-ratio are the same but have differet sigs, ad that this secod order term may have a substatial impact o the size distortio of the t-ratio test. Usig the asymptotic expasio of the large t-ratio, a modified t-ratio statistic which does ot iclude terms of order O( /2 ) i the expasio is proposed. I Sectio 2, the model ad t-tests with may istrumets are explaied, ad a large t-test is defied. I Sectio 3, large- asymptotic expasios of the ull distributios of the t-ratio statistic are provided both i the cases of ormal ad o-ormal disturbaces. Some Mote Carlo experimets are provided i Sectio 4, ad coclusios are provided i Sectio 5. All derivatios of theorems are provided i Appedices. 2. The Model ad t-tests with May Istrumets Let a sigle structural equatio be y = Y 2 β + Z γ + u, (2.) where y ad Y 2 are ad G matrices, respectively, of observatios of the 3

5 edogeous variables, Z is a matrix of observatios of the exogeous variables, β ad γ are colum vectors with G ad ukow parameters, ad u is a colum vector of disturbaces. We assume that (2.) is the first equatio i a simultaeous system of G + liear stochastic equatios relatig G + edogeous variables ad ( = + 2 ) exogeous variables. The reduced form of y = (y Y 2 ) is defied as Y = ZΠ + V = (Z Z 2 ) Π Π 2 + (v V 2 ), (2.2) where Z is a matrix of istrumetal variables, Π = (π Π 2 ) ad Π 2 = (π 2 Π 22 ) are ( + G ) ad 2 ( + G ) matrices, respectively, of the reduced form coefficiets, ad (v V 2 ) is a ( + G ) matrix of disturbaces. The rows of V are idepedetly distributed, each row havig mea 0 ad (osigular) covariace matrix Ω = ω 2 ω 2 Ω 22. (2.3) I order to relate (2.) ad (2.2), we postmultiply (2.2) by (, β ), the u = v V 2 β, γ = π Π 2 β, ad π 2 = Π 22 β. (2.4) The matrix (π 2 Π 22 ) is of rak G ad so is Π 22. The compoets of u are idepedetly ormally distributed with mea 0 ad variace σ 2, which is defied to be ω 2β ω 2 + β Ω 22 β. We defie, for ay full colum matrix F, P F = F (F F ) F, P F = I F (F F ) F. (2.5) The LIML estimator of (β γ ) is (ˆβ LI ˆγ LI) satisfyig y y Y P 2 Z (y Y 2 Z ) ˆλ Y P 2 Z (y Y 2 Z ) Z Z 4 ˆβ LI ˆγ LI = 0, (2.6)

6 where ˆλ is the smallest root of y Y P 2 Z (y Y 2 Z ) λ Z y Y 2 Z P Z (y Y 2 Z ) = 0. (2.7) The TSLS estimator of (β γ ) is (ˆβ T S ˆγ T S) satisfyig Y 2 P Z (y Y 2 Z ) Z ˆβ LI = 0. (2.8) ˆγ LI Uder the covetioal (fixed ) asymptotics, both LIML ad TSLS estimators are cosistet ad have the same asymptotic distributios. Let i be a (G + ) colum vector of zeros, apart from its ith elemet which is uity. The stadard t-ratio for testig H 0 : i β γ = 0, (2.9) is t k = i s ˆQ k ii (k) ˆβ k ˆγ k, k = LIML, T SLS, (2.0) where ˆβ ad ˆγ ca be the LIML ad TSLS estimators, s 2 k is a esimator of σ 2 that is give as s 2 k = G (y Y 2 ˆβk Z ˆγ k ) (y Y 2 ˆβk Z ˆγ k ), (2.) ad ˆQ ii (k) is the ith diagoal elemet i the matrix, where ˆQ = Y 2Z(Z Z) Z Y 2 (k )Y P 2 Z Y 2 Y 2Z Z Y 2 Z Z. (2.2) Here, k = for the TSLS estimator, ad k = + ˆλ for the LIML estimator, ad (s 2 ˆQ ii ) is a cosistet estimator of the asymptotic variace of (ˆβ the ull hypothesis of the test. 5 ˆγ )i uder

7 t LIML 0. t TSLS N(0,) Figure : The ull distributios of t LIML ad t T SLS whe the umber of excluded istrumets is 30 ( = 00, 2 = 30, α = 0.5, δ 2 = 30) However, whe the umber of istrumets is large classical ormal asymptotic approximatios may provide poor approximatios to the fiite-sample distributios of IV statistics. Whe the umber of the istrumets is large, the TSLS estimator ca be extremely biased. Aderso et al (2005) show that for 2 = 0 ad 2 = 30, the media of the TSLS (ad GMM) estimators ca be lower tha -.0 ASD(asymptotic stadard deviatio) s. O the other had, the LIML estimator has larger variaces tha the asymptotic variace based o the stadard large sample theory. Figure icludes the empirical ull distributios of the (stadard) t-ratio associated with the LIML ad TSLS estimators whe the umber of the excluded istrumets is 30. Bekker (994) poited out that the large- asymptotic thoery may be suited better to applicatios, where the umber of the (excluded) istrumets ( 2 ) is allowed to icrease with the umber of observatios (). I this paper we cosider the same situatios, that is, the umber of the (excluded) istrumets ( 2 ) is allowed to icrease proportioally with the umber of observatios ():, 6

8 where we defied q =. / = c + O( ), (0 c < ) (2.3) /q = c 2 + O( ), (0 c 2 < ) Uder the large- asymptotics, the asymptotic distributios of the LIML ad TSLS estimators are rather differet. The LIML estimator is cosistet ad asymptotic ormal while the TSLS estimators eve lose cosistecy. The LIML estimator attais the asymptotic lower boud whe the umber of istrumets is large. See uitomo(982) ad Aderso ad uitomo (2006). For this reaso, i this paper, we focus o t-tests based o the LIML estimator. We defie D 2 as follows: ad assume that I D 2 = Π 2 Π 22 0 (2.4) D 2Z ZD 2 p Q, (2.5) where Q is a osigular costat matrix. Uder the sequece (2.3), Aderso ad uitomo (2006) have proved uder certai coditios ˆβ LI β d N(0 Ψ), ˆγ LI γ where Ψ = σ 2 Q + c ( + c 2 )Q Ω 22σ 2 0 q 2 q 2σ 4 Q 0 0 +Q [(Ξ 3 + Ξ 3) + ηγ 4 ]Q, which is idetical to the Bekker (994) variace i the case of the ormal disturbaces. Here we have used the otatios that Ξ 3 = plim D 2 z i [( + c 2 )a () ii i= Γ 4 = E(u 2 i w 2i w 2i) σ 2 E[w 2i w 2i], η = ( + c 2 ) 2 plim a ()2 ii c 2 2, i= c 2 ]E[u 2 i w 2i], q 2 = σ 2 (ω 2 β Ω 22 0 ), w 2i = (v 2i 0 ) u i q 2, Aderso ad uitomo (2006) provided the results oly o ˆβ usig differet otatios. 7

9 ad a () ii = z i(z Z) z i. Ψ ca be estimated cosistetly by ˆΨ = ˆσ 2 ˆQ + ( + ˆλ) ˆQ ˆΩ 22ˆσ ˆq 2ˆq 2ˆσ 4 ˆQ + ˆQ [( ˆΞ 3 + ˆΞ 3) + ˆη ˆΓ 4 ] ˆQ, where ˆΩ = q Y P Z Y, ˆσ 2 = q ˆb Y P Z Y ˆb, ˆq 2 = ˆσ 2 q Y P 2 Z Y ˆb, ˆΞ 3 = [ ˆD () 2z i ( + ˆλ)a ii ˆλ] ˆΠ 2 û i2ŵ 2i, ˆD2 = i= i= ˆΠ 22 0 ˆΓ 4 = û 2 i ŵ2iŵ 2i ˆσ 2 ŵ 2i ŵ i= 2i, ˆη = ( + ˆλ) 2 i= i= I, a ()2 ii ˆλ 2, û i = y i y 2i ˆβ z iˆγ, ad ŵ 2i = ((y 2i ( ˆΠ 2 ˆΠ 22)z i ), 0) û iˆq 2. Here ˆΠ is the OLS estimator of Π i (2.2), ad we have used the otatio that ˆb = (, β LI). The large t-ratio for testig H 0 is give by t large = i ˆΨii ˆβ LI ˆγ LI, (2.6) where ˆΨ ii is the i-th diagoal elemet i the matrix ˆΨ. The estimate of the asymptotic variace depeds o the estimates of the third ad fourth order momets of the distributios of the disturbaces, which make it complicated. However, we will see that these terms seem to have little effects eve whe the distributios of the disturbaces are deviated from the ormal distributio i Sectio 4. See Aderso ad uitomo (2006) for further discussios. 3. Asymptotic Expasios of the Distributios of the Large t-ratio Uder H 0 3. The Case of Normal Disturbaces 8

10 I order to explore the fiite sample properties of the large t-test, asymptotic expasios of the ull distributios of the large t-ratio are derived uder the large- asymptotics i this sectio. We cosider the case of ormal disturbaces first. Whe the rows of V are ormally distributed, a cosistet estimator of the asymptotic variace of the LIML estimator is give by ˆΨ = ˆσ 2 ˆQ + ( + ˆλ) ˆQ q Y P 2 Z Y 2ˆσ 2 Y P q 2 2 Z Yˆbˆb Y PZ Y ˆQ. (3.) A ew assumptio is ecessary for the expasio. Assumptio There exists a costat positive defiite matrix Q = plim ˆQ s.t. ˆQ = Q + Op ( ). (3.2) ad B. The followig theorem is obtaied. The derivatio is provided i Appedices A Theorem Whe the rows of V are ormally distributed, the asymptotic expasio of the distributio of the large t-ratio (2.6) uder the sequece (2.3) is give by P {t large ξ} = Φ(ξ) Ψii (i Ψq 2 )ξ 2 φ(ξ) + O( ), (3.3) where ξ is a (G + ) vector ad Φ(ξ) ad φ(ξ) are the cdf ad the desity fuctio of the stadard ormal distributio, respectively. As i the case with the LIML estimator (see Appedix A), whe c = 0, this asymptotic expasio is idetical to the result uder the stadard large sample theory up to O( /2 ). See Morimue (989). Whe G =, we have a simple expressio of the expasio of the large t-ratio for testig H 0 : β = 0 as follows. 9

11 Corollary Whe G =, the asymptotic expasio of the distributio of the large t-ratio (2.6) for testig H 0 : β = 0 uder the sequece (2.3) is give by P (t large ξ) = Φ(ξ) + α µ ηξ 2 φ(ξ) + o(µ ). (3.4) Here, we use the otatios η = + ( + ν2 ), ν 2 µ = lim 2 τ 2 τ 2, τ 2 µ = lim 2 q µ 2 = σ2 Ω Π 22A 22. Π 22, A 22. = Z 2Z 2 Z 2Z (Z Z ) Z Z 2, α = (ω 22 β ω 2 )/ ad L = 2., L Ω, From Corollary ad A. (i Appedix A), we fid that the absolute values of the secod terms of the asymptotic expasio of the LIML estimator ad large t-ratio are the same but have differet sigs. This implies that the distributios of the LIML estimator ad large t-ratio are skewed i opposite directios. We will later see that this secod order term may have a substatial impact o the size distortio of the t-test. We fid from the asymptotic expasio (3.3) that there exists a simple adjustmet of the t-ratio statistic which does ot iclude terms of order O( /2 ) i the expasio. We propose a adjusted t-ratio as t adj = t large ˆΨii (i ˆΨˆq2 )t 2 large, (3.5) where ˆΨ is defied by (3.), ad ˆq 2 is a estimator of q 2 where ω 2, Ω 22, ad β are cosistetly estimated by q Y 2 P Z y, q Y 2 P Z Y 2, ad ˆβ LI, respectively. 3.2 The Case of No-ormal Disturbaces I order to ivestigate the effects of the ormality assumptio for disturbaces, the asymptotic expasio of the distributio of the large t-ratio uder H 0 is derived i the case of o-ormal disturbaces. Oe coveiet class of uderlyig disturbaces is the elliptically cotoured distributio, which cotais may importat distributios icludig the multivariate ormal distributio, the multivariate t distributio ad the uiform distributio o the sphere i R p. 0

12 Whe the rows of V are followed by the class of elliptically cotoured distributio EC(Ω) 2, the asymptotic variace of the LIML estimator Ψ is give by Ψ = σ 2 Q + {c ( + c 2 ) + ηκ}q Ω 22σ 2 0 q 2 q 2σ 4 Q, (3.6) 0 0 where κ = (E(u 4 i )/σ 4 3)/3. (Aderso ad uitomo (2006)) Hece, the large t-ratio for testig H 0 is give by t large = i ˆβ LI, (3.7) ˆΨ ˆγ ii LI where ˆΨ is a cosistet estimator of Ψ usig ˆη = ( + ˆλ) 2 i= a ()2 ii ˆλ 2 ad ˆκ = ( i= (y i y 2i ˆβ z iˆγ)4 3)/3 for estimatig η ad κ i (3.6), respectively. ˆσ 4 We obtai the ext result. The derivatio is provided i Appedix C. Theorem 2 Let the rows of V be followed by the class of elliptically cotoured distributio EC(Ω). I additio to Assumptio, we assume that i= a () ii = plim i= a () ii + O p ( ). The asymptotic expasio of the ull distributio of the large t-ratio (3.7) uder the sequece (2.3) is give by P {t large ξ} = Φ(ξ) Ψ ii (i Ψ q 2 )ξ 2 φ(ξ) + O( ). (3.8) From Theorem 2, whe the disturbaces are followed by the class of the elliptically cotoured distributio, the asymptotic expasio of the distributio of the large t-ratio uder H 0 has the same form as that i the case of ormal disturbaces with Ψ replaced by Ψ. 4. Mote Carlo Experimets Empirical distributios by Mote Carlo studies are obtaied i order to examie the quality of the precedig asymptotic approximatios to the fiite sample distributios of the LIML estimator ad t-ratio statistics. We cosidered models with 2 The precise defiitio of elliptically cotoured (EC) distributio has bee give by Sectio 2.7 of Aderso (2003).

13 two edogeous variables, i.e., G =. I this case, the distributios of the LIML estimator ad the t-ratios for a coefficiet o edogeous variables deped oly o the key parameters used by Aderso et al (982), which are 2, the umber of excluded exogeous variables;, the umber of degrees of freedom i ˆΩ; the ocetrality parameter associated with (2.); ad δ 2 = Π 22A 22. Π 22 ω 22, (4.) α = ω 22β ω 2 Ω /2 ρ =, (4.2) ( ρ 2 ) /2 where ρ is a correlatio betwee u ad v 2. The umerator of the ocetrality parameter δ 2 represets the additioal explaatory power due to y 2i over z i i the structural equatio, ad its deomiator is the error variace of y 2i. Hece, the ocetrality parameter δ 2 determies how well the equatio is defied i the simultaeous equatios system. We use the DGP y = y 2 β (0) + Z γ (0) + u, (4.3) ad y 2 = ZΠ (0) 2 + V 2, (4.4) where =, Z N(0, I I ), (u, V ) N(0, Σ I ), Σ = ρ, ad the ρ true values of parameters β (0) = γ (0) = 0. We have cotrolled the values of δ 2 by choosig a real value of c ad settig ( + 2 ) vector Π (0) 2 = c(,, ). The umber of repetitios i each experimet is 20,000. The accuracy of our simulatio method has bee carefully examied by uitomo ad Matsushita (2003a). 4. Distributios of t-ratios o β Uder H 0 2

14 The empirical small sample distributios are calculated for a stadardized form of the LIML estimator Π 22A 22. Π 22 ê β = ( σ ˆβ β), ad the three types of t-ratios t LIML, the large t-ratio (t large ), ad the adjusted large t-ratio (t adj ) uder H 0. The ull hypothesis H 0 was imposed so that the true coefficiet is zero. I Tables -3, the 5, 0, 50, 90, ad 95 percetiles, ad the observed sizes at the 0% ad 5% asymptotic critical values are tabulated. I Figure 2, graphs of N(0, ), ad empirical ull distribuitios of t, t large, t adj are give i the case of = 00, 2 = 30, α =, ad δ 2 = 30. From the tables, the distributio of t LIML is close to the stadard ormal distributio whe α ad 2 are small. As α icreases, a slight asymmetry is observed; ad as 2 icreases, the tails become log, which causes a large differece betwee actual ad omial sizes. For give α, 2, ad, the size distortio become small as δ 2 icreases. For give α, δ 2, ad, the size distortio icreases with 2. The distributio of the large t-ratio (t large ) is closer to the stadard ormal distributio tha t LIML irrespective of whether 2 is small or large. This implies that the large asymptotics are more accurate tha the stadard large sample asymptotics, which agrees with the results reported i Bekker (994). However, the distributio is still skewed whe α is large. It is ofte the case i umerous applicatios that the sig of the parameters is kow from the ecoomic theory ad that the oe-sided test is used. I such cases, the size distortio of the large t-test ca be rather large. The distributio of the adjusted large t-ratio (t adj ) is the closest to the stadard ormal distributio i all the cases. It is close to beig symmetric eve whe α ad 2 are large ad the differece betwee the omial ad real sizes is small i all cases icludig the may weak istrumets cases (δ 2 / 2 = ; see Moreira (2003)). We ote that the small sample distributios of the t-ratio statistics are rather differet from that of the (stadardized) LIML estimator. The distributios of t LIML ad t large have log left tails for α > 0 (ad log right tails for α < 0), while the 3

15 Table : Distributios of t ratios uder H 0 : = 30, 2 = 3, δ 2 = 30 α = 0. α = ormal ê β t t large t adj ê β t t large t adj X X MEDN X X P (t < z 05 ) P (t < z 0 ) P (t > z 90 ) P (t > z 95 ) distributio of the LIML estimator has a log right tail for α > 0 (ad a log left tail for α < 0). The abovemetioed observatios agree with the asymptotic expasios of the cdfs of ê β ad t large i Appedix A ad Sectio 3, respectively. I summary, it may be stated that the distributio of t adj is closest to the stadard ormal distributio i all the cases. The distributio of t LIML is skewed ad extremely deviated from the ormal distributio, particularly whe α ad 2 are large. The distributio of t large is closer to the ormal distributio tha t LIML ; however, it is still skewed ad the size distortio ca be large. However, the differece betwee t adj ad the stadard ormal distributio is small i all cases except whe δ 2 is too small. The actual size of t adj is close to the omial size. 4.2 Power Compariso We coduct power comparisos of the large t-ratio statistic with the Aderso- Rubi (AR) statistic (Aderso ad Rubi (949)), the statistic (leibege(2002)), ad the coditioal likelihood ratio (CLR) statistic (Moreira(2003)). 3 3 We do ot report power results for the likelihood ratio (LR) test ad the stadard t-ratio test because their size properties appear to be rather poor i the situatio cosidered here. 4

16 Table 2: Distributios of t ratios uder H 0 : = 00, 2 = 30, δ 2 = 30 α = 0. α = ormal ê β t t large t adj ê β t t large t adj X X MEDN X X P (t < z 05 ) P (t < z 0 ) P (t > z 90 ) P (t > z 95 ) Table 3: Distributios of t ratios uder H 0 : = 00, 2 = 50, δ 2 = 50 α = 0. α = ormal ê β t t large t adj ê β t t large t adj X X MEDN X X P (t < z 05 ) P (t < z 0 ) P (t > z 90 ) P (t > z 95 )

17 t t* t* adj N(0,) Figure 2: The ull distributios of t, t large, ad t adj : = 00, 2 = 30, α =, δ 2 = 30 Aderso-Rubi (AR) Test Aderso ad Rubi (AR) statsitic is give by AR = (, β 0 )Y (P Z P Z )Y (, β 0 ) (, β 0 )Y P Z Y (, β 0 ) /( ). (4.5) Because, uder the ull hypothesis, we have AR = u (P Z P Z )u u P Z u/( ), (4.6) the ull distributio of the AR statistic does ot deped o δ 2. Thus the AR test is oe of the testig procedures which are robust to weak istrumets. Uder either the stadard large sample asymptotics or weak-istrumet asymptoics, AR d χ 2 ( 2 ) uder the ull hypothesis. Score-type Test Defie the statistics S = (P Z P Z )Y b 0 (b 0 Ωb 0 ) /2 (4.7) 6

18 ad T = (P Z P Z )Y Ω β 0 (β 0, I G )Ω β 0 /2, (4.8) I G I G ad Ŝ ad ˆT deote S ad T evaluated with ˆΩ = Y P Z Y /( ) replacig Ω, where b 0 = (, β 0 ). lieberge (2002) proposed the statistic = Ŝ ˆT ( ˆT ˆT ) ˆT Ŝ. (4.9) leiberge showed that uder either the stadard large sample asymptotics or weak-istrumet asymptoics, robust to the weak istrumets. Coditioal Likelihood Ratio (CLR) Test d χ 2 (G ) uder the ull hypothesis, i.e. The likelihood ratio (LR) statistic for testig H 0 : β = β 0, whe Ω is kow, is give by LR = b 0Y (P Z P Z )Y b 0 b 0Ωb 0 mi b b Y (P Z P Z )Y b b. (4.0) Ωb Moreira (2003) showed that the LR statistic is a fuctio of S ad T defied i (4.7) ad (4.8), ad that, i the fixed-istrumets ad ormal-distubaces model with kow Ω, if its critical value is computed from the coditioal distributio give T this coditioal likelihood ratio (CLR) test is similar (i.e. fully robust to weak istrumes). Moreira (2003) suggested computig the ull distributio by Mote Carlo simulatio or umerical itegratio. I practice, Ω is ukow. However, Ω ca be cosistetly estimated by ˆΩ = Y P Z Y /( ) uder the weak-istrumet asymptotics, ad the coditioal likelihood ratio (CLR) test based o the plug-i value of Ω ca be show to be asymptotically robust to weak istrumets uder the geeral coditios (stochastic istrumets ad oormal disturbaces. ) We geerate 5,000 datasets from DGP (4.3) ad (4.4) for various values of β ad report size-corrected power curves at the 5% sigificace level. We also use 5,000 realizatios each of χ 2 () ad χ 2 ( 2 ) radom variables to simulate the critical 7

19 values of Moreira s CLR statistic. Figures 3-6 display the power curves i the case i which 2 = 3. Figures 7-0 display the power curves i the case of may (weak) istrumets 2 = 30 ad δ 2 = 30. Our results are similar to the results of leiberge (2002), Moreira(2003), ad Guggeberger ad Smith (2005): (i) The power of the AR test decreases substatially whe the umber of istrumets icreases, (ii) The CLR test is usually more powerful tha the AR ad the tests, ad (iii) The most importat fidig of our experimets is power curve of the large t-ratio uiformly domiates the power curves of the other tests. Occasioally, their differeces are rather large. 4.3 Effects of Normality Sice the distributios of estimators ad t-ratio statistics deped o the distributios of the disturbaces, we have ivestigated the effects of the o-ormality of disturbaces. We calculated a large umber of cases i which the distributios of disturbaces are skewed (χ 2 (3)) ad have log tails (t(3)). We have chose the case of = 00, 2 = 30, α =, ad δ 2 = 30 ad reported the 5, 0, 50, 90, ad 95 percetiles of the ull distributios ad the observed sizes at the 0% ad 5% asymptotic critical values of t, t large, t adj, ad t large i Table 4. From these experimets, the size properties of the three t-ratio statistics, t, t large, t adj, which are derived uder the assumptio of ormal disturbaces, are approximately valid eve if the distributios of disturbaces are deviated from ormal. The power curves chage slightly whe the distributios of the disturbaces have log tails. Figures -2 cotai the power curves whe the distributios of disturbaces are t(3) distributios. However, the large t-test cotiues to uiformly domiate the others i these cases. 5. Coclusios Whe the umber of istrumets is large, the ull distributio of the stadard t-ratio (t LIML ) is skewed ad extremely deviated from the ormal distributio. The 8

20 Table 4: Distributios of t ratios uder H 0 (The Cases of No-ormal Disturbaces): = 00, 2 = 30, δ 2 = 30 = 00, 2 = 30, δ 2 = 30, α = u i = (χ 2 (3) 3)/ 6 u i = t(3) ormal t t large t adj t large t t large t adj t large X X MEDN X X P (t < z 05 ) P (t < z 0 ) P (t > z 90 ) P (t > z 95 ) ull distributio of a large t-ratio (t large ) is closer to the ormal distributio, but it cotiues to be skewed ad the size distortio ca be large, particularly for the oe-sided test. I order to explore the fiite sample properties of the large t-ratio, we derived a asymptotic expasio of the ull distributio both i the cases of the ormal ad o-ormal disturbaces. We proposed a adjusted large t-ratio (t adj ) from the asymptotic exapasio. The actual size of t adj is show to be close to the omial size. We also have foud that the power of the large t-ratio test domiates the AR test, the -test, ad the coditioal LR test. It may be stated that the large t-test should be used whe the umber of istrumets is large (except whe the istrumets are too weak). Whe we kow the sig of the parameter from the ecoomic theory, the use of the modified large t-ratio statistic is recommeded as a more accurate test procedure. 9

21 AR CLR large t Figure 3: Power of tests: = 30, 2 = 3, α = 0., δ 2 = AR CLR large t Figure 4: Power of tests: = 30, 2 = 3, α =, δ 2 = 30 20

22 AR CLR large t Figure 5: Power of tests: = 30, 2 = 3, α = 0., δ 2 = AR CLR large t Figure 6: Power of tests: = 30, 2 = 3, α =, δ 2 = 0 2

23 AR 0. CLR large t Figure 7: Power of tests: = 00, 2 = 30, α = 0., δ 2 = AR CLR large t Figure 8: Power of tests: = 00, 2 = 30, α =, δ 2 = 30 22

24 AR CLR large t Figure 9: Power of tests: = 00, 2 = 30, α = 0., δ 2 = AR 0. CLR large t Figure 0: Power of tests: = 00, 2 = 30, α = 0., δ 2 = 0 23

25 AR 0. CLR large t Figure : Power of tests: = 00, 2 = 30, α = 0., δ 2 = 30, u i = t(3) AR CLR large t Figure 2: Power of tests: = 00, 2 = 30, α =, δ 2 = 30, u i = t(3) 24

26 APPENDIX A. Asymptotic expasio of the distributio of the LIML estimator I this sectio we give the asymptotic expasio of the distributio of the LIML estimator ê LI = uder the large- asymptotics. ê β = ˆβ LI β ê γ ˆγ LI γ (A.) Theorem A. Whe the rows of V are ormally distributed, the asymptotic expasio of the joit distribuito of ê LI uder the sequece (2.3) is give by P {ê LI ξ} = Φ Ψ (ξ) + (q 2ξ)ξφ Ψ (ξ) + O( ), (A.2) where ξ is a (G + ) vector ad Φ Ψ (ξ) ad φ Ψ (ξ) are the cdf ad the desity fuctio of the multivariate ormal distributio with mea 0 ad covariace matrix Ψ, respectively. We ote that whe c = 0, this asymptotic expasio is idetical to the result uder the stadard large sample theory up to O( /2 ). See Fujikoshi et al (982). Whe G =, itegratig (A.2) with respect to the last elemets of ξ, the asymptotic expasio of the margial joit distributio of a stadardized statistic ê β = Π 22A 22. Π 22 ( σ ˆβ LI β) η (A.3) is derived. Here, we use the otatios η = + τ 2 ( + ν2 τ 2 ), ν 2 = lim µ 2 q, τ 2 = lim µ 2 L, µ2 = σ2 Ω Π 22A 22. Π 22, A 22. = Z 2Z 2 Z 2Z (Z Z ) Z Z 2, ad L = 2. 25

27 Corollary A. Whe G =, the asymptotic expasio of the distributio of ê β uder the sequece (2.3) is give by where α = (ω 22 β ω 2 )/ Ω. P {ê β ξ} = Φ(ξ) α µ ηξ 2 φ(ξ) + o(µ ), (A.4) This expasio of the distributio of ê β is idetical to the result of Morimue (983) up to O( /2 ). The derivatio of the asymptotic expasio is provided ext. Derivatio of Theorem A.: I order to derive the asymptotic expasio of the distributio of the LIML estimator, we derive the stochastic expasio of the LIML estimator ê LI first. The LIML estimator is defied by (2.6), which is rewritte as follows: Y (P Z ˆλ P Z )(Y, Z )ê LI Z = Y (P Z ˆλ P Z )(Y, Z ) Z β. γ (A.5) Defiig D = (D D 2 ) = π π 2 I Π 2 Π 22 0, (A.6) we ca write Y Z (P Z ˆλ P Z )(Y, Z ) = {ZD + (V 0)} (P Z ˆλ P Z ){ZD + (V 0)} = D Z ZD + D Z (V 0) + V ZD + 0 V 0 (P Z ˆλ P Z )(V 0). (A.7) We defie E ad E 2 such that V 0 P Z (V 0) = 26 Ω E, (A.8)

28 ad V q 0 P Z (V 0) = Ω q E 2. (A.9) We use the followig otatios: W 2 = (V 2 0) uq 2, (A.0) ad q 2 = σ 2 (ω 2 β Ω 22 0 ). By substitutig (A.7), (A.8), ad (A.9) ito (A.5) ad puttig ê LI = e (0) + e () + o p ( /2 ), (A.) ˆλ = λ (0) + λ () + o p ( /2 ), (A.2) we ca determie each e (i) ad λ (i) successively, which is give as follows: λ (0) = c 2, λ () = c 2 σ 2 { c ( u P Z u) e (0) = Q [ D 2Z u + c2 c W 2P Z u (A.3) q u P Z u}, (A.4) c c c 2 W P q 2 Z u], (A.5) e () = Q [{ D 2Z c (V 2 0) + W 2P Z (V 2 0) (A.6) c c 2 q W 2 P Z (V 2 0)}e (0) + W 2ZD 2 e (0) c λ () Ω 22 0 q c 2 q 2σ 2 e (0) c c 2 λ () q W 2 P Z u]. Each λ (i) is obtaied by premultiplyig (, β, γ ) to (A.5). Each e (i) is obtaied by usig the last G + rows of (A.5). It should be oted that W 2 ad u are ucorrelated whe {v i } are idepedetly distributed. I light of this fact, we ote that the Corish-Fisher expasio of W 2P Z u ad q W P 2 Z u ca be writte as W 2P Z u = X 0 + O p ( ) (A.7) q W 2 P Z u = Y 0 + O p ( ), (A.8) 27

29 where X 0 ad Y 0 are distributed idepedetly as both N(0 σ 2 C 2 ), where C 2 = E[w 2i w 2i]. Hece, we ca rewrite ê LI as ê LI = ẽ (0) + e () + o p ( /2 ), (A.9) where ẽ (0) x = Q [ D 2Z u + c X 0 + c c 2 Y 0 ]. (A.20) We derive a asymptotic expasio of the distributio of ê LI by ivertig the characteristic fuctio of ê LI up to order /2 : C(t) = E[exp(it x)] + E[it E(e () x)exp(it x)] + O( ), (A.2) where t = (t i ) is a (G + ) vector of real variables ad i 2 =. The coditioal expectatio of e (), give the first order term x, is calculated as E(e () x) = x(x q 2 ) + O p ( /2 ). (A.22) The probability P (ê ξ) is approximated to the order /2 by the Fourier iverse trasformatio of the characteristic fuctio (A.2). The iverse trasformatio of the first term is Φ Ψ (ξ). We also use the ext Fourier Iversio formula that was developed by Fujikoshi et al (982): for ay polyomials h( ) ad g( ), ( ) F [h( it)e(g(x)exp(it x))] x=ξ = h g(ξ)φ Ψ (ξ), (A.23) ξ where / ξ = ( / ξ,, / ξ G + ). The, we have P (ê LI ξ) = Φ Ψ (ξ) + (q 2ξ)ξφ Ψ (ξ) + O( ), (A.24) where ξ is a (G + ) vector ad, Φ Ψ (ξ) ad φ Ψ (ξ) are the multivariate ormal cdf ad desity fuctio with mea 0 ad covariace matrix Ψ, respectively. B. Asymptotic expasio of the ull distributio of the large t-ratio statistic 28

30 I order to derive theasymptotic expasio of the ull distributio of the large t-ratio statistic, we eed to expad stochastically each term of ˆΨ as follows: ˆQ = Q + [ D 2Z (V 2 0) + V 2 ZD 2 (B.) 0 + c J 2E J 2 + c c 2 J 2E 2 J 2 λ () Ω 22 0 ] + O p ( ). 0 0 Here, we have used the ( + G + ) (G + ) choice matrix J 2 = (0 I G + ). Hece, ˆQ = Q Q BQ + O p ( ), (B.2) where B = D 2Z (V 2 0) + V 2 0 ZD 2 + c J 2E J 2 c c 2 J 2E 2 J 2 λ () Ω q ˆb Y P Z Y ˆb = {b 0 J 2 e (0) β J 2e () β + } (B.3) {Ω + c2 [ q( c q V P Z V Ω)]}{b 0 = σ 2 + [ 2e (0) β J 2Ωb 0 + J 2 e (0) β J 2e () β + } c2 c qb 0( q V P Z V Ω)b 0 ] + O p ( ), where we have used the otatio b 0 = (, β ). Similarly, q Y P 2 Z Y ˆbˆb Y P Z Y = Ωb 0 b 0Ω + [ Ωb 0 e (0) β J c2 2Ω + Ωb 0 b 0 q( c q V P Z V Ω) ΩJ 2 e (0) c2 β b 0Ω + q( c q V P Z V Ω)b 0 b 0Ω] + O( ) (B.4) 29

31 The, we have ˆΨ = Ψ + Ψ () + O p ( ), (B.5) where Ψ () = Q [ 2e (0) q 2 σ 2 + Q BQ σ 2 c2 c q( q u PZ u σ 2 )] +c ( + c 2 )Q AQ c ( + c 2 )Q BQ Ω 22 0 q 2 q 2σ 4 Q 0 0 c ( + c 2 )Q Ω 22 0 q 2 q 2σ 4 Q BQ, 0 0 A = e (0) β q 2σ 2 c2 + q 22 0 ( 0 0 c 0 0 q u PZ u σ 2 ) c2 + J c 2E 2 σ 2 + q 2 σ 2 e (0) 22 0 β 0 0 c2 q c 2 q( q u P Z (V 2, 0) q 2σ 2 )σ e (0) β q 2σ 2 c2 2 q( 0 0 c q Z u q 2 σ 2 )q 2σ 2. 0 The, the iequality {t ξ} is approximated as x i + t () ξ + O p ( ), (B.6) where x i = i ẽ (0) Ψii (B.7) t () = i e () Ψ () ii x i. (B.8) Ψii 2 Ψ ii The first order term x i is distributed as the stadard ormal radom variable, ad the expectatio of t () coditioal upo x i is calculated as 30

32 E(t () x i ) = Ψii (i Ψq 2 )x 2 i + O p ( /2 ). (B.9) The asymptotic expasio of the distributio is derived by usig the same formula as (A.23). P (t ξ) = Φ(ξ) Ψii (i Ψq 2 )ξ 2 φ(ξ) + O( ), (B.0) where Φ(ξ) ad φ(ξ) are the stadard ormal cdf ad desity fuctio, respectively. The validity of the expasio is give by the same method as that i Appedix C i Fujikoshi et al (982). The radom variables that appear i our aalyses are x = D 2Z u, x 2 = D 2Z W 2, w = (u P Z u σ 2 ), w 2 = q (u P Z u σ 2 ), w 3 = W 2P Z u, w 4 = q W P 2 Z u, w 5 = W 2P Z W 2, ad w 6 = q W P 2 Z W 2. We use the space J where each elemet of x i, wherei =, 2, is i the iterval ( 2c log, 2c log ) ad c is a stadard deviatio of each radom variable; each elemet of w i, where i =,, 6, is i the iterval ( 2log, 2log ). The, P (J ) = o( 2 ), which is proved by Aderso (974). We see that each elemet of e (j) ad t (j) is a homogeeous polyomial of degree j + i the elemets of x i ad w i. The remaider terms of (A.5) ad (B.6) are of the order O( ) uiformly i J. Therefore, the aalysis subsequet to (C.3) i Fujikoshi et al (982) is applicable. C. Derivatio of Theorem 3 Aderso ad uitomo (2006) showed that the limitig distributio of e (0) uder the sequece (2.3) is N(0, Ψ ) whe the disturbaces are followed by the elliptically cotoured distributio. Moreover, we otice that E[w 2i w 2iw 2i u i ] = 0 ad E[w 2i w 2iw 2i u i 3 ] = 0 i this case. Usig these facts, we ca calculate, i the same maer as that i the case of ormal disturbaces, that x i = i ẽ (0) Ψ ii t () = i e () Ψ 2 ii 3 Ψ () ii Ψ ii x i, (C.) (C.2)

33 where Ψ () = Q [ 2e (0) q 2 σ 2 + Q BQ σ 2 c2 c q( q u PZ u σ 2 )] κ = +{c ( + c 2 ) + κη}q AQ {c ( + c 2 ) + κη}q BQ σ2 Ω 22 0 q 2 q 2σ 4 Q 0 0 {c ( + c 2 ) + κη}q σ2 Ω 22 0 q 2 q 2σ 4 Q BQ 0 0 +κ ηq σ2 Ω 22 0 q 2 q 2σ 4 Q, 0 0 3σ 2 [ 4 u 3 i w 2ie (0) 4 u 3 i z i= id 2 e (0) i= + ( u 4 i E(u 4 i )) 2E(u4 i ) c2 q( σ 2 c q u P Z u σ 2 )], i= ad the first order term x i is distributed as the stadard ormal radom variable. Usig Lemma 4.3 give i uitomo ad Matsushita (2005) ad the fact that ay odd momets of the elliptically cotoured distributio is 0, the expectatio of t () coditioal upo x i is calculated as E(t () x i ) = (i Ψ q Ψ 2 )x 2 i + O p ( /2 ). (C.3) ii Hece, we ca derive a asymptotic expasio of the desity fuctio of the large t-ratio by ivertig the characteristic fuctio of t up to O( /2 ), which ca be writte as C(t) = E[exp(itx i )] + E[itE(t () x i )exp(itx i )] + O( ). (C.4) Sice x i is asymptotically ormal, we ca ivert this characteristic fuctio followig the same discussio as that i Sectio 4 i uitomo ad Matsushita (2005). The, we obtai the result. 32

34 Refereces [] Aderso, T.W. (974), A Asymptotic Expasio of the Distributio of the Limited Iformatio Maximum Likelihood Estimate of a Coefficiet i a Simultaeous Equatio System, Joural of the America Statistical Associatio, Vol. 69, [2] Aderso, T.W. (2003), A Itroductio to Multivariate Statistical Aalysis, Joh-Wiley, 3rd Editio. [3] 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, [4] 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, [5] Aderso, T.W., N. uitomo, ad T. Sawa (982), Evaluatio of the Distributio Fuctio of the Limited Iformatio Maximum Likelihood Estimator, Ecoometrica, Vol. 50, [6] Aderso, T.W., N. uitomo, ad Y. Matsushita (2005), A New Light from Old Wisdom: Alterative Estimatio methods of Simultaeous Equatios ad Microecoometric Models, Discussio Paper CIRJE-F- 32, Graduate School of Ecoomics, Uiversity of Tokyo. ( [7] Aderso, T.W., ad N. uitomo (2006), O Asymptotic Properties of the LIML Estimator with possibly May Istrumets, Preprit [8] Adrews, D. W.., M. J. Moreira, ad J. H. Stock (2006), Optimal Two- Sided Ivariat Similar Tests for Istrumetal Variables Regressio with Weak Istrumets, Ecoometrica, Vol. 74,

35 [9] Agrist, J. D. ad A. rueger (99), Does Compulsory School Attedace Affect Schoolig ad Earigs, Quarterly Joural of Ecoomics, Vol. 06, [0] Bekker, P.A. (994), Alterative Approximatios to the Distributios of Istrumetal Variables Estimators, Ecoometrica, Vol. 63, [] 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, [2] Fujikoshi, Y.,. Morimue, N. uitomo, ad M. Taiguchi (982), Asymptotic Expasios of the Distributios of the Estimates of Coefficiets i a Simultaeous Equatio System, Joural of Ecoometrics, Vol. 8, 2, [3] Guggeberger, P. ad R.J. Smith (2005), Geeralized Empirical Likelihood Tests uder Partial, Weak ad Strog Idetificatio, Ecoometric Theory, Vol. 2. [4] Hase, C., J. Hausma, ad W..Newey (2006), Estimatio with May Istrumetal Variables, Upublished Mauscript. [5] leiberge, F. (2002), Pivotal Statistics for Testig Structural Parameters i Istrumetal Variables Regressio, Ecoometrica, Vol.70-5, [6] leiberge, F. (2005), Testig Parameters i GMM without assumig that they are idetified, Ecoometrica, Vol.73, [7] uitomo, N. (980), Asymptotic Expasios of Distributios of Estimators i a Liear Fuctioal Relatioship ad Simultaeous Equatios, Joural of the America Statistical Associatio, Vol. 75, [8] uitomo, N. (982), Asymptotic Efficiecy ad Higher Order Efficiecy of the Limited Iformatio Maximum Likelihood Estimator i Large Ecoomertric 34

36 Models, Techical Report No. 365, Istitute for Mathmeatical Studies i the Social Scieces, Staford Uiversity. [9] uitomo, N. ad Y. Matsushita (2003a), Fiite Sample Distributios of the Empirical Likelihood Estimator ad GMM Estimator, Discussio Paper CIRJE-F-200, Graduate School of Ecoomics, Uiversity of Tokyo. ( [20] uitomo,n. ad Y. Matsushita (2005), Asymptotic Expasios of the Distribuitos of Semi-Parametric Estimators i A Liear Simultaeous Equatios System, Preprit. [2] Moreira, M. (2003), A Coditioal Likelihood Ratio Test for Structural Models, Ecoometrica, Vol. 7-4, [22] Morimue,. (983), Approximate Distribiutios of k-class Estimators whe the Degree of Overidetificatio is Large Compared With Sample Size, Ecoometrica, Vol.5-3, [23] Morimue,. (989), t Test i a Structural Equatio, Ecoometrica, Vol.57-6, [24] Staiger, D. ad J. Stock (997), Istrumetal Variables Regressio with Weak Istrumets, Ecoometrica, Vol. 65,