Cox-type Tests for Competing Spatial Autoregressive Models with Spatial Autoregressive Disturbances

Transcription

1 Cox-type Tests for Competig Spatial Autoregressive Models with Spatial Autoregressive Disturbaces Fei Ji a,, Lug-fei Lee a a Departmet of Ecoomics, The Ohio State Uiversity, Columbus, OH 430 USA Abstract I this paper, we cosider the Cox-type tests of o-ested hypotheses for spatial autoregressive (SAR models with SAR disturbaces. We formally derive the asymptotic distributios of the test statistics. I cotrast to regressio models, we show that the Cox-type ad J-type tests for o-ested hypotheses i the framework of SAR models are ot asymptotically equivalet uder the ull hypothesis. The Cox test i o-spatial settig has bee foud ofte to have large size distortio, which ca be removed by the bootstrap. Cox-type tests for SAR models with SAR disturbaces may also have large size distortio. We show that the bootstrap is cosistet for Cox-type tests i our framework. Performaces of the Cox-type ad J-type tests as well as their bootstrapped versios i fiite samples are compared via a Mote Carlo study. These tests are of particular iterest whe there are competig models with differet spatial weights matrices. Usig bootstrapped p-values, the Cox tests have relatively high power i all experimets ad ca outperform J-type ad several other related tests i some cases. Keywords: Specificatio, Spatial autoregressive model, No-ested, Cox test, J test, QMLE JEL classificatio: C, C, C5, R5. Itroductio There are three geeral approaches i testig o-ested hypotheses: the cetered log-likelihood ratio procedure, kow as the Cox test (Cox, 96, 96; the comprehesive model approach, which ivolves costructig artificial geeral models icludig o-ested models as special cases (Cox, 96; Atkiso, 970; ad the ecompassig approach that tests directly the ability of oe model to explai features of a alterative model (Deato, 98; Dastoor, 983; Mizo ad Richard, 986; Gourieroux ad Mofort, 995. I a cotributio related to the ecompassig approach, Gourieroux et al. (983 exted the Wald We are grateful to the editor ad two aoymous referees for their helpful commets. Correspodig author. Tel.: ; fax: addresses: ji.0@osu.edu (Fei Ji, lee.777@osu.edu (Lug-fei Lee For the defiitio ad overviews of o-ested hypotheses, see McAleer ad Pesara (986, Gourieroux ad Mofort (994, Pesara ad Weeks (00, Pesara ad Dupleich Ulloa (008, amog others. Preprit submitted to Regioal Sciece ad Urba Ecoomics March 8, 03

2 ad score tests to o-ested hypotheses based o the differece betwee two estimators for the alterative model. The comprehesive model approach suffers from the Davies s problem (Davies, 977, which ca be circumveted i various ways. Davidso ad MacKio (98 s J test ca be see as a way to deal with the problem. These well-established procedures may also be very useful for model specificatios i spatial ecoometrics. There are may spatial ecoometric models, e.g., spatial autoregressive models, spatial movig average models (Cliff ad Ord, 98 ad spatial error compoets models (Kelejia ad Robiso, 993, that caot est other models as special cases. I additio, spatial ecoometric models usually ivolve spatial weights matrices which are assumed to be exogeous. As ecoomic theories are ofte ambiguous about spatial weights, we may costruct spatial weights matrices i differet ways, which also lead to o-ested models. The J test, as the most widely used procedure for testig o-ested hypotheses due to its simplicity (McAleer, 995, has bee discussed i spatial ecoometrics by several authors, while other procedures have seldom bee focused o. Aseli (984 illustrates the use of the J test for spatial autoregressive (SAR models with a empirical example ad Aseli (986 presets Mote Carlo results of the J-type tests for SAR models where oly a itercept term is icluded as the exogeous variable. Kelejia (008 formally exteds the J test to SAR models with SAR disturbaces (SARAR models, for short. Piras ad Lozao- Gracia (0 preset some Mote Carlo evidece i support of Kelejia s spatial J test. Burridge (0 proposes to improve Kelejia s spatial J test by usig parameter estimates costructed from the likelihood based momet coditios. Kelejia ad Piras (0 modify Kelejia (008 s spatial J test so that available iformatio is used i a more effective way ad thus may have higher power i fiite samples. Liu et al. (0 exted Kelejia (008 s spatial J test to differetiate betwee models with a o-row-ormalized spatial weights matrix versus a row-ormalized oe i a social-iteractio model. No formal results o other o-ested procedures, as far as we are aware of, have bee derived for spatial ecoometric models. I this paper, we derive asymptotic distributios of the Cox-type tests for SARAR models ad compare them with spatial J test statistics. It is of iterest to derive the Cox-type test statistics. For regressio models, it has bee established that the Cox ad J statistics are asymptotically equivalet uder the ull hypothesis (Atkiso, 970; Davidso ad MacKio, 98; Gourieroux ad Mofort, 994. For the SARAR models, we shall show that the Cox statistics ad the proposed spatial J test statistics i Kelejia (008 ad Kelejia ad Piras (0 are, i geeral, ot asymptotically equivalet uder the ull hypothesis. The differet ways that the Cox-type tests use available iformatio might lead to distict size ad power properties. For compariso purposes, we also preset the exteded Wald ad exteded score tests (Gourieroux et al., 983 for the SARAR models as supplemets (i Appedix B. For the o-spatial settig, may Mote Carlo experimets (see, e.g., Godfrey ad Pesara 983 have See Aseli (984 for a geeral discussio of applyig tests of o-ested hypotheses i spatial ecoometrics.

3 show that the Cox ad J tests ca have large size distortio ad typically reject a true ull hypothesis too frequetly. Horowitz (994 cosiders the use of the bootstrap i ecoometric testig ad fids that it ca overcome the well-kow problem of the excessive size of variats of the iformatio matrix test. Fa ad Li (995 ad Godfrey (998 have suggested bootstrappig the J test ad other o-ested hypothesis tests. Davidso ad MacKio (00 provides a theoretical aalysis of why bootstrappig the J test ofte works well. Burridge ad Figleto (00 umerically demostrate that Kelejia (008 s spatial J test is excessively liberal i some leadig cases ad the bootstrap approach is superior to the asymptotic test. For spatial ecoometric models, Ji ad Lee (0 have show that the bootstrap is i geeral cosistet for statistics that may be approximated by a liear-quadratic form of disturbaces. 3 Usig the result, we show that the bootstrap is cosistet for Cox-type tests i our framework. We compare the fiite sample performaces of various tests as well as their bootstrapped versios by a Mote Carlo study. Our Mote Carlo experimets show that although the Cox-type tests have larger size distortios tha the J-type tests i some cases, the bootstrap ca essetially remove size distortios of both types of tests. The bootstrapped Cox-type tests have relatively high power i all experimets ad outperform the bootstrapped J-type ad several other tests i some cases. The rest of the paper is laid out as follows. Sectio formally derives the asymptotical distributios of the Cox-type test statistics. Sectio 3 shows that the Cox-type ad J-type tests for SARAR models are ot asymptotically equivalet uder the ull hypothesis, ad also briefly compares the two types of tests. Sectio 4 shows that the bootstrap is cosistet for Cox-type tests. Sectio 5 compares the performaces of various test statistics as well as their bootstrapped versios i fiite samples by a Mote Carlo study. Sectio 6 illustrates the use of Cox-type tests with a housig data set. Fially, Sectio 7 cocludes. Some assumptios, expressios, lemmas ad proofs are collected i the appedices.. Cox-type Tests We derive the Cox-type tests for SARAR models i this sectio. The settig of the o-ested testig problem is as follows. A SARAR model as the ull hypothesis H 0 is tested agaist aother SARAR model as the alterative hypothesis H : H 0 : y = λ W y + X β + u, u = ρ M u + ɛ, ( 3 Cosistecy of the bootstrap for a statistic meas that the bootstrap ca provide a cosistet estimator for the asymptotic distributio of the statistic. O the questio that whether the bootstrap ca provide asymptotic refiemets, i.e., whether the bootstrap ca be more accurate tha the first-order asymptotic theory, oly prelimiary results are available. Ji ad Lee (0 establish the Edgeworth expasio for a liear-quadratic form with ormal disturbaces, which ca be used to show the asymptotic refiemets of the bootstrap for a liear-quadratic form. The for a statistic that ca be approximated by a liear-quadratic form, with proper regularity coditios o the remaider term, the bootstrap ca provide asymptotic refiemets. For a liear-quadratic form with o-ormal disturbaces, the Edgeworth expasio has ot be established. 3

4 H : y = λ W y + X β + u, u = ρ M u + ɛ, ( where is the sample size, y is a -dimesioal vector of observatios, W j ad M j are spatial weights matrices with zero diagoals, X j is a k j matrix of exogeous variables, elemets of a -dimesioal vector of disturbaces ɛ j are i.i.d. with mea zero ad fiite variace σj, ad θ j = (λ j, ρ j, β j, σ j for j =, are vectors of parameters to be estimated. Deote S j (λ j = I λ j W j ad R j (ρ j = I ρ j M j with I beig a idetity matrix. Let the true parameter vector of the model ( be θ 0, S = S (λ 0 ad R = R (ρ 0 for short. The X ad X may have differet dimesios. The W j ad M j are i geeral differet, but could be the same i empirical applicatios. A particularly iterestig case i practice is the oe i which we have differet spatial weights matrices W vs W or M vs M i the two models. Let L j (θ j be the log likelihood fuctio of the model (j, for j =,, as if the disturbaces were ormally distributed: L j (θ j = l(π l σ j + l S j (λ j + l R j (ρ j σj [S j (λ j y X j β j ] R j(ρ j R j (ρ j [S j (λ j y X j β j ]. (3 Let ˆθ j be the correspodig quasi-maximum likelihood estimator (QMLE by maximizig L j (θ j. The idea of the Cox-type tests is to modify the log-likelihood ratio [L (ˆθ L (ˆθ ] so that it is approximately cetered at zero uder the ull hypothesis, ad the test whether the modified statistic after beig properly scaled is sigificatly differet from zero. 4 As the test statistics ivolve the QMLEs ˆθ ad ˆθ, we first ivestigate their properties, ad the derive the Cox-type test statistics with the QMLEs. For a correctly specified first order SAR model without spatially correlated disturbaces, Lee (004a has proved that the QMLE is cosistet uder suitable regularity coditios. We ca exted the aalysis to SARAR models. Whe we estimate the alterative model, geerally it might have a differet umber of parameters ad/or variables from that of the data geeratig process (DGP, let aloe the cosistecy to the true values of the DGP. We use the so-called pseudo-true values to study the behavior of the QMLE for the alterative model. 5 For the model (, we defie the pseudo true value θ, to be the vector that maximizes E L (θ, ad we shall show that / (ˆθ θ, is asymptotically ormal. With the pseudotrue values, we ca derive the asymptotic distributio of the Cox-type test statistics by usig the cetral limit theorem for liear-quadratic forms ɛ A ɛ σ0 tr(a + b ɛ (Kelejia ad Prucha, 00, where ɛ is a -dimesioal vector of i.i.d. disturbaces with mea zero ad variace σ0, ad the elemets of the 4 Sice the data geeratig process is ot assumed to have ormally distributed disturbaces ad we will costruct the tests with the cetered log quasi-maximum likelihood ratio, the tests correspod to Aguirre-Torres ad Gallat (983 s geeralized, distributio-free Cox tests. 5 For the defiitio of pseudo-true values, see, e.g., Sawa (978 ad White (98. The pseudo-true values are ofte used for o-ested hypothesis testig problems, see, amog others, Gourieroux et al. (983 ad Gourieroux ad Mofort (994. 4

5 matrix A ad -dimesioal vector b are all o-stochastic. 6 Similar to that i Lee (004a, the cosistecy of ˆθ ca be established by ivestigatig the cocetrated log likelihood fuctio L (φ = max β,σ L (θ with φ = (λ, ρ. For i, j =,, let L j (θ j ; θ i be the expected value of L j (θ j whe the model (i with parameter θ i geerates the data. Thus, i particular, L (θ ; θ 0 = E L (θ ad L (θ ; θ 0 = E L (θ. Deote L j (φ j ; θ 0 = max βj,σ j L j (θ j ; θ 0 with φ j = (λ j, ρ j for j =,. We make the followig assumptios for the cosistecy of ˆθ. Assumptio. {ɛ,i } s i ɛ = (ɛ,,..., ɛ,, i =,...,, are i.i.d. with mea zero ad variace σ0. The momet E(ɛ 4+ζ,i for some ζ > 0 exists. Assumptio. The elemets of X are uiformly bouded costats, X has full colum rak k, ad lim X X exists ad is osigular. Assumptio 3. Matrices S ad R are osigular. Assumptio 4. {W } ad {M } have zero diagoals. The sequeces of matrices {W }, {M }, {R } ad {S } are bouded i both row ad colum sum orms (for short, UB.7 Assumptio 5. {S (λ } is bouded i either row or colum sum orm uiformly i λ i a compact parameter space Λ, ad {R (ρ } is bouded i either row or colum sum orm uiformly i ρ i a compact parameter space ϱ. The true λ 0 is i the iterior of Λ ad the true ρ 0 is i the iterior of ϱ. Assumptio 6. The limit lim X R (ρ R (ρ X exists ad is osigular for ay ρ ϱ, ad the sequece of the smallest eigevalues of R (ρ R (ρ is bouded away from zero uiformly i ρ. 8 Assumptio 7. Either (i lim [l σ 0S R R S l σ,a(φ S (λ R (ρ R (ρ S (λ ] exists ad is ozero for ay φ φ 0, where σ,a(φ = σ 0 tr[r S S (λ R (ρ R (ρ S (λ S R ], or (ii lim (Q X β 0, X (Q X β 0, X exists ad is osigular, ad for ay ρ ρ 0, lim [l σ 0S R R S l σ,a(λ 0, ρ S R (ρ R (ρ S ] exists ad is ozero, where Q = W S. 6 I Kelejia ad Piras (0, the pseudo-true values are ot explicitly discussed for spatial J tests. This is because their tests are based o two-stage least squares (SLS estimators, which have closed forms. Thus, by assumig that some matrices ivolvig the estimators for the alterative model coverge to positive defiite matrices i probability, there is o eed to explicitly cosider the pseudo-true values. 7 A sequece of matrices {A = [a,ij ]} is bouded i row sum orm if there is a costat c such that sup i j= a,ij < c for all, ad is bouded i colum sum orm if there is a costat c such that sup j i= a,ij < c. See Hor ad Johso ( Let µ,ρ be the smallest eigevalue of R (ρ R (ρ. The the secod part of the assumptio meas that there is some costat c > 0 such that if ρ ϱ µ,ρ > c for all. 5

6 Assumptios 5 are similar to those i Lee (004a, except for the additioal coditios o R (ρ which resemble those o S (λ. I practice, the λ ad ρ are typically assumed to be i the iterval (, such that S (λ ad R (ρ are positive, while for the theoretical purpose, the parameter space ca be take to be the compact iterval cotaied i (, so that the cosistecy of the estimator would still hold. 9 Note that R (ρ is liear i ρ, a sufficiet coditio for the first part of Assumptio 6 is that the limit of X [X, (M + M X, M M X ] exists ad has full colum rak. 0 The secod part of Assumptio 6 is required to guaratee the uiform covergece of [L (φ L (φ ; θ 0 ] to zero i probability. As R (ρ R (ρ is positive semi-defiite, its eigevalues are o-egative. The assumptio further limits the eigevalues to be strictly positive for all. Assumptio 7 provides sufficiet coditios for global idetificatio, where (i is related to the uiqueess of the variace-covariace (VC matrix of y ad (ii states that a part of the idetificatio ca be from the asymptotically o-multicolliearity of Q X β 0 ad X. The first part of (ii does ot hold if X cotais a vector of oes ad W is a matrix of equal weights. Propositio. Uder H 0 ad Assumptios 7, ˆθ θ 0 = o P (. The asymptotic distributio of ˆθ ca be derived by applyig the mea value theorem to the first order coditio L(ˆθ θ = 0 at the true θ 0 : ( (ˆθ θ 0 = L ( θ L (θ 0 θ θ, (4 θ 9 To make S (λ positive, the admissible iterval for λ is (/µ,mi, /µ,max, where µ,mi ad µ,max are, respectively, the miimum ad maximum real eigevalue of W. If W with o-egative elemets is row ormalized, the µ,max = ad µ,mi < 0. Thus the iterval is (/µ,mi,, where /µ,mi. The admissible iterval for ρ is similar, thus we oly focus o the admissible iterval for λ. The cocetrated quasi log likelihood fuctio over is L (φ = [l(π + ] l ˆσ (φ + l S (λ + l R (ρ, where ˆσ (φ = y S (λ R (ρ H (ρ R (ρ S (λ y with H (ρ = I R (ρ X [X R (ρ R (ρ X ] X R (ρ, from (A.. By the proof of Propositio 3, ˆσ (φ σ (φ ; θ 0 = o P (, where σ (φ ; θ 0 = σ 0 tr[r S S (λ R (ρ R (ρ S (λ S R ] + (X β 0 S S (λ R (ρ H (ρ R (ρ S (λ S X β 0 is bouded away from zero. The l ˆσ (φ is bouded i probability. I the case that µ,max =, whe λ approaches, l S (λ approaches mius ifiity, thus L (φ approaches mius ifiity i probability, which implies that L (φ at a λ very close to will be smaller tha its value at some λ i the iterior of (, i probability oe. Similarly, whe /µ,mi =, L (φ approaches mius ifiity i probability as λ approaches. Whe /µ,mi <, S (λ at is positive ad fiite. Thus the iterval for λ ca be take to be (, i practice, while it makes o harm to assume the parameter space to be compact. This view is i Amemiya (985, p. 08. I this paper, the QMLE is proved to be cosistet oly for a compact parameter space. 0 Whe X cotais a vector of oes ad M is a matrix of equal weights, X [X, (M + M X, M M X ] doest ot have full colum rak, but the first part of Assumptio 6 may still hold i this case. The coditio is equivalet to that the limit [Q X β 0 ] M X Q X β 0 exists ad is o-zero whe the limit of X X exists ad is osigular, where M X = I X (X X X. Let W = (l l I /(, where l is a -dimesioal vector of oes. The M X W k = ( k M X. Thus M X Q X β 0 = 0 ad [Q X β 0 ] M X Q X β 0 = 0. 6

7 where θ is betwee ˆθ ad θ 0. L I the above equatio, every elemet of (θ 0 θ is a liear-quadratic form of the disturbaces ɛ, thus the cetral limit theorem i Kelejia ad Prucha (00 is applicable. 3 The term L ( θ θ θ ca be show (see the proof of Propositio 4 to be equal to E( L (θ 0 θ θ plus a term covergig to zero i probability. The followig assumptio is eeded for the limit of Σ, = E( L (θ 0 θ θ to exist ad be osigular. Assumptio 8. The limit lim L(φ 0;θ 0 φ φ Propositio. Uder H 0 ad Assumptios 8, exists ad is osigular. (ˆθ θ 0 d N ( 0, lim (Σ, Ω,Σ,, (5 where Ω, = E( L (θ 0 L (θ 0 θ θ ad Σ, = E( L (θ 0 θ θ. I the case that ɛ,i s are ormally distributed, (ˆθ θ 0 d N(0, lim Σ,. The Ω, geerally ivolves the third ad fourth momets of the disturbaces if they are ot ormally distributed, thus it has a form more complicated tha that of Σ,. Whe ɛ,i s are ormally distributed, the iformatio matrix equality holds, i.e., Σ, = Ω,, so the VC matrix has a simpler form. For the alterative model (, the followig assumptios are made for the covergece of ˆθ θ, to zero i probability uder the ull hypothesis of the model (. Deote S = S ( λ, ad R = R ( ρ, for short. Assumptio 9. The elemets of X are uiformly bouded costats, X has full colum rak k, ad lim X X exists ad is osigular. Assumptio 0. Matrices S ad R are osigular. Assumptio. {W } ad {M } have zero diagoals. The sequeces of matrices {W }, {M }, {R } ad {S } are UB. Assumptio. {S (λ } is bouded i either row or colum sum orm uiformly i λ i a compact parameter space Λ, ad {R (ρ } is bouded i either row or colum sum orm uiformly i ρ i a compact parameter space ϱ. Assumptio 3. The limit lim X R (ρ R (ρ X exists ad is osigular for ay ρ ϱ, ad the sequece of the smallest eigevalues of R (ρ R (ρ is bouded away from zero uiformly i ρ. The mea value theorem is applicable to a fuctio but ot a vector-valued mappig. So θ ca be differet for each row of the Hessia matrix. 3 The expressios for L (θ 0 ad some other terms i the text are collected i Appedix A. θ 7

8 Assumptio 4. For η > 0, there exists κ > 0 such that, whe φ φ, > η, ( L ( φ, ; θ 0 L, (φ ; θ 0 > κ for ay large eough. Assumptio 5. The limit of tr[r S S R R S S R ] or (X β 0 S S R H R S S X β 0 exists ad is o-zero. Assumptios 9 3 are similar to those for the estimatio of the model (. With a misspecified model beig estimated, it is ot straightforward to fid primitive idetificatio coditios, so Assumptio 4 is imposed. Assumptio 5 implies that { σ,}, the sequece of pseudo true values for σ, is bouded away from zero by (A.4, which is ecessary to prove the uiform covergece of ( L (φ L (φ ; θ 0 to zero i probability o Λ ϱ. Without this assumptio, L (φ ; θ 0 ca be arbitrarily large. Propositio 3. Uder H 0 ad Assumptios 4, 9 5, ˆθ θ, = o P (. The asymptotic distributio for ˆθ θ, ca be derived by a expasio of the first order coditio that L(ˆθ θ = 0 at θ, : (ˆθ θ (, = where θ is betwee ˆθ ad θ,. Notig that E L( θ, θ every elemet of L( θ, Sice L ( θ θ θ E L ( θ, θ θ L ( θ L ( θ, θ θ, (6 θ = 0 ad L( θ, θ = L( θ, θ E L( θ, θ, θ ca be writte as a liear-quadratic form of the vector of disturbaces ɛ. = E L ( θ, θ θ is osigular i the limit. Assumptio 6. The limit lim L( φ,;θ 0 φ φ Propositio 4. Uder H 0 ad Assumptios 4, 9 6, where Σ, = E( L ( θ, θ θ + o P (, we make the followig assumptio which guaratees that exists ad is osigular. (ˆθ θ, d N ( 0, lim (Σ, Ω,Σ,, (7 ad Ω, = E( L ( θ, θ L ( θ, θ. With asymptotic distributios of the estimators, we are ow ready to derive the Cox-type test statistics. As metioed earlier, the Cox-type tests are based o the recetered log likelihood ratio L (ˆθ L (ˆθ. Thus we eed to fid a expressio for the asymptotic mea of the ratio. Because of the results i Propositios ad 4, we shall show that / [L (ˆθ L (ˆθ ] = / [L ( θ, L (θ 0 ]+o P (. The leadig order term of / [L ( θ, L (θ 0 ] is the expected value / [E L ( θ, E L (θ 0 ], which ca be show by applyig Chebyshev s iequality, as L ( θ, E L ( θ, ad L (θ 0 E L (θ 0 are both liear-quadratic forms of ɛ. The E L ( θ, ivolves the ukow parameters θ, ad θ 0 because a expectatio is take, ad E L (θ 0 ivolves θ 0. Except for ˆθ, aother estimate for θ, ca be the 8

9 vector that maximizes L (θ ; ˆθ. Deote θ (θ = max θ L (θ ; θ. The differece betwee θ (ˆθ ad ˆθ is expected to be small uder the ull hypothesis, sice they are maximizers of two fuctios whose differece is small i probability. 4 Hece, we ivestigate the asymptotic distributio of the statistic [[L (ˆθ L (ˆθ ] [ L ( θ (ˆθ ; ˆθ L (ˆθ ; ˆθ ]], or [ [L (ˆθ L (ˆθ ] [ L (ˆθ ; ˆθ L (ˆθ ; ˆθ ] ], uder H 0. But ote that L (ˆθ = L (ˆθ ; ˆθ, 5 so essetially the tests are based o the statistics [ L (ˆθ L ( θ (ˆθ ; ˆθ ], (8 or [ L (ˆθ L (ˆθ ; ˆθ ]. (9 As (ˆθ θ (ˆθ = O P ( by Propositio 7 i Appedix B, a secod order Taylor expasio implies that [ L (ˆθ ; ˆθ L ( θ (ˆθ ; ˆθ ] = (ˆθ θ (ˆθ L (ˇθ ; ˆθ (ˆθ θ θ θ (ˆθ = o P (, where ˇθ is betwee ˆθ ad θ (ˆθ. Thus, (8 ad (9 are asymptotically equivalet. Note that L ( θ (ˆθ ; ˆθ L (ˆθ ; ˆθ, so the expressio i (8 is smaller tha that i (9. The origial versio of the Cox test is based o (8, while (9 correspods to Atkiso (970 s versio. As show i the proof of Propositio 5, we have [ L (ˆθ L ( θ (ˆθ ; ˆθ ] = [L ( θ, L ( θ, ; θ 0 ] C,Σ, L (θ 0 + o P (, θ (0 where C, = L ( θ,;θ 0 θ. The secod term o the r.h.s. of (0 appears as we estimate θ 0 by ˆθ. The first term o the r.h.s. of (0 ca be writte as a liear-quadratic form of ɛ ad elemets of L(θ0 θ are also of such forms, so the asymptotic distributios of the Cox-type test statistics follow by applyig the cetral limit theorem for liear-quadratic forms. Let σ c, be the variace of [L ( θ, L ( θ, ; θ 0 ] C,Σ, L (θ 0 θ, the σc, = ( [L [, C,Σ, ] var ( θ, L ( θ, ; θ 0, L (θ 0 ] [, C,Σ, ], ( θ 4 The exteded Wald test costructs a asymptotic χ statistic usig the asymptotic ormality of / [ θ (ˆθ ˆθ ], ad the exteded score test costructs a asymptotic χ statistic usig the asymptotic ormality of the score vector L ( θ (ˆθ. Appedix B presets those tests to supplemet the Cox-type tests. θ 5 This ca be see from (A. ad (A. with the estimators plugged i. 9

10 where var( deotes the VC matrix of a radom vector. I the case that ɛ,i s are ormal, C, = E ( L ( θ, L(θ0 θ ad the iformatio matrix equality that Σ, = Ω, ca be applied, so σ c, = var[l ( θ, L ( θ, ; θ 0 ] C,Σ, C,. ( The σ c, ivolves θ,, θ 0, ad also ɛ,i s third ad fourth momets µ 3 ad µ 4 if ɛ,i is o-ormal. Let ˆσ co, ad ˆσ ca, be, respectively, cosistet estimators of σ c, used i Cox ad Atkiso s versios. The ˆσ co, ˆσ ca, may be obtaied, e.g., by replacig θ 0 s i σ c, with ˆθ s, µ 3 ad µ 4 s with the third ad fourth sample momets of the residuals from the quasi-maximum likelihood (QML estimatio, ad θ, s with either θ (ˆθ s or ˆθ s. 6 Propositio 5. Uder H 0 ad Assumptios 6, the Cox-type test statistics Cox o = /[ L (ˆθ L ( θ (ˆθ ; ˆθ ] /ˆσco,, (3 ad Cox a = /[ L (ˆθ L (ˆθ ; ˆθ ] /ˆσ ca,, (4 are asymptotically stadard ormal, if σ c, is bouded away from zero. Sice L ( θ (ˆθ ; ˆθ L (ˆθ ; ˆθ as oted earlier, Cox o Cox a asymptotically uder H 0. We shall digest a little bit more o the two versios of the Cox test uder the alterative hypothesis. Let θ 0 be the true parameter of the model ( which geerates the data, ad θ, be the pseudo true value of the model (. Uder the alterative hypothesis, [ (L (ˆθ L (ˆθ ( L ( θ (ˆθ ; ˆθ L (ˆθ ; ˆθ ] = [ ( L (θ 0 ; θ 0 L ( θ, ; θ 0 ( L ( θ (ˆθ ; ˆθ L (ˆθ ; ˆθ ] + o P (, (5 ad [( L (ˆθ L (ˆθ ( L (ˆθ ; ˆθ L (ˆθ ; ˆθ ] = [( L (θ 0 ; θ 0 L ( θ, ; θ 0 ( L (ˆθ ; ˆθ L (ˆθ ; ˆθ ] + o P (. (6 By Jese s iequality (the iformatio iequality, L (θ 0 ; θ 0 L ( θ, ; θ 0, L (ˆθ ; ˆθ L ( θ (ˆθ ; ˆθ ad L (ˆθ ; ˆθ L (ˆθ ; ˆθ, so the leadig order terms of (5 ad (6 are o-egative. The Cox tests thus have oe-sided critical regios such that we reject the ull hypothesis if the Cox statistics are greater tha the critical value u α, where u α is the ( α quatile of the stadard ormal distributio 6 Note that for Cox a below, if we use ˆθ for θ, i σ c,, the there is o eed to compute θ (ˆθ. I the Mote Carlo study, for Cox o, we use θ (ˆθ for θ, ; for Cox a, we use ˆθ. 0

11 for the chose level of sigificace α. If the leadig order terms of (5 ad (6 are bouded away from zero, ad ˆσ co, ad ˆσ ca, are stochastically bouded uder the alterative hypothesis, the the Cox tests are cosistet. From (5 ad (6, the two Cox-type test statistics are geerally ot asymptotically equivalet uder the alterative hypothesis. 3. Relatioship ad Compariso betwee the Cox-type ad J-type Tests I this sectio, we first ivestigate whether there is a equivalece relatioship betwee the Cox ad J-type tests for SARAR models ad the shortly compare these two types of tests. To ivestigate the relatioship betwee the Cox ad J-type tests for SARAR models, we start from a short review o establishig the asymptotic equivalece of the Cox ad J tests for uivariate regressios uder the ull hypothesis, ad the examie whether a similar relatioship of these two types of tests for SARAR models would exist or ot. Cosider the problem of testig a oliear uivariate regressio model agaist aother oe: H 0 : y i = f i (X,i, β + ɛ,i, ɛ,i s are i.i.d. N(0, σ, θ = (β, σ, (7 H : y i = f i (X,i, β + ɛ,i, ɛ,i s are i.i.d. N(0, σ, θ = (β, σ, (8 where y i s are observatios o a depedet variable, X,i s ad X,i s are vectors of exogeous variables, ad θ ad θ are vectors of parameters. To test H 0 agaist H by the J test (Davidso ad MacKio, 98, the followig compoud model is cosidered: y i = ( τf i (X,i, β + τf i (X,i, β + ɛ,i. (9 As β disappears from the model whe τ = ad β disappears whe τ = 0, the compoud model suffers from Davies s problem (Davies, 977. The J test circumvets the problem by substitutig a estimator ˆβ of β from H ito (9 ad the estimatig τ ad β joitly. The t statistic for τ = 0, which is asymptotically stadard ormal, is the J test statistic. Davidso ad MacKio (98 has proved that the J test is asymptotically equivalet to the Cox test uder H 0. Gourieroux ad Mofort (994 ote that the Cox test statistic is asymptotic equivalet to a score test statistic for η = 0 uder H 0, computed as if a estimator ˆθ of θ from H was determiistic, i a model with the followig probability desity fuctio l η (y, X, θ l η (y, X, ˆθ l η (y, X, θ l η (y, X, ˆθ dy ( (π (σ η (ˆσ η exp = (π (σ η (ˆσ η exp ( η σ η σ y f (X, β y f (X, β η ˆσ η ˆσ y f (X, ˆβ y f (X, ˆβ, (0 dy where y = (y,..., y, X j = (X j,,..., X j,, f j (X j, β j = ( f j (X j,, β,..., f j (X j,, β j for j =, ;... deotes the Euclidea vector orm; ad l (y, X, θ ad l (y, X, θ are, respectively,

12 the likelihood fuctios of H 0 ad H. The asymptotic equivalece of the J ad Cox tests is ot surprisig, sice (0 is the likelihood fuctio of the regressio model 7 ( ηˆσ y i = ησ + ( f i (X,i, β + ηˆσ ησ + ( f i (X,i, ˆβ + ξ i, ( ηˆσ where ξ i s are i.i.d. N ( 0, σ ˆσ /[ησ + ( ηˆσ ], which is the same as (9 after reparameterizatio. Give the equivalece result o the models (7 ad (8, it is temptig to just use the J-type tests but igore the Cox-type tests for other models. However, o such equivalece result exists for SARAR models. For the SARAR models ( ad (, the spatial J test, as described i Kelejia ad Piras (0, is obtaied by augmetig the spatial Cochrae-Orcutt trasformed ull model ησ R (ρ y = λ R (ρ W y + R (ρ X β + ɛ to the model R (ρ y = λ R (ρ W y + R (ρ X β + αr (ρ S (λ X β + ɛ, ( or R (ρ y = λ R (ρ W y + R (ρ X β + αr (ρ (λ W y + X β + ɛ, (3 as both S (λ X β ad (λ W y + X β are predictors of y with some estimator for θ plugged i. I the first step of the spatial J test, we ca get a estimator ˆρ of ρ 0 from the ull model ad a estimator ˆθ of θ from the alterative model. The R (ˆρ y, R (ˆρ W y, R (ˆρ X, ad the predictors R (ˆρ S (ˆλ X ˆβ or R (ˆρ (ˆλ W y + X ˆβ, ca be computed. After that, ( ad (3 ca be estimated by SLS i order to costruct a t statistic to test whether α is equal to zero or ot. We call the J test statistic based o ( J ad the other J. The Mote Carlo study i Kelejia ad Piras (0 shows similar fiite sample results for J ad J. For computatioal coveiece, they suggest the use of J. Let the likelihood fuctios of the models ( ad ( still be deoted by l (y, X, θ ad l (y, X, θ, respectively. The compoud model with a probability desity fuctio correspodig to (0 is l η (y, X, θ l η (y, X, ˆθ l η (y, X, θ l η (y, X, ˆθ = c (σ dy η ˆσ ( η (ˆσ η exp η σ R (ρ [S (λ y X β ] R (ˆρ [S (ˆλ y X ˆβ ] S (λ R (ρ η S (ˆλ R (ˆρ η, (4 where c oly depeds o. The score test for η = 0 i (4, computed as if ˆθ is o-stochastic, ca be show to be asymptotically equivalet to the Cox test uder H 0. The score test is based o the asymptotic 7 See Atkiso (970, amog others.

13 distributio of the score [l l (y, X, ˆθ l l (y, X, ˆθ [l l (y, X, ˆθ l l (y, X, ˆθ ]l (y, X, ˆθ dy ], where ˆθ is from H 0. The asymptotic variace of (5 is computed as if ˆθ were determiistic. (5 is equal to the umerator of Atkiso (970 s versio of the Cox test statistic. To derive the asymptotic distributio of (5, as oted i (D. ad (D., ˆθ ca be replaced by the o-stochastic pseudo true value θ,. Oce the aalytical form of the asymptotic variace for (5 is foud, θ, may be substituted by ˆθ to approximate the asymptotic variace. Thus the score test for η = 0 deduced from (4 is asymptotically equivalet to the Cox test uder H 0. O the other had, (4 is ot equivalet to (, (3 or ay other simple combiatios of the models ( ad (. The expoet i (4 writte i the quadratic form is equal to (A y A b (A y A b plus a term ot ivolvig y, where A = η σ ad b = η S σ (λ R (ρ R (ρ X β + model with i.i.d. ormal disturbaces would be (5 S (λ R (ρ R (ρ S (λ + η S ˆσ (ˆλ R (ˆρ R (ˆρ S (ˆλ η ˆσ S (ˆλ R (ˆρ R (ˆρ X ˆβ. The correspodig A y = A b + u, (6 which is ot liear i parameter ad does ot correspod to ay simple liear combiatio of the origial models. I particular, this model is very differet from the compoud models ( ad (3 (or the oe i Kelejia (008. Therefore, the Cox-type ad J-type tests for SARAR models caot be show to be asymptotically equivalet uder the ull hypothesis by showig that the expoetial compoud model (4 is equivalet to ( or (3. It seems ot to be surprisig that there is o such a equivalece relatioship because of the spatial depedece. The origial J-type tests i Kelejia ad Piras (0 employ the geeralized spatial SLS (GSSLS proposed i Kelejia ad Prucha (998 to estimate the ull ad alterative models, ad the SLS to estimate the augmeted model. Sice the GSSLS or SLS oly uses liear istrumets, which is less efficiet tha the QML or the GMM which uses both liear ad quadratic momets, the power ca be low due to the estimatio method, especially whe the variatio i exogeous variables caot explai much of the variatio i the depedet variable. We may estimate the ull, alterative ad augmeted models by the GMM or QML for the J-type tests, which is computatioal more ivolved. For the estimatio of the ull ad alterative models i the J-type tests, a advatage of the geeralized spatial SLS is that it ca be robust to ukow heteroskedasticity while the QML is ot. 8 The Cox-type tests are built upo the QMLEs of the ull ad alterative models, which ivolve oliear objective fuctios, thus idetificatio coditios are eeded. The J-type tests oly ivolve the GSSLS ad SLS, where a idetificatio coditio is oly 8 The GMM ca also be robust to ukow heteroskedasticity, see Li ad Lee (00. 3

14 eeded for the spatial error depedece parameter. 9 Also related to the oliear objective fuctios, the QML eeds the compact parameter space assumptio while the GSSLS does ot eed that assumptio. 4. Cosistecy of the Bootstrap for Cox-type Tests I this sectio, we show that the bootstrap is cosistet for Cox-type tests. procedure is as follows: 0 The bootstrap testig (i Compute the QML estimator (ˆλ, ˆρ, ˆβ ad the correspodig residual vector e = R (ˆρ [S (ˆλ y X ˆβ ] for the model (. Compute the Cox-type test statistics. (ii Draw a -dimesioal vector e of radom samples from the residuals i e usig samplig with replacemet ad geerate data y accordig to y = S (ˆλ [X ˆβ + R (ˆρ e ]. (iii Compute various test statistics usig the data y. (iv Repeat (ii ad (iii s times, ad obtai the bootstrapped p-values. (v The bootstrap tests cosist i rejectig the ull hypothesis if the bootstrapped p-value is smaller tha the chose level of sigificace ad ot rejectig otherwise. Usig y, we have the estimators ˆθ, ˆθ ad θ (ˆθ, correspodig to the estimators ˆθ, ˆθ ad θ (ˆθ respectively. Deote the bootstrapped versios of ˆσ co,, ˆσ ca,, Cox o, Cox a by, respectively, ˆσ co,, ˆσ ca,, Cox o, Cox a. Let P be the probability distributio iduced by the bootstrap samplig process. From (0, the Cox-type test statistics ca be approximated by a liear-quadratic form of disturbaces, thus we ca apply a theorem i Ji ad Lee (0, who establish that the bootstrap is cosistet for spatial ecoometric statistics that ca be approximated by a liear-quadratic form. The result is based o the uiform covergece of the distributio for a liear-quadratic form to the ormal distributio. The cosistecy result for Cox-type test statistics eeds a stroger assumptio o the disturbaces amely, the existece of eighth momet tha assumed earlier, for o-ormal disturbaces. Oe reaso of the stroger assumptio is that the umerators for the Cox-type tests geerally ivolve estimators of the fourth momets of the disturbaces. The stroger coditio is eeded for the rate of covergece of the estimators. Assumptio 7. {ɛ,i } s i ɛ = (ɛ,,..., ɛ,, i =,...,, are i.i.d. with mea zero ad variace σ 0, ad the momet E(ɛ 8,i exists. 9 The idetificatio coditio is ot explicitly stated i Kelejia ad Piras (0. They assume istead the high level coditio that the limits of some matrices ivolvig parameter estimates for the alterative model have osigular probability limits. 0 The resamplig procedure above has bee used by Burridge ad Figleto (00. For the Cox-type tests, as they are oe-sided tests, the bootstrapped p-value is the percetage of test statistics calculated from the bootstrapped samples that are greater tha the correspodig test statistic obtaied i (i. For two-sided tests, the bootstrapped p-value is the equal-tail bootstrapped p-value which is equal to times the smaller oe of the percetages of test statistics that are greater ad o-greater tha the test statistic i (i (MacKio,

15 Propositio 6. Uder H 0 ad Assumptios 7, sup x P (Cox o x P(Cox o x = o P ( ad sup x P (Cox a x P(Cox a x = o P (. 5. Mote Carlo Study We compare the fiite sample size ad power properties of the tests derived i this paper with those of the spatial J tests (Kelejia ad Piras, 0 with Mote Carlo experimets. I additio, we also compare them with a test derived from a comprehesive model. For the SARAR models ( ad (, a atural comprehesive model for them is y = λ W y + λ W y + X β + X,a β a + u, u = ρ M u + ρ M u + ɛ, (7 where X,a cotais the variables i X that are differet from ay i X, ad β a is the correspodig parameter vector. We test whether λ, ρ ad β a are joitly zero with a Lagragia multiplier (LM test. Deote the correspodig test statistic by Aug. I the experimets, the spatial weights matrix i the spatial error process is set to be the same as that i the spatial lag equatio for the two SARAR models, ad the two models either have the same spatial weights matrix or the same exogeous variable matrix. For the J test statistics J ad J, first estimate the model ( to obtai ˆρ by the geeralized spatial SLS, as described i Kelejia ad Prucha (998, with istrumetal variables [X, W X, W X ] LI, where LI deotes the liear idepedet colums of a matrix, the estimate the model ( with istrumetal variables [X, W X, W X ] LI to obtai y s predictors, ad fially ( ad (3 are estimated with the istrumetal variables [X, W X, W X, W X, W X, W W X, W W X ] LI whe X = X but W W ; or [X, X, W X, W X, W X, W X ] LI whe W = W but X X. As a alterative, we first estimate the model ( by the QML to derive the predictor S (ˆλ X ˆβ or (ˆλ W y + X ˆβ, ad the estimate ( ad (3 by the GMM with both liear ad quadratic momets. Deote the J tests with the alterative estimatio methods as J a ad J a respectively. The liear istrumets for J a ad J a are the same for J ad J, ad the matrices for the quadratic momets iclude differet matrices of W, W, W tr(w I /, W tr(w I /, W W tr(w W I / ad W W tr(w W I /. Note that for the exteded Wald ad score tests, we use the asymptotic chi-square critical values with degrees of freedom equal to the umber of parameters i the alterative model to evaluate the empirical size ad power. Note that our GMM approach estimates ρ joitly with λ ad β i ( ad (3. This is differet from the origial approach i Kelejia ad Piras (0 where ρ is first estimated i the model ( ad the the estimate is plugged ito the augmeted model. The GMM estimatio of ( ad (3 ivolvig quadratic momets with a iitial estimate of ρ plugged i would geerate a complicated variace-covariace matrix because a part of the variace-covariace would be from the estimatio error of ρ s estimator. 5

16 Table : Sets of Experimets Experimets H 0 H Set I W a, X a W b, X a Set II W c, X a W b, X a Set III W c, X b W c, X a The experimetal desig is based o former Mote Carlo studies of spatial models (see, e.g., Aseli ad Florax 995, Kelejia ad Prucha 999, Arraiz et al. 00 ad Kelejia ad Piras 0. We cosider three differet spatial weights matrices W a, W b ad W c : W a is geerated accordig to the rook criterio, W b is geerated accordig to the quee criterio ad W c is a block diagoal matrix with the diagoal blocks beig the cotiuity matrix for 49 eighborhoods i Columbus, OH from Aseli (988. We use row ormalized matrices. Two exogeous variable matrices X a ad X b are used: X a cotais a vector of oes ad a vector of radom samples draw from the stadard ormal, ad X b cotais a vector of oes, a variable draw from the uiform distributio U(0,, ad a variable equal to times the secod variable plus / times a variable draw from the chi-square distributio with degrees of freedom. For X b, the correlatio coefficiet betwee the secod ad third variables is 0.5. The three sets of experimets cosidered are show i Table. For each set of experimets, the disturbaces are draw from either the stadard ormal or a ormalized chi-square (χ (3 3/ 6 with mea zero ad variace oe. The true parameter vector is either (0.5, 0.5 or (0.5, correspodig to X a, ad either (0.5,, 0.5 or (0.5, 4, correspodig to X b, leadig to the ratio of the variace of Xβ with the sum of the variace of Xβ ad that of the error terms to be equal to 0. ad 0.8, respectively. 3 Deote this ratio by R. Whe the ull ad alterative models geerate the data, i.e., whe the empirical size ad power are cosidered, λ i the ull model ad λ i the alterative model, or, ρ i the ull model ad ρ i the alterative model, are the same, takig value of 0. or 0.8. Deote the two parameters by λ ad ρ respectively i the reported tables. I total, we have 3 = 48 experimets for each sample size. We cosider a small size = 98 ad a large sample size = The omial level of sigificace is set to 5% ad the umber of Mote Carlo repetitios is 000. For = 98, bootstrapped tests of various test statistics are also implemeted. 5 We set the umber of resamplig s to 99, leadig to a stadard error of the bootstrapped p-value beig equal to.5%. The Mote Carlo results for = 98 are reported i Tables 7. Usig the asymptotic p-values, J, J, 3 This kid of Mote Carlo settig for spatial models follows from Lee (007 ad Lee ad Liu (00. 4 For = 98, the W a ad W b are first geerated o a 0 0 grid, the the last two rows ad last two colums are deleted, ad fially they are row-ormalized to have row su by dividig each elemet i a row by the sum of all elemets i that row. The W a ad W b for = 59 are similarly derived. 5 For = 59, implemetig bootstrap tests for all statistics with 000 repetitios takes too log, so bootstrap tests are ot implemeted. 6

17 Aug ad Score geerally have small size distortios while other statistics have large size distortios i some cases. The empirical sizes of J deviate from the omial oe by o more tha 3 percetage poits i all experimets, the empirical size of J ca be as large as 9.7% as show i Table 3, Aug i experimet set III ad Score i experimet sets II ad III with chi-square disturbaces sigificatly uder-reject the true ull hypothesis. The J a ad J a almost have o size distortio i experimet set III, but have large size distortio i the first two sets of experimets. The empirical size of J a ca be over 40% whe R = 0. i experimet set I. The Wald have empirical sizes larger tha 50% i may cases. The size distortio of Cox o ad Cox a is o more tha 3.7 percetage poits i experimet set I, but the size of Cox o ca be as large as 0.% i experimet set II ad 30.% i experimet set III, ad the size of Cox a ca be as large as 3.% i experimet set II ad.6% i experimet set III. The empirical sizes based o the bootstrapped critical values show that the bootstrap removes the size distortio of various statistics i most cases. We thus compare the empirical powers of differet statistics based o the bootstrapped p-values. Several patters for the empirical powers of the bootstrapped tests ca be summarized as follows: oe of the tests ca domiate the rest of tests i power i all experimets, but the Cox-type statistics usually have high powers compared to other statistics ad domiate other oes i some cases; i most cases of all experimets, J a is more powerful tha J ; i most cases, J a is more powerful tha J i experimet sets II ad III, but less powerful i experimet set I; J is more powerful tha J i almost all cases. We ow ivestigate the results for experimet set I with ormal disturbaces i some detail, ad briefly summarize results for other experimets. Table presets the results for experimet set I with ormal disturbaces. The powers of Cox o ad Cox a are similar, which are the highest amog all the test statistics, ad the powers of other statistics are sigificatly lower i most cases. Takig the case with R = 0.8, λ = 0. ad ρ = 0.8 as a example, Cox o ad Cox a have powers higher tha 90%, Aug has a power of 73.7%, Score has a power of 5.5%, but the powers of the rest statistics are all below %. I all cases except the oe with R = 0., λ = 0. ad ρ = 0., J has a higher power tha J. Whe R = 0.8, λ = 0.8 ad ρ = 0.8, J has a power of 84.0%, while J has a power of oly 5.6%. 6 Table 3 presets the results for experimet set I with chi-square disturbaces. Chagig the distributios of the disturbaces from ormal to chi-square has ot led to big chages i the results. For experimet set II, Tables 4 ad 5 show that, J a, Aug, Score, Cox o ad Cox a have similar magitude of power, amog which Cox a has the highest power i most cases, ad other statistics have sigificatly lower powers. For experimet set III, all statistics, except J ad Wald i some cases, have powers close or equal to 00%. The Wald has very low power compared to other test statistics. The empirical size ad power based o the asymptotic p-values for = 59 are reported i Tables I the Mote Carlo study of Kelejia ad Piras (0, their Mote Carlo desig has produced high powers for the J tests, where i geeral J is also relatively more powerful tha J, but due to their high power, their differeces seem small. 7

18 Most statistics have o sigificat size distortio with a sample size of 59, except for Wald, Cox o ad Cox a i some cases, which have much smaller size distortio compared to that with a sample size of 98. The Wald still has sigificat size distortio for all experimets. For experimet set I, Cox o ad Cox a have empirical sizes close to the omial level. For experimet set II, Cox o ad Cox a have large size distortio oly whe λ = 0. ad ρ = 0.. For experimet set III, Cox o ad Cox a still have large distortio i some cases. For example, whe R = 0., λ = 0., ρ = 0.8 ad the disturbaces are ormal, Cox o ad Cox a with = 59 have empirical sizes equal to 7.% ad 7.3% respectively, smaller tha the sizes.9% ad.6% for = 98. All the statistics have powers close or equal to 00% with the large sample size except for J, J ad J a. For experimet set I, whe R = 0. ad λ = 0., J ad J have very low powers, less tha 7%, ad J a has powers lower tha 76% with ρ = 0. ad lower tha 4% with ρ = 0.8. For experimet set II, whe R = 0. ad λ = 0., J, J ad J a have powers lower tha 60%. All statistics i experimet set III have powers close or equal to 00%. Note that with = 59, J a may still have slightly lower power tha Cox o ad Cox a, e.g., i experimet set I with R = 0., λ = 0.8, ρ = 0. ad chi-square disturbaces, J a has a power of 98.5%, while both Cox o ad Cox a have a power of 00%. The Cox-type tests are computatioally more ivolved tha the J-type tests, especially for large sample sizes. 7 First, the Cox-type tests are based o the QMLEs. However, with the developmet of more advaced computers ad computatioal techiques 8, the QMLE ca be efficietly computed. A further computatioal problem i calculatig the Cox-type test statistics after derivig the QMLEs is o the traces ivolvig the iverses S (ˆλ ad R (ˆρ or o the product of S (ˆλ ad a vector (see Appedix A. LeSage ad Pace (009, pp. 0 3 have discussed some techiques i computig such terms. Those approaches may make the computatio practically easier. 6. Empirical Illustratio We illustrate the use of the Cox-type tests with the housig data set i Harriso ad Rubifeld (978. Pace ad Gilley (997 added logitude-latitude coordiates for cesus tracts to the data set. With the augmeted data set, LeSage (999, pp estimates a SARAR model, where the depedet variable is the studetized log of media housig prices for each of the 506 cesus tracts, the explaatory variables iclude 3 covariates, ad the spatial weights matrix for both the spatial lag ad the spatial error depedece is a first order cotiguity matrix (call it W foc. We create a row-ormalized spatial weights matrix based o 5 earest eighbors (call it W 5, where the elemets correspodig to a cesus tract s five earest 7 For Experimet Set I with the sample size of = 59, whe R = 0.8, λ = 0., ρ = 0. ad the disturbaces are ormal, Computig J, J, J a, J a, Cox o ad Cox a oce take, respectively, 0.3, 0.3, 7.8, 7.8, 0.6 ad 7.6 secods o average, usig Matlab o a desktop computer with Itel Core i7-600 processor ad 8 gigabyte memory. 8 See, e.g., Pace ad LeSage (009 ad Smirov ad Aseli (009. 8

19 Table : Empirical size ad power for experimet set I with ormal disturbaces ad = 98 Asymptotic Bootstrap Asymptotic Bootstrap Size Power Size Power Size Power Size Power R =0., λ=0., ρ=0. R =0., λ=0., ρ=0.8 J J J a J a Aug Wald Score Cox o Cox a R = 0., λ = 0.8, ρ = 0. R = 0., λ = 0.8, ρ = 0.8 J J J a J a Aug Wald Score Cox o Cox a R = 0.8, λ = 0., ρ = 0. R = 0.8, λ = 0., ρ = 0.8 J J J a J a Aug Wald Score Cox o Cox a R = 0.8, λ = 0.8, ρ = 0. R = 0.8, λ = 0.8, ρ = 0.8 J J J a J a Aug Wald Score Cox o Cox a All empirical sizes ad powers are expressed as percetages with the sig % beig omitted. The Asymptotic ad Bootstrap mea that the reported empirical size ad power are computed by usig, respectively, the asymptotic ad bootstrapped p-values. 9

20 Table 3: Empirical size ad power for experimet set I with chi-square disturbaces ad = 98 Asymptotic Bootstrap Asymptotic Bootstrap Size Power Size Power Size Power Size Power R =0., λ=0., ρ=0. R =0., λ=0., ρ=0.8 J J J a J a Aug Wald Score Cox o Cox a R = 0., λ = 0.8, ρ = 0. R = 0., λ = 0.8, ρ = 0.8 J J J a J a Aug Wald Score Cox o Cox a R = 0.8, λ = 0., ρ = 0. R = 0.8, λ = 0., ρ = 0.8 J J J a J a Aug Wald Score Cox o Cox a R = 0.8, λ = 0.8, ρ = 0. R = 0.8, λ = 0.8, ρ = 0.8 J J J a J a Aug Wald Score Cox o Cox a All empirical sizes ad powers are expressed as percetages with the sig % beig omitted. The Asymptotic ad Bootstrap mea that the reported empirical size ad power are computed by usig, respectively, the asymptotic ad bootstrapped p-values. 0