GEL estimation and tests of spatial autoregressive models

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1 GEL estimatio ad tests of spatial autoregressive models Fei Ji a ad Lug-fei Lee b a School of Ecoomics, Shaghai Uiversity of Fiace ad Ecoomics, ad Key Laboratory of Mathematical Ecoomics (SUFE), Miistry of Educatio, Shaghai 00433, Chia b Departmet of Ecoomics, The Ohio State Uiversity, Columbus, OH 430, USA February 9, 07 Abstract This paper cosiders the geeralized empirical likelihood (GEL) estimatio ad tests of spatial autoregressive (SAR) models by explorig a iheret martigale structure. The GEL estimator has the same asymptotic distributio as the geeralized method of momets estimator explored with same momet coditios for estimatio, but circumvets a first step estimatio of the optimal weightig matrix with a prelimiary estimator, ad thus ca be robust to ukow heteroskedasticity ad o-ormality. While a geeral GEL removes the asymptotic bias from the prelimiary estimator ad partially removes the bias due to the correlatio betwee the momet coditios ad their Jacobia, the empirical likelihood as a special member of GELs further partially removes the bias from estimatig the secod momet matrix. We also formulate the GEL overidetificatio test, Mora s I test, ad GEL ratio tests for parameter restrictios ad o-ested hypotheses. While some of the covetioal tests might ot be robust to o-ormality ad/or ukow heteroskedasticity, the correspodig GEL tests ca. Keywords: Spatial autoregressive, martigale, empirical likelihood, higher order asymptotic bias, ukow heteroskedasticity, o-ormality, robustess, over-idetificatio test, Mora s I test for spatial depedece, J test JEL classificatio: C, C3, C4, C, C5 Itroductio I this paper, we cosider empirical likelihood (EL) ad geeralized EL (GEL) estimatio ad tests of the popular spatial autoregressive (SAR) model with spatially depedet data. The EL approach is itroduced i Owe (99) for idepedet sample observatios. It ca be iterpreted as a oparametric maximum likelihood ad a geeralized miimum cotrast estimatio method (Kitamura, 007). The class of GEL estimators icludes the Correspodig author. Tel.: ; fax: addresses: ji.fei@sufe.edu.c (F. Ji), lee.777@osu.edu (L.-F. Lee). Helpful reviews iclude, amog others, Hall ad La Scala (990), Owe (00), Kitamura (007) ad Che ad Keilegom (009).

2 EL, the expoetial tiltig (ET) of Kitamura ad Stutzer (997) ad Imbes et al. (998), ad the cotiuous updatig geeralized method of momets (GMM) of Hase et al. (996). With idepedet sample observatios, the EL ad GEL ca have various advatages over other methods as show i the literature. They ca be robust agaist distributioal assumptios but may still have good properties aalogous to the parametric likelihood procedure i estimatio ad testig. As alteratives to the two-step optimal GMM estimator which usually requires a first step estimatio of a optimal weightig matrix with a prelimiary estimator, the EL ad GEL estimators are oe-step estimators. They are cosistet ad have the same asymptotic distributio as the two-step optimal GMM estimator by usig same momet coditios, but ivariat to parameter-depedet liear trasformatios of momet coditios, ad have improved high order properties (Imbes et al., 998; Owe, 00; Newey ad Smith, 004). I particular, Newey ad Smith (004) show that, for i.i.d. data, the GEL estimator has o asymptotic bias from estimatio of the Jacobia or the prelimiary estimator, ad the EL further removes a bias compoet from estimatio of the secod momet matrix. I fiite samples, while the two-step optimal GMM ca have large bias (e.g., Altoji ad Segal, 996), the GEL estimators are observed to perform better tha the GMM estimator (e.g., Hase et al., 996; Imbes, 997; Ramalho, 00; Mittelhammer et al., 005; Newey et al., 005). The EL ad GEL ca also be applied to testig problems. A oparametric aalog of the parametric likelihood ratio statistic follows a asymptotic chi-squared distributio uder the ull. A EL ratio test ad cofidece regio are ofte Bartlett correctable (Corcora, 998; DiCiccio et al., 99; Lazar ad Myklad, 999), ad EL tests are Bahadur efficiet (Otsu, 00) ad have optimality properties i terms of large deviatios (Kitamura, 00). The EL ad GEL have origially bee cosidered for idepedet data. Later o, there are attempts to geeralize them for time series data (e.g., Kitamura, 997). For time series, some authors have studied the EL for models with martigale structures. Myklad (995) geeralizes the EL defiitio for i.i.d. data to models with martigale structures ad itroduces the cocept of dual likelihood, ad Chuag ad Cha (00) develop the EL for autoregressive models with iovatios that form a martigale differece sequece. But the EL ad GEL approaches have ot be cosidered for estimatio ad testig with spatially depedet data. These motivate our ivestigatio of the use of EL ad GEL for estimatio ad hypothesis testig with spatial data. We realize that may popular spatial ecoometric models ad hece spatially correlated variables ca be characterized by martigale processes uder proper filtratios. The importace of martigale processes for spatial radom variables has bee recogized by Kelejia ad Prucha (00). They develop a cetral limit theorem (CLT) for liear-quadratic forms of idepedet disturbaces by explorig the martigale structure of a liear-quadratic form. This CLT ca be applied to a large class of spatial ecoometric models such as the SAR model, the spatial error (SE) model, the spatial movig average model, the spatial Durbi model, the spatial error compoets model, ad the SAR model with SAR disturbaces (SARAR model). Various estimatio methods for the SARAR model, which icludes the SAR ad SE models as special cases, have bee proposed i the literature, e.g., the geeralized spatial two-stage least squares (GSSLS) estimatio (Kelejia ad Prucha, 998), the quasi maximum likelihood (QML) estimatio (Lee, 004), ad the GMM estimatio (Lee, I the time series literature, quadratic statistics have log bee writte as martigales. See, e.g., Hall ad Heyde (980).

3 007). 3 The GSSLS estimates the equatio by the two stage least squares (SLS), thus it is computatioally simple, but ca be asymptotically iefficiet compared to the QML. Although beig relatively efficiet, the QML may be computatioally itesive for large sample sizes, especially for SAR models with high order spatial lags. The GMM ca be computatioally simpler tha the QML ad ca be as efficiet as the QML. 4 The GMM may employ ot oly liear momets i disturbaces but also quadratic oes. Quadratic momets ca be motivated from the QML ad Mora s I test (Mora, 950), which capture spatial depedece. I the presece of ukow heteroskedasticity, by selectig quadratic matrices with zero diagoals, the quadratic momets ca obtai robust estimates (Kelejia ad Prucha, 00; Li ad Lee, 00). Liu ad Yag (05) propose to modify the QML scores to obtai estimators robust to ukow heteroskedasticity. We cosider the GEL estimatio of the SARAR model uder both homoskedasticity ad ukow heteroskedasticity i this paper. For spatial data, origial sample observatios are ot martigale differeces, so the EL ad GEL caot be applied directly to them. However, as oted i Kelejia ad Prucha (00), liear-quadratic forms of idepedet disturbaces ca be writte as a sum of martigale differeces. For liear ad quadratic momets, treatig each martigale differece as if it was a data observatio, we ca set up EL ad GEL objective fuctios to derive correspodig estimates ad relevat test statistics. We show that, for spatial data, the GEL estimatio with momet coditios ca remove the asymptotic bias from the prelimiary estimator ad partially remove the asymptotic bias due to the correlatio betwee momet coditios ad their Jacobia. The EL further partially removes the bias from estimatio of the secod momet matrix. This coclusio is cosistet with that i Aatolyev (005) for statioary time series models uder mixig coditios. I the evet that oly liear momets are used, the EL has the ability to completely remove the asymptotic bias from estimatio of the secod momet matrix. We also cosider test statistics i the GEL framework. The GEL objective fuctio (with proper ormalizatio) evaluated at the GEL estimator is a overidetificatio test statistic that ca be used to test for validity of momet coditios. Tests of parameter restrictios ca be coveietly implemeted with GEL ratio statistics. The popular Mora s I test for spatial depedece formulated with a GEL ratio is robust to ukow heteroskedasticity. I additio, we employ the GEL ratio statistic to costruct a spatial J test for competig SARAR models (Kelejia, 008; Kelejia ad Piras, 0). Ulike origial spatial J tests based o the SLS or GSLS estimatio, the spatial J test with a GEL ratio coveietly employs quadratic momets i additio to liear oes to obtai more efficiet estimators for testig. These tests do ot ivolve estimatio of variaces ad are robust to ukow heteroskedasticity. For testig with quadratic momets, GEL tests are also robust to o-ormality i the sese that (higher order) momet parameters do ot eed to be evaluated. As far as we kow, this may be the first paper that explores the GEL estimatio ad tests of models with spatial data. 5 3 Due to edogeeity of the spatial lag i a SAR model, the least squares estimator is oly cosistet i certai cases (Lee, 00). 4 The GMM estimator with properly chose momets ca be as efficiet as the QML estimator for the SARAR model with ormal disturbaces, but it ca be more efficiet tha the QML estimator for the SARAR model with o-ormal disturbaces (Liu et al., 00; Lee ad Liu, 00). 5 Although we oly focus o the SARAR model, by explorig martigale structures, other spatial ecoometric models may be possibly 3

4 This paper is orgaized as follows. Sectio itroduces the SARAR model, ad the GEL ad GMM estimatio based o its martigale structure. Sectio 3 shows the cosistecy ad asymptotic ormality of the GEL estimator ad compare its asymptotic bias with that of the GMM estimator. Sectio 4 ivestigates test statistics i the GEL framework. Sectio 5 reports some Mote Carlo results, which demostrate desirable fiite sample performace of GEL estimators ad test statistics. Sectio 6 cocludes. All lemmas ad proofs are collected i appedices. The SARAR model ad GEL estimatio Cosider the SARAR model: Y = κw Y + X β + U, U = τm U + V, () where is the sample size, Y is a vector of observatios o the depedet variable, X is a k x matrix of exogeous variables with parameter vector β, W = (w,ij ) ad M = (m,ij ) are ostochastic spatial weights matrices with zero diagoals, κ ad τ are scalar spatial depedece parameters, ad V = (v i ) is a vector of idepedet disturbaces with mea zero ad fiite variaces. I this paper, we cosider two cases o the variace of v i. I the first case v i s are homoskedastic, ad i the secod case v i s have heteroskedastic variaces with ukow form. Let S (κ) = I κw, ad R (τ) = I τm, with I beig the idetity matrix, ad (κ 0, τ 0, β 0) be the true value of (κ, τ, β ). As a equilibrium model, Y has the reduced form Y = S (X β 0 + R V ), where S = S (κ 0 ) ad R = R (τ 0 ) are assumed to be ivertible. The X is assumed to be ostochastic for coveiece, as i Kelejia ad Prucha (998) ad Lee (004). 6 If the disturbaces v i s i model () are i.i.d. with mea 0 ad variace σ 0, the momet vector for a GMM estimatio ca be g (θ) = [V (θ)p V (θ) σ tr(p ),..., V (θ)p,kp V (θ) σ tr(p,kp ), V (θ)q ], () where V (θ) = R (τ)[s (κ)y X β], with θ = (κ, τ, β, σ ) beig a k θ -dimesioal vector for k θ = k x + 3, P l for l =,..., k p are ostochastic matrices, ad Q is a k q matrix of istrumetal variables (IV) with full colum rak k q. Without loss of geerality, assume that P l, for l =,..., k p, are symmetric ad liearly idepedet. 7 The quadratic momets are valid sice E(V P l V ) = σ 0 tr(p l ). The IV matrix Q may cosist of idepedet colums of X, W X ad so o, ad P i s ca be fuctios of W ad M such as W, M, W ad M. The total umber of elemets i g (θ) is k g = k p + k q, which is greater tha or equal to k θ. As each momet coditio of g (θ) at the true parameter vector θ 0 is either liear or quadratic i V, we may also cosider a more geeral vector momet coditios of liear-quadratic forms, which are liearly idepedet, such as Ξ = [V A V σ 0 tr(a ) + b V,..., V A p V σ 0 tr(a p ) + b pv ] studied. 6 Alteratively, X ca be stochastic with fiite momets of certai order. 7 If P l is ot symmetric, replacig it with (P l + P l )/ does ot chage the value of the momet vector. 4

5 for some fiite p, where A r = (a r,ij ) for r =,..., p are symmetric matrices ad b r = (b r,i ) for r =,..., p are vectors. We ca rewrite Ξ as a sum of martigale differeces. Specifically Ξ = ξ i, where i i ξ i = [a,ii (vi σ0) + v i a,ij v j + b,i v i,..., a p,ii (vi σ0) + v i a p,ij v j + b p,i v i ] j= j= is a p-dimesioal colum vector. Cosider the σ-fields F 0 = {, Ω}, F i = σ(v,..., v i ), i. As F,i F i ad E(ξ i F,i ) = 0, {ξ i, F i, i, } forms a martigale differece array. Thus ξ i s are ucorrelated ad the variace of Ξ is E(ξ iξ i ). Let ϕ = (ξ,..., ξ ) be a p matrix of martigale differeces. The, the variace of Ξ is E(ϕ ϕ ). The momet vector g (θ) i () ca be equivalet to Ξ above whe relevat quadratic matrices ad liear fuctios icludig zeros are properly chose. Let Q = [Q,..., Q ], V (θ)p l V (θ) σ tr(p l ) = ω l,i(θ) for l =,..., k p, where with v j (θ) beig the jth elemet of V (θ), ad i ω l,i (θ) = p l,ii [vi(θ) σ ] + v i (θ) p l,ij v j (θ) (3) j= g i (θ) = [ω,i (θ),..., ω,kp,i(θ), Q iv i (θ)]. (4) The g (θ) = g i(θ). The quadratic momets ivolve the variace parameter σ due to (3) i order that g (θ) ca be decomposed ito a sum of g i (θ) s i (4), where g i (θ 0 ) for i =,...,, are martigale differeces. Thus the variace of g (θ 0 ) is E[g i(θ 0 )g i (θ 0)]. Our quadratic momets ivolvig the estimatio of σ are i lie with those i Kelejia ad Prucha (998, 999). 8 I the case that there is ukow heteroskedasticity, we may select all P l s to have zero diagoals i order to derive valid momet coditios, as i Kelejia ad Prucha (00) ad Li ad Lee (00). Such P i s ca be W, M, W diag(w), M diag(m) ad so o, where diag(a) for a square matrix A deotes a diagoal matrix formed by the diagoal elemets of A. Let the momet vector be g (θ) = [V (θ)p V (θ),..., V (θ)p,kp V (θ), V (θ)q ], (5) where P l s have zero diagoals, ad V (θ) is the same as above, but θ = (κ, τ, β ) would ot cotai σ so that θ is k θ -dimesioal for k θ = k x +. 9 The ω l,i (θ) ad g i (θ) ca still have the forms i (3) ad (4), as the first term o the r.h.s. of (3) is zero. 8 Note that E(V P l V ) σ 0 tr(p l) = E{V [P l tr(p l )I /]V } + [E(V V ) tr(p l )/ σ 0 tr(p l)] = 0, where E{V [P l tr(p l )I /]V } = 0 as P l tr(p l )I / is a matrix with a zero trace. Lee (00) ad Lee (007) use quadratic momets of the form E{V [P l tr(p l )I /]V } = 0 to formulate the GMM estimatio, which do ot ivolve σ. The zero trace quadratic matrices would ot be appropriate to be used here due to the required martigale differece property. However the two sets of quadratic momets ca be asymptotically equivalet as show i Liu et al. (00) for GMM estimatio. As we show i Appedix A, i the case that v i s are ormal, for the GMM i Lee (00, 007) ad that cosidered here, there are momet vectors with which the resultig GMM estimators are as efficiet as the ML estimator. 9 This is proper because a sigle σ would ot be meaigful with heteroskedastic errors. 5

6 We cosider the GEL estimator: ˆθ,GEL = arg mi θ Θ sup λ Λ(θ) ρ(λ g i (θ)), (6) where Λ (θ) = {λ : λ g i (θ) V, i =,..., } for a ope iterval V cotaiig 0, ad ρ(v) is a twice cotiuously differetiable cocave fuctio of a scalar v o V. 0 Deote ρ k (v) = dk ρ(v) dv k ad ρ k = ρ k (0) for k = ad. As log as ρ 0 ad ρ < 0, without loss of geerality, we may let ρ = ρ = (Newey ad Smith, 004). The EL is a special case of the GEL with ρ(v) = l( v) for v < (Qi ad Lawless, 994; Smith, 997); the ET is a special case with ρ(v) = e v (Kitamura ad Stutzer, 997; Smith, 997); ad the cotiuous updatig GMM is a special case with a quadratic ρ(v) = (v + ) (Newey ad Smith, 004). To study large sample properties of the GEL estimator, we assume formally the followig regularity coditios. Assumptio. Either (i) v i s are i.i.d. with mea zero, variace σ 0 ad E( v i 4+ι ) < for some ι > 0; or (ii) v i s are idepedet with mea zero ad variaces σ i s, ad sup sup i E( v i 4+ι ) <. Assumptio. The elemets of X are uiformly bouded costats, X has full colum rak, ad lim X X exists ad is osigular. Assumptio 3. (i) W ad M have zero diagoals; (ii) S ad R are osigular; ad (iii) the sequeces of matrices {W }, {M }, {S } ad {R } are bouded i both row ad colum sum orms. Assumptio 4. θ 0 is i the iterior of a compact parameter space Θ i the k θ -dimesioal Euclidea space. Assumptio 5. ρ(v) is cocave o V, twice cotiuously differetiable i a eighborhood of zero, ad ρ = ρ =. We shall cosider both homoskedastic ad heteroskedastic cases, so Assumptio gives geeral coditios to allow both cases. Assumptios (i) ad 4 are the same as those i Lee (007); ad the additioal coditios o M are similar to those o W. Assumptio (ii) for the heteroskedastic case is the same as that i Li ad Lee (00). The existece of momets higher tha the fourth order i Assumptio is eeded for the applicatio of the CLT o liear-quadratic forms as i Kelejia ad Prucha (00). I Assumptio, explaatory variables are assumed to be costats for coveiece ad multicolliearity is ruled out. Assumptio 3 restricts the degree of spatial depedece to be maageable. Assumptio 4 is a stadard assumptio o extremum estimatio. Assumptio 5 is a smoothess coditio o ρ( ) as i Newey ad Smith (004). We have the iterest to compare asymptotic properties of GEL estimatio with GMM estimatio. Let Ω (θ) = g i(θ)g i (θ), the var[ g (θ 0 )] = E[Ω (θ 0 )]. Deote Ω = E[Ω (θ 0 )], which ca be estimated by Ω ( θ ) with some iitial cosistet estimator θ. With Ω ( θ ), we cosider the followig feasible optimal GMM (FOGMM) estimator: ˆθ,GMM = arg mi θ Θ g (θ)ω ( θ )g (θ), (7) 0 I practice, λ ca be chose from R kg. If for some θ, for ay λ, there exists some i such that λ g i (θ) falls out of the domai of ρ( ), it is theoretically appropriate to set the GEL objective fuctio at θ to ifiity. If ot, but λ g i (θ) falls out of the domai of ρ( ) for some i ad λ, the the λ is ot the solutio of the problem. This is because ˆλ = O p( / ) by Propositio 3., ad with probability approachig oe, λ g i (θ) V for all i, θ Θ ad λ ζ, where ζ is a positive umber, by Lemma C.0. 6

7 We shall compare this FOGMM estimator with the GEL estimator. For these estimators, Ω is required to be osigular i the limit. The osigularity of Ω will be guarateed by the liear idepedece of the liearquadratic momet coditios. I the limit, we just require such liear idepedece properties ot to vaish. Assumptio 6. lim Ω exists ad is osigular. For the iitial estimator θ for the FOGMM, oe may suppose that it is derived from mi θ Θ g (θ)ĵ g (θ), where Ĵ is a k g k g weightig matrix. Followig Newey ad Smith (004), we assume that Ĵ satisfies the followig assumptio. Assumptio 7. Ĵ = J + / ξ J + O p ( ), where J is a ostochastic positive defiite matrix, lim J is osigular, ξ J = O p () ad E(ξ J ) = 0. 3 Large sample properties of estimators I this sectio, we ivestigate the cosistecy ad asymptotic ormality of the GEL estimator, ad compare its asymptotic bias of some higher orders with that of the FOGMM estimator. 3. Cosistecy ad asymptotic distributio For the GEL estimatio, it is coveiet to preset results o asymptotic properties i both the homoskedastic ad heteroskedastic cases together, though θ ad other terms below may have differet expressios i the two cases. Uder the idetificatio assumptios ad i Appedix A, the followig propositio establishes the cosistecy of ˆθ,GEL ad related probability orders of the momet vector ad the correspodig GEL estimate ˆλ,GEL of λ. Propositio 3.. Uder Assumptios (i),, 3, 5, 6 ad i the homoskedastic case, or uder Assumptios (ii),, 3, 5, 6 ad i the heteroskedastic case, ˆθ,GEL p θ0, ad g (ˆθ,GEL ) = O p ( / ); furthermore, ˆλ,GEL = arg max λ Λ(ˆθ,GEL) ρ(λ g i (ˆθ,GEL )) exists with probability approachig oe (w.p.a..), ad ˆλ,GEL = O p ( / ). With the cosistecy of the GEL estimator, its asymptotic distributio ca be derived as usual. Let Ḡ = E( g(θ0) θ ), γ = (θ, λ ), ad γ 0 = (θ 0, 0 kg ). Furthermore, deote Σ = (Ḡ ad D = Ω Ω Ḡ(Ḡ Ω Ḡ) Ḡ Ω. Assumptio 8. lim Ḡ has full rak. Ω Ḡ), H = (Ḡ Ω Ḡ) Ḡ Ω, As usual, Assumptio 8 rules out fuctioally depedet momets. The ext propositio shows that ˆγ,GEL = (ˆθ,GEL, ˆλ,GEL ) is asymptotically ormal. Propositio 3.. Uder Assumptios (i), 6, 8 ad i the homoskedastic case, or uder Assumptios (ii), 6, 8 ad i the heteroskedastic case, (ˆγ,GEL γ 0 ) d N ( 0, lim diag( Σ, D )), where diag( Σ, D ) is the block diagoal matrix formed by Σ ad D. 7

8 We see that the GEL estimator ˆθ,GEL of θ 0 has the same asymptotic distributio as the GMM estimator ˆθ,GMM i (7) (see Propositios A. ad A. i Appedix A). 3. Stochastic expasio ad high order asymptotic bias To study high order asymptotic biases of the GMM ad GEL estimators, we shall first derive Nagar-type expasios (Nagar, 959) of a -cosistet estimator ˆγ of γ 0 with the form (ˆγ γ 0 ) = ξ + / ψ + O p ( ), (8) where ξ = O p (), E(ξ ) = 0 ad ψ = O p (). High order bias of the estimator ˆγ ca be computed as E(ψ ). For the FOGMM estimator ˆθ,GMM, followig Newey ad Smith (004), a auxiliary parameter vector ˆλ,GMM = Ω ( θ )g (ˆθ,GMM ) ca be defied to make the derivatio of its correspodig Nagar-type expasio easier. With ˆλ,GMM, the first order coditio for the FOGMM estimator ˆθ,GMM ca be writte as ( G ) (ˆθ,GMM )ˆλ,GMM 0 =. (9) g (ˆθ,GMM ) + Ω ( θ )ˆλ,GMM The stochastic expasio requires the existece of higher order momets of disturbaces. Assumptio 9. sup sup i E v i 8 <. Propositio 3.3. For the FOGMM estimator ˆγ,GMM = (ˆθ,GMM, ˆλ,GMM ), uder Assumptios (i), 4, 6 9 ad i the homoskedastic case, or uder Assumptios (ii), 4, 6 9 ad i the heteroskedastic case, the expasio (8) holds. The explicit forms of ξ ad ψ for the asymptotic expasio of ˆγ,GMM are rather complex, but ca be foud i Appedix D i the proof of that propositio. A similar expasio for the GEL estimator ˆθ,GEL ca also be derived as i Appedix D. The expasio requires further smoothess coditio o ρ(v). Assumptio 0. ρ(v) is four times cotiuously differetiable with Lipschitz fourth derivative i a eighborhood of zero. Propositio 3.4. For the GEL estimator ˆγ,GEL, uder Assumptios (i), 6, 8 0 ad i the homoskedastic case, or uder Assumptios (ii), 6, 8 0 ad i the heteroskedastic case, the expasio (8) holds. With the above two propositios, we ca compute the asymptotic biases of the FOGMM ad GEL estimators with the form E(ψ ). Let Ω = Ω (θ 0 ), G = G (θ 0 ), Ḡ (l) = E( G(θ0) θ l ), g = g (θ 0 ), g i = g i (θ 0 ), g (l) i = g i(θ 0) θ l, ad e kθ,l be the lth colum of the k θ k θ idetity matrix, where θ l deotes the lth elemet of θ. Sice the GMM estimators i both homoskedastic ad heteroskedastic cases have bee studied i the literature, we relegate their cosistecy ad asymptotic distributio results to Appedix A ad omit their proofs. 8

9 Propositio 3.5. Uder Assumptios (i), 4, 6 9 ad i the homoskedastic case, or uder Assumptios (ii), 4, 6 9 ad i the heteroskedastic case, the bias of the FOGMM estimator ˆθ,GMM is B I + B G + B Ω + B, J where B I = H E(G H g ) kθ H l= Ḡ (l) Σ e kθ,l, B G = Σ E(G D g ), B Ω = H E(Ω D g ) ad B J = k θ l= H [E(g i g (l) i + g (l) i g i )] D Ω ( H J H ) e kθ,l with H J = (Ḡ I Propositio 3.5, B I J Ḡ) Ḡ J. is the asymptotic bias for a GMM estimator with the optimal liear combiatio Ḡ Ω g (θ 0 ) of empirical momets g (θ 0 ); B G arises from estimatig Ḡ; B Ω arises from estimatig the secod momet matrix Ω with the empirical variace Ω ; ad B J arises from the choice of the iitial GMM estimator. For the latter, if J is a scalar multiple of Ω, the B J = 0 as H = H J. With exact idetificatio, D = 0; thus, B G = B Ω = B J = 0. Let G i = gi(θ0) θ = [g () i,..., g(k θ) i ]. Propositio 3.6. Uder Assumptios (i), 6, 8 0 ad i the homoskedastic case, or uder Assumptios (ii), 6, 8 0 ad i the heteroskedastic case, the bias of the GEL estimator ˆθ,GEL is B I + B G B G + B Ω + ρ 3 BΩ, where B G = Σ E(G i D g i ), ρ 3 = d3 ρ(0) dv 3 is the third order derivative of ρ(v) evaluated at v = 0, ad B Ω = H E(g i g i D g i ). Sice g i (θ 0 ) s are ot idepedet across i, B G B G ad B Ω B Ω i geeral. Thus, ulike the case with i.i.d. data, the bias of the GEL estimator does ot reduce to B I + B Ω + ρ3 B Ω ad does ot reduce further to B I for the EL with ρ 3 =. The GEL oly partially removes the asymptotic bias from the correlatio betwee G (θ 0 ) ad g (θ 0 ). This coclusio is similar to that i Aatolyev (005) for statioary time series models with mixig coditios. Whe g (θ) oly cotais liear momets, g i becomes Q i v i. The, with oly IV estimatio, B Ω = B Ω ad the bias of the EL estimator reduces to B I + B G B G, i.e., the EL does ot have a bias from estimatio of the secod momet matrix Ω. If further E(v 3 i ) = 0 for i =,...,, because BΩ = B Ω = 0, B Ω is removed from the bias of the FOGMM estimator ad B Ω + ρ3 B Ω is removed from the bias of ay GEL estimator, ot just the EL estimator. Corollary 3.. Whe g (θ) = Q V (θ), the bias of the EL estimator reduces to B I + B G B G, ad the bias of the FOGMM estimator is B I + B G + B Ω + B J, where B Ω = H Q iq i D Q i E(v 3 i ). 4 Test statistics I this sectio, we ivestigate several popular test statistics for SAR models i the GEL framework, icludig the parameter restrictio test, overidetificatio test, Mora s I test ad spatial J test. As show below, a iterestig aspect of those test statistics is their robustess to ukow heteroskedasticity as log as their momet coditios are valid, while covetioal test statistics without takig ito accout carefully their heteroskedastic variaces for relevat evaluatio might ot be robust. Furthermore, GEL test statistics based o quadratic momets with zero diagoal quadratic matrices ca be robust to o-ormal distributios, while covetioal test statistics might ot be so if higher order momets are ot properly take ito accout. 9

10 4. Test for parameter restrictios We may test for parameter restrictios i the GEL framework. Let θ = (α, φ ), where α is a k α subvector of θ, e.g., α might be a vector of spatial depedece parameters κ ad/or τ i (). Suppose that we are iterested i testig whether the true value of α is equal to zero or more geerally a kow costat vector c α. Let θ = (c α, φ ) be the restricted GEL estimator with the restrictio α = c α imposed, ad λ = arg max λ Λ( θ ) ρ(λ g i ( θ )). By the max-mi characterizatio of the saddle poit of the GEL objective fuctio, ρ( λ g i ( θ )) ρ(ˆλ g i ( θ )) ρ(ˆλ g i (ˆθ )). The we have the followig GEL ratio test. Propositio 4.. Suppose that Assumptios 6 ad 8 are satisfied. The, give Assumptios (i) ad for the homoskedastic case, or Assumptios (ii) ad for the heteroskedastic case, uder the ull hypothesis H 0 : α 0 = c α, [ ρ( λ g i ( θ )) ρ(ˆλ d g i (ˆθ ))] χ (k α ). The GEL ratio test is asymptotically equivalet to the distace differece test i the GMM framework (Doald et al., 003). But it does ot ivolve estimatio of a optimal weightig of momets as i the GMM distace differece test. The GEL ratio has a similarity to a classical likelihood ratio statistic. As log as the momet vector g (θ) is valid, this test statistic ca be formulated ad is robust to ukow heteroskedasticity. These latter ad distributio-free features are more attractive tha those of a likelihood ratio test statistic. I a likelihood ratio test, the likelihood fuctio eeds to be properly specified to take ito accout heteroskedasticity ad distributios of sample observatios. For this GEL, oe relies oly o momets ad does ot eed to have the proper formulatio of heteroskedastic variaces ad distributios of disturbaces. I the regard of ukow heteroskedasticity, it has a computatioal advatage over a Wald test as the latter would require the use of a robust variace estimate as i White (980). To uderstad power properties of this test statistic, we ivestigate its power uder a local alterative sequece. Suppose that the true value of α is subject to a Pitma drift α = c α + / d α, where d α is a k α vector of costats, the the GEL ratio statistic ca be show to be asymptotically distributed with a ocetral chi-squared distributio, which is the same as that for a distace differece test i the GMM framework (Newey ad West, 987). Let Ḡα = E( g(θ0) α ), Ḡ φ = E( g(θ0) φ ), Dφ = Ω Ω Ḡφ(Ḡ φ Ω Ḡφ) Ḡ φ Ω, ad χ (a, b) be a ocetral chi-squared distributio with a degrees of freedom ad a ocetrality parameter b. Propositio 4.. Suppose that Assumptios 6 ad 8 are satisfied. The, give Assumptios (i) ad for the homoskedastic case, or Assumptios (ii) ad for the heteroskedastic case, uder the Pitma drift α = c α + / d α, [ ρ( λ g i ( θ )) ρ(ˆλ g i (ˆθ ))] d χ (k α, lim d αḡ α D φ Ḡ α d α ). 0

11 4. Overidetificatio test Like the GMM, a properly ormalized GEL objective fuctio at the GEL estimator (ˆθ, ˆλ ) ca provide a overidetificatio test of momet coditios. The test statistic [ ρ(ˆλ g i (ˆθ )) ρ(0)] is o-egative as ρ(0) is the restricted value of ρ(λ g i (ˆθ )) with the restrictio λ = 0 while ρ(ˆλ g i (ˆθ )) is a urestricted maximum for λ. Propositio 4.3. Suppose that Assumptios 6 ad 8 are satisfied. The uder Assumptios (i) ad i the homoskedastic case, or Assumptios (ii) ad i the heteroskedastic case, [ ρ(ˆλ d g i (ˆθ )) ρ(0)] χ (k g k θ ), where the umber of momets k g is ot less tha the umber of parameters k θ. This GEL overidetificatio test is asymptotically equivalet to the GMM overidetificatio test. I geeral, misspecificatio of a SAR model may come from differet sources which give misspecified momet coditios. The overidetificatio test will be able to detect those misspecificatios. If oe believes that misspecificatio might come oly from a particular source, the the overidetifcatio test might detect it. However, for a specific directio of departure, it is desirable to desig more power test statistics. I a subsequet sectio, we cosider a o-ested test, amely, a J-test, for SAR models with differet specified spatial weights matrices. Before that, we cosider a test of spatial depedece, the well-kow Mora s I statistic. 4.3 Mora s I test Mora s I test is a popular test for spatial depedece. I practice, the least squares (LS) residual vector ˆV = [I X (X X ) X ]Y from the regressio of Y o X i the regressio model Y = X β + V is ofte used ad the test is based o the asymptotic distributio of ˆV W ˆV. After ormalizatio with a proper stadard error, a asymptotically ormal distributio of the ormalized statistic is used for testig. Such a test has a ull hypothesis that v i i V are idepedet but ot spatially correlated. Here we show that such a test of spatial depedece ca be coveietly implemeted i the GEL framework. Such a GEL test ca be robust agaist disturbaces with ukow heteroskedasticity, while there is o eed to estimate the asymptotic variace of ˆV W ˆV. Let g i = v i i j= (w,ij + w,ji )v j ad ĝ i = ˆv i i j= (w,ij + w,ji )ˆv j, where ˆv i is the ith elemet of ˆV, for i =,...,, ad ˆΛ = {λ : λĝ i V, i =,..., }. 3 Propositio 4.4. Suppose that i the regressio model Y = X β 0 + V with zero mea idepedet disturbaces v i s, W is a ostochastic matrix with a zero diagoal ad bouded row ad colum sum orms, ad lim E(g i ) 0. Uder Assumptios, ad 5, [ max λ ˆΛ ] ρ(λĝ i ) ρ(0) = ( ) ( ) ĝ i + op () d χ (). ĝi Kelejia ad Prucha (00) propose a geeralized Mora s I test that cover the SARAR models ad limited depedet variable models. Qu ad Lee (0, 03) have cosidered the use of geeralized residuals for the costructio of locally most powerful LM tests for the spatial Tobit model. 3 Note that ĝ = g = 0 by the covetio of the summatio otatio. We defie ĝ ad g for coveiece.

12 The GEL test statistic ca use the estimated ĝ i istead of the true g i, because ˆV W ˆV with the OLS estimated ˆV has the same asymptotic distributio as V W V due to a orthogoality property. Note that the GEL Mora s I test statistic is robust to ukow heteroskedasticity. A covetioal Mora s I test would eed to evaluate the asymptotic variace of the statistic V W V uder the ull. A robust Mora s I test ca be computed as ( ĝ i ) ( ĝi), give i the above propositio, if we use ĝ i to estimate the variace of ĝi. A GEL versio of Mora I s test ca bypass such calculatios as the GEL takes cares of ukow heteroskedasticity iterally. For the local power of Mora s I test, we cosider the alterative model beig a SE model, Y = X β+u, U = τ U + V, where the spatial error depedece parameter is subject to the Pitma drift τ = / d τ. Propositio 4.5. Suppose that Y = X β 0 +U, U = / d τ W U +V, where d τ is a costat, W is a ostochastic matrix with a zero diagoal ad bouded row ad colum sum orms, ad lim E(g i ) 0. Uder Assumptios, ad 5, [ max λ ˆΛ ρ(λĝ i ) ρ(0)] d χ (, lim { E[(V W V ) ]} { d τ E[V (W + W )W V ] } ). We may compare this GEL Mora s I test with the parameter restrictio test for spatial error depedece i the SE model based o the momet vector [V W V, V X ]. By Propositios 4. ad 4.5, these test statistics have the same asymptotic distributio uder the same Pitma drift. The above GEL Mora s I test uses the estimated momet coditio ˆV W ˆV, which relies o the ull model beig a liear regressio model. If the model is a SARAR model (), ad the test is for spatial depedece i disturbaces, the with cosistetly estimated residual vector ˆV such as the estimated residuals from a SLS or QML estimated SAR equatio, ˆV W ˆV may ot have the same asymptotic distributio as V W V ad the test statistic would ot be asymptotically chi-squared distributed. Neither would the GEL test versio. This problem occurs due to the issue that the cosistet estimator used to costruct the momets for testig has a impact o the asymptotic distributio of the momets. 4 To overcome this problem i the GEL framework, we may cosider a correspodig C(α)-type statistic as suggested i Ji ad Lee (06). Let θ = (α, φ ), where α is the spatial error depedece parameter τ ad the test is o whether α 0 = 0. Deote ˆθ = (0, ˆφ ) for ay -cosistet estimator ˆφ of φ 0. Istead of the momet g (θ) = V (θ)m V (θ), 5 where V (θ) = (I τm )[(I κw )Y X β], we may use the momet g (θ) = g (θ) g(θ) φ ( g(θ) φ ) g (θ), where g (θ) is a (k θ ) vector of liear ad quadratic momets. As g (θ) ad g (θ) are liear ad quadratic momets, g (θ) ca be writte as g (θ) = g i(θ), where g i (θ) = g,i (θ) g(θ) φ ( g(θ) φ ) g,i (θ) with g,i (θ 0 ) s ad g,i (θ 0 ) s beig martigale differeces. I place of the estimated momet g (ˆθ ), we cosider the alterative g (ˆθ ). By the mea value theorem, we ca see that g (ˆθ ) has the same asymptotic distributio as g (θ 0 ). 4 For Mora s I test, the orthogoality holds because (Y X ˆβ) W (Y X ˆβ) = (Y X β 0 ) W (Y X β 0 ) + o p() due to ˆβ beig the least squares estimator. 5 Model () ca allow for differet spatial weights matrices i the spatial lag ad spatial error processes, eve though i practice they are usually the same. The spatial weights matrix i the spatial error process is M, so we have the quadratic momet V (θ)mv(θ).

13 Propositio 4.6. For model () with τ 0 = 0, suppose that Assumptios 3 ad 5 hold, ad lim E(g i (θ 0)) [ ] 0. The max λ Λ(ˆθ ) ρ(λg i(ˆθ )) ρ(0) = ( g i (ˆθ )) ( g i(ˆθ )) + o p () d χ (). The test statistic is readily available with the GEL estimate of λ. It is robust to ukow heteroskedasticity if quadratic matrices i the quadratic momets of g (θ) have zero diagoals. The above GEL test ca use ay -cosistet estimator ˆθ. However, it is desirable to choose g (θ) ad its momet estimator ˆθ = (0, ˆφ ) such that g (0, ˆφ ) = 0. Because with such momets, the estimated momet vector g (ˆθ ) is exactly the same estimated momet g (ˆθ ) ad we do ot chage the basic momets g (θ) for testig. However, the idividual g i (ˆθ ) ad g,i (ˆθ ) are differet eve their summatios over i are the same. The direct use of g,i (ˆθ ) i a GEL test would ot overcome the impact of ˆθ o the asymptotic distributio of that GEL test statistic while the former ca, because g (θ) has a orthogoality property while g (θ) does ot. 4.4 Spatial J test Empirical researchers ofte face the problem o how to specify ecoometric models. I spatial ecoometrics, sice a ecoomic theory may be ambiguous o spatial weights matrices, their specificatios are frequetly challeged. Thus we may have possible specificatios of SAR models with differet spatial weights matrices. For testig ad model selectio, SARAR models with differet spatial weights matrices are o-ested. A popular testig procedure is based o the spatial J test (Kelejia, 008; Kelejia ad Piras, 0). 6 I this sectio, we formulate the spatial J test i the GEL framework. Suppose that we are iterested i testig model () agaist a alterative SARAR model: Y = κ W Y + X β + U, U = τ M U + V, (0) where W, M, X ad V have similar meaigs to those i model (). 7 The J test is origiated i Davidso ad MacKio (98) ad is based o whether the alterative model ca sigificatly improve the predictio of the depedet variable vector Y. Let ˆκ ad ˆβ be, respectively, estimators of κ ad β i (0), which are cosistet if model (0) was the true model. The ˆκ ad ˆβ ca be the QML, GMM or eve GEL estimators. 8 A predictor of Y from the alterative model ca be either Ŷ = ˆκ W Y + X ˆβ usig the mai equatio of (0) or Ŷ = (I ˆκ W ) X ˆβ usig the reduced form of Y uder (0). The differece of usig the two versios has bee discussed i Kelejia ad Piras (0). As Y is o the right had side of the first predictio versio, that Ŷ would be edogeous, while the secod oe is exogeous. The spatial J test for () is based o a augmeted model: Y = κw Y + X β + ηŷ + U, U = τm U + V, () 6 Cox-type tests for SARAR models are developed i Ji ad Lee (03). Delgado ad Robiso (05) propose o-ested tests i a geeral spatial, spatio-temporal or pael data cotext. 7 While it is possible to test oe model agaist several alteratives simultaeously, we oly cosider oe alterative model for simplicity. 8 Large sample properties of the GEL estimators ˆκ ad ˆβ are preseted i Appedix B uder regularity coditios for misspecified models. 3

14 where Ŷ is added i the ull model () to predict Y. We test whether the coefficiet η is sigificatly differet from zero or ot. If it is, we do ot reject the alterative model; otherwise, we reject it. I Kelejia ad Piras (0), the spatial J test uses the GSSLS to estimate the augmeted model. 9 Whe Ŷ is exogeous, it ca be used directly as a extra IV for W Y. For the versio that Ŷ is edogeous, the extra IVs would be eeded for Ŷ. The GSSLS uses oly liear IV momets but does ot utilize quadratic momets for the mai equatio of (). Thus it may lead to a relatively iefficiet estimator ad a less powerful test (Ji ad Lee, 03). Here as a geeralizatio, we cosider the GEL estimatio of model () with both liear ad quadratic momets. For the augmeted model (), let V (ϑ) = R (τ)[s (κ)y X β ηŷ], where ϑ = (θ, η). The momet vector ca be g (ϑ) = [V (ϑ)p V (ϑ) σ tr(p ),..., V (ϑ)p,kp V (ϑ) σ tr(p ), Q V (ϑ)] i the homoskedastic case, ad g (ϑ) = [V (ϑ)p V (ϑ),..., V (ϑ)p,kp V (ϑ), Q V (ϑ)] where each P l, l =,..., k p, has a zero diagoal i the heteroskedastic case. Defie g i (ϑ) such that g (ϑ) = g i(ϑ). Uder the ull, g i (ϑ 0 ) s are martigale differeces, where ϑ 0 = (θ 0, 0). The GEL estimator is ˆϑ = arg mi ϑ Θ max λ Λ(ϑ) ρ(λ g i (ϑ)), where Λ (ϑ) = {λ : λ g i (ϑ) V, i =,..., } ad Θ is the parameter space of ϑ. With the idetificatio ad regularity coditios i Appedix B, the spatial J test statistic ca be formulated as a GEL ratio. This GEL test is essetially a test of the parameter restrictio that η = 0 i (). It differs from the oe i the precedig Sectio 4. i that here Ŷ o the right had side of () is a geerated regressor. As the followig propositio will show, the iitial estimate i Ŷ does ot have a asymptotic impact o the GEL statistic uder the ull. Propositio 4.7. Suppose that Assumptios 6, 8, 3 ad 4 hold ad ϑ 0 is i the iterior of the compact parameter space Θ. The, uder Assumptios (i) ad i the homoskedastic case, or Assumptios (ii) ad i the heteroskedastic case, [ ρ(ˆλ g i (ˆθ )) max λ Λ( ˆϑ ) ρ(λ g i ( ˆϑ ))] d χ (), where (ˆθ, ˆλ ) is the GEL estimator for model (), i.e., it is the restricted GEL estimator with the restrictio η = 0 imposed. 5 Mote Carlo I this sectio, we report Mote Carlo results o the GEL estimator ad test statistics cosidered i this paper. The data geeratig process is the SARAR model () or its restricted form with κ = 0 ad/or τ = 0. There are three exogeous variables i X : a itercept term, a variable radomly draw from the stadard ormal distributio N(0, ) ad a variable from the uiform distributio U[0, ]. The true value β 0 of β = (β, β, β 3 ) 9 Sice the origial spatial J test uses the GSSLS to estimate the augmeted model, the mai equatio of () is trasformed by pre-multiplyig it with (I ˆτ M ) before estimatio, where ˆτ is a cosistet estimator of τ 0 (Kelejia ad Piras, 0). 4

15 is [0.5, 0.5, 0.5]. The disturbaces v i s are radomly draw from the ormal distributio N(0, σ 0) i the homoskedastic case, or N(0, σ 0c i ) i the heteroskedastic case, where c i is the the umber of ozero elemets i the ith row of the spatial weights matrix W, ad σ 0 is chose such that R var(x β 0 )/[var(x β 0 ) + σ ] is either 0.4 or 0.8, where σ is the average variace of all v i s. We set the two spatial weights matrices W ad M to be the same. For GEL estimatio ad tests other tha the spatial J test, W is based o the circular world matrix i Arraiz et al. (00). For the circular world matrix, spatial uits are equally spaced o a circle. Oe third of them are coected to te earest eighbors ad the rest are coected to two earest eighbors. For the spatial J test, the ull ad alterative models oly differ i W ; specifically, the circular matrix ad the oe based o the quee criterio are tested agaist each other. These matrices are ormalized to have row sums equal to oe. For the estimatio of model (), i the homoskedastic case, we use the momet vector [V V σ 0, V W V, V W V σ 0 tr(w ), V (X, W X, W X )], where X is a submatrix of X that excludes the itercept term so that the IV matrix (X, W X, W X ) oly cotais oe itercept; i the heteroskedastic case, we use the momet vector [V W V, V (W diag(w ))V, V (X, W X, W X )]. For the spatial J test, the ull ad alterative models are estimated with momet vectors similar to the above oes. To estimate the augmeted model (), if Ŷ = (I ˆκ W ) X ˆβ is used as the augmeted explaatory variable, Ŷ is added to the IV matrix i the above momet vectors; o the other had, if Ŷ = ˆκ W y + X ˆβ is the augmeted explaatory variable, W [X, X 3 ] is added to the IV matrix. The omial size of various tests is The umber of Mote Carlo repetitios for each case is, 000. Table reports biases, stadard errors, ad root mea square errors (RMSE) of the GMM, EL ad ET estimators i the homoskedastic case. 0 The GMM estimator is a FOGMM estimator where i the first step the idetity matrix is used as the weightig matrix to derive a cosistet estimator θ ad i the secod step Ω ( θ ) is used as the weightig matrix. The biases of the EL ad ET estimators are smaller tha those of the GMM estimator except for some cases, mostly for τ. For the compariso of the EL ad ET, except for the variace parameter σ, they have similar biases i most cases ad either the EL or the ET would domiate each other. For σ, the bias of the GMM estimator is sigificatly larger tha that of the ET estimator, while the latter is larger tha that of the EL estimator. I terms of stadard errors, the ET estimator performs better tha the EL estimator, ad the GMM estimator geerally performs the worst. Sice stadard errors of estimates domiate biases for parameters other tha σ, the RMSEs display a order i magitude similar to that of stadard errors. For σ, the EL estimator has the smallest RMSE, ad the ET estimator has a smaller RMSE tha that of the GMM estimator. As the sample size icreases from 44 to 400, biases geerally decrease, ad stadard errors decrease approximately at the theoretical rate. Table shows summary statistics of the estimators i the heteroskedastic case. The biases are small ad their patter are similar to those i the homoskedastic case. The EL estimator is observed to have larger stadard errors ad RMSEs tha those of the ET estimator. For the parameters κ, τ ad the itercept β, the ET estimator 0 We do ot cosider the cotiuous updatig GMM estimator because it is ofte observed to possess multiple modes ad thus geerally cosidered to be less desirable tha the EL ad ET estimators (Hase et al., 996; Imbes et al., 998). 5

16 geerally has the smallest stadard errors ad RMSEs, eve though for the parameters β ad β 3 of regressors, the GMM estimator has the smallest stadard errors ad RMSEs i some cases. Table 3 reports coverage probabilities (CP) of 95% cofidece itervals for parameters i the SARAR model (). I the homoskedastic case, for = 44, the GMM CPs are below 95%, ad those for σ are much smaller tha 95%; the EL ad ET CPs are closer to 95% tha GMM oes, ad those for σ are about te percetage poits higher tha correspodig GMM CPs. The ET CPs are higher tha EL oes except for σ. With a larger sample size = 400, the CPs are closer to 95%, but the patters are similar. I the heteroskedastic case, the EL ad ET CPs are still closer to 95% tha GMM oes i geeral, though the differeces are smaller. For Mote Carlo studies o hypothesis testig, ie tests are cosidered i the homoskedastic case: PT gmm, PT el ad PT et deote parameter restrictio tests implemeted with, respectively, the GMM distace differece, EL ratio ad ET ratio based o the momet vector [V V σ 0, V W V, V W V σ 0 tr(w ), V (X, W X, W X )] ; OT gmm, OT el ad OT et deote, respectively, the GMM, EL ad ET overidetificatio tests based o the momet vector [V W V, V X ] ; Mora deotes Mora s I test with a robust variace estimator, ad Mora el ad Mora et deote, respectively, EL ad ET Mora s I tests. For the latter three tests, OLS residuals are used to formulate test statistics. I the heteroskedastic case, the above tests are also cosidered, amog which parameter restrictio tests are based o the momet vector [V W V, V (W diag(w ))V, V (X, W X, W X )] robust to ukow heteroskedasticity. I additio, we cosider two tests which do ot take ito accout ukow heteroskedasticity: the GMM parameter restrictio test PT gmm based o the momet vector [V V σ 0, V W V, V W V σ 0 tr(w ), V (X, W X, W X )] ad covetioal Mora s I test Mora. Table 4 presets empirical sizes of tests for τ 0 = 0 i a SE model. PT el ad PT et have relatively large sizes for small sample cases ad have improved sizes for the larger sample size = 400. As expected, PT gmm ad Mora have large size distortios ad the distortios do ot improve with the larger sample size = 400. Other tests have relatively small size distortios. Powers of these tests except PT gmm ad Mora are preseted i Table 5. Their powers are geerally similar for differet valid tests, but are higher for the homoskedastic model tha those of the heteroskedastic model. R does ot have much impact o powers. These tests are powerful i cases with a larger τ 0 ad a larger sample size i the data geeratig process (DGP). Test results o τ 0 = 0 i the SARAR model () are reported i Tables 6 ad 7. Parameter restrictio tests are based o momet coditios similar to those for the SE model. Overidetificatio tests are based o the momet vector [V W V, V (W diag(w ))V, V (X, W (I ˆκ W ) X ˆβ )], where ˆκ ad ˆβ are the FOGMM estimator of the SAR model as described above. To compute Mora s I tests, we use the SLS estimator ˆφ of φ = (κ, β ) with the IV matrix Q = [X, W X, W X ] for the SAR model. The test statistics employ the momet coditio g (θ) = g (θ) g(θ) φ ( g(θ) φ ) g (θ), where g (θ) = V (θ)w V (θ) ad g (θ) = Z Q (Q Q ) Q V (θ) with Z = [W Y, X ]. Thus g (ˆθ ) = 0, where ˆθ = (0, ˆφ ). Whe = 44, the size distortios of parameter restrictio tests are larger tha those of overidetificatio tests, ad those of Mora s I tests are smallest; whe = 400, all sizes are geeral close to the omial 5%. Differet versios of parameter 6

17 restrictio tests have similar powers. So are differet versios of overidetificatio tests ad those of Mora s I tests. Parameter restrictio tests are more powerful tha overidetificatio tests, ad the latter oes are geerally more powerful tha Mora s I tests. With larger R, sample sizes, ad τ 0 i the DGP, all tests ted to be more powerful. Tables 8 ad 9 report empirical sizes ad powers of spatial J tests for the SARAR model (). GMM deotes the spatial J test implemeted with the GMM distace differece test usig the predictor Ŷ = ˆκ W Y +X ˆβ, ad GMM uses Ŷ = (I ˆκ ) X ˆβ. Correspodigly, we have EL ad ET ratio tests EL, EL, ET ad ET. The EL, EL, ET ad ET have relatively larger size distortios for a small sample size, but are reasoably adequate for a larger sample size. Powers of these tests are similar. With larger R, κ 0 ad sample sizes, these tests are more powerful. [Tables 9 about here.] 6 Coclusio By explorig the martigale structure of the SARAR model, this paper cosiders its GEL estimatio ad tests. We show that the GEL estimator is cosistet ad has the same asymptotic ormal distributio as the optimal GMM estimator based o the same momet coditios. But the GEL avoids a first step estimatio of the optimal weightig matrix with a prelimiary estimator ad ca be robust to ukow heteroskedasticity without the computatio of possibly higher order momet parameters of disturbaces. A geeral GEL is free from the asymptotic bias of the prelimiary estimator ad partially removes the bias due to the correlatio betwee the momet coditios ad their Jacobia. A EL further partially removes the bias from estimatig the secod momet matrix. We also ivestigate the GEL overidetificatio test, Mora s I test, GEL ratio tests for parameter restrictios ad o-ested hypotheses. These tests do ot ivolve estimatio of variaces ad higher order momet parameters, ad ca be robust to ukow heteroskedasticity. Our Mote Carlo results show that GEL estimators ad tests perform well compared with GMM estimators ad tests whe the latter GMM estimates ad tests take ito accout properly their variaces ad/or momet parameters of disturbaces. The GMM tests are ot robust while GEL tests are much better to deal with extra complexity of spatial regressio models. I a future research, it is of iterest to ivestigate various optimality properties of EL tests for the SARAR model as i Kitamura (00) ad Otsu (00), ad their Bartlett correctability. The latter is expected by Myklad (995). However, Bartlett correctability is based o Edgeworth expasios, for which it is ot kow how to show geeral poitwise results o martigales. For SAR models, a smoothed (istead of poitwise) asymptotic expasio based o martigales i Myklad (993) is show i Ji ad Lee (03). 7