Specification and Estimation of Spatial Autoregressive Models with Autoregressive and Heteroskedastic Disturbances

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1 Specificatio ad Estimatio of Spatial Autoregressive Models with Autoregressive ad Heteroskedastic Disturbaces Harry H. Kelejia ad Igmar R. Prucha Departmet of Ecoomics Uiversity of Marylad, College Park, MD Revised September 2005 Revised May 2007 Abstract Oe importat goal of this study is to develop a methodology of iferece for a widely used Cliff-Ord type spatial model cotaiig spatial lags i the depedet variable, exogeous variables, ad the disturbace terms, while allowig for ukow heteroskedasticity i the iovatios. We first geeralize the geeralized momets (GM) estimator suggested i Kelejia ad Prucha (1998,1999) for the spatial autoregressive parameter i the disturbace process. We prove the cosistecy of our estimator; ulike i our earlier paper we also determie its asymptotic distributio, ad discuss issues of efficiecy. We the defie istrumetal variable (IV) estimators for the regressio parameters of the model ad give results cocerig the joit asymptotic distributio of those estimators ad the GM estimator uder reasoable coditios. Much of the theory is kept geeral to cover a wide rage of settigs. We ote the estimatio theory developed by Kelejia ad Prucha (1998, 1999) for GM ad IV estimators ad by Lee (2004) for the quasi-maximum likelihood estimator uder the assumptio of homoskedastic iovatios does ot carry over to the case of heteroskedastic iovatios. The paper also provides a critical discussio of the usual specificatio of the parameter space. Key Words: Spatial depedece, heteroskedasticity, Cliff-Ord model, two-stage least squares, geeralized momets estimatio, asymptotics JEL Classificatio: C21, C31 Correspodig Author: Igmar R. Prucha, Tel.: , prucha@eco.umd.edu. 1

2 1 Itroductio 1 I recet years the ecoomics literature has see a icreasig umber of theoretical ad applied ecoometric studies ivolvig spatial issues. 2 While this icrease i iterest i spatial models i ecoomics is relatively recet, spatial models have a log history i the regioal sciece ad geography literature. 3 Oe of the most widely refereced model of spatial iteractios is oe that was put forth by Cliff ad Ord (1973, 1981). This model is a variat of the model cosidered by Whittle (1954). I its simplest (ad origial) form the model oly cosiders spatial spillovers i the depedet variable, ad specifies the edogeous variable correspodig to a cross sectioal uit i terms of a weighted average of edogeous variables correspodig to other cross sectioal uits, plus a disturbace term. This model is typically referred to as a spatial autoregressive model, the weighted average is typically referred to as a spatial lag, the correspodig parameter as the autoregressive parameter, ad the matrix cotaiig the weights as the spatial weights matrix. A geeralized versio of this model also allows for the depedet variable to deped o a set of exogeous variables ad spatial lags thereof. A further geeralizatio allows for the disturbaces to be geerated by a spatial autoregressive process. Cosistet with the termiology developed by Aseli ad Florax (1995) we refer to the combied model as a spatial autoregressive model with autoregressive disturbaces of order (1, 1), for short SARAR(1, 1). We ote that this model is fairly geeral i that it allows for spatial spillovers i the edogeous variables, exogeous variables ad disturbaces. Somewhat surprisigly, eve though the SARAR(1, 1) model has bee a modelig tool for may years, util recetly there has bee a lack of formal results cocerig estimatio methods for this model. Oe method that has bee employed to estimate this model is the (quasi) maximum likelihood (ML) 1 Our thaks for very helpful discussios ad suggestios are owed to Irai Arraiz, Badi Baltagi, Peter Egger, David Drukker, Beedikt Pötscher, ad Paulo Rodrigues, ad to semiar participats at Pesylvaia State Uiversity, Sigapore Maagemet Uiversity, ADRES Coferece o Networks of Iovatios ad Spatial Aalysis of Kowledge Diffusio i Sait- Etiee, Texas A&M Uiversity, SUNY Albay, Uiversity of Isbruck, Syracuse Uiversity ad Kasas Uiversity. Also, we gratefully ackowledge fiacial support from the Natioal Sciece Foudatio through grat SES ad the Natioal Istitute of Health through the SBIR grat 1 R43 AG Some recet applicatios of spatial models are, e.g., Audretsch ad Feldma (1996), Baltagi, Egger ad Pfaffermayr (2005), Bell ad Bockstael (2000), Besley ad Case (1995), Betrad, Luttmer ad Mullaiatha (2000), Case (1991), Case, Hies, ad Rose (1993), Cohe ad Morriso Paul (2004), Haushek et al. (2003), Holtz-Eaki (1994), Sacrerdote (2001), Shroder (1995), Topa (2001). Cotributios to the theoretical ecoometric literature iclude, e.g., Baltagi ad Li (2001a,b, 1999), Baltagi, Sog, Jug ad Koh (2005), Baltagi, Sog ad Koh (2003), Bao ad Ullah (2003), Coley (1999), Das, Kelejia ad Prucha (2003), Driscol ad Kraay (1998), Kapoor, Kelejia ad Prucha (2004), Kelejia ad Prucha (2005, 2004, 2002, 2001, 1999, 1998), Koriotis (2005), Lee (2005, 2004, 2003, 2002), Pikse ad Slade (1998), Pikse, Slade, ad Brett (2002), Yag (2005). 3 See, e.g., Aseli (1988), Beett ad Hordijk (1986), Cliff ad Ord (1973, 1981), ad Cressie (1993) ad the refereces cited therei. 2

3 procedure, where the likelihood fuctio correspods to the ormal distributio. Formal results cocerig the asymptotic properties of the ML estimator have bee established oly recetly i a importat cotributio by Lee (2004). Give that the likelihood fuctio ivolves the determiat of a matrix whose dimesios deped o the sample size ad a ukow parameter, there ca be sigificat difficulties i the practical computatio of this estimator especially if the sample size is large, as it might be if the spatial uits relate to couties, sigle family houses, etc. I part because of this Kelejia ad Prucha (1999) itroduced a geeralized momets (GM) estimator for the autoregressive parameter of the disturbace process that is simple to compute ad remais computatioally feasible eve for large sample sizes. I Kelejia ad Prucha (1998) we used that GM estimator to itroduce a geeralized spatial two stage least squares estimator (GS2SLS) for the regressio parameters of the spatial SARAR(1, 1) model that is agai simple to compute, ad demostrated its cosistecy ad asymptotic ormality. 4 All of the above estimators for the SARAR(1, 1) model were itroduced ad their asymptotic properties were derived uder the assumptio that the iovatios i the disturbace process are homoskedastic. The lack of a estimatio theory that allows for heteroskedasticity, ad the lack of correspodig joit hypothesis tests for the presece of spatial depedecies i the edogeous variables, exogeous variables ad/or disturbaces, is a serious shortcomig. Spatial uits are ofte heterogeeous i importat characteristics, e.g., size, ad hece the homoskedasticity assumptio may ot hold i may situatios (coditioally ad ucoditioally). It is readily see that if the iovatios are heteroskedastic, the ML estimator cosidered i Lee (2004) is icosistet, ad the asymptotic distributio give i Kelejia ad Prucha (1998) for the GS2SLS estimator is ot appropriate. Oe importat goal of this study is therefore to develop a methodology of iferece for the SARAR(1, 1) model that allows for heteroskedastic iovatios. I developig this theory we will adopt a modular approach such that much of the theory ot oly applies to the SARAR(1, 1) model, but ca also be utilized i differet settigs i future research. I more detail, i this paper we itroduce a ew class of GM estimators for the autoregressive parameter of a spatially autoregressive disturbace process that allows for heteroskedastic iovatios. Our GM estimators are agai computatioally simple eve i large samples. We determie their cosistecy; ulike i our earlier paper we also determie, uder reasoably geeral coditios, their asymptotic distributio. Loosely speakig, i derivig those results we essetially oly maitai that the disturbaces are 1/2 -cosistetly estimated (where is the sample size) ad that the estimator of the model parameters employed i estimatig the disturbaces is asymptotically liear i the iova- 4 The formulatio of the GS2SLS estimator is based o a approximatio of the ideal istrumets. Recetly Lee (2003) ad Kelejia, Prucha, ad Yuzefovich (2004) exteded the aalysis to iclude the use of ideal istrumets. Das, Kelejia ad Prucha (2003) aalyzed the small sample properties of the GS2SLS (as well as those of other estimators). They fid that i may situatios the loss of efficiecy due to the approximatio of the ideal istrumets is mior. 3

4 tios. As a result the methodology developed i this paper covers a wide rage of (liear ad oliear) models ad estimators, i additio to the SARAR(1, 1) model ad estimators specific for that model. We furthermore derive results cocerig the joit distributio of the GM estimators ad estimators of the regressio parameters to facilitate joit tests. While the results are preseted for the case of two step estimatio procedures where the spatial autoregressive parameter ad the regressio parameters are estimated i separate steps, the aalysis ca be readily adapted to oe step procedures where all parameters are estimated i a sigle (but umerically more ivolved) step. The geeral theory is the applied to develop iferece methodology for the SARAR(1, 1) model. I particular, we use the GM estimator i costructig a GS2SLS estimator for the regressio parameters of the SARAR(1, 1) model ad demostrate the cosistecy ad asymptotic ormality of this estimator. We also provide results cocerig the joit distributio of the GM estimator ad the GS2SLS estimator, which permits, amog other thigs, testig the joit hypothesis of the absece of spatial spillovers stemmig from the edogeous variables, exogeous variables or disturbaces. Aother cocer with the existig literature o Cliff-Ord type models, icludig i the above cited literature o the SARAR(1, 1) models, is the specificatio of the parameter space for spatial autoregressive parameters. I virtually all of the literature it is assumed that the parameter space for autoregressive parameters is the iterval ( 1, 1), or a subset thereof. Oe may cojecture that this traditioal specificatio of the parameter space received its motivatio from the time series literature. However, as discussed i detail below, choosig the iterval ( 1, 1) as the parameter space for the autoregressive parameter of a spatial model is ot atural i the sese that the spatial autoregressive parameter always appears i those models i product form with the spatial weights matrix. Hece equivalet model formulatios are obtaied by applyig a (arbitrary) scale factor to the autoregressive parameter ad its iverse to the weights matrix. Of course, applyig a scale factor to the autoregressive parameter leads to a correspodig re-scalig of its parameter space. I this paper we therefore allow for a more geeral specificatio of the parameter space. Eve if a scale factor is used that results i the parameter space beig the iterval ( 1, 1), this scale factor ad correspodigly the autoregressive parameter will the typically deped o the sample size. I cotrast to the existig literature we thus allow for the parameters to deped o the sample size. Our discussio of the parameter space ad possible ormalizatios of the spatial weights matrix also poits out potetial pitfalls with the frequetly used approach of row-ormalizig the spatial weights matrix. The paper is orgaized as follows: The geeralized SARAR(1, 1) model is specified ad iterpreted i Sectio 2. This sectio also cotais a discussio of the parameter space of the autoregressive parameter. I Sectio 3 we defie ad establish the large sample properties of our suggested GM estimators for the autoregressive parameter of a spatially autoregressive disturbace process. I this sectio we also provide results cocerig the joit large sample distributio of the GM estimators ad a wide class of estimator of the regressio parameters. 4

5 We also develop HAC type estimators for the large sample variace-covariace matrix of the suggested estimators. Sectio 4 cotais results relatig to the suggested istrumetal variable estimators of the regressio parameters of the SARAR(1, 1) model ad their joit large sample distributio with GM estimators. Cocludig remarks are give i the Sectio 5. Techical details are relegated to the appedices. It proves helpful to itroduce the followig otatio: Let A with N be some matrix; we the deote the (i, j)-th elemet of A as a ij,. Similarly, if v is a vector, the v i, deotes the i-th elemet of v. A aalogous covetio is adopted for matrices ad vectors that do ot deped o the idex, i which case the idex is suppressed o the elemets. If A is a square matrix, the A 1 deotes the iverse of A. If A is sigular, the A 1 should be iterpreted as the geeralized iverse of A. At times it will also be helpful to deote the geeralized iverse more explicitly as A +. With a i., ad a.i, we deote the i-th row ad colum of A, respectively, ad with a i. ad a.i those of A 1. If A is a square symmetric oegative matrix, the A 1/2 deotes the uique symmetric ad oegative square root of A.IfA is osigular, the A 1/2 deotes (A 1 ) 1/2. Further, we say the row ad colum sums of the (sequece of) matrices A are bouded uiformly i absolute value if there exists a costat c A < (that does ot depedet of ) such that max 1 i j=1 X a ij, c A ad max 1 j X a ij, c A for all N holds. As a poit of iterest, we ote that the above coditio is idetical to the coditio that the sequeces of the maximum colum sum matrix orms ad maximum row sum matrix orms of A are bouded; cp. Hor ad Johso (1985, pp.294-5). For defiiteess, let A be some vector or matrix, the kak =[Tr(A 0 A)] 1/2. We ote that this orm is submultiplicative, i.e., kabk kakkbk. 2 Model I this sectio we specify the geeralized SARAR(1, 1) model ad discuss the uderlyig assumptios. 2.1 Specificatio Suppose a cross sectio of spatial uits is observed, ad the iteractios betwee those spatial uits ca be described by the followig model: y = X β + λ W y + u (1) = Z δ + u ad u = ρ M u + ε, (2) 5

6 with Z =[X, W y] ad δ = β 0 0.,λ Here y deotes the 1 vector of observatios of the depedet variable, X deotes the k matrix of o-stochastic (exogeous) regressors, W ad M are o-stochastic matrices, u deotes the 1 vector of regressio disturbaces, ε is a 1 vector of iovatios, λ ad ρ are ukow scalar parameters, ad β is a k 1 vector of ukow parameters. The matrices W ad M are typically referred to as spatial weights matrices, ad λ ad ρ are typically called spatial autoregressive parameters. The aalysis allows for W = M, which will frequetly be the case i applicatios. All quatities are allowed to deped o thesamplesize. The vectors y = W y ad u = M u are typically referred to as spatial lags of y ad u, respectively. We ote that all quatities are allowed to deped o the sample size ad so some of the exogeous regressors may be spatial lags of exogeous variables. Thus the model is fairly geeral i that it allows for spatial spillovers i the edogeous variables, exogeous variables ad disturbaces. The spatial weights matrices ad the autoregressive parameters are assumed to satisfy the followig assumptio. Assumptio 1 (a) All diagoal elemets of W ad M are zero. (b) λ ( a λ, a λ ), ρ ( a ρ, a ρ ) with 0 <a λ, a λ a λ < ad 0 <a ρ, a ρ a ρ <. (c) The matrices I λw ad I ρm are osigular for all λ ( a λ, aλ ), ad ρ ( a ρ, aρ ). Assumptio 1(a) is clearly a ormalizatio rule. Assumptio 1(b) cocerig the parameter space of λ ad ρ will be discussed i the ext subsectio. Assumptio 1(c) esures that y ad u are uiquely defied by (1) ad (2) as y = (I λ W ) 1 X β +(I λ W ) 1 u, (3) u = (I ρ M ) 1 ε. As remarked i the Itroductio, spatial uits are ofte heterogeeous i importat characteristics, e.g., size. For that reaso it is importat to develop a estimatio theory that allows for the iovatios to be heteroskedastic. Therefore, we maitai the followig set of assumptios with respect to the iovatios. Assumptio 2 The iovatios {ε i, :1 i, 1} satisfy Eε i, =0, E(ε 2 i, )=σ2 i, with 0 <aσ σ 2 i, aσ <, adsup 1 i, 1 E ε i, 4+η < for some η>0. Furthermore, for each 1 the radom variables ε 1,,...,ε, are totally idepedet. The above assumptio also allows for the iovatios to deped o the sample size, i.e., to form triagular arrays. We ote that eve if the iovatios do ot deped o, the elemets of y ad u would still deped o i light of (3) sice the elemets of the iverse matrices ivolved would geerally deped o. We maitai the followig assumptio cocerig the spatial weights matrices. 6

7 Assumptio 3 The row ad colum sums of the matrices W, M, (I λ W ) 1 ad (I ρ M ) 1 are bouded uiformly i absolute value. Give (3), Assumptio 2 implies that Eu =0, ad that the VC matrix of u is give by Eu u 0 =(I ρ M ) 1 Σ (I ρ M ) 1 where Σ = diag(σ 2 i, ). This specificatio allows for fairly geeral patters of autocorrelatio ad heteroskedasticity of the disturbaces. It is readily see that the row ad colum sums of products of matrices, whose row ad colum sums are bouded uiformly i absolute value, are agai uiformly bouded i absolute value; see, e.g., Kelejia ad Prucha (2004), Remark A.1. Because of this, Assumptios 2 ad 3 imply that the row ad colum sums of the variacecovariace (VC) matrix of u (ad similarly those of y ) are uiformly bouded i absolute value, thus limitig the degree of correlatio betwee, respectively, the elemets of u (ad of y ). That is, makig a aalogy to the time series literature, these assumptios esure that the disturbace process ad the process for the depedet variable exhibit a fadig memory Parameter Space for a Autoregressive Parameter Assumptio 1(b) defies the parameter space for the autoregressive parameters. I discussig this assumptio we focus o W ad λ. (A aalogous discussio applies to M ad ρ.) I the existig literature relatig to Cliff-Ord models the parameter space for the autoregressive parameter is typically take to be the iterval ( 1, 1) ad the autoregressive parameter is assumed ot to deped o the sample size. However, i applicatios it is typically foud that for uormalized spatial weights matrices, I λw is sigular for some values of λ ( 1, 1). To avoid this situatio, may applied researchers ormalize each row of their spatial weights matrices i such a way that I λw is o-sigular for all λ ( 1, 1). We ow discuss the implicatios of various ormalizatios of the spatial weight matrix. Suppose c deotes a scalar ormalizatio factor. Clearly, this ormalizatio factor may deped o the sample size. For example, some of our results below relate to the case i which c correspods to the maximal row or colum sum of the absolute values of the elemets of W. Give such a ormalizig factor, a equivalet specificatio of model (1) for y is obtaied if λ W is replaced by λ W where λ = c λ ad W = W /c. It is importat to observe that eve if λ ad its correspodig parameter space do ot deped o, λ ad its implied parameter space will deped othesamplesizeasaresultof the ormalizatio of the spatial weights matrix. 6 It is for this reaso that we allow i Assumptio 1 for the elemets of the spatial weights matrices, ad the 5 Of course, the extet of correlatio is limited i virtually all large sample aalysis; see, e.g., Amemiya (1985), ch. 3,4, ad Pötscher ad Prucha (1997), ch. 5,6. 6 The parameter space for λ is give by ( c a λ,c a λ ). 7

8 autoregressive parameters ad the correspodig parameter spaces to deped o. Of course, Assumptio 1 also covers the case where the true data geeratig process correspods to a model where autoregressive parameters do ot deped o. Assumptio 1 defies the parameter space for λ as a iterval aroud zero such that I λw is osigular for values λ i that iterval. The followig trivial lemma gives bouds for that iterval. Lemma 1 Let τ deote the spectral radius of W ; i.e., τ =max{ ν 1,,..., ν, } where ν 1,,...,ν, deote the eigevalues of W.TheI λw is osigular for all values of λ i the iterval ( 1/τ, 1/τ ). 7 Clearly, if we select ( 1/τ, 1/τ ) as the parameter space for λ,theall eigevalues of λ W are less tha oe i absolute value. Thus if we iterpret (1) as a equilibrium relatioship, the this choice of the parameter space rules out ustable Nash equilibria. Of course, we obtai a equivalet specificatio of the model if istead of workig with W we work with the ormalized weights matrix W = W /τ ad select the iterval ( 1, 1) as the parameter space for λ = λ τ. Assumptio 1 is sufficietly geeral to cover both cases. For large sample sizes the computatio of the eigevalues o W is difficult. The followig lemma gives boudaries, which are simple to compute, for a (sub)set of values of λ for which I λw is osigular. Lemma 2 Let τ =mi{ max 1 i j=1 X w ij,, max 1 j X w ij, }. The τ τ ad cosequetly I λw is osigular for all values of λ i the iterval ( 1/τ, 1/τ ). The above lemma suggests ( 1/τ, 1/τ ) as a alterative (although somewhat more restrictive) specificatio of the parameter space. Of course, we obtai a equivalet model specificatio if we ormalize the spatial weights matrix by τ ad if we choose ( 1, 1) as the parameter space for the autoregressive parameter. Sice the spectral radius is boud by ay matrix orm, other orms i place of the maximum absolute row ad colum sum orms ca be used, but τ is especially easy to compute. Rather tha to ormalize W by τ or τ, i much of the empirical literature the spatial weights matrices are ormalized such that each row sums to uity. 7 I some of the spatial literature the followig closely related claim ca be foud: I λw is osigular for all values of λ i the iterval (1/ν,mi,1/ν,max ),whereν,mi ad ν,max deote the smallest ad largest eigevalue of W, respectively. This claim is correct for the case i which all eigevalues of W are real ad ν,mi < 0 ad ν,max > 0. Sice, e.g., the eigevalues of W eed ot be real, this claim does ot hold i geeral. 8

9 The motivatio for this ormalizatio is that if W is row-ormalized the I λw is osigular for all values of λ i the iterval ( 1, 1); this ca be readily cofirmed via Lemma 2. However, this ormalizatio is quite differet tha those described above i that i row-ormalizig a matrix oe does ot use a sigle ormalizatio factor, but rather a differet factor for the elemets of each row. Therefore, i geeral, there exists o correspodig re-scalig factor for the autoregressive parameter that would lead to a specificatio that is equivalet to that correspodig to the u-ormalized weights matrix. Cosequetly, uless theoretical issues suggest a row-ormalized weights matrix, this approach will i geeral lead to a misspecified model. The above discussio provides the motivatio for our specificatio that the autoregressive parameters may deped o. Furthermore, sice some of the regressors may be spatial lags, we allow all of the model parameters to deped o the sample size. 3 GM Estimator for the Autoregressive Parameter ρ I the followig we itroduce a class of GM estimators for ρ that ca be easily computed, ad prove their cosistecy ad asymptotic ormality uder a set of geeral assumptios. We ote that the discussio i this sectio oly maitais model (2) for the disturbaces u, but ot ecessarily (1) for y.thustheresults will also be useful i other settigs such as cases where y is determied by a oliear model; see, e.g. Kelejia ad Prucha (2001, p. 228). The estimators put forth below geeralize the GM estimator for ρ itroduced i Kelejia ad Prucha (1999). I cotrast to that earlier paper we ow allow for heteroskedastic iovatios ε i,, ad optimal weightig of the momet coditios. We also do ot cofie the parameter space for ρ to be the iterval ( 1, 1), adallow ρ to deped o. I our earlier paper we oly demostrated the cosistecy of the estimator. I the followig we also derive the asymptotic distributio of the cosidered estimators. 3.1 Defiitio of the GM Estimator for ρ I the followig let eu deote some predictor of u. Furthermore, for otatioal coveiece let u = M u ad u = M u = M 2 u, ad correspodigly, eu = M eu,ad e u = M 2 eu. Similarly, let ε = M ε. It is readily see that uder Assumptios 1 ad 2 we have the followig momet coditios: 1 Eε 0 ε = 1 Tr M diag (Eε 2 i,) M 0 ª, (4) 1 Eε 0 ε = 0. It proves coveiet to rewrite these coditios as 1 ε 0 E A 1, ε ε 0 =0 (5) A 2, ε 9

10 with A 1, = M 0 M diag(m 0.i,m.i, ), A 2, = M. Uder Assumptios 1 ad 3 it is readily see that the diagoal elemets of A 1, ad A 2, are zero ad that the row ad colum sums of A 1, ad A 2, are bouded uiformly i absolute value; see, e.g., Remark A.1 i Kelejia ad Prucha (2004). Our GM estimators for ρ are based o these momets. Specifically, ote that i light of (2) ε =(I ρ M )u = u ρ u ad so ε = u ρ u. Substitutig these expressios ito (4) or (5) yields the followig two equatio system: γ Γ α =0 (6) where α =[ρ,ρ 2 ] 0 ad the elemets of Γ = γ ad γ rs, r,s=1,2 = 0 γ 1,,γ 2, are give by o γ 11, = 2 1 E u 0 u Tr[M [diag (u i,u i, )] M 0 ] (7) = 2 1 Eu 0 M 0 A 1, u, γ 12, = 1 E u 0 u + Tr M diag (u 2 i,) M 0 o = 1 Eu 0 M 0 A 1, M u, γ 21, = 1 E(u 0 u + u 0 u ) = 1 Eu 0 M 0 (A 2, + A 0 2,)u, γ 22, = 1 Eu 0 u = γ 1, = 1 Eu 0 M 0 A 2, M u, 1 E u 0 u Tr M diag (u 2 i,) M 0 ª = 1 Eu 0 A 1, u, γ 2, = 1 Eu 0 u = 1 Eu 0 A 2, u. Now let e Γ = eγ rs, r,s=1,2 ad eγ = eγ 1,, eγ 2, 0 deote correspodig estimators for the elemets of Γ ad γ, which are obtaied from the above expressios for the elemets of Γ ad γ by suppressig the expectatios operator, ad replacig the disturbaces u, u,adu by their predictors eu, eu,ad e u, respectively. The, the empirical aalog of the relatioship i (6) is eγ Γ e α = υ, (8) where υ ca be viewed as a vector of regressio residuals. Our GM estimators of ρ,sayeρ,areowdefied as weighted oliear least squares estimators based o (8). That is, let Υ e be a 2 2 symmetric positive semidefiite (momets) weightig matrix; the eρ is defied as 10

11 ( eρ = eρ ( Υ e )= argmi eγ e ρ Γ ρ [ a ρ,a ρ ] ρ 2 0 eυ eγ e Γ ρ ρ 2 ). (9) We ote that the objective fuctio for eρ remais well defiedeveforvalues of ρ for which I ρ M is sigular, which allows us to take the optimizatio space for eρ to be ay compact iterval that cotais the true parameter space. For computatioal efficiecy it is best to use the formulae for the elemets of e Γ ad eγ correspodig to the first expressio o the r.h.s. of (7) ad to compute eu ad e u recursively as eu = M eu ad e u = M eu. I this fashio oe ca avoid the computatio of M 2, i.e., the computatio of the product of two matrices. We ow relate the above estimator for the autoregressive parameter to the GM estimator itroduced i Kelejia ad Prucha (1999). Uder homoskedasticity, σ 2 = σ 2 i, ad so E[ 1 ε 0 ε ]=σ 2, E[ 1 ε 0 ε ]=σ 2 1 Tr{M M 0 }, ad E[ 1 ε 0 ε ]=0. These three momet coditios uderlie the GM estimator suggested i Kelejia ad Prucha (1999). Substitutig the first of these momet coditios ito the secod yields E[ 1 ε0 ε ] = E[ 1 ε 0 ε ] 1 Tr{M M 0 }, (10) E[ 1 ε0 ε ] = 0, which is clearly a special case of (4) uder homoskedasticity. It is ot difficult to see that the GM estimator suggested i our previous paper ca be viewed as beig based h o the two momet coditios i (10) with Υ e = diag(υ, 1) ad υ =1/ 1+ 1 Tr{M M 0 } i Cosistecy of the GM Estimator for ρ To establish cosistecy of eρ we postulate the followig additioal assumptios. Assumptio 4 : Let eu i, deote the i-th elemet of eu. We the assume that eu i, u i, = d i., where d i., ad are 1 p ad p 1 dimesioal radom vectors. Let d ij, be the j-th elemet of d i.,. The, we assume that for some δ>0 E d ij, 2+δ c d < where c d does ot deped o, ad that 1/2 k k = O p (1). 8 If we rewrite the momet coditios i (10) i the form correspodig to (5), the A 1, = M 0 M 1 Tr(M M 0 ) I ad A 2, = M. 11

12 Assumptio 5 (a)the smallest eigevalue of Γ 0 Γ is uiformly bouded away from zero. 9 (b) Υ e Υ = o p (1), whereυ are 2 2 o-stochastic symmetric positive defiite matrices. (c) The largest eigevalues of Υ are bouded uiformly from above, ad the smallest eigevalues of Υ are uiformly bouded away from zero. Assumptio 4 implies 1 P kd i.,k 2+δ = O p (1), which was maitaied i Kelejia ad Prucha (1999), ad so is slightly stroger tha their assumptio. Assumptio 4 should be satisfied for typical liear spatial models where eu i is based o 1/2 -cosistet estimators of regressio coefficiets, d i., deotes the i-th row of the regressor matrix, ad deotes the differece betwee the parameter estimator ad the true parameter values. I the ext sectio we will actually demostrate that Assumptio 4 holds for the estimated residuals of model (1) based o a istrumetal variable procedure. Assumptio 4 should also be satisfied for typical o-liear models provided the respose fuctio is differetiable i the parameters, ad the derivatives are (uiformly over the parameter space) bouded by some radom variable with bouded 2+δ momets; compare Kelejia ad Prucha (1999). Assumptio 5 esures that the smallest eigevalue of Γ 0 Υ Γ is uiformly bouded away from zero ad will be sufficiet to permit us to demostrate that ρ is idetifiably uique w.r.t. the ostochastic aalogue of the objective fuctio of the GM estimator. This aalogue is give by the fuctio i curly brackets o the r.h.s. of (9) with eγ, e Γ ad e Υ replaced by γ, Γ ad Υ. Uder homoskedasticity ad e Υ = Υ specified as at the ed of the previous subsectio this assumptio is i essece equivalet to Assumptio 5 i Kelejia ad Prucha (1999). Clearly Assumptio 5 requires Γ to be osigular, or equivaletly that [tr (M 0 A 1, M S u, ),tr(m 0 A 2, M S u, )] 0 is liearly idepedet of tr M 0 (A 1, + A 0 1,)S u,,tr M 0 (A 2, + A 0 2,)S u, 0, which is readily see by observig that Eu 0 M 0 A i, M u = tr (M 0 A i, M S u, ) ad 1 Eu 0 M 0 (A i, + A 0 i, )u = tr M 0 (A i, + A 0 i, )S u, where Su, =(I ρ M ) 1 Σ (I ρ M ) 1. It is ot difficult to see that this liear idepedece coditio is a aalogue to idetificatio coditios postulated i Lee (2007), Assumptio 5(b), relatig to quadratic forms uder homoskedasticity. We ote that while Assumptio 5 should be satisfied i may settigs, it does ot cover situatios where all elemets of the spatial weights matrix coverge to zero uiformly as - see Lee (2004) - sice i this case the elemets of Γ would ted to zero. O the other had, Assumptio 5 does ot geerally 9 That is, λ mi (Γ 0 Γ ) λ γ > 0 where λ γ does ot deped o. More specifically, i geeral Γ depeds o M, ρ, σ 2 1,,...σ2,. Deotig this depedece as Γ = Γ (M,ρ,σ 2 1,,...σ2,) the assumptio should be uderstood as to postulate that if λ mi Γ (M,ρ,σ 2 1,,...σ 2,) 0 Γ (M,ρ,σ 2 1,,...σ 2,) > 0. I this sese the assumptio allows for λ γ to deped o the sequece of spatial weights matrices M, ad o the true values of the autoregressive parameters ρ ad variaces. 12

13 rule out settigs where M is row ormalized, there is a icreasig umber of ozero elemets i each row, ad the row sums of the absolute values of the o-ormalized elemets are uiformly bouded. The vector of derivatives (multiplied by mius oe) of the momet coditios (6) w.r.t. ρ is give by J = Γ [1, 2ρ ] 0. As expected, the limitig distributio of the GM estimator eρ will be see to deped o the iverse of J 0 Υ J. Assumptio 5 also esure that J 0 Υ J is osigular. Because of the equivalece of matrix orms it follows from Assumptio 5 that the elemets of Υ ad Υ 1 are O(1). We ca ow give our basic cosistecy result for eρ. Theorem 1 Let eρ = eρ ( Υ e ) deote the GM estimator defied by (9). The, provided the optimizatio space cotais the parameter space, ad give Assumptios 1-5, p eρ ρ 0 as. Clearly the coditios of the theorem regardig Υ e ad Υ are satisfied for eυ = Υ = I 2. I this case the estimator reduces to the oliear least squares estimator based o (8). This estimator ca, e.g., be used to obtai iitial cosistet estimates of the autoregressive parameter. Choices of Υ e that lead to a efficiet GM estimator for ρ (but require some cosistet iitial estimate of ρ ) will be discussed below i cojuctio with the asymptotic ormality result. 3.3 Asymptotic Distributio of the GM Estimator for ρ Let D = [d 0 1.,,...,d 0.,] 0 where d i., is defied i Assumptio 4 so that eu u = D. To establish the asymptotic ormality of eρ we eed some additioal assumptios. Assumptio 6 For ay real matrix A whose row ad colum sums are bouded uiformly i absolute value 1 D 0 A u 1 ED 0 A u = o p (1). Asufficiet coditio for Assumptio 6 is, e.g., that the colums of D are of the form π +Π ε, where the elemets of π are bouded i absolute value ad the row ad colum sums of Π are uiformly bouded i absolute value; see Lemma C.2. This will ideed be the case i may applicatios. I the ext sectio we will verify that this assumptio holds for the model give by (1) ad (2), ad where D equals the (egative of the) desig matrix Z. Assumptio 7 Let be as defied i Assumptio 4. The 1/2 = 1/2 T 0 ε + o p (1), where T is a p dimesioal real ostochastic matrix whose elemets are uiformly bouded i absolute value. 13

14 As remarked above, typically deotes the differece betwee the parameter estimator ad the true parameter values. Assumptio 7 will be satisfied by may estimators. I the ext sectio we verify that this assumptio ideed holds for the cosidered istrumetal variable estimators for the parameters of model (1). It may be helpful to provide some isight cocerig the variace of the limitig distributio of the GM estimator 1/2 (eρ ρ ) give below. To that effect we ote that a ispectio of the derivatio of this limitig distributio i Appedix C shows that it depeds o the limitig distributio of the (properly ormalized) vector of quadratic forms 1 v = 1/2 2 ε0 (A 1, + A 0 1,)ε + a 0 1,ε 1 2 ε0 (A 2, + A 0 2,)ε + a 0 (11) 2,ε where for r =1, 2 the matrices A r, are defied i (5), ad where the 1 vectors a r, are defied as with a r, = T α r, (12) α r, = 1 E D 0 (I ρ M 0 )(A r, + A 0 r,)(i ρ M ) u. From (11) ad (12) we see that, i geeral, the limitig distributio of 1/2 (eρ ρ ) will deped o the limitig distributio of 1/2 via the matrix T, uless α r, =0. Clearly, if D is ot stochastic, the α r, =0. Withi the the cotext of model (1) ad with D equal to the (egative of the) desig matrix Z this would be the case if the model does ot cotai a spatial lag of the edogeous variable. Observig further that the diagoal elemets of the matrices A r, are zero it follows from Lemma A.1 that the VC matrix of the vector of quadratic forms i (11) is give by Ψ =(ψ rs, ) where for r, s =1, 2 ψ rs, =(2) 1 tr A r, + A 0 r, Σ As, + A 0 s, Σ + 1 a 0 r,σ a s,. (13) We ow have the followig result cocerig the asymptotic distributio of eρ. We ote that the theorem does ot assume covergece of the matrices ivolved. Theorem 2 (Asymptotic ormality) Let eρ be the weighted oliear least squares estimators defied by (9). The, provided the optimizatio space cotais the parameter space, give Assumptios 1-7, ad give that λ mi (Ψ ) c Ψ > 0, we have 1/2 (eρ ρ) =(J 0 Υ J ) 1 J 0 Υ Ψ 1/2 ξ + o p (1) (14) where 1 J = Γ, (15) 2ρ ξ = Ψ 1/2 d v N(0, I2 ) 14

15 Furthermore 1/2 (eρ ρ )=O p (1) ad Ω ρ (Υ )=(J 0 Υ J ) 1 J 0 Υ Ψ Υ J (J 0 Υ J ) 1 cost > 0. (16) The above theorem implies that the differece betwee the cumulative distributio fuctio of 1/2 (eρ ρ ) ad that of N 0, Ω ρ coverges poitwise to zero, which justifies the use of the latter distributio as a approximatio of the former. 10 Remark 1. Clearly Ω ρ (Ψ 1 )=(J 0 Ψ 1 J ) 1 ad Ω ρ (Υ ) Ω ρ (Ψ 1 ) is leads positive semi-defiite. Thus choosig Υ e as a cosistet estimator for Ψ 1 to the efficiet GM estimator. Such a cosistet estimator will be developed i the ext subsectio. As discussed i the proof of the above theorem, the elemets of Ψ are uiformly bouded i absolute value ad hece λ max (Ψ ) c Ψ for some c Ψ <. Sice by assumptio also 0 <c Ψ λ mi(ψ ) it follows that the coditios o the eigevalues of Υ postulated i Assumptio 5 are automatically satisfied by Ψ 1 We ote that Ψ is, i geeral, oly idetical to the VC matrix of the momet vector i (5) if a r, =0. The terms ivolvig a r, reflect the fact that the GM estimator is based o estimators of the disturbaces u ad ot o the true disturbaces. As oted above, J equals the vector of derivatives (multiplied by mius oe) of the momet coditios (6) w.r.t. ρ, ad thus Ω ρ (Υ ) has the usual structure, except that here Ψ is ot idetical to the VC matrix of the momet vector. Remark 2. From (11), (12), (14) ad (15) we see that 1/2 (eρ ρ ) depeds liearly o a vector of liear quadratic forms i the iovatios ε plus a term of order o p (1). This result is helpful i establishig the joit distributio of eρ with that of estimators of some of the other model parameters of iterest. I particular, it may be of iterest to derive the joit limitig distributio of 1/2 ad 1/2 (eρ ρ ).ByAssumptio7, 1/2 is asymptotically liear i ε ad hece the joit limitig distributio ca be readily derived usig the CLT for liear quadratic forms give i Appedix A. We will illustrate this below withi the cotext of IV estimators for model (1). We ext itroduce a cosistet estimator for Ω ρ. For this purpose let ej = e Γ 1 2eρ (17) where e Γ is defied above by (7) ad the discussio after that equatio. We ext defie a HAC type estimator for Ψ whose elemets are defied by (13). For this purpose let eσ = diag,..., (eε 2 i,) 10 This follows from Corollary F4 i Pötscher ad Prucha (1997). Compare also the discussio o pp i that referece. 15

16 with eε =(I eρ M ) eu. We furthermore eed to specify a estimator for a r, = T α r,. The matrix T itroduced i Assumptio 7 will i may applicatios be of the form T = F P with F = H or F =(I ρ M 0 ) 1 H, (18) where H is a real ostochastic p matrix of istrumets, ad P is a real ostochastic p p matrix, with p as i Assumptio 7. I the ext sectio we will cosider istrumetal variable estimators for the parameters of model (1) ad (2). I that sectio we will see that if correspods to these istrumetal variable estimators, the the matrix T will ideed have the above structure, ad where P ca be estimated cosistetly by some estimator e P. We ow defie our estimator for T as 11 et = e F e P with e F = H or e F =(I eρ M 0 )+ H. (19) I light of (12) it ow seems atural to estimate a r, by with ea r, = e T eα r,. (20) eα r, = 1 D 0 (I eρ M 0 )(A r, + A 0 r,)(i eρ M ) eu. GivetheaboveweowitroducethefollowigHACtypeestimatorΨ e = ( ψ e rs, ) where for r, s =1, 2 h Ar, eψ rs, =(2) 1 tr + A 0 i eσ r, As, + A 0 s, eσ + 1 ea 0 e r,σ ea s,. (21) Furthermore, based o e Ψ we defie the followig estimator for Ω ρ : eω ρ =( e J 0 e Υ e J ) + e J 0 e Υ e Ψ e Υ e J ( e J 0 e Υ e J ) +. (22) The ext theorem establishes the cosistecy of e Ψ ad e Ω ρ. Theorem 3 : (VC matrix estimatio) Suppose all of the assumptios of Theorem 2 hold ad that additioally all of the fourth momets of the elemets of D are uiformly bouded. Suppose furthermore that (a) that the elemets of the ostochastic matrices H are uiformly bouded i absolute value,(b) sup ρ < 1 ad the row ad colum sums of M are bouded uiformly i absolute value by, respectively, oe ad some fiite costat (possibly after a reormalizatio of the weights matrix ad parameter space as discussed i Sectio 2.2), ad (c) e P P = o p (1) with P = O(1). The eψ Ψ = o p (1), e Ψ 1 Ψ 1 = o p (1), e Ω ρ Ω ρ = o p (1). 11 The reaso for usig the geeralized iverse is that ρ defied by (9) is ot forced to lie i the parameter space, ad thus I ρ M may be sigular (where the probability of this evet goes to zero as the sample size icreases). 16

17 The hypothesis of zero spatial correlatio i the disturbaces, i.e., H 0 : ρ = 0, ca ow be tested i terms of N h0, Ω e i ρ. Remark 3. We ote that the above theorem also holds if eρ is replaced by ay other estimator eρ with 1/2 (eρ ρ )=O p (1). IcaseF = H coditio (b) ca be dropped. The cosistecy result for Ψ e 1 verifies that this estimator for Ψ 1 caideedbeuseditheformulatioofaefficiet GM estimator, as discussed after Theorem 2. Remark 4. The modules uderlyig the derivatio of Theorems 2 ad 3 ca be readily exteded to cover a wider class of estimators. A crucial uderlyig igrediet is the CLT for vectors of liear quadratic forms give i Appedix A, which was used to establish the limitig distributio of the vector of liear quadratic forms (11); that CLT is based o Kelejia ad Prucha (2001). We emphasize that while i (5) we cosider two momet coditios, all of the above results geeralize trivially to the case where the GM estimator for ρ correspods to m momet coditios 1 E[Eε 0 A 1, ε,...,ε 0 A m, ε ] 0 =0, (23) where the diagoal elemets of A r, are zero ad the row ad colum sums of A r, are bouded uiformly i absolute value. The focus of this paper is o two-step estimatio procedures, which is motivated by their computatioal simplicity, geerality of the first step (where residuals may come from oliear models) ad, sice at least uder homoskedasticity, Mote Carlo experimets suggest that very little efficiecy is lost, see, e.g., Das, Kelejia ad Prucha (2003). Give istrumets H, oe-step GMM estimators for all parameters, i.e., ρ,λ,β,ofthesarar(1, 1) model could be defied by augmetig those momet coditios (23) by the coditios 1 EH 0 ε = 1 E[h 0.1,ε,...,h 0.p,ε ] 0 =0. (24) with ε =(I ρ M )(y X β λ W y ). The limitig distributio of the stacked momet vector follows immediately from the CLT for vectors of liear quadratic forms. Theorem 3 establishes cosistet estimatio of the VC matrix of the vector of (ormalized) liear quadratic forms (11). Estimatio of the VC matrix of the vector of (ormalized) liear quadratic forms correspodig to the stacked momet coditios (23) ad (24), is aalogous. I fact, i this case estimatio simplifies i that the compoets of the vector are either quadratic or liear, ad the elemets of the liear terms h r, are observed Several moths after completig this paper we became aware of a paper by Li ad Lee (2005) that cosiders a SAR(1) model with ukow heteroskedasticity. That paper is complemetary to this oe, with a focus ad extesive discussio of oe-step estimatio procedures for that model amog other thigs. That paper does ot discuss specificatio issues regardig the parameter space of the autoregressive parameters, which are cosidered i this paper. 17

18 3.4 Joit Asymptotic Distributio of GM Estimator for ρ ad Estimators of Other Model Parameters I the followig we discuss how the above results ca be exteded to obtai the joit asymptotic distributio of the GM estimator for ρ ad of other estimators that are asymptotically liear i the iovatios ε, i.e., that are of the form cosidered i Assumptio 7. As remarked above, the IV estimators for the regressio parameters of model (1) ad (2) cosidered i the ext sectio will be of this form. Based o the joit distributio it will the be possible to test joit hypothesis cocerig ρ ad other model parameters. I the followig we will give results cocerig the joit asymptotic distributio of eρ ρ ad as cosidered i Assumptios 4 ad 7 i cojuctio with the estimatio of the disturbaces u. Clearly, i geeral it will be of iterest to have available results ot oly cocerig the joit asymptotic distributio of eρ ρ ad, but also cocerig other estimators, say, that are of the geeral form cosidered i Assumptio 7. To avoid havig to itroduce further otatio we give our result i terms of eρ ρ ad, but the commet o what chages would be eeded i the formulae to accommodate other estimators i place of. The discussio assumes that T = F P ad T e = F e P e as defied i the previous subsectio. I light of the above discussio we expect that the joit limitig distributio of 1/2 (eρ ρ ) ad 1/2 will deped o the limitig distributio of 0. 1/2 F 0 ε, v 0 Observig agai that the diagoal elemets of the matrices A r, are zero it follows from Lemma A.1 that the VC matrix of this vector of liear ad liear quadratic form is give by Ψ, Ψ Ψ, = ρ, Ψ 0 (25) ρ, Ψ with Ψ, = 1 F 0 Σ F, Ψ ρ, = 1 F 0 Σ [a 1,, a 2, ] ad where Ψ is defied by (13). We shall also employ the followig estimator for Ψ, : # " eψ, Ψ ρ, e eψ, = eψ 0 (26) ρ, eψ with Ψ e, = 1 F e0 e ΣF e, Ψ ρ, e = 1 F e0 e Σ [ea 1,, ea 2, ] ad where Ψ e is defied by (21). We ow have the followig result cocerig the joit limitig distributio of 1/2 (eρ ρ ) ad 1/2. Theorem 4 Suppose all of the assumptios of Theorem 3 hold, ad λ mi (Ψ, ) c Ψ > 0. The 1/2 P 0 1/2 = 0 (eρ ρ ) 0 (J 0 Υ J ) 1 J 0 Ψ 1/2,ξ, + o p (1),(27) Υ ξ, = Ψ 1/2, h 1/2 F 0 ε, v 0 i 0 d N(0, Ip +2). 18

19 Furthermore let P 0 Ω, = 0 P 0 0 (J 0 Υ J ) 1 J 0 Υ Ψ, 0 Υ J (J 0 Υ J ) 1 (28), " # " # ep 0 eω, = 0 ep 0 0 ( J e0 e Υ J e ) + J e0 Υ e eψ, 0 Υ e J e ( J e0 e Υ J e ) + (29) the e Ψ, Ψ, = o p (1), e Ω, Ω, = o p (1), adψ, = O(1), Ω, = O(1). The above theorem implies that the differece betwee the joit cumulative distributio fuctio of 1/2 [ 0, (eρ ρ )] 0 ad that of N [0, Ω, ] coverges poitwise to zero, which justifies the use of the latter distributio as a approximatio of the former. The theorem also states that e Ω, is a cosistet estimator for Ω,. Remark 5. The above result geeralizes readily to cases where we are iterested i the joit distributio betwee eρ ρ ad some other estimator, say,,where 1/2 = 1/2 T 0 ε + o p (1), T = F P ad T e = F e e P, assumig that aalogous assumptios are maitaied for this estimator. I particular, the results remai valid, but with Ψ, = 1 F 0 Σ F, Ψ ρ, = 1 F 0 Σ [a 1,, a 2, ], Ψ, e = 1 F e 0 e Σ F e, Ψ ρ, e = 1 F e 0 e Σ [ea 1,, ea 2, ], ad P, P e replaced by P, P e. 4 Istrumetal Variable Estimator for δ As remarked, the cosistecy ad asymptotic ormality results developed i a importat paper by Lee (2004) for the quasi-ml estimator for the SARAR(1,1) model defied by (1) ad (2) uder the assumptio of homoskedastic iovatios do ot carry over to the case where the iovatios are heteroskedastic. I fact, uder heteroskedasticity the limitig objective fuctio of the quasi-ml estimator would geerally ot be maximized at the true parameter values, ad therefore the quasi-ml estimator would be icosistet. Also, the asymptotic ormality results developed by Kelejia ad Prucha (1998), Kelejia, Prucha ad Yuzefovich (2004) ad Lee (2003) for istrumetal variable (IV) estimators of the SARAR(1,1) model do ot carry over to the case where the iovatios are heteroskedastic. I the followig we provide results cocerig the asymptotic distributio of IV estimators allowig the iovatios to be heteroskedastic. More specifically, we will show that the cosidered IV estimators satisfy certai coditios such that their asymptotic distributio ca be readily obtaied via Theorem 4. We also allow for a more geeral defiitio of the parameter space of the spatial autoregressive parameters to avoid certai pitfalls discussed i Sectio 2. 19

20 4.1 Istrumets It is evidet from (3) that i geeral W y will be correlated with the disturbaces u, which motivates the use of IV estimatio procedures. We maitai the followig assumptios w.r.t. the k regressor matrices X, ad the p istrumet matrices H. Assumptio 8 : The regressor matrices X have full colum rak (for large eough). Furthermore, the elemets of the matrices X are uiformly bouded i absolute value. Assumptio 9 : The istrumet matrices H have full colum rak p k+1 (for all large eough). Furthermore, the elemets of the matrices H are uiformly bouded i absolute value. Additioally H is assumed to, at least, cotai the liearly idepedet colums of (X, M X ). Assumptio 10 : The istrumets H satisfy furthermore: (a) Q HH = lim 1 H 0 H is fiite, ad osigular. (b) Q HZ = p lim 1 H 0 Z ad Q HMZ = p lim 1 H 0 M Z are fiite ad have full colum rak. Furthermore, let Q HZ (ρ )=Q HZ ρ Q HMZ, the the smallest eigevalue of Q 0 HZ (ρ )Q 1 HH Q HZ (ρ ) is bouded away from zero uiformly i. (c) Q HΣH = lim 1 H 0 Σ H is fiite ad osigular. The above assumptios are similar to those maitaied i Kelejia ad Prucha (1998, 2004), ad Lee (2003), ad so a discussio which is quite similar to those give i those papers also applies here. Regardig the specificatio of the istrumets H observe first that E(W y )=W (I λ W ) 1 X β = X i=0 λ i W i+1 X β (30) provided that the characteristic roots of λ W are less tha oe i absolute value; compare Lemma 1 ad 2 cocerig the choice of the parameter space for λ. The istrumet matrices H will be used to istrumet Z =(X, W y ) ad M Z =(M X, M W y ) i terms of their predicted values from a least squares regressio o H,i.e., Z b = P H Z ad \M Z = P H M Z with P H = H (H 0 H ) 1 H 0. Towards approximatig the ideal istrumets E(Z )=(X, W E(y )) ad E(M Z )=(M X, M W E(y )) it seems reasoable, i light of (30), to take H to be a subset of the liearly idepedet colums of (X, W X, W 2 X,...,W q X, M X, M W X,,...,M W q X ) (31) 20

21 where q is a pre-selected fiite costat. 13 We ote that if H is selected as i (31) it follows from Assumptios 3 ad 8 that its elemets will be bouded i absolute value as postulated i Assumptio 9. Assumptio 9 esures that X ad M X are istrumeted by themselves. Fially we ote that the assumptio that H has full colum rak could be relaxed at the expese of workig with geeralized iverses. 4.2 Defiitio, Cosistecy ad Asymptotic Normality Towards estimatig the model (1) ad (2) we propose a three step procedure. I the first step the model is estimated by two stage least squares (2SLS) usig the istrumets H. I the secod step the autoregressive parameter, ρ,is estimated usig the geeralized momets estimatio approach from Sectio 3 based o the 2SLS residuals obtaied via the first step. I the third step, the regressio model i (1) is re-estimated by 2SLS after trasformig the model via a Cochrae-Orcutt-type trasformatio to accout for the spatial correlatio. More specifically, the first step 2SLS estimator is defied as: eδ =( b Z 0 Z ) 1 b Z 0 y, (32) where b Z = P H Z =(X, \W y ) ad \W y = P H W y. I the secod step we estimate ρ by the GM procedure defiedby(9)basedothe2sls residuals eu = y Z e δ. We deote the GM estimator agai as eρ. The ext lemma shows that various assumptios maitaied i Sectio 3 w.r.t. the estimator of the regressio parameters ad estimated residuals are automatically satisfied by the 2SLS estimator e δ ad the correspodig residuals. Lemma 3 : Suppose Assumptios 1-3 ad 8-10 hold, ad sup kβ k <. Let D = Z, the the fourth momets of the elemets of D are uiformly bouded, Assumptio 6 holds, ad: (a) 1/2 ( e δ δ )= 1/2 T 0 ε + o p (1) with T = F P ad where P = Q 1 HH Q HZ[Q 0 HZQ 1 HH Q HZ] 1, F = (I ρ M 0 ) 1 H. (b) 1/2 T 0 ε = O p (1). (c) P = O p (1) ad e P P = o p (1) for ep = ( 1 H 0 H ) 1 ( 1 H 0 Z ) [( 1 Z 0 H )( 1 H 0 H ) 1 ( 1 H 0 Z )] 1, 13 I Kelejia, Prucha, ad Yuzefovich (2004), who cosidered the case of homoskedastic iovatios, the istrumets were determied more geerally by takig q as a fuctio of the sample size, i.e., q,suchthatq as. Their Mote Carlo results suggest that q =2may be sufficiet for may applicatios. 21