Heteroskedasticity of Unknown Form in Spatial Autoregressive Models with Moving Average Disturbance Term

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1 City Uiversity of New York (CUNY) CUNY Academic Works Ecoomics Workig Papers CUNY Academic Works 4 Heteroskedasticity of Ukow Form i Spatial Autoregressive Models with Movig Average Disturbace Term Osma Doga CUNY Graduate Ceter Follow this ad additioal works at: Part of the Ecoomics Commos This Workig Paper is brought to you by CUNY Academic Works. It has bee accepted for iclusio i Ecoomics Workig Papers by a authorized admiistrator of CUNY Academic Works. For more iformatio, please cotact AcademicWorks@gc.cuy.edu.

2 CUNY GRADUATE CENTER PH.D PROGRAM IN ECONOMICS WORKING PAPER SERIES Heteroskedasticity of Ukow Form i Spatial Autoregressive Models with Movig Average Disturbace Term Osma Doga Workig Paper, Revised Ph.D. Program i Ecoomics CUNY Graduate Ceter 365 Fifth Aveue New York, NY 6 December 3 Revised December 4 4 by Osma Doga. All rights reserved. Short sectios of text, ot to exceed two paragraphs, may be quoted without explicit permissio provided that full credit, icludig otice, is give to the source.

3 Heteroskedasticity of Ukow Form i Spatial Autoregressive Models with Movig Average Disturbace Term Osma Doga CUNY Graduate Ceter Workig Paper No. December 3 JEL No. C3, C, C3 ABSTRACT I this study, I ivestigate the ecessary coditio for cosistecy of the maximum likelihood estimator (MLE) of spatial models with a spatial movig average process i the disturbace term. I show that the MLE of spatial autoregressive ad spatial movig average parameters is geerally icosistet whe heteroskedasticity is ot cosidered i the estimatio. I also show that the MLE of parameters of exogeous variables is icosistet ad determie its asymptotic bias. I provide simulatio results to evaluate the performace of the MLE. The simulatio results idicate that the MLE imposes a substatial amout of bias o both autoregressive ad movig average parameters. Osma Doga Ph.D. Program i Ecoomics City Uiversity of New York 365 Fifth Aveue New York, NY 6 ODoga@gc.cuy.edu

4 Itroductio The spatial depedece amog the disturbace terms of a spatial model is geerally assumed to take the form of a spatial autoregressive process. The spatial model that has a spatial lag i the depedet variable ad a autoregressive process i the disturbace term is kow as the SARAR model. The mai characteristic of a autoregressive process is that the effect of a locatio-specific shock trasmits to all other locatios with its effects gradually fadig away for the higher order eighbors. The spatial autoregressive process may ot be appropriate if there is strog evidece of localized trasmissio of shocks. That is, the autoregressive process is ot the correct specificatio whe the effects of shocks are cotaied withi a small regio ad are ot trasmitted to other regios. A alterative to a autoregressive process is a spatial movig average process, where the effect of shocks are more localized. Haiig (978), Aseli (988) ad more recetly Hepple (3) ad Figleto (8a,b) cosider a spatial movig average process for the disturbace terms. The spatial model that cotais a spatial lag of the depedet variable ad a spatial movig average process for the disturbace term is kow as the SARMA model. I the literature, various estimatio methods have bee proposed (Das, Kelejia, ad Prucha, 3; Hepple, 995a; Kelejia ad Prucha, 998, 999, ; Lee, 4, 7a; Lee ad Liu, ; LeSage ad Pace, 9; Lesage, 997; Liu, Lee, ad Bolliger, ). The ML method is the best kow ad most commo estimator used i the literature for both SARAR ad SARMA specificatios. Lee (4) shows the first order asymptotic properties of the MLE for the case of SARAR(,). The geeralized method of momet (GMM) estimators are also cosidered for the estimatio of the spatial models. Kelejia ad Prucha (998, 999) suggest a two step GMM estimator for the SARAR(,) specificatio. Oe disadvatage of the two-step GMME is that it is usually iefficiet relative to the MLE (Lee, 7c; Liu, Lee, ad Bolliger, ; Prucha, 4). To icrease efficiecy, Lee (7a), Liu, Lee, ad Bolliger () ad Lee ad Liu () formulate oe step GMMEs based o a set of momet fuctios ivolvig liear ad quadratic momet fuctios. I this approach, the reduced form of spatial models motivates the formulatio of momet fuctios. The reduced equatios idicate that the edogeous variable, i.e., the spatial lag term, is a fuctio of a stochastic ad a o-stochastic term. The liear momet fuctios are based o the orthogoality coditio betwee the o-stochastic term ad the disturbace term, while the quadratic momet fuctios are formulated for the stochastic term. The, the parameter vector is estimated simultaeously with a oe-step GMME. Lee (7a) shows that the oe-step GMME ca be asymptotically equivalet to the MLE whe disturbace terms are i.i.d. ormal. I the case where disturbaces are simply i.i.d., Liu, Lee, ad Bolliger () ad Lee ad Liu () suggest a oe-step GMME that ca be more efficiet tha the (quasi) MLE. Figleto (8a,b) exteds the two-step GMME suggested by Kelejia ad Prucha (998, 999) for spatial models that have a movig average process i the disturbace term, i.e., SARMA(,). Baltagi ad Liu () modify the momet fuctios cosidered i Figleto (8a) i the maer of Arold ad Wied (), ad suggest a GMME for the case of SARMA(,). The spatial movig average parameter i both Figleto (8a) ad Baltagi ad Liu () is estimated by a o-

5 liear least squares estimator (NLSE). The asymptotic distributio for the NLSE of the spatial movig average parameter is ot provided i either Figleto (8a) or Baltagi ad Liu (). Recetly, Kelejia ad Prucha () ad Drukker, Egger, ad Prucha (3) provide a basic theorem regardig the asymptotic distributio of their estimator uder fairly geeral coditios. The estimatio approach suggested i Kelejia ad Prucha () ad Drukker, Egger, ad Prucha (3) ca easily be adapted for the estimatio of the SARMA(,) ad SARMA(,) models. Fially, although the Kelejia ad Prucha approach i Figleto (8a) ad Baltagi ad Liu () has computatioal advatages, it may be iefficiet relative to the ML method. I the presece of a ukow form of heteroskedasticity, Li ad Lee () show that the MLE for the cases of SARAR(,) may ot be cosistet as the log-likelihood fuctio is ot maximized at the true parameter vector. They suggest a robust GMME for the SARAR(,) specificatio by modifyig the momet fuctios cosidered i Lee (7a). Likewise, Kelejia ad Prucha () modify the momet fuctios of their previous two-step GMME to allow for a ukow form of heteroskedasticity. The spatial movig average model itroduces a differet iteractio structure. Therefore, it is of iterest to ivestigate implicatios of a movig average process for estimatio ad testig issues. I this paper, I ivestigate the effect of heteroskedasticy o the MLE for the case of SARMA(,) ad SARMA(,) alog the lies of Li ad Lee (). The aalytical results show that whe heteroskedasticity is ot cosidered i the estimatio, the ecessary coditio for the cosistecy of the MLE is geerally ot satisfied for both SARMA(,) ad SARMA(,) models. For the SARMA(,) specificatio, I also show that the MLE of other parameters is also icosistet, ad I determie its asymptotic bias. My simulatio results idicate that the MLE imposes a substatial amout of bias o spatial autoregressive ad movig average parameters. However, the simulatio results also show that the MLE of other parameters reports a egligible amout of bias i large samples. The rest of this paper is orgaized as follows. I Sectio, I specify the SARMA(,) model i more detail ad list assumptios that are required for the asymptotic aalysis. I Sectio 3, I briefly discuss implicatios of spatial processes proposed for the disturbace term i the literature. Sectio 4 ivestigates the ecessary coditio for the cosistecy of the MLE of autoregressive ad movig average parameters. Sectio 5 provides expressios for the asymptotic bias of the MLE of parameters of the exogeous variables. Sectio 6 cotais a small Mote Carlo simulatio. Sectio 7 closes with cocludig remarks. Model Specificatio ad Assumptios I this study, the followig first order SARMA(,) specificatio is cosidered: Y = W Y + X + u, u = " M " (.) Figleto (8a) ad Baltagi ad Liu () do ot compare the fiite sample efficiecy of their estimators with the MLE. 3

6 where Y is a vector of observatios of the depedet variable, X is a k matrix of o-stochastic exogeous variables, with a associated k vector of populatio coefficiets, W ad M are spatial weight matrices of kow costats with zero diagoal elemets, ad " is a vector of disturbaces. The variables W Y ad M " are kow as the spatial lag of the depedet variable ad the disturbace term, respectively. The spatial effect parameters ad are kow as the spatial autoregressive ad movig average parameters, respectively. As the spatial data is characterized with triagular arrays, the variables i (.) have subscript. The model specificatios with 6=, 6=,ad =, 6= are kow, respectively, as SARMA(,) ad SARMA(,) i the literature. Let be the parameter space of the model. I order to distiguish the true parameter vector from other possible values i, the model is stated with the true parameter vector =, with =(, ). For otatioal simplicity, we deote S ( ) = (I W ), R ( ) = (I M ), G ( ) = W S ( ), H ( ) =M R ( ), X ( ) =R ( )X,adG ( )=R ( )G ( )R ( ). Also,atthe true parameter values (, ), we deote S ( )=S, R ( )=R, G ( )=G, H ( )=H, X ( )=X,adG ( )=G. The model i (.) is cosidered uder the followig assumptios. Assumptio. The elemets " i of the disturbace term " are distributed idepedetly with mea zero ad variace i,ade " i < for some >4 for all ad i. The elemets of the disturbace term have momets higher tha the fourth momet. existece momets coditio is required for the applicatio of the cetral limit theorem for the quadratic form give i Kelejia ad Prucha (). I additio, the variace of a quadratic form i " exists ad is fiite whe the first four momets are fiite. The Fially, Liapuov s iequality guaratees that the momets less tha are also uiformly bouded for all ad i. Assumptio. The spatial weight matrices M ad W are uiformly bouded i absolute value i row ad colum sums. Moreover, S, S ( ), R ad R ( ) exist ad are uiformly bouded i absolute value i row ad colum sums for all values of ad i a compact parameter space. The uiform boudedess of terms i Assumptio is motivated to cotrol spatial autocorrelatios i the model at a tractable level (Kelejia ad Prucha, 998). 3 Assumptio also implies that the model i (.) represets a equilibrium relatio for the depedet variable. By this assumptio, the reduced form of the model becomes feasible as Y = S X + S R ". The uiform boudedess of S ( ) ad R ( ) i Assumptio is oly required for the MLE, ot for the GMME (Liu, Lee, ad Bolliger, ). Whe W is row ormalized, a closed subset of iterval (/ mi, ), where mi is the smallest eigevalue of W, ca be cosidered as the parameter space for. Aalogously, a closed subset of (/ mi, ), where mi is the smallest eigevalue of M, ca be the parameter space of (LeSage ad Pace, 9, p.8). 4 The ext assumptio states the regularity coditios for the exogeous variables. See Kelejia ad Prucha (). 3 For a defiitio ad some properties of uiform boudedess, see Kelejia ad Prucha (). 4 There are some other formulatios for the parameter spaces i the literature. For details see Kelejia ad Prucha 4

7 Assumptio 3. The matrix X is a k matrix cosistig of costat elemets that are uiformly bouded. It has full colum rak k. Moreover,lim! X X ad lim! X ( )X ( ) exist ad are osigular for all values of i a compact parameter space. 3 Spatial Processes for the Disturbace Term I the literature, there are three mai parametric processes to model spatial autocorrelatio amog disturbace terms: (i) spatial autoregressive process (SAR), (ii) spatial movig average process (SMA), ad (iii) spatial error compoets model (SEC). The implied covariace structure is differet uder each specificatio. I this sectio, I describe the trasmissio ad the effect of shocks uder each specificatio. The SAR process is specified as u = M u + ", (3.) where u is a vector of regressio disturbaces, ad " is a vector of i.i.d. iovatios with variace. Uder the assumptio of a equilibrium, i.e., R is ivertible, the reduced from of (3.) is u = R " with the covariace matrix of E u u = = R R. Note that eve if the iovatios are homoskedastic, the diagoal elemets of are ot equal suggestig heteroskedasticity for the regressio disturbaces. A expasio of (I M ) for < yields (I M ) = P j= j M j = I + M + M +. Hece, the SAR specificatio of the disturbace term implies that a shock at locatio i is trasmitted to all other locatios. The first term I implies that the shock at locatio i directly affects locatio i, ad through other terms deoted by the powers of M affects higher order eighbors. Evetually, the shock feeds back to locatio i through the itercoectedess of eighbors. Note that < esures that the magitude of the trasmitted shock decreases for the higher orders of eighbors. As a result, the SAR specificatio allows researchers to model global trasmissio of shocks where the full effect of a shock to locatio i is the sum of iitial shock ad the feedback from other locatios. If a more localized spatial depedece is cojectured for a ecoomic model, the a spatial movig average process (SMA) specificatio is more suitable (Figleto, 8a,b; Haiig, 978; Hepple, 3). The SMA process is specified as u = " M ", (3.) where is the spatial movig average parameter. The reduced form does ot ivolve a iverse of a square matrix. Hece, the trasmissio of a shock emaated from locatio i is limited to its immediate eighbors give by the ozero elemets i the ith row of M. Uder this specificatio, the covariace matrix of u is = R R = I (M + M)+ M M. The spatial () ad LeSage ad Pace (9). Note that the parameter spaces for ad are ot required to be compact. As show i (4.3a) ad (4.3b), the MLE of these parameters is a OLS type estimator, hece boudedess is eough for the parameter spaces. 5

8 covariace is limited to ozero elemets of M + M ad M M. I compariso with the SAR specificatio, the rage of covariace iduced by the SMA model is much smaller. Kelejia ad Robiso (993) suggest aother specificatio which is called the spatial error compoets (SEC) model. This specificatio is similar to the SMA process i the sese that the implied covariace matrix does ot ivolve a matrix iverse. Formally, the SEC model is give by u = M " +, where " is a vector of regioal iovatios, whereas is a vector of locatioal iovatios. Assumig that " ad are idepedet, the variace-covariace matrix becomes = I + "M M, which idicates that the spatial correlatio i a SEC specificatio is eve more localized. There have bee some direct attempts to parametrize the covariace matrix of u, rather tha defiig a process for the disturbace term. For example, Besag (974) cosiders a coditioal first-order autoregressive model (CAR()) such that the covariace matrix of u takes the form of = (I M ), where M is assumed to be a symmetric cotiguity matrix. This covariace structure implies a process of u =(I M ) / ". As i the case of the SAR process, a shock i a locatio is trasmitted to all other locatios, but ow with a smaller amplitude. Aother example is = (I + M ), where M is assumed to be symmetric (Hepple, 995b; Richardso, Guiheeuc, ad Lasserre, 99). I this case, the spatial correlatio is restricted to first order eighbors, i.e., o-zero elemets of M. The elemets of ca also be specified through a covariace geeratig fuctio. For example, i Ripley (5), the covariace geeratig fuctio is defied i terms of distace betwee two locatios i such a way that the resultig covariace is always o-egative defiite. Let d ij be the distace betwee locatio i ad j, ad ij, be the covariace betwee these two locatios. The, the covariace geeratig fuctio is defied by 8 apple >< cos ( d ij ij, = ) d ij ( d ij ) /, if d 4 ij apple >:, otherwise. Ituitively, ij, is proportioal to the itersectio area of two discs of commo radius cetered o locatios i ad j. The covariace geeratig fuctio i (3.3) depeds o the sigle parameter, ad has a fairly liear egative relatioship with d ij (Richardso, Guiheeuc, ad Lasserre, 99; Ripley, 5). Aother covariace geeratig fuctio family, first itroduced by Whittle i 954, is a two parameter fuctios defied i terms of Gamma ad Bessel fuctios. This family has the followig specificatio: (3.3) ij, = ( ) ( dij ) K ( d ij ), (3.4) where K ( ) is the modified Bessel fuctio, ad ( ) is the stadard Gamma fuctio. The parameters >ad > are respectively kow as a shape parameter ad a spatial parameter. The spatial parameter determies how far the spatial correlatio will stretch. For the special 6

9 case, where =, this covariace geeratig fuctio gives a expoetial decayig spatial correlatios (Richardso, Guiheeuc, ad Lasserre, 99). There is also a more geeral expoetial covariace geeratig fuctio that depeds o two parameters. This fuctio is specified by ij, = exp( d ij ), where ad are parameters eed to be estimated. This fuctio also exhibits expoetial decay for the spatial correlatios. I the literature, there are some other covariace geeratig fuctio families. However, the majority of these fuctios do ot ecessarily esure that is a positive defiite matrix (Haiig, 987; Richardso, Guiheeuc, ad Lasserre, 99). The formal properties of the MLE for spatial models that have a covariace structure determied by a parametric fuctio are ivestigated i a early study by Mardia ad Marshall (984). I this study, the authors state coditios uder which the MLE is cosistet ad has asymptotic ormal distributio. I this study, the spatial model specified i (.) is cosidered. The iteractio betwee the spatial autoregressive process ad the movig average process for this model iduces a complicated patter for the trasmissio of a locatio specific shock. Uder Assumptio, the reduced form of the model is give by Y = S X + S R ". The last term i the reduced form ca be writte as S R " = " M " + P l l= W " l P M l l= W " l. I this represetatio, the higher power of W does ot have zero diagoal elemets, which i tur implies that the total effect of a regio specific shock also cotais the feedback effects passed through other locatios. The correspodig expressio i the case of SARAR(,) specificatio is give by S R " = P l l= W l P k= k M " k. Agai, the iduced patter ivolves the iteractio of two weight matrices ad two parameters. Followig Figleto (8a), I illustrate the trasmissio patter for a shock uder each specificatio by usig a rook weight matrix over a 5 5 lattice. Figure shows the impact of a shock emaated from the uit located at the ceter of lattice. 5 I the case of SAR ad SARAR(,), the effect of shock is more vigorous over the whole lattice. For the SMA specificatio, the shock is oly trasmitted to the immediate uits as show i Figure (b). I cotrast, the effect of the shock gradually dies out uder the SARMA(,) model. 5 For easy compariso, we set =.9 for SAR, =.9 for SMA, (, )=(.5,.9) for SARAR(,) ad (, )=(.5,.9) for SARMA(,). The disturbace of the uit located at the ceter of the lattice is icreased by 3. 7

10 . Figure : The Effect of a Shock (a) The effect of a shock: SAR (b) The effect of a shock: SMA (c) The effect of a shock: SARAR(,) (d) The effect of a shock: SARMA(,) 8

11 4 The MLE of ad The log-likelihood fuctio for the model i (.) uder the assumptio that the disturbace terms of the model are i.i.d. ormal with mea zero ad variace where = ca be writte as l L () = l( ) l( )+l S ( ) l R ( ) (S ( )Y X ) R ( )R ( )(S ( )Y X ), (4.),. The first order coditios with respect to ad are respectively give l L () X ( )R ( )(S ( )Y X ), l L = + 4 " ( )" ( ), (4.b) where " ( ) =R ( )(S ( )Y X ). The solutios of the first order coditios for a give yield the MLE of ad : ˆ( )= X( )X ( ) X ( )R ( )S ( )Y, (4.3a) ˆ( ) = " ( )" ( ). Cocetratig the log-likelihood fuctio by elimiatig l L ( ) = l( ) l " ( )" ( ) S ( ) R ( )! gives the followig equatio: (4.3b) (4.4) The above represetatio is useful for explorig the role of the Jacobia terms S ( ) ad R ( ) i the ML estimatio. The MLE of is the extremum estimator obtaied from the maximizatio of (4.4). I a equivalet way, the MLE of ca be defied by ˆ = argmi ( " ( )" ( ) S ( ) R ( ) ) I the special case, where S ( ) = R ( ) =, the MLE is the NLSE obtaied from the miimizatio of " ( )" ( ), i.e., ˆ NLSE, = argmi " ( )" ( ). It is clear that the Jacobia terms S ( ) ad R ( ) play a role of a weight (or a pealty) o " ( )" ( ). The pealty is a fuctio of the autoregressive parameters ad the spatial weight matrices, which ca be defied as f (,, W,M )= S ( ) R ( ). For the SARAR(,) specificatio, the last term i (4.4) is give by l " ( )" ( ), where " ( ) =R ( )(S ( )Y X ). Therefore, i the case S ( ) R ( ) (4.5) 9

12 of SARAR(,), the MLE of is give by ( ) " ˆ = argmi ( )" ( ) S ( ) R ( ) (4.6) It is hard to make ay geeral statemet about the effects ad magitudes of the pealty fuctios i both cases. Hepple (976) illustrates that the Jacobia term imposes a substatial pealty for the SARAR(,) specificatio. To illustrate the effect of pealty fuctios for the case of SARMA(,) ad SARAR(,), I use a distace based weight matrix for a sample of 9 coutries such that each coutry is coected to every other coutry. The elemets of the weight matrices are specified by 8 < if i = j, w ij = m ij = d : ij if i 6= j, P 9 j= d ij where d ij betwee coutries i ad j is measured by the great-circle distace betwee coutry capitals. 6 Figure shows the surface plots of pealty fuctios over a grid of spatial parameters. (4.7) Figure : The pealty fuctios for the dese weight matrix (a) The pealty fuctio for SARMA(,) (b) The pealty fuctio for SARAR(,) For the SARAR(,) specificatio, the value of the pealty fuctio decreases as the parameter combiatio (, ) moves away from (, ) i ay directio as show i Figure (b). 7 O the 6 d ij = R arccos cos logitude i logitude j cos(latitude i)cos(latitude j)+si(latitude i)si(latitude j), where R is the Earth s radius. 7 For SARAR(,), the pealty fuctio is f(,, W,M )= S ( ) R( ).

13 other had, there is o such mootoic decrease i the pealty fuctio uder the SARMA(,) specificatio as illustrated i Figure (a). The pealty fuctio of SARMA(,) obtais relatively larger values whe there is strog spatial depedece i the disturbace term, i.e., whe is ear or. I cotrast, the pealty fuctio has smaller values whe there is strog spatial depedece i the depedet variable. This patter idicates that the sum " ( )" ( ) is pealized as moves toward to either or. I the case of SARAR(,), this sum gets larger as (, ) moves toward (±, ±) i ay directio, suggestig that the solutio of the miimizatio problem is restricted to the regio (, ) (+, +). Fially, i a small eighborhood of (, ), the surface plots i Figure idicate that the pealty fuctios take values aroud, suggestig that the parameter estimates from the MLE ca be similar to those from the NLSE uder both specificatios. Next, I ivestigate the effect of heteroskedasticity o the MLE for the case of SARMA(,). I assume that the true data geeratig process is characterized by Assumptio. More explicitly, the MLE ˆ( ) ca be writte as ˆ( )= Y S ( )R ( )M ( )R ( )S ( )Y, (4.8) where M ( ) = (I P ( )) is a projectio type matrix with P ( ) = X ( ) X( )X ( ) X ( ). Substitutig R ( )S ( )Y = R ( )X + " ito ˆ( ) ad usig the fact that X( )M ( ) = k ad M ( )X ( ) = k,themleˆ( ) ca be writte as ˆ( )= " M ( )". (4.9) At, theprobabilitylimitofˆ( ) is plim ˆ( )=plim!! " " plim! " X X X X". (4.) For the first term o the right had side, we have " " = P i= i +o p() by Chebyshev s Weak Law of Large Numbers. The secod term vaishes by virtue of Lemma (4) i Appedix 8., ad Assumptio 3. Therefore, we have ˆ( )= X i= i + o p (). (4.) The result i (4.) idicates that the average of the idividual variaces is asymptotically equivalet to ˆ( ). Cocetratig out ad from the log-likelihood fuctio i (4.) yields l L ( )= (l( ) + ) l ˆ( )+l S ( ) l R ( ). (4.)

14 The MLE ˆ ad ˆ are extremum estimators obtaied from the maximizatio of (4.). The first order coditios of (4.) with respect to ad are l L l L ( = ) ˆ( tr (G ( ) +tr(h ( )), ˆ( (4.3a) (4.3b) where G ( )=W S ( ) ad H ( ) =M R ( ). For the cosistecy of ˆ ad ˆ, the l L coditio is plim ( =. Below, I ivestigate the probability limit of the followig l L ( ) B ( ) " ( ) " M tr (G ) C + tr (H A. (4.4) ) Uder Assumptio, both H ad G are uiformly bouded i absolute value i row ad colum sums. Therefore, tr (H ) ad tr (G ) i (4.4) are of order O(). With these results for tr (H ) ad tr (G ), a coveiet result for the probability limit of (4.4) ca be obtaied, which is stated i the followig propositio. Propositio. Uder Assumptios through 3, we l L ( ) Cov(G,ii, i ) + o p () C Cov(H,ii, i ) A, (4.5) + o p () where Cov G,ii, i is the covariace betwee the diagoal elemets of G, {G,, G,,...,G, }, ad the idividual variaces {,,..., }. Similarly, Cov H,ii, i deotes the covariace betwee diagoal elemets of H, {H,,H,,...,H, }, ad the idividual variaces {,,..., }. Proof. See Appedix 8.. The above propositio idicates that the MLE of the spatial autoregressive ad movig average parameters is ot cosistet as log as the covariace terms i (4.5) are ot zero. Notice that, whe the disturbace terms are homoskedastic, the covariace terms i (8.) are zero. I the special case where W = M ad =,wehaves = R ad G = H so that G = R G R = 8 For these results, we use the derivative rule give by =tr ( ). For a proof, see l R( ad Magus (5, p.37). Also ote the commutative property of R ( )M = M R ( ) =H ( ).

15 R H R = R M R R = H. Hece, the ecessary coditio for the cosistecy of ˆ is idetical to the oe for ˆ. The result i Propositio idicates that the cosistecy of the MLE depeds o the specificatio of weight matrices. It is of iterest to ivestigate specificatios that yield zero covariaces. A obvious case is whe there is o variatio i the diagoal elemets of G ad H. The, the ecessary coditio for the cosistecy of ˆ ad ˆ is ot violated, eve if the disturbaces are heteroskedastic. For example, there is o variatios i the diagoal elemets of G ad H whe W ad M are block-diagoal matrices with a idetical sub-matrix i the diagoal blocks ad zeros elsewhere. This type of block diagoal weight matrix ca be see i social iteractio scearios where a block represets a group i which each idividual is equally affected by the members of the group (Lee, 7b; Lee, Liu, ad Li, ). Suppose that there are R groups each of which has m members so that = mr. If we assig equal weight to each member of a group, the W = M = I R B m, where B m = m l m l m I m,adl m is a m-dimesioal colum vector of oes. For this set up, there is o variatio i the diagoal elemets of G ad H, therefore Cov G,ii, i =Cov H,ii, i =. There is also o variatio i the diagoal elemets of G ad H whe the circular world weight matrices cosidered i Kelejia ad Prucha (999) are employed. I these weight matrices, the order of observatios is importat sice the observatios are related to some uits i frot ad to some i back. As a example cosider a ahead ad behid weight matrix where each observatio is related to the oe immediately after ad immediately before it. For this sceario, we also have Cov G,ii, i =Cov H,ii, i =. The circular world weight matrices ca be adjusted to create some variatio i the diagoal elemets of G ad H. For example, Kelejia ad Prucha (7) costruct a differet versio i which the first ad the last oe third of the sample observatios have 5 eighbors i frot ad 5 i back, while the middle third oly has eighbor i frot ad i back. Uder this sceario, the Mote Carlo results i Kelejia ad Prucha (7) show that the MLE is sigificatly biased for the case of SARAR(,). 5 The MLE of I the previous sectio, I showed that the cosistecy of the MLE of the spatial autoregressive ad movig average parameters is ot esured. I this sectio, I ivestigate the cosistecy of the MLE of. The result i (4.3a) idicates that the MLE ˆ(ˆ) is also icosistet, sice it is based o the icosistet estimators ˆ ad ˆ. The asymptotic bias of ˆ(ˆ) ca be determied from (4.3a). By usig S ( )=S +( ˆ( )= + X( )X ( ) ) W,theMLE ˆ( ) ca be writte as X ( )R X ( )R " +( ) X( )X ( ) ( )R ( )G X +( ) X( )X ( ) X ( )R ( )G R ", (5.) 3

16 Uder Assumptio 3, the term X ( )X ( ) is uiformly bouded i absolute value i row ad colum sums. By Lemma (5) of Appedix 8., terms ivolvig " i (5.) vaish i probability. Thus, ˆ( )= +( ) X( )X ( ) X ( )R ( )G X + o p () (5.) The asymptotic bias of ˆ(ˆ) follows from (5.), which is give by ˆ X(ˆ )X (ˆ ) X (ˆ )R (ˆ )G X. This result shows that the asymptotic bias of ˆ(ˆ) depeds o weight matrices ad the regressors matrix, ad is ot zero uless spatial parameters are cosistet. Note that the bias is the OLS estimator obtaied from the artificial regressio of R (ˆ )G X o X(ˆ ). For the special case of ˆ = + o p (), wehave ˆ( )= + o p (). I this case, there is o asymptotic bias ad the icosistecy of ˆ has o effect o ˆ(ˆ). The specificatio with =i (.) is called the spatial movig average model (SARMA(,) or SMA). For the SARMA(,) specificatio, the log-likelihood fuctio simplifies to l L () = l( ) l( ) l R ( ) (Y X ) R ( )R ( )(Y X ), (5.3) where =, with =,. For a give value of, the first order coditios yield ˆ( ) = X( )X ( ) X ( )R ( )Y ˆ( ) = " ( )" ( ), where " ( ) =R Y X. The ecessary coditio for the cosistecy of the MLE ˆ ca be obtaied from (4.5). From the secod row of (4.5), we l L ( = Cov(H,ii, i ) + o p (), which implies that the MLE ˆ is icosistet. Substitutio of Y = X + R " ito ˆ( ) yields ˆ( ) = + X( )X ( ) X ( )R ( )". (5.4) The variace of X( )X ( ) X ( )D Appedix 8.. The, Chebyshev s iequality implies that ˆ( ) = o asymptotic bias eve though ˆ is icosistet. ( )" i (5.4) has a order of O( ) by Lemma (5) of + o p (). Hece, ˆ(ˆ ) has For the spatial autoregressive model, where = i (.), the result i (4.5) simplifies l L ( = Cov(G,ii, i ) + o p (). The term X( )X ( ) X ( )R ( )G X i (5.) 4

17 simplifies to XX X G X so that ˆ( )= +( ) XX X G X + o p (), (5.5) The result i (5.5) is the exact result stated i Li ad Lee () for the case of SARAR(,). I collect the above results for the MLE of i the followig propositio. Propositio. Cosider the model i (.) uder Assumptios through 3, the. For the SARMA(,) model, we have ˆ( )= +( ) X( )X ( ) X ( )R ( )G X + o p (). (5.6). For the SARMA(,) model, where =,wehave ˆ( ) = + o p (). 3. For the SARMA(,) model, where =,wehave ˆ( )= +( ) XX X G X + o p (). (5.7) I Sectio 4 ad 5, I showed that the MLE of ad is geerally icosistet whe heteroskedasticity is preset i the model. Besides its computatioal burde, the cosistecy of MLE is ot esured. I the ext sectio, I cofirm these large sample results through a Mote Carlo simulatio. 6 Mote Carlo Simulatio I this sectio, the fiite sample properties of the MLE are ivestigated through a Mote Carlo experimet for the cases of (i) SARMA(,), ad (ii) SARMA(,). For both models, we assume heteroskedastic iovatios i the data geeratig processes. 6. Desig There are two regressors ad o itercept term such that X =[x,,x, ] ad =(, ), where x, ad x, are idepedet radom vectors that are geerated from a Normal(,). We cosider =, 5, ad let W = M ad set =(, ) for all experimets. For the spatial autoregressive parameters (, ), we employ combiatios from the set B =(.6,,,,.6) to allow for weak ad strog spatial iteractios. The row ormalized spatial weight matrix is based o the small group iteractio sceario described i Li ad Lee (). I this sceario, the weight matrix is a block diagoal matrix where each block represets a group iteractio. The size of each block is determied by the group size, which are determied by a radom draw from Uiform(5,5). Let {g,...,g G } be the set of 5

18 groups, where G is the total umber of groups. Deote the size of each group by m i for i =,...,G. The, the block for group i is give by B i = m i l mi l mi I mi, where l mi is the m i vector of oes. The, W = M = Diag (B,...,B G ). 9 The observatios i a group have the same variace, ad I use the group size to create heteroskedasticity. If the group size is greater tha 35, we set the variace of that group equal to its size raised to.4 power; otherwise we let the variace be the square of the iverse of the group size. The, the i-th elemet of the iovatio vector " is geerated accordig to " i = i i, where i is the stadard error for the i-th observatio ad i s are i.i.d. Normal(,). I use the followig expressios to measure the level of sigal-to-oise i our step up (Pace, LeSage, ad Zhu, ): R SARMA(,) = tr RS S X S S X +tr R, (6.) RS S R R SARMA(,) = tr RR X X +tr(rr ), (6.) where is the diagoal covariace matrix of the disturbace terms. This set-up yields a R value close to.55. For each specificatio, the Mote Carlo experimet is based o repetitios. 6. Simulatio Results The simulatio results are preseted i Appedix 8.3 ad 8.4. I each table, the empirical mea (Mea), the bias (Bias), the empirical stadard error (Std.), ad the root mea square error (RMSE) of parameter estimates are preseted ext to each other. First, we cosider the simulatio results for the SARMA (,) model. The simulatio results are preseted i Table of Appedix 8.3. The MLE imposes almost o bias o ad i all cases. The movig average parameter has substatial amout of bias whe =, buttheamout of bias decreases as the sample size icreases. Despite this, the MLE imposes sigificat amout of bias o whe = 5 ad = i cases where the true value of is ozero. Overall, the simulatio results are cosistet with our large sample results. That is, the MLE of ad is cosistet, while the MLE of is icosistet i the presece of heteroskedasticity. Now, we tur to the simulatio results for the case of SARMA(,). First, we cosider the simulatio results for ad. Table shows the estimatio results for =. The MLE imposes substatial amout of bias o both parameters i all cases. The amout of bias for is relatively larger whe there exists a strog egative spatial depedece i the depedet variable. There is a similar patter for, where the amout of bias ad RMSE is, i geeral, larger for the 9 Here, Diag (B,...,B G) deotes the block diagoal matrix i which the diagoal blocks are m i m i matrices B is. 6

19 cases of high egative spatial depedece i both depedet variable ad disturbace term. The patter that we see for ad shows itself for the estimatio results of ad. That is, the reported biases ad RMSEs are relatively larger for ad, whe there are strog spatial depedece i the model. Table 3 cotais the simulatio results whe = 5. The same patter that I described for ad is also prevalet i Table 3. The MLE still imposes substatial amout of bias o ad. The oticeable improvemet i the estimatio results for ad suggests that these parameters are less affected by the icosistecy of the MLE of ad, whe the sample size is moderately large. The estimatio results i Table 4 for ad are also cosistet with this claim. That is, whe the sample size is large, i.e., =, the MLE imposes trivial bias o ad i most cases. O the other had, the estimatio results i Table 4 show that the MLE imposes sigificat bias o ad, which i tur implies the icosistecy of the MLE for these parameters. I ow evaluate the fiite sample efficiecy measured by RMSE of the MLE through the surface plots give i Appedix 8.5. Figure 3 shows the surface plots of RMSEs for ad. It is clear from the surface plots that the MLE has higher RMSEs whe strog spatial depedece exists i the model. The surface plots i Figure 4 are for ad. These surface plots idicate that the MLE of these parameters has higher RMSEs whe there exists strog egative spatial depedece i both the depedet variable ad disturbace term. 7 Coclusio I this study, I show that the MLE of the spatial autoregressive ad movig average parameters for the SARMA(,) specificatio is geerally icosistet i the presece of heteroskedastic disturbaces. The aalytical results idicate that the cocetrated log-likelihood fuctio is ot maximized at the true parameter values whe heteroskedasticity is ot cosidered i the estimatio. The ecessary coditio for the cosistecy of the MLE depeds o the specificatio of spatial weight matrices. We also show that the MLE of the parameters of the exogeous variables is icosistet, ad we state the expressio for the correspodig asymptotic bias. The Mote Carlo results show that the MLE imposes substatial amout of bias o the spatial autoregressive ad movig average parameters i all cases for all sample sizes whe the spatial weight matrix has o-idetical blocks o the diagoals. The simulatio results also show that the icosistecy of the spatial autoregressive ad movig average parameters has almost o effect o the estimates of parameters of the exogeous variables for cases where the sample size is large. 7

20 8 Appedix 8. Some Useful Lemmas Lemma. Let A, B ad C be matrices with (i, j)th elemets respectively deoted by a,ij, b,ij ad c,ij. Assume that A ad B have zero diagoal elemets, ad C has uiformly bouded row ad colum sums i absolute value. Let q be vector with uiformly bouded elemets i absolute value. Assume that " satisfies Assumptio with covariace matrix deoted by =Diag{,..., }.The, () E " A " " B " = () E (" C " ) = + = X i= X c,ii i= j= X i= (3) Var (" C " )= = X i= X i= j= X a,ij (b,ij + b,ji )! X E " 4 i 3 i 4 + c,ii i X c,ij (c,ij + c,ji ) c,ii E " 4 i X i= i i= j i j =tr A B + B 3 i 4 +tr ( C )+tr C C + C C, c X,ii E(" 4 i ) 3 i 4 + i= j= X c,ij (c,ij + c,ji ) i c,ii E(" 4 i ) 3 i 4 +tr C C + C C. (4) E " C " = O(), Var " C " = O(), " C " = O p (). (5) E (C " )=, Var (C " )=O(), C " = O p (), Var qc " = O(), qc " = O p (). Proof. For (), (), (3), (4) ad (5) see Lemmas A. through A.4 i Li ad Lee () ad Lemma i Doga ad Suleyma (3). Lemma. Cosider M =(I P ),wherep = X (XX ) X uder Assumptio 3. Assume that " satisfies Assumptio with covariace matrix deoted by =Diag{,..., }.The, () M ad P are uiformly bouded i absolute value i both row ad colum sums. () Var (P " )=O, P " = o p (), Var (" P " )=O," P " = O p (). (3) Elemets of P are O. j Proof. The proof is similar to the proof of Lemma 3 i Doga ad Suleyma (3). Hece, it is 8

21 omitted. 8. Proof of Propositio For the probability limit of terms i (4.4), the partial ( ) required, which are give by ( = apple R ( )M X ( ) X( )X ( ) X ( apple X ( ) X( )X ( ) apple X ( ) X( )X ( ) X ( )H( )X ( ) X( )X ( ) X ( ) X( )M R ( ) + apple + X ( ) X( )X ( ) X ( )H ( )X ( ) X( )X ( ) X ( ) are apple = Y S ( )H( )R ( )M ( )R ( )S ( )Y apple Y S ( )R ( )P ( )HM ( )R ( )S ( )Y. = apple Y S ( )R ( )M ( )R ( )W Y. First, the probability limit of the first row i (4.4) is ivestigated: ( ) " M =plim! " M G " +plim " M "! where we use X M = k. For the secod term o the r.h.s. of (8.), we have plim! " M G X " M ", (8.) " M R G X " M " =, (8.) sice the umerator coverges i probability to zero by Lemma (5) ad Lemma (), ad for the term i the deomiator we have " M " = P i= i + o p() as show i (4.). The overall result is zero sice P i= i is uiformly bouded for all by Assumptio. As for the first term o the r.h.s of (8.), we have plim! " M G " " M " =plim! " G " " M " plim! " X XX X G ". (8.3) " M " We first evaluate the last term i (8.3). The umerator of this term teds to zero i probability as goes to ifiity by Lemma (4) ad Assumptio 3. Hece, the last term i (8.3) vaishes. 9

22 Now, we retur to the first term i the r.h.s. of (8.3). By Lemma (4), Var " G " = O = o(). The, the Chebyshev iequality implies that plim! " G " E " G " =plim P! " G " i= G.ii i =. Hece, " G " = " M " P i= G,ii P i= i i These results imply the followig oe:! ( ) i= = G.ii " M P i= Now, we retur to the first term i the secod row of (4.4): ( ) " M + o p (). (8.4) = plim! i +plim! i + o p (). (8.5) Y S H R M R S Y " M " Y S R P HM R S Y. (8.6) " M " Each term is hadled separately below by usig R S Y = X + ", S Y = X + R ", XM = k ad M X = k.notethat Y S R P HM R S Y = X P HM " + " P HM ". By Lemma (5) ad Lemma (), X H M " = o p (). By Lemma (4) ad Assumptio 3, we have " P H " = o p (). For the remaiig term, by Lemma, we have " P HM " = o p (). Hece, the secod term o the r.h.s. of (8.6) vaishes. The first term o r.h.s. of (8.6) ca be writte as plim! Substitutig M = I Y S R M H R S Y = plim " M "! " M H X " M " X X X X ito (8.7) yields plim! " M H " " M ". (8.7) plim! Y S R M H R S Y = plim " M "! +plim! " H " " M " plim! " M H X " M " " X X X X H ". (8.8) " M " By Lemma (5) ad (4.), the secod term o the r.h.s of (8.8) vaishes. The third term vaishes by Lemma (4) ad (4.). The probability limit of the remaiig term ca be foud by the Chebyshev iequality. By Lemma (4), we have Var " H " = O( )=o(). Hece, plim! " H " E( " H " ) =plim! " H " P i= H,ii i =. Combiig these results, we get the

23 followig result for the first term i the first row of ( ) = " M P i= H,ii P i= i i + o p (). (8.9) By combiig the results i (8.5) ad (8.9), we l L ( ) P i= G.ii i P i= i P i= H,ii P i= i i tr(g )+o p () C tr(h A. (8.) ) + o p () For the otatioal simplificatio, deote H = tr(h ) = P i= H,ii, G = tr(g ) = P i= G,ii, ad = P l L ( ) = i. The, (8.) ca be writte i a more coveiet form as P i= G.ii G i P i= H,ii H tr(g G )+o p () C A i + o p () cov(g,ii, i ) + o p () C cov(h,ii, i ) A. (8.) + o p () Note that tr(g G) =.

24 8.3 Simulatio Results for SARMA(,) Table : Simulatio Results for SARMA(,) = (Mea)[Bias](Std.)[RMSE] (Mea)[Bias](Std.)[RMSE] (Mea)[Bias](Std.)[RMSE] -.6 (.987)[-.3](.9)[.9] (.9)[.9](.8)[.8] (-.45)[.95](.68)[.648] - (.)[-.](.)[.] (.998)[-.](.3)[.3] (-.5)[.95](.63)[.637]. (.)[.](.4)[.4] (.8)[.8](.9)[.9] (.)[.](.565)[.574] (.)[.](.7)[.7] (.993)[-.7](.)[.] (.434)[4](86)[.49].6 (.996)[-.4](.)[.] (.998)[-.](.)[.] (.7)[.](.4)[] =5 -.6 (.6)[.6](.83)[.83] (.995)[-.5](.8)[.8] (-.65)[-.5](77)[8] - (.)[.](.83)[.83] (.)[.](.84)[.84] (-54)[-.54](88)[9]. (.998)[-.](.84)[.84] (.)[.](.8)[.8] (.7)[.7](.93)[.93] (.5)[.5](.85)[.85] (.998)[-.](.8)[.8] (46)[.46](.89)[.94].6 (.997)[-.3](.78)[.78] (.)[.](.8)[.8] (.65)[.5](.95)[.8] = -.6 (.)[-.](.58)[.58] (.)[.](.59)[.59] (-.68)[-.8](.84)[.96] - (.999)[-.](.59)[.59] (.998)[-.](.58)[.58] (-4)[-.4](.74)[.77]. (.)[-.](.57)[.57] (.4)[.4](.59)[.59] (.)[.](.9)[.9] (.998)[-.](.58)[.58] (.)[.](.58)[.58] (3)[.3](.5)[.8].6 (.)[.](.57)[.57] (.999)[-.](.57)[.57] (.63)[.3](.7)[.78]

25 8.4 Simulatio Results for SARMA(,) Table : Simulatio Results for SARMA(,): = (Mea)[Bias](Std.)[RMSE] (Mea)[Bias](Std.)[RMSE] (Mea)[Bias](Std.)[RMSE] (Mea)[Bias](Std.)[RMSE] (-.583)[-.983](4.6)[4.374] (.874)[-.6](4)[64] (.898)[-.](5)[65] (-.73)[7](.98)[.34] (-.79)[-.9](4.346)[4.56] (.848)[-.5](7)[.4] (.847)[-.53](6)[9] (-.78)[.](.997)[.4] -.6. (-.794)[-.94](4.355)[4.56] (.867)[-3](57)[8] (.865)[-5](53)[78] (.)[.](.934)[.934] -.6 (-.44)[-.84](3.687)[3.773] (.839)[-.6](79)[.4] (.85)[-.49](8)[.4] (.64)[-.36](.79)[.7] (-.59)[.9](.8)[.8] (.76)[-.4](.455)[.55] (.76)[-.4](.455)[.55] (.47)[-](4)[66] (-.97)[-.67](3.75)[3.33] (.9)[-.88](5)[37] (.97)[-.93](4)[37] (-.59)[4](.8)[.89] - - (-.3)[-.83](3.497)[3.594] (.88)[-.8](5)[7] (.88)[-.9](6)[8] (-6)[.64](.96)[.9] -. (-.335)[-.35](3.84)[3.977] (.857)[-.43](6)[88] (.86)[-9](67)[93] (-.5)[-.5](.86)[.86] - (-.45)[-.745](3.364)[3.445] (.84)[-.6](99)[.43] (.835)[-.65](.4)[.433] (.)[-.8](.79)[.74] -.6 (-.574)[-.74](.873)[.893] (.768)[-](.466)[.5] (.758)[-.4](.459)[.59] (.436)[-.64](9)[.43]. -.6 (-.45)[-.45](.57)[.69] (.94)[-.96](54)[67] (.898)[-.](5)[65] (-.9)[8](.7)[.784]. - (-.69)[-.69](3.3)[3.99] (.93)[-.97](37)[5] (.889)[-.](4)[58] (-.8)[.9](.77)[.778].. (-.834)[-.834](3.74)[3.8] (.84)[-.59](83)[.45] (.857)[-.43](9)[.46] (-.79)[-.79](.84)[.88]. (-.45)[-.45](.3)[.78] (.839)[-.6](.47)[.438] (.838)[-.6](.4)[.44] (8)[-.6](.59)[.593]..6 (-.78)[-.78](.68)[.4] (.768)[-](.469)[.53] (.763)[-7](.463)[.5] (.4)[-.89](49)[97] -.6 (.68)[-](.49)[.448] (.938)[-.6]()[7] (.95)[-.49](7)[] (-84)[.6](.543)[.585] - (-.57)[-.457](.74)[.] (.93)[-.97](44)[58] (.9)[-.98](45)[59] (-.79)[.](.63)[.63]. (-.)[-.5](.3)[.94] (.867)[-3](76)[99] (.864)[-6](8)[.44] (-.6)[-.6](.66)[.679] (-.3)[-.53](.7)[.69] (.89)[-.8](.437)[.473] (.83)[-.87](.43)[.47] (.95)[-.5](.6)[.654].6 (-.)[-](.735)[.8] (.659)[-4](.58)[.6] (.657)[-43](.53)[.69] (9)[-.7](8)[.468] (.4)[-.78](.7)[.733] (.98)[-.9]()[] (.98)[-.9]()[] (-.584)[.6](46)[46].6 - (76)[-.4](.58)[.6] (.976)[-.4](.53)[.54] (.965)[-.35](.55)[.57] (-.5)[-.](9)[9].6. (.7)[-3](.84)[.95] (.96)[-.39](.94)[.96] (.945)[-.55](.9)[.97] (-.4)[-.4](86)[.564].6 (.5)[-.448](.345)[.48] (.9)[-.79](6)[35] (.9)[-.8](35)[44] (-.86)[-.586](.45)[.79].6.6 (.59)[-.44](.767)[.884] (.8)[-.98](.436)[.479] (.8)[-.](.43)[.476] (-.59)[-.659](.44)[.779] 3

26 Table 3: Simulatio Results for SARMA(,): = 5 (Mea)[Bias](Std.)[RMSE] (Mea)[Bias](Std.)[RMSE] (Mea)[Bias](Std.)[RMSE] (Mea)[Bias](Std.)[RMSE] (-3.5)[-.45](7.86)[7.687] (.94)[-.86](.57)[.7] (.9)[-.89](.56)[.7] (-.4)[-.44](.9)[.97] (-.95)[-.35](7.3)[7.57] (.96)[-.84](.53)[.67] (.98)[-.8](.54)[.67] (-.75)[-.45](.93)[.96] -.6. (-.77)[-.7](5.677)[5.797] (.953)[-.47](.3)[.8] (.949)[-.5](.4)[.] (-)[-](.47)[.43] -.6 (-.977)[-77](3.577)[3.597] (.985)[-.5](.4)[.43] (.98)[-.8](.4)[.4] (3)[.3](.84)[.84] (-.667)[-.67](9)[.54] (.3)[.3](.88)[.88] (.6)[.6](.85)[.85] (.69)[.9](.87)[.87] (-.985)[-.685](4.89)[4.44] (.979)[-.](.63)[.65] (.975)[-.5](.6)[.64] (-.68)[-.8](.)[.] - - (-.53)[-.3](5.64)[5.34] (.953)[-.47](.87)[.93] (.96)[-.4](.89)[.93] (-.577)[-.77](.47)[.498] -. (-.96)[-.896](4.6)[4.689] (.97)[-.8](.74)[.77] (.968)[-.3](.7)[.74] (-.55)[-.55](.84)[.93] - (-.457)[-.57](.797)[.84] (.996)[-.4](.3)[.3] (.994)[-.6](.)[.] ()[.](.449)[.449] -.6 (-.46)[-.6](.)[.66] (.)[-.](.8)[.8] (.6)[.6](.8)[.8] (.557)[-.43]()[9]. -.6 (.4)[.4](.86)[.87] (.998)[-.](.9)[.9] (.998)[-.](.86)[.86] (-7)[.9](.468)[.5]. - (-.)[-.](.89)[.] (.994)[-.6](.3)[.3] (.994)[-.6](.9)[.9] (-33)[-.33](.73)[.74].. (-.5)[-.5](.87)[.89] (.996)[-.4](.)[.] (.995)[-.5](.)[.] (-.75)[-.75](.68)[.685]. (-.77)[-.77](.73)[.735] (.996)[-.4](.85)[.85] (.998)[-.](.87)[.87] (.98)[-.](8)[8]..6 (-.53)[-.53](.53)[.96] (.987)[-.3](6)[7] (.989)[-.](5)[6] (.5)[-.79](.97)[] -.6 (7)[.7](.4)[.4] (.3)[.3](.84)[.84] (.)[.](.8)[.8] (-.43)[.7](.)[.63] - (.8)[-.7](.73)[.88] (.3)[.3](.86)[.86] (.998)[-.](.83)[.83] (-3)[-.3](.7)[.73]. (7)[-.63](.75)[.734] (.998)[-.](.86)[.87] (.997)[-.3](.86)[.86] (-.74)[-.74](.48)[.444] (.99)[-.](.)[4] (.996)[-.4](.)[.] (.996)[-.4](.)[.] (.6)[-.84](6)[7].6 (.45)[-.55](.94)[.] (.96)[-.39](.)[.4] (.958)[-.4](.9)[] (.587)[-.3](.5)[.5] (.545)[-.55](.86)[.] (.998)[-.](.8)[.8] (.)[-.](.84)[.84] (-.65)[-.5](.)[.5].6 - (.486)[-.4](.8)[.4] (.998)[-.](.8)[.8] (.)[.](.8)[.8] (-.583)[-.83](.3)[].6. (.4)[-.89](.9)[.9] (.)[.](.83)[.83] (.997)[-.3](.8)[.8] (-.49)[-.49](.4)[.55].6 (4)[-.76](.88)[.9] (.7)[.7](.89)[.9] (.3)[.3](.9)[.9] (-44)[-.644](.)[.674].6.6 (.88)[-](.59)[5] (.943)[-.57](.53)[.59] (.94)[-.59](.53)[.6] (-.7)[-.67](87)[.774] 4

27 Table 4: Simulatio Results for SARMA(,): = (Mea)[Bias](Std.)[RMSE] (Mea)[Bias](Std.)[RMSE] (Mea)[Bias](Std.)[RMSE] (Mea)[Bias](Std.)[RMSE] (-3.449)[-.849](8.68)[9.77] (.97)[-.93](.83)[.98] (.96)[-.94](.8)[.97] (-.33)[-.73](.487)[.59] (-4.5)[-3.55](9.648)[.8] (.877)[-]()[34] (.88)[-.]()[35] (-.35)[-.835](.798)[.9] -.6. (-.675)[-.75](6.)[6.4] (.957)[-.43](.)[.5] (.958)[-.4](.99)[.4] (-.48)[-.48](.666)[.67] -.6 (-.65)[-.5](.4)[.4] (.99)[-.9](.9)[.93] (.99)[-.9](.93)[.93] (5)[.5](.568)[.57] (-.68)[-.8](.95)[.6] (.7)[.7](.59)[.6] (.7)[.7](.57)[.58] (.595)[-.5](.54)[.55] (-.698)[-98](3.63)[3.653] (.983)[-.7](.8)[.9] (.985)[-.5](.9)[.9] (-.64)[-.4](.5)[.53] - - (-.69)[-.39](6.83)[6.4] (.95)[-.48](.4)[.] (.954)[-.46](.4)[.9] (-.74)[-.44](.839)[.883] -. (-.89)[-.59](4.86)[4.] (.98)[-.9](.4)[.43] (.98)[-.8](.4)[.43] (-.3)[-.3](.4)[.45] - (-85)[-.85](.45)[.48] (.)[-.](.73)[.73] (.999)[-.](.74)[.74] ()[-.](35)[35] -.6 (-.476)[-.76](.69)[.44] (.5)[.5](.58)[.58] (.7)[.7](.58)[.58] (.54)[-.76](.5)[]. -.6 (.9)[.9](.866)[.87] (.998)[-.](.64)[.64] (.)[-.](.67)[.67] (-6)[9](73)[.443]. - (-.96)[-.96](.58)[.5] (.996)[-.4](.83)[.83] (.995)[-.5](.78)[.78] (-3)[-.3](.575)[.575].. (-.)[-.](.87)[.9] (.997)[-.3](.7)[.7] (.994)[-.6](.7)[.7] (-.63)[-.63](.55)[.59]. (-.74)[-.74](.8)[.95] (.997)[-.3](.6)[.6] (.999)[-.](.6)[.6] (.6)[-.4](.69)[.74]. -.6 (.68)[-](.49)[.448] (.938)[-.6]()[7] (.95)[-.49](7)[] (-84)[.6](.543)[.585] -.6 (4)[.4](.9)[.99] (.)[.](.6)[.6] (.)[.](.59)[.59] (-.49)[.7](.8)[.8] - (.5)[-.49](.)[.] (.)[-.](.63)[.63] (.4)[.4](.6)[.6] (-38)[-.38](.5)[.57]. (.74)[-.6](.5)[.78] (.)[.](.58)[.58] (.998)[-.](.59)[.59] (-.88)[-.88](.9)[.96] (.5)[-.75](.45)[.63] (.999)[-.](.6)[.6] (.999)[-.](.58)[.58] ()[-.77](.7)[.8].6 (.74)[-.6](.47)[.49] (.996)[-.4](.97)[.97] (.99)[-.8](.97)[.97] (.69)[.9](.48)[.48] (.56)[-.38](.55)[.67] (.)[.](.59)[.59] (.)[.](.59)[.59] (-.668)[-.68](.66)[.95].6 - (.496)[-.4](.58)[.9] (.)[.](.59)[.59] (.)[.](.58)[.58] (-.59)[-.9](.7)[.99].6. (.47)[-.83](.58)[.9] (.998)[-.](.6)[.6] (.)[.](.6)[.6] (-.495)[-.495](.7)[.5].6 (4)[-.76](.6)[.8] (.4)[.4](.6)[.6] (.)[.](.6)[.6] (-7)[-.67](.4)[.68].6.6 ()[-.8](.76)[3] (.977)[-.3](.75)[.77] (.975)[-.5](.76)[.77] (.7)[-.593](.46)[.75] 5

28 8.5 Surface Plots of RMSEs for SARMA(,) Figure 3: RMSE of ad λ ρ RMSE (a),= λ ρ RMSE (b),= λ ρ RMSE (c),= λ ρ RMSE (d),= λ ρ RMSE (e),= λ ρ RMSE (f),= 6

29 Figure 4: RMSE of 4.5 ad RMSE.5.5 RMSE ρ.6.6 λ.6 ρ.6.6 λ.6.4 (a), = (b), = RMSE RMSE ρ.6.6 λ ρ.6.6 λ (c), =5 (d), = RMSE RMSE ρ.6.6 λ.6.6 ρ.6.6 λ.6.5 (e), = (f), = 7

30 Refereces Abadir, Karim M. ad Ja R. Magus (5). Matrix Algebra. Ecoometric Exercises. New York: Cambridge Uiversity Press. Aseli, Luc (988). Spatial ecoometrics: Methods ad Models. New York: Spriger. Arold, Matthias ad Domiik Wied (). Improved GMM estimatio of the spatial autoregressive error model. I: Ecoomics Letters 8., pp Baltagi, Badi H. ad Log Liu (). A improved geeralized momets estimator for a spatial movig average error model. I: Ecoomics Letters 3.3, pp Besag, Julia (974). Spatial Iteractio ad the Statistical Aalysis of Lattice Systems. I: Joural of the Royal Statistical Society. Series B (Methodological) 36., pp Das, Debabrata, Harry H. Kelejia, ad Igmar R. Prucha (3). Small Sample Properties of Estimators of Spatial Autoregressive Models with Autoregressive Disturbaces. I: Papers i Regioal Sciece 8, pp. 6. Doga, Osma ad Taspiar Suleyma (3). GMM Estimatio of Spatial Autoregressive Models with Autoregressive ad Heteroskedastic Disturbaces. Workig Papers. City Uiversity of New York Graduate Ceter, Ph.D. Program i Ecoomics. Drukker, David M., Peter Egger, ad Igmar R. Prucha (3). O Two-Step Estimatio of a Spatial Autoregressive Model with Autoregressive Disturbaces ad Edogeous Regressors. I: Ecoometric Reviews 3.5-6, pp Figleto, Berard (8a). A geeralized method of momets estimator for a spatial model with movig average errors, with applicatio to real estate prices. I: Empirical Ecoomics 34 (), pp (8b). A Geeralized Method of Momets Estimator for a Spatial Pael Model with a Edogeous Spatial Lag ad Spatial Movig Average Errors. I: Spatial Ecoomic Aalysis 3., pp Haiig, R. P. (978). The Movig Average Model for Spatial Iteractio. I: Trasactios of the Istitute of British Geographers. New Series 3., pp. 5. Haiig, Robert (987). Tred-Surface Models with Regioal ad Local Scales of Variatio with a Applicatio to Aerial Survey Data. I: Techometrics 9.4, pp Hepple, Leslie W. (976). A Maximum Likelihood Model for Ecoometric Estimatio with Spatial Series. I: Theory ad practice i regioal sciece. Ed. by Ia Masser. Lodo papers i regioal sciece 6. Lodo: Pio Limited. (995a). Bayesia techiques i spatial ad etwork ecoometrics:. Model compariso ad posterior odds. I: Eviromet ad Plaig 7.3, pp (995b). Bayesia techiques i spatial ad etwork ecoometrics:. Computatioal methods ad algorithms. I: Eviromet ad Plaig 7.4, pp (3). Bayesia ad maximum likelihood estimatio of the liear model with spatial movig average disturbaces. School of Geographical Scieces, Uiversity of Bristol, Workig Papers Series. 8