GMM Estimation of Spatial Autoregressive Models with Autoregressive and Heteroskedastic Disturbances

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1 City Uiversity of New York (CUNY) CUNY Academic Works Ecoomics Workig Papers CUNY Academic Works 203 GMM Estimatio of Spatial Autoregressive Models with Autoregressive ad Heteroskedastic Disturbaces Osma Doga CUNY Graduate Ceter Süleyma Taşpıar 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 GMM Estimatio of Spatial Autoregressive Models with Autoregressive ad Heteroskedastic Disturbaces Osma Doga Süleyma Taşpıar Workig Paper Ph.D. Program i Ecoomics CUNY Graduate Ceter 365 Fifth Aveue New York, NY 006 December 203 This research was supported, i part, by a grat of computer time from the City Uiversity of New York High Performace Computig Ceter uder NSF Grats CNS ad CNS We would like to thak Wim Vijverberg for isightful ad istructive commets o this work. 203 by Osma Doga ad Süleyma Taşpıar. 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 GMM Estimatio of Spatial Autoregressive Models with Autoregressive ad Heteroskedastic Disturbaces Osma Doga ad Süleyma Taşpıar CUNY Graduate Ceter Workig Paper No. December 203 JEL No. C3, C2, C3 ABSTRACT We cosider a spatial ecoometric model cotaiig a spatial lag i the depedet variable ad the disturbace term with a ukow form of heteroskedasticity i iovatios. We first prove that the maximum likelihood (ML) estimator for spatial autoregressive models is geerally icosistet whe heteroskedasticity is ot take ito accout i the estimatio. We show that the ecessary coditio for the cosistecy of the ML estimator of spatial autoregressive parameters depeds o the structure of the spatial weight matrices. The, we exted the robust geeralized method of momet (GMM) estimatio approach i Li ad Lee (200) for the spatial model allowig for a spatial lag ot oly i the depedet variable but also i the disturbace term. We show the cosistecy of the robust GMM estimator ad determie its asymptotic distributio. Fially, through a comprehesive Mote Carlo simulatio, we compare fiite sample properties of the robust GMM estimator with other estimators proposed i the literature. Osma Doga Süleyma Taşpıar Ph.D. Program i Ecoomics Ph.D. Program i Ecoomics City Uiversity of New York City Uiversity of New York 365 Fifth Aveue 365 Fifth Aveue New York, NY 006 New York, NY 006 ODoga@gc.cuy.edu STaspiar@gc.cuy.edu

4 Itroductio Spatial ecoometric models that have a log history i regioal sciece ad geography has bee receivig attetio i ecoomics i recet years. Spatial ecoometric models allow regressio specificatios through which spatial depedece amog observatios ca be icorporated i ecoomic aalysis ad i the estimatio of models. The spatial depedece is a special form of cross-sectioal depedece amog observatios determied by locatios of observatios i space. The estimatio of models with spatial depedece requires special estimatio techiques. There are three mai estimatio approaches: (i) the maximum likelihood (ML) estimatio method, (ii) the geeralized method of momet (GMM/IV) estimatio method, ad (iii) the Bayesia Markov Chai Mote Carlo (MCMC) estimatio method. For may spatial model specificatios, the ML estimatio has bee the most widely used techique ad has ofte bee the oly techique that is implemeted (Aseli, 988; LeSage ad Pace, 2009). However, formal results cocerig the asymptotic properties of the (quasi) ML estimator have recetly bee established i Lee (2004) oly for pure spatial ad spatial autoregressive models. The ML estimatio ca ivolve a sigificat computatioal difficulty due to the presece of the determiat of a matrix i the likelihood fuctio, whose dimesios deped o the sample size. (Das, Kelejia, ad Prucha, 2003; Kelejia ad Prucha, 998, 200). Several solutios have bee suggested to overcome the computatioal burde of the ML method (Barry ad Pace, 999; LeSage ad Pace, 2004, 2007; Ord, 975; Pace ad Barry, 997a,b; Smirov ad Aseli, 200). The GMM ad IV estimators have the advatage that they do ot require ay distributioal assumptio for the disturbace term ad remai to be computatioally more feasible tha ML estimatio. I the literature, differet kids of two stage least squares (2SLS) estimators correspodig to the differet set of istrumetal variables have bee suggested (Aseli, 988; Kelejia ad Prucha, 998, 2007, 200; Lee, 2003, 2007a). The spatial structure of regressio equatios motivate the selectio of the istrumets which are usually costructed from the exogeous variables ad spatial weight matrices. Despite its computatioal simplicity, the 2SLS estimator is iefficiet relative to the ML estimator. The iefficiecy arises because the 2SLS estimator focuses oly o the determiistic part of the edogeous variable (i.e., the spatial lag term) ad the iformatio i the stochastic part is ot used i the estimatio. Kelejia ad Prucha (998, 200) propose a multi-step estimatio method that ivolves a combiatio of IV ad GMM estimatio for the spatial model that has a spatial autoregressive process i the depedet variable ad disturbace term (for short SARAR(,)). This kid of model specificatio is ofte referred as the Kelejia-Prucha Model (Elhorst, 200). I the first step, the iitial estimates of the parameters of the exogeous variable ad the autoregressive parameter of the spatial lag of the depedet variable are estimated by the 2SLS estimator. I the secod step, residuals from the first step are used to estimate the autoregressive parameter of the spatial lag of the disturbace term by the GMM estimator. I the fial step, the parameters are re-estimated by the For the Bayesia MCMC approach, see Lesage (997), Paret ad Lesage (2007) ad LeSage ad Pace (2009). 2

5 2SLS estimator after trasformig the model via a Cochrae-Orcut type trasformatio to accout for the spatial correlatio. However, the estimatio approach i Kelejia ad Prucha (998) is iefficiet relative to the ML estimatio (Prucha, Forthcomig 202). The extesive Mote Carlo results i Das, Kelejia, ad Prucha (2003) demostrate that the differece betwee fiite sample efficiecy, measured with root mea squared errors (RMSE), betwee the ML ad the GMM ad IV estimators of Kelejia ad Prucha (998, 999) is very small. Drukker, Egger, ad Prucha (202) cosider the specificatio SARAR(,) where they allow for edogeous regressors i additio to spatial lag of the depedet variable. The estimatio approach ivolves several steps ad is a extesio of GMM/IV estimatio method of Kelejia ad Prucha (998, 999). To icrease the efficiecy of the GMM estimator, Lee (2007a, 2007b), Li ad Lee (200), Liu, Lee, ad Bolliger (200), ad Lee ad Liu (200b) suggest sets of momet fuctios that are liear ad quadratic i the disturbace term for the GMM estimatio. I this approach, the liear momet fuctios are based o the determiistic part of the spatial lag term ad the quadratic momet fuctios are costructed for exploitig the stochastic part of the spatial lag variable (i.e., the edogeous variable). The quadratic momet fuctios are chose i a such way that the GMM estimator is asymptotically equivalet to the ML estimator whe disturbaces are i.i.d. ormal. Whe disturbaces are simply i.i.d., Liu, Lee, ad Bolliger (200) ad Lee ad Liu (200b) show that the oe step GMM estimator (joit GMM estimator) is more efficiet tha the quasi ML estimator, respectively for the case of a SARAR(,) ad a SARAR(p,q). Most of the estimatio methods metioed above are valid uder the assumptio that the disturbace terms of the spatial models are i.i.d. I may regressio applicatios, heteroskedasticity might be preset. 2 I the presece of ukow heteroskedasticity, the ML ad GMM estimators are geerally ot cosistet. The ML estimator is icosistet if the heteroskedasticity is ot icorporated ito the estimatio. For a SARAR(,0), Li ad Lee (200) shows that the likelihood fuctio is ot maximized at the true parameter values i the presece of the ukow heteroskedasticity. The GMM estimators are also icosistet sice the momet fuctios are ofte desiged uder the assumptio that disturbaces are i.i.d. Hece, the orthogoality coditios for the momet fuctios might ot be satisfied. To hadle ukow heteroskedasticity, Kelejia ad Prucha (200) exted their estimatio approach by modifyig the momet fuctios for the case of a SARAR(,). Badiger ad Egger (20) exted the robust estimatio approach i Kelejia ad Prucha (200) to the case of SARAR(p,q). Likewise, Li ad Lee (200) suggest a oe-step robust GMM estimator for the model with oly spatial depedece i the depedet variable. 3 I the preset study, the oe-step robust GMM estimatio approach suggested by Li ad Lee (200) is exteded to the spatial model with a spatial autoregressive process i both the depedet variable ad the disturbace term uder the assumptio that there is ukow form of heteroskedasticity i the disturbace term. We show that the ML estimator might ot be cosistet i the presece of the ukow heteroskedasticity, as the probability limits of the first order co- 2 For a example, see the empirical applicatio i Li ad Lee (200). 3 For a robust 2SLS estimator of SARAR(,0), see Aseli (2007). 3

6 ditios evaluated at the true parameter values are geerally ot zero. We show that the ecessary coditio for the cosistecy of the ML estimator of spatial autoregressive parameters depeds o the structure of the spatial weight matrices. The, a robust GMM estimator is derived from a set of momet fuctios that are composed of both liear ad quadratic momet fuctios. The cosistecy of the estimator is established ad its asymptotic distributio is determied. Fiite sample properties are compared with that of other estimators through a comprehesive Mote Carlo simulatio. This paper is orgaized i the followig way. I Sectio 2, the theoretical motivatio for the case of a SARAR(,) is provided alog with the model assumptios ad their implicatios. I Sectio 3, the GMM estimators that have bee suggested i the literature are reviewed. I Sectio 4., we show the icosistecy of the ML estimator i the presece of ukow heteroskedasticity. We determie the asymptotic bias of the parameters of the exogeous variables. I Sectio 4.2, a robust GMM estimatio method is cosidered for the case of a SARAR(,). The idetificatio coditios are determied. The mai large sample properties of the robust GMM estimator are stated i three propositios. The Mote Carlo simulatios are carried out i Sectio 5. Sectio 6 closes with cocludig remarks. 2 The Model Specificatio ad Theoretical Motivatio I the literature, spatial depedece i regressio specificatios is categorized i two broad categories kow as spatial lag ad spatial error models. The spatial lag model icludes fuctioal forms i which a depedet variable at a poit i space depeds o depedet variables of surroudig locatios. The equilibrium outcome of theoretical ecoomic models of iteractig spatial uits motivates this kid of specificatio. I spatial error models, cross-sectioal correlatios amog error terms are icorporated ito the specificatio ad estimatio of models. Measuremet error i data usually teds to vary systematically over space, which causes spatial depedece amog error terms of a specificatio. 4 I this study, the followig first order SARAR(,) specificatio is cosidered: Y = λ 0 W Y + X β 0 + u, u = ρ 0 M u + ε, (2.) where Y is vector of depedet variable, X is k matrix of ostochastic exogeous variables, W ad M are spatial weight matrices of kow costats with zero diagoal elemets, ad ε is vector of disturbaces (or iovatios). The variables W Y ad M u are kow respectively as spatial lag of the depedet variable ad the disturbace term. The spatial effect parameters λ 0 ad ρ 0 are kow as the spatial autoregressive parameters. The above specificatio is fairly geeral i the sese that it allows for spatial spillovers i the depedet variable, exogeous variables ad disturbaces. 5 As the spatial data is characterized with triagular arrays, 4 For the motivatio of model specificatios, see Aseli (988); Aseli (2007) ad LeSage ad Pace (2009) 5 Elhorst (200) ames the model with spatial spillovers i the depedet variable, exogeous variable ad distur- 4

7 the variables i (2.) have subscript. 6 Let Θ be the parameter space of the model. I order to distiguish the true parameter vector from other possible values i Θ, the model is stated with the true parameter vector θ 0 = (ρ 0, ς 0 ) with ς 0 = (λ 0, β 0 ). For the otatioal simplicity, we deote S (λ) = (I λw ), R (ρ) = (I ρm ), G (λ) = W S (λ) ad H (ρ) = M R (ρ). Also, at the true parameter values (ρ 0, λ 0 ), we deote S (λ 0 ) = S, R (ρ 0 ) = R, G (λ 0 ) = G, H (ρ 0 ) = H ad Ḡ = R G R. Next, assumptios that are required for the asymptotic properties of estimators are elaborated ad the their iterpretatios are cosidered for (2.). Assumptio : The elemets ε i of the disturbace term ε are distributed idepedetly with mea zero ad variace σ 2 i, ad E ε i ν < for some ν > 4 for all ad i. This assumptio allows idepedet ad heteroskedastic disturbaces. The elemets of the disturbace term have momets higher tha the fourth momet. This coditio is specifically required for the applicatio of the cetral limit theorem for the quadratic form give i Kelejia ad Prucha (200) for the GMM estimator. I additio, the variace of a quadratic form i ε exists ad is fiite whe the first four momets are fiite. 7 Fially, Liapuov s iequality guaratees that the momet less tha ν are also uiformly bouded for all ad i. Assumptio 2: The spatial weight matrices M ad W are uiformly bouded i absolute value i row ad colum sums. Moreover, S, S (λ), R ad R (ρ) exist ad are uiformly bouded i absolute value i row ad colum sums for all values of ρ ad λ i a compact parameter space. I the literature, weight matrices are usually treated as exogeous ad fixed. Lee (2004, 2007b) formulate the weight matrix as a fuctio of the sample size. Accordig to this formulatio, the sequece of weight matrix {W } is uiformly bouded i both row ad colum sums ad its elemets w,ij s are O( h ). The sequece {h } ca be bouded or diverget with the property that h lim 0 = 0, which implies that h is allowed to diverge oly at a rate slower tha that of. This formulatio provides a explicit way that describes how the spatial weight matrix W is expadig as the sample size icreases. For example, assume that a ecoomy cosists of r regios ad each regio is populated by k agets. The, the total umber of observatios from this ecoomy is = rk. I additio, i each regio each aget is equally affected by other agets of the same regio. There is o iteractio amog regios. Deote the row ormalized spatial weight matrix of a regio by C k which is give by k (l kl k I k) where l k is a k dimesioal vector of oes. The, the spatial weight matrix W for this ecoomy is block diagoal W = I r C k. Each elemet i bace term as the Maski Model. He states that that the parameter estimates caot be iterpreted i a meaigful way for this kid of model sice the edogeous ad exogeous effects caot be distiguished from each other. See also Aseli (2007). 6 See Kelejia ad Prucha (200). 7 For the variace of the quadratic form i ε, see Lemma 2 (3). 5

8 a diagoal block is give by k, so that w,ij = O( k ). The, h = k k r = O( r ). Assume that the icrease i is geerated by the icrease of both r ad k. The, the fractio h teds to zero, as h diverges to ifiity. This kid of spatial weight matrix is used for large group iteractios scearios which have importat implicatios for the covergece rate of estimators (Lee, 2004). 8 For large group iteractios for which lim h 0, cosistecy of estimators might ot be available. As a example, Kelejia ad Prucha (2002) ad Yuzefovich, Kelejia, ad Prucha (2006) cosider a row ormalized spatial weight matrix that has equal weights for all observatios. The spatial weight matrix is formulated as W = l l I where each off-diagoal elemet is. I that case, w,ij = O( ) ad lim h = lim =. With this specificatio, Kelejia ad Prucha (2002) show that OLS, 2SLS ad ML estimators are icosistet for spatial autoregressive models. I this study, we assumed that h is bouded. The uiform boudedess of the terms i Assumptio ad 2 is motivated to cotrol spatial autocorrelatios i the model at a tractable level (Kelejia ad Prucha, 998). 9 Assumptio 2 also implies that the model i (2.) represets a equilibrium relatio for the depedet variable. By this assumptio, the reduced form of the model becomes feasible as Y = S X β 0 + S R ε. Fially, the statemet of Assumptio 2 is assumed to hold at the true ad arbitrary autoregressive parameter vector. The uiform boudedess of S (λ) ad R (ρ) is required for the ML estimator ot for the GMM estimator (Liu, Lee, ad Bolliger, 200). I the literature, the parameter space for spatial autoregressive parameters λ 0 ad ρ 0 is restricted to the iterval (, ), whe spatial weight matrices are row ormalized. 0 I that case, matrices S ad R are osigular. More geeral parameter spaces have also bee cosidered i the literature. Let ν j for j =,..., be eigevalues of W. The spectral radius of W is defied by τ = max j ν j. The, S is osigular for all values of λ 0 i the iterval ( τ, τ ). However, the computatio of eigevalues ivolves computatioal difficulties, ad becomes umerically ustable for spatial weight matrices with more tha 000 observatios (Smirov ad Aseli, 200). Aother formulatio for the parameter space base o the maximum row ad colum sums of spatial weight matrices is also cosidered i the literature. Deote R i ad C j respectively as ith row sum ad jth colum sum of W i absolute value. Let the maximum row sum be give by R = max i j= w ij = max i R i. Likewise, the maximum colum sum is defied by C = max j i= w ij = max j C j. Let m = max{c, R}. The, S is osigular for all values of λ 0 i the iterval ( m, m ).2 The followig assumptios are the usual regularity coditios required for the GMM estimator. Throughout this study, the vector of momet fuctios cosidered for the GMM estimator is i the form of g(θ 0 ) = ( ε P ε,..., ε P m ε, ε Q ). The momet fuctios ivolvig 8 For examples of this kid of weight matrices, see Case (99, 992). 9 For a defiitio ad some properties of uiform boudedess see Kelejia ad Prucha (200). 0 Kelejia ad Prucha (200) states that the iterval (, ) is ot atural i the sese that equivalet model formulatio are possible by applyig a arbitrary scale factor to autoregressive parameters ad its iverse to weight matrices ad therefore the parameter space will deped o the scalig factor. Elhorst, Lacombe, ad Piras (202) outlie a simple procedure for fidig the parameter space for models with multiple spatial weights matrices. 2 For a proof of this result see Kelejia ad Prucha (2007). 6

9 costat matrices P j for j =,..., m are kow as quadratic momet fuctios. The last momet fuctio Q ε is the liear momet fuctio, where the full colum rak matrix Q is k with k k +. The matrices P j s ad Q are chose i such way that orthogoality coditios of populatio momet fuctios are ot violated. Let P be the class of costat matrices with zero trace ad P 2 be class of costat matrices with zero diagoal elemets. 3 The quadratic momet fuctios ivolvig matrices from these both classes satisfy the orthogoality coditios whe disturbace terms are i.i.d. As it will be show, whe disturbace terms are merely idepedet, matrices from the class P \ P 2 ca ot be used to form quadratic momet fuctios. 4 Assumptio 4 states regularity coditios for these matrices ad the last assumptio characterizes the parameter space. Assumptio 3: The regressors matrix X is a k matrix cosistig of uiformly bouded costat elemets. It has full colum rak of k. Moreover, lim X X exists ad is osigular. Assumptio 4: Elemets of IV matrix Q are uiformly bouded. P j for j =,..., m is uiformly bouded i absolute value i row ad colum sums. Assumptio 5: The parameter space Θ is a compact subset of R k+2 ad θ 0 is i the iterior of Θ. 3 GMM Estimatio of Spatial Autoregressive Models The GMM estimatio approach depeds o the momet fuctios that are derived from the structure of the model. The edogeous variable W Y o the right had side of the model is give more explicitly by W Y = W S X β 0 + W S R ε = G X β 0 + G R ε where G = W S = W (I λ 0 W ) exists by Assumptio 2. Thus, W Y is a fuctio of a ostochastic term G X β 0 ad a stochastic term G R ε. Lee (200a, 2007a), Liu, Lee, ad Bolliger (200), Lee ad Liu (200b) ad Li ad Lee (200) form momet fuctios based o stochastic ad o-stochastic terms. The o-stochastic term is istrumeted by Q = (R G X β 0, R X ), which forms the liear momet fuctio Q ε. The liear momet matrix Q is costructed from the expectatio of Z = (W Y, X ). Give cosistet iitial estimates of λ 0, ρ 0 ad β 0, the IV matrix Q becomes available. Lee (2003) shows that the 2SLS estimator with Q is best i the sese that its asymptotic variace covariace matrix is the smallest amog the class of 2SLS estimators based o liear momet coditios. The stochastic part G R ε of W Y is istrumeted by P j ε, where P j P ad/or P j P 2 for j =,..., m. I this case, the quadratic momet is i the form of ε P j ε ad the orthogoality (or populatio momet) coditio is satisfied whe disturbaces are simply i.i.d. I that case, E(ε P j ε ) = tr(p j E(ε ε )) = 0 for P j s from either P or P Note that P 2 is a subclass of P, i.e., P 2 P. 4 Here, P \ P 2 deotes set-theoretic differece of P ad P 2. 5 tr( ) returs the sum of the diagoal elemets of a iput matrix. For both 7

10 stochastic ad o-stochastic term, the IVs are costructed i a such way that they are correlated with W Y but ucorrelated with ε. 6 The cosistecy of the GMM estimator does ot deped o a particular P j but the asymptotic variace-covariace matrix is a fuctio of P j s. asymptotic efficiecy of estimators eeds to be cosidered. Therefore, for the selectio of P j s, the Liu, Lee, ad Bolliger (200) ad Lee ad Liu (200b) provide the best selectio of P j P i the case of a SARAR (,) ad SARAR(p,q), respectively. 7 I the case of SARAR (,) with i.i.d ormal iovatios, the best selectio is () P = (R G R tr(r G R )I ), ad (2) P 2 = (H tr(h )I ). Let g (θ) = ( ε (θ)p ε (θ), ε (θ)p 2 ε (θ), ε ) (θ)q be the set of sample momet fuctios. Liu, Lee, ad Bolliger (200) show that give the set of momet fuctio g (θ), ay other momet fuctios that ca be added to this set is redudat. They also show that the ML estimator is characterized by the set of momet fuctios g (θ), therefore, the GMM estimator based o these momet fuctios is asymptotically equivalet to the ML estimator. Whe the iovatios are simply i.i.d, Liu, Lee, ad Bolliger (200) suggest aother best set of quadratic momet fuctios so that the optimal GMM estimator is asymptotically more efficiet tha the quasi ML estimator. Whe disturbace terms are idepedet ad heteroskedastic, some matrices P j with zero trace property caot be used i the formatio of the quadratic momet fuctios. Let Σ = Diag(σ 2,..., σ2 ) be the diagoal variace matrix of the disturbace terms. If P j (P \ P 2 ) for ay j =,..., m, the the covariace E(ε P j ε ) = tr(p E(ε ε )) = tr(p j Σ ) 0. O the other had, P j with zero diagoal property is still available for the formatio of the quadratic momets, sice tr(p E(ε ε )) = tr(p j Σ ) = 0 for ay P j P 2. Thus, the class of matrices with zero diagoal elemets provides robustess for the heteroskedasticity. Li ad Lee (200) exted the GMM estimatio method i Lee (200a, 2007a) to SARAR(,0) that has a ukow form of heteroskedasticity i iovatios. The quadratic momet fuctios are based o the class P 2. Let ς 0 = (λ 0, β 0 ) be the parameter vector of the model, Li ad Lee (200) suggest the set of momet fuctios g (ς) = ( ε (ς)p ε (ς), ε (ς)q 2 ), where P = (G Diag(G )) P 2 ad Q 2 = ( G X β 0, X ). 8 The optimal robust GMM estimator derived from mi ς Θ g (ς)ˆω g (ς) is cosistet ad asymptotically ormally distributed. Here, ˆΩ is a estimate of var(g (ς 0 )) = Ω based o a iitial cosistet estimator of ς 0. For the heteroskedastic case, the best selectio of P j is ot available. Li ad Lee (200) suggest that the selectio from P 2 for the simply i.i.d case ca be used for the case of idepedetly distributed disturbace terms. Thus, the cosistet estimates of ( G Diag(G ) ) ad ( G X β 0, X ) are used i g (ς) for the robust optimal GMM estimator. The computatioally simple two-step GMM estimatio approach i Kelejia ad Prucha (998, 999) for the case of a SARAR(,) is based o two quadratic momet matrices from P : () P = v(m M tr(m M )) with v = +( tr(m M )) 2, ad (2) P 2 = M. Whe the iovatios 6 Note that cov(q, ε ) = 0 ad cov(p jε, ε ) = 0. 7 Liu, Lee, ad Bolliger (200) also cosider the best GMM estimatio for the case of a SARAR(,0) ad a SARAR(0,). 8 Diag( ) is a operator that creates a matrix from the diagoal elemets of a iput matrix. 8

11 are heteroskedastic, the orthogaality coditio of the quadratic momet fuctio based o P is violated, therefore Kelejia ad Prucha (200) cosider a quadratic momet matrix from the class P 2. I that case, the first momet is formed with P = (M M Diag(M M )). The liear momet coditios i Kelejia ad Prucha (998) are based o the liearly idepedet colums of the set Q 3 = (X, W X, W 2 X,..., M W X, M W 2 X,...). The IV matrix Q 3 provides a approximatio for E(Z ) ad E(M Z ). For the illustratio of two-step GMM estimatio approach of Kelejia ad Prucha (200), let g (ρ, ς) = ( ε (θ)p ε (θ), ε (θ)p 2 ε (θ) ) be the set of sample momet fuctios, ad let ς be a iitial cosistet estimator based o the istrumet matrix Q 3. The optimal GMM estimator of ρ 0 is defied as ˆρ = argmi ρ g (ρ, ς ) ˆΨ g (ρ, ς ), where ˆΨ is a estimator of the variace matrix of the limitig distributio of the ormalized sample momet g (ρ, ς ). 9 The estimator ˆρ is used for the two step GMM estimator of ς 0, which is based o the liear istrumetal matrix Q 3. Let g 2 (ˆρ, ς) = Q 3 ε (ˆρ, ς) be the sample momet fuctio, where ε (ˆρ, ς) = R (ˆρ )S (λ)y R (ˆρ )X β. The the optimal two-step GMM estimator of ς 0 is defied by ˆς = argmi ς g 2 (ˆρ, ς)υ g 2 (ˆρ, ς), where Υ = ( Q 3 Q 3). 20 As illustrated, the estimatio approach i Kelejia ad Prucha (998), Kelejia ad Prucha (200) ad Drukker, Egger, ad Prucha (202) is characterized by a sequetial two-step GMM estimatio method. 2 The sequetial GMM estimatio is motivated by computatioal simplicity as the ML estimatio ivolves sigificat computatioal burde for the large samples. I additio, the Kelejia-Prucha methodology also does ot ivolve the computatio of the iverse of the matrix S i the GMM framework. A possible disadvatage of the two-step GMM approach is that the resultig estimators may be iefficiet relative to the joit GMM estimator (oe step GMM estimator) derived by usig the complete set of momet fuctios with a optimal weight matrix (Lee, 2007b; Lee ad Liu, 200b) Estimatio Approach uder Ukow Heteroskedasticity I this sectio, we cosider GMM ad ML estimatio of spatial autoregressive models with heteroskedastic disturbaces. I the first subsectio, the ecessary coditio for the cosistecy of the ML estimator is studied. The results show that the ML estimator of autoregressive parameters is geerally icosistet whe heteroskedasticity is ot icorporated ito estimatio. The ext subsectio covers a robust GMM estimatio method for a spatial model with spatial depedece i the depedet variable ad i the disturbace term. The results idicate that the robust GMM estimator is cosistet ad asymptotically ormally distributed. 9 For the explicit form of ˆΨ, see Arraiz et al. (200). Note that ς ca be updated by usig the weight matrix I for a iitial first step ˆρ. 20 The estimator ˆς has bee called the feasible geeralized spatial two-stage least squares (FGS2SLS) estimator. 2 For the descriptio of the estimatio steps, see Arraiz et al. (200) ad Drukker, Egger, ad Prucha (202). 22 For a differet approach of the GMM estimatio method, see Coley (999). 9

12 4. The Icosistecy of Maximum Likelihood Estimator Li ad Lee (200) show that the MLE is icosistet for the case of a SARAR(,0). I this sectio, we show that the ML estimator is also icosistet for the spatial model i (2.) whe there is a ukow form of heteroskedasticity i the iovatio terms. Let ζ = (θ, σ 2 ) with θ = (ρ, λ, β ). The log likelihood of the model i (2.) uder the assumptio that disturbaces are i.i.d. N(0, σ0 2 ) is give by l L (ζ) = 2 l(2π) 2 l(σ2 ) + l S (λ) + l R (ρ) (4.) 2σ 2 [S (λ)y X β] R (ρ)r (ρ) [S (λ)y X β]. For otatioal simplicity, let R (ρ)x = X (ρ), M (ρ) = (I P (ρ)) with P (ρ) = X (ρ)[ X (ρ) X (ρ)] X (ρ), ad δ = (ρ, λ). Note that X (ρ) M (ρ) = 0 k ad M (ρ) X (ρ) = 0 k. The solutio of the first order coditios for β ad σ 2 yields the followig ML estimators. 23 ˆβ (δ) = [ X (ρ) X (ρ) ] X (ρ)r (ρ)s (λ)y ˆσ 2 (δ) = ε (θ)ε (θ) = Y S (λ)r (ρ) M (ρ)r (ρ)s (λ)y. (4.2a) (4.2b) For a give value of δ, the ML estimators ˆβ (δ) ad ˆσ (δ) 2 ca be see as OLS estimators from the regressio equatio R (ρ)s (λ)y = R (ρ)x β +ε. Substitutio of R (ρ)s (λ)y = R (ρ)x β + ε ito ˆσ (δ) 2 yields ˆσ (δ) 2 = M ε (ρ)ε. For the asymptotic argumet of this sectio, we modify Assumptio 3 i the followig way. Assumptio 3 : The exogeous variables 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 R (ρ)r (ρ)x exist ad are osigular for all values of ρ i Θ. The compact parameter space cotais ρ 0 by Assumptio 5, therefore the modified assumptio also requires a fiite ad osigular limit for the term X R R X. With this ew assumptio, orders of certai terms ca be obtaied via the asymptotic aalysis give i Appedix B. At δ 0, the probability limit of ˆσ 2 (δ 0 ) is plim ˆσ 2 (δ 0 ) = plim ε ε plim 2 ε X [ X X ] X ε. (4.3) I (4.3), the first term o the right had side coverges to i= σ2 i by Chebyshev Weak Law of Large umbers. The secod term vaishes i probability so that the average of variaces of the disturbace terms is asymptotically equivalet to ˆσ 2 (δ 0 ), amely, ˆσ 2 (δ 0 ) = i= σ2 i + o p() The first order coditios from (4.) are give i Appedix B. 24 For the asymptotic argumet see Appedix B. 0

13 The cocetrated log-likelihood fuctio is obtaied by substitutig ˆβ (δ) ad ˆσ 2 (δ) ito (4.): l L (δ) = 2 (l(2π) + ) 2 l(ˆσ2 (δ)) + l S (λ) + l R (ρ). (4.4) The MLE ˆδ = (ˆλ, ˆρ ) is the extremum estimator derived from the cocetrated log-likelihood fuctio. The first order coditios of the cocetrated log-likelihood fuctio with respect to ρ ad λ are give by l L (δ) ρ l L (δ) λ = ˆσ (δ) 2 2ˆσ (δ) 2 tr(h (ρ)), (4.5) ρ = ˆσ (δ) 2 2ˆσ (δ) 2 tr(g (λ)), (4.6) λ where G (λ) = W S (λ) ad H (ρ) = M R (ρ). The cosistecy of the MLE ˆδ requires that the first order coditios evaluated at the true parameter value δ 0 coverges i probability to zero i.e., plim δ 0 is l L (δ 0 ) δ l L (δ 0 ) = δ = 0. This ecessary coditio for the cosistecy of the ML estimator of i= H,iiσi 2 i= σ2 i i= Ḡ.iiσ 2 i i= σ2 i tr(h ) + o p () tr(g. (4.7) ) + o p () Deote σ 2 = i= σ2 i, H = tr(h ) = i= H,ii, ad Ḡ = tr(ḡ) = i= Ḡ,ii where Ḡ = R G R. The, (4.7) ca be writte i a more coveiet form 25 l L (δ 0 ) = δ = i= [H,ii H ] [σ2 i σ2 ] + o σ 2 p () σ 2 tr(ḡ G ) + o p () i= [Ḡ.ii Ḡ ] [σi 2 σ2 ] cov(h,ii, σ 2 i ) + o σ 2 p (). (4.8) + o σ 2 p () cov(ḡ,ii, σ 2 i ) The above equatio shows that the ML estimators ˆλ ad ˆρ are icosistet uless cov(h,ii, σi 2 ) = 0 σ 2 ad cov(ḡ,ii, σi 2 ) = 0. The icosistecy of ˆλ σ 2 ad ˆρ depeds o the covariace betwee variaces of elemets of the disturbace terms ad diagoal elemets of H ad Ḡ. It is obvious that whe l L (δ 0 ) δ ε is homoskedastic, the trivial case of ρ 0 = λ 0 = 0. is o p () as σ 2 i = σ2 for i =,...,. This result also holds for Ituitively, the result i (4.8) idicates that the cocetrated log-likelihood fuctio is ot maximized at the true parameter vector whe disturbace terms have 25 Note that σ 2 = i= σ2 i is the average of the variace of the disturbace terms, ad tr(ḡ G) = 0.

14 ukow heteroskedasticity. The ML estimator of β 0 i (4.2a) is also icosistet, sice it is a fuctio of icosistet estimators ˆλ ad ˆρ. Explicitly, ˆβ (ˆδ ) = β 0 + (λ 0 ˆλ )D (ˆρ ) X Ḡ X β 0 + (λ 0 ˆλ )(ρ 0 ˆρ )D (ˆρ ) X H s Ḡ X β 0 + (λ 0 ˆλ )(ρ 0 ˆρ ) 2 D (ˆρ ) X H H Ḡ X β 0 + o p () (4.9) where D (ˆρ ) = [ X (ˆρ ) X (ˆρ )]. 26 The above result shows that the asymptotic bias of ˆβ (ˆδ ) depeds o weight matrices ad the regressors matrix, ad is ot zero uless autoregressive parameters are cosistet. For the special case of ˆλ = λ 0 + o p (), the icosistecy of ˆρ has o effect o the asymptotic bias of ˆβ (ˆδ ), so that ˆβ (ˆδ ) = β 0 + o p (). For the spatial autoregressive model, where ρ 0 = 0 i (2.), the result i the secod row of (4.8) simplifies to l L (λ 0 ) λ = cov(g,ii, σi 2 ) +o σ 2 p () sice Ḡ = G. The term D (ˆρ ) X Ḡ X β 0 i (4.9) simplifies to (X X ) X G X β 0 so that ˆβ (ˆλ ) = β 0 + (λ 0 ˆλ )(X X ) X G X β 0 + o p (), which is the exact result stated i Li ad Lee (200). The cocetrated log-likelihood fuctio is oliear i δ, which makes it hard to make ay geeral coclusio about the asymptotic bias of the MLE ˆδ = (ˆλ, ˆρ ). For the spatial autoregressive model, Li ad Lee (200) ivestigate the asymptotic bias of ˆβ (ˆλ ) for a case of group iteractios, where W is assumed to be a block-diagoal matrix such that each block has differet umber of uits ad each uit is equally affected by the other uits. Li ad Lee (200) shows that whe covariates are i.i.d with mea zero for all blocks, the asymptotic bias of the itercept is larger tha those of other coefficiets, ad the bias of all coefficiets are egatively related to the average block size. The specificatio i (2.) with λ 0 = 0 is called the special error model (SEM or SARAR (0,)) i the literature (LeSage ad Pace, 2009). For this model, the ecessary coditio for the cosistecy of the ML estimator of ρ 0 is ot satisfied, sice the result i the first row of (4.8) is geerally ot zero. The MLE of β 0 for the SEM is give by ˆβ (ρ) = D (ρ) X (ρ)r (ρ)y for a give ρ, which is the OLS estimator from the artificial regressio R (ρ)y = R (ρ)x β + ε. Substitutig Y = X β 0 + R ε ito ˆβ (ρ) yields ˆβ (ρ) = β 0 + D (ρ) X (ρ)r (ρ)r ε. Uder Assumptio 3, it ca be show that ˆβ (ˆρ ) = β 0 + o p (). 27 This result idicates that uder ukow form of heteroskedasticity, the MLE ˆβ (ˆρ ) has o asymptotic bias, eve whe the MLE ˆρ is icosistet. The spatial model specificatio with β 0 = 0 ad ρ 0 = 0 i (2.) is kow as the pure spatial autoregressive model i the literature. The MLE estimator of λ 0 for this kid of model is also icosistet uder heteroskedastic disturbaces. The first order coditio of the cocetrated log-likelihood l L (λ 0 ) λ = fuctio of this model with respect to λ is ˆσ (λ) Y W S 2 (λ)y tr(g (λ)), where ˆσ (λ) 2 = ε (λ)ε (λ) with ε (λ) = S (λ)y. At λ 0, ˆσ (λ 2 0 ) = i= σ2 i + o p(). The, l L (λ 0 ) λ = cov(g,ii, σi 2 ) + o σ 2 p () by the same asymptotic argumet applied i the derivatio of (4.8). This result is the same as with the oe obtaied i Li ad Lee (200) for the case of a 26 For the asymptotic argumet, see Appedix B. 27 For the asymptotic argumet, see Appedix B. 2

15 SARAR(,0). I the special case, where the spatial weight matrices are the same ad the true parameter values λ 0 ad ρ 0 are equal, the covariace terms i (4.8) are equal. 28 I this special case, the result i (4.8) simplifies to l L (δ 0 ) δ = cov(g,ii, σi 2 ) + o σ 2 p (), which is the ecessary coditio stated i Li ad Lee (200) for a spatial model with oly a spatial lag i the depedet variable. Despite this result, the asymptotic bias of the MLE ˆβ (ˆδ ) will ot simplify to the oe derived for a spatial model with a spatial lag i the depedet variable. A atural questio is that uder what coditios the covariace terms i (4.8) are zero. A obvious case is whe both Ḡ ad H have diagoal elemets that are equal. The, the ecessary coditio for the cosistecy of ˆλ ad ˆρ is ot violated, eve if the disturbaces are heteroskedastic. As a example, cosider a circular world weight matrix with equal diagoal elemets that relate each uit to the uits i frot ad i back. I that case, both H ad Ḡ have equal diagoal elemets. Aother case arises, whe the weight matrices W ad M are block-diagoal matrices with a idetical submatrix i the diagoal blocks ad zeros elsewhere. This is a special case of group iteractios example i Lee (200a) where all group sizes are equal ad each eighbor of the same uit has equal weight. I this sectio, we have show that the ML estimators for autoregressive spatial models are geerally icosistet whe heteroskedasticity is preset i the disturbace terms. computatioal burde, the cosistecy of ML estimator is ot esured. 4.2 Robust GMM Estimatio of SARAR(,) Besides its I this sectio, the robust GMM estimatio method suggested by Li ad Lee (200) is exteded for the model i (2.). We cosider the set of populatio momet fuctios g (θ 0 ) = ( ε P ε,..., ε P m ε, ε Q ) where Pj P 2 for j =,..., m. This set defies the orthogoality coditios that are cosidered for the estimatio. Throughout this sectio, we assume that the model i (2.) satisfies Assumptios through Assumptios 5. First, we discuss the idetificatio of the parameter vector θ 0 i the GMM framework ad state coditios for the idetificatio. The, we determie the large sample properties of the robust GMM estimator. 29 The idetificatio of parameters i a GMM framework requires lim E(g (θ 0 )) = For ay value of the parameter vector θ Θ, cosider the expectatio of the set of momet fuctios 28 I this case, W = M ad R = S so that H = W S = G ad Ḡ = SGS = S W S S = G. 29 The argumets provided here is geeral, ad issues about the selectio of paticular P j ad Q are preseted i the fial part of Sectio See Lemma 2.3 i Newey ad McFadde (994). 3

16 i (4.0): E(ε (θ)p ε (θ)) E(ε (θ)p 2 ε (θ)) E(g (θ)) =.. (4.0) E(ε (θ)p m ε (θ)) E(Q ε (θ)) From (2.), ε (θ) ca be writte i terms of the model parameters i the followig way: ε (θ) = R (ρ)s (λ)s R ε + R (ρ)[s (λ)s X β 0 X β] = K (δ)k ε + R (ρ)k (ς), (4.) where K (δ) = R (ρ)s (λ), K = R S, k (ς) = [S (λ)s X β 0 X β], ad ς = (λ, β ) are itroduced for otatioal simplicity. Substitutig (4.) ito (4.0) ad takig expectatio yield k (ς)r (ρ)p R (ρ)k (ς) + tr(σ K K (δ)p K (δ)k ) k (ς)r (ρ)p 2 R (ρ)k (ς) + tr(σ K K (δ)p 2 K (δ)k ) E(g (θ)) =.. (4.2) k (ς)r (ρ)p m R (ρ)k (ς) + tr(σ K K (δ)p m K (δ)k ) Q R (ρ)k (ς) where Σ = Diag(σ 2 i,..., σ2 ). The idetificatio of the parameter vector θ 0 ca be verified from E(g (θ)) = 0; i.e., θ 0 is idetified if θ 0 is the uique solutio of E(g (θ)) = 0. The term Q R (ρ)k (ς) i (4.2) ca be writte more explicitly as ( ) Q R (ρ)k (ς) = Q β 0 β R (ρ) (X, G X β 0 ) = 0. (4.3) λ 0 λ The uique solutio of (4.3) is (β 0, λ 0 ) if the matrix, Q R (ρ)(x, G X β 0 ) = (Q R (ρ)x, Q R (ρ)g X β 0 ) has full colum rak k + for each possible value of ρ Θ by the virtue of Lemma of Appedix A. Sice the liear IV matrix Q has colum rak greater tha or equal to k +, this rak coditio is equivalet to the fact that the matrix (X, G X β 0 ) has full colum rak k +. Uder this rak coditio, the remaiig momet equatios i E(g (θ)) are for the idetificatio of ρ 0. To this ed, K (δ) is decomposed i the followig way 3 K (δ) = R (ρ)s (λ) = (R (ρ ρ 0 )M )(S (λ λ 0 )W ) = K + (ρ 0 ρ)m S + (λ 0 λ)r W + (ρ 0 ρ)(λ 0 λ)m W. (4.4) 3 K (δ) ca be decomposed by usig idetities S (λ) = S (λ λ 0)W ad R (ρ) = R (ρ ρ 0)M. 4

17 Cosider the momet equatio with P j, k (ς)r (ρ)r (ρ)p j k (ς) + tr(σ K K (δ)p j K (δ)k ). Sice β 0 ad λ 0 are idetified from the rak coditio of the last momet equatio, the first term i the jth momet equatio is zero ad K (δ) term reduces to K + (ρ 0 ρ)m S. The remaiig term i the jth momet equatio ca be explicitly writte as tr(σ K K (δ)p j K (δ)k ) = (ρ 0 ρ)tr(pjm s R Σ ) + (ρ 0 ρ) 2 tr(r M P j M R Σ ) = 0, (4.5) where Pj s = P j + P j. There are two roots for ρ i (4.5). The first root is the true parameter value ρ 0, ad the secod root is ρ = ρ 0 + tr(pj s M R Σ ) tr(r M P j M R Σ ). (4.6) There are three cases i which ρ 0 is the uique root. If tr(pj s M R Σ ) = 0 ad the deomiator is ot zero, the ρ 0 is the uique root. If the umerator is ot zero but the deomiator is zero, the the secod root is ot defied. I both cases, ρ 0 is uiquely idetified. If there is more tha oe matrix for the quadratic momet equatios, the there is aother case i which ρ 0 ca be uiquely idetified. The coditio for this case is that the fractio i (4.6) must be differet for each P j for j =,..., m so that the secod root does ot exist, tr(pi s M R Σ ) tr(r M P i M R Σ ) tr(p s j M R Σ ) tr(r M P j M R Σ ) for all i j. (4.7) Whe the rak coditio for Q R (ρ)k (ς) = 0 fails the β 0 ad λ 0 are ot idetified separately from the last momet equatio i E(g (θ)). I this case, the colum rak of the matrix (X, G X β 0 ) is less tha k +. This implies that there exists a costat vector v such that X v = G X β 0. Usig this relatio i (4.3) Q R (ρ)k (ς) = Q R (ρ) [X (β β 0 ) + X v(λ λ 0 )] = Q R (ρ)x [(β β 0 ) + v(λ λ 0 )] = 0. (4.8) The regressors matrix X has full colum rak k by Assumptio 3; therefore, the matrix Q R (ρ)x i the above equatio has full colum rak k for each ρ Θ. This implies that all solutios of (4.8) satisfies the relatio β = β 0 v(λ λ 0 ) by virtue of Lemma Appedix A. This idicates that β 0 ad λ 0 are ot separately idetified from this momet equatio ad that oly oce λ 0 is idetified the idetificatio of β 0 will be feasible. The remaiig momet equatios i (4.2) are fuctios of δ = (ρ, λ). Hece, these momet fuctios may provide idetificatio for the parameter vector δ 0. I this case, these momet equatios are simplified to tr(σ K K (δ)p j K (δ)k ) = 0 for j =,..., m (sice, k (ς) = X [(β β 0 ) + v(λ λ 0 )] = 0 at β = β 0 v(λ λ 0 )). Lee (200b) 5

18 makes the observatio that these remaiig momet equatios correspods to the momet equatios of the followig process: Y = λ 0 W Y + u, u = ρ 0 M u + ε. (4.9) For the above process, ε (θ) = R (ρ)s (λ)y = R (ρ)s (λ)(s R ε ) = K (δ)k ε. Thus, the K (δ)p j K (δ)k ). expectatio of the jth quadratic momet is E(ε (θ)p j ε (θ)) = tr(σ K Therefore, the idetificatio of δ 0 ca be ivestigated from (4.9). Whe M = W, the reduced form of (4.9) is Y = (ρ 0 + λ 0 )W Y ρ 0 λ 0 W 2 Y + ε. The idetificatio of δ 0 is ot possible from this process sice λ 0 ad ρ 0 ca ot be distiguished from each other (Aseli, 988, p. 88). Thus, oly uder the coditio that M W, the idetificatio issue ca be ivestigated from the equatio tr(σ K tr(σ K K (δ)p j K (δ)k ) = 0. This equatio ca be explicitly writte as K (δ)p j K (δ)k ) = tr ( Σ (I + (ρ 0 ρ)m S K + (λ 0 λ)r W K + (ρ 0 ρ)(λ 0 λ)m W K ) P j (I + (ρ 0 ρ)m S K +(λ 0 λ)r W K I order to simplify the otatio, let us itroduce the followig variables: + (ρ 0 ρ)(λ 0 λ)m W K ) ) = 0. (4.20) α ρ,j = tr(σ P s j H ), α λ,j = tr(σ P s jḡ), α ρ 2,j = tr(σ H P j H ), α λ 2,j = tr(σ Ḡ P j Ḡ ), α ρλ,j = tr(σ P s j H Ḡ + Σ Ḡ P s j H ), α ρ 2 λ,j = tr(σ Ḡ H P s j H ), α ρλ 2,j = tr(σ Ḡ H P s jḡ) ad α ρ 2 λ 2,j = tr(σ Ḡ H P j H Ḡ ). Usig these variables, the equatio (4.20) simplifies to tr(σ K K (δ)p j K (δ)k ) = α ρ,j (ρ 0 ρ) + α λ,j (λ 0 λ) + α ρ 2,j(ρ 0 ρ) 2 + α λ 2,j(λ 0 λ) 2 + α ρλ,j (ρ 0 ρ)(λ 0 λ) + α ρ 2 λ,j(ρ 0 ρ) 2 (λ 0 λ) + α ρλ 2,j(ρ 0 ρ)(λ 0 λ) 2 + α ρ 2 λ 2,j(ρ 0 ρ) 2 (λ 0 λ) 2 = 0 for j =,..., m. (4.2) The above system of equatios ca be writte i matrix form i the followig way α ρ, α λ, α ρ 2, α λ 2, α ρλ, α ρ 2 λ, α ρλ 2, α ρ 2 λ 2, α ρ,2 α λ,2 α ρ 2,2 α λ 2,2 α ρλ,2 α ρ 2 λ,2 α ρλ 2,2 α ρ 2 λ 2, α ρ,m α λ,m α ρ 2,m α λ 2,m α ρλ,m α ρ 2 λ,m α ρλ 2,m α ρ 2 λ 2,m ρ 0 ρ λ 0 λ (ρ 0 ρ) 2 (λ 0 λ) 2 (ρ 0 ρ)(λ 0 λ) = 0. (ρ 0 ρ) 2 (λ 0 λ) (ρ 0 ρ)(λ 0 λ) 2 (ρ 0 ρ) 2 (λ 0 λ) 2 (4.22) 6

19 By Lemma i Appedix A, the system i (4.22) has a uique solutio at δ 0 if colums of the above matrix do ot have a liear combiatio with oliear o-zero costat coefficiets of the form α ρ c + α λ c 2 + α ρ 2c 2 + α λ 2c α ρλ c c 2 + α ρ 2 λc 2 c 2 + α ρλ 2c c α ρ 2 λ 2c2 c 2 2 = 0, (4.23) where αs represet the colum vectors of the above matrix ad c ad c 2 are arbitrary ozero costat coefficiets. With this coditio, ρ 0 ad λ 0 are uiquely idetified from the system i (4.22). Oce λ 0 is idetified, the idetificatio of β 0 follows from the last momet fuctio i (4.2). Assumptio 6 summarizes coditios for the idetificatio of the parameter vector θ 0 from the set of momet fuctios i g (θ) for sufficiet large. The similarity of this assumptio with Assumptio 5 i Liu, Lee, ad Bolliger (200) is revealig: the mai differece is that the idetificatio coditios ow ivolve covariace matrix Σ. 32 Assumptio 6: For the idetificatio of the parameter vector θ 0 Θ, oe of the followig cases is assumed. Case(): (i) The limitig matrix lim Q R (X, G X β 0 ) has full colum rak k + for each ρ Θ, Case(2): (ii) The limitig value lim tr(p j s H Σ ) 0 for some j, ad the limitig vector [lim tr(p s H Σ ),..., lim tr(p mh s Σ )] is liearly idepedet of the limitig vector [ lim tr(h P H Σ ),, lim tr(h P m H Σ )]. (i) The limitig matrix lim Q R X has full colum rak k for each ρ Θ, (ii) W M, (iii) The vector αs defied above do ot have a liear combiatio with some oliear ozero costat coefficiets c ad c 2 i the form of α ρ c +α λ c 2 +α ρ 2c 2 +α λ 2c2 2 +α ρλc c 2 + α ρ 2 λc 2 c 2 + α ρλ 2c c α ρ 2 λ 2c2 c2 2 = 0. The first coditio i Case() esures the idetificatio of β 0 ad λ 0 from the liear momet fuctio. The secod coditio i Case() provides the idetificatio for ρ 0 from the quadratic momet fuctios. I Case(2), the quadratic momet fuctios esures the idetificatio of ρ 0 ad λ 0 uder the coditio of W M. Oce λ 0 is idetified, the idetificatio of β 0 follows from the first coditio i Case(2). 32 See also the idetificatio assumptios i Lee ad Liu (200b) ad Li ad Lee (200), which have a similar structure. 7

20 Let Ω = E[g (θ 0 )g (θ 0 )]. By usig Lemma 2 i Appedix A, the variace covariace matrix of the set of momet fuctios ca be obtaied. tr(σ P (P Σ + Σ P )) tr(σ P (P mσ + Σ P m )) 0 tr(σ P 2 (P Σ + Σ P )) tr(σ P 2 (P mσ + Σ P m )) 0 Ω =.... tr(σ P m (P Σ + Σ P )) tr(σ P m (P mσ + Σ P m )) Q Σ Q (4.24) The variace-covariace ( matrix Ω has the same structure as the oe i Li ad Lee (200). Let Γ = E g(θ0 ) ). A straightforward applicatio of matrix calculus yields (4.25). Elemets of Γ θ are fuctios of matrices that are uiformly bouded i absolute value i row ad colum sums so that the order of elemets is either O() or O(), which i tur implies that Γ is bouded. tr(σ H P s ) tr(σḡ P s ) 0 tr(σ H P 2 s ) tr(σḡ P2 s ) 0 Γ =.... (4.25) tr(σ H P m) s tr(σ Ḡ Pm) s 0 0 Q R G X β 0 Q R X Let Ψ Ψ be a arbitrary o-stochastic weightig matrix for the GMM objective fuctio. The weightig matrix plays the role of a metric by which the sample momet fuctios are made as close as possible to zero. Assume that Ψ coverges to a costat matrix Ψ 0 that has full rak, ad lim Ψ Γ exists ad has full rak (Hase, 982). 33 The followig propositio shows that the geeric GMM estimator based o the set of momet fuctios g (θ 0 ) = ( ε P ε,..., ε P m ε, ε ) Q with geeral P j s ad Q is cosistet ad has a asymptotic ormal distributio. 34 Propositio. Suppose P j P 2 for j =,..., m ad Q is liear IV matrix. Uder Assumptios -6, the estimator ˆθ derived from the objective fuctio mi θ Θ g (θ)ψ Ψ g (θ) is a cosistet robust GMM estimator (RGMME) of θ 0. It has a asymptotic ormal distributio, amely (ˆθ θ 0 ) d N(0, Υ), (4.26) where Υ = lim [ Γ Ψ Ψ Γ ] Γ Ψ Ψ Ω Ψ Ψ Γ [ Γ Ψ Ψ Γ ]. 33 For our case, matrices {Ψ } have dimesios equal or bigger tha k + 2. Let Q be (k + ) liear IV matrix. I that case, g : R R k+2 R m+k+, where (m + k + ) is the umber of orthogoality coditios. The, matrices {Ψ } are dimesioed (m + k + ) (m + k + ). See Assumptio 2.5 i Hase (982, p.033). 34 The details of the proofs for all propositios are give i Appedix C. 8

21 The estimator i Propositio is the geeric GMM estimator cosidered i Hase (982). 35 The variace-covariace matrix of the RGMME of Propositio is a fuctio of ukow terms Γ ad Ω. As usual, cosistet estimates of these terms ca be obtaied from a iitial cosistet estimator of θ 0. I the followig propositio, cosistet estimators for Γ ad Ω are give. Propositio 2. Let ˆε i be the residual of the model based o cosistet iitial estimates of θ 0 ad deote ˆΣ = Diag(ˆε 2, ˆε2 2,..., ˆε2 ). The, uder the assumed regularity coditios, () ˆΩ Ω = o p (), (2) ˆΓ Γ = o p (). The proof of Propositio 2 utilizes the facts that quadratic momet matrices are uiformly bouded i absolute value i row ad colum sums ad disturbace terms have uiformly bouded fourth momets. These two properties esure that the elemets ivolvig the trace operator i ˆΩ ad ˆΓ coverge i probability to the correspodig elemets of Ω ad Γ. The remaiig elemet i ˆΩ is Q ˆΣ Q. The asymptotic argumet for this term is i lie with that of White (980). Uder certai regularity coditios, White (980) shows that X ˆΣ X coverges almost surely to X Σ X, where ˆε i is a cosistet estimate of ε i. I Propositio, the GMM estimator is derived from the objective fuctio with a arbitrary weightig matrix. It is clear that differet choices of weightig matrices give rise to GMM estimators with differet asymptotic covariace matrices. The optimal estimator is the oe that has a asymptotic covariace matrix at least as small as that of ay other GMM estimator. Hase (982) shows that the optimal GMM estimator is based o the weightig matrix Ψ Ψ = Ω. This matrix plays a promiet role for the optimal GMM estimator uder the followig regularity coditio. Assumptio 7: The limitig matrix lim Ω exists ad is osigular. I (4.24), otice that the terms i Ω are fuctios of matrices that are uiformly bouded i absolute value i row ad colum sums. For example, a geeric term is tr(σ P i (P j Σ + Σ P j ) which has a order of O(). Therefore, Ω is order of O() which implies that Ω is bouded. Propositio 2 yields a cosistet estimator ˆΩ for this optimal weightig matrix. The ext propositio shows that the optimal GMM estimator based o the weightig matrix ˆΩ is cosistet ad asymptotically ormal. Propositio 3. Uder Propositio 2 ad Assumptio -7, the optimal robust GMM estimator 35 The structure of the asymptotic variace-covariace matrix Υ is also geeric ad it has the same structure with the oe give i Theorem 3. of Hase (982, see p. 042). 9