Essays On Robust Estimators For Non-Identically Distributed Observations In Spatial Econometric And Time Series Models

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1 City Uiversity of New York CUNY CUNY Academic Works Dissertatios, Theses, ad Capstoe Projects Graduate Ceter Essays O Robust Estimators For No-Idetically Distributed Observatios I Spatial Ecoometric Ad Time Series Models Suleyma Taspiar Graduate Ceter, City Uiversity of New York How does access to this work beefit you? Let us kow! Follow this ad additioal works at: Part of the Ecoomics Commos Recommeded Citatio Taspiar, Suleyma, "Essays O Robust Estimators For No-Idetically Distributed Observatios I Spatial Ecoometric Ad Time Series Models" 204. CUNY Academic Works. This Dissertatio is brought to you by CUNY Academic Works. It has bee accepted for iclusio i All Dissertatios, Theses, ad Capstoe Projects by a authorized admiistrator of CUNY Academic Works. For more iformatio, please cotact deposit@gc.cuy.edu.

2 ESSAYS ON ROBUST ESTIMATORS FOR NON-IDENTICALLY DISTRIBUTED OBSERVATIONS IN SPATIAL ECONOMETRIC AND TIME SERIES MODELS by Süleyma Taşpıar A dissertatio submitted to the Graduate Faculty i Ecoomics i partial fulfillmet of the requiremets for the degree of Doctor of Philosophy, The City Uiversity of New York 204

3 c 204 Süleyma Taşpıar All Rights Reserved ii

4 This mauscript has bee read ad accepted for the Graduate Faculty i Ecoomics i satisfactio of the dissertatio requiremet for the degree of Doctor of Philosophy. Dr. Wim Vijverberg Date Chair of Examiig Committee Dr. Merih Uctum Date Executive Officer Dr. Wim Vijverberg Dr. Merih Uctum Dr. Fracesc Ortega Supervisory Committee THE CITY UNIVERSITY OF NEW YORK iii

5 Abstract ESSAYS ON ROBUST ESTIMATORS FOR NONIDENTICALLY DISTRIBUTED OBSERVATIONS IN SPATIAL ECONOMETRIC AND TIME SERIES MODELS by Süleyma Taşpıar Adviser: Professor Wim Vijverberg This thesis proposal cosists of three essays o the estimatio methods ad applicatios of spatial ecoometric models ad oe essay o the geeralized autoregressive coditioally heteroskedastic GARCH-type models i fiacial time series. The first essay discusses the heteroskedasticity robust geeralized method of momets estimator RGMME for the spatial models that allow for spatial depedece i both the depedet variable ad the disturbace term SARAR,. First, we show that the maximum likelihood estimator MLE is geerally icosistet i the presece of ukow heteroskedasticity. The, we exted robust GMM approach i Li ad Lee 200 to SARAR,. The large sample properties are rigorously studied ad preseted for the RGMME. Through a comprehesive Mote Carlo study, we compare the fiite sample properties of the RGMME with some other estimators proposed i the literature. The secod essay focuses o the GMM estimatio of the spatial autoregressive models which impose a movig average process for the disturbace term SARMA. We exted the best GMM estimator BGMME of Liu et al. 200 to the SARMA models ad provide the best set of istrumets for the SARMA, ad the SARMA0, specificatios. The large sample properties are rigorously studied ad preseted for the BGMME. The fiite sample properties are ivestigated through a extesive Mote Carlo study. To cofirm our results from the Mote Carlo study, we replicate the results for the SARMA, specificatio i Behres et al. 202 i a empirical illustratio. The third essay ivestigates the effect of foreig direct ivestmet FDI o ecoomic growth through a spatially augmeted Solow growth model. The curret literature o the relatioship betwee FDI ad ecoomic growth uses caoical cross-coutry growth regressio specificatios that are derived from the textbook Solow growth model for closed ecoomies. iv

6 We claim that these specificatios caot reflect the relatioship betwee ecoomic growth ad FDI, because they model each coutry as a isolated islad that does ot iteract with the rest of the world. O the other had, a spatially augmeted Solow growth model allows for techological iterdepedece amog coutries through spatial exteralities. The modified growth model yields regressio specificatios that properly accout for spatial autocorrelatios. We costruct a pael of 85 coutries for the period ad estimate the modified specificatios with the tools from spatial ecoometrics. Our fidigs idicate that FDI iflows have a sigificat positive effect o the growth rate of host coutries. The fial essay proposes a flexible distributio for the maximum likelihood estimatio of the GARCH-type time series models. The ew distributio ca better accout for the potetial skewess ad leptokurticity i the drivig oise sequece. We study the large sample properties of the ew estimator followig the methodology preseted i Fracq ad Zakoïa To ivestigate the fiite sample properties of the ew estimator, we first coduct a Mote Carlo study. Furthermore, to test the relative out-of-sample predictive power of the ew estimator, we test for its predictio power o two data sets usig the methods described i White 2000 ad Hase et al v

7 Ackowledgemets This dissertatio would ot have bee possible without the ecouragemet ad support of Dr. Wim Vijverberg, Dr. Merih Uctum, ad Dr. Fracesc Ortega. 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 , CNS ad ACI-263. vi

8 Cotets Preface List of Tables iv xi List of Tables xi List of Figures xiii List of Figures xiii GMM Estimatio of Spatial Autoregressive Models with Autoregressive ad Heteroskedastic Disturbaces with Osma Doğa. Itroductio The Model Specificatio ad Theoretical Motivatio GMM Estimatio of Spatial Autoregressive Models Estimatio Approach uder Ukow Heteroskedasticity The Icosistecy of Maximum Likelihood Estimator Robust GMM Estimatio of SARAR, Mote Carlo Experimets Desig Simulatio Results Coclusio Appedix Some Useful Lemmas The Icosistecy of the ML Estimator Proof of Mai Propositios vii

9 .7.4 Simulatio Results A Efficiet GMM Estimator for A Spatial Cliff-Ord-Type Model with Movig Average Disturbaces Itroductio Spatial Depedece Specificatios for the Disturbace Term Model Specificatio ad Assumptios Estimatio of the SARMA0, FGLS ad GMM estimatio Oe-Step GMM Estimatio Oe-Step GMM Estimatio of SARMA, Mote Carlo Simulatios Desig Results Empirical Illustratio The Model ad Ecoometric Specificatio Data ad the Results Coclusio Appedix Lemmas Local Idetificatio of the Parameters Aalytical derivatio of best quadratic momet matrix P Proof of Propositios Dual Gravity Model Mote Carlo Tables SARMA Results The Effect of FDI o Ecoomic Growth: A Spatial Ecoometric Approach with Osma Doğa Itroductio Modified Solow Growth Model ad Cross-Coutry Regressio Specificatios. 5 viii

10 3.3 Descriptio of Data Estimatio Approach Estimatio of Spatial Dyamic Pael Data Model Iterpretatio of Parameter Estimates Empirical Results Estimatio Results for No-spatial Pael Data Models Estimatio Results for SDPD Model Robustess Exercises Coclusio Appedix Spatially Augmeted Growth Model With Huma Capital Data ad Sample of Coutries Robustess Exercises for No-spatial Models Margial Effects for Models i Table Mote Carlo Results for Bias Corrected QMLE GTλ: A flexible distributio for GARCH-type models Itroductio GTλ Family GTλ-GARCH Large Sample Results Fiite Sample Properties Empirical Applicatios Predictio RC ad SPA tests: Backgroud Realized Variace Data Coclusio Appedix Large Sample Results Empirical Results ix

11 Bibliography 256 total 27 x

12 List of Tables. λ 0, β 0, β 20, β 30, ρ 0 = 0.8, 0.7, 0.4,.2, λ 0, β 0, β 20, β 30, ρ 0 = 0.3, 0.7, 0.4,.2, λ 0, β 0, β 20, β 30, ρ 0 = 0, 0.7, 0.4,.2, λ 0, β 0, β 20, β 30, ρ 0 = 0.3, 0.7, 0.4,.2, λ 0, β 0, β 20, β 30, ρ 0 = 0.8, 0.7, 0.4,.2, λ 0, β 0, β 20, β 30, ρ 0 = 0.3, 0.7, 0.4,.2, 0.3, = λ 0, β 0, β 20, β 30, ρ 0 = 0.8, 0.7, 0.4,.2, 0.3, = SARMA Results: λ, = SARMA Results: λ, = SARMA Results: ρ, = SARMA Results: ρ, = SARMA Results: β, = SARMA Results: β, = SARMA Results: β 2, = SARMA Results: β 2, = SMA Results: β ad β SMA Results: ρ SARMA regressios Nospatial Model Estimatios Bias Corrected QML Estimatio Results Bias Corrected QML Estimatio Results Uified Approach Margial Effects for Models i Table xi

13 3.5 Bias Corrected QML Estimatio Results for SDPD Model Bias Corrected QML Estimatio Results Uified Approach Margial Effects for Models i Table Margial Effects for Models i Table Variable Defiitios ad Descriptive Statistics Nospatial Model Estimatios Margial Effects for Models i Table SDPD Mote Carlo Results SDPD Mote Carlo Results Uified Approach Mote Carlo Results: = {00, 250} Mote Carlo Results: = {500, 000} Estimatio Results LM test results for serial correlatio i squared log returs LM test for serial correlatio i estimated squared iovatios Miimum Loss SPA ad RC Tests for log exchage returs SPA ad RC Tests for log stock returs Estimatio Results: DM-US $ Estimatio Results: DM-US $ Estimatio Results: IBM Estimatio Results: IBM xii

14 List of Figures. Weight Matrices e = : RMSE of λ ad ρ e = 2: RMSE of λ ad ρ FDI Iflows ad Stocks The GTλ desity for λ, λ 2 = { 0.3, 0, 0.3} Feasible parameter regios Specificatios for the scaled iovatios QQ plots for log returs of selected fiacial market variables Estimated volatility: DM-$ Returs Estimated volatility: IBM Stock Returs Relative loss from GTλ-GARCH with µ t = m 0 for DM-US $ Relative loss from GTλ-GARCH with µ t = 0 for IBM xiii

15 GMM Estimatio of Spatial Autoregressive Models with Autoregressive ad Heteroskedastic Disturbaces with Osma Doğa

16 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. Author Keywords: Spatial autoregressive models, Ukow heteroskedasticity, Robustess, GMM, Asymptotics, MLE JEL classificatio codes: C3, C2, C3

17 . 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, 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. Kelejia ad Prucha, 998, Das et al., 2003, Kelejia ad Prucha, 200. Several solutios have bee suggested to overcome the computatioal burde of the ML method Ord, 975, Pace ad Barry, 997b,a, Barry ad Pace, 999, Simirov ad Aseli, 200, LeSage ad Pace, 2004, 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, Lee, 2003, 2007a, Kelejia ad Prucha, 2007, 200. The spatial structure of regressio equatios motivate the selectio of the istrumets which are usually costructed from the exogeous variables ad spatial weight ma- For the Bayesia MCMC approach, see Lesage 997, Paret ad Lesage 2007 ad LeSage ad Pace

18 trices. 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 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 et al 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 et al. 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,b, Li ad Lee 200, Liu et al. 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 idepedet ad idetically distributed i.i.d. with a ormal desity. 4

19 Whe disturbaces are simply i.i.d., Liu et al. 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 SARARp,q. Most of the estimatio methods metioed above are valid uder the assumptio that the disturbace terms are i.i.d. I may regressio applicatios, heteroskedasticity is likely to 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 SARARp,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 coditios 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 2 For a example, see the empirical applicatio i Li ad Lee For a robust 2SLS estimator of SARAR,0, see Aseli

20 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,. where Y is a vector of observatios for the depedet variable, X is a k matrix of ostochastic exogeous variables, W ad M are spatial weight matrices of kow costats with zero diagoal elemets, ad ε is a 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 4 For the motivatio of model specificatios, see Aseli 988, 2006 ad LeSage ad Pace

21 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, the variables i. 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 G = R G R. Next, assumptios that are required for the asymptotic properties of estimators are elaborated ad the their iterpretatios are cosidered for.. 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, 5 Elhorst 200 ames the model with spatial spillovers i the depedet variable, exogeous variable ad disturbace 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 See Kelejia ad Prucha For the variace of the quadratic form i ε, see Lemma

22 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. diverget with the property that lim 0 h The sequece {h } ca be bouded or = 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 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, 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 et al 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 h lim = 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 Assumptios. ad.2 is motivated to cotrol spatial autocorrelatios i the model at a tractable level Kelejia ad Prucha, Assumptio.2 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 8 For examples of this kid of weight matrices, see Case 99, For a defiitio ad some properties of uiform boudedess see Kelejia ad Prucha

23 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. S 200. The uiform boudedess of λ ad R ρ is required for the ML estimator ot for the GMM estimator Liu et al., 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 Simirov 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 gθ 0 = ε P ε,..., ε P m ε, ε Q. The momet fuctios ivolvig 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 r with r 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 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 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 Note that P 2 is a subclass of P, i.e., P 2 P. 9

24 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. The IV matrix Q has r k + liearly idepedet colums ad its elemets are uiformly bouded. P j matrices 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 G = W S X β 0 +W S R ε = G X β 0 +G R ε where = W I λ 0 W exists by Assumptio.2. Thus, W Y is a fuctio of a o-stochastic term G X β 0 ad a stochastic term G R ε. Lee 200a, 2007a, Liu et al. 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. 4 Here, P \ P 2 deotes set-theoretic differece of P ad P 2. 0

25 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 Pj Eε ε = 0 for P j s from either P or P 2. 5 For both 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. Therefore, for the selectio of P j s, the asymptotic efficiecy of estimators eeds to be cosidered. Liu et al. 200 ad Lee ad Liu 200b provide the best selectio of P j P i the case of a SARAR, ad SARARp,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 θp ε θ, ε θp 2 ε θ, ε θq be the set of sample momet fuctios. Liu et al. 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 et al. 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 disturbaces 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 = D σ 2,..., σ2 be the diagoal variace matrix of the disturbace terms. If Pj P \ P 2 for ay j =,..., m, the the covariace E ε P j ε = tr P Eε ε = tr P j Σ 0. O the other had, Pj with zero diagoal property is still available for the formatio of the quadratic momets, sice tr P Eε ε = tr P j Σ = 0 for ay P j P 2. Thus, the class of matrices with zero diagoal elemets provides robustess for 5 tr returs the sum of the diagoal elemets of a iput matrix. 6 Note that cov Q, ε = 0k+ ad covp jε, ε = 0. 7 Liu et al. 200 also cosider the best GMM estimatio for the case of a SARAR,0 ad a SARAR0,.

26 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 ςp ε ς, ε ςq 2, where P = G D G P2 ad Q 2 = [ G X β 0, X ]. 8 The optimal robust GMM estimator derived from mi ς Θ g ςˆω 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 DG 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 trm M with v = trm 2, ad 2 P 2 = M M. Whe the iovatios 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 D 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 θp ε θ, ε θp 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. optimal GMM estimator of ρ 0 is defied as ˆρ = arg mi ρ g ρ, ς + The ˆΨ g ρ, ς, where ˆΨ is a estimator of the variace matrix of the limitig distributio of the ormalized sample 8 D is a operator that creates a matrix from the diagoal elemets of a iput matrix. 2

27 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 λy R ˆρ X β. The the optimal two-step GMM estimator of ς 0 is defied by ˆς = arg mi ς g 2ˆρ, ς Υ g 2 ˆρ, ς, where Υ = Q 3 Q As illustrated, the estimatio approach i Kelejia ad Prucha 998, Kelejia ad Prucha 200 ad Drukker et al. 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 twostep 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 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 et al For a differet approach of the GMM estimatio method, see Coley

28 .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. 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. uder the assumptio that disturbaces are i.i.d. N0, σ0 2 is give by l L ζ = 2 l2π 2 lσ2 + l S λ + l R ρ.2 2σ 2 [ S λy X β ] R ρr ρ [ S λy X β ]. For otatioal simplicity, let R ρx = X ρ, ad M ρ = I P ρ, where P ρ = X ρ [ X ρx ρ ] X ρ, ad δ = ρ, λ. Note thatx ρm ρ = 0 k ad M ρx ρ = 0 k. The solutio of the first order coditios for β ad σ 2 yields the followig ML estimators. 23 ˆβ δ = [ X ρx ρ ] X ρr ρs λy.3a ˆσ 2 δ = ε θε θ = Y S λr ρm ρr ρs λy..3b 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 ρs λy = R ρx β + ε. Substitutio of R ρs λy = R ρx β + ε ito ˆσ δ 2 yields ˆσ δ 2 = ε M ρε. For the asymptotic argumet of this sectio, we modify Assumptio.3. Assumptio.6. 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 ρr ρx 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 23 The first order coditios from.2 are give i Appedix

29 give i Appedix.7.2. From.3b, we have plim ˆσ δ 2 0 = plim ε ε plim 2 ε X X X X ε..4 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 δ 2 0, amely, ˆσ δ 2 0 = i= σ2 i + o p. 24 The cocetrated loglikelihood fuctio is obtaied by substitutig ˆβ δ ad ˆσ δ 2 ito.2: l L δ = 2 l2π + 2 lˆσ2 δ + l S λ + l R ρ..5 The MLE ˆδ = ˆλ, ˆρ is the extremum estimator derived from the cocetrated loglikelihood fuctio. The first order coditios of the cocetrated loglikelihood 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,.6 ρ = ˆσ δ 2 2ˆσ δ 2 tr G λ,.7 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 proba- l L bility to zero i.e., plim δ 0 δ = 0. This ecessary coditio for the cosistecy of the ML estimator of δ 0 is l L δ 0 = δ i= H,iiσi 2 i= σ2 i i=g.ii σ 2 i i= σ2 i tr H + op tr..8 G + op Deote σ 2 = i= σ2 i, H = tr H = i= H,ii, ad G = tr G = i= G,ii 24 For the asymptotic argumet see Appedix

30 where G = R G R. The,.8 ca be writte i a more coveiet form 25 l L δ 0 = δ = i= i= H,ii H σ i 2 σ2 + o σ 2 p G.ii G σi 2 σ2 σ 2 tr G G + op cov H,ii, σi 2 + o σ 2 p,.9 + o σ 2 p cov G,ii, σ 2 i which shows that the ML estimators ˆλ ad ˆρ are icosistet uless cov H,ii, σi 2 = 0 σ 2 ad cov G,ii, σ 2 i σ 2 = 0. The icosistecy of ˆλ ad ˆρ depeds o the covariace betwee variaces of elemets of the disturbace terms ad diagoal elemets of H ad G. It is obvious that whe ε is homoskedastic, l L δ 0 δ is o p as σi 2 = σ2 for i =,...,. This result also holds for the trivial case of ρ 0 = λ 0 = 0. Ituitively, the result i.9 idicates that the cocetrated loglikelihood fuctio is ot maximized at the true parameter vector whe disturbace terms have ukow heteroskedasticity. The ML estimator of β 0 i.3a 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 G X β 0 + λ 0 ˆλ ρ0 ˆρ D ˆρ X H s G X β 0 + λ 0 ˆλ ρ0 ˆρ 2D ˆρ X H H G X β 0 + o p,.0 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., the result i the secod row of.9 simplifies to l L λ 0 λ = cov G,ii, σi 2 + o σ 2 p sice G = G. The term D ˆρ X G X β 0 i.0 simplifies to X X X G X β 0 so that ˆβ ˆλ = β 0 + λ 0 25 Note that σ 2 = i= σ2 i is the average of the variace of the disturbace terms, ad tr G G = For the asymptotic argumet, see Appedix

31 ˆλ X X X G X β 0 + o p, which is the exact result stated i Li ad Lee 200. The cocetrated loglikelihood 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. with λ 0 = 0 is called the special error model SEM or SARAR 0, i the literature LeSage ad Pace, 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.9 is geerally ot zero. The MLE of β 0 for the SEM is give by ˆβ ρ = D ρx ρr ρy for a give ρ, which is the OLS estimator from the artificial regressio R ρy = R ρx β + ε. Substitutig Y = X β 0 + R ε ito ˆβ ρ yields ˆβ ρ = β 0 + D ρx ρr ρr ε. Uder Assumptio.6, 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. 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 loglikelihood fuctio of this model with respect to λ is l L λ 0 λ = ˆσ λ Y W S 2 λy tr G λ, where ˆσ λ 2 = ε λε λ with ε λ = S λy. At λ 0, ˆσ λ 2 0 = i= σ2 i + o l L p. The, λ 0 λ = cov G,ii, σi 2 + o p by the same asymptotic argumet applied i the derivatio of.9. This result is the same as with the oe obtaied i Li ad Lee 200 for the case of a SARAR,0. I the special case, where the spatial weight matrices are the same ad the true parameter σ 2 values λ 0 ad ρ 0 are equal, the covariace terms i.9 are equal. 28 I this special case, 27 For the asymptotic argumet, see Appedix I this case, W = M ad R = S so that H = W S G. 7 = G ad G = S G S = S W S S =

32 the result i.9 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.9 are zero. A obvious case is whe both G 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 G 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. Besides its computatioal burde, the cosistecy of ML estimator is ot esured..4.2 Robust GMM Estimatio of SARAR, I this sectio, the robust GMM estimatio method suggested by Li ad Lee 200 is exteded for the model i.. For the estimatio, we cosider the set of populatio momet fuctios g θ 0 = ε P ε,..., ε P m ε, ε Q where P j 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. satisfies Assumptios..6. 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 = 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

33 0. 30 For ay value of the parameter vector θ Θ, cosider the expectatio of the set of momet fuctios i.: E ε θp ε θ E g θ E ε θp 2 ε θ =... E ε θp m ε θ E Q ε θ From., ε θ ca be writte i terms of the model parameters i the followig way: ε θ = R ρs λs R ε + R ρ [ S λs X β 0 X β ] = K δk ε + R ρk ς,.2 where K δ = R ρs λ, K = R S, k ς = [ S λs X β 0 X β ], ad ς = λ, β are itroduced for otatioal simplicity. Substitutig.2 ito. ad takig expectatio yield k ςr ρp R ρk ς + tr Σ K k ςr ρp 2 R ρk ς + tr Σ K Eg θ =. k ςr ρp m R ρk ς + tr Σ K Q R ρk ς K δp K δk K δp 2 K δk K δp m K δk..3 where Σ = D σ 2 i,..., σ2. The idetificatio of the parameter vector θ0 ca be verified from E g θ = 0, i.e., θ 0 is idetified if θ 0 is the uique solutio for Eg θ = 0. The term Q R ρk ς i.3 ca be writte more explicitly as Q R ρk ς = Q R ρ [ X, G X β 0 ] [ β 0 β λ 0 λ ] = 0 r..4 The uique solutio of.4 is β 0, λ 0 if the matrix, [ Q R ρx, Q R ρg X β 0 ] 30 See Lemma 2.3 i Newey ad McFadde

34 has full colum rak k + for each possible value of ρ Θ by the virtue of Lemma. of Appedix.7.. 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 ρs λ = R ρ ρ 0 M S λ λ 0 W = K + ρ 0 ρm S + λ 0 λr W + ρ 0 ρλ 0 λm W..5 Cosider the terms with P j, k ςr ρr ρp j k ς + tr Σ K K δp j K δk. 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 δp j K δk = ρ0 ρ tr P s j M R Σ + ρ 0 ρ 2 tr R M P j M R Σ = 0,.6 where P s j = P j + P j. There are two roots for ρ i.6. The first root is the true parameter value ρ 0, ad the secod root is ρ = ρ 0 + tr P s j M R Σ tr R M P j M R..7 Σ There are three cases i which ρ 0 is the uique root. If tr P s j 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.7 must be differet for each P j for j =,..., m so that the secod root 3 K δ ca be decomposed by usig idetities S λ = S λ λ 0 W ad R ρ = R ρ ρ 0 M. 20