Essays in Spatial Econometrics: Estimation, Specification Test and the Bootstrap. Dissertation

Transcription

1 Essays i Spatial Ecoometrics: Estimatio, Specificatio Test ad the Bootstrap Dissertatio Preseted i Partial Fulfillmet of the Requiremets for the Degree Doctor of Philosophy i the Graduate School of The Ohio State Uiversity By Fei Ji, M.A. Graduate Program i Ecoomics The Ohio State Uiversity 203 Dissertatio Committee Lug-fei Lee, Advisor Stephe Cosslett Robert de Jog

2 Copyright by Fei Ji 203

3 Abstract This dissertatio cosists of three chapters coverig the followig topics i spatial ecoometrics: estimatio, specificatio ad the bootstrap. I Chapter, we first geeralize a approximate measure of spatial depedece, the AP LE statistic (Li et al., 2007), to a spatial Durbi (SD) model. This geeralized AP LE takes ito accout exogeous variables directly ad ca be used to detect spatial depedece origiatig from either a spatial autoregressive (SAR), spatial error (SE) or SD process. However, that measure is ot cosistet. Secodly, by examiig carefully the first order coditio of the cocetrated log likelihood of the SD (or SAR) model, whose first order approximatio geerates the AP LE, we costruct a momet equatio quadratic i the autoregressive parameter that geeralizes a origial estimatio approach i Ord (975) ad yields a closed-form cosistet root estimator of the autoregressive parameter. With a specific momet equatio costructed from a iitial cosistet estimator, the root estimator ca be as efficiet as the MLE uder ormality. Furthermore, whe there is ukow heteroskedasticity i the disturbaces, we derive a modified AP LE ad a root estimator which ca be robust to ukow heteroskedasticity. The root estimators are computatioally much simpler tha the quasi-maximum likelihood estimators. I Chapter 2, we cosider the Cox-type tests of o-ested hypotheses for spatial autoregressive (SAR) models with SAR disturbaces. We formally derive the asymptotic distributios of the test statistics. I cotrast to regressio models, we show that the Coxtype ad J-type tests for o-ested hypotheses i the framework of SAR models are ot asymptotically equivalet uder the ull hypothesis. The Cox test i o-spatial settig has bee foud ofte to have large size distortio, which ca be removed by the bootstrap. Cox-type tests for SAR models with SAR disturbaces may also have large size distortio. We show that the bootstrap is cosistet for Cox-type tests i our framework. Performaces ii

4 of the Cox-type ad J-type tests as well as their bootstrapped versios i fiite samples are compared via a Mote Carlo study. These tests are of particular iterest whe there are competig models with differet spatial weights matrices. Usig bootstrapped p-values, the Cox tests have relatively high power i all experimets ad ca outperform J-type ad several other related tests i some cases. Chapter 3 is cocered with the use of the bootstrap for spatial ecoometric models. We show that the bootstrap for spatial ecoometric models ca be studied based o liear-quadratic (LQ) forms of disturbaces. By provig the uiform covergece of the cumulative distributio fuctio for LQ forms to that of a ormal distributio, we show that the bootstrap is geerally cosistet for test statistics that ca be approximated by LQ forms, which iclude Mora s I, Cox-type ad spatial J-type test statistics. Possible asymptotic refiemets of the bootstrap for spatial ecoometric models may be studied based o some asymptotic expasios for LQ forms. We discuss two cases: whe the disturbaces are ormal, we directly show the existece of Edgeworth expasios for LQ forms ad apply the result to show that the bootstrap for Mora s I ca provide asymptotic refiemets; whe the disturbaces are ot ormal, we show the existece of a oe-term asymptotic expasio of LQ forms based o martigales, which sheds light o the secod-order correctess of the bootstrap for LQ forms. iii

5 Dedicatio Dedicated to my wife, Yuqi Wag, ad my parets, Chagrog Ji ad Guifag Ya iv

6 Ackowledgemets I would like to express my deepest gratitude to my advisor, Lug-fei Lee, for his ivaluable guidace, ecouragemet ad geerous help i all aspects of my life. He is always willig to sped his precious time with me o research issues ad givig suggestios o other matters. His great ethusiasm ad rigorous attitudes towards research are life-log wealth for me. I am grateful to my dissertatio committee members, Professor Robert de Jog ad Professor Stephe Cosslett, for their valuable commets ad ecouragemet. I would also like to thak Professor Jaso Blevis for servig o my third year paper committee ad givig valuable suggestios. I thak Professor Lucia Du for her solicitude ad helpful advice ad Professor Daeho Kim for his woderful suggestios o my job market paper presetatio. May fellow classmates ad frieds have bee helpig ad supportig me, which I wish to thak here. My special thaks go to Feg Guo, Ya Bao ad Feg Zhou. Last but ot the least, I would like to thak my wife, Yuqi Wag, ad my parets, Chagrog Ji ad Guifag Ya, for their love, support ad uderstadig. v

7 Vita Jue B.S., Ecoomics, Huazhog Uiversity of Sciece ad Techology December M.A., Ecoomics, The Ohio State Uiversity 2008 Preset Graduate Teachig Associate, The Ohio State Uiversity Fields of Study Major Field: Ecoomics vi

8 Table of Cotets Abstract ii Dedicatio iv Ackowledgemets v Vita vi List of Tables x List of Figures xi Chapter : Approximated Likelihood ad Root Estimators for Spatial Iteractio i Spatial Autoregressive Models Itroductio A Root Estimator A Root Estimator: Homoskedastic Case A Root Estimator: Heteroskedastic Case Mote Carlo Study Coclusio Chapter 2: Cox-type Tests for Competig Spatial Autoregressive Models with Spatial Autoregressive Disturbaces Itroductio Cox-type Tests Relatioship ad Compariso betwee the Cox-type ad J-type Tests Mote Carlo Study Empirical Illustratio Coclusio Appedix 2. Notatios ad Expressios vii

9 Appedix 2.2 The Exteded Wald ad Exteded Score Tests Appedix 2.2. The Exteded Wald Test Appedix The Exteded Score Test Chapter 3: O the Bootstrap for Spatial Ecoometric Models Itroductio Statistics i Spatial Ecoometrics ad LQ Forms Cosistecy of the Bootstrap Mora s I Spatial J-type Tests Cox-type Tests Asymptotic Refiemets Normal Disturbaces No-ormal Disturbaces Coclusio Appedix 3. Assumptios Appedix 3.. Assumptios for Mora s I Appedix 3..2 Assumptios for the Spatial J Tests: J Appedix 3..3 Assumptios for the Spatial J Tests: J Refereces Appedix A: Lemmas Appedix A.: Geeral Lemmas Appedix A.2: Lemmas for Chapter Appedix A.3: Lemmas for Chapter Appedix A.3.: Lemmas for the SARAR Model Appedix A.3.2 Lemmas for the Spatial J Tests: J Appedix A.3.3 Lemmas for the Spatial J Tests: J Appedix A.3.4 Lemmas for Cox-type Tests Appedix B: Proofs Appedix B.: Proofs for Chapter Appedix B.2: Proofs for Chapter viii

10 Appedix B.3: Proofs for Chapter ix

11 List of Tables Table. Compariso of the computig time, bias, STD ad RMSE of the QMLE, RE sar ad AP LE sar whe the sample size is large Table 2. Sets of Experimets Table 3. Empirical size ad power for experimet set I with ormal disturbaces ad = Table 4. Empirical size ad power for experimet set I with chi-square disturbaces ad = Table 5. Empirical size ad power for experimet set II with ormal disturbaces ad = Table 6. Empirical size ad power for experimet set II with chi-square disturbaces ad = Table 7. Empirical size ad power for experimet set III with ormal disturbaces ad = Table 8. Empirical size ad power for experimet set III with chi-square disturbaces ad = Table 9. Empirical size ad power computed usig asymptotic p-values for experimet set I with = Table 0. Empirical size ad power computed usig asymptotic p-values for experimet set II with = Table. Empirical size ad power computed usig asymptotic p-values for experimet set III with = Table 2. Testig results with a housig data set (Whether H 0 is rejected or ot) x

12 List of Figures Figure. Compariso of the bias, STD ad RMSE of the QMLE, RE sar, AP LE sar, ACME sar, RE sd, AP LE sd ad ACME sd whe the DGP is the SAR model uder homoskedasticity Figure 2. Compariso of the bias, STD ad RMSE of the QMLE, RE sd, AP LE se, AP LE sd ad ACME sd whe the DGP is the SE model uder homoskedasticity Figure 3. Compariso of the bias, STD ad RMSE of the QMLE, RE sar, AP LE sar, ACME sar, RE sd, AP LE sd ad ACME sd whe the DGP is the SAR model uder heteroskedasticity (HD-) Figure 4. Compariso of the bias, STD ad RMSE of the QMLE, RE sar, AP LE sar, ACME sar, RE sd, AP LE sd ad ACME sd whe the DGP is the SAR model uder heteroskedasticity (HD-2) Figure 5. Compariso of the bias, STD ad RMSE of the the QMLE, RE sd, AP LE se, AP LE sd ad ACME sd whe the DGP is the SE model uder heteroskedasticity (HD-) 2 Figure 6. Compariso of the bias, STD ad RMSE of the QMLE, RE sd, AP LE se, AP LE sd ad ACME sd whe the DGP is the SE model uder heteroskedasticity (HD-2) xi

13 Chapter : Approximated Likelihood ad Root Estimators for Spatial Iteractio i Spatial Autoregressive Models. Itroductio Li et al. (2007) propose a closed-form measure of spatial depedece, a approximate profilelikelihood estimator (AP LE), based o a pure spatial autoregressive (SAR) model. Their Mote Carlo experimets for spatial weights matrices defied accordig to a secod-order eighborhood structure o toroidal lattices show that the AP LE provides a better assessmet of the stregth of spatial depedece for data geerated by the pure SAR model tha alterative measures such as Mora s I (Mora, 950). Thus, the AP LE provides a better measure of spatial depedece tha Mora s I for exploratory aalyses. It has bee show i Martellosio (200) that Mora s I has zero power to detect spatial correlatio i a SAR model whe the autoregressive coefficiet is large ad close to oe. Li et al. (202) geeralize the AP LE statistic to the spatial error (SE) model to accout for exogeous variables. As both the SAR ad SE models are costraied forms of the more geeral spatial Durbi (SD) model, a approximate measure for spatial depedece of iterest should accout for exogeous variables directly ad provide a good approximatio to the autoregressive parameter i the SD model. This approximate measure ca be used to detect spatial depedece origiatig from either the SAR, SE or SD model. I this chapter, we exted the AP LE to the SD model, which, similarly to Li et al. (2007), is based o a first order approximatio to the first order coditio of the cocetrated log likelihood of the SD model. The origial AP LE as well as the exteded AP LE from the first order approximatios are ot cosistet for the autoregressive parameter due to a systematic bias. Higher order approximatios of the first order coditio may geerate more accurate measures of the autoregressive parameter, but they ivolve multiple roots ad geerally do ot yield closed-form solutios.

14 Treatig the first order coditio differetly, we obtai a momet equatio quadratic i the autoregressive parameter that geerates a closed-form root estimator. Our proposed root estimator geeralizes a estimator origiated i Ord (975) i a geeral settig. For the quadratic momet equatio, coditios uder which oe of the roots is cosistet will be specified. With a iitial cosistet estimator, a momet equatio ca be desiged to geerate a secod step root estimator which is asymptotically as efficiet as the maximum likelihood estimator (MLE) uder ormality. Oce a estimate of the autoregressive parameter is available, other parameters i a SD model may be estimated by least squares (LS) after applyig a spatial filter to the data o the depedet variable. The modified AP LE ad the root estimator ca be used as measures of spatial depedece or simple estimators for the autoregressive parameter i a SD or SAR model, as the SD model ests the SAR model. The proposed estimators ca further be exteded to possess some robust properties. The origial AP LE i Li et al. (2007) has ot accouted for possible heteroskedasticity i the disturbaces. Li et al. (202) argue that a valid trasformatio ca be applied to the SE model, so the exteded AP LE may be calculated with the trasformed data. This is so whe the heteroskedastic variace has a kow fuctioal form. However, if we do ot kow the form of heteroskedasticity, the data could ot be properly trasformed. A misspecified trasformatio ca lead to errors i iferece. With ukow heteroskedasticity, we may adjust the first order coditio to derive a modified AP LE statistic, which we call a approximate cocetrated momet estimator (ACME), ad we ca also adjust the momet equatio to derive a root estimator that is robust to ukow heteroskedasticity. Existig estimatio methods for SAR (SD) models do ot have a closed form ad are usually computatioally ivolved. 2 The MLE or quasi-maximum likelihood estimator (QM- LE) does ot have a closed form (Aseli, 988; Lee, 2004a). The computatio ivolves the evaluatio of the log-determiat of a square matrix with dimesio equal to the sample size at differet parameter values, so it might be computatioally demadig whe the I this situatio, we ca also easily modify our root estimates to accommodate the kow heteroskedasticity because after proper trasformatio, it results i a SAR model with homogeous disturbaces. 2 Because of the correlatio of the spatially lagged depedet variable with disturbaces, the LS estimator is oly cosistet for a subclass of models (Lee, 2002). 2

15 sample size is large. 3 Some empirical applicatios may create large matrices, for example, the US Cesus Bureau collects data at over 250,000 cesus block group locatios ad the Home Mortgage Disclosure Act data have over 00 millio observatios. Because of the computatioal burde of the MLE, eve with sample sizes that might ot be too large, researchers may tur to less efficiet estimatio methods such as the two stage least squares (2SLS) proposed by Kelejia ad Prucha (998). 4 For example, Helms (202) uses the 2SLS estimatio whe the sample size is 6,638. Lee (2007a) cosiders the geeralized method of momets (GMM) estimatio, which combies the quadratic momets that capture the correlatio across the spatial uits with the liear momets used i the 2SLS approach. Compared to the QMLE, the GMM estimator is computatioally simpler ad it ca be as efficiet as the MLE uder ormality. 5 Lee (2007b) proposes a computatioally simpler GMM for the estimatio of SAR models. The method reduces the GMM estimatio of a vector of parameters ito oliear estimatio of oly the autoregressive parameter. It ca reduce the computatioal burde substatially ad it may be as efficiet as the joit GMM estimator uder certai coditios. But it still does ot geerate a closed-form solutio ad searchig over a parameter space is ecessary. Eve though the GMM avoids computig log-determiats of matrices, searchig over a parameter space with large matrices ivolved 3 Various techiques ad simplificatios have bee proposed to tackle this problem, see, for example, Marti (993), Griffith ad Soe (995), Pace (997), Pace ad Barry (997a,b), Barry ad Pace (999), Griffith (2000), Smirov ad Aseli (200), Pace ad LeSage (2004), Pace ad LeSage (2009) ad Smirov ad Aseli (2009). Eve with these techiques ad simplificatios, the computatio ca be still timecosumig. Alterative simplificatios ofte lead to less accurate estimates. We ote that Pace ad LeSage (2009) propose a samplig approach to estimate the log determiat. Their Mote Carlo study shows that the approach ca be very fast i estimatig the log determiat. Give that the log determiat eeds to be evaluated may times at differet parameter values, the actual time of computig a MLE or QMLE may be much loger. 4 Their model is more geeral oe with both a spatial lag of the depedet variable ad a SAR process i the disturbaces. While the autoregressive parameter for the spatial lag of the depedet variable is estimated by 2SLS, the autoregressive parameter i the disturbace process is estimated by GMM with three momet equatios. 5 Liu et al. (200) ad Lee ad Liu (200) cosider the efficiet GMM estimatio of the regular ad high order SAR models with properly modified momet equatios. Their estimator is as efficiet as the MLE uder ormality ad is more efficiet tha the QMLE otherwise. 3

16 could still be computatioally itesive. Our root estimator is asymptotically as efficiet as the MLE uder ormality sice the desiged secod step momet equatio automatically combies the liear ad quadratic momet coditios i a efficiet way. 6 For SAR models with ukow heteroskedasticity, Li ad Lee (200) study the GMM estimatio where liear ad quadratic momet equatios ivolvig both the autoregressive parameter ad parameters for other exogeous variables are used. 7 Our (robust) root estimator is obtaied with a properly modified ad combied momet equatio quadratic i the autoregressive parameter. Thus, for the closed-form root estimator (see Eq. (.24)), o searchig over a parameter space is eeded. Because of the closed form, the root estimator requires little programmig effort. Our Mote Carlo study shows that the root estimator has similar fiite sample performace as the QMLE uder ormality ad the robust root estimator performs well uder ukow heteroskedasticity. Computig the root estimates oly takes slightly loger time tha computig the AP LE, which is much faster tha computig the QMLE. As the computatioal burde of both the modified AP LE ad the root estimate is miimal, they ca be applied to SAR, SE or SD models for huge data sets. The rest of the chapter is orgaized as follows. Sectio.2 itroduces related models ad develops the AP LE ad ACME; Sectio.3 establishes the cosistecy ad asymptotic distributio of our root estimators i both the homoskedastic ad heteroskedastic cases; Sectio.4 presets some Mote Carlo results; Sectio.5 cocludes. Some lemmas ad proofs are collected i the appedices. 6 Both the modified GMM ad our root estimator reduce the estimatio to that of oly the autoregressive parameter, which might lead to better fiite sample performace whe a bias correctio might be costructed ad applied to this sigle estimate, compared to the case whe a complete vector of parameters are estimated joitly ad the the bias correctio is applied to this vector of estimates. 7 Kelejia ad Prucha (200) also cosider the specificatio ad estimatio of the SAR model with SAR disturbaces that has heteroskedastic iovatios. As i Kelejia ad Prucha (998), the autoregressive parameter for the spatial lagged depedet variable is estimated by 2SLS ad the autoregressive parameter i the disturbace process is estimated by GMM with multiple momet equatios. 4

17 .2 The Models, AP LE ad ACME I this sectio, we itroduce the related models, ad the derive the AP LE for the SD model whe ɛ i s are i.i.d., ad the ACME whe ɛ i s may be oly idepedet but with differet ad ukow variaces. A SAR model is specified as y = ρw y + X β + ɛ, (.) where is the sample size, y is a -dimesioal vector of observatios, W is a spatial weights matrix with a zero diagoal, X is a k matrix of exogeous variables, ɛ = (ɛ,..., ɛ ) with ɛ i s beig idepedet with mea zero, ad ρ is a autoregressive parameter. If the spatial depedece is i the disturbaces istead, we have a SE model which is y = X β + u, u = ρw u + ɛ. (.2) Let I deote the -dimesioal idetity matrix. Pre-multiplyig both sides of Eq. (.2) by (I ρw ) yields y = ρw y + X β + W X ( ρβ) + ɛ, (.3) which is a costraied form of the SD model 8 y = ρw y + X β + W X γ + ɛ. (.4) That is, γ i the SD model (.4) is required to be equal to mius ρ times β for the SE process. A regressio model with the SAR process is just the SD model (.4) with γ = 0, so it is also a costraied form of the SD model. Without the costraits, the SD model may also have a iterest of its ow. The W X as regressors may capture exterality arisig form eighbors characteristics (see, e.g., LeSage ad Pace 2009, p. 30). If W is row-ormalized ad X cotais a itercept term, i.e., X = [l, X ], where l is a -dimesioal colum vector of oes ad X is a (k ) matrix, the W X will geerate a colum vector of oes as W X = [l, W X ]. Coefficiets o these two colum vectors of oes should be collected together. If W is ot row-ormalized, the colums of 8 We use this termiology followig LeSage ad Pace (2009). 5

18 X ad W X are i geeral liearly idepedet. I this case, for l i X, W l is the vector of row sums which is a extra regressor. 9 To make later arrative easier, we write the SD model as y = ρw y + Z θ + ɛ, (.5) where Z = [X, W X ] or Z = [X, W X ], depedig o whether both X ad W X cotai a colum vector of oes or ot, ad θ is the correspodig vector of coefficiets. The Z is d with d = 2k or d = 2k. The AP LE ad ACME are derived for the SD model (.5). Our root estimators are also stated with the settig of Eq. (.5). Whe a SAR model rather tha a more geeral SD model is cosidered, just take Z to be X. Let the true parameters of ρ ad θ be ρ 0 ad θ 0. Whe ɛ i s are i.i.d. (0, σ2 ), the true parameter for σ 2 is σ 2 0 ; whe there is ukow heteroskedasticity, E(ɛ ɛ ) = Diag(σ 2,..., σ2 ) = Σ, where Diag(a ) deote a diagoal matrix with the diagoal elemets beig those of the vector a. Let S (ρ) = I ρw ad G (ρ) = W S (ρ). Deote S = S (ρ 0 ) ad G = G (ρ 0 ) for short. Whe ɛ i s are i.i.d. with variace σ 2, the log likelihood fuctio for the model (.5) is L (ρ, θ, σ 2 ) = 2 l(2πσ2 ) + l S (ρ) 2σ 2 [S (ρ)y Z θ] [S (ρ)y Z θ]. Maximizig the fuctio with a fixed ρ, we obtai the QMLEs for θ ad σ 2 as: ˆθ = (Z Z ) Z S (ρ)y, (.6) ˆσ 2 = y S (ρ)m Z S (ρ)y, (.7) where M Z = I Z (Z Z ) Z. Eqs. (.6) ad (.7) are just like the LS estimators after the spatial filter S (ρ) has bee applied to y. Substitutig these expressios ito the log likelihood fuctio, we have the cocetrated log (or profile) likelihood fuctio of ρ: L (ρ) = 2 [l(2π/) + ] + l S (ρ) 2 l[y S (ρ)m Z S (ρ)y ]. The first order coditio for the maximizatio of the cocetrated log likelihood fuctio is: y S (ρ)m Z W y y S (ρ)m Z S (ρ)y tr[g (ρ)] = 0, (.8) 9 If elemets i W are 0 or as i a etwork, a row sum refers to a outdegree. So i such a case, the outdegrees of idividuals form a explaatory variable. 6

19 where tr(a ) deotes the trace of a square matrix A. Multiplyig both sides by y S (ρ)m Z S (ρ)y yields y S (ρ)m Z W y y S (ρ)m Z S (ρ)y tr[g (ρ)] = 0. (.9) Similar to the derivatio of the AP LE i Li et al. (2007), a approximate measure of ρ ca be obtaied from a first order approximatio of the left-had side of Eq. (.9). Note that tr[g (ρ)] = tr[w (I +ρw +... )] ρ tr(w 2 ) as W has a zero diagoal, the approximatio yields y APLE sd = M Z W y y W M Z W y + y M tr(w Z y 2), (.0) For the coveiece of later referece, we also write dow the AP LE for the SAR model as y APLE sar = M X W y y W M X W y + y M tr(w X y 2), (.) where M X = I X (X X ) X. Furthermore, for a pure SAR process, M Z = I. As y W y = y [(W +W )/2]y ad tr(w 2 ) = λ λ, where λ is the vector of W s eigevalues, Eq. (.0) would reduce to which is that give i Li et al. (2007). y [(W + W )/2]y, y W W y + y y λ λ Usig the same approach, Li et al. (202) derive the AP LE for the SE model as APLE se = y M X [(W + W )/2]M X y A, (.2) where A = y M X W W M X y y M X (W + W )(I M X )(W + W )M X y + y M tr(w X y ) 2. The AP LE for the SAR model i Eq. (.) ad that for the SE model i Eq. (.2) have differet forms. Alteratively, we could use the AP LE based the SD model i Eq. (.0) as a approximate measure of spatial depedece origiatig from either the SAR, SE or SD model. Eq. (.9) ca be rewritte as [ y S (ρ) G (ρ) tr[g (ρ)] I ]M Z S (ρ)y = 0. (.3) Whe there is ukow heteroskedasticity, the expectatio of the left-had side of the above equatio over at the true parameters ρ 0, θ 0, σ 2,..., σ2 does ot coverge to zero 7

20 i geeral, sice { [ E y S = { E G tr(g ) (Z θ 0 + ɛ ) [ G tr(g ) I ] M Z S y } = {[ tr G tr(g ] } ) I M Z Σ = {[ tr G tr(g ] } ) I Σ + o() = [ G tr(g ) I ]ii σ2 i + o(), i= } I ]M Z (Z θ 0 + ɛ ) (.4) by Lemma i Appedix C. Uder ukow heteroskedasticity, we may modify Eq. (.3) ito the followig equatio y S (ρ) [ G (ρ) Diag[G (ρ)] ] M Z S (ρ)y = 0, (.5) which is a valid momet equatio because the zero diagoal of G (ρ) Diag[G (ρ)] implies that the expectatio of the left-had side of the equatio over at ρ 0 coverges to zero. Takig a first order Taylor expasio of the left-had side of Eq. (.5) with ρ ad settig it to zero yield a modified AP LE statistic, which we call ACME: ACME sd = For the SAR model, the ACME is ACME sar = For a pure SAR process, Eq. (.6) simplifies to y M Z W y y W M Z W y + y Diag(W 2 )M Z y. (.6) y M X W y y W M X W y + y Diag(W 2 )M X y. (.7) y W y y W W y + y Diag(W 2 )y. (.8) Eqs. (.6) (.8) ca be used as approximate measures of ρ whe ukow heteroskedasticity exists. Eqs. (.0) ad (.6) (or Eqs. (.) ad (.7)) oly differ i the secod terms of their deomiators..3 A Root Estimator.3. A Root Estimator: Homoskedastic Case Eq. (.3) also motivates a exteded GMM root estimator for ρ of the SD model whe ɛ i s are i.i.d.. The matrix G (ρ) I tr[g (ρ)]/ i Eq. (.3) has a zero trace. Not 8

21 accoutig for the ρ s i the matrix, Eq. (.3) is quadratic i ρ. Replacig the matrix with ay costat matrix P satisfyig tr(p M Z ) = 0 (or tr(p ) = 0) 0, a cosistet GMM root estimator ca be derived by solvig the equatio g (ρ) = y S (ρ)p M Z S (ρ)y = 0, (.9) because the expectatio of g (ρ 0 ) is zero: E[g (ρ 0 )] = E[(Z θ 0 + ɛ ) P M Z (Z θ 0 + ɛ )] = σ0 2 tr(p M Z ) = 0. (.20) The P = G I tr(g )/ or P = G I tr(g M Z )/ is expected to geerate a root estimator that is asymptotically as efficiet as the MLE uder ormality sice Eq. (.9) with P = G I tr(g )/ is essetially the first order coditio of the cocetrated log likelihood fuctio Eq. (.3), eve though there is a sigle momet equatio. The form of the momet equatio automatically combies the liear ad quadratic momets i a way such that the root estimator ca be efficiet uder ormality, ulike Lee (2007a) or Lee (2007b), where liear momets are used together with the quadratic momets as a system with optimum weightig by the iverse of their variace-covariace matrix. This is ot surprisig because the sigle momet equatio is motivated from the first order coditio of the cocetrated log likelihood fuctio. Oce a cosistet estimator of ρ is available, Eqs. (.6) ad (.7) ca be used to calculate estimates for β ad σ 2, respectively. To establish the cosistecy of the root estimator, the followig regularity coditios are assumed. Assumptio.. ɛ i s i ɛ = (ɛ,..., ɛ ) are i.i.d. (0, σ0 2 ) ad the momet E( ɛ4+η i ) exists for some η > 0. 0 We still have a cosistet estimator if tr(p ) = 0 istead of tr(p M Z ) = 0. This is so because for the expectatio of the left-had side of Eq. (.9) at ρ 0, the additioal term divided by is σ2 0 tr[p Z (Z Z ) Z ] = σ2 0 tr[z P Z (Z Z ) ] = O( d ), which is ot exactly zero but coverges to zero as goes to ifiity. However, usig a matrix P such that tr(p M Z ) = 0 might have better small sample properties. Give ay matrix A, such a P matrix ca be costructed as P = A tr(am Z ) d I. For a pure SAR process, Eq. (.9) will be reduced to y S (ρ)p S (ρ)y = 0. I Ord (975), based o the motivatio of a modified LS estimatio, he cosidered the quadratic momet y S (ρ)w S (ρ)y = 0, but dismissed it i favor of the MLE i terms of efficiecy. The Eq. (.9) with a class of P provides a geeral framework icludig the Ord s momet equatio. Oe ca overcome the relative iefficiecy of the Ord s momet estimator by the selectio of a efficiet P as above. 9

22 Assumptio.2. Matrices {W } ad {S } are bouded i both row ad colum sum orms (Hor ad Johso, 985). The diagoal elemets of W are zero. Assumptio.3. Elemets of X are uiformly bouded costats, Z has full colum rak ad lim Z Z exists ad is osigular. Assumptio.4. Costat -dimesioal square matrices {P = [p,ij ]} which satisfy tr(p M Z ) = 0 are bouded i both row ad colum sum orms. The existece of a momet higher tha the fourth order of the disturbaces i Assumptio 2. is eeded for the applicatio of the cetral limit theorem for liear ad quadratic forms (Kelejia ad Prucha, 200). The boudedess i row ad colum sum orms of a sequece of matrices i Assumptio.2 origiated i Kelejia ad Prucha (998, 999, 200). Assumptio.3 is required for coveiece, as i Lee (2004a). As P is ofte geerated from W, it is reasoable to assume that {P } are bouded i both row ad colum sum orms. The quadratic momet equatio Eq. (.9) has two roots i geeral. Uder certai coditios, oe of the roots is cosistet. Let B s = B + B for ay -dimesioal square matrix B. Propositio.. Uder Assumptios 2..4, if (Z θ 0 ) P M Z G (Z θ 0 ) + 2 σ2 0 tr(p s G s ) were o-egative, the cosistet root for ρ 0 of Eq. (.9) would be ˆρ = b b 2 4a c 2a, (.2) where a = y W P M Z W y, b = y (P M Z ) s W y ad c = y P M Z y ; but if (Z θ 0 ) P M Z G (Z θ 0 ) + 2 σ2 0 tr(p s G s ) were egative, the cosistet root would be ˆρ 2 = b + b 2 4a c 2a, (.22) whe lim [(Z θ 0 ) G P M Z G (Z θ 0 ) + σ 2 0 tr(g P G )] 0. I the case that lim [(Z θ 0 ) G P M Z G (Z θ 0 ) + σ 2 0 tr(g P G )] = 0, ˆρ 3 = c /b is the uique cosistet root if lim [(Z θ 0 ) P M Z G (Z θ 0 ) + 2 σ2 0 tr(p s G s )] 0. The coditios that lim [(Z θ 0 ) G P M Z G (Z θ 0 ) + σ 2 0 tr(g P G )] 0 ad lim [(Z θ 0 ) P M Z G (Z θ 0 ) + 2 σ2 0 tr(p s G s )] 0 guaratee that a / ad b / do 0

23 ot coverge to zero i probability, respectively. Let H (ρ) = G (ρ) tr(g (ρ)m Z ) d M Z ad f(p ) = (Z θ 0 ) P M Z G (Z θ 0 )+ 2 σ2 0 tr(p s G s ) = (Z θ 0 ) P M Z H (Z θ 0 )+ 2 σ2 0 tr(p s H s ) with H = H (ρ 0 ). 2 The sig of f(p ) depeds o the correlatio betwee P ad H. If P = H, the f(h ) 0 ad Eq. (.2) is the cosistet root whe a / o P (). By cotiuity, f ( H (ρ) ) is o-egative whe ρ is close to ρ 0. I empirical applicatios, ρ 0 is ofte positive, the P = H (0.5) or P = H (0) = W tr(w M Z ) d M Z could geerate a cosistet root estimator of the form Eq. (.2). Give P, the scalars a, b ad c are products of vectors ad matrices, so the computatioal cost of Eq. (.2) or (.22) is miimal. The asymptotic distributio of the cosistet root ˆρ ca be derived from a first order expasio of g (ˆρ ) = 0 at ρ 0. As g (ρ 0 ) is quadratic i the disturbaces, the cetral limit theorem for liear ad quadratic forms is applicable. Propositio.2. The cosistet root ˆρ i Propositio. has the asymptotic distributio that (ˆρ ρ 0 ) D N ( 0, Ω ), where Ω = V ρ Σ 2 ρ with { V ρ = lim σ 2 0 (Z θ 0 ) P M Z P (Z θ 0 ) + 2 E(ɛ 3 i)(z θ 0 ) P M Z Diag(P M Z )l + [E(ɛ 4 i) 3σ0] 4 p 2,ii + 2 σ4 0 tr(pp s ) s } i= ad Σ ρ = lim [(Z θ 0 ) P M Z G (Z θ 0 ) + 2 σ2 0 tr(p s G s )] beig assumed to exist ad be o-zero. The V ρ i the above propositio is the limit of the variace of g (ρ 0 ), so it is geerally positive. Whe E(ɛ 3 i ) = E(ɛ4 i ) 3σ4 0 = 0, e.g., ɛ i s are i.i.d. ormal, the asymptotic variace of ˆρ reduces to Ω = lim σ2 0 (Zθ 0) P M Z P (Zθ 0)+ 2 σ4 0 tr(p sp s). The, by applyig the Cauchy [(Z θ 0 ) P M Z G (Z θ 0 )+ 2 σ2 0 tr(p sgs )]2 iequality, H is the best P matrix such that the asymptotic variace of this cosistet 2 Note that usig P = G tr(g M Z ) M d Z ad P = G tr(g M Z ) I d geerate the same root estimator. We use H for arrative coveiece, but we may use G tr(g M Z ) I d whe calculatig a root estimate.

24 root estimator is the smallest. As poited out earlier, with the best P (= H ) matrix, the cosistet root estimator has the form (b b 2 4a c )/(2a ) whe a / o P (). Propositio.3. Whe E(ɛ 3 i ) = E(ɛ4 i ) 3σ4 0 = 0, suppose that lim {(Z θ 0 ) G M Z G (Z θ 0 ) + 2 σ2 0[tr(G s G s ) tr2 (G s )]} exists ad is o-zero, the best root estimator is ˆρ b, = b b 2 4a c 2a, (.23) where a = y W H M Z W y, b = y (H M Z ) s y ad c = y H M Z y, i the sese that (ˆρ b, ρ 0 ) D N(0, Ω b ) with Ω b Ω, where Ω b = σ0{ 2 [ lim (Z θ 0 ) G M Z G (Z θ 0 ) + ( 2 σ2 0 tr(g s G s ) tr2 (G s ) )]}. Whe E(ɛ 3 i ) = E(ɛ4 i ) 3σ4 0 = 0, the asymptotic variace Ω b for the best root estimator i the above propositio is the same as that for the QMLE (Lee, 2004a). Whe the coditio E(ɛ 3 i ) = E(ɛ4 i ) 3σ4 0 = 0 does ot hold, the root estimator ˆρ b, may lose efficiecy. Note that o matter whether the coditio holds or ot, ˆρ b, i the above propositio is the cosistet root estimator whe H is used as the P matrix. As H ivolves the ukow parameter ρ 0, it ca be estimated by usig a iitial cosistet estimator for ρ 0. A estimated H would geerate a root estimator with the same limitig distributio as ˆρ b,. Propositio.4. Suppose that ˆρ is a -cosistet estimator of ρ 0, ad The the root estimator lim [(Z θ 0 ) G H M Z G (Z θ 0 ) + σ0 2 tr(g H G )] 0. ρ b, = ˆb ˆb2 4â ĉ 2â, (.24) where â = y W H (ˆρ )M Z W y, ˆb = y ( H (ˆρ )M Z ) sw y ad ĉ = y H (ˆρ )M Z y, is cosistet ad has the same limitig distributio as ˆρ b,. 2

25 A iitial cosistet estimator ˆρ may be derived by usig H (0) as the P matrix. Based o H (ˆρ ), Eq. (.24) is the best root estimator whe E(ɛ 3 i ) = E(ɛ4 i ) 3σ4 0 = 0.3 We shall use the otatio RE sd for this root estimator based o the SD model. Replacig Z with X everywhere above, we obtai the root estimator RE sar specifically for the SAR model. Note that the expressio for H (ρ) ivolves a matrix iverse (I ρw ), which is computatioally itesive for large sample sizes. 4 If ρw < with a matrix orm, the we have the expasio (I ρw ) = I +ρw +ρ 2 W ad (I ρw ) [I + ρw + + ρ r W] r = ρ r+ W r+ (I ρw ) < ρw r+ /( ρw ). I the secod step of computig a root estimate, we may start from usig a few term approximatio (I + ˆρ W + + ˆρ r W r )W tr[(i+ˆρw + +ˆρ r W r )W M Z ] d M Z of H (ˆρ ) i Eq. (.24). If the chage of the root estimate i absolute value is smaller tha a chose tolerace level, we ca stop ad report the estimate; otherwise, we may use (r + ) term approximatio of H (ρ) ad also use the ewly computed estimate from the last step i computig a ew ˆρ usig Eq. (.24). We could use more ad more terms to approximate H (ρ) util the tolerace criterio is met. This procedure turs out to be very efficiet i our Mote Carlo study..3.2 A Root Estimator: Heteroskedastic Case Whe there is ukow heteroskedasticity i disturbaces, from Eq. (.4), the expectatio of the left-had side of the momet equatio Eq. (.9) over is tr(p M Z Σ ), which geerally does ot coverge to zero eve if tr(p M Z ) = 0. I order to derive a cosistet root estimator from solvig Eq. (.9), we require P M Z to have a zero diagoal, so that 3 Because a iitial cosistet estimator ˆρ, e.g., derived with H (0)[= W tr(w M Z ) M d Z ], has a closed form expressio, the feasible two step root estimator will also have a closed form expressio as the aalytical expressio of the iitial ˆρ ca be substituted ito H (ˆρ ) i its derivatio. Li et al. (2007) have emphasized o closed form statistics for exploratory aalyses. They propose the AP LE but dismiss Ord s quadratic root estimator because of the eed to solve the quadratic momet equatio (as well as its possible iefficiecy). They have overlooked possible aalytical solutios of a quadratic equatio. 4 The best GMM estimator i Lee (2007a) or Lee (2007b) also ivolves this matrix iverse. 3

26 the expectatio of the left-had side of Eq. (.9) at ρ 0 would be zero. 5 Assumptio.5. The costat -dimesioal square matrices {P }, which satisfy that P M Z has a zero diagoal, are bouded i both row ad colum sum orms. We make the followig assumptio about the ukow heteroskedasticity. Assumptio.6. ɛ i s i ɛ = (ɛ,..., ɛ ) are idepedet (0, σi 2 ) ad the momets E ɛ 4+η i for some η > 0 exist ad are uiformly bouded for all ad i. The cosistet root is described i the followig propositio. The regularity coditios are similar to those i Propositio. after takig ito accout the heteroskedastic variace matrix Σ. Propositio.5. Uder Assumptios.2,.3,.5 ad.6, if (Z θ 0 ) P M Z G (Z θ 0 ) + tr(σ P s G ) were o-egative, the cosistet root would be ˆρ = b b 2 4a c 2a, (.25) where a = y W P M Z W y, b = y (P M Z ) s W y ad c = y P M Z y ; if tr(σ P s G ) + (Z θ 0 ) P M Z G (Z θ 0 ) were egative, the cosistet root would be ˆρ 2 = b + b 2 4a c 2a, (.26) whe lim [(Z θ 0 ) G P M Z G (Z θ 0 ) + tr(σ G P G )] 0. I the case that lim [(Z θ 0 ) G P M Z G (Z θ 0 ) + tr(σ G P G )] = 0, ˆρ 3 = c /b is the uique cosistet root if lim [(Z θ 0 ) P M Z G (Z θ 0 ) + tr(σ P s G )] 0. 5 As i the homoskedastic case, if P, istead of P M Z, is required to have a zero diagoal, the we ca still obtai a cosistet GMM estimator, sice the expectatio of the momet equatio over at ρ 0 coverges to zero as goes to ifiity i this case. We require P M Z to have a zero diagoal so that better small sample properties may be obtaied. Let P = [P,..., P ] ad M Z = [M,..., M ], where P i ad M i are -dimesioal vectors, the P M Z has a zero diagoal meas that P satisfies P im i = 0, i =,...,. So we have may choices of the P matrix. I particular, give ay -dimesioal square matrix A, we may let P = A Diag(A M Z )[Diag(M Z )] or P = A Diag(A M Z )[Diag(M Z )] M Z, if every diagoal elemet of M Z is o-zero. I the case that some diagoal elemets of M Z are zero, we may simply let P = A Diag(A ), or adjust A to be A such that the correspodig diagoal elemets of A M Z are zero ad let P = A Diag(A M Z )[Diag(M Z )] or P = A Diag(A M Z )[Diag(M Z )] M Z, where B deotes a geeralized matrix iverse for a matrix B. 4

27 The coditios that lim [(Z θ 0 ) G P M Z G (Z θ 0 ) + tr(σ G P G )] 0 ad lim [(Z θ 0 ) P M Z G (Z θ 0 ) + tr(σ P s G )] 0 are equivalet to oe-zero probability limits of a / ad b /, respectively. Propositio.6. The cosistet root ˆρ i Propositio.5 has the asymptotic distributio that (ˆρ ρ 0 ) D N(0, Ω), where Ω = V ρ Σ 2 ρ with V ρ = lim [(Z θ 0 ) P M Z Σ M Z P (Z θ 0 ) + tr(σ P Σ P s )] ad Σ ρ = lim [(Z θ 0 ) P M Z G (Z θ 0 ) + tr(σ P s G )] beig assumed to exist ad be o-zero. Note that V ρ is the limit of the variace of g (ρ 0 ). As cotrary to the homogeous variace case, the third ad fourth momets of o-ormal disturbaces do ot play a role i the asymptotic variace of the estimator due to the desig of P M Z havig a zero diagoal. Sice the asymptotic variace of the cosistet root i the above propositio ivolves ukow heteroskedasticity terms, the best selectio of the matrix P may be uavailable. If oe of the diagoal elemets of M Z is zero, a possible choice of P i practice might be the cosistetly estimated G Diag(G M Z )[Diag(M Z )]. To get a estimator for G, we may first derive a iitial cosistet estimator ˆρ for ρ 0 based o the momet equatio y S (ρ) { W Diag(W M Z )[Diag(M Z )] } M Z S (ρ)y = 0. The usig G (ˆρ ) Diag[G (ˆρ )M Z ][Diag(M Z )] as the P matrix i the momet equatio, we derive the root estimator ˆρ. We shall call this root estimator RE sd. The root estimator specifically for the SAR model is deoted by RE sar..4 Mote Carlo Study We coduct some Mote Carlo experimets to ivestigate fiite sample performaces ad computig times of the QMLE ad various versios of RE, AP LE ad ACME. The DGP is either the SAR or SE model. For the QMLE, the likelihood fuctio is derived as follows: igore (ukow) heteroskedasticity eve though there might be ad form the likelihood based o the SAR model if the DGP is the SAR process or based o the SE model if the 5

28 DGP is the SE process. For RE sd, i the homoskedastic case, the iitial estimator ˆρ is the root ˆρ of Eq. (.9) with P = W tr(w M Z ) d I, ad RE sd deotes the correspodig root estimator Eq. (.24) i Propositio.4 with P = G (ˆρ ) tr[g (ˆρ)M Z ] d I ; i the heteroskedastic case, the iitial estimator is the root ˆρ of Eq. (.25) with P = W Diag(W M Z )[Diag(M Z )], ad RE sd deotes the correspodig root estimator ˆρ of Eq. (.25) with P = G (ˆρ ) Diag[G (ˆρ )M Z ][Diag(M Z )]. The root estimate RE sar specific for the SAR model is derived similarly. We cosider three differet spatial weights matrices W, W 2 ad W 3. The W is the circular world matrix cosidered i Arraiz et al. (200). Specifically, each of the first /3 ad last /3 rows except the first ad last rows oly has two o-zero elemets, 6 which are i the positios (i, i ) ad (i, i + ) ad are equal to 0.5. For the first row, the o-zero elemets are i the positios (, 2) ad (, ) ad they are equal to 0.5; for the last row, the o-zero elemets are i the positios (, ) ad (, ) ad they are also equal to 0.5. Each of the middle /3 rows has 0 o-zero elemets, which are i the positios (i, i 5),..., (i, i ), (i, i + ),..., (i, i + 5) ad are equal to 0.. The W 2 ad W 3 are geerated accordig to, respectively, the quee ad rook criteria o regular m m grids, leadig to a sample size of = m 2. We use the row-ormalized W 2 ad W 3. The exogeous variable matrix X cosists of a itercept term, a exogeous variable draw from the ormal distributio N(3, ), ad the third oe draw from the uiform distributio U(, 2). The true parameter vector correspodig to these exogeous variables is β 0 = (0.8, 0.2,.5). The desig of exogeous variables ad correspodig parameters has bee used i Li ad Lee (200). For the homoskedastic case, the error terms are radomly draw from the ormal distributio N(0, ). For the heteroskedastic case, two desigs of heteroskedasticity are cosidered: Heteroskedasticity desig (HD-): For W, the stadard deviatio (STD) is equal to a costat times the umber of o-zero elemets i each row; 7 for W 2 ad W 3, the STD is equal to a costat times the absolute value of the secod exogeous 6 Whe /3 is ot a iteger, the smallest iteger larger tha /3 is take. 7 This desig is oe used i Arraiz et al. (200). 6

29 variable. 8 The costats are chose such that the average STD is equal to 0.5. Heteroskedasticity desig 2 (HD-2): For W, the STD is equal to a costat times the iverse of the umber of o-zero elemets i each row; for W 2 ad W 3, the STD is equal to a costat times the iverse of the absolute value of the secod exogeous variable. Agai the costats are chose to make the average STD be equal to 0.5. We calculate various measures of the autoregressive coefficiet by focusig o oegative ρ 0 values, as this is usually the case i empirical applicatios. For each case of ρ 0, the umber of repetitios is Figs. 6 compare the mea, STD ad root mea square error (RMSE) of the QMLE ad differet versios of RE, AP LE ad ACME. For these figures, we have = 400. Figs. ad 2 are the case whe there is o ukow heteroskedasticity i the disturbaces. Whe the DGP is the SAR process, from Fig., the QMLE ad RE sar have similarly small bias (i absolute value) for differet spatial weights matrices ad ρ 0 s, while the biases of AP LE sar, ACME sar, AP LE sd ad ACME sd are oly small whe ρ 0 is close to zero ad geerally icrease as ρ 0 icreases. I terms of bias, AP LE sar has ot show a advatage over AP LE sd, though AP LE sar is based o the DGP. The AP LE sar ad AP LE sd have similar bias for W ad W 2, but AP LE sar has large bias for large ρ 0 s i the case of W 2. The QMLE, RE sar, AP LE sar ad ACME sar have similar STD that is smaller tha those of RE sd, AP LE sd ad ACME sd, which is expected sice the latter oes are based o the more geeral SD model. It is oted that for W, the bias of ACME sd is sigificatly larger tha that of other statistics. The RMSEs of differet statistics show similar patters as their biases. Whe the DGP is the SE process, the bias, STD, RMSE of the QMLE, RE sd, AP LE se, AP LE sd ad ACME sd are plotted i Fig. 2. The biases of statistics other tha AP LE se have similar patters as the correspodig oes i Fig.. The AP LE se have ot show a advatage over AP LE sd i terms of smaller bias, which is obvious for W 2 for which AP LE se usually has larger bias tha AP LE sd. The STDs of all statistics are very similar. The AP LE se, AP LE sd ad ACME sd have larger RMSEs tha 8 For W 2 ad W 3, (m 2) 2 rows would have the same umber of o-zero elemets, which is approximately 00[(m 4)/m]% of the total umber of rows. If the same heteroskedasticity desig as for W is used, there would be little heteroskedasticity. 7

30 the QMLE ad RE sd for W ad W 3, while all statistics have similar RMSEs for W 2. Figs. 3 6 show the results whe there is ukow heteroskedasticity i the disturbaces. Figs. 3 ad 4 correspod to the DGP beig the SAR process but with differet desigs of heteroskedasticity, ad Figs. 5 ad 6 correspod to the DGP beig the SE process with differet variaces. I geeral, RE sar ad RE sd have the smallest bias i Figs. 3 ad 4 ad RE sd has the smallest bias i Figs. 5 ad 6. Sice the QMLE has igored the heteroskedasticity, it may geerate large bias i some cases, e.g., its bias is close to 0.2 whe ρ 0 = 0.6 for W i Fig. 5. I most figures, however, the QMLE has relatively small bias. The statistics derived from homoskedastic models AP LE sar, AP LE se ad AP LE sd geerally have relatively small bias whe ρ 0 is small ad relatively large bias whe ρ 0 is large. We ote that for W i Fig. 5, both AP LE se ad AP LE sd have very large bias for positive ρ 0 s. I Figs. 3 ad 4, like AP LE sar ad AP LE sd, ACME sar ad ACME sd have large bias for large ρ 0 s; i Figs. 5 ad 6, ACME sd have large bias for large ρ 0 s except for the case with W i Fig. 5, where the bias of ACME sd is smaller tha those of the QMLE, AP LE se ad AP LE se. The STDs ad RMSEs i Figs. 3 ad 4 are similar to the correspodig oes i Fig., ad the STDs ad RMSEs i Figs. 5 ad 6 are similar to the correspodig oes i Fig. 2. Table compares the computig times ad fiite sample properties of differet statistics whe the sample size is large. The DGP is the SAR model. We focus o the QMLE, RE sar ad AP LE sar, as computig the other statistics above are expected to take similar time. To compute RE sar, we use the procedure described i the last paragraph of subsubsectio.3. which starts from usig a two term approximatio (I + ˆρ W + ˆρ 2 W 2 )W tr[(i +ˆρ W +ˆρ2 W 2)W M Z ] d M Z of H (ˆρ ) i Eq. (.24). The tolerace criteria for RE sar ad the QMLE are both set to be The reported results are from Matlab o a desktop computer with Itel Core i processor ad 8 gigabyte memory. For the same sample size ad spatial weights matrix i the DGP, while computig a AP LE sar takes about the same time for differet ρ 0 s, computig the QMLE ad RE sar take more time whe ρ 0 becomes larger. For moderate values of ρ 0, computig RE sar oly takes slightly loger time tha computig the AP LE sar ad is at least 8 times faster tha computig the QMLE. The bias, STD ad RMSE have the same patter as we have see i Fig.. 8

31 Bias Bias with W ρ 0 STD STD with W ρ 0 RMSE QMLE RE sar APLE sar ACME sar RMSE with W RE sd APLE sd ACME sd ρ 0 0 Bias with W 2 0. STD with W 2 0. RMSE with W 2 Bias 0.05 STD 0.05 RMSE ρ 0 ρ 0 ρ 0 Bias Bias with W 3 STD STD with W 3 RMSE RMSE with W ρ ρ ρ 0 Figure : Compariso of the bias, STD ad RMSE of the QMLE, RE sar, AP LE sar, ACME sar, RE sd, AP LE sd ad ACME sd whe the DGP is the SAR model uder homoskedasticity. Bias Bias with W ρ 0 STD STD with W ρ 0 RMSE QMLE RE sd APLE se RMSE with W APLE sd ACME sd ρ Bias with W 2 0. STD with W 2 0. RMSE with W 2 Bias STD 0.05 RMSE ρ ρ ρ Bias with W STD with W 3 RMSE with W 3 Bias ρ 0 STD ρ 0 RMSE ρ 0 Figure 2: Compariso of the bias, STD ad RMSE of the QMLE, RE sd, AP LE se, AP LE sd ad ACME sd whe the DGP is the SE model uder homoskedasticity. 9