GMM and 2SLS Estimation of Mixed Regressive, Spatial Autoregressive Models. Lung-fei Lee* Department of Economics. Ohio State University, Columbus, OH

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1 GMM ad 2SLS Estimatio of Mixed Regressive, Spatial Autoregressive Models by Lug-fei Lee* Departmet of Ecoomics Ohio State Uiversity, Columbus, OH October 200 Abstract The GMM method ad the classical 2SLS method are cosidered for the estimatio of mixed regressive, spatial autoregressive models These methods have computatioal advatage over the covetioal quasimaximum likelihood method The proposed GMM estimators are show to be cosistet ad asymptotically ormal Withi certai classes of GMM estimators, best oes are derived The proposed GMM estimators improve upo the 2SLS estimators ad are applicable eve if all regressors are irrelevat A best GMM estimator may have the same limitig distributio as the ML estimator (with ormal disturbaces) The GMM ca be exteded to the estimatio of high-order spatial autoregressive models without additioal computatioal complexity Key Words: Spatial ecoometrics, spatial autoregressive models, regressio, high-order spatial lags, 2SLS, GMM, ML, efficiecy ClassiÞcatio: C3, C2, R5 Correspodig address: Lug-fei Lee, Departmet of Ecoomics, Ohio State Uiversity, Columbus, OH lßee@ecoohio-stateedu * I am grateful to the Natioal Sciece Foudatio, Ecoomics Program for Þacial support for my research uder the grat umber SES-0380

2 Itroductio I this paper, we propose a geeral GMM estimatio framework for the estimatio of mixed regressive, spatial autoregressive (MRSAR) models The GMM estimatio for those models may be relatively computatioal simpler tha the maximum likelihood (ML) or quasi maximum likelihood (QML) methods i a geeral settig The GMM estimator may be asymptotically relatively efficiet tha the two-stage least squares (2SLS) estimator The GMM method by Hase (982) has broad applicatios i empirical studies i various ecoomic Þelds Hase s GMM method goes beyod the oliear two-stage least squares (2SLS) method of Amemiya (974) as it icorporates oliear momet coditios beyod those geerated by orthogoality of exogeous regressors with disturbaces i a model The GMM method has bee oted for its possible use for the estimatio of spatial autoregressive (SAR) models i the presece of exogeous regressors, eg, Aseli (988, 990), Lad ad Deae (992), Kelejia ad Robiso (993), Kelejia ad Prucha (997, 998), ad Lee (999a), amog others However, those GMM methods are 2SLS methods as their momet coditios are based o exogeous regressors (ad spatial weights matrices) i the model ad all the istrumetal variables (IV) used are geerated from them There are two issues o the 2SLS approaches The Þrst oe is that the proposed 2SLS estimators are iefficiet relative to the ML estimator The 2SLS estimators have bee show to be cosistet ad asymptotically ormally distributed (Kelejia ad Prucha 998), but they do ot possess ay efficiet properties With a careful selectio of IVs, the best 2SLS (B2SLS) estimator i Lee (999a) is show to be the best oe withi the class of IV estimators where the istrumetal variables are fuctios of exogeous variables i the model A spatial autoregressive (SAR) model with exogeous regressors ad spatial lags has bee termed a MRSAR model i order to distiguish it from pure SAR processes, where exogeous regressors are abset Eve though a pure SAR process ca be regarded as a special case of a MRSAR model, exogeous regressors play a distiguished role i the available 2SLS methods For the estimatio of both pure SAR processes ad MRSAR models with a sigle spatial lag ad a row-ormalized spatial weights matrix, the ML method has bee popular sice a tractable computatioal device has bee suggested by Ord (975) By comparig the limitig variace matrix of the 2SLS or B2SLS estimator with that of the ML estimator, the 2SLS ad B2SLS estimators are less efficiet relative to the ML estimator (whe the disturbaces are ormally distributed) This is less satisfactory as it is kow for the estimatio of a covetioal liear 2

3 simultaeous equatio model, the 2SLS estimator is asymptotically efficiet as the limited iformatio ML (LIML) estimator (see, eg, Amemiya 985) The secod issue is that the 2SLS methods would ot be cosistet whe all the exogeous regressors i the MRSAR model are really irrelevat Furthermore, it is ot possible to test the joit sigiþcace of all the exogeous regressors based o those IV estimators (Kelejia ad Prucha 998) O the cotrary, the ML estimator of a MRSAR is cosistet ad its limitig distributio ca be used for testig the joit sigiþcace of regressio coefficiets These show that the 2SLS approaches are less satisfactory tha the ML estimator i some of their statistical properties However, the 2SLS approaches are computatioally simpler tha the ML approach ad are distributioally free For pure SAR processes, a method of momets (MOM) has bee itroduced i Kelejia ad Prucha (999a) Their MOM method is computatioally simpler tha the ML method Their MOM estimator is cosistet but is ulikely to be efficiet relative to the ML estimator Recetly, i Lee (200), a GMM estimatio framework was proposed Withi that GMM framework, a best GMM estimator ca have the same limitig distributio of the ML or QML estimator While the proposed GMM estimatio is iterestig for pure SAR processes, it eeds to be exteded for the estimatio of the MRSAR model I this paper, we cosider the possible geeralizatio by takig ito accout the presece of possible exogeous regressors i a MRSAR model We suggest a combiatio of the momets i the 2SLS framework with some modiþed momet fuctios origiated for the estimatio of pure SAR processes The resultig estimatio method is i the GMM framework We show that the resultig GMM estimator ca be asymptotically relatively more efficiet tha those of the 2SLS ad B2SLS estimators The best GMM estimator ca be made available ad it ca be efficiet relative to the ML estimator With this GMM framework, oe may also test the joit sigiþcace of all the possible exogeous regressors We show that this GMM method ca be easily exteded to the estimatio of MRSAR models with high-order spatial lags This paper is orgaized as follows I Sectio 2, we cosider the estimatio of a Þrst-order MRSAR model We discuss the momet fuctios that ca be used i additio to the momets based o the orthogoality of exogeous regressors with the model disturbace Cosistecy ad asymptotic distributio of the GMM estimators will be derived The GMM estimator ad the 2SLS estimator are compared The best selectio of momet fuctios ad optimal IVs will be discussed ad its possible efficiecy property is The asymptotic distributio of their MOM estimator has ot bee established i their article They have provided Mote Carlo results to demostrate that their MOM estimator is oly slightly iefficiet relative to the QML estimator uder various distributios 3

4 derived All the proofs of the results are collected i the appedices Sectio 3 geeralizes the method to the estimatio of MRSAR model with higher order spatial lags Sectio 4 geeralizes the method to the estimatio of the MRSAR model with a SAR disturbace As the asymptotic aalysis of cosistecy ad the derivatio of asymptotic distributios of estimators for the Þrst-order MRSAR model ca be easily exteded to the correspodig GMM estimates of the geeralized spatial models, details are omitted Much attetio i Sectios 3 ad 4 are devoted to the idetiþcatio problem ad the selectio of the best momet fuctios ad IV matrices Coclusios are draw i Sectio 5 2 GMM Estimatio of the MRSAR Model The MRSAR model differs from a pure SAR process i the presece of exogeous regressors X as explaatory variables i the model: Y = λw Y + X β + ², (2) where X is a k dimesioal matrix of ostochastic exogeous variables, W is a spatial weights matrix of kow costats with a zero diagoal, ad the disturbaces ² i, i =,,,ofthe-dimesioal vector ² are iid (0, σ 2 ) SpeciÞcally, we assume that Assumptio The ² i are iid with zero mea, variace σ 2 ad that a momet of order higher tha the fourth exists Assumptio 2 The elemets of X are uiformly bouded costats, X has the full rak k, ad lim X0 X exists ad is osigular The W Y i (2) is called a spatial lag ad its coefficiet is supposed to represet the spatial effect due to the ißuece of eighborig uits o a sigle spatial uit The mai iterest i estimatio of the model is, i geeral, the parameters λ ad β I order to distiguish the true parameters from other possible values i the parameter space, we deote λ 0, β0 0,adσ2 0 as the true parameters which geerate a observed sample Let θ =(λ, β 0 ) 0 For ay possible value λ, deotes (λ) =I λw At λ 0, S = S (λ 0 ) for simplicity This model is supposed to be a equilibrium model The structural equatio (2) implies the reduced form equatio that Y = S X β 0 + S ² (22) It follows that W Y = G X β 0 + G ² where G = W S, ad W Y is correlated with ² because, i geeral, E((G ² ) 0 ² ) = σ 2 0 tr(g ) 6= 0 There are some regularity coditios o W ad S eeded so that the spatial correlatios betwee uits ca be maageable The followig assump- 4

5 tio 3 is origiated i the works of Kelejia ad Prucha, eg, Kelejia ad Prucha (998) A sequece of square matrices {A },wherea =[a,ij ], is said to be uiformly bouded i row sums if there exists a costat c such that max i=,, P j= a,ij c for all It is said to be uiformly bouded i colum sums if max j=,, P i= a,ij c for all The k A k r = max i=,, P j= a,ij ad k A k c =max j=,, P i= a,ij are both matrix orms (Hor ad Johso 985) The uiform boudedess of matrices is equivalet to the boudedess of a sequece of orms of matrices Assumptio 3 The spatial weights matrices {W } ad {S } are uiformly bouded i both row ad colum sums Based o the ormal distributioal assumptio o ², the ukow parameters λ, β ad σ 2 ca be estimated by the ML (or QML) method (Ord 975) The ML method is based o a oliear optimizatio which ivolves the computatio of the Jacobia of S (λ), ie, the absolute value of the determiat of S (λ), at each possible value of λ durig a optimizatio search For the case that W is row-ormalized ad the correspodig spatial weights matrix before row ormalizatio is symmetric, the eigevalues of W are all real (istead of complex values) As the Jacobia of S (λ) is solely a fuctio of λ ad eigevalues of W, Ord (975) poits out that computatio of the Jacobia of S (λ) ca be easily updated as the eigevalues of W eed to be computed oly oce With a geeral spatial weights matrix ad a large data set, Smirov ad Aseli (999) discuss the attractability of usig a characteristic polyomial approach However, both approaches ca be tractable oly for a autoregressive model with a sigle spatial lag i (2) For models with high-order lags, both Ord s device ad the characteristic polyomial approaches will ot be implemetable For a computatioal poit of view, a 2SLS method remai the simplest Kelejia ad Prucha (998) suggest the use of W X, W 2X, etc, together with X itself as IVs i a 2SLS or a oliear 2SLS (N2SLS) framework for estimatig θ As oe ca see from the reduced form equatio (22), the structural equatio (2) is ot strictly a liear structural equatio i the classical liear simultaeous equatio framework Istead, (2) might be regarded as a structural equatio i a oliear simultaeous equatio framework For the latter, there is a issue o costructig the best N2SLS (Amemiya, 975) Lee (999a) shows that for the spatial model i (2), the best N2SLS (B2SL) correspods to use G X ad X as IV matrices These 2SLS ad B2SL estimators are computatioally simple ad have closed form expressios They ca, i geeral, be applicable to models with a geeral spatial weights matrix, ad ca be easily exteded to the 5

6 estimatio of MRSAR models with higher-order spatial lags However, the 2SLS ad B2SL methods have the limitatio i that some regressors of X must have sigiþcat coefficiets so that really valid istrumetal variables ca be geerated from them As the 2SLS ad B2SL estimators are based o β 0 6=0,itisimpossible to test the joit sigiþcace of X with those estimators Furthermore, by comparig the limitig variace matrices of the ML ad 2SLS estimators (see, eg, Aseli 988; Aseli ad Bera 998), either the 2SL estimator or the B2SL estimator have the same limitig distributio as the ML estimator I this paper, we suggest to icorporate some other momet coditios i additio to those based o X i a GMM framework for the estimatio of (2) Let Q be a m matrix of IVs costructed as fuctios of X ad W i a 2SLS approach Deote ² (θ) =S (λ)y X β for ay possible value θ The momet fuctios correspod to the orthogoality coditios of X ad ² are Q ² (θ) Let P be the class of costat matrices P with tr(p )=0 AsubclassP 2 of P cosistig of P with a zero diagoal is also iterestig By selectig matrices P,,P m from either P or P 2,wesuggesttheuse of the momet fuctios (P j ² (θ)) 0 ² (θ) i additio to Q 0 ² (θ) toformasetofmometfuctios For aalytical tractability, the matrices i P or P 2 are assumed to have the uiformly boudedess properties as W : Assumptio 4 The matrices P j sfromp or P 2 are uiformly bouded i both row ad colum sums, ad elemets of Q are uiformly bouded With the selected matrices P sadivmatricesq, the set of momet fuctios form a vector ² 0 (θ)p ² (θ) g (θ) =(P ² (θ),,p m ² (θ),q ) 0 ² (θ) = ² 0 (23) (θ)p m ² (θ) Q 0 ² (θ) for estimatio i the GMM framework At θ 0, g (θ 0 )=(² 0 P ²,,² 0 P m²,² 0 Q ) 0, which has a zero mea because E(Q 0 ² )=Q 0 E(² )=0adE(² 0 P j ² )=σ 2 0tr(P j )=0forj =,,m The ituitio behid these momet fuctios ad the IV matrix Q is as follows The IV variables i Q,which,igeeral, should iclude X, ca be used as IV variables for W Y ad X i (2) As W Y = G X β 0 + G ², P j ², which is ucorrelated with ², may be used as aother IV for W Y as log as P j ² ad G ² are correlated For ay possible value θ, E(g (θ)) = d 0 (θ)p d (θ)+σ0 2 tr(s0 S0 (λ)p S (λ)s ) d 0 (θ)p m d (θ)+σ0tr(s 2 0 Q 0 d (θ) 6 S 0 (λ)p m S (λ)s ), (24)

7 where d (θ) =S (λ)s X β 0 X β = X (β 0 β)+(λ 0 λ)g X β 0 because S (λ) =S +(λ 0 λ)w For these momet fuctios to be useful, they have to be sufficiet to idetify the true parameter θ 0 of the model The idetiþcatio of θ 0 is related partly to the correlatio of the IVs, Q ad P j ²,withW Y ad X i (2) I the GMM framework, the idetiþcatio coditio requires the uique solutio of the model equatios E(g (θ)) = 0 at θ 0 (as goes to iþity) (Hase 982) The momet equatios correspod to Q are Q 0 d (θ) =(Q 0 X,Q 0 G X β 0 )((β 0 β) 0, λ 0 λ) 0 = 0 They will have a uique solutio at θ 0 if (Q 0 X,Q 0 G X β 0 ) has a full colum rak, ie, rak (k +) Thissufficiet rak idetiþcatio coditio implies the ecessary rak coditio that (X,G X β 0 ) has a full colum rak (k +) I additio, i order to have the sufficiet rak coditio, Q eeds to have a rak at least as large as (k + ) This requires that the umber of liearly idepedet IVs m is at least equal to (k + ) This is ideed the case for the existig 2SLS estimators for the estimatio of (2) because the umber of explaatory variables i (2) is (k +) I additio to X, a IV is eeded for W Y The sufficiet rak coditio requires Q to be correlated with W Y ThisissobecauseE(Q 0 W Y )=Q 0 G X β 0 Uderthesufficiet rak coditio, θ 0 ca thus be idetiþed via Q 0 d (θ) = 0 Uder such a circumstace, θ ca be estimated by the 2SLS method with Q as IVs, ad the IV fuctios ² 0 (λ)p j² is used to provide extra momet coditios, which may improve the asymptotic efficiecy of the correspodig GMM estimate The ecessary full rak coditio of (X,G X β 0 )ispossibleolyifg X β 0 is ot liearly depedet o X This rak coditio will ot hold, i particular, if β 0 happes to be zero There are other cases whe this depedece ca occur (see, eg, Kelejia ad Prucha 998) If (X,G X β 0 ) does ot have a full rak (k + ), its rak will be k, ad there will exist a costat vector c such that G X β 0 = X c The, d (θ) =X (β 0 β +(λ 0 λ)c) adq 0 d (θ) =Q 0 X (β 0 β +(λ 0 λ)c) The correspodig momet equatios Q 0 d (θ) = 0 will have may solutios but the solutios are all described by the relatio that β = β 0 +(λ 0 λ)c as log as Q 0 X has a full colum rak k Uder this sceario, β 0 ca be idetiþed oly if λ 0 is idetiþable I tur, this implies d (θ) = 0 ad the remaiig momet equatios of E(g (θ)) = 0 i (24) will be simpliþed to tr(s 0 S(λ)P 0 j S (λ)s )=0forj =,,m, each of which is a quadratic fuctio similar to that of the momet equatios of the pure SAR process This ca be uderstood as follows Because W Y = G X β 0 + G ², it follows that Y = λ 0 (G X β 0 + G ² )+X β 0 + ² = λ 0 (G X β 0 )+X β 0 + S ² = X (β 0 + λ 0 c)+s ² Let u = S ² The u = λ 0 W u + ² is a SAR process The idetiþcatio of λ 0 thus comes from 7

8 the SAR process, ad the idetiþcatio of β 0 comes from the idetiþcatio of (β 0 + λ 0 c)asthecoefficiet vector of X ad the idetiþed λ 0 I Lee (200), we have show that, for each of the momet equatios of the pure SAR process, there is a sigle solutio λ 0 if tr(p s j G ) 6= 0buttr(G 0 P jg ) = 0 There ca be two solutios, amely λ 0 ad (λ 0 + tr(p s j G )/tr(g 0 P j G )) where A s deotes the sum (A + A 0 )foray square matrix A, iftr(g 0 P j G ) 6= o The commo set of solutios from these m momet equatios will be a sigleto if tr(p s j G) tr(g 0 PjG) 6= tr(p s l G ) tr(g 0 P lg ) for a pair of j ad l More geerally, the set of momet equatios i the limit (as goes to iþity) has a uique solutio at λ 0 if lim (tr(p s G ),,tr(p s mg )) 0 is liearly idepedet of lim (tr(g0 P G ),,tr(g 0 P m G )) 0 Alteratively, as the correspodig GMM miimizatio is equivalet to a oliear least square estimatio of a certai regressio fuctio with oliear coefficiets, idetiþcatio ca also be addressed i terms of such a oliear regressio fuctio 2 I summary, idetiþcatio of λ 0 is feasible from these IV fuctios whe several distict P sareused With λ 0 idetiþed, β 0 is idetiþable from Q 0 d (θ) = 0 The followig assumptio summarizes the idetiþcatio of θ 0 from the momet equatios E(g (θ)) = 0 for sufficiet large Assumptio 5 Either (i) lim Q0 (G X β 0,X ) has the full rak (k+),or(ii)lim Q0 X has the full rak k, lim tr(p jg ) 6= 0for some j, ad lim (tr(p s G ),,tr(p s m G )) 0 liearly idepedet of lim (tr(g0 P G ),,tr(g 0 P m G )) 0 For ay square matrix A, letvec D (A) =(a,,a ) 0 deote the colum vector formed with the diagoal elemets of a matrix A The, E(Q 0 ² ² 0 P ² )=Q 0 P i= P j= p,ije(² ² i ² j )=Q 0 vec D(P )µ 3, is where µ 3 = E(² 3 i )isthethirdmometof² i Let µ 4 = E(² 4 i ) be the fourth momet of ² i Lemma A6 shows that E(² 0 P j² ² 0 P l² ) = (µ 4 3σ 4 0 )vec0 D (P j)vec D (P l )+σ 4 0 tr(p jp s l ) It follows that var(g (θ 0 )) = Ω where Ω = µ (µ4 3σ0 4)ω0 mω m µ 3 ωmq 0 µ 3 Q 0 + V ω m 0, (25) with ω m =[vec D (P ),,vec D (P m )] ad tr(p P) s tr(p P s m) 0 V = σ0 4 tr(p m P s ) tr(p mpm s ) 0 = σ Q 0 σ Q 0 2 µ m 0 0 Q 0 σ0 2 Q, (26) where m =[vec(p 0 ),,vec(p 0 m)] 0 [vec(p s ),,vec(p s m)], by usig tr(ab) =vec(a 0 ) 0 vec(b) foray coformable matrices A ad B Whe ² is ormally distributed, Ω is simpliþed to V because µ 3 =0 2 Such idetiþcatio coditios have bee observed for the MOM i Kelejia ad Prucha (999a) 8

9 ad µ 4 =3σ 4 0 If P sarefromp 2, Ω = V also, because vec D (P j ) = 0 for all j =,,m I geeral, from (23), Ω is osigular if ad oly if both matrices (vec(p ),,vec(p m )) ad Q have full colum raks This is so, because Ω would be sigular if ad oly if P m j= a jp j =0adQ b =0forsome costat vector (a,,a m,b 0 ) 6= 0 Whe the selected P sadq satisfy these rak requiremets, Ω will be ivertible As elemets of P j sadq are uiformly bouded by Assumptio 4, it is apparet that ω0 m Q, ω0 m ω m ad Q0 Q are of order O() Furthermore, because P j Pl s is bouded i row or colum sums, tr(p jp s l )=O() Cosequetly, Ω = O() It is thus meaigful to impose the followig covetioal regularity coditio o the limit of Ω : Assumptio 6 The limit of Ω exists ad is a osigular matrix The followig propositio provides the asymptotic distributio of a GMM estimator with a liear trasformatio of the momet equatios, a g (θ), where a is a matrix with a full row rak greater tha or equal to the ukow of parameters (k +) The a is assumed to coverge to a costat matrix a 0 which has also a full row rak This correspods to the Hase s GMM settig, which illustrates the optimal weightig issue i the GMM framework As usual for oliear estimatio, the parameter space Θ of θ is assumed to be a compact set Assumptio 7 The θ 0 is i the iterior of the parameter space Θ, which is a compact subset of R k+ Propositio 2 Suppose that P j for j =,,m,arefromp ad Q is a k x IV matrix so that lim a E(g (θ)) = 0 has a uique root at θ 0 i Θ The, the GMM estimator ˆθ derived from mi θ Θ g 0 (θ)a 0 a g (θ) is a cosistet estimator of θ 0,ad (ˆθ θ) D N(0, Σ), where Σ = lim ( D0 )a0 a ( D ) ( D0 )a0 a ( Ω )a 0 a ( D ) ( D0 )a0 a ( ) D, (27) where µ σ 2 D = 0 tr(p s G ) σ0 2tr(P m s G ) (G X β 0 ) 0 0 Q 0 0 XQ 0, (28) uder the assumptio that lim a D exists ad has the full rak (k +) The rak coditios i Assumptio 5 imply that D of (28) has rak (k +) for large Thelastk x rows of D i (28) form the k x (k + )-dimesioal matrix Q 0 [G X β 0,X ] If G X β 0 is ot multicolliear with X, it shall be reasoable to assume that Q 0 [G X β 0,X ] ad the limit of Q0 [G X β 0,X ]tohave full colum rak (k +), whe Q is chose to properly correlate with W Y ad X i the model (2) ad has rak k x (k + ) This is ideed the rak coditio for the 2SLS estimatio of (2) with Q 9

10 beig the IV matrix The rak coditios i Assumptio 5 is more geeral tha the rak coditio of the 2SLS as it allows G X β 0 to be liearly depeded o X The latter icludes the case of all exogeous variables X beig really irrelevat i the MRSAR model The combiatio of the coditios that the limitig matrix of Q0 X has the rak k ad that tr(p s j G ) is ozero i the limit for some j is sufficiet to imply that the limit of D has rak (k + ) The momet fuctio with P sithiscasehelpsthe idetiþcatio ad estimatio of θ However, it should be oted that there is a limitatio o the coditio that lim tr(p s j G ) 6= 0wheG X β 0 is liearly depeded o X This coditio has ruled out some iterestig spatial weights structures, i particular, that of Case (99) The spatial sceario cosidered by Case (99) has may eighborhoods for each spatial uit I that case, tr(p s j G ) will vaish as the umber of eighborig uits teds to iþity (Lee 200) Whe G X β 0 ad X are colliear, the spatial weights case of Case (99) has ot bee covered by Propositio 2 3 Fially, we ote that, i the presece of Assumptio 5, the extra coditios that lim a E(g (θ)) = 0 has a uique root at θ 0 ad the limit of a D has full rak (k + ) are simply to elimiate the bad choice of a liear combiatio of a g (θ) which may result i the loss of idetiþcatio from the origial momet fuctio g The a does ot play a practical role i GMM estimatio but it provides a framework to address the optimal weightig matrix ad the correspodig optimal GMM estimator with a give set of momet fuctios g (θ) Assumptio 5 cotais the essetial idetiþcatio coditios From Propositio 2, with the momet fuctios g (θ) i (23), the optimal choice of a weightig matrix a 0 a is Ω by the geeralized Schwartz iequality If P s are selected from the subclass P 2 or ² i sare ormally distributed, Ω i (25) will be reduced to the simpler matrix V i (26) These variace matrices ca be used to form the optimal GMM objective fuctio with g (θ) However, as differet from the pure SAR process i Lee (200), the ukow parameter σ 2 ivolved i V is o loger a simple proportioal factor, ad two additioal parameters, amely, the third ad fourth order momets µ 3 ad µ 4 of ² i are also i Ω So the optimal GMM objective fuctio shall be formulated with a two-step feasible approach by estimatig cosistetly σ 2,aswellasµ 3 ad µ 4 They ca be doe by usig estimated residuals of ² 3 We suspect that the GMM estimator ˆλ of λ 0 i this situatio might have a slower rate of covergece tha the GMM estimator ˆβ of β 0 This has bee the case for the ML estimators (see, Lee (999b)) The above aalysis is ot powerful eough to reveal this possible feature More sophisticated aalysis might have to look at the characteristic of the cocetrated GMM objective fuctio by cocetratig out β The cocetrated likelihood approach i Lee (999b) is much simpler as the ML estimate of β give λ is simply a least-square type estimate, which has a simple closed form expressio O the cotrary, the GMM estimate of β give λ is a result of implicit optimizatio which does ot have a simple closed form expressio The cocetratio approach may be worthy of cosideratio i a future research project 0

11 from a iitial cosistet estimate of θ 0 4 TheΩ ca the be cosistetly estimated as ˆΩ The followig propositio shows that the feasible optimum GMM estimator with a cosistetly estimated ˆΩ has the same limitig distributio of the optimum GMM estimator based o Ω With the optimum GMM objective fuctio, a overidetiþcatio test is available Propositio 22 Suppose that ( ˆΩ ) ( Ω ) = o P (), the the feasible optimal GMM estimator ˆθ o, derived from mi θ Θ g 0 (θ)ˆω g (θ) based o g (θ) i (23) with P j sfromp has the asymptotic distributio Furthermore, (ˆθo, θ 0 ) D N(0, ( lim D0 Ω D ) ) (29) g 0 (ˆθ )ˆΩ g (ˆθ ) D χ 2 ((m + k x ) (k +)) (20) The asymptotic optimal GMM estimator ˆθ o, i Propositio 22 ca be compared with the 2SLS estimator of the equatio (2) With Q as the IV matrix, the 2SLS estimator of θ 0 is ˆθ 2sl, =[Z 0 Q (Q 0 Q ) Q 0 Z ] Z 0 Q (Q 0 Q ) Q 0 Y, (2) where Z =(W Y,X ) The asymptotic distributio of ˆθ 2sl, has bee derived i Kelejia ad Prucha (998) as (ˆθ2sl, θ 0 ) D N(0, σ 2 0 lim { (G X β 0,X ) 0 Q (Q 0 Q ) Q 0 (G X β 0,X )} ), (22) uder the assumptio that the limitig matrix (G X β 0,X ) has the full colum rak (k +) The 2SLS estimator ca be derived from the optimizatio problem: mi θ ² 0 (θ)q0 (Q Q 0 ) Q ² (θ) The 2SLS approach is therefore a special case of the GMM estimatio i Propositio 2 with a =(0, (Q 0 Q ) /2 )ad a g (θ) =(Q 0 Q ) /2 Q 0 ² (θ) It follows from Propositio 22, the optimal GMM estimator ˆθ o, shall be efficiet relative to the 2SLS estimator ˆθ 2sl, It may also be revealig by comparig directly the asymptotic variace of the optimal GMM estimator i (29) with that of the 2SLS estimator i (22) The compariso becomes simpler ad revealig for the case with P sfromp 2 or the case with µ 3 = 0, which icludes the ormal disturbaces case ad, more geeral, disturbaces with a symmetric distributio Whe µ 3 =0or P sfromp 2, (25) is reduced to a diagoal block matrix Whe µ 3 =0, µ (µ4 3σ0 Ω = 4)ω0 m ω m + m 0 0 σ0q 2 0 Q 4 The detailed proof is straightforward ad is omitted here, (23)

12 It follows that, as D i (28) is a lower block diagoal matrix, µ D 0 Ω D Cm [(κ = 4 3)ωm 0 ω m + m ] Cm µ (G X β 0 ) 0 σ0 2 X 0 Q (Q 0 Q ) Q 0 (G X β 0,X ), µ where C m =[tr(pg s ),,tr(pmg s m 0 )] whe µ 3 =0 WithP sfromp 2, Ω = µ DΩ 0 Cm D = m C0 m σ 2 0 µ (G X β 0 ) 0 X 0 (24) 0 σ0q 2 0 ad Q Q (Q 0 Q ) Q 0 (G X β 0,X ) (24) 0 From the precisio matrix i (24), it is clear that the additioal momets costructed with P sprovide additioal iformatio to the GMM estimator The 2SLS approach might ot be a valid procedure whe all the exogeous variables i X are really irrelevat This is easily see whe the irrelevacy correspods to β 0 =0 Wheβ 0 = 0, the limitig variace of the 2SLS estimator i (22) would be iþity ad ˆβ 2sl, would ot be cosistet More geerally, the 2SLS estimator ˆβ 2sl, would be problematic whe G X β 0 ad X are liearly depedet A example for the latter is a model where the oly relevat variable i X is the itercept term l ad W is row-ormalized (Kelejia ad Prucha 998) I that case, G X β 0 = G l β 0 = W S l β 0 = β 0 λ 0 W l = β 0 λ 0 l because S l = λ 0 l ad W l = l However, eve whe G X β 0 ad X are liearly depedet, the GMM approach may still work because of the additioal momet fuctios with P s For example, whe β 0 = 0, (24) becomes µ DΩ 0 Cm [(κ 4 3)ωm 0 D = ω m + m ] Cm X σ Q (Q 0 Q ) Q 0, X which ca be osigular The asymptotic distributio of the GMM estimator i (29) ca be used to formulae a Wald statistic for testig the sigiþcace of all exogeous variables while that of the 2SLS estimator ca ot Withi the 2SLS estimatio framework, by the geeralized Schwartz iequality applied to the asymptotic variace of ˆθ 2ls, i (22), the best IV matrix Q will be (G X β 0,X ) Lee (999a) shows that, with a iitial cosistet estimate θ of θ 0, G X β 0 ca be cosistetly estimated by W S ( λ )X β Withthis feasible best IV matrix, the feasible B2SL estimator has the same limitig distributio as the exact B2SL estimator Usig the best IV matrix for Q i the GMM framework, the resultig GMM estimator shall be efficiet relative to the B2SL estimator Whe µ 3 =0orP sfromp 2,theefficiet gai ca be easily quatiþed from (24) With µ 3 =0,thebestIVmatrix(G X β 0,X ) implies that D 0 Ω D = µ Cm [(κ 4 3)ωm 0 ω m + m ] Cm σ0 2 (G X β 0,X ) 0 (G X β 0,X ); 2

13 ad with P sfromp 2, µ D 0 Ω D Cm = m C0 m σ0 2 (G X β 0,X ) 0 (G X β 0,X ) (25) These are the asymptotic precisio matrices of the resultig GMM estimators, while that of the B2SL estimate is (G σ0 2 X η 0,X ) 0 (G X β 0,X ) I coclusio, the B2SL estimator of the model (2) ca be improved upo by addig momet fuctios with P j sfromp or P 2 There is a related questio o whether (G X β 0,X ) ca be the best IV matrix i the class of matrices Q with give P j s The aswer ca be affirmativeforcaseswherethemometsq 0 ² dootiteractwiththemomets² 0 P j² via their correlatio The covariace of Q 0 ² ad ² 0 P j ² is µ 3 Q 0 ω m, which ca be zero whe µ 3 =0orω m =0 The remaiig issue is o the best selectio of P s For the pure SAR process, the best P withi the subclass P 2 is (G Diag(G )) regardless of the distributio of the disturbaces Whe the disturbaces of ² are ormally distributed, the best P withi the more geeral class P is available ad it is the matrix (G tr(g) I ) These have bee show i Lee (200) These results ca be exteded to the MRSAR model Whe the disturbace ² is ormally distributed or P sarefromp 2, D 0 Ω D has the expressio i (25) Because tr(p j Pl s )= 2 tr(p j s P l s ), m ca be rewritte as m = 2 Whe P j sarefromp 2, tr(pp s ) s tr(pp s m) s tr(pm s P s ) tr(p m s P m s ) = 2 [vec(p s ) vec(p s m)] 0 [vec(p s ) vec(p s m)] tr(p s j G )=tr(p s j (G Diag(G ))) = 2 tr(p s j (G Diag(G )) s )= 2 vec0 ((G Diag(G )) s )vec(p s j ) for j =,,m, ad, therefore, the geeralized Schwartz iequality implies that C m mc 0 m 2 vec0 ((G Diag(G )) s )vec((g Diag(G )) s )=tr((g Diag(G )) s G ) Thus, i the subclass P 2,(G Diag(G )) s ad together with (G X β 0,X )providethesetofbestiv fuctios Similarly, for the case where ² is N(0, σ 2 0 I ), because, for ay P j P, tr(p s jg )=tr(p s j(g tr(g ) I )) = 2 tr(p s j(g tr(g ) for j =,, m, the geeralized Schwartz iequality implies that C m m C0 m 2 vec0 ((G tr(g ) I ) s )vec((g tr(g ) 3 I ) s )= 2 vec0 ((G tr(g ) I ) s )vec(p s j), I ) s )=tr((g tr(g ) I ) s G )

14 Hece, (G tr(g) I )ad(g X β 0,X ) provide the best IV fuctio withi P ad the IV matrix for the MRSAR model For all those cases, i the evet that G X β 0 is liearly depeded o X, G X β 0 is redudat ad the best IV matrix ca simply be X The best IV fuctio ivolves the matrix G, ad the best IV matrix ivolves both G ad β 0 i G X β 0 As G = W S (λ 0), G has to be estimated Þrst for a feasible GMM estimatio with these best IV fuctio ad matrices With iitial cosistet estimates ˆλ, ˆβ of λ 0 ad β 0, G ca be estimated as Ĝ = W S (ˆλ ), ad G X β 0 by ĜX ˆβ For the feasible best GMM estimatio with P,the correspodig estimated V is ˆV =ˆσ 4 ad the vector of momet fuctios is Ã! tr((ĝ tr(ĝ) I ) s Ĝ ) 0 0 ˆσ (ĜX 2 ˆβ,X ) 0, (26) (ĜX ˆβ,X ) ĝ b, (θ) =(² 0 (θ)(ĝ tr(ĝ) I )² (θ),² 0 (θ)ĝx ˆβ,² 0 (θ)x ) 0 (27) The feasible best GMM estimator with P will be derived from mi θ Θ gb, 0 (θ) ˆV g b,(θ) To accommodate the possibility that G X β 0 may be liearly depeded o X,somemodiÞcatios are eeded For this latter, eve though (ĜX ˆβ,X ) 0 (ĜX ˆβ,X ) might be umerically osigular with a Þite sample, it will be a ill-coditioed matrix This will reßect o the weightig matrix ˆV If G X β 0 ad X are liearly depedet, Ĝ X ˆβ would be redudat ad ca be dropped from the momet fuctios ad correspodig weightig matrix The proper weightig fuctio ˆV ad the momet fuctio g b, (θ) should simply be ad ˆV =ˆσ 4 Ã! tr((ĝ tr(ĝ) I ) s Ĝ ) 0 0 ˆσ X 0, (28) X 2 ) ĝ b, (θ) =(² 0 (θ)(ĝ tr(ĝ) I )² (θ),² 0 (θ)x ) 0 (29) To be complete, oe has to decide o usig either the weightig matrix ad the momet fuctios i (26) ad (27), or the oes i (28) ad (29) for the GMM estimatio We suggest that oe may regress the -dimesioal vector ĜX ˆβ o the k matrix X to check whether there is a serious liear depedecy A decisio ca be based o the estimated variace of the residuals of this artiþcal regressio Let ˆσ 2 a, = (ĜX ˆβ ) 0 (I X (X 0 X ) X 0 )(ĜX ˆβ ) (220) 4

15 We may expect that, i the evet that G X β 0 is liearly depeded o X,ˆσ a, 2 would coverge i probability to zero; otherwise, it would coverge to a positive costat Oe may use ˆV ad g b, (θ) i(26)ad (27) whe ˆσ 2 ζ where {ζ } is a certai sequece of positive costats; otherwise, shift to those i (28) ad (29) The criterio sequece {ζ } shallbecarefullyselectedwewattobesurethatwheg X β 0 ad X are liearly idepedet, the probability of ˆσ 2 a, ζ teds to oe as goes to iþity; otherwise, the probability of ˆσ 2 a, < ζ goes to oe whe G X β 0 ad X are liearly depedet I the followig lemma, it is show that ˆσ 2 would coverge i probability to zero at the rate O( )wheg X β 0 is liearly depeded o X This suggests that ζ shall be selected as a sequece covergig to zero at a rate slower tha Lemma 2 Suppose that ζ = 2 η for some costat η > 0 The, ) if G X β 0 = X c for some costat vector c for sufficietly large, lim P (ˆσ 2 a, < ζ )=; 2) if (G X β 0 ) 0 (I X (X 0 X ) X )(G X β 0 ) is bouded alway from zero, lim P (ˆσ 2 a, < ζ )=0 The above Lemma has ruled out the case that while G X β 0 ad X c are liearly idepedet for ay, G X β 0 ad X ted to be early perfectly colliear for large i the sese that lim (G X β 0 ) 0 (I X (X 0 X ) X )(G X β 0 ) = 0 This is a case which will be hard to hadle without kowig the rate of covergece of this zero sequece This is a pedagogical case which will be ruled out Assumptio 8 lim if (G X β 0 ) 0 (I X (X 0 X ) X )(G X β 0 ) 6= 0, wheever G X β 0 ad X are liearly idepedet for large eough For the best GMM estimatio with the subclass P 2, the weightig matrix ad its momet fuctios shall be ˆV 2 =ˆσ 4 µ tr(( Ĝ Diag(Ĝ)) s Ĝ ) 0 0 ˆσ (ĜX 2 ˆβ,X ) 0, (220) (ĜX ˆβ,X ) ad ĝ 2b, (θ) =(² 0 (θ)(ĝ Diag(Ĝ))² (θ),² 0 (θ)ĝx ˆβ,² 0 (θ)x ) 0 (22) For the case G X β 0 ad X are liearly depedet, they shall be modiþed to ˆV 2 =ˆσ 4 µ tr(( Ĝ Diag(Ĝ)) s Ĝ ) 0 0 ˆσ X 0 2 X, (222) ad ĝ 2b, (θ) =(² 0 (θ)(ĝ Diag(Ĝ))² (θ),² 0 (θ)x ) 0 (223) 5

16 The followig propositio summarizes the results ad shows that the feasible best GMM estimator has the same limitig distributio as the best GMM estimator Propositio 23 Suppose that ˆλ is a -cosistet estimate of λ 0, ˆβ is a cosistet estimate of β 0, ad ˆσ 2 is a cosistet estimate of σ 2 0 LetĜ = W S (ˆλ ) Withi the class of GMM estimators derived with P 2, the best GMM estimator ˆθ 2b, is the cosistet root derived from mi θ Θ ĝ2b, 0 (θ) ˆV 2 ĝ2b,(θ), ad it has the limitig distributio that (ˆθ 2b, θ 0 ) D N(0, Σ 2b ) where Σ 2b = lim à tr((g Diag(G )) s G )+ σ 2 0 σ 2 0 (G X β 0 ) 0 (G X β 0 ) σ 2 0 (G X β 0 ) 0 X X 0 (G X β 0 ) σ 2 0 X 0 X! (224) I the evet that ² N(0, σ 2 0 I ), withi the broader class of GMM estimators derived with P,the best GMM estimator ˆθ b, is the cosistet root derived from mi θ Θ ĝb, 0 (θ) ˆV ĝb,(θ), adithasthe limitig distributio that (ˆθ b, θ 0 ) D N(0, Σ b ) where à tr((g tr(g ) I ) s G )+ σ Σ b = lim 0 2 σ 2 0 (G X β 0 ) 0 (G X β 0 ) σ 2 0 (G X β 0 ) 0 X X 0 (G X β 0 ) σ 2 0 X 0 X! (225) We ote that i additio to the best matrix (Ĝ tr(ĝ ) I )or(ĝ Diag(Ĝ)), it is possible to icorporate additioal IV fuctios with other P s (such as those with W, etc), i the best GMM estimatio framework The asymptotic efficiecy of the resulted best GMM estimatio would ot be improved but they might help idetiþcatio i isolatig the cosistet root This ca be valuable advice especially for the special case that X does ot help idetiþcatio of λ 0 5 Whe ² is N(0, σ 2 0I ), the parameters of the model (2) ca be estimated by the ML method The asymptotic variace of the MLE is tr(g 2 )+tr(g 0 G )+ σ AsyVar(λ, β, σ )= σ0 2 (G X β 0 ) 0 (G X β 0 ) (X σ G X β 0 ) 0 tr(g ) σ0 2 X 0 G X β 0 σ 2 0 X 0 X 0 tr(g ) σ σ 4 0 (see, eg, p256 of Aseli ad Bera (998)) By a familiar expressio for the iverse of a partitioed matrix, the asymptotic variace of the MLE of θ is AsyV ar(λ, β) = à tr(g 2 )+tr(g 0 G )+ σ0 2 σ0 2 5 See Lee (200) for the pure SAR process case (G X β 0 ) 0 (G X β 0 ) 2 tr2 (G ) σ 2 0! (XG 0 X β 0 ) 0 X 0 G X β 0 σ 2 0 X 0 X 6

17 As tr(g 2 )+tr(g 0 G ) 2 tr2 (G )=tr(g s G ) 2 tr2 (G )=tr((g tr(g) I ) s G ), from Propositio 23, we see that the GMM estimator ˆβ b, has the same limitig distributio as the MLE of θ 0 3 Extesio : MRSAR Models with Higher Order Spatial Lags The GMM framework itroduced ca be easily exteded to the estimatio of MRSAR models with high-order spatial lags Cosider the estimatio of a MRSAR model with p-order spatial lags: Y = px λ j W j Y + X β + ², (3) j= where elemets ² i, i =,,,of² are iid(0, σ 2 0), ad W j sarep distict spatial weights matrices For this model, let λ =(λ,, λ p ) 0 ad θ =(λ 0, β 0 ) 0 Deote S (λ) =I P p j= λ jw j, S = S (λ 0 ), ad G j = W j S for j =,,p Furthermore, let ² (θ) =S (λ)y X β As for the Þrst-order case, the spatial weights matrices W j, j =,,p, ad the correspodig S of this model are assumed to be uiformly bouded i both row ad colum sums Uder this assumptio ad similar assumptios i the previous sectio, the asymptotic aalyzes of the GMM estimatio ca be easily exteded to this model For GMM estimatio of the model (3), the momet fuctios are ² 0 (θ)p j ² (θ), j =,,m,where P j s are costat matrices from either P or P 2,adtheIVmatrixQ cosists of fuctios of X ad W s The empirical momet fuctios ca be g (θ) =(² 0 (θ)p ² (θ),,² 0 (θ)p m² (θ),² 0 (θ)q ) 0 Atθ 0, g (θ 0 )=(² 0 P ²,,² 0 P m ²,² 0 Q ) 0 ad, hece, E(g (θ 0 )) = 0 At a feasible value of θ, let It follows that d (θ) =S (λ)s X β 0 X β = E(g (θ)) = px (λ 0j λ j )G j X β 0 + X (β 0 β) (32) j= d 0 (θ)p d (θ)+σ0tr(s 2 0 S(λ)P 0 S (λ)s ) d 0 (θ)p md (θ)+σ0 2 tr(s0 S0 (λ)p ms (λ)s Q 0 d (θ) ) (33) The momet equatio Q 0 d (θ) =Q 0 [P p j= (λ 0j λ j )G j X β 0 + X (β 0 β)] = 0 of (33) has the uique solutio (λ 0, β 0 0) if ad oly if the matrix Q 0 (G X β 0,,G p X β 0,X ) has a full colum rak (p + k) I this case, λ 0 ad β 0 ca be idetiþed from this rak coditio This rak coditio requires ecessarily that (G X β 0,,G p X β 0,X ) has a full rak (p + k) If this ecessary rak coditio is satisþed, the IV matrix Q will be useful if it is chose so that Q 0 (G X β 0,,G p X β 0 ) has the full colum rak Ituitively, this requires that the chose Q should be correlated with (W Y,,W p Y ) This is because (3) implies that W j Y = G j X β 0 + G j ² for j =,,p 7

18 If (G X β 0,,G p X β 0,X ) does ot have a full colum rak, it is likely that some G l X β 0 depeds liearly o X CosiderÞrst the case that G X β 0 is liearly depeded o the others I that case, there exists a costat vector a ad costats c j for j =2,,p, such that G X β 0 = P p j=2 c jg j X β 0 + X a Suppose that the submatrix Q 0 [G 2 X β 0,,G p X β 0,X ] has a full colum rak (p ) + k It follows from (32) that the momet equatio Q 0 d (θ) =Q 0 [ px G j X β 0 (λ 0j λ j + c j (λ 0 λ )) + X (β 0 β +(λ 0 λ )a)] = 0 j=2 will have all its solutios satisfyig the relatioships that (λ j λ 0j )=c j (λ 0 λ ), j =2,,p; (β β 0 )=a(λ 0 λ ) (34) I this situatio, β 0 ad λ 0j, j =2,,p,areidetiÞable oce λ 0 is idetiþed Uder this situatio, d (θ) = 0 ad, hece, the idetiþcatio of λ 0 based o E(g (θ)) = 0 from (33) will be reduced to that tr(s 0 S(λ)P 0 j S (λ)s )=0,j =,,m, has a uique solutio at λ 0 For ay P matrix from P, as S (λ)s = I + P p j= (λ 0j λ j )G j, tr(s 0 S(λ)P 0 S (λ)s )= px (λ 0j λ j )tr(g 0 jp)+ s j= Substitutig the solutios (34) ito (35), oe has px j= l= px (λ 0j λ j )(λ 0l λ l )tr(g 0 jp G l ) (35) tr(s 0 S0 (λ)p js (λ)s )=(λ λ 0 )tr(s Pj s )+(λ λ 0 ) 2 tr(s P j s 0 ), j =,,m, where s = P p k= c kg 0 k with c = The λ 0 is a commo root of these momet equatios There will be o other commo roots if the two vectors (tr(s P s ),,tr(s P s m )) ad (tr(s P s 0 ),,tr(s P m s 0 )) are liearly idepedet The latter is the idetiþcatio coditio for λ 0 uder this situatio I the evet that [G 2 X β 0,,G p X β 0,X ] also does ot have a full colum rak because G 2 X β 0 is liearly depeded o [G 3 X β 0,,G p X β 0,X ], the there exists costats c lj, j =3,,pad vectors a l such that G l X β 0 = P p j=3 G jx β 0 c lj + X a l for l =, 2 If Q 0 [G 3 X β 0,,G p X β 0 ]hasafull colum rak, it follows that the momet equatios Q 0 d(θ) = 0 are reduced to px Q 0 { G j X β 0 [(λ 0 λ )c j +(λ 02 λ 2 )c 2j +(λ 0j λ j )]+X [(λ 0 λ )a +(λ 02 λ 2 )a 2 +(β 0 β)]} =0, j=3 which have the solutios that λ j = λ 0j +(λ 0 λ )c j +(λ 02 λ 2 )c 2j for j =3,,pTheβ 0 ad λ 0j for j =3,,p, will be idetiþable oce λ 0 ad λ 02 are idetiþed The idetiþcatio of λ 0 ad λ 02 will deped 8

19 o the m momet equatios that tr(s 0 S0 (λ)p js (λ)s )=0forj =,,m The same logic shall be applicable to situatios where more terms of G j X β 0 s are liearly depeded o others ad X For the sceario that all G j X β 0, j =,,p, are liearly depeded o X,theidetiÞcatio of λ will deped solely o the m algebraic equatios tr(s 0 S(λ)P 0 j S (λ)s )=0 Withsufficiet umber of distict P j s, the idetiþcatio of λ is possible as it ca be the uique iteractio of the algebraic (solutio) curves i the space of λ 6 Thevariacematrixofg (θ 0 )isvar(g (θ 0 )) = Ω where Ω has the same expressio as that i (25) The derivative vector of g (θ) with respect to θ, whereθ =(λ,, λ p, β 0 ) 0 is It follows that g (θ) (W Y ) 0 P s ² (θ) (W p Y ) 0 P s ² (θ) X 0 P s ² (θ) = (W Y ) 0 Pm² s (θ) (W p Y ) 0 Pm² s (θ) XP 0 m² s (θ) (36) Q 0 W Y Q 0 W py Q 0 X E(g (θ 0 )) 0 σ0 2tr(P s G ) σ0 2tr(P s G p) 0 = σ0tr(p 2 mg s ) σ0tr(p 2 mg s p ) 0 Q 0 G X β 0 Q 0 G px β 0 Q 0 X (37) Whe P s are selected from P 2 or ² is ormally distributed, Ω will be reduced to V i (26) ad, hece, µ E(g 0 (θ0)) Ω E(g (θ 0)) A B = A0 0 + C,where σ0 2 tr(p s G ) tr(pmg s ) A = tr(pg s p ) tr(pmg s p ), B = tr(p P s ) tr(p Pm) s, tr(p m P s ) tr(p m P s m) ad C =(G X β 0,,G p X β 0,X ) 0 Q (Q 0 Q ) Q 0 (G X β 0,,G p X β 0,X ) From this asymptotic precisio matrix, the geeralized Schwartz iequality implies that the best selectio of Q will be the matrix (G X β 0,,G p X β 0,X ) 7 The best selectios of P sfromp 2 are (G j Diag(G j )) for j =,,p,adwhe² is ormally distributed, the best selectios of P from P are (G j tr(g j) I ) for j =,,p 4 Extesio 2: MRSAR Models with SAR Disturbaces The MRSAR model with SAR disturbaces is Y = λw Y + X β + u, u = ρm u + ², (4) 6 Sufficiet (ad ecessary) idetiþcatio coditios derived for the pure high-order SAR process ca be foud i Lee (200) 7 From this, we see also that if the high order MRSAR model is estimated by a 2SLS method, the best 2SLS estimator shall be based o this IV matrix 9

20 where ² i, i =,,,of² are iid with zero mea ad variace σ0 2 The model (4) icorporates both spatial lag W Y ad spatial correlated disturbaces u i a regressio equatio The W ad M are spatial weights matrices Let θ =(ρ, λ, β 0 ) 0 For this model, deote S (λ) =I λw, R (ρ) =I ρm, F (ρ, λ) =R (ρ)s (λ), u (θ) =S (λ)y X β,ad² (θ) =R (ρ)u (θ) At the true θ 0, S = S (λ 0 ), R = R (ρ 0 )adf = R S for simplicity Furthermore, deote G = W S, H = M R The reducedformequatioofthemodel(4)is Y = S X β 0 + S u = S X β 0 + F ² (42) I order for the asymptotic aalysis i Sectio 2 to be applicable, the spatial weights matrices W, M, ad the correspodig S ad R of this model are eeded to be uiformly bouded i both row ad colum sums The GMM method for the estimatio of this model (4) ca be based o the IV fuctios P j ² (θ), j =,,m,wherep j P,adIVmatrixQ costructed from X, W ad M The vector of empirical momets for estimatio ca be g (θ) = (P ² (θ)) 0 (P m ² (θ)) 0 Q 0 ² (θ) = (P R (ρ)u (θ)) 0 (P m R (ρ)u (θ)) 0 Q 0 R (ρ)u (θ) (43) At θ 0, g (θ 0 )=(² 0 P ²,,² 0 P m ²,² 0 Q ) 0 ad E(g (θ 0 )) = 0 For ay feasible θ, themodel(4) implies that E(g (θ)) = d 0 (θ)r(ρ)p 0 R (ρ)d (θ)+σ0tr(f 2 0 F(ρ, 0 λ)p F (ρ, λ)f ) d 0 (θ)r0 (ρ)p mr (ρ)d (θ)+σ0 2tr(F 0 Q 0 R (ρ)d (θ) where d (θ) =S (λ)s X β 0 X β = X (β 0 β)+g X β 0 (λ 0 λ) F 0 (ρ, λ)p mf (ρ, λ)f ), (44) Cosider Þrst the idetiþcatio of θ from the momet equatio E(g (θ)) = 0 of (44) The momet µ equatio Q 0 R (ρ)d (θ) = 0 is explicitly Q 0 R β0 β (ρ)(x,g X β 0 ) = 0, which has the uique λ 0 λ solutio (β 0, λ 0 )forβ ad λ if (Q 0 R (ρ)x,q 0 R (ρ)g X β 0 ) has full colum rak (k+) for each possible ρ i its parameter space It is ecessary that Q has a rak equal to or greater tha (k +), ad (X,G X β 0 ) has full colum rak (k +) Whe β 0 ad λ 0 are idetiþed, the remaiig momet equatios from (44) become tr(r 0 R 0 (ρ)p j R (ρ)r ) = 0, j =,,m, which are for the idetiþcatio of ρ 0 With a similar argumet for the pure SAR process, these remaiig momet equatios will idetify ρ 0 as if u were observable for the spatial process u = ρu + ² 20

21 For the case that (X,G X β 0 ) does ot have a full colum rak, because X has a full colum rak k, there exists a costat vector c 6= 0suchthatG X β 0 = X cwheq 0 R (ρ)x has the full colum rak k for each possible ρ, the momet equatios 0 = Q 0 R (ρ)d (θ) =Q 0 R (ρ)x ((β 0 β)+(λ 0 λ)c) imply that all their solutios satisfy the relatio β = β 0 +(λ 0 λ)c This idicates that oce λ 0 is idetiþed, β 0 will be idetiþable Furthermore, the relatio of solutios implies that d (θ) =X ((β 0 β)+(λ 0 λ)c) =0ad the remaiig momet equatios are simpliþed to tr(f 0 F 0 (ρ, λ)p jf (ρ, λ)f )=0,j =,,mthese momet equatios correspod to similar momet equatios of a pure SAR process with SAR disturbaces, Y = λ 0 W Y + u, u = ρ 0 M u + ² For this process, ² (θ) =R (ρ)s (λ)y = F (ρ, λ)y = F (ρ, λ)f ² ad, hece, E(² 0 (θ)p j ² (θ)) = σ 2 0tr(F 0 F(ρ, 0 λ)p j F (ρ, λ)f ) For this pure SAR process, idetiþcatio of ρ 0 ad λ 0 separately would ot be possible if W = M This pure SAR process implies the trasformed process Y = ρ 0 M Y + λ 0 W Y ρ 0 λ 0 M W Y + ² This trasformed process is a high-order SAR process with spatial lags, M Y, W Y ad M W Y, ad with oliear costraits o the spatial effect parameters, amely, ρ 0, λ 0 ad ρ 0 λ 0 IfM = W, the trasformed equatio would be reduced to Y =(ρ 0 +λ 0 )W Y ρ 0 λ 0 W 2 Y +² I this situatio, the ρ 0 ad λ 0 would ot be distiguished from each other eve though they might be locally idetiþable So without some other restrictios such as the iequality costrait ρ 0 > λ 0 or λ 0 > ρ 0,the ρ 0 ad λ 0 could oly be locally idetiþed This idetiþcatio issue is similar to that of a dyamic AR() process with AR() disturbaces i time series 8 Eve whe W is differet from M, each of these momet equatios is a secod degree algebraic equatio i ρ ad λ ad its solutio i the plae of (λ, ρ) caeither be a ellipse, a parabola, a hyperbola, two lies, or oe poit It is complex ad ot illumiatig to explore the solutio set for each equatio Istead, idetiþcatio coditios ca be derived by ivestigatig some characteristics of the momet equatios tr(f 0 F 0 (ρ, λ)p jf (ρ, λ)f )=0forj =,,mas F (ρ, λ) =F +(ρ 0 ρ)m S +(λ 0 λ)r W +(ρ 0 ρ)(λ 0 λ)m W, it implies that F (ρ, λ)f = I +(ρ 0 ρ)k ρ +(λ 0 λ)k λ +(ρ 0 ρ)(λ 0 λ)k ρλ,wherek ρ = M S F, K λ = R W F ad K ρλ = M W F Let a ρ,j = tr(pj s K ρ), a λ,j = tr(pj s K λ), a ρ 2,j = tr(k 0 ρp j K ρ ), a λ2,j = tr(k 0 λ P jk λ ), a ρλ,j = tr(p s j K ρλ + K 0 ρp s j K λ), a ρ2 λ,j = tr(k 0 ρλ P s j K ρ), 8 It is oted that the idetiþcatio of the MRSAR model with relevat X may ot ecessarily require W to be distict from R This is so for a dyamic AR() regressio equatio with AR() disturbaces 2

22 a ρλ 2,j = tr(k 0 λ P s j K ρλ) ada ρ 2 λ 2,j = tr(k 0 ρλ P jk ρλ ) It follows that tr(f 0 F 0 (ρ, λ)p jf (ρ, λ)f )=a ρ,j(ρ 0 ρ)+a λ,j (λ 0 λ)+a ρ 2,j(ρ 0 ρ) 2 + a λ 2,j(λ 0 λ) 2 + a ρλ,j (ρ 0 ρ)(λ 0 λ)+a ρ 2 λ,j(ρ 0 ρ) 2 (λ 0 λ)+a ρλ 2,j(ρ 0 ρ)(λ 0 λ) 2 + a ρ 2 λ 2,j(ρ 0 ρ) 2 (λ 0 λ) 2, for j =,,m It is apparet that (ρ 0, λ 0 ) is a commo solutio of these m momet equatios Let a ρ be the m-dimesioal vector with a ρ,j as its jth elemet Similarly, a λ,etc,aredeþed The ecessary ad sufficiet coditio for this equatio system to have the uique solutio at (ρ 0, λ 0 )isthatthevectorsa s do ot have a liear combiatio with oliear coefficiets i the form that a ρ δ + a λ δ 2 + a ρ 2δ 2 + a λ 2δ a ρλ δ δ 2 + a ρ2 λδ 2 δ 2 + a ρλ 2δ δ a ρ2 λ 2δ2 δ 2 2 =0 (45) for some costats δ ad δ 2 with (δ, δ 2 ) 6= 0 This coditio is a ecessary ad sufficiet coditio for the idetiþcatio of the SAR process with SAR disturbaces A sufficiet idetiþcatio coditio is that the a s are liearly idepedet As the a s together will form a matrix of dimesio m 8, i order to have this sufficiet idetiþcatio coditio, the umber of P s has to be at least 8 Weaker sufficiet coditios are available If there exists a solutio ρ ot equal to ρ 0,oehasδ 6= 0 i (45) This will imply that either a ρ or a 2 ρ will be liearly depedet o all other seve a s So it is sufficiet to idetify ρ 0 if either a ρ or a 2 ρ are liearly idepedet of the other seve a s With ρ 0 beig idetiþed, (45) becomes a λ δ 2 + a λ 2δ2 2 =0 The λ 0 will be idetiþed if a λ ad a λ 2 are ot liearly idepedet This set of sufficiet coditios is apparetly weaker By symmetric argumets, a similar set of sufficiet coditios ca be stated for the idetiþcatio of λ 0 Þrst ad the the idetiþcatio of ρ 0 As a geeral priciple, the true (λ 0, ρ 0 )maybeidetiþable whe sufficiet distict momet equatios are used ad their solutio sets itersect oly at the true parameter vector 9 Cosider ow the asymptotic distributio of a GMM estimator for the model (4) As g (θ 0 ) = (² 0 P ²,,² 0 P m ²,² 0 Q ), it has zero mea ad its variace is var(g (θ 0 )) = Ω,whereΩ has the same expressio i (25) The derivatives of g (θ) i(43)withrespecttoρ, λ, adβ are g (θ) 0 u 0 (θ)m 0 P s R (ρ)u (θ) (W Y ) 0 R 0 (ρ)p s R (ρ)u (θ) u 0 (θ)r0 (ρ)p s R (ρ)x = u 0 (θ)mp 0 mr s (ρ)u (θ) (W Y ) 0 R 0 (ρ)pmr s (ρ)u (θ) u 0 (θ)r 0 (ρ)pmr s (ρ)x Q 0 M u (θ) Q 0 R (ρ)w Y Q 0 R (ρ)x 9 As the GMM estimatio with those momet fuctios ca be rewritte i a oliear least squares estimatio framework with oliearity oly i parameters, sufficiet idetiþcatio coditio ca also be derived from the correspodig oliear regressio equatio 22