GMM Estimation of the Spatial Autoregressive Model in a System of Interrelated Networks

Transcription

1 GMM Estimation of the Spatial Autoregressive Model in a System of Interrelated etworks Yan Bao May, Introduction In this paper, I extend the generalized method of moments framework based on linear and quadratic moment conditions for the estimation of the first-order spatial autoregressive model introduced by Lee 2001; 2007 to estimate the SAR model in a system of interrelated networks The SAR model in a system of interrelated networks can be used to describe a market situation with several chain stores competing against one another Each chain is represented by a vector The strategy of a store in the chain does not only involve coordination with the other stores in the same chain, but also competition against opponent stores in the other chains Thus, we need to set up a SAR equation for each chain These equations need to be estimated simultaneously because each equation contains spatial lag terms for every chain in the market This setting complicated the estimation of the model in two ways First, all the spatial lag terms have different coefficients This makes it hard to implement the method of maximum likelihood even when the disturbances are normally distributed because the Jacobian transformation becomes very complicated Ord s 1975 device cannot be extended into this model Second, the disturbances are assumed to be correlated across equations Because of this assumption, the estimation method developed by Lee and Liu 2010 cannot be directly applied in this model Although this model can be rewritten into the form of a high order SAR model like in their paper, the heteroskedasticity and correlation affect the efficiency of their GMM estimator 1

2 There has been some attempt at the estimation of this model in the literature Kelejian and Prucha 2004 use 2SLS and 3SLS methods to estimate a very general SAR model with simultaneous equations In addition, Tao 2005 studies the identification problem of a similar model and also applies the 2SLS method to school district data The problem with these esmation methods, however, is that they do not have efficiency properties On the other hand, the GMM approach of Lee yields efficient estimators Therefore, we apply the GMM approach in this model in light of Lee s work I consider a simpler two-equation SAR model in this paper Similar to Lee s approach, I set up linear and quadratic moment functions and derive the identification conditions for the parameters Then I show the existence of the best GMME BGMME within the class of GMMEs based on linear and quadratic moment conditions The proposed BGMME is asymptotically as efficient as the MLE when the disturbances are independently and normally distributed It is not as efficient as the MLE when the disturbances are interrelated, but still more efficient than other linear estimators such as the 2SLSE By inspecting the first order conditions of the log likelihood function, I find that we need additional moments for the variance parameters in order to achieve efficiency This finding is intuitive in that the variance parameters cannot be estimated efficiently in separation of the estimation of the main equations due to the complicated error structure With the inclusion of the new moments, the resulting GMM estimator achieves the efficiency of the ML estimator The paper is organized as follows In Section 2, I introduce a SAR model with two simultaneous equations and point out the difficulty in implementing the maximum likelihood estimation Then I derive the identification condition of the model and propose a GMM estimation approach in Section 3 In Section 4 I investigate the consistency and asymptotic distribution of the GMMEs In Section 5 I derive the best selection of moment functions and optimal IVs The results are extended to multiple-equation systems in Section 7 Then I provide some Monte Carlo results for the comparison of finite sample properties of estimators in Section 6 Section 8 concludes 2

3 2 The MRSAR Model in a System of Interrelated etworks We start from the basic MRSAR model introduced by Anselin 1988, which has the following form: Y n λw n Y n + X n β + ɛ n The interpretation here is that every individual is affected by its neighbors in the space, and the i, jth element of W n represents a measure of distance between each pair of y i, y j Thus, by introducing the spatial lag term, W n Y n, into the model, we can capture the neighboring effects in the coefficient λ However, in market competitions between chain stores, we may face a more complicated situation Specifically, we may consider supermarkets and gas stations For example, the pricing strategy of a Shell gas station does not only depend on those of nearby Shell gas stations, but also depends on the strategies of Mobil, BP or other rival gas stations in the vicinity Similarly, we may consider the interactions between supermarket branches in a MSA as a star network, but meanwhile the stores also engage in price and/or quantity competition with rival retailer chains, which means that there is correlation between different star networks Unlike in a simple social interaction model, we have to incorporate the correlation between networks into the model Another distinctive feature of the model is that different networks have different spatial autoregressive coefficients The mathematical form is the following For simplicity, I only consider a market with two competitors in the model, which fits conveniently in the case of a duopoly Y 1,m λ 11 W 11,m Y 1,m + λ 12 W 12 Y 2,n + X 1,m β 1 + ɛ 1,m, Y 2,n λ 22 W 22,n Y 2,n + λ 21 W 21 Y 1,m + X 2,n β 2 + ɛ 2,n Here, Y 1,m represents the vector of prices or quantities of Wal-Mart stores and Y 2,n represents the vector of prices or quantities of K-Mart stores or, in Ohio s case, Meijer stores W 11,m, W 22,n are the weighting matrices for Wal-Mart and K-Mart stores respectively, designated in terms of distances between stores Similarly, W 12 and W 21 denote the distances betweens rival stores Unlike W 11,m and W 22,n, they are m n and n m matrices rather than square matrices when m n 3

4 Due to symmetry in this example, we have W 12 W 21 However, this condition can be relaxed in a more general setting We will drop this assumption in the rest of the paper In addition, X 1,m and X 2,n are exogenous variables which may be the same or contain different components There are three complications to this model First, the exogenous variables also have different effects on Y 1,m and Y 2,n Secondly, we suspect that λ 11 λ 22 This is because each competitor has its own cost structure, distribution channels and market strategies It is not reasonable to assume that they share the same network structure Finally, the coefficients for the cross terms W 12 Y 2,n and W 21 Y 1,m are also different than each other It is actually an interesting empirical question to compare λ 12 and λ 21 and determine which retailer has a larger impact on its rival than the other way around On the other hand, we can also assume that λ 11 λ 22 It is very easy to extend the results in this case This model has many potential applications Regarding the competition between chain stores, the model describe a Cournot competition if the Y s represent quantities and a Bertrand competition if the Y s represent prices Furthermore, in the special case when m n, the model becomes a simultaneous equations system and can be used to describe the demand and supply system as an example We are tempted to follow the standards in the literature and assume that the disturbances of the vector ɛ 1,m are iid 0, σ1 2 and the disturbances of the vector ɛ 2,n are iid 0, σ2 2 However, due to the correlation between ɛ 1,m and ɛ 2,n, we need a more general and complicated error structure Thus, we let ɛ 1,m u 1 ι m + v 1,m and ɛ 2,n u 2 ι n + v 2,n Here, u 1 and u 2 are considered as group effects while v 1,m and v 2,n are simply white noises The white noises v 1,m are assumed to be iid 0, σ 2 v 1 and v 2,n are assumed to be iid 0, σ 2 v 2, and they are not correlated with u 1, u 2 and the X s In this paper, we treat the group effects as random effects I will leave the fixed effect model to future work Thus, we assume that u 1 and u 2 are not correlated with the X s We also assume that they have zero mean, variances σ 2 u 1 and σ 2 u 2 and a covarince σ 12 Therefore, the covariance matrix of the disturbances is Σ σ2 u 1 ι m ι m + σv 2 1 I m σ 12 ι n ι m σ 12 ι m ι n σ 2 u 2 ι n ι n + σ 2 v 2 I n In practice, however, we cannot consistently estimate the variances of the group effects σ 2 u 1 and 4

5 σ 2 u 2 when the number of groups is small In this context, we may treat them as fixed effects and estimate the model with dummy variables In this paper, we do not make specific assumption on the variance matrix but opt for a general structure of Σ with unknown parameters η 1,, η t ow we can write this model in the matrix form as follows Y 1,m λ 11W 11,m λ 12 W 12 Y 1,m + X 1,m 0 β 1 + ɛ 1,m λ 21 W 21 λ 22 W 22,n Y 2,n 0 X 2,n β 2 ɛ 2,n Y 2,n This equation is not in the form of a spatial autoregressive model since the four weights matrices all have different coefficients Thus, the standard estimation procedure is not applicable here However, we can rewrite the equation above as Y 1,m Y 2,n λ 11 W 11,m 0 Y 1,m + λ 12 0 W 12 Y 1,m 0 0 Y 2,n 0 0 Y 2,n +λ Y 1,m + λ Y 1,m W 21 0 Y 2,n 0 W 22,n Y 2,n + X 1,m 0 β 1 + ɛ 1,m 0 X 2,n β 2 ɛ 2,n By changing notations, this model can be simplified into Y λ 11 W 1 Y + λ 12 W 2 Y + λ 21 W 3 Y + λ 22 W 4 Y + X β + ɛ This equation is exactly in the form of a high order spatial autoregressive model Lee and Liu 2010 show that this model can be estimated efficiently by the generalized method of moments Thus, it may seem as if their estimation procedure can be directly used in this model However, that is not the case The difficulty in using Lee and Liu s method lies in the assumption of autocorrelation While Lee and Liu assume that there is no correlation between the disturbances, we relaxed this assumption here by allowing for a more general error structure Even if we let the correlations be zero, we still have to deal with the problem that the disturbances in the two equations have different variances whereas Lee and Liu assume homoskedasticity in their model 5

6 Therefore, we have to handle the model as it is and find new ways to estimate it Thus, we come back to the original equation: where Y β 2 Y 1,m Y 2,n, W λ ɛ 2,n Y W λy + X β + ɛ, λ 11W 11,m λ 12 W 12 λ 21 W 21 λ 22 W 22,n β 1 and ɛ ɛ 1,m Here, varɛ Σ, X Let λ λ 11, λ 12, λ 21, λ 22, θ λ, β and η η 1,, η t true parameters that generate the observed sample S S λ 0 The structural equation implies the reduced form equation: X 1,m 0 0 X 2,n, β Denote θ 0 and η 0 as the Then denote S λ I W λ and Y S 1 X β 0 + ɛ From this reduced form equation, we can see that the spatial autoregressive terms in the structural system are endogenous in general If ɛ is normally distributed, the log likelihood function of this model is ln L n ln2π 1 2 ln Σ + ln S λ 1 2 Y W λy X β Σ 1 Y W λy X β If we want to implement the maximum likelihood method to estimate the model, we need to put restrictions on the parameters to guarantee that the determinant of S λ is positive, ie, S λ > 0 Let be any matrix norm Then a necessary condition is λ 11 W 11,m λ 12 W 12 λ 21 W 21 λ 22 W 22,n < 1 When all the spatial weights matrices W ij are row-normalized such that W ij 1 for i 1, 2 and j 1, 2, a possible parameter space for λ can be those satisfying λ 11 + λ 12 < 1 and λ 21 + λ 22 < 1 If the weights matrices are not row-normalized, then the parameter space can be 6

7 such that λ 11 + λ 12 < [max W 11,m, W 12 ] 1 and λ 21 + λ 22 < [max W 21, W 22,n ] 1 for some matrix norm Then S λ is invertible and S λ > 0 on this convex parameter space With the appropriate parameter space, it is still not an easy task to implement the maximum likelihood estimation in this model because of the complexity of S λ The evaluation of the determinant of S λ remains an issue because all the spatial weights matrices have different unknown coefficients, which makes W λ undiagonalizable in general Therefore, we need to consider other estimation methods which does not require the computation of the determinant of S λ Following Lee and Liu 2010, the generalized method of moments seems like a viable option 3 Identification and GMM Estimation The regularity assumptions for GMM estimation specified in Lee 2007 need to be modified to fit the current model Assumption 1 The elements of ɛ 1,m and ɛ 2,n have zero mean and covariance matrix Σ The fourth moment exists for both series Assumption 2 The elements of X 1,m and X 2,n are uniformly bounded constants Moreover, X 1,m and X 2,n have full ranks k 1 and k 2, and lim m 1 m X 1,m X 1,m and lim n 1 n X 2,n X 2,n exist and are nonsingular Assumption 3 The spatial weights matrices {W ij } i 1, 2, j 1, 2 and {S 1 } are uniformly bounded in both row and column sums in absolute value Let Q be a m + n k x matrix of IVs constructed as functions of X 1,n, X 2,n and W ij,n s Q can be decomposed into Q 1,m and Q 2,n which correspond with the two equations respectively in the system, so Q Q 1,m, Q 2,n Denote ɛ θ S λy X β and ɛ ɛ θ 0 The moment functions correspond to the orthogonality conditions of X and ɛ are Q ɛ θ We can also use additional moment functions ɛ θp ɛ θ suggested by Lee 2006 However, due to the general error structure in this model, we require the P s to have the property trσp 0 rather than trp 0 We need to place the following restriction on these matrices Assumption 4 The matrices P s with trσp 0 are uniformly bounded in both row and column sums in absolute value, and elements of Q are uniformly bounded With the selected matrices P j s j 1,, r and IV matrix Q, the set of moment functions 7

8 for a vector g θ Q, P 1ɛ θ,, P rɛ θ ɛ θ, which will be used for the estimation of θ First consider the identification of θ 0 with these moments At θ 0, g θ 0 Q, P 1 ɛ,, P r ɛ ɛ has a zero mean At a feasible value of θ, let d θ S λs 1 X β 0 X β As was defined, S λ I W λ Thus, S 1 S λs 1 I + [W λ 0 W λ]s 1 I +W λ 0 +[W λ 0 ] 2 +, and hence, Therefore, we have d θ [W λ 0 W λ]s 1 X β 0 + X β 0 β It follows that EQ ɛ θ Q d θ and Eɛ θp j ɛ θ E[d θ + S λs 1 ɛ P j d θ + S λs 1 ɛ ] d θp j d θ + E[ɛ S 1 S λp j S λs 1 ɛ ] d θp j d θ + tr[eɛ ɛ S 1 S λp j S λs 1 ] d θp j d θ + tr[σ 0 S 1 S λp j S λs 1 ] Thus, we have Eg θ Q d θ d θp 1d θ + tr[σ 0 S 1 S λp 1S λs 1 ] d θp rd θ + tr[σ 0 S 1 S λp rs λs 1 ] 8

9 We can further analyze the first moment equation Q d θ Q [W λ 0 W λs 1 X β 0 + X β 0 β] Q 1,m Q λ 11,0 λ 11 W 11,m λ 12,0 λ 12 W 12 S11 S 12 X 2,n β 0 λ 21,0 λ 21 W 21 λ 22,0 λ 22 W 22,n S 21 S 22 +Q X β 0 β λ 11,0 λ 11 Q 1,m W 11,m + λ 21,0 λ 21 Q 2,n W 21 λ 12,0 λ 12 Q 1,m W 12 + λ 22,0 λ 22 Q 2,n W 22,n +Q X β 0 β S11 S 12 S 21 S 22 X β 0 Q 1,m[λ 11,0 λ 11 G 11 11X 1,m β 1,0 + G 12 11X 2,n β 2,0 + λ 12,0 λ 12 G 21 12X 1,m β 1,0 +G 22 12X 2,n β 2,0 ] + Q 2,n[λ 21,0 λ 21 G 11 21X 1,m β 1,0 + G 12 21X 2,n β 2,0 +λ 22,0 λ 22 G 21 22X 1,m β 1,0 + G 22 22X 2,n β 2,0 ] + Q 1,mX 1,m β 1,0 β 1 +Q 2,nX 2,n β 2,0 β 2 Here, G kl ij W ijs kl and S kl is the corresponding block in S 1 Denote H 11 G X 1,mβ 1,0 +G X 2,nβ 2,0, H 12 G X 1,mβ 1,0 +G X 2,nβ 2,0, H 21 G X 1,mβ 1,0 + G X 2,nβ 2,0 and H 22 G X 1,mβ 1,0 + G X 2,nβ 2,0 Thus, the moment equation Q d θ 0 has the unique solution λ 0, β 0 if and only if the matrix Q 1,n H 11,n, Q 1,n H 12,n, Q 2,n H 21,n, Q 2,n H 22,n, Q 1,n X 1,n, Q 2,n X 2,n has a full column rank k 1 + k 2 +4 Equivalently, we require Q H, X to have a full rank k 1 +k 2 +4 In this case, λ 0 and β 0 can be identified from this rank condition A necessary condition is that the matrix H, X has a full rank k 1 + k H 21, H 22, X 2,n to have full column ranks Here, we define Equivalently, we require both the matrices H 11, H 12, X 1,m and H H 1, H 2, H 3, H 4 H 11 H H 21 H 22 On the other hand, if the rank condition is not satisfied, then we need the additional moments P 1 ɛ θ,, P r ɛ θ ɛ θ to identify λ 0 and β 0 Suppose that the matrix H 11, H 12, X 1,m has a column rank k p and the matrix H 21, H 22, X 2 has a column rank k q with p, q {1, 2} As X 1,m and X 2,n both have full ranks, we can form a k p-column basis and a 9

10 k q-column basis with X 1,n, X 2,n and other linearly independent columns Then the other columns in these two matrices can all be expressed as linear combinations of these two bases Without loss of generality, we assume here that H 1,p+1,, H 12, X 1 has full rank k p and also that H 2,q+1,, H 22, X 2 has full rank k q Then there exist constant vectors a 1l, a 2l and constants c 1lj, c 2lj such that H 1l 2 jp+1 H 1jc 1lj + X 1 a 1l for l 1, 2 and H 2l 2 jq+1 H 2jc 2lj + X 2 a 2l for l 1, 2 Hence, the linear moment equations Q d θ 0 are reduced to Q 1 { 2 jp+1 H 1j[ p l1 λ 1l,0 λ 1l c 1lj + λ 1j,0 λ 1j ] + X 1 [ p l1 λ 1l,0 λ 1l a 1l + β 1,0 β 1 ]} + Q 2 { 2 jq+1 H 2j[ q l1 λ 2l,0 λ 2l c 2lj + λ 2j,0 λ 2j ] + X 2 [ q l1 λ 2l,0 λ 2l a 2l + β 2,0 β 2 ]} 0, which have all their solutions satisfying λ 1j λ 1j,0 + λ 2j λ 2j,0 + p λ 1l,0 λ 1l c 1lj, β 1 l1 q λ 2l,0 λ 2l c 2lj, β 2 l1 p λ 1l,0 λ 1l a 1l + β 1,0, for j p + 1,, 2, l1 q λ 2l,0 λ 2l a 2l + β 2,0, for j q + 1,, 2 l1 From these conditions, we know that β 0 and λ 0 are identifiable as long as λ 11,, λ 1p and λ 21,, λ 2q are identified In this situation, it is clear that d θ 0, so the identification problem is reduced to tr[σ 0 S 1 S λp js λs 1 ] 0, for j 1,, r As we have already computed, S λs 1 any square matrix A Thus, I + [W λ 0 W λ]s 1 Let AS A + A for tr[σ 0 S 1 S λp j S λs 1 ] tr{σ 0 [I + W λ 0 W λs 1 ] P j [I + W λ 0 W λs 1 ]} trσ 0 P j + tr{σ 0 [W λ 0 W λs 1 ] P S j} +tr{σ 0 [W λ 0 W λs 1 ] P j [W λ 0 W λs 1 ]} We will compute this expression by parts First of all, we already assume that trσ 0 P j 0, so the first term disappears 10

11 ext, we compute Σ 0 [W λ 0 W λs 1 ] Pj S Σ 0 11,0 λ 11 W 11 λ 12,0 λ 12 W 12 S11 S 12 λ 21,0 λ 21 W 21 λ 22,0 λ 22 W 22 S 21 S 22 Pj S Σ 0 λ 11,0 λ 11 G λ 12,0 λ 12 G λ 11,0 λ 11 G λ 12,0 λ 12 G λ 21,0 λ 21 G λ 22,0 λ 22 G λ 21,0 λ 21 G λ 22,0 λ 22 G Obviously, all the elements in the resulting matrix are linear terms regarding the λ ij,0 λ ij s i, j {1, 2} Using the linear dependence relations and proper definitions of the α 1l,j s and α 2l,j s, we can rewrite the second term into tr{σ 0 [W λ 0 W λs 1 ] Pj S } p l1 λ 1l,0 λ 1l α 1l,j + q l1 λ 2l,0 λ 2l α 2l,j, for j 1,, r Lastly, we need to open up the third term: P S j Σ 0 [W λ 0 W λs 1 ] P j [W λ 0 W λs 1 ] λ 11,0 λ 11 G λ 12,0 λ 12 G λ 11,0 λ 11 G λ 12,0 λ 12 G P j λ 21,0 λ 21 G λ 22,0 λ 22 G λ 21,0 λ 21 G λ 22,0 λ 22 G λ 11,0 λ 11 G λ 12,0 λ 12 G λ 11,0 λ 11 G λ 12,0 λ 12 G λ 21,0 λ 21 G λ 22,0 λ 22 G λ 21,0 λ 21 G λ 22,0 λ 22 G By taking the trace of the matrix above, the third term is essentially the sum of a number of quadratic terms with regards to the λ ij,0 λ ij s Once again, we can use the linear dependence relations and proper definitions of the α 1lk,j s, α 2lk,j s and γ lk,j s to rewrite the third term into tr{σ 0 [W λ 0 W λs 1 ] P j [W λ 0 W λs 1 ]} p p q q λ 1l,0 λ 1l λ 1k,0 λ 1k α 1lk,j + λ 2l,0 λ 2l λ 2k,0 λ 2k α 2lk,j l1 k1 + p l1 k1 l1 k1 q λ 1l,0 λ 1l λ 2k,0 λ 2k γ lk,j, for j 1,, r 11

12 Combining all three terms together, we get the moment equation we need: tr[σ 0 S 1 S λp j S λs 1 ] p q p p λ 1l,0 λ 1l α 1l,j + λ 2l,0 λ 2l α 2l,j + λ 1l,0 λ 1l λ 1k,0 λ 1k α 1lk,j l1 + q l1 q λ 2l,0 λ 2l λ 2k,0 λ 2k α 2lk,j + l1 k1 l1 k1 l1 k1 p q λ 1l,0 λ 1l λ 2k,0 λ 2k γ lk,j 0, for j 1,, r Apparently, λ 0 is a common solution of these m moment equations Let α 1l be the m- dimensional vector with α 1l,j as its jth element Also define α 2l, α 1lr, α 2lr and γ lr in a similar way Therefore, the necessary and sufficient condition for this equation system to have a unique solution at λ 0 is that the vectors α s and γ s do not have a linear combination with nonlinear coefficients in the form that p δ 1l α 1l + l1 q δ 2l α 2l + l1 p l1 r1 p δ 1l δ 1r α 1lr,j + q l1 r1 q δ 2l δ 2r α 2lr,j + p l1 r1 q δ 1l δ 2r γ lr,j 0, for some nonzero constants δ 11,, δ 1p and δ 21,, δ 2q This condition is necessary and sufficient for the identification of the model The sufficient conditions for the identification of θ 0 from the moment equations lim Eg θ 0 are summarized in the following assumption Assumption 5 We must have that either i lim 1 Q [H, X ] has full rank k 1 + k 2 + 4, or ii lim m 1 m Q 1 [H 1,p+1,, H 12, X 1 ] has full rank k p for some 0 p 2, lim n 1 n Q 2 [H 2,q+1,, H 22, X 2 ] has full rank k q for some 0 q 2, and the vectors α s and γ s do not have a linear combination with nonlinear coefficients in the form that p l1 δ 1lα 1l + q l1 δ 2lα 2l + p p l1 r1 δ 1lδ 1r α 1lr,j + q q l1 r1 δ 2lδ 2r α 2lr,j + p q l1 r1 δ 1lδ 2r γ lr,j 0, for some nonzero constants δ 11,, δ 1p and δ 21,, δ 2q Let Ω varg θ 0 The variance matrix Ω involves variances and covariances of linear and quadratic forms of ɛ Throughout the paper we assume normality of the disturbances Then EQ ɛ ɛ Q Q Σ 0Q Obviously, ɛ Σ 1/2 0 ɛ follows a standard normal distribution Thus, we can show that EQ ɛ ɛ P ɛ EQ Σ1/2 0 ɛ ɛ Σ1/2 0 P Σ 1/2 0 ɛ 0 In addition, Eɛ P jɛ ɛ P lɛ Eɛ Σ1/2 0 P j Σ 1/2 0 ɛ ɛ Σ1/2 0 P l Σ 1/2 0 ɛ trσ1/2 0 P j Σ 1/2 0 Pl S Σ1/2 0 12

13 trσ 0 P j Σ 0 Pl S It follows that Ω Q Σ 0Q trσ 0 P 1 Σ 0 P1 S trσ 0P 1 Σ 0 Pr S 0 trσ 0 P r Σ 0 P1 S trσ 0P r Σ 0 Pr S Q Σ 0Q 0 0 r, where r 1 2 [vecσ1/2 0 P1 S Σ1/2 0,, vecσ 1/2 0 Pr S Σ1/2 0 ] [vecσ 1/2 0 P1 S Σ1/2 0,, vecσ 1/2 0 Pr S Σ1/2 0 ] Then we impose the following conventional regularity condition on the limit of 1 Ω : Assumption 6 The limit of 1 Ω exists and is a nonsingular matrix Let g θ g 0 θ, g 1 θ,, g m θ Then we have g 0 θ θ Q W 11Y 1 W 12 Y X W 21 Y 1 W 22 Y 2 0 X 2 Hence, the score is Eg 0 θ 0 θ Q H 11 H X H 21 H 22 0 X 2 Q H 1 H 2 H 3 H 4 X In addition, we have g j θ λ 11 g j θ λ 12 g j θ λ 21 g j θ λ 22 g j θ β 1 g j θ β 2 W 11 Y 1 0 W 12 Y 2 0 P S jɛ θ, P S jɛ θ, 0 W 21 Y 1 P S jɛ θ, 0 W 22 Y 2 Pjɛ S θ, X 1 0 Pjɛ S θ, 0 X 2 P S jɛ θ, 13

14 for j 1,, m First compute Eg j θ 0 [ ] E W λ 11 Y 1 0 Pjɛ S θ 0 11 [ ] tr PjE S ɛ θ 0 W 11 Y 1 0 tr Pj S σ2 1,0 G11 + σ 12,0G σ 12,0 G 11 tr PjΣ S 0 G11 0 G σ2 2,0 G trp S jσ 0 G 1 trg 1 Σ 0 P S j trσ 0 P S jg 1 Here we define G 1 G11 G Similarly, G and G G G It follows that D Eg θ 0 θ G21 12 G , G G G Q H 1 Q H 2 Q H 3 Q H 4 Q X trσ 0 P1 S G 1 trσ 0 P1 S G 2 trσ 0 P1 S G 3 trσ 0 P1 S G 4 0 trσ 0 Pr S G 1 trσ 0 Pr S G 2 trσ 0 Pr S G 3 trσ 0 Pr S G Consistency and Asymptotic Distributions The following proposition provides the asymptotic distribution of a GMME with a linear transformation of the moment equations, a g θ, where a is a matrix with a full row rank greater than or equal to k 1 + k The a is assumed to converge to a constant full rank matrix a 0 As usual, we assume that the parameter space is a compact convex set including θ 0 in its interior Assumption 7 θ 0 is in the interior of the parameter space Θ, which a compact convex subset of R k 1+k 2 +4 Proposition 1 Under Assumptions 1-5, suppose that g θ is the moment function such that lim a Eg θ 0 has a unique root at θ 0 in Θ Then, the GMME ˆθ derived from 14

15 min θ Θ g θa a g θ is a consistent estimator of θ 0, and ˆθ θ 0 D 0, Φ, where [ D ] 1 Φ lim a D D [ a a Ω a a D D ] 1 a a D a, under the assumption that lim 1 a D exists and has the full rank k 1 + k From Proposition 1, the optimal choice of a weighting matrix a a is Ω 1 by the generalized Schwartz inequality Since Ω contains unknown parameters σ 2 u 1, σ 2 u 2, σ 12, σ 2 v 1 and σ 2 v 2, the optimal GMM objective function should be formulated with a consistent estimator of the covariance matrix These paramters can be consistently estimated by using estimated residuals ɛ consistent estimate of θ 0 Then Ω can be consistently estimated as ˆΩ from an initial The following propositiion shows that the feasible optimal GMME with a consistently estimated ˆΩ has the same limiting distribution as that of the optimal GMME based Ω With the optimal GMM objective function, an overidentification test is available Proposition 2 Under Assumptions 1-6, suppose that ˆΩ / 1 Ω / 1 o p 1, then the feasible optimal GMME ˆθ fo, derived from min θ Θ g θa a g θ has the asymptotic distribution Furthermore, ˆθfo, θ 0 D 1 0, lim n D Ω 1 D 1 g ˆθ ˆΩ 1 g ˆθ D χ 2 k x + r k 1 + k Efficiency and the BGMME From the previous sections, we know that D Ω 1 D A B 1 A C, 15

16 where A B trσ 1/2 0 P1 S Σ1/2 0 Σ 1 0 G 1Σ 1/2 0 trσ 1/2 0 Pr S Σ1/2 0 Σ 1 0 G 1Σ 1/2 trσ 1/2 0 P1 S Σ1/2 0 Σ 1 0 G 4Σ 1/2 0 trσ 1/2 0 Pr S Σ1/2 trσ 1/2 0 P 1 Σ 1/2 0 P1 S Σ1/2 0 trσ 1/2 0 P 1 Σ 1/2 0 Pr S Σ1/2 trσ 1/2 0 P r Σ 1/2 0 P1 S Σ1/2 0 trσ 1/2 0 P r Σ 1/2 0 Pr S Σ1/2 C H, X Q Q Σ 0Q 1 Q H, X 0 0 Σ 1 0 G 4Σ 1/ , and From this asymptotic precision matrix, the generalized Schwartz inequality implies that the best selection of Q will be the matrix Σ 1 0 H, X As for the best selection of the P s, we need to inspect A B 1 A We already know that we can rewrite B as B 1 2 [vecσ1/2 0 P S 1 Σ1/2 0,, vecσ 1/2 0 P S r Σ1/2 0 ] [vecσ 1/2 0 P S 1 Σ1/2 0,, vecσ 1/2 0 P S r Σ1/2 0 ] On the other hand, we can compute, trσ 1/2 1 2 tr 0 PjΣ S 1/2 0 Σ 1 0 G kσ 1/2 0 Σ 1/2 0 PjΣ S 1/2 0 [ 1 2 vec Σ 1/2 0 Σ 1 0 G kσ 1/2 0 trg k [ 1 2 vec Σ 1/2 0 Σ 1 0 [ Σ 1/2 0 Σ 1 0 G kσ 1/2 0 trσ1/2 0 Σ 1 0 G kσ 1/2 G k trg k I I ] S 0 vecσ 1/2 0 P S jσ 1/2 0 ] S I ] S Σ 1/2 0 vecσ 1/2 0 PjΣ S 1/2 0 for j 1, r and k 1, 4 Therefore, the generalized Schwartz inequality implies that A B 1 A 1 2 [vecσ1/2 0 Σ 1 0 Γ 1 S Σ 1/2 0,, vecσ 1/2 0 Σ 1 0 Γ 4 S Σ 1/2 0 ] [vecσ 1/2 0 Σ 1 0 Γ 1 S Σ 1/2 0,, vecσ 1/2 0 Σ 1 0 Γ 4 S Σ 1/2 0 ] trσ 0 Σ 1 0 Γ 1 S G 1 trσ 0 Σ 1 0 Γ 1 S G 4, trσ 0 Σ 1 0 Γ 4 S G 1 trσ 0 Σ 1 0 Γ 4 S G 4 where Γ k G k trg k /I, for k 1,, 4 Hence, the best selection of the P s will be the matrices Σ 1 0 [G k trg k /I ] for k 1,, 4 16

17 In practice, with initial consistent estimates of λ, β and the variances, we can get consistent estimates ˆΣ, Ĥ and the Ĝ s The following proposition summarizes the results above Proposition 3 Let Q Σ 1 0 H, X and P i Σ 1 0 [G k trg k /I ] for i 1,, 4 Under Assumptions 1-3, the consistent root ˆθ b, derived from min θ Θ g θω 1g θ, where Ω varg θ 0 and g θ Q, P 1 ɛ θ,, P 4 ɛ θ ɛ θ, is the BGMME with the asymptotic distribution ˆθ b, θ 0 D 0, Φ 1 b and 1 Φ b lim Φ 11 H Σ 1 0 X X Σ 1 0 H X Σ 1 0 X, Φ 11 H 1 Σ 1 0 H 1 + trσ 0 Σ 1 0 Γ 1 S G 1 H 1 Σ 1 0 H 4 + trσ 0 Σ 1 0 Γ 1 S G 4 H 4 Σ 1 0 H 1 + trσ 0 Σ 1 0 Γ 4 S G 1 H 4 Σ 1 0 H 4 + trσ 0 Σ 1 0 Γ 4 S G 4 In order to study the efficiency of the BGMME, we write out the log likelihood function of the structural equation: ln L n ln2π 1 2 ln Σ + ln S λ 1 2 Y W λy X β Σ 1 Y W λy X β The first order conditions are ln L λ 11 H 1Σ 1 ɛ θ + ɛ θσ 1 G 1 ɛ θ trg 1 0, ln L λ 12 H 2Σ 1 ɛ θ + ɛ θσ 1 G 2 ɛ θ trg 2 0, ln L λ 21 H 3Σ 1 ɛ θ + ɛ θσ 1 G 3 ɛ θ trg 3 0, ln L λ 22 H 4Σ 1 ɛ θ + ɛ θσ 1 G 4 ɛ θ trg 4 0, ln L β X Σ 1 ɛ θ 0, and ln L η i 1 2 tr Σ 1 Σ η i 12 ɛ θ Σ 1 η i ɛ θ 0, for i 1, t, 17

18 where the η i is the ith element in the vector η, namely one of the variance parameters The similarity of the best GMM moments and the likelihood equations is revealing However, the efficiency of the BGMME may be affected by the dependence between the disturbances because of the generalized form of the covariance matrix Due to the complicated error structure, it is hard to solve the first order conditions with respect to the variance parameters Thus, we cannot substitute the variance matrix in the first order conditions regarding λ and β and get the first order conditions for the concentrated likelihood function This gives us some hint at the additional moments we may use to get efficient estimates First, we need to compare the variance matrices of the MLE and the BGMME By inverting the Fisher Information and following Anselin and Bera 1998, we can get the asymptotic variance of the MLE ˆθ ml,, ˆη ml, : AsyV arˆθ ml,, ˆη ml, V 11 H Σ 1 X V 13 X Σ 1 H X Σ 1 X 0 V 13 0 V 33 1, H where V 11 H + Ṽ11 with H 1 Σ 1 H 1 H 1 Σ 1 H 4 and H 4 Σ 1 H 1 H 4 Σ 1 H 4 trσ 1/2 Σ 1 G 1 Σ 1/2 Σ 1/2 Σ 1 G 1 S Σ 1/2 trσ 1/2 Σ 1 G 1 Σ 1/2 Σ 1/2 Σ 1 G 4 S Σ 1/2 Ṽ 11 trσ 1/2 Σ 1 G 4 Σ 1/2 Σ 1/2 Σ 1 G 1 S Σ 1/2 trσ 1/2 Σ 1 G 4 Σ 1/2 Σ 1/2 Σ 1 G 4 S Σ 1/2, V 13 tr Σ 1/2 Σ 1 G 1 Σ 1/2 Σ1/2 Σ 1 η 1 Σ 1/2 tr Σ 1/2 Σ 1 G 4 Σ 1/2 Σ1/2 Σ 1 η 1 Σ 1/2 1 4 [vecσ1/2 0 Σ 1 0 G 1 S Σ 1/2 0,, vecσ 1/2 0 Σ 1 0 G 4 S Σ 1/2 0 ] [vecσ 1/2 0 Σ 1 η 1 S Σ 1/2 0,, vecσ 1/2 0 Σ 1 η t S Σ 1/2 0 ] 1 4 F D, tr Σ 1/2 Σ 1 G 1 Σ 1/2 Σ1/2 Σ 1 η t Σ 1/2 tr Σ 1/2 Σ 1 G 4 Σ 1/2 Σ1/2 Σ 1 η t Σ 1/2 18

19 and V 33 tr Σ tr Σ 1/2 Σ 1 η 1 1/2 Σ 1 η t Σ 1/2 Σ1/2 Σ 1 η 1 Σ 1/2 Σ 1/2 Σ1/2 Σ 1 η 1 Σ 1/2 tr Σ1/2 Σ 1 η 1 tr Σ 1/2 Σ 1 η t 1 4 [vecσ1/2 0 Σ 1 S Σ 1/2 0,, vecσ 1/2 0 Σ 1 S Σ 1/2 0 ] η 1 η t [vecσ 1/2 0 Σ 1 η 1 S Σ 1/2 0,, vecσ 1/2 0 Σ 1 η t S Σ 1/2 0 ] 1 4 D D, Σ 1/2 Σ1/2 Σ 1 η t Σ 1/2 Σ 1/2 Σ1/2 Σ 1 η t Σ 1/2 with F [vecσ 1/2 0 Σ 1 0 G 1 S Σ 1/2 0,, vecσ 1/2 0 Σ 1 0 G 4 S Σ 1/2 0 ] and D [vecσ 1/2 0 Σ 1 η 1 S Σ 1/2 0,, vecσ 1/2 0 Σ 1 η t S Σ 1/2 0 ] From the inverse of a partitioned matrix, the asymptotic variance of the MLE ˆθ ml, is AsyV arˆθ ml, V 11 V 13 V H Σ 1 X X Σ 1 H X Σ 1 X 1 Here V 13 V 1 33 V F DD D 1 D F If the disturbances are iid, then V 13 V 1 33 V F veci vec I F Thus, trg Ṽ 11 V 13 V33 1 V 13 Ṽ trg 1 trg 1 trg 4 trg 4 trg 1 trg 4 trg 4 trg 1 ΣΣ 1 Γ 1 S trg 1 ΣΣ 1 Γ 4 S trg 4 ΣΣ 1 Γ 1 S trg 4 ΣΣ 1 Γ 4 S Hence, the asymptotic variance of the MLE would be the same as that of the BGMME with Φ 11 V 11 V 13 V33 1 V 13 However, when the disturbances are not iid, these two matrices are not equal and there is no clear relationship between them Thus, the BGMME is not as efficient as the MLE since the variance of the MLE attains the lower bound of Fisher Information evertheless, it is more efficient than any other estimator in the specified class such as the 2SLS estimator and the 19

20 3SLS estimator The following proposition summarizes these results and shows that the feasible BGMME has the same limiting distribution as the BGMME Proposition 4 Suppose ˆθ and ˆη are consistent estimates of θ 0 and η 0 Then the problem min θ Θ ĝ 1 θˆω ĝ θ, where ĝ θ ˆQ, ˆP 1 ɛ θ,, ˆP 4 ɛ θ ɛ θ, yields the feasible BGMME ˆθ fb, which has the same limiting distribution as that of ˆθb, This estimator is the best in the class, but generally not as efficient as the MLE In the special case where the disturbances are iid, the feasible BGMME is efficient This result is clearly different than what we see in the previous work of Lee and Liu From the analysis of the first order conditions above, we know that the problem arises from the complicated structure of the variance matrix In order to achieve the efficiency of the MLE, we need to estimate not only λ and β, but also the variance parameters η simultaneously Following this idea, we develop a new class of moment functions g θ Q ɛ θ, ɛ θp 1 ɛ θ trp 1 Σθ,, ɛ θp r ɛ θ trp r Σθ With this modification, we no longer require trp i 0 for i 1,, The identification conditions only need to be slightly modified due to the similar form of moment functions Denote ϑ λ 11, λ 12, λ 21, λ 22, η, β The score function D now becomes Eg θ 0 ϑ Q H 1 Q H Q X trσ 0 P1 S G 1 trσ 0 P1 S G Σ 4 trp 0 Σ 1 η 1 trp 0 1 η t 0 trσ 0 Pr S G 1 trσ 0 Pr S G Σ 4 trp 0 Σ r η 1 trp 0 r η t 0 Q H 1 Q H Q X trσ 0 P1 S G 1 trσ 0 P1 S G Σ 4 trp 1 Σ 1 0 Σ 0 η 1 Σ 0 trp 1 Σ η t Σ 0 0 trσ 0 Pr S G 1 trσ 0 Pr S G Σ 4 trp r Σ 1 0 Σ 0 η 1 Σ 0 trp r Σ η t Σ

21 The variance matrix of the moment functions is slightly changed to Ω Q Σ 0Q trσ 0 P 1 Σ 0 P1 S trσ 0P 1 trσ 0 P 1 trσ 0 P1 S Σ 0Pr S trσ 0P 1 Σ 0 Pr S 0 trσ 0 P r Σ 0 P1 S trσ 0P r trσ 0 P 1 trσ 0 P r Σ 0 Pr S trσ 0P r trσ 0 P r Using similar steps in the previous case and exploiting the symmetry of the variance matrix, we can get the asymptotic precision matrix of the GMME for λ, η and β: lim D 1 Ω D à B 1 à C, where à A, A + with trσ 1/2 0 P 1 Σ 1/2 0 Σ 1 0 η 1 S Σ 1/2 0 trσ 1/2 0 P r Σ 1/2 0 Σ 1 0 η 1 S Σ 1/2 0 A + 1 2, trσ 1/2 0 P 1 Σ 1/2 0 Σ 1 0 η t S Σ 1/2 0 trσ 1/2 0 P r Σ 1/2 0 Σ 1 0 η t S Σ 1/2 0 trσ 1/2 0 P 1 Σ 1/2 0 P1 S Σ1/2 0 trσ 1/2 0 P 1 Σ 1/2 0 Pr S Σ1/2 0 B, and C H, 0, X Q Q Σ 0 trσ 1/2 0 P r Σ 1/2 0 P1 S Σ1/2 0 trσ 1/2 0 P r Σ 1/2 0 Pr S Σ1/2 0 Again, by the generalized Schwartz inequality, the best selection of Q will be the matrix Σ 1 0 H, X just as before As for the best selection of the P s, it is not hard to figure out that we still need the matrices P i Σ 1 0 G i for i 1,, 4 In addition, we need some moments with P j Σ 1 η j for j 1,, t If we substitute these matrices in the asymptotic precision matrix, we can see that it is equal to the asymptotic precision matrix of MLE, which means that this GMME is efficient The following proposition summarizes the results above Proposition 5 Let Q Σ 1 0 H, X, P i Σ 1 0 G k for i 1,, 4 and P j Σ 1 η j j 1,, t Under Assumptions 1-3, the consistent root ˆθ b, derived from min θ Θ g θω 1g θ, where Ω varg θ 0 and g θ Q ɛ θ, ɛ θp 1 ɛ θ trp1 Σθ,, ɛ θp 4 ɛ θ trp4 Σθ, ɛ θp 1 ɛ θ trp1 Σθ,, ɛ θp t ɛ θ trpt Σθ, is the best GMME for λ, β and η This GMME is efficient for 21

22 We will refer to this estimator as the EGMME in the Monte Carlo study The asymptotic properties of the feasible EGMME are also easy to obtain 6 Multiple etwork Systems So far we have only considered a system with two networks In reality, however, we often needs to deal with multiple networks That is what we will briefly discuss in this section A spatial autoregressive model with R interrelated networks can be formulated as follows Y 1,n1 λ 11 W 11 Y 1,n1 + λ 12 W 12 Y 2,n2 + + λ 1R W 1R Y 1,nR + X 1,n1 β 1 + ɛ 1,n1, Y 2,n2 λ 21 W 21 Y 1,n1 + λ 22 W 22 Y 2,n2 + + λ 2R W 1R Y 1,nR + X 2,n2 β 2 + ɛ 2,n2, Y R,nR λ R1 W R1 Y 1,n1 + λ R2 W R2 Y 2,n2 + + λ RR W RR Y 1,nR + X R,nR β 1 + ɛ R,nR As we see in previous sections, the structure of the variance matrix does not affect the optimal selection of the moment functions Therefore, we do not need to place any restriction on the variance matrix, but only assumes that it contains t unknown parameters η 1,, η t In practice, we can allow the disturbances to have a panel data-like structure if the number of networks is large For example, we can assume that Eɛ r,nr ɛ r,n r σ 2 uι nr ι n r +σ 2 vi nr for r 1,, R and Eɛ r,nr ɛ s,n s 0 if r s These assumptions allow us to get consistent estimators of σ 2 u and σ 2 v We can conveniently extend the results in previous sections to the multiple network systems Define all the G and H matrices in a similar fashion as in the previous sections Then the EGMME can be obtained according to the following proposition Proposition 6 Let Q Σ 1 0 H, X, P i Σ 1 0 G k for i 1,, R 2 and P j Σ 1 η j for j 1,, t Under Assumptions 1-3, the consistent root ˆθ b, derived from min θ Θ g θω 1g θ, where Ω varg θ 0 and g θ Q ɛ θ, ɛ θp 1 ɛ θ trp1 Σθ,, ɛ θp R 2, ɛ θ trpr 2, Σθ, ɛ θp 1 ɛ θ trp1 Σθ,, ɛ θp t ɛ θ trpt Σθ, is the best GMME for λ, β and η This GMME is efficient The asymptotic properties of the feasible EGMME are also easy to obtain 22

23 7 Monte Carlo Study in progress The model in the Monte Carlo study is specified as Y 1,m λ 11 W 11 Y 1,m + λ 12 W 12 Y 2,n + X 1,m β 1 + ɛ 1,m, Y 2,n λ 21 W 21 Y 1,m + λ 22 W 22 Y 2,n + X 2,n β 2 + ɛ 2,n, where x 1i and x 2i are independently generated variables x 1i s are iid 50, 1 and x 2i s are iid 50, 15 There is heteroskedasticity in the disturbances across the two networks: ɛ 1i s are iid 0, 1 and ɛ 2i s are iid 0, 2 For the ease of simulation we assume that the fixed effects are zero The weights matrices W s are randomly generated and row normalized The sample size of the first network is either 100, 200 or 400 and the sample size of the second network is 80 percent that of the first network The estimation methods considered are the 1 2SLS the 2SLS method with IV s W 11 X 1, W 2 11 X 1, W 11 W 12 X 2 for W 11 Y 1, W 12 X 2, W 12 W 21 X 1, W 12 W 22 X 2 for W 12 Y 2, W 21 X 1, W 21 W 11 X 1, W 21 W 22 X 2 for W 21 Y 1, W 22 X 2, W 22 W 21 X 1, W 2 22 X 2 for W 22 Y 2 and X 1, X 2 ; 2 BGMM the best GMM approach by using Q ˆΣ 1 Ĥ, X for the linear moments and P i ˆΣ 1 [Ĝk trĝk/i ] for i 1,, 4 for the quadratic moments with initial consistent estimates for λ, β and η; 3 EGMM the efficient GMM approach by using Q ˆΣ 1 Ĥ, X for the linear moments and P i ˆΣ 1 Ĝ k for i 1,, 4 and P j initial consistent estimates for λ, β and η ˆΣ 1 η j for j 1, 2 for the quadratic moments with The number of repetitions is 300 in this Monte Carlo experiment The regressors are randomly redrawn for each repetition In each case, we report the bias, the standard deviation and the root mean squared error of the empirical distributions of the estimates The following table reports the results where λ 11,0 06, λ 12,0 02, λ 21,0 01, λ 22,0 05, β 1,0 1 and β 2,

24 2SLS BGMM EGMM BiasSD[RMSE] BiasSD[RMSE] BiasSD[RMSE] 180 λ [0999] [0681] [0660] λ [1561] [1066] [1032] λ [1687] [1048] [0948] λ [2566] [1593] [1440] β [0107] [0106] [0106] β [0135] [0135] [0135] 360 λ [1087] [0590] [0589] λ [1681] [0916] [0922] λ [1715] [0981] [0872] λ [2622] [1488] [1813] β [0073] [0072] [0072] β [0097] [0097] [0096] 720 λ [3018] [0544] [0505] λ [4667] [0844] [0779] λ [3643] [0955] [0889] λ [5608] [1460] [1363] β [0056] [0052] [0052] β [0068] [0065] [0065] From this table, we have several findings First of all, the bias and the standard deviation generally become smaller as the sample size grows larger for the BGMMEs and the EGMMEs On the other hand, the 2SLSEs have large biases and standard deviations, and thus provide poor performances Also notice that the bias and the standard deviation for the β s are very small in every case This happens because X 1 and X 2 are exogenous regressors Secondly, the BGMM and EGMM approaches yield estimates with smaller bias and smaller standard deviation compared with the 2SLS approach for almost every variable and every sample 24

25 size In the few situations where this is not the case, the BGMME and EGMME still have the smaller root mean squared errors These results show that the GMM approaches almost always outperform the 2SLS approach in estimating the SAR model in a system of interrelated networks These findings are in line with the conclusions in the previous sections Finally, when we compare the BGMME and the EGMME, it seems that the EGMME generally outperforms the BGMME In most cases the EGMME has a smaller standard deviation than the BGMME Once again, the Monte Carlo simulations support the conclusion that the EGMME is more efficient than the BGMME This suggests that we estimate the variance parameters together with the λ s and β s when error term is not iid 8 Conclusion In this paper, I introduce a very general spatial autoregressive model with multiple equations This model can be used to analyze a wide range of empirical problems which involve the interactions between individuals in the same network and across different networks Just like in the high order SAR models, we cannot estimate this model by maximum likelihood for practical reasons While previous research uses the 2SLS method, it does not possess desirable efficiency properties Therefore, this paper aims at developing GMM estimation approaches in this framework that have the same efficiency property as MLE I extend the GMM method in the previous literature for the high order SAR model This approach improves upon 2SLS method by incorporating quadartic moment conditions However, this GMME is generally not as efficient as the MLE in this framework due to the complex structure of the disturbances although it is still the best estimator in the defined class Because of the discrepancy I look for a new class of estimators by modifying the structure of the quadratic moments and including new moments for the variance parameters The resulting new GMME are asymptotically as efficient as the MLE under normality I also present some evidence from Monte Carlo experiments that the GMMEs outperform the 2SLSE in finite sample Furthermore, the new EGMME generally has smaller standard deviation than the BGMME Therefore, the simulation results support the theoretical conclusions in this paper 25

26 Appendix in progress Proof of Proposition 1 This proof is similar to the one in Lee and Liu 2010 The difference between this model and a high order SAR model lies in the structure of the error term While Lee and Liu assume that the disturbances are iid, I do not place any structural assumption on the disturbances However, dropping the iid assumption does not affect the validity of the original proof This is because the variance parameters are all included in Ω, which is the variance matrix of the moment functions The proof only involves operations on Ω rather than the variance parameters Therefore, Proposition 1 still holds when we have a generalized error structure Proof of Proposition 2 Once again, the proof is similar to the one in Lee and Liu 2010 Since the original proof only involves operations on Ω, we can directly use it in the new context The generalized Schwartz inequality implies that the optimal weighting matrix for a a in Proposition 1 is 1 Ω 1 The rest of the proof follows References Anselin, L 1988 Spatial Econometrics: Methods and Models, Kluwer Academic Publishers, Dordrecht Kelejian, H H and Prucha, IR 2004 Estimation of simultaneous systems of spatially interrelated cross sectional equations, Journal of Econometrics 118: Lee, L F 2001 Generalized method of moments estimation of spatial autoregressive processes Manuscript, Department of Economics, Ohio State University Lee, L F 2007 GMM and 2SLS estimation of mixed regressive, spatial autoregressive models, Journal of Econometrics 137: Lee, L F and Liu, X 2010 Efficient GMM estimation of high order spatial autoregressive models with autoregressive disturbances, Econometric Theory 26: Ord, J 1975 Estimation methods for models of spatial interaction, Journal of the American Statistical Association 70: Tao, J 2005 Spatial econometrics: models, methods and applications PhD thesis, Department of Economics, Ohio State University 26