E cient GMM estimation of spatial dynamic panel data models with xed e ects

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1 E cient GMM estimation of satial dynamic anel data models with xed e ects Lung-fei Lee Deartment of Economics Ohio State University l ee@econ.ohio-state.edu Jihai Yu Guanghua School of Management Peking University jihai.yu@gmail.com First draft: Aril, Second draft: August, Third draft: Setember, Fourth draft: June, 3 This draft: February, Abstract In this aer we derive the asymtotic roerties of GMM estimators for the satial dynamic anel data model with xed e ects when n is large, and T can be large, but small relative to n. The GMM estimation methods are designed with the xed individual and time e ects eliminated from the model, and are comutationally tractable even under circumstances where ML aroach would be either infeasible or comutationally comlicated. The ML aroach would be infeasible if the satial weights matrix is not row-normalized while the time e ects are eliminated, and would be comutationally intractable if there are multile satial weights matrices in the model; also, consistency of the MLE would require T to be large and not small relative to n if the xed e ects are jointly estimated with other arameters of interest. The GMM aroach can overcome all these di culties. We use exogenous and redetermined variables as instruments for linear moments, along with several levels of their neighboring variables and additional quadratic moments. We stack u the data and construct the best linear and quadratic moment conditions. An alternative aroach is to use searate moment conditions for each eriod, which gives rise to many moments estimation. We show that these GMM estimators are nt consistent, asymtotically normal, and can be relatively e cient. We comare these aroaches on their nite samle erformance by Monte Carlo. JEL classi cation: C3; C3; R5 Keywords: Satial autoregression, Dynamic anels, Fixed e ects, Generalized method of moment, Many moments We would like to thank the co-editor, Professor Hsiao Cheng, of this journal and two anonymous referees for helful comments.

2 Introduction Recently, there is a growing literature on satial anel and dynamic anel models. By including satial e ects into static or dynamic anel models, we can take into account the cross section deendence from contemoraneous or lagged cross section interactions. Kaoor et al. (7) extend the method of moments estimation to a satial anel model with error comonents. Baltagi et al. (7) consider the testing of satial and serial deendence in an extended error comonents model, where serial correlation on each satial unit over time and satial deendence across satial units are in the disturbances. Su and Yang (7) study a dynamic anel data model with satial error and random e ects. These anel models secify satial correlations by including satially correlated disturbances and have emhasized on error comonents. In the xed e ects setting, Korniotis () estimates a time-sace recursive model, where individual time lag and satial time lag are resent, by the least square dummy variable (LSDV) regression aroach. Yu et al. (, ) and Yu and Lee () study the quasi maximum likelihood (QML) estimation for, resectively, the stable, satial cointegration, and unit root satial dynamic anel data (SDPD) models, where individual time lag, satial time lag and contemoraneous satial lag are all included. For the stable SDPD model with xed e ects, the asymtotics of the QML estimation in Yu et al. () is develoed under T! where T cannot be too small relative to n. In emirical alications, we might have data sets where n is large while T is relatively small. Under this circumstance, in the literature of dynamic anels without satial interactions, the maximum likelihood estimator (MLE) of the autoregressive coe cient of a linear dynamic anel model, which is also known as the within estimator, is biased and inconsistent when n tends to in nity but T remains nite (Nickell, 9; Hsiao, 9). This bias is due to the incidental arameter roblem in Neyman and Scott (9). By taking time di erences to eliminate individual xed e ects in the dynamic equation, the estimation method of instrumental variables (IV) is oular (see Anderson and Hsiao, 9; Arellano and Bond, 99; Arellano and Bover, 995; Blundell and Bond, 99; Bun and Kiviet,, etc). This motivates our study of generalized method of moments (GMM) estimation of the SDPD model in order to cover the scenario that both n and T can be large, but T is small relative to n. The case of a nite T will also be considered. In this aer, we investigate the GMM estimation of an SDPD model with ossibly high order satial lags. The inclusion of high order satial lags can allow satial deendence from di erent interactions characteristics such as geograhical contiguity and economic interaction. Comared to QML estimation, the GMM estimation has the following merits for the The reason for focusing on the asymtotic with T! instead of a nite T is that, in this framework, we have the best IV or best GMM estimation with roer designs of IVs and moment conditions. This might not be ossible for a xed e ects model when T is assumed to be nite. In addition, a high order satial lag model can be regarded as a general case of the rst order satial lag model with satial disturbances. To see this, for a cross sectional SAR model Y n = W ny n + X n + U n where U n = M nu n + V n,

3 SDPD model: () GMM has a comutational advantage over MLE, because GMM does not need to comute the determinant of the Jacobian matrix in the likelihood function for a satial model, which is esecially inconvenient for MLE when n is large or the model has high order satial lags; 3 () some GMM methods can be alied to a short SDPD model and is free of asymtotic bias, while ML estimation of the SDPD model requires a large T and a bias correction rocedure is needed to eliminate the asymtotic bias. For a nite T case, an initial seci cation for the rst time eriod observations would also be needed in order to formulate a likelihood function. 5 (3) with carefully designed moment conditions, the GMM estimate can be more e cient than the QML estimate when the true distribution of the disturbances are not normal and has a nonzero degree of excess kurtosis; () GMM is also alicable for the SDPD model with time e ects and non-row-normalized satial weights matrices. 7 Comared to dynamic anel data models where serial correlation occurs in the time dimension, the SDPD model may have correlation in the time dimension as well as satial correlation across units. In one aroach, we stack u the data and use moment conditions where the IVs have a xed column dimension for all the eriods. In another, we can use searate moment conditions for each time eriod, which result in many moments. Those many moments not only come from time lags, but are also designed for satial lags. We focus on the design of estimation methods that can have some asymtotic e cient roerties. Normalized with remultilication of (I n M n), we have Y n = ( W n + M n M nw n)y nt + (I n M n)x n + V n after re-arrangement. This is a high order satial lags model with satial weights matrices W n, M n, and M nw n and constrained coe cients. 3 For a rst order SAR model where the satial weights matrix is diagonalizable, the determinant of the Jacobian term can be comuted by its eigenvalues (see Ord, 975). If the satial weights matrix is not diagonalizable or we have some higher order satial lags, the Ord device might not be alicable. We note that, P to construct the best instruments for the GMM in Section 3.., we need to inverse the n n matrix S n() = I n j= jw nj in () (the matrix inversion is also involved in obtainin the information matrix of ML estimation). This will cause a comutation burden if n is large. However, unlike the comutation of the determinant in ML estimation that is reeated in the arameter search, the matrix inverse comutation needs to be obtained only once given a consistent estimate of arameter vector so that the comutation burden is less severe (we can use ower series exansion to comute the matrix inverse if necessary). 5 Elhorst () has develoed an ML estimation using the initial value aroximation in Bhargava and Sargan (93), which does not have much bias from their Monte Carlo results. Due to the multile dimension search in the nonlinear variance matrix function, the ML estimation in Elhorst () is comutationally comlicated; also, it has a larger bias than the GMME. In Yu et al. (), the consistency of ML estimator is derived under large n and large T. The MLE can have satisfactory nite samle results after the bias correction from the Monte Carlo simulation. Both Yu et al. () and Elhorst () work well under a rst order SDPD model. It is ossible to eliminate the time e ects by taking cross sectional di erence, but the resulting equation would not have an SAR reresentation and, therefore, one cannot set u a likelihood function for estimation. The MLE will have an additional incidental arameter roblem if time e ects need to be estimated in addition to the individual e ects. 7 Bell and Bockstael () argue that, based on some underlying economic story, it is not necessarily to always row-normalize the satial weights matrix. In some cases, row-standardizing changes the total imact of neighbors across observations, although it does not change the relative deendence among all neighbors of any given observation. They use real estate roblem to argue that row-standardizing will attach too much weight to the neighbors of remote houses. In social interaction and network literatures, when the social interaction is seci ed as an SAR model, the measure of centrality in Bonacich (97) comes out naturally in the reduced form equation. When the indegrees (the sums of each row) of the sociomatrix have a non-zero variation, so does the Bonacich centrality measure, which hels to identify the various interaction e ects. Therefore, in emirical alications, sometimes a satial weights matrix without row-normalization would be aroriate. For estimation rocedure in satial econometrics, Kelejian and Prucha (7) consider imlications on the arameter sace of the SAR model when the satial weights matrix is not row-normalized.

4 asymtotic distributions of IV estimators with a nite number of moments are roerly centered at the true arameter vector. In the many moment aroach, normalized asymtotic distributions of IV estimates might not be roerly centered or an IV estimator might not be consistent due to the many IV moments (but not directly due to the xed e ects). In contrast to the asymtotics in Yu et al. () where there are ratio conditions on how T and n go to in nity in order that ML estimates can be consistent or their normalized asymtotic distributions are roerly centered, such ratio conditions may no longer be needed in the roosed GMM estimation with a nite number of moments in the resent aer. In the many IVs estimation method, the ratio condition concerns about the number of IVs or moments relative to the total samle size nt, but not directly the ratio of T and n. However, if the total number of IVs is essentially a function of T, then n and T ratio conditions would aear; but in that case, the ratio condition requires that T shall be small relative to n. Thus, the many IVs aroach is comlementary to the QML aroach. In other words, the roosed estimation methods can be alied to some scenarios where the T is small relative to n, while the QML method might not be so, in theory. The aer is organized as follows. Section introduces the model and discusses moment conditions. Section 3 investigates the consistency and asymtotic distribution of various GMM estimators, and we discuss the asymtotic e ciency of the roosed estimators. Monte Carlo results for various estimators are rovided in Section. Section 5 concludes the aer and summarizes the contributions. Some lemmas and roofs are collected in the Aendices. 9 The Model and Moment Conditions. The Model The model under consideration is an SDPD model with both individual and time e ects Y nt = X j= jw nj Y nt + Y n;t + X j= jw nj Y n;t +X nt +c n + t l n +V nt ; t = ; ; ; T, () where Y nt = (y t ; y t ; ; y nt ) and V nt = (v t ; v t ; ; v nt ) are n column vectors, and v it s are i:i:d: across i and t with zero mean and variance. The W nj is an n n satial weights matrix for j = ; ;, which is nonstochastic and generates the deendence of y it s across satial units. If, () is a high order SAR structure. The W nj s may or may not be row-normalized. X nt is an n k x matrix of nonstochastic regressors, c n = (c ; ; ; c n; ) is an n column vector of individual e ects, and t is a scalar time e ect. The initial values in Y n are assumed to be observable. We imose the normalization l nc n = where l n is However, for the case with multile satial weights matrices, when T is not really small, in order to accommodate satial exansions, the IVs might be too many in order to be ractical. This nite samle issue is resented in the Monte Carlo section. 9 Proofs for lemmas and more Monte Carlo results are rovided in a sulement le available on request. 3

5 an n vector of ones to avoid the un-identi cation of c i; and t because c i; + t = (c i; + ) + ( t ) for an arbitrary. To avoid the incidental arameter roblem in estimating (), the individual e ect c n and time e ect t shall be eliminated. Let [F T;T ; T l T ] be the orthonormal matrix of the eigenvectors of J T = (I T T l T l T ), where F T;T is the T (T ) eigenvectors matrix corresonding to the eigenvalues of one and l T is the T -dimensional vector of ones. The n T matrix of deendent variables [Y n ; Y n ; ; Y nt ] can be transformed into the n (T ) matrix [Y n; Y n; ; Y n;t ] = [Y n; Y n ; ; Y nt ]F T;T ; and, also, (; ) [Y (; ) (; ) n ; Y n ; ; Y n;t ] = [Y n; Y n ; ; Y n;t ]F T;T. It is imortant to note that Y (; ) n;t and Y n;t are not equal. The V nt and X nt are de ned similarly for t = ; :::; T. As l T F T;T =, it follows that [c n ; ; c n ]F T;T = and individual e ects are eliminated by the orthonormal transformation. Among these orthonormal transformations, the forward orthogonal di erence (FOD) transformation (also known as the Helmert transformation) is found to be convenient. After the FOD transformation F T;T to eliminate the individual e ects, () becomes Ynt = X jw nj Y (; ) nt + Y j= n;t + X (; ) jw nj Y j= n;t + X nt + tl n + Vnt, t = ; ; T, () where Vnt = ( T t T t+ ) [V nt T t P T h=t+ V (; ) nh] and Y n;t = ( T t T t+ ) P T [Y n;t T t h=t Y nh] deend on current and future variables, but not on the ast ones, and [ ; ; ; T ; ] = [ ; ; ; T ]F T;T can be considered as transformed time e ects. As (V n; ; V n;t ) = (F T;T I n)(v n; ; V nt ) and F T;T F T;T = I T, the v it s are uncorrelated where v it is the ith element of V nt. The time e ects t can be further eliminated in the following equation: J n Y nt = X j= jj n W nj Y nt + J n Y (; ) n;t +X (; ) jj n W nj Y j= n;t +J nxnt +J n Vnt, t = ; ; T, where J n = I n n l nl n because J n l n =. Estimation of (3) by the ML method have two issues. First, Anderson and Hsiao (9) roose to use the rst di erence to eliminate the individual e ects, where lagged values of the deendent variable can be used as IVs, and the resulting disturbances have serial correlation. Instead of rst di erence, Arellano and Bover (995) use the FOD for the data transformation, where the resulting disturbances are still uncorrelated if they are originally i.i.d. Even if we can estimate the time e ects when n is large under the MLE, we will have the incidental arameter roblem with a large T and the asymtotic distribution of the estimators might not be roerly centered if T does not increase faster than n. See Lee and Yu () for details under a rst order SDPD model (Theorems 3 and ), and also Hahn and Kuersteiner () and Alvarez and Arellano (3) under a dynamic anel data setting. The ML estimation of () will incur incidental arameter roblem as time e ects need to be estimated when T is large and W nj s are not row-normalized. When T is small, we need to secify the initial observation rocess to write down the likelihood function. For these reasons, we focus on the GMM estimation of the following (3). Alvarez and Arellano (3) investigate a random e ects ML estimator where there is no exogenous variable. They have derived the likelihood function of the random comonent model as a roduct of two likelihood functions where one is for the between equation. The likelihood for the xed e ect model is a ratio of the two likelihoods. When exogenous variables are resent and correlated with the individual e ects, it is not obvious on how to derive the likelihood function. Instead, Elhorst () studies an ML aroach by an initial eriod aroximation similar to Bhargava and Sargan (93). (3)

6 when W nj is not row-normalized, (3) would not have a well-de ned SAR structure for J n Y nt. 3 Second, in the resence of time lags variables, the regressors in (3) are correlated with the disturbances after data transformation by F T;T. For these reasons, a likelihood function could not be formed directly from (3). We roose GMM estimation of (3), which does not require an SAR form for J n Y nt and it can be free of asymtotic bias as will be shown. For asymtotic analysis of GMM estimates of (3), the reduced form of () is needed. Denote = ( ; ; ) and = ( ; ; ). At true arameter values, let = ( ; ; ) and = ( ; ; ). For the reduced form from (), by denoting S n () = I n P j= jw nj, S n S n ( ) and A n = Sn ( I n + P j= jw nj ), we have Ynt (; ) = A n Y n;t + S n (Xnt + tl n + Vnt): For each satial lag W nj Ynt for j = ; ;, by de ning G nj W nj Sn, we have W nj Y nt = G nj (Z nt + tl n ) + G nj V nt; () where = ( ; ; ) and Znt (; ) = [Y n;t ; W (; ) ny n;t ; X nt] is the redetermined regressors in () with (; ) (; ) W n Y n;t = (W n Y n;t ; ; W (; ) ny n;t ). In general, for a vector (or matrix) b n with n rows, we denote W s nb n with s being an integer as a matrix consisting of vectors (or matrices) W s n W s n W s nb n where s, s, ; s are nonnegative integers such that s + s + + s = s. For examle, W nb n = [W nb n ; ; W n W n b n ; W n W n b n ; ; W n W n b n ; ; W n W n b n ; ; W nb n ].. Moment Conditions (; ) For the estimation equation (3), we note that Y n;t is correlated with Vnt. For this reason, IVs are (; ) (; ) needed for Y n;t and W nk Y n;t for each t (and also for W nj Ynt). In addition to all strictly exogenous variables X ns for s = ; ; T, the time lag variables Y n ; ; Y n;t can also be used to construct IVs for (; ) Y n;t as in the literature of dynamic anel data models (Alvarez and Arellano, 3, etc). Corresondingly, we may use W nk X ns for s = ; ; T and W nk Y ns for s = ; ; t as IVs for W nk Y for the estimation of (3), we need to nd IVs for the regressors J n [W n Ynt; (; ) Y n;t ; W ny (; ) n;t (; ) n;t. Therefore, ]: (5) For notational urose, we denote I t as the -algebra sanned by (Y n; ; Y n;t ), conditional on (X n ; ; X nt ; c n ; ; ; T ). The best theoretical IV for (5) shall be its conditional mean conditional on I t. We might use the estimated conditional mean (so that we have a nite number of moments 3 When W nj is row-normalized, we have J nw nj = J nw nj J n as W nj l n = l n and J nl n =, which imlies that (3) may have J nynt as a deendent variable in an SAR form. For the SDPD model with row-normalized weights matrix, Lee and Yu () has formulated a artial likelihood aroach for the estimation with the time e ects removed but not the individual e ects. The artial likelihood aroach estimates both the common arameters of interest as well as the individual e ects. 5

7 in Section 3.), or use levels of neighboring variables (satial ower series exansion) of redetermined and exogenous variables to aroximate the best IV (so that we have many moments in Section 3.). When the satial weights matrices are row-normalized, the time e ects will not in uence the conditional mean of (5) (see Lee and Yu, ). However, when W nj s are not row-normalized, the conditional mean of (5) will be in uenced by the time e ects. For the linear moments, we can stack u the data and construct moment conditions. An IV matrix for (5) can take the form J n Q nt where Q nt has a xed column dimension q greater than or equal to k x + +. For examle, Q nt could be Let V n;t [Y n;t ; W n Y n;t ; W ny n;t ; X nt; W n X nt; W nx nt]: () () = (Vn(); ; Vn;T ()) where Vnt() = S nt ()Ynt Z nt t l n with = (; ) and = (; ; ). IV estimation corresonds to the linear moments Q n;t J n;t V n;t () where Q n;t = (Q n; :::; Q n;t ) and J n;t = I T J n. Here, even though we have the transformed time e ect t l n in V nt(), it will be eliminated in the moment conditions by J n. In addition to the linear moments, due to the satial correlation in the DGP, quadratic moments can cature correlations and may increase the e ciency of estimates. 5 The vector P nl V nt can be uncorrelated with J n V nt in (3) for an n n nonstochastic matrix P nl satisfying the roerty tr(p nl J n ) =, while it may correlate with G nj V nt in (). Denoting P nl;t = I T P nl where P nl satis es tr(p nl J n ) =. The quadratic moments are V n;t ()J n;t P nl;t J n;t V n;t () for l = ; ; ; m so that we can have m such quadratic moments. For analytical tractability, we assume that P nl is uniformly bounded in both row and column sums in absolute value (for short, UB). These settings rovide general frameworks in which one may discuss the best designs of Q nt and P nl;t. For the aroach with nite number of moments, the moment conditions would be Vn;T ()J n;t P n;t J n;t Vn;T () g nt () = Vn;T ()J n;t P nm;t J n;t Vn;T () C A : (7) Q n;t J n;t Vn;T () Even though time e ects in () can be eliminated with the J n remultilication so that the estimation equation (3) is free of time e ects for its regressors, the time e ects comonent in the DGP of W nj Y nt cannot be eliminated if W nj is not row-normalized as seen from (). 5 The use of quadratic moments is motivated by the the likelihood function of the SAR model under normality disturbances (Lee, 7), as well as the Moran test statistic (Moran, 95). We say a (sequence of n n) matrix P n is uniformly bounded in row and column sums if su n kp nk < and su n kp nk <, where kp nk su in P n j= j ij;nj is the row sum norm and kp nk = su jn P n i= j ij;n j is the column sum norm.

8 For the many moment aroach, denoting h nt = (Y n ; ; Y n;t ; X n ; ; X nt ; l n ), we can use the IV matrix J n H nt with H nt = (h nt ; W n h nt ; ; W n n h nt ). () The column dimension of h nt is t = k x T + t + and that of H nt can be t ( n ) where n is the order of satial ower series exansion. Here, in addition to the satial ower series exansion of the lagged and exogenous variables, we also include the satial ower series exansion of the n vector l n. When the W nj s are not row-normalized, the satial ower series exansion of l n as roducts with W n and its high orders will aroximate the time e ects comonent G nj l n in the conditional mean of the satial lag in (). On the other hand, if W nj s are row-normalized, neither l n nor its roduct with W n would matter because W nj l n = l n and J n l n =. Combined with the quadratic moments, for the many moment aroach, we have where Diag(H n ; ; H n;t g nt () = Vn;T ()J n;t P n;t J n;t Vn;T (). Vn;T ()J n;t P nm;t J n;t Vn;T () C A ; (9) Diag(H n ; ; H n;t ) J n;t Vn;T () ) is a block diagonal matrix with diagonal blocks H nt s. For (7), the column dimension of Q nt is xed and is the same for all t. For (9), the column dimension of H nt might increase in t. The latter aroach requires careful analysis due to the many moments issue when T!. 3 Asymtotic Proerties of GMME In the following, Section 3. derives the consistency and asymtotic distribution of GMM estimators when we use a nite number of moment conditions where T can be nite or large. Under the framework of T being large, otimal moment conditions can be designed. Section 3. derives the asymtotic roerties of GMM estimators when we use many moment conditions. For our analysis of the asymtotic roerties of estimators, we make the following assumtions. Assumtion. W nj is a nonstochastic satial weights matrix with zero diagonals for j = ; ;. Assumtion. The disturbances fv it g, i = ; ; ; n and t = ; ; ; T; are i:i:d: across i and t with zero mean, variance and E jv it j + < for some >. Assumtion 3. S n () is invertible for all, where the arameter sace is comact and is in the interior of. Assumtion. W nj is UB for j = ; ; and jj P j= jw nj jj <. Also, Sn () is UB, uniformly in. 7

9 P T P n Assumtion 5. X nt, t and c n are nonstochastic with su n;t nt t= i= jx it;lj + < for l = P T ; ; k, su T T t= j tj + P n < and su n n i= jc nij + < for some >, where x it;l is the P T (i; l) element of X nt and c i is the ith element of c n. Also, lim n! n(t ) t= X ntj n Xnt exists and is nonsingular. Assumtion. Y n = P h h= Ah ns n (c n + X n; h + h; l n + V n; h ), where h could be nite or in - nite. Assumtion 7. P h= abs(ah n) is UB where [abs(a n )] ij = ja n;ij j. The zero diagonal assumtion on W nj hels the interretation of the satial e ect as self-in uence shall be excluded in ractice. Assumtion rovides regularity assumtions for v it. Assumtion 3 guarantees that the model is an equilibrium one. Also, the comactness of the arameter sace is a condition for theoretical analysis. 7 In Assumtion, when jj P j= jw nj jj <, Sn can be exanded as an in nite series in terms of W nj. In many emirical alications of satial issues, each of the rows of W nj sums to, which ensures that all the weights are between and. The uniform boundedness assumtion in Assumtion is originated by Kelejian and Prucha (99, ) and also used in Lee (, 7). That W nj and Sn () are UB is a condition that limits the satial correlation to a manageable degree. From Lee and Liu (), the arameter sace for could be one satisfying P j= j jj < (max j=; ; kw nj k ). When exogenous variables X nt are included in the model, an emirical moment of a higher than second order restriction is imosed as in Assumtion 5. So is for the individual e ects and time e ects. The second emirical moment restrictions are useful for some samle statistics to be bounded in our asymtotic analysis. The higher than the second moment restrictions are used in a central limit theorem for a linear and quadratic form in Kelejian and Prucha (). The remaining art of Assumtion 5 oints out that the regressors of X nt are asymtotically linearly indeendent. Assumtion seci es the initial condition so that the rocess may start from a nite or in nite ast. The h can be arbitrary to cature the origin of the rocess from the ast and it does not need to be seci ed in the estimation or asymtotic analysis. Assumtion 7 combines the absolute summability condition and the UB condition of the ower series of A n, which is essential for 7 The invertible S n makes sure the we have an equilibrium system. To construct the best instruments in Section 3.., invertiblity of estimated S n is also needed. For the arameter sace, comactness is relatively stronger than the boundedness assumtion for convenience. Therefore, if one takes this as the arameter sace of interest, for the case of = 3, the true values of the js are not allowed be all.5 or all -.5 when W nj s are row-normalized. Elhorst et al. () discussed the arameter sace of the two order satial lag model for stationarity. They nd that that the rectangular with vertices ( jw ;min j ; jw ;min j ), ( jw ;min j ; ), w ;max ( ; w ;max jw ;min j ) and ( ; ) would be too broad, while the circle with j w ;max w j + j j < (max j=; ; kw nj k ;max ) would be too restrictive. They argue that, according to Hele (995), the exact boundaries for the curves connecting the four coordinates can only be determined by a numerical search. In Elhorst et al. (), they have develoed a rocedure to decide the arameter sace of js of a cross sectional SAR model under the "stationary" condition, where the arameter sace of js will be decided by the largest and smallest eigenvalues of each satial weights matrix in a comlicated way. In the satial dynamic anel data model, the j s, and js shall then satisfy the absolute summability condition in Assumtion 7. Therefore, the arameter sace for the high order satial dynamoc anel data model is comlicated.

10 the analysis in this aer, because it limits the deendence over time series and across satial units Asymtotic Proerties of GMME with Finite Moments 3.. Consistency and Asymtotic Distribution of GMME For the moment conditions in (7), the identi cation requires that lim n! n(t ) g nt () = should have a unique solution. From the linear moments, because Y nt = S n (Z nt + tl n + V nt) and J n l n =, we have J n Vnt() = J n [S n ()Ynt Z nt] = J n [S n ()Sn (Znt + tl n ) Znt + S n ()Sn Vnt] where S n ()S n = I n P j= ( j j )G nj. Denote Z n;t = (Z n; ; Z n;t ), L nj;t = G nj (Z nt + tl n ); L nt = [L n;t; L n;t; ; L n;t] and L n;t = [L n; ; L n;t ] : () We have Q n;t J n;t V n;t () = Q n;t J n;t [Z n;t ( )+L n;t ( )+S n;t ()S where S n;t = I T S n. As lim n! n(t ) Lemma (iv), the unique solution of lim n! n(t P T n;t V n;t ] t= Q ntj n S n ()Sn Vnt = uniformly in from ) g nt () = at requires that lim n! n(t ) Q n;t J n;t [L n;t ; Z n;t ](( ) ; ( ) ) = should have a unique solution : That lim n! n(t ) Q n;t J n;t [L n;t ; Z n;t ] has the full column rank k z +, where k z = + k x +, is a su cient condition. Because Z n;t consists of time and satial time lags, this condition will, in general, be satis ed as long as = because time and satial time lags can be used to construct relevant IVs. Assumtion. The n q IV matrix Q nt is redetermined such that E(Q nt ji t ) = Q nt, its column dimension is xed for all n and t. Let C nt be an n column vector from Q nt. The E[jC nt;i j + ] for some > is bounded uniformly in all i = ; :::; n and all n and t. Additionally, lim n! n(t ) Q n;t J n;t Q n;t is of full rank q and lim n! n(t ) Q n;t J n;t [L n;t ; Z n;t ] is of full rank k z +. As in Hansen s GMM setting (9), one considers a linear transformation of the moment conditions, a nt g nt (), where a nt is a matrix with its number of rows greater than or equal to (k z + ) and a nt is assumed to converge in robability to a constant full rank matrix a. estimation, we need the variance matrix of the moment conditions. For the otimal GMM (OGMM) Let vec D (A) be the column vector formed by diagonal elements of any square matrix A; vec(a) the column vector formed by stacking the columns of A; and A s = A + A. For the variance matrix of the moment conditions in (7), let! nm;t = 9 In this aer, we focus only on the stable dynamic model setting, but not unit root or related issues. Because S n()sn = I n +( )G n, the lim n! n(t ) P T t= Q nt BnV nt = in Lemma imlies uniform convergence in for those terms because the arameter sace of is bounded and aears linearly or in a quadratic form in those relevant terms. 9

11 [vec D (J n;t P n;t J n;t ); ; vec D (J n;t P nm;t J n;t )] and mn;t = [vec(j n;t P n;t J n;t ); ; vec(j n;t P nm;t J n;t )] [vec(j n;t P s n;t J n;t ); ; vec(j n;t P s nm;t J n;t )]: Denote as the fourth moment of v it. The variance matrix of the moments in (7) can be aroximated by nt = + n(t ) n(t ) nm;t mq qm n(t ) Q n;t J n;t Q n;t ( 3 )! nm;t! nm;t. qm qq When v it is normally distributed, the second comonent of nt will be zero because 3 =. For the otimal GMM, nt is used in lace of a nt a nt. Also, as shown in Aendix C., we nt (^ nt ) n(t = D nt + R nt + O, nt where D nt is O() seci ed in (3) and R nt is O T seci ed in (33). Denote DnT = D nt + R nt. Thus, when T is large, the comonent R nt will disaear and D nt will be reduced to D nt asymtotically. Throughout this aer, we assume a large number of satial units n, while the time eriod T could be either large or small. The articular interest in this aer is for the case that T can be large, but small relative to n, as the estimation of such a case has not been exlicitly covered in the satial anel literature. Theorem rovides the consistency and asymtotic distributions of GMM estimates. The results are valid with either a nite T or T!. For the OGMM estimation, we assume that Q n;t redundant IV s so that nt is invertible.! () does not contain Theorem Suose we use the moment conditions in (7). Assume that a lim n! n(t ) g nt () = has a unique root at in. Under Assumtions -, as n!, the GMME ^ nt derived from min g nt ()a nt a nt g nt () is consistent and n(^nt ) d! N(; lim n! T (D nt a nt a nt D nt ) D nt a nt a nt nt a nt a nt D nt (D nt a nt a nt D nt ) ): Here, nt is not exactly the variance matrix of the moment conditions but a consistent estimate. While the elements involving the quadratic moment conditions take the exectation form, the elements involving the linear moment conditions take the regular form without exectations or robability limit (because the Q nt s are functions of redetermined variables). By Lemma (i), the exact variance matrix of the linear moment is n(t ) P T t= E(Q ntjnqnt), which has the same limit as n(t ) Q n;t J n;t Q n;t in nt. The reason we use such a nt is due to its simlicity. As we see from the Monte Carlo results, the estimation of standard deviation of estimates using nt in () is close to the emirical standard deviation. Due to the deviation from the time average to eliminate individual e ects, the third moment 3 is irrelevant in the variance matrix in () even though v it is not normally distributed. Therefore, the variance matrix nt of the moment conditions (7) is block diagonal. See Lemma and its roof in Aendix B.

12 Also, the otimal GMM estimator (OGMME) ^ o;nt derived from min g nt () nt g nt () has n(^o;nt ) d! N(; lim n! T (D nt nt D nt ) ). () Suose that ^ nt nt = o (), then the feasible OGMME 3 derived from min g nt ()^ nt g nt () has the same asymtotic distribution as (). When T is large, () will become n(t )(^ o;nt ) d! N(; lim n;t! (D nt nt D nt ) ) where the D nt is reduced to D nt. The OGMME can be comared with the SLSE alied to (3) using IV matrix Q n;t : ^sl;nt = (W n;t Y n;t ; Z n;t ) M JQ;nT (W n;t Y n;t ; Z n;t ) (3) (W n;t Y n;t ; Z n;t ) M JQ;nT Y n;t, where W n;t Y n;t = [W n;t Y n;t ; ; W n;t Y n;t ] with W nj;t = I T W nj and M JQ;nT = J n;t Q n;t (Q n;t J n;t Q n;t ) Q n;t J n;t. The ^ sl;nt is consistent and asymtotically normal with the limiting variance matrix lim n! n(t ) (L n;t ; Z n;t ) M JQ;nT (L n;t ; Z n;t ). The e ciency of the OGMME ^ o;nt relative to the SLSE is aarent due to the additional quadratic mm mq moments. For the SLS, a nt = qm ( Q n;t J n;t Q n;t ) =, which gives zero weight to the quadratic moments. For otimal GMM, we have " a nt = nm;t + ( 3 )! nm;t! # = nm;t mq qm ( Q n;t J n;t Q n;t ) = : 3.. The Best Linear and Quadratic Moment Conditions under Large T As the quadratic and linear moment conditions of Vn;T do not interact with each other (see Lemma ), nt in () is block diagonal. When T is nite, the choice of the best linear moments might not be obvious. When T is large, the R nt comonent in D nt = D nt +R nt will disaear and the best linear moments would be tractable. Esecially, under the large T setting, we can comare the asymtotic variance matrix of the 3 The otimal weighting matrix involves the true arameter and, which can be consistently estimated using the initial GMM ^ nt. Denote ^r nt = S n(^)y nt Z nt^nt where is the rst di erence and ^r nt = Sn(^)Y nt Znt^ nt. The can be estimated from ^r nt by ^ = n(t ) P T t= ^r nt Jn^r nt and the can be estimated from ^rnt by ^ = ^S n(t ) 3^ where ^S = P n i= P T t= f[jn^rnt] ig. See the sulement le for the estimation of in detail. From the detailed analysis involved, the individual e ects rovide information about the best IV for the model estimation. When T becomes large, those individual e ects can be consistently estimated and the best IV can be based on such information. However, with nite T, the individual e ects cannot be consistently estimated, so the best IVs might not be available.

13 GMM with the ML estimates which are derived by using a direct estimation aroach with the individual e ects being estimated. 5 for GMM estimation when T is large. Thus, in this section, we will investigate the best linear and quadratic moments Under large T, the recision matrix of the OGMME from Theorem is D nt nt D nt = n(t ) (L n;t ; Z n;t ) M JQ;nT (L n;t ; Z n;t ) + O + n(t ) T Cm;nT ( 3! nm;t! nm;t + mn;t ) C m;nt kz kz kzk z!, () where the O( T ) term is related to R nt in D nt. As is derived in Aendix D, the best quadratic moment matrix is where P nj = (G nj tr(g njj n) n J n ) + b n P nj;t = I T Pnj, for j = ; ; (5) h i tr(g diag(j n G nj J n ) njj n) n I n with b n = ( n n ) ( n n + 3 and =, diag(a) denotes the diagonal matrix formed by diagonal elements of a square matrix A. Thus, Pnj is the best within the class of matrices such that tr(p njj n ) =. With such a set of best quadratic moments, the derived GMME is e cient relative to any nite number of quadratic moments used for estimation. When V nt is normally distributed so that = 3, we have b n = and the best quadratic matrix is reduced to I T (G nj tr(g njj n) n J n ) for j = ; ;. For linear moments, the best IV matrix is to choose the conditional mean E(W n Y nt; Z ntji t (; ) the main comonent is E(Y T t S n n ) n ), where n;t ji t ). While this ideal IV matrix might not be directly available, one may P T h=t T hx nh, V ~ n;tt = design an aroximated sequence for it. For that urose, de ne X ~ n;tt = T t S n P T h=t T hv nh, ~ tt = T t S n P T h=t T h h and t = c T t (I n A n T t T t ) where j = P j h= Ah n and c T t = ( T t T t+ ). As is derived in Lemma, the best IV matrix can be obtained from H nt in where n;t = H nt + [ t(i n A n ) Sn t (; ) Y X t s= V ns c T t ~ Vn;tT ]; () H nt = t[y n;t (I n A n ) X t S n t (S ny ns Z ns s l n )] c T t Xn;tT ~ c T t ~ tt l n. (7) s= (; ) Thus, the best theoretically IV J n E(Y n;t ji t ) can be aroximated by redetermined variables u to the eriod t and exogenous variables u to the eriod T via J n H nt. Even though t(i n A n ) Sn P t s= V ns in () is in I t but cannot be observed, it can be ignored. Indeed, it can be small t 5 Such an ML estimation method has been considered in detail in Lee and Yu (). That E(W ny nt ; Z nt ji t ) is the best IV can be seen from the asymtotic variance comonent of a GMM estimator due to the IVs in Theorem.

14 as long as t is far from the initial eriod. Thus, the aroximation can be accurate for those t s far away from the initial eriod t =. Hence, we may use J n H nt as a desirable IV for J n Y (; ) n;t. For t =, we may simly take H n = Y n c T Xn;T ~ c T ~ T l n. For these IVs with t s close to the initial eriod t =, the aroximations yield valid IVs but might not be adequate. However, as T is large, the segment with early observations is short relative to the later segment of observations; asymtotically, these IVs are adequate. Therefore, the best IV for J n Znt may be taken as J n K nt where K nt (H nt ; W n H nt ; Xnt). Also, from (), the best IV for J n W nj Ynt is J n G nj (K nt + tl n ). This suggests that we may use J n Q nt as an IV matrix for J n (W n Ynt; Znt) where Q nt = (G n (K nt + tl n ); K nt ), () and G n (K nt + tl n ) = [G n (K nt + tl n ); ; G n (K nt + tl n )]. For feasible IVs, there are two sources of unobservables, namely, c n and [ t ; ; T ; ] in (7), while other unknown arameters can be estimated consistently from Theorem. After Q nt is remultilied by J n if W nj s are row-normalized, the time e ects comonents ~ tt l n in (7) and tl n in () will be eliminated as they are roortional to l n ; however, they may not be eliminated when W nj s are not rownormalized. Anyhow, as n is large, the time e ects for each eriod can be estimated consistently. Denoting ^r nt = S n (^ nt )Y nt Z nt^nt as the estimate for r nt = c n + t l n (= S n Y nt Z nt V nt ). With l nc n = imosed, the estimate for t is ^ t = n l n^r nt for t = ; ; T. It follows that ~ tt l n in (7) and tl n in () can be estimated by using the estimated ^ s for s = t; ; T version for () is. By lugging them back to (7) and () along with the estimated IVs, the feasible ^Q nt = ( ^G n ( ^K nt^ + ^ t l n ); ^K nt ) (9) where ^G n, ^K nt, ^ and ^ t are feasible counterarts constructed with initial consistent estimates as described. Assumtion 9. The robability limit of nt; = n(t ) (L n;t ; Z n;t ) J n;t (L n;t ; Z n;t ) is nonsingular. Theorem Suose we use the moment conditions in (7) where Q nt takes the secial form ^Q nt in (9) and ^P nj;t is estimated from (5) for j = ; :::;.7 Under Assumtions -9, as both n and T tend to in nity, the feasible best GMME (BGMME) ^ b;nt derived from min g nt ()^ nt g nt (), where ^ nt nt = o (), 7 The P nj;t involves the true arameter, and, where can be estimated from Theorem and the moments arameters and can be consistently estimated as exlained in footnote 3. 3

15 has n(t )(^ b;nt )! d N ; b where b = lim n(t ) Cb ;nt kz n;t! kz kzk z with C b ;nt = (I T G nj ). + lim n;t! nt;, () tr(g n;t J n;t P s n;t J n;t ) tr(g n;t J n;t P s n;t J n;t )..... tr(g n;t J n;t P s n;t J n;t ) tr(g n;t J n;t P s n;t J n;t ) C A and G nj;t = For the QML estimator (QMLE) in Yu et al. (), it has an O(=T ) bias, which can be eliminated but requires the condition that T 3 n!. From Theorem, the BGMME does not have a bias term with such an order. Under normality of V nt, the BGMME and MLE have the same asymtotic variance. However, when V nt is not normally distributed, the BGMME with the best IV and best quadratic moment matrix in (5) can be more e cient than the QMLE. This e ciency comes from the quadratic moment in the GMM estimation incororates kurtosis of the disturbances. 9 We note that when T is nite, the roosed BGMME is still consistent and asymtotic normal. However, its limiting variance matrix is not the inverse of b, but the inverse of c;t + r;t where c;t lim n(t ) Cb ;nt kz n! kz kzk z + lim n! n(t ) (L n;t ; Z n;t ) M JQ;nT (L n;t ; Z n;t ); with M JQ;nT = J n;t Q n;t (Q n;t J n;t Q n;t ) Q n;t J n;t, and r;t is the comonent associated with R nt so that r;t = lim = T n! R nt lim n! + T lim n! nt R nt + RnT nt D nt + DnT nt R nt b m [C ] b m b m[c ] b mz b mz[c ] b m b mz[c ] + b b mz T lim m [C ] C;nT b n! b mz[c ] C;nT b C b ;nt [C ] b m C;nT b [C ] b mz kz kzk z kz kzk z where C = 3! nm;t! nm;t + mn;t is related to the variance matrix of the best quadratic moments, and b m and b mz are de ned from (33). The SDPD model considered in Yu et al. () has one satial lag, but the argument for the bias order of the QMLE will be alicable to the general case with multile satial lags. 9 The IV moments use rst moment roerty of the disturbances to set u an estimation framework, and quadratic moments use the second moment roerty. Thus, the best GMM is referred to the best choice of linear and quadratic moments. Comared with QMLE, some linear combinations of the rst and second moments of GMME characterize the rst order conditions of QMLE. Such linear combinations give rise to the e cient GMM estimator under the normal disturbances, but they are not the best combinations under other distributions rather than normal. Other GMM estimators based on additional moments higher than the rst two might be ossible to rovide more e cient estimate than the BGMME currently established in our aer. For the SAR model for cross section data, this has been investigated in Liu et al. (). ()

16 Hence, for the GMM estimation using nite number of moments, we can follow the following two stes. Ste : Obtain an initial consistent estimate ~ with the linear IV in () and quadratic matrices I T (W nj trw nj n J n) and I T (W nj tr(w nj ) n J n ) for j = ;. Ste : By using ~, obtain the best linear IV from (9) and the best quadratic matrices from (5). Then imlement the GMM estimation. 3. Asymtotic Proerties of GMME with Many Moments When T is nite, the IVs from all the available time lags variables may imrove, in rincile, the asymtotic e ciency of the estimators. When T is moderate or large, however, the many moment issue will aear. In the literature on IV and GMM estimation with many moment conditions, e.g., in nonlinear simultaneous equations models or conditional moments restrictions models, many moments decrease the variances of the IV or GMM estimates, but increase their biases (see Bekker, 99; Donald and Newey, ; Chao and Swanson, 5; Han and Phillis,, etc). In a simle dynamic anel data model with xed e ects, when T is moderately large, but small relative to n, Alvarez and Arellano (3) study the many IV estimation and its asymtotic roerties. In this section, we use the moment condition in (9) where the dimension of H nt might increase with t (and also increase with n, where n is the order of satial ower series exansion of G nj ). We investigate the asymtotic roerties of the SLS and GMM estimators for this aroach. 3.. Consistency, Asymtotic Normality and E ciency of SLSE For the many moment aroach, we can use the IV matrix H nt = (h nt ; W n h nt ; ; W n n h nt ) () motivated by (7), 3 where, if W nj s are not row-normalized, h nt = (Y n ; ; Y n;t ; X n ; ; X nt ; l n ) with its column dimension t = k x T + t + ; otherwise, l n shall be droed from h nt if W nj s are row-normalized. The n (resectively, t ) needs to increase as n (resectively, t and T ) increases in order to rovide adequate aroximation to the best theoretical IV. Therefore, the dimension of H nt is K t = t ( n ). The choice of many moments might have a trade-o between the bias and variance of the GMM estimate, i.e., a larger number of moments might increase the bias of an IV estimator but decrease its variance. 3 3 There are some technical di culties in the resence of many IVs which involve estimated arameters in the literature, which is also true for our model. Hence, it is desirable to avoid it by using IVs which do not involve estimated arameters. 3 For a simle dynamic anel data without exogenous variables and time e ects, we have K t = t = t so that the total number of IVs is equal to T (T )=. With exogenous variables and satial lags, the total number of IVs would be much larger. 5

17 The SLS estimate of (3) with many moments is XT ^sl;nt = XT (W nynt; Znt) M nt (W n Ynt; Znt) (W nynt; Znt) M nt Ynt ; (3) t= t= where M nt = J n H nt (HntJ n H nt ) + HntJ n. When T and n are nite, the ^ sl;nt is consistent and asymtotically normal with the limiting variance matrix lim n! n(t ) t= (L nt; Znt) M nt (L nt; Znt). X T When T or n is large, we need to consider the many moment issue as those many moments will introduce bias to the estimates. In the following, we will focus on the asymtotic analysis of the many moment case where both T and n are large. 3 Assumtion. Both T and n! as n!. Assumtion seci es that, many moments in H nt come out not only from the satial ower series exansion ( n! ) but also from the inclusion of lagged values (T! ). As we use a nite number of quadratic moment conditions in the SDPD model, we ay secial attention to the linear moments. The additional quadratic moment conditions will not comlicate the asymtotic analysis as the two sets of moments do not interact with each other. The J n f nt where f nt = E(W n Ynt; ZntjI t ) is the best IV for J n (W n Ynt; Znt). From () and (), we have (W n Ynt; Znt) = f nt + u nt where f nt = [G n (E(ZntjI t ) + tl n ); E(ZntjI t )]; () with E(ZntjI (; ) t ) = [E(Y n;t ji t ); W n E(Y t u nt = [G n ( nt + V nt); nt ]; (5) (; ) n;t ji t ); X nt] and nt = c T t [ ~ V n;tt ; W n ~ Vn;tT ; nkx ]: P t s= Y ns, From () and (7), for t, the f nt can be aroximated by the variables: Y n;t, t P t s= Z ns, the exogenous variables afterwards (X nt ; ; X nt ), the n vector l n, and their roducts with satial ower series exansions. 33 As the elements in H nt contain satial series of h nt, the many moments via () come out from both satial and time dimensions. The f nt can be well aroximated by some linear combination of H nt when t is far from the initial eriod. 3 When only T is large but n is not large, the asymtotic variance matrix of the ^ sl;nt is not clear. On the other hand, when T is nite, we do not need to use satial exansion for G nj s. With an initial consistent estimate, one can estimate G nj s and construct a nite number of IVs with them. However, we might not use estimated G nj s due to technical di culty when T becomes large and there are many moments due to the time dimension. With both T and n being large, we can follow the asymtotic setting in Donald and Newey () where the many moments would aroximate the conditional mean of endogenous regressors. 33 For t =, f n can be aroximated by the satial ower series exansion of Y n ; X n ; :::; X n;t and l n.

18 Thus, the SLSE in (3) can be written as n(t )(^sl;nt T " # P ) = (f nt + u nt ) TP M nt (f nt + u nt ) (f nt + u nt ) M nt Vnt. n(t ) t= n(t ) t= () P T From Lemma 5, lim n;t! n(t ) t= f ntf nt = lim n;t! nt; is the robability limit of the rst comonent inside the inverse in (). As u nt and V nt are correlated, the second comonent in () has a non-zero mean. Denote and with C nt t = T b ;j = b ;j = n(t ) P T t= b ; = b ;k = P T t t h= hah n. n(t ) P T t= [tr(m ntg nj )]; T + t tr(m ntcnt tsn G nj ( I n + P j= W nj j )); P T t= n(t ) T + t tr(m ntcnt tsn ) P T t= n(t ) T + t tr(m ntcnt tsn W nk ) Theorem 3 Suose we use many linear moments in () and n; T!. Under Assumtions -7, 9- P T t= and Kt n(t )!, the SLS ^ sl;nt in () is consistent and P T! n(t )(^sl;nt ) [ ^H] t= Kt d (' + ' ) + O! N(; lim n;t! nt; ), (7) n(t ) where ^H P T = n(t ) t= (W nynt; Znt) M nt (W n Ynt; Znt), P! T ' = (b ; ; ; b ; ; (kx++)) t= = O K t ; n(t ) and! ' = (b ; ; ; b ; ; b ; ; b ; ; ; b ; ; kx ) X T K = O t : n(t ) t= (T + t)(t t) Consequently, P T t= (i) if Kt!, then n(t )(^ sl;nt )! d N(; n(t ) lim n;t! P T t= Kt n(t )! c where c is a ositive nite constant and (ii) if n(t )(^sl;nt ) [ ^H] d '! N(; lim n;t! nt; ); nt; ); maxfkt:t=; ;T g P T t= Kt! as T!, then (iii) let ^ sl;nt = ^ sl;nt ^H ^' n(t ) be a bias corrected estimate, where ^' is estimated ' with ^ sl;nt. N(; lim n;t! nt; ). Then, under the setting in (i), or (ii) and P T t= Kt n(t ))!, n(t )(^ sl;nt ) d! 7