Spatial panels: random components vs. xed e ects

Transcription

1 Spatial panels: random components vs. xed e ects Lung-fei Lee Department of Economics Ohio State University l eeecon.ohio-state.edu Jihai Yu Department of Economics University of Kentucky jihai.yuuky.edu January 8, Abstract This paper investigates spatial panel data models with spatially and serially correlated disturbances. We rst discuss the short panels and consider their estimation by both xed e ects and random e ects speci cations. With a between equation properly de ned, the di erence of the random vs. xed e ects models can be highlighted. We show that the random e ects estimate is a pooling of the within and between estimates. A Hausman type speci cation test and also an LM test are proposed for the testing of the random components speci cation vs. the xed e ects speci cation. We then discuss the case for long panels with time e ects included. After the time e ects are eliminated, we develop the xed e ects and random e ects estimates, and show that the within estimate is asymptotically as e cient as the random e ects estimate. JEL classi cation: C3; C3; R5 Keywords: Spatial autoregression, Panel data, Random components, Fixed e ects, Maximum likelihood estimation, Pooling

2 Introduction Panel data with spatial interactions is of interest as it enables researchers to take into account the dynamic and spatial dependence and also control for the unobservable heterogeneity. Anselin (988) provides a panel regression model with error components and spatial autoregressive (SAR) disturbances. Baltagi et al. (3) consider the speci cation test for spatial correlation in that panel regression model. Kapoor et al. (7) have a di erent speci cation of error components and SAR structure in the overall disturbance and suggest a method of moments (MOM) estimation of their model; and Fingleton (8) adopts a similar approach to estimate a spatial panel model with SAR dependent variables but with random components and a spatial moving average (SMA) structure in the overall disturbance. In an attempt to nest the Anselin (988) and Kapoor et al. (7) models, Baltagi et al. (7a) suggest an extended model. As an alternative to the random e ects speci cation, Lee and Yu () consider the xed e ects speci cation. The xed e ects model has the advantage of robustness in that the xed e ects are allowed to depend on included regressors in the model. It can also provide a uni ed model framework as the di erent random e ects models in Anselin (988), Kapoor et al. (7) and Baltagi et al. (7a) reduce to the same xed e ects model, when the random e ects are conditioned upon as xed parameters. Lee and Yu () investigate the estimation of xed e ects spatial panel data models with a partial likelihood approach, where the xed e ects may also include time e ects in addition to individual e ects. In the present paper, we are interested in exploring the econometric contents of random e ects spatial panels vs. xed e ects ones. As there are various random components speci cations in spatial panels, we put forward a generalized speci cation in order to nest the existing random e ects spatial panels as special cases. Such a generalized model is motivated by Baltagi et al. (7a) but goes beyond their generalization to additionally incorporate serial correlated disturbances and SMA structure. Thus, the generalized spatial panel data model in this study incorporates SAR disturbances and error components in Kapoor et al. (7), the generalized spatial error components in Baltagi et al. (7a), the additional serially correlated disturbances in Baltagi et al. (7b), and also the spatial panel model with SMA in Fingleton (8). This generalized spatial model is not necessarily more valuable than a parsimonious model in an empirical application; but it provides a common speci cation for our analysis without restriction to a particular existing model. In addition, we may compare the various special models within this generalized framework. While Kapoor et al. (7) and Fingleton (8) consider the MOM estimation and Baltagi et al. (7a, 7b) emphasize on the testing of correlations, we investigate the maximum likelihood (ML) and quasi-maximum likelihood (QML) approaches for estimation and testing. The estimation under the xed e ects speci cation as well as

3 the random components speci cation will be considered. The use of the likelihood functions are revealing as they can clearly highlight the di erent econometric contents of a random components speci cation vs. its xed e ects counterpart; also, it can provide conventional Hausman type and Lagrangian multiplier (LM) tests of the random e ects vs. xed e ects. For the random e ects speci cation in the linear regression panel data models, as is shown in Maddala (97), the GLS estimates of the regression coe cients are weighted averages of the within and between estimates. Hence, by pooling the within and between estimators, the GLS estimate will be more e cient relative to the within estimator under the random e ects speci cation. In the present paper, we show that, with a properly de ned between equation, the random e ects estimate, which is a GLS estimate, can also be interpreted as a pooling of within and between estimates for spatial panel data models. For the linear regression panel data model, the within estimate will be consistent under both random e ects and xed e ects speci cations. The random e ects estimator is consistent and can be more e cient under the random e ects speci cation; however, it would be inconsistent if the e ects were correlated with the regressors or the spatial weights matrix. Hausman (978) has proposed a test of correlation of random e ects with regressors for the panel regression model. For the Kapoor et al. (7) type panel data model, Mutl and Pfa ermayr (8) have considered a Hausman type test based on two stage least square (SLS) estimates. In the likelihood framework, it is natural to consider the Hausman type test via the ML estimates. The di erence of the log likelihood functions of the random components and xed e ects models is highlighted by the likelihood function of the between equation. The di erences of various random components speci cations would also be revealed via the content of the between equation. This paper is organized as follows. Section presents the general model speci cation and discusses the estimation for nite T case. We rst consider the estimation of the xed e ects model, followed by the random e ects model. We demonstrate that the random e ects model can be decomposed into a within equation and a between equation, and the estimate of the random e ects model can be regarded as the pooling of the estimates of within and between equations. The distinctions of various random components speci cations are also captured by the between equation. Section 3 investigates the Hausman and LM tests for the random e ects speci cation. The between equation provides the role to determine the proper degrees of freedom for the Hausman test. Section 4 discusses the large T case where additional time e ects are There are studies in the literature on dynamic panel models with spatial interactions. Korniotis (9) considers a model that individual time lag and spatial time lag are present but not contemporaneous spatial lag. Elhorst (5) and Su and Yang (7) study a dynamic panel data model with spatial error and random e ects. For the case with the additional spatial lag in panel data, Yu et al. (7, 8) and Yu and Lee () consider, respectively, the spatial cointegration, stable, and unit roots models in the time dimension. However, the dynamic models involve an initial value problem, which needs special treatment. Panel data models without dynamics do not, in general, have such an issue. Hence, estimation methods and asymptotic properties of estimators can be di erent for the static and dynamic models.

4 introduced. We show that the within estimate is asymptotically as e cient as the random e ects estimate when T is large. Section 5 provides some Monte Carlo results on performance of the estimates and test statistics. Section 6 concludes. A list of notations and some algebra are collected in the Appendices. The General Spatial Panel Model: Finite T Case Consider the following model: Y nt = W n Y nt + X nt + n + U nt, t = ; :::; T; (.) U nt = W n U nt + (I n + M n )V nt, n = 3 W n3 n + (I n + 3 M n3 )c n, V nt = V n;t + e nt, t = ; :::; T, where n is an n-dimensional vector of individual e ects 3 with spatial interactions, U nt is the SAR error which is also serially correlated, and c n and e nt are independent with i.i.d. elements such that c ni s i:i:d:(; c) and e it s i:i:d:(; e). As usual in the literature, we assume stationarity so that V n (; e I n ) and is independent with e nt for t = ; :::; T. In addition, U nt and n are allowed to incorporate SMA features. This is a generalized spatial panel model which incorporates spatial correlation, heterogeneity and serial correlation in disturbances. It nests the various spatial panels existing in the literature. Speci cally, the generalized random components model in (.) nests the following models (where the abbreviation of the models is from Baltagi et al. 9): () Kapoor et al. (7) SAR-RE: = 3, = 3 = and W n = W n3. () Fingleton (8) SMA-RE: = 3 =, = 3 and M n = M n3. (3) Anselin (988) RE-SAR: 3 =, = and 3 =. (4) Anselin et al. (8) RE-SMA: =, 3 = and 3 =. (5) Baltagi et al. (7a) Generalized RE-SAR: = and 3 =. In addition, it also includes (6) SARMA-RE where n +U nt = (I n W n ) (I n + M n )(c n +V nt ) with = 3 and = 3. Baltagi et al. (3, 7a, 7b) have considered the tests of spatial and/or serial correlations. Kapoor et al. (7) consider the estimation of their SAR-RE model by MOM. Fingleton (8) modi es the MOM Intermediate steps and detailed proofs for the Propositions are in a supplement le to this paper (available on request). 3 When n is random, they can be assumed to have zero means. In that situation, we may introduce an overall constant term. Note that in the random e ects model, we may also have individual or time invariant regressors z n with coe cient b. In short panels, time e ects will not cause the incidental parameter problem and they can be treated as regressors. However, for long panels (see Section 4), to avoid the incidental parameter problem, we may eliminate them before the estimation, or allow them in the regression by the random e ects speci cation. 3

5 estimation method for his SMA-RE model. Parent and LeSage (8) apply the Markov Chain Monte Carlo method to the linear panel regression model where the spatially and serially correlated disturbances are present. The product of the quasi-di erence over time and the spatial transformation is called the spatial-time lter in Parent and LeSage (8). In the present paper, as additional serial correlation in the disturbances V nt is allowed, the model incorporates spatial-time lter in the disturbances. We will consider the ML estimation under both xed e ects and random e ects speci cations of the individual e ects in n. For the xed e ects speci cation, n is allowed to depend on the regressors and elements of the spatial weights matrices. For the random e ects speci cation, n is assumed to be independent with all the regressors and the spatial weights matrices. The parameter subvector = (,,,,, e) can be estimated from both xed e ects and random e ects models. The remaining parameters in = (b ; 3, 3, c) can only be estimated under the random e ects speci cation.. Fixed Individual E ects In this section, individual e ects in n are assumed to be xed parameters. The model is Y nt = W n Y nt + X nt + n + U nt, t = ; :::; T; (.) U nt = W n U nt + (I n + M n )V nt, V nt = V n;t + e nt, t = ; :::; T. Because elements of n are xed parameters, their spatial structure would be irrelevant. Thus, the random components speci cation originated in Anselin (988) and that of Kapoor et al. (7) would yield the same xed e ects model once the random e ects are conditioned upon. For the standard linear panel data model with serial correlation, relevant model speci cation and estimation methods have been summarized in Hsiao (3) and Baltagi (8). For the panel model with serially correlated disturbances, Kiefer (98) and Bhargava et al. (98) investigate the relevant issues under the xed e ects speci cation and recognize the possible inconsistency of the estimates when T is nite. 4 As contrary to the xed e ects panel regression model with independent disturbances, in the presence of serial correlated disturbances or dynamic e ects, the ML estimates of the regression coe cients would su er from the incidental parameter problem. This section will consider estimation methods without the incidental parameter problem. For that purpose, we shall rst eliminate the xed e ects before estimation. One may regard the resulting equation as a within equation. To eliminate the individual e ects, we suggest the use of rst di erence rather than the deviation from 4 The incidental parameter problem is also well-known for dynamic panel models with xed e ects (Nickell, 98). 4

6 time mean. 5 This is so because serial correlation in time series can be better dealt with in a recursive fashion. For any n vector Z nt, denote Z nt = Z nt Z n;t as the rst di erence. We have Y nt = W n Y nt + X nt + U nt, t = ; :::; T; U nt = W n U nt + (I n + M n )V nt, where V nt = V n;t + e nt, t = 3; :::; T and V n = e n ( )V n. The Y nt with t = ; :::; T form a spatial panel system with SARMA disturbances and serially correlated ARMA disturbances but with neither xed e ects nor time invariant factors. To handle the autocorrelated disturbances, a quasi-di erence (Cochrane-Orcutt transformation) can be applied; however, the serial MA correlation remains. Let Z nt; = Z nt Z n;t be the quasi-di erence so that Z nt; = Z nt Z n;t. After quasi-di erence for t = 3; :::; T, the above equation becomes 6 Y nt; = W n Y nt; + X nt; + U nt; with (.3) U nt; = W n U nt; + (I n + M n )e nt, t = 3; :::; T; Y n = W n Y n + X n + U n with U n = W n U n + (I n + M n )V n, V n = e n ( )V n, where e nt becomes a MA process. For the covariance matrix of e d n;t = (V n; e n3; :::; e nt ), we have Var(V n ) = + ei n, Cov(V n ; e n3 ) = ei n and Cov(V n ; e nt ) = for t = 4; :::; T. 7 Hence, the estimation equation would be (.3), and the variance matrix of e d n;t is eh T I n where H T is a (T ) (T ) matrix as H T = B In this paper, we will rely on the rst di erence to eliminate the individual e ects, as the di erence from time mean transformation J T, where J T = I T T l T lt with l T being the T -dimensional vector of ones, might not be easy to deal with the dynamic feature in the disturbances. To see that, denote [F T;T ; p l T T ] as the eigenvector matrix of J T. With the transformation [U n ; U n ; :::; U nt ]F T;T = [Un ; U n ; :::; U nt ] for t = ; :::; T, the individual e ects are eliminated; however, it is rather complicated to deal with the serial correlation of Vnt in U nt. We can alternatively make the quasi-di erence for t = ; :::; T and rescale by ( ) for t = of (.). After the rst di erence to eliminate the transformed individual e ects ( ) n, the result is the same. This is so, because the two operations are both linear ones. If we assume that V n is xed, Var(V n ) = Var(e n ) = e In, Cov(V n; e n3 ) = e In and Cov(V n; e nt) =, for t = 4; :::; T. Also, when V n is assumed to be random, it could be in general (; e In) in Baltagi and Li (99). If these were the selected speci cations, the (,) entry of H T would be modi ed accordingly.. C A 5

7 In the vector form, denote Y nt = (Yn; Yn; :::; YnT ), Q T () = B.....,.. C A... and Q T = Q T ( ). For the rst di erence, we transform Y nt into Y d n;t = (Y n; :::; Y nt ). For the additional quasi transformation, we have Y d n;t ; = (Q T I n )Y d n;t and Xd n;t ; = (Q T I n )X d n;t. Hence, (.3) can be written in the vector form as Y d n;t ; = (I T W n )Y d n;t ; + X d n;t ; + (I T S n B n)e d n;t, (.4) where S n = I n W n and B n = (I n + M n ). With S n = I n W n, we denote S n;t ; = I T S n, S n;t ; = I T S n, B n;t ; = I T B n, Q n;t = Q T I n and H n;t = H T I n. The likelihood function for Y d n;t is L nt (Yn;T d ) = () n(t )= js n;t ; ( )j js n;t ; ( )j jb n;t ; ( )j (.5) j eh n;t ()j = jq n;t ()j exp e d n;t ( )H n;t ()ed n;t ( ), e where e d n;t ( ) = B n;t ; ( )S n;t ; ( ) S n;t ; ( )Q n;t ()Y d n;t Q n;t ()X d n;t, jh n;t ()j = jh T ()j n, js n;t ;j ( j )j = js nj ( j )j T for j = ;, jb n;t ; ( )j = jb n ( )j (T ) and jq n;t ()j = jq T ()j n =. Thus, the log likelihood of this within equation (under the speci cation with normal disturbances) is ln L w;nt ( ) = n(t ) ln e + (T ) (ln jsn ( )j + ln js n ( )j ln jb n ( )j) n ln jh T ()j e d n;t ( )H n;t ()ed n;t ( ), (.6) e where its score and information matrix are in Appendix B.. For the evaluation of this likelihood function, it involves the inverses of H T () and B n ( ) and their determinants as well as those of S n ( ) and S n ( ). The computation of the determinants of S n ( ) and S n ( ) would be the same as that for a conventional SAR model for cross section data (this is also true for B n ( ) of a SMA model). 8 For the matrix H T (), 8 If W n, W n and M n are diagonalizable, they can be evaluated with their eigenvalues and eigenvectors (see, e.g., Ord 975). For example, as M n = q n nqn where n is the diagonal eigenvalue matrix and q n is the corresponding eigenvector matrix, we have B n = qn(in + n) qn. 6

8 its inverse and determinant have closed form expressions (see, Hsiao et al., or Appendix A in the present paper). Hence, the computation of this log likelihood for the within equation is tractable and not more complicated than that for a typical spatial model with cross sectional data. For our asymptotic analysis of the estimators, we make the following assumptions. Assumption. W nj for j = ; ; 3 and M nj for j = ; 3 are nonstochastic spatial weights matrices with zero diagonals. Assumption. The disturbances fe it g, i = ; ; :::; n and t = ; 3; :::; T; are i:i:d: across i and t with zero mean, variance e and E je it j 4+ < for some > ; also, they are independent with V n (; e I n ). Assumption 3. S nj ( j ) for j = ; ; 3 and B nj ( j ) for j = ; 3 are invertible for all j j and j j, and P, where j and j are compact intervals and P is a compact subset in ( is in the interior of j, j is in the interior of j, and is in the interior of P. ; ). Furthermore, j Assumption 4. W nj for j = ; ; 3 and M nj for j = ; 3 are uniformly bounded in both row and column sums in absolute value (for short, UB). 9 Also S nj ( j) and B nj ( j) are UB, uniformly in j j and j j. Assumption 5. n is large, where T is nite. Assumption 6. The elements of nk x matrix of regressor X nt are nonstochastic and bounded, uniformly in P T n and t. Also, under the asymptotic setting in Assumption 5, the limit of n(t ) t= X d n;t ; H n;t X d n;t ; exists and is nonsingular where X d n;t ; = B n;t ; S n;t ;X d n;t ;. The zero diagonal assumption helps the interpretation of the spatial e ect, as self-in uence shall be excluded in practice. In many empirical applications, each of the rows of W nj (and M nj ) sums to, which ensures that all the weights are between and. In this section, our estimation and analysis for the model do not require the feature of row-normalization. We note that the spatial weights matrices W nj s and M nj s are written in general notations; but they may or may not be the same in practice. Assumption speci es an i.i.d. assumption for e it. If there were unknown heteroskedasticity in e nt, the MLE would not be consistent. Methods such as the generalized method of moments (GMM) in Lin and Lee (9) and that in Kelejian and Prucha (9) would be designed for that situation. Invertibility of S nj ( j ) and B nj ( j ) in Assumption 3 guarantees that we have a valid reduced form. Also, compactness of parameter spaces is a convenient condition for theoretical analysis on nonlinear functions. When W nj (and M nj ) is row-normalized, a compact subset of (-,) has often been taken as the parameter space for j (and j ) in 9 We say a (sequence of nn) matrix P n is uniformly bounded in row and column sums in absolute value if sup n kp nk < and sup n kp nk <, where kp nk = sup in P n j= jp ij;nj is the row sum norm and kp nk = sup jn P n i= jp ij;n j is the column sum norm. Due to the nonlinearity of j and j in the reduced form of the model, compactness of j and j is needed. However, the compactness of and e is not necessary because the and e estimates given j and j are least squares type estimates. 7

9 theory. Assumption 4 is originated by Kelejian and Prucha (998, ) and also used in Lee (4, 7). That W nj, M nj, S nj ( j) and B nj ( j) are UB is a condition to limit the spatial correlation to a manageable degree. Assumption 5 is assumed for the short panel data case (in this section). Subsequently, we shall consider the case with both n and T being large. The case with a nite n but a large T is of less interest as the incidental parameter problem would not occur in this model and spatial models are mainly designed for large n cases. When exogenous variables X nt are included in the model, it is convenient to assume that they are uniformly bounded as in Assumption 6. If X nt is allowed to be stochastic and unbounded, appropriate moment conditions can be imposed instead. We also make the following assumptions in order to establish the consistency and asymptotic distribution of the estimates. Assumption 7. Either (a) the conditions (B.3) and (B.4) in Appendix B. hold or (b) the condition (B.5) holds. Assumption 8. The limit of the information matrix (B.) is nonsingular. Assumption 7 speci es the identi cation conditions of the model. The part (a) of Assumption 7 represents the possible identi cation of and through the deterministic part of the reduced form equation of (.4), and the identi cation of ; ; and e from the SAR process of U nt in (.4). The part (b) of Assumption 7 states the identi cation through the SAR process of the reduced form of disturbances of Y nt in (.4). Assumption 8 is for the nonsingularity of the limits of the information matrix for the within equation. Proposition Under Assumptions -7(a);or Assumptions -6,7(b) and 8, the within estimate ^ w of under the xed e ects speci cation from (.6) is consistent and asymptotically normal: p n(^w ) d! N(; lim n! ;nt ( ;nt + ;nt ) ;nt ) where ;nt in (B.) is the information matrix, and ;nt in (B.) is related to the fourth and third moments of e it ( ;nt = under normality). Proof. The consistency will follow from typical steps by showing () uniform convergence; () uniform equicontinuity and (3) identi cation uniqueness. The asymptotic distribution follows from the linear and quadratic CLT in Kelejian and Prucha () and its extension to panels in Yu et al. (8). The details for the proof (and proofs for other propositions) are provided in a separate le available upon request. We do not need j = 3 for Assumptions, 3 and 4 under the xed e ects model. 8

10 . Random Individual E ects In this section, we consider the random e ects speci cation of the individual e ects n. Additionally, we allow the SAR and SMA features in n, which could be considered as the permanent (global and local) spillover e ects as described in Baltagi et al. (7a). In a random e ects model with n having a zero mean, we shall allow the possible presence of time invariant regressors z n for generality. Hence, the model is (.) with the extended main equation: Y nt = z n b + W n Y nt + X nt + n + U nt, t = ; :::; T; (.7) where b is the coe cient vector for the nk z matrix z n. This is a spatial random components model, which generalizes and uni es the existing spatial panel models in the literature. Denote V nt = [V n; V n; :::; V nt ], and other variables accordingly. The equation (.7) can be written in the vector form with the nt observations as Y nt = l T z n b + (I T W n )Y nt + X nt + l T S n3 B n3c n + (I T S n B n)v nt. (.8) The V nt in (.8) can be transformed by the Prais-Winsten transformation p P T () = B......,.. C A... into serially uncorrelated disturbances where P T = P T ( ). Hence, denoting Y nt; = (P T I n )Y nt, X nt; = (P T I n )X nt and e nt = (P T I n )V nt, we have Y nt; = P T l T z n b + (I T W n )Y nt; + X nt; + P T l T S n3 B n3c n + (I T S n B n)e nt. For this transformed model, denoting l T = ( q + ; ; :::; ) and d = l T l T = + + (T ), the variance of the transformed total disturbance is nt; because P T l T = ( d l T l T Z n + I T = l T l T [( ) cs n3 B n3b n3s n3 ] + I T [ es n B nb ns n ], ) l T l T. This variance matrix can be a form of spectral decomposition as nt; = e Sn B nbns n where Zn = [d ( ) csn3 B n3bn3s n3 + d l T l T esn B nbns n ]. Hence, by Lemma. in Magnus (98), nt; = d l T l T Z n + I T d l T l T j nt; j = j es n B nb ns n jt jz n j. 9 SnB n B n S n e,

11 Denote P nt = P T I n and S nt; = I T S n. The likelihood of Y nt in (.8) is L nt (Y nt ) = () nt= j nt; j = js nt; ( )j jp nt ()j exp nt;() nt; nt;(), (.9) where nt; () = S nt; ( )(P T () I n )Y nt (P T () I n )X nt P T ()l T z n b, jp nt ()j = jp T ()j n = ( ) n= and js nt; ( )j = js n ( )j T. Hence, the log likelihood of the random components model (under normality) is ln L r;nt () = nt ln() ln j nt;()j + T ln js n ( )j + n ln where its score and information matrix are in Appendix B.. nt;() nt; () nt;(), (.) Assumption 9. c n and e nt are i.i.d. and independent of X nt and z n. Also, c n (; ci n ) is independent with e nt. Assumption. The elements of n k x matrix of regressor X nt and n k z matrix z n are nonstochastic and bounded, uniformly in n and t. P T t= Z nt nt; Z nt exists and is nonsingular where Z nt = [l T z n ; X nt ]. nt Assumption. (B.) holds. Also, under the asymptotic setting in Assumption 5, the limit of Either (a) the conditions (B.8) and (B.9) in Appendix B. hold or (b) the condition Assumption. The limit of the information matrix (B.6) is nonsingular. Part (a) of Assumption 9 represents the possible identi cation of, b and through the deterministic part of the reduced form equation of (.8), and the identi cation of rest of the parameters from the overall disturbances in (.8). Part (b) of Assumption 9 states the identi cation through the SAR process of the reduced form of disturbances of Y nt in (.8). Assumption is a condition for the nonsingularity of the limits of the information matrix for the random equation. Proposition Under Assumptions -5, 9,, (a), or Assumptions -5, 9,, (b) and, the QML estimates ^ r = (^ r; ^ r) of from (.) of the random e ects model are consistent and asymptotically normal: p n(^r ) d! N(; lim n! ;nt ( ;nt + ;nt ) ;nt ) where ;nt in (B.6) is the information matrix, and ;nt in (B.7) is related to the fourth and third moments of e it and c i ( ;nt = under normality)..3 The Between Equation The likelihood of the random e ects model can be written as a product of two likelihoods one is the within likelihood from (.4), and the other can be obtained from the rst period observation as shown below.

12 For the within model, we have n(t the remaining Y n, which is ) sample observations in the form Y nt for t = ; :::; T. Consider now Y n = z n b + W n Y n + X n + n + U n. (.) The covariance of the disturbances in the rst period and those in the within model is Cov(V n ; e d n;t ) = e + [e T I n] where e T = (; ; :::; ). As e T H T = [ + (T ) + ] (T ; T ; :::; ), we have V n = = e (e T + ( eh T ) I n )e d n;t + V ~ n ((T )V n + P T t=3 T (T ) (T + t)e nt) + V ~ n, where ~ V n is the residual vector uncorrelated with V n and e nt for t = 3; :::; T. For ~ V n, it has zero mean and its variance is Var( ~ V n ) = = e e I n I n ( e + ) (e T I n )(( eh T ) I n )(e T I n ) e ( + ) (e T H T e T )I n = I n, (.) where = e ( )(T (T ) ). Thus, the conditional likelihood of V n, conditional on e d n;t (under normality), can be constructed. From (.), we have where ~ V nt = ~ V n and can be simpli ed as S n ~ YnT = z n b + ~ X nt + n + S n B n ~ V nt (.3) ~V nt = [T (T ) ] [V n + ( )(V n + + V n;t ) + V nt ]; (.4) and similarly for ~ Y nt and ~ X nt. We may interpret (.3) as a between equation which captures the cross sectional variation across spatial units, as the outcomes for each unit have been properly aggregated over time. When =, we see that ~ Y nt = Y nt, ~ XnT = X nt and ~ V nt = V nt are time averages, and (.3) becomes S n YnT = z n b + X nt + n + S n B n V nt in the familiar between equation form. From this between equation, its overall disturbances consist of random e ects n as well as a spatial correlated disturbances. Any speci c spatial structure on n is captured in this between equation. As the within equation does not involve n, identi cation of the spatial structure of n will solely depend on the between equation. This between equation highlights the main distinction of the random components model and the within equation.

13 The variance matrix of the overall disturbances in (.3) is n = Var( n + S n B n ~ V nt jy d n;t ) = cs n3 B n3b n3s n3 + S n B nb ns n, and the likelihood function of the between equation from (.) is L n (Y n jyn;t d ) = () n= j n ()j = js n ( )j exp n()n () n(), (.5) where n () = S n ( ) ~ Y nt () ~ XnT () z n b. Hence, the log likelihood is ln L b;n () = n ln() ln j n()j + ln js n ( )j n() n () n(), (.6) with its score and information matrix in Appendix B.3. Consider the identi cation of model parameters from the between equation. The parameters b,, and will be identi ed from the structural equation in (.3); and the variance matrix n will be the sole source for the identi cation of, 3,, 3, e and c. As n is the sum of two variance matrices of the spatial error components, the identi cation of those parameters from the between equation would crucially depend on the distinctive structures of the two processes. For the error components of the type introduced by Kapoor et al. (7) and Fingleton (8), the two spatial processes are similar and have S n3 B n3 = S n B n. Under such a speci cation, n will be reduced into a single piece and it is apparent that only the sum, c +, can be identi ed but not the individual parameters c and e from the between equation. However, the identi cation of c in the random e ects model is possible as the identi cation of e can be from the within equation. That is, e can be identi ed from the within equation as well as the random e ects model but may not be so solely from the between equation. Such a kind of irregularity will have implications on the construction of Hausman-type speci cation test (Section 3.) for random components vs. xed e ects speci cations. For the general model with two di erent spatial processes for the error components, due to the complicated variance structures in n, those parameters might not be easily estimated from the between equation, as that equation is essentially a cross sectional spatial one. These might be seen from matrix expansions of n under di erent speci cations as described below. The followings are some heuristic arguments. General random speci cation with W n3 = W n = W n : From n, all the four parameters ( ; 3 ; c; ) can be identi ed as long as 6= 3. However, a close look at its nonlinearity might reveal some weak identi cation scenario. Assume that j j j < for j = ; 3 where W n is row-normalized. Thus, their inverses can be expanded into series: S nj ( j) = I n + j W n + jw n + 3 jw 3 n + O( 4 j) :

14 It follows that S nj ( j)s nj ( j) = I n + j (W n +W n)+ j(w n +W n +W n W n) + 3 j(w 3 n +W 3 n +W n W n +W nw n) +O( 4 j). Therefore, we have n () = ( c + )I n + ( c 3 + )(W n + W n) + ( c 3 + )(W n + W n + W n W n) +( c )(W 3 n + W 3 n + W n W n + W nw n) + O((j 3 j + j j) 4 ). Because there are four parameters in ( ; 3 ; c; ), at least the coe cients of the four leading terms are needed in order to identify them. However, if the values and 3 are small, then high order coe cients would also be small. Under this situation, the estimates would intuitively be more di cult to be estimated. In addition, the di culty presents itself also for the case with and 3 being close to each other because the above expansion can be rewritten as n () = I n + ( c( 3 ) + )(W n + W n) + ( c( 3 ) + )(W n + W n + W n W n) +( c( ) + 3 )(W 3 n + W 3 n + W n W n + W nw n) + O((j 3 j + j j) 4 ), where = c +. When the di erence of 3 and is small, c would not be easily estimated, and nor is in consequence. From these, we would expect that the estimates of 3 and c could be possible in the random e ects likelihood simply due to the fact that and have been consistently estimated from the within equation. General random speci cation (W n3 6= W n ): For such a general case, we have n () = ( c + )I n + c 3 (W n3 + W n3) + (W n + W n) + c 3(W n3 + W n3 + W n3 W n3) + (W n + W n + W n W n) + O((j 3 j + j j) 3 ). Here, the four parameters can be identi ed from the expansion up to the second order. Thus, when the weights matrices W n and W n3 can be distinguished from each other, the estimation of the between equation might be easier. Anselin s speci cation: For this speci cation, n () = ci n + S n ( )S n ( ) = ( c + )I n + ( c 3 + )(W n + W n) + ( c 3 + )(W n + W n + W n W n) +( c )(W 3 n + W 3 n + W n W n + W nw n) + O((j 3 j + j j) 4 ). 3

15 Hence, the expansion is needed for up to the second order only. KKP s speci cation: Here, we have 3 =. Hence, the parameters and (but c and can not be separately identi ed) can be recovered from the expansion up to the rst order n () = I n + (W n + Wn) + O( ). From this, it seems that the between equation of the KKP model would be easier to be estimated than the Anselin s speci cation and the generalized ones..4 Pooling of Estimates The likelihood function of the random e ects model has all the unknown parameters in = ( ; ), but that of the xed e ects model has only. The excluded parameters in would appear in the likelihood function of the between equation. Without loss of generality, assume that can be identi ed from both the within and between equations. In order to compare the e ciency of the estimates of the two models, one simple approach is to use the concentrated likelihood function L c r;nt ( ) (L c nt (Y nt )) of the random e ects model with concentrated out, and compare it with the likelihood function L w;nt ( ) (L nt (Y d n;t the within equation. Similarly, one can have the concentrated likelihood L c b;n ( ) (L c n(y n jy d n;t )) of )) of the between equation. Thus, the concentrated likelihood functions of the random e ects model and the between equation have the same common set of parameters as that of the likelihood function of the within equation. The random e ects estimate of can be interpreted asymptotically as a weighted average of the within and between estimates, closely analogous to Maddala (97) for the panel regression model as shown below. For the case T being nite, the within estimator ^ w would be p n-consistent and its asymptotic distribution would follow from p n(^ w ) = ( n ln L nt (Y d n;t ) ) p ln L nt (Y d n;t ) n + o p (), and the between estimator ^ b would have p n(^ b ) = ( ln L c n (YnjYd n;t ) n ) ln L c n p (YnjYd n;t ) n + o p (). On the other hand, for the ML estimator based on the likelihood L c nt (Y nt ) of the random components model, one has p n(^ r ) = ( n ln L c nt (Y nt ) ) ln L p c nt (Y nt ) n + o p (). Because L c nt (Y nt ) = L nt (Y d n;t )L c n(y n jy d n;t ); we have p ln L c nt (Y nt ) n = p ln L nt (Y d n;t ) n + p ln L c n (YnjYd n;t ) n and ln L c nt (Y nt ) n = n ln L nt (Y d n;t ) + If some of them can only be identi ed and estimated in one but not the other, we may consider the subset of common parameters which can be identi ed in both equations. In that case, the relevant concentrated within and between likelihood functions will be used instead. 4

16 n ln L c n (YnjYd n;t ). Hence, p n(^r ) = where A nt; = n ln L c nt (Y nt ) ln L nt (Y pn n;t d ) + ln L c n(y n jyn;t d p )! + o p () n = A nt; p n(^w ) + A nt; p n(^b ) + o p (), (.7) n ln L c nt (Y nt ) ln L nt (Y d n;t ) n and A nt; = The A nt; and A nt; are weights because A nt; + A nt; = I k, where k n ln L c nt (Y nt ) n ln L c n (YnjYd n;t ). is the dimension of. Hence, the random e ects estimate of can be interpreted as pooling the within and between estimates. We note that the weighting above is valid even though the likelihood functions are quasi ones. This is so because the preceding derivations depend only on the forms of the likelihood functions and their algebraic relations, but not the distribution of the disturbances. However, when the likelihoods are only quasi likelihoods, the weights A nt; and A nt; are not necessarily interpreted as ratios of the precision matrices of the within and between estimates relative to that of the random e ects estimates. From the relation of the second derivatives of a concentrated log likelihood function with those of the original log likelihood (see, e.g., Amemiya 985, p.7, Eq.(4..53)), an alternative expression is that p n(^ r ) = B nt; p n(^w ) + B nt; p n(^b ) + o p (), where and B nt; = ln L r;nt ln L r;nt ln L r;nt ln L r;nt! ln L w;nt B nt; = ln L r;nt ln L b;n ln L r;nt! ln L r;nt ln L r;nt ln L b;n! ln L b;n ln L b;n. Again, we note that these are valid even for quasi-likelihoods. 3 Testing 3. Hausman s Test The likelihood decomposition provides a useful device for a Hausman type test of random e ects speci - cation against the xed e ects speci cation where the individual e ects could be correlated with exogenous regressors. 3 Under the null hypothesis that the individual e ects c n are independent of the regressors, 3 Instead of the ML approach, if the main equation is estimated by the SLS method, Hausman test statistics based on the coe cients of the main equations can be constructed as in Mutl and Pfa ermayr (8). 5

17 the MLE ^ r of the random e ects model is consistent and asymptotically e cient (assuming the likelihood function is correctly speci ed, in this case, normal disturbances). However, under the alternative hypothesis that c n is correlated with the regressors, ^ r is inconsistent. The within estimator ^ w is consistent under both the null and alternative hypotheses. Such a null hypothesis can be tested with a Hausman type statistic n(^ r ^w ) ^+ n (^ r ^w ) by comparing the two estimates ^ r and ^ w, where ^ n is a consistent estimate of the limiting variance matrix of p n(^ r ^w ) under the null hypothesis, and ^ + n is its generalized inverse. This test statistics will be asymptotically distributed, and its degrees of freedom is the rank of the limiting matrix of n (see, e.g., Ruud, ). Under normality, as the likelihood function is correctly speci ed, ^ r is asymptotically e cient. Because ^r is asymptotically e cient, the di erence of the inverses of the information matrices of the within equation and the random e ects model, evaluated at ^ r, provides a consistent estimate of n under the null. However, the rank of n needs special attention, as it is the degrees of freedom of the test statistic. Suppose that B is the limiting variance matrix of p n(^ w ), and the limiting variance matrix of p n(^ r ) is written as (B + C) for some nonnegative de nite matrix C. B If C happens to be positive de nite, (B + C) = B (B + C ) B is a positive de nite matrix. Hence, in the case that C is positive de nite, the generalized inverse is an inverse, and the degrees of freedom of the test is the number of common parameters in, i.e., k. However, if C is only positive semi-de nite but not positive de nite, the degrees of freedom could be smaller. Suppose that C is a positive semi-de nite matrix of dimension k, where k = k, but it has rank m, where < m < k. Let = diagf ; ; k g be the diagonal matrix of the eigenvalues of C in the metric of B. That is, there exists a nonsingular matrix Q such that B = QQ and C = QQ, where all these eigenvalues in are non-negative and the number of positive eigenvalues corresponds to the rank of C (see, e.g., Proposition 6 in Dhrymes, 978). 4 Let = where consists of the positive eigenvalues of C with m entries. Then, has the rank m. B (B + C) = (QQ ) (QQ +QQ ) = Q [I k (I k + ) ]Q = Q m diag ; ; ; ; ; Q + + m For the Hausman test, we need to consider the di erence of the information matrix of the within likelihood 4 To obtain Q, we can follow the following four steps: () Let R be the cholesky decomposition of B such that R R = B ; () Let A = RCR ; (3) Find eigenvector matrix P and eigenvalues matrix of A such that P AP = ; (4) Let Q = R P. By doing so, we see that QQ = R P P (R ) = R (R ) = B and QQ = R P P AP P (R ) = R A(R ) = R RCR (R ) = C. 6

18 and that of the random e ects likelihood. As ln L r;nt ln L r;nt ln L r;nt ln L r;nt A = ln L w;nt + ln L b;n ln L b;n ln L b;n ln L b;n A (3.) from the likelihood decomposition, the asymptotic variance of the estimate ^ r of from the random e ects likelihood would be ln L w;nt E( ) + E( and that of the within estimate ^ w is C = E( n ln L b;n ) E( ln L b;n ) E( n ln L b;n )[E( ln L b;n )] E( ln L b;n ) : n E( ln L w;nt. n )o The matrix C in this case would be ln L b;n ) E( n ln L b;n ) E( n ln L b;n ): (3.) Therefore, the Hausman test statistics can be computed as + n(^ r ^w ) Qdiag ; ; + m ; ; ; Q (^ r ^w ). (3.3) m and its degrees of freedom is the rank m of C, where Q is the eigenvector of C in the metric of B = E( ln L w;nt ). If can be identi ed and estimated from the between equation, C would have full rank k. This would be the case with the exception of the KKP type random components speci cation. The KKP model speci cation Here we consider the likelihood function for the KKP type model, which has = 3 and = 3 with W n = W n3 and M n = M n3 imposed. It is convenient to consider the reparameterization = c+. Thus, is used instead of c in the likelihood of the between equation. For this model, = ( ; ; ; ; ; e) and = (b ; ). Note that the parameter e is in L w;nt but not in L b;n ; on the other hand, is in L b;n but not in L w;nt. Thus, by denoting = ( ; ; ; ; ), we have and B = E( C = E( ln L w;nt ln L b;n ) E( ln L w;nt e ) E( ) E( ln L b;n ) E( The rank of this C is the number of parameters of, which is k ln L w;nt e ) E( ln L w;nt ) ln L b;n ) E( ln L b;n ).. This is intuitively appealing because even though e is estimated in the within equation and the random components model, it is not estimated by the between equation. Thus, there are no two separate estimates for e from the within and between equations to be compared with. 7

19 3. LM Test Mutl and Pfa ermayr (8) consider the Hausman s test of the xed e ects vs. random e ects in the KKP spatial panel with SLS estimates. In order to investigate the power of the test, they specify a model similar to Mundlak (978) where the individual e ects depend on the time average of the regressors. For the general spatial panel model, we can have the speci cation c n = X nt + n, (3.4) where n would be assumed to be independent with X nt and are i.i.d. N(; I n). With the speci cation in (3.4), an alternative test can be based on the LM approach. equation as we have derived will become The between S n ~ YnT = z n b + ~ X nt + S n3 B n3 X nt + S n3 B n3 n + S n B n ~ V nt. (3.5) If =, this between equation would be the one in (.3). This suggests that we may consider = as the null hypothesis and (3.5) will be useful for the construction of the test. Under the null, all the parameters of the restricted random e ects model can be estimated by the ML. The LM test evaluated at the restricted MLE can be constructed from the random e ects log likelihood or, equivalently, the sum of the log likelihood functions of the within and extended between equations. The log likelihood function for this extended between equation is ln L b;n () = n ln() ln j n()j + ln js n ( )j n() n () n(), where n () = S n ( ) ~ Y nt () ~ XnT () z n b S n3 B n3 X nt with extended to include. For the random e ects model, its likelihood is similar to (.), where nt () = S nt; ( )Y nt X nt l T (z n b + S n3 B n3 X nt ). For the rst order derivatives of the random e ects model evaluated at the restricted estimates ^ r, they are zero except that with respect to. Thus, the LM test statistics would be! 3 ln L r;nt (^ r ) 5 where 4 E ln L r;nt (^ r ) ; ln L r;nt (^ r ) ; ln L r;nt (^ r ) = ln L b;n(^ r ) = (l T S n3 (^ 3;r )B n3 (^ 3;r ) X nt ) nt (^ r ) nt (^ r ) = (S n3 (^ 3;r )B n3 (^ 3;r ) X nt ) n (^ r ) n (^ r ) 8

20 and [( E ln L r;nt (^ r) ) ] ; is the diagonal block in ( E ln L r;nt (^ r) ) corresponding to the entries. The E ln L r;nt (^ r) = [E ln L w;nt (^ r) + E ln L b;n (^ r) ] would be similar to (B.6) but with additional deriv- atives involving in L b;n. Here, one would be interested in comparing the type I error and power of this LM test vs. those of the Hausman-type test. The comparison is reported in the Monte Carlo section (see Tables 6-7). 4 The General Spatial Panel Model: Large T Case With Time E ects For the large T case, if the time e ects are present, they might cause the incidental parameter problem. Hence, it is desirable to eliminate them before the estimation. Section 4. considers the xed e ects estimation where both the individual and time e ects are eliminated. Section 4. considers the random e ects estimation, where the time e ects are eliminated Fixed E ects The model is Y nt = W n Y nt + X nt + n + t l n + U nt, t = ; :::; T; (4.) U nt = W n U nt + B n V nt, V nt = V n;t + e nt, t = ; :::; T; where t is the time e ect at period t, and the individual e ects are in n. Similar to Section, we can use the rst di erence to eliminate the individual e ects n and the quasi-di erence to handle the serial correlation in V nt. For the time e ects, let (F n;n, pn l n ) be the orthonormal matrix of J n = I n n l nl n, where F n;n corresponds to the eigenvalues of ones and p n l n corresponds to the eigenvalue zero; then, F n;n l n =. By premultiplying (4.) with F n;n equation can provide a well de ned model in terms of F n;n, the time e ects are eliminated. In order that the nal transformed Y nt, we assume that W nj and M nj are rownormalized. This is so because, under row-normalization, we have F n;n W nj = F n;n W nj F n;n F n;n as F n;n W nj l n = F n;n l n = and I n = F n;n F n;n + n l nl n. From the transformed system, we can derive a partial likelihood approach for the estimation. 6 Denote W F n ;j = F n;n W nj F n;n and M F n ; = 5 We do not consider a random time e ects speci cation because we would like to leave its stochastic time series structure unspeci ed. 6 This approach has been considered in our earlier work, Lee and Yu (). 9

21 F n;n M n F n;n as the transformed spatial weights matrices. Then, F n;n Y nt; = W F n ;F n;n Y nt; + F n;n X nt; + F n;n U nt; (4.) with F n;n U nt; = W F n ;F n;n U nt; + B F n ;F n;n e nt, t = 3; :::; T; F n;n Y n = W F n ;F n;n Y n + F n;n X n + F n;n U n with F n;n U n = W F n ;F n;n U nt + B F n ;F n;n V n and V n = e n ( )V n, where B F n ; = I n + M F n ; = F n;n B n F n;n. Therefore, by denoting Y df n ;T ; = (Q T Fn;n )Yn;T d, (4.) can be written in the vector form of dimension (n )(T ) as Y df n ;T ; = (I T W F n ;)Y df n ;T ; + X df n ;T ; + (I T S F; n ; BF n ;)e df n ;T, (4.3) where Sn F ; = I n Wn F ; = Fn;n S n F n;n and e df n ;T = (I T Fn;n )e d n;t. Here, the elements of e df n ;T are uncorrelated in the spatial dimension because E(edF n ;T edf n ;T ) = eh T I n using F n;n F n;n = I n. Let Yn df ;T = (I T Fn;n )Yn;T d be an (n )(T ) column vector of transformed observations. By denoting S F n ;T ;j = I T S F n ;j, BF n ;T ; = I T B F n ; and Q n ;T = Q T I n, the likelihood of (4.3) is L nt (Y df n ;T ) = () (n )(T )= js F n ;T ;( )j js F n ;T ;( )j jb F n ;T ;( )j j eh n ;T ()j = jq n ;T ()j exp e df n ;T ( )H n ;T ()edf n ;T ( ) e where e df n ;T ( ) = B F; n ;T ; ( )S F n ;T ; ( )(S F n ;T ( )Q n ;T ()Y df n ;T Q n ;T ()X df n ;T ), H n ;T () = H T ()I n and jq n ;T ()j = jq T ()j n =. Also, js F n ;j ( j)j = j ji n j W nj j, (S F n ;j ( j)) = F n;n S nj ( j)f n;n, and similarly for B F n ;j. The rst and second order derivatives of this partial likelihood function would be similar to Section., but with premultiplication by F n;n. Assumption. W nj for j = ; ; 3 and M nj for j = ; 3 are row-normalized nonstochastic spatial weights matrices with zero diagonals. Assumption 5. Both n and T are large. Assumption 6. The elements of nk x matrix of regressor X nt are nonstochastic and bounded, uniformly in P T n and t. Also, under the asymptotic setting in Assumption 5, the limit of (n )(T ) t= X df n ;T ; H n;t X df n ;T ; exists and is nonsingular where X df n ;T ; = B ;F n ;T ; SF n ;T ; XdF n ;T ;. Assumption 7. Either (a) the conditions (C.3) and (C.4) in Appendix C. hold or (b) the condition (C.5) holds. (4.4),

22 Assumption 8. The limit of the information matrix (C.) is nonsingular. Assumption has the row-normalization of the spatial weights matrices so that the transformed equation still has a SAR representation. Assumption 5 speci es that we are interested in the large panels. Under this situation, the introduction of the time e ects will cause the incidental parameter problem and we need to eliminate the time e ects before estimation. Assumption 7 and 8 are modi cation of Assumption 7 and 8 to take into account the transformation by F n;n. Proposition 3 Under Assumptions,, 3, 4, 5, 6, 7 (a); or Assumption,, 3, 4, 5, 6, 7 (b) and 8, the estimates under the xed e ects speci cation from (4.4) are consistent and asymptotically normal: p (n )(T )(^w ) d! N(; lim n! ;F ;nt (F ;nt + F ;nt ) ;F ;nt ) where F ;nt in (C.) is the information matrix, and F ;nt moments of e it ( F ;nt = under normality). in (C.) is related to the fourth and third 4. A Mixed Case In this section we will consider the estimation of random individual e ects model with random individual e ects but with time e ects eliminated. Under such a speci cation, as z n is time invariant but individual variant, z n does not include the constant term, because the constant term will be eliminated along with t. Hence, the model is Y nt = z n b + W n Y nt + X nt + n + t l n + U nt, t = ; :::; T; (4.5) n = 3 W n3 n + B n3 c n, U nt = W n U nt + B n V nt, V nt = V n;t + e nt, t = ; :::; T: Denote Y F n ;T; = (P T F n;n )Y nt, X F n ;T; = (P T F n;n )X nt and e F n ;T = (P T F n;n )V nt. We have Y F n ;T; = (P T l T z F n b ) + (I T W F n)y F nt; + X F n ;T; + P T l T S F; n ;3 BF n ;3c n + (I T S F; n ; BF n ;)e F n ;T. By F n;n F n;n = I n, the variance matrix of the error components is F n ;T; = l T l T [( ) cs F; n ;3 BF n ;3B F n ;3S F; n ;3 ] + I T [ es F; n ; BF n ;B F n ;S F; n ; ]. By denoting Y F n ;T = (I T F n;n )Y nt, P n ;T = P T I n, and S F n ;T; = I T S F n ;, the likelihood of this random e ects model is L nt (Y F n ;T ) = () (n )T= j F n ;T;j = js F n ;T;( )jjp n ;T ()jexp F n ;T;() n ;T; F n ;T;(), (4.6)