Estimation of xed e ects panel regression models with separable and nonseparable space-time lters

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1 Estimation of xed e ects panel regression models with separable and nonseparable space-time lters Lung-fei Lee Department of Economics Ohio State University lee.777osu.edu Jihai Yu Guanghua School of Management Peking University jihai.yugmail.com First draft: February, 23 Second draft: October, 23 This draft: March, 24 Abstract This paper considers a quasi-maximum likelihood estimation for a linear panel data model with time and individual xed e ects, where the disturbances have dynamic and spatial correlations which might be spatially stable or unstable. We rst consider both separable and nonseparable space-time lters for the stable model. The separable space-time lter is subject to a parametric restriction which results in relative computational simplicity. In contrast to the spatial econometrics literature, we expose economic restrictions imposed by the separable space-time lter model and explore computational tractability of the nonseparable lter model. Throughout this paper, the e ect of initial observations is taken into account, which results in an exact likelihood function for estimation. This is important when the span of time periods is short. We then investigate spatial unstable cases, where we propose to apply a spatial di erencing to all variables in the regression equation as a data transformation, which may eliminate unstable or explosive spatial components in order to achieve a robust estimator. For estimates of the parameters in both the regression part and the disturbance process, they are p nt -consistent and asymptotically centered normal regardless of whether T is large or not and whether the process is stable or not. JEL classi cation: C3; C23; R5 Keywords: Spatial autoregression, Space-time lter, Panel data, Spatial cointegration, Explosive roots, Fixed e ects, Time e ects, Quasi-maximum likelihood estimation, Exact likelihood function We would like to thank participants of 8th International Conference on Panel Data (Paris, 22), International Conference on Policy Analysis Using Modern Econometric Methods (Beijing, 22), st China Meeting of the Econometric Society (Beijing, 23), seminars at South Western University of Finance and Economics (Chengdu, 23), two anonymous referees and the coeditor, Professor Peter M. Robinson, of this journal for helpful comments.

2 Introduction Panel regression models can be augmented with serial correlation or spatial dependence so as to control for time and spatial dependence in addition to heterogeneity in panels. These spatio-temporal interactions can be speci ed in dependent variables as in Su and Yang (27), Yu et al. (28), Elhorst (2) among others. They can also be speci ed in the error components such as in Elhorst (24), Baltagi et al. (27), Parent and LeSage (2, 22) and Lee and Yu (22). Empirical applications of these spatio-temporal dependences can be found in habit formation (Korniotis, 2), growth convergence of countries and regions (Ertur and Koch, 27; Mohl and Hagen, 2), regional markets (Keller and Shiue, 27), labor economics (Lottmann, 22), public economics (Revelli, 2; Franzese, 27) and other areas of study. The current paper focuses on a panel regression model with serially and spatially correlated disturbances. In the spatial literature, the serial and spatial correlations in disturbances have often been separated. Baltagi et al. (27) consider testing serial and spatial correlations in a model where these correlations are separable in disturbances. Parent and LeSage (2, 22) term it a space-time lter and consider Bayesian estimation. However, this separability imposes restrictions on spatial and time dynamics. It also rules out the possibility of spatial cointegration (Yu et al., 22) as well as cases in which serial and spatial correlations may operate only through time, which is known as di usion. In terms of forecasting, using spatial correlations improves forecasting performance, as noted in Giacomini and Granger (24) and Longhi and Nijkamp (27). Indeed, Giacomini and Granger (24) conclude that ignoring spatial correlation leads to highly inaccurate forecasts in their application. The separable model does not involve any spatial e ect in the best linear unbiased predictor formula, but it is present under the nonseparable model. From the estimation point of view, ignoring the mixing feature of space-time di usion might result in estimating a misspeci ed dynamic process. The estimates of parameters in the misspeci ed disturbances may be subject to estimation bias and lead to inaccurate statistical inference when a mixing of time and space features is present in the data generating process (DGP). The current paper investigates asymptotic properties of the quasi-maximum likelihood estimator (QMLE) for a panel regression model with dynamic and spatial correlations, where the separable space-time lter is a special case. The estimation of such a general space-time dynamic process has been argued (e.g., in Parent and LeSage, 2, 22) as unwise due to additional complexity in computation relative to the separable space-time lter for Bayesian estimation. Thus, we pay special attention to the computational issue of the nonseparable space-time model with the QML approach. In the estimation, we also consider the treatment of initial period observations, which is important when the number of time periods is small. This results in the formulation of an exact likelihood function. Indeed, the computational issue is raised due to the initial

3 period observation. If the initial period observation were treated as exogenously given, the QML estimation would be conditional on the initial period observation. Due to the recursive nature of a dynamic process, the evaluation of the conditional likelihood would be computationally simple (Yu et al. 28; Parent and LeSage 2 and 22). The conditional approach, however, is not proper in general for a short spatial panel. For the disturbances in a panel regression model with a nonseparable space-time lter, they may have not only serial correlation and spatial correlation, but also possible space-time unstable or explosive features in certain circumstances, e.g., the study of market integration in Keller and Shiue (27). For the general space-time dynamic process, some absolute summability conditions can be imposed if the process is stable in both space and time dimensions. However, a general nonseparable process might have spatial cointegration or explosive features when the eigenvalues in the DGP of the disturbances have some unit roots or even explosive ones (Holly et al. 2; Yu et al. 22). This calls for a treatment in the estimation because ordinary least square (OLS) or least square dummy variable (LSDV) estimates of the regression equation would not be consistent if there is non-stability in the disturbances. Thus, in addition to the analysis of the general stable space-time dynamic process and that of a separable space-time lter, this paper also concerns possible space-time nonstationarity in the disturbances. We propose the use of time di erencing and spatial di erencing transformations to handle space-time nonstationarity in estimation. The spatial di erencing transformation can be applied regardless of whether disturbances are spatially stable or not. With this data transformation, common inference can be performed and is robust without nonstandard asymptotic properties for estimates. In addition to providing asymptotic properties of QML estimation and testing for a spatial panel with both separable and nonseparable space-time lters, the current paper has the following contributions to the literature. First, we consider the stochastic spatial time process in the disturbances. In Yu et al. (28, 22), the model considered has spatial and dynamic lags in the main regression. Its particular feature is the presence of xed individual e ects and/or exogenous variables that will generate time trend components in the nal form of the dynamic model. With spatial nonstationarity, the time trend is a dominant feature for the asymptotic distribution of an estimator. The model in the current paper has a stochastic trend term but not a dominant time term. This di erence could make asymptotic distribution of estimators di erent. The current paper provides some estimation methods which can overcome the spatial nonstationary in terms of stochastic trend which may be present in a spatial nonstationarity process. To make the paper complete, we consider not only spatial nonstationarity but also stable and explosive cases. Thus, the current paper is di erent from our previous research on spatial cointegration for a spatial dynamic panel data (SDPD) See Baltagi et al. (28) for detailed discussions. 2

4 model. Second, we consider the estimation based on a likelihood function that is exact in the sense that we do not need to approximate the initial observation. Lee and Yu (2b, 2) present the spatial di erence transformation for the SDPD model but the estimation is conditional on the initial observation, and the consistency of those estimators requires that T tend to in nity. Furthermore, when T and n tend to in nity at the same rate, asymptotic biases in estimators exist. In the current paper, the initial condition is generated by the process itself and our estimation method is applicable to the xed T situation. The estimators are shown to be consistent and asymptotically normal without T tending to in nity, and there are no asymptotic biases. Therefore, the analysis in the current paper can be applied to both xed T and large T cases. Third, although there are some discussions on the use of the spatial di erence transformation in Lee and Yu (2b, 2), rigorous justi cations on asymptotic properties of estimators are not provided. The current paper provides rigorous analysis; in particular, we extend the inverse matrix of a generalized rst di erence matrix to the block matrix form (Hsiao et al, 22). As a result, we can justify the uniform boundedness property of the involved matrix in order to relate the analysis to those established by Kelejian and Prucha (998) for spatial econometric models. The rest of the paper is organized as follows. Section 2 introduces the model and discusses the separable and nonseparable space-time lters. Economic implications of separable and nonseparable lters and distinctive characteristics of stability and space-time unstable features are presented. Section 3 studies the panel regression model with the separable space-time lter structure and its estimation, 2 and Section 4 investigates estimation of the general nonseparable time and space correlations in disturbances. Section 5 discusses the use of the spatial di erence operator that can eliminate possible nonstationary features in the data and studies the resulting asymptotic properties of QMLE. Section 6 investigates the nite sample performance by Monte Carlo simulations of the QML estimates and classical tests for the constraint which characterizes the space-time lter. Section 7 concludes. Some algebraic derivations and proofs are collected in the Appendices. 2 A Panel Regression Model with a General Space-Time Filter 2. The Model Consider the model Y nt = X nt + c n + t l n + U nt ; t = ; ; T () U nt = W n U nt + U n;t + W n U n;t + V nt, 2 While Parent and LeSage (2) suggest the implementation of a Bayesian approach, we study the classical QML estimation and investigate asymptotic properties of the estimates. 3

5 where Y nt = (y t ; y 2t ; ; y nt ) and V nt = (v t ; v 2t ; ; v nt ) are n column vectors, and v it s are i:i:d: across i and t with zero mean and variance 2. W n is an n n nonstochastic spatial weights matrix, 3 and X nt is an n k matrix of nonstochastic regressors. For the regression part, we have individual e ects c n and time e ects t, where l n is an n-dimensional vector of ones. 4 process. 5 The disturbances U nt s follow the SDPD captures contemporaneous spatial interactions e ect, is the conventional time dynamic e ect of a Markov type, and may be interpreted as an e ect of spatial di usion that takes place over time. In practical applications, the spatial weights matrix W n is usually a row-normalized non-negative matrix with zero diagonals so that W n l n = l n. Row-normalization ensures that all the spatial weights are between and, and a weighting operation can be interpreted as an average of the neighboring values. The data [Y nt ; X nt ] for all the n spatial units are available for t = ; 2; ; T. 6 De ne S n S n ( ), where S n () = I n W n and I n is the n-dimensional identity matrix. Then, presuming that S n is invertible and denoting A n = Sn ( I n + W n ), U nt in () can be rewritten as U nt = A n U n;t + S n V nt. (2) By xing the number of spatial units but moving along the time dimension, (2) is a VAR process. The 3 The spatial lag W nu nt and the spatial time lag W nu n;t use the same spatial weights matrix W n. This is reasonable if the spatial weight matrices are time invariant in the sampling periods. In general, they can be associated with di erent spatial weights matrices such as those time varying ones. However, the features of spatial cointegration and explosive roots are hard to track when we have di erent spatial weights matrices. This is so because the eigenvalues of the process would not be straightforward when those spatial weights matrices are not diagonalizable with the same eigenvectors. 4 In the spatial panel data literature, Anselin (988) and Kapoor et al. (27) are di erent in terms of the location of individual e ects, i.e., whether c n are in the regression equation or the disturbances. For (), it is the Anselin model. As discussed in the supplement le (available in these two models may have some related features after our data transfromation under the xed e ects speci cation. Our proposed estimation methods can handle both of them in the same way. 5 The current paper focuses on lters so that the SDPD process is in the disturbances. The model () can be extended to include the spatial dynamics in the main regression (see Yu et al. 28, 22). When T is short, we can use GMM estimation similar to Kelejian and Prucha (998), or use MLE with initial observation approximation as in Elhorst (25) and Su and Yang (27) where the information of X nt for t = ; ; T will be used in the approximation. The MLE in the current paper can also be applied to such a general model after the initial observation approximation with X nt for t = ; ; T is adopted if its approximation error can be properly taken care of. 6 The current paper covers the case of a typical balanced panel data. In some empirical applications, we have to face some missing data situations. Although numerous methods are proposed in the literature for the missing data (see Baltagi and Song, 26 for a review on panel data), some simple methods such as listwise deletion are unsuitable for spatial models. This is so because we have to consider the interactions among unobserved variables as well as those between the observed and unobserved ones. Omission of relevant units would lead to misspeci cation of an underlying spatial weights matrix of a population. See LeSage and Pace (24), Kelejian and Prucha (2b) and Wang and Lee (23a) for cross sectional spatial models. For spatial panel data model, missing data can occur in both time and cross sectional dimensions. Wang and Lee (23b) investigate the missing data issue in spatial panel models when we have randomly missing data in the dependent variable. For the current paper with both dynamic and spatial features and exact likelihood estimation, the analysis of missing data would not be easy to handle, and will be even more complicated if missing data are also in independent variables. We expect that (i) when the missing is random and the missing unit links with some other units, by ignoring the missing spatial units and treating the remaining ones as a complete population, an SAR model would be subject to misspeci cation and it will cause bias in estimation because such missing units would distort the W n structure. (ii) only when the missing is random and the missing unit has no links with other units, then it is a sample size reduction; (iii) when the missing is non-random, we have to take into account the mechanism (the selection rule) that causes the missing (Verbeek, 996); with spatial dependence, this situation would be more complicated if the missing unit has interaction with observed units. 4

6 stationarity of this VAR will depend on the eigenvalues of A n. If all the eigenvalues of A n are less than in absolute value, the VAR process is covariance stationary; otherwise, it might be nonstationary (see, e.g., Hamilton, 994). In the spatial and time dynamic model, as both n and T are relevant, when all the eigenvalues of A n in (2) are smaller than in absolute value, we may call it the stable case. For nonstationarity due to space-time interactions, the case of spatial cointegration with + + = is of interest (Yu et al. 22). We shall discuss the di erences of the stable and unstable models in more detail in a subsequent subsection. 2.2 Separable and Nonseparable Space-time Filters There is some interest in whether the space and time dynamics should interact or not. The general process () allows the space and time dynamics to be interacted. For the general spatial and time dynamic disturbances U nt in (), by denoting L n as the time shift operator such that L n U nt = U n;t, one has the representation [(I n W n ) ( I n + W n )L n ]U nt = V nt with (I n W n ) ( I n + W n )L n being a general space-time lter. There is, however, a special case where the space dynamic is separated from the time dynamic. When =, the space-time lter can be decomposed into a product of the space lter (I n W n ) and the time lter (I n L n ), i.e., (I n W n ) ( I n + W n )L n = (I n W n )(I n L n ) = (I n L n )(I n W n ). An alternative way to see the di erence of the general space-time lter and the separable one is that the general lter can be rewritten into the form: (I n W n ) ( I n + W n )L n = (I n W n )(I n L n ) ( + )W n L n, (3) where the absence of the last term on the interaction of W n and L n gives rise to the separable space-time lter. For the case =, the model becomes Y nt = X nt + c n + t l n + U nt ; (4) U nt = W n U nt + e nt with e nt = e n;t + V nt. From (4), we see that U nt is a spatial autoregressive (SAR) process with its noise term e nt being a VAR of order (VAR()). In place of (4), we also have the representation that U nt = U n;t + e nt with e nt = W n e nt + V nt : (5) From (5), U nt is a VAR() process with its noise vector e nt being an SAR process. The I n L n is a time lter and I n W n is a spatial lter. With these lters, we have from (4) (I n L n )(I n W n )U nt = V nt, and from (5), (I n W n )(I n L n )U nt = V nt. The spatial and time lters are apparently commutative 5

7 because they are functioning in di erent dimensions. The separable space-time lter model as in (4) is introduced in Baltagi et al. (27). Parent and LeSage (2, 22) consider the estimation of such a separable lter model in a Bayesian framework. They propose to use MCMC sampling methods that involve generating sequential samples from the complete set of conditional posterior distributions, because direct evaluation of the joint posterior distribution involves multidimensional numerical integration and is not computationally feasible. They show that the separability restriction can greatly simplify the computational task in estimation. Under the assumption that the disturbances are normally distributed, Parent and LeSage (22) apply the MCMC method to obtain random draws from parameters posterior distributions. They show that a Bayesian estimation of the general model () with a nonseparable space-time lter is too complicated to be computationally attractive. Under the MCMC setting, they argue that the computational gains are great enough to justify some inaccuracy in the parameter estimation. While the separable space-time lter model may be computationally simpler, the constraint = may rule out interesting spatial and time dynamic interactions. Under the QML estimation framework, the computational gains from reduction in the search dimension might not be great enough for the signi cant bias of the separable lter model when the true model is nonseparable (see Section 6 for Monte Carlo results). With the separable space-time lter =, if there were no contemporaneous spatial interaction or serial correlation, i.e., = or =, the spatial time lag W n U n;t would have no e ect because of the implied =. This would rule out di usion and also the habit formation model in Korniotis (2), which does not have contemporaneous spatial interaction. Furthermore, if both and are positive, the e ect of the spatial time lag W n U n;t must be negative. Because both and are less than one in absolute value, the would also be less than both and in magnitude. The separable space-time lter also rules out the spatial cointegration phenomenon. For the separable space- lter speci cation with =, it implies that ( + + ) = ( )( ). Under j j < and j j <, the sum + + cannot be, so that the separable space-time lter rules out the spatial cointegration case. Additionally, in the forecasting of a panel regression model with spatially and serially correlated disturbances, the nonseparable model incorporates spatial e ect in the best linear unbiased predictor formula, while the separable model does not involve any spatial e ect. This causes inaccuracy in forecasting for the separable model when the true DGP is nonseparable (see Section 2.4 for details). In contrast to the separable space-time lter, some researchers have proposed the speci cation of the disturbance process of U nt in () without the W n U n;t term but the inclusion of only contemporary space lag W n U nt and time dynamic U n;t in the space-time process (e.g., Elhorst 22, eq.(b)). Elhorst (28) 6

8 regards the inclusion of both W n U nt and U n;t as a way of introducing space and serial correlation interaction and criticizes the separable space-time lter speci cation. From his Monte Carlo results, Elhorst (28) argues that serial and spatial error correlations are better not modeled independently from one another as in the separable case, because an estimation of the separable model results in estimators ine ciency and the probability of ending up with a nonstationary model would be larger. We take Elhorst s (28) argument a step further and propose a nonseparable model that encompasses both the separable model and the model with joint serial and spatial correlation in Elhorst (28). We argue that the exclusion of the possible spatial time lag W n U n;t in Elhorst (28, 22) has ignored the possibility of a separable space-time lter and in consequence dismissed its possible merit. It also rules out the habit formation model in Korniotis (2). Therefore, to encompass both a separable space-time lter and a nonseparable space-time lter, the inclusion of both contemporary spatial lag W n U nt and time-spatial lag W n U n;t is needed. 2.3 Stability, Spatial-cointegration, and Spatial-explosion The space-time dynamic process of U nt in () may have di erent stochastic properties depending on the parameter values of, and. For this space-time process, we consider the situation that W n is row-normalized from an original symmetric matrix (Ord 975). In this case, the spatial weights matrix W n is diagonalizable with real eigenvalues, i.e., there exists real eigenvalues and eigenvector decomposition such that W n = n $ n n, where $ n is the diagonal matrix of eigenvalues and n is the matrix consisting of eigenvectors. Thus, as A n = S n ( I n + W n ), the eigenvalues matrix of A n is D n = (I n $ n ) ( I n + $ n ) such that A n = n D n n. With n xed, the VAR process U nt is covariance stationary when all the eigenvalues in D n are less than in absolute value. Denote $ n = diagf$ n ; ; $ nn g and D n = diagfd n ; ; d nn g. We have d ni = + $ni $ ni. As W n is row-normalized, all the eigenvalues are less than or equal to in absolute value by the spectral radius theorem, where it de nitely has some eigenvalues being. Let m n be the number of unit eigenvalues of W n and let the rst m n eigenvalues of W n be the unity. The D n can be decomposed into two parts, one corresponding to the unit eigenvalues of W n, and the other corresponding to the eigenvalues of W n smaller than. De ne J n = diagf m n ; ; ; g with mn being an m n vector of ones and ~ D n = diagf; ; ; d n;mn+; ; d nn g. As J n ~ D n =, we have A h + n = h nj n n + ~ A h n where ~ A h n = n ~ D h n n (6) for any h = ; 2;. Depending on the value of +, we have di erent stable or unstable cases. For all those cases, we require that eigenvalues of ~ An are smaller than in absolute value, i.e., jd ni j < for i = m n + ; ; n. 7

9 The following discussion will investigate those conditions to satisfy jd ni j < for i = m n + ; ; n. Also, we need S n to be invertible and its determinant js n j to be positive. Because W n is row-normalized and has a zero diagonal, all j$ ni j with its maximum $ n;max = and its minimum value $ n;min satisfying $ n;min <. Thus, we need $ n;min < < for the invertibility of S n and its determinant js n j being positive. The stable case has all the eigenvalues of A n being less than in absolute values, which corresponds to + < and jdni j < for i = m n + ; ; n. As d ni = + $ni $ ni, by regarding d s as a function of $, we have $ ( + $ ) = + $ ( $). Thus, there are three di erent situations: 2 (i) + > if and only if d ni has the same increasing order as $ ni. (ii) + =, i.e., separable space-time lter, if and only if d ni is a constant (under this case, d ni = ). (iii) + < if and only if d ni has the decreasing order of $ ni. With these characteristics, the region of parameter values for stability is the convex set B s =f(; ; ) : + + < ; + ( )$ n;min > ; + > ; + ( + )$ n;min < g (7) with vertices (; ; ), ( $ n;min ; ; $ n;min ), ( $ n;min ; ; $ n;min ), and (; ; ). We note that the condition jd ni j < for all i happens to include the condition $ n;min < <, so that $ n;min < < is not explicitly expressed in (7). The separable space-time lter case is a special stable case in that all the eigenvalues d ni take the same value and are not related to the eigenvalues of W n. The unstable cases that are of interest are the spatial cointegration case when + + = and the explosive case when + + >. Denote Wn u = n J n n. For unstable cases, we assume that the process in U nt has started from t = m period where m >. Then, for t, U nt can be decomposed into a sum of a possible stable part and a possible unstable or explosive part (see Appendix C for derivations): where and U u nt = W u n U nt = U u nt + U s nt, (8) t+ + X h t + U n; + V n;t h! ; ( ) h= U s nt = X t+m h= ~A h ns n V n;t h : The U u nt can be an unstable component when + =, which occurs when + + = and 6=. In this case, U u nt has a stochastic trend, which is the feature of a random walk process. When + + > and <, it implies + > and, hence, U u nt can be explosive. The U s nt can be a stable component 8

10 unless + + is much larger than. If the sum + + were too big, some of the eigenvalues d ni in U s nt might become larger than. More on these will be discussed subsequently. We term spatial cointegration for the situation that + + =, which has a revealing error correction model (ECM) representation. Denote U nt = U nt U n;t as a di erence in time. As A n I n = S n [( + )W n ( )I n ], the U nt process in () has the ECM representation For the spatial cointegration case, U nt = S n [( + )W n ( )I n ]U n;t + S n V nt : U nt = ( )S n (W n I n )U n;t + S n V nt : In this situation, (I n W n ) is a cointegrating matrix for U nt because W n W u n = W u n and (I n W n )U nt = (I n W n )U s nt which depends only on the stable component. Thus, U nt is spatially cointegrated. The cointegration rank of I n W n equals to n m n, which is the number of eigenvalues of W n di erent from. For our estimation theory, we require also that all the eigenvalues d ni of A n which correspond to eigenvalues $ ni with j$ ni j < have the property jd ni j <. This ensures that all eigenvalues of A n (except ) are less than in absolute value. Combining the two features of + + = and jd ni j < for i = m n + ; ; n, the corresponding parameter range turns out to be (; ; ) : + + = ; < < ; < ; < + ( )( + $ n;min ) $ n;min $ n;min for the spatial cointegration case. Because j$ n;min j <, this does not seem to have imposed much restriction. The explosive case corresponds to + + = + a where a >. As jj < is assumed, the explosive feature refers to cases with spatial and time e ects mixed together; thus, the total value of spatial e ects + plays a role. To satisfy the condition for the eigenvalues jd ni j < whenever j$ ni j <, the corresponding parameter region will depend on the excess level a of explosiveness. This condition for the explosive case will be satis ed by the parameters in the set ( " #) + + = + a; $ (; ; ) : n;min < < ; jj < ; < + ( ) (+$n;min) ( $ + a n;min) ( $ ; a ; () n;min) ( $ n;(2) ) < ( + ); a > which depends on the excess explosive level a, where $ n;(2) refers to the largest eigenvalue of W n which is less than. Thus, the e ective constraint is on the total spatial value of + to be positive and not too small relative to the excess explosive level a. (9) 9

11 2.4 Forecasting of Separable and Nonseparable Models The separable and nonseparable models have di erent e ects on the forecasting in panel data models. In this section, we show that the separable model does not involve any spatial e ect in the best linear unbiased predictor formula, but it is present under the nonseparable model. For a linear regression model y t = x t + t where = ( ; 2 ; ; T ) has the covariance matrix, Goldberger (962) showed that the best linear unbiased predictor (BLUP) for y T + is ^y T + = x T + ^GLS +! ^ GLS, () where ^ GLS and ^ GLS are the GLS estimator and residuals, and! is the row vector of covariance of T + with. The x T + ^GLS estimates the expected value of y T + given x T +, and! ^ GLS utilizes the interdependence of disturbances to predict future disturbance at T +. Baltagi et al. (22) investigate the BLUP for a panel regression model with spatial error components. For the xed e ects model, the second component! in () does not show up because it is a static process, and x i;t + ^GLS + ^c i, where ^c i is the estimated xed e ect for ith unit, is used to replace x i;t + ^GLS for the rst component in (). The prediction formula does not involve the spatial weights matrix W n and turns out to be the same as the panel regression model with classical error components and without spatial e ect. With a dynamic feature in the disturbances, the spatial e ect (hence W n ) will play a role in the prediction formula in general. Under the panel regression with the nonseparable space-time lter in the disturbances, the spatial weights matrix W n would be present in the second component! in (). However, under the separable case, W n will not play a role in!, even though the dynamic feature is still present in the disturbance. For the xed e ects panel regression model with the general space-time lter, the correlation of future and current disturbances comes from the space and time dynamic feature. For the prediction of Y n;t +, we need and c ni for the rst component in (), and also the correlation term for the second component in (). With an estimated ^ nt, we have the resulting ^r nt = Y nt X nt^nt. The estimate for c n is 7 ^c n = P T T t= ^r nt. Thus, the predictor for Y i;t + for the model () can be ^Y n;t + = X n;t + ^nt + ^c n + [EU n;t + U nt ][E(U nt U nt )] ^U nt;gls, (2) where U nt = (U n; ; U nt ). Here, the time e ect T + does not appear because the future time e ect has zero mean expectation due to the restriction l T T =. 7 We impose l T T = so that the intercept term is absorbed in c n. We can alternatively impose l n c n = so that the estimated zero mean individual e ect is ^c n = T P T t= Jn^rnt, and the estimated intercept term would be T l T ^ T where ^ t = n l n^r nt.

12 The third term in (2) can be simpli ed. From U n;t + = A nu nt + S n (V n;t + + A n V n;t A n V n;t + ), we have EU n;t + U nt = A neu nt U nt. Therefore, using EU nt U nt = (; ; ; I n) EU nt U nt, we have [EU n;t + U nt ][E(U nt U nt )] ^U nt = (; ; ; A n) ^U nt = A n ^U nt. (3) Based on the residual ^U nt, we can predict U n;t +, the period ahead value of U nt. The matrix A n involves the spatial weights matrix W n under the nonseparable case. Under the separable case where =, we have A n = (I n W n ) ( I n + W n ) = I n, which does not involve the spatial weights matrix. Hence, [EU n;t + U nt ][E(U nt U nt )] ^U nt = ^U nt. (4) Therefore, under the separable case, only the self time lag is relevant and the contemporaneous and spatial time lags are irrelevant in the prediction formula. 8 3 Estimating the Panel Regression Model with a Separable Space- Time Filter 3. A Separable Space-Time Filter and the Likelihood Function In this section, we consider the classical QML estimation approach for the model () with the separable space-time lter in (4) or (5). In a subsequent section, we shall compare the estimation for the model with a nonseparable space-time lter. To eliminate the time e ects in the panel regression equation, we can use the deviation of the group mean transformation J n = I n n l nl n. By using J n l n =, the time e ects t l n will be eliminated. However, the variance matrix of the transformed disturbances J n V nt is equal to 2 J n so that the elements of J n V nt are correlated. Also, J n is singular with rank (n ) as J n is an orthogonal projector with trace (n ). Hence, there is a linear dependence among the elements of J n V nt. An e ective estimation method shall eliminate the linear dependence in sample observations. This can be done with the eigenvalues and eigenvectors decomposition as in the theory of generalized inverses for the estimation of linear regression models (see, e.g., Theil (97), Ch. 6). Let (F n, l n = p n) be the orthonormal matrix of eigenvectors of J n where F n 8 In addition to forecasting, the impulse responses (as in VAR) illustrate di erent implications by the separable lter vs. the nonseparable one. Also, the e ect of a random shock to the system is transitory if the system is stable; however, for the spatial cointegration case, there is a component of permanent e ect which does not diminish over time. See the supplement le for details.

13 corresponds to the eigenvalues of ones and l n = p n corresponds to the eigenvalues of zeros. W n = F nw n F n, Y nt = F ny nt and other variables accordingly, we have By denoting Y nt = X nt + c n + U nt; (5) U nt = W nu nt + e nt with e nt = e n;t + V nt, where V nt is an n dimensional disturbance vector with zero mean and variance matrix 2 I n. 9 where For any n vector Z nt, denote Z nt = Z nt Z n;t as the rst di erence. From (5), we have Y nt = X nt + U nt, (6) U nt = W nu nt + e nt for t = 2; ; T; e nt = e n;t + V nt for t = 3; ; T; e n2 = ( )e n + V n2, and V nt is an MA process. Here, e n2 represents the initial observation. When T is large, we may ignore the initial period observations and treat them as exogenously given as, for example, in Yu et al. (28), because ignoring information conveyed by the initial period should have little e ect on the estimates. It will simplify the likelihood function and its computation. However, when T is xed, the information in the rst period should not be easily discarded. In panel models where the spatial dependence is present in the disturbances rather than the dependent variables, treatment of the initial period observations has been found to be important (Su and Yang, 27). De ne B n;t = Q T Sn where Sn = Sn( ) with Sn() = I n Wn and Q T = Q T ( ) with Q T () = B..... (7).. C A... being a quasi-di erence transformation matrix over time of dimension (T ) (T ). By de ning Y n;t = (Y n2; ; Y nt ), X n;t (6) can be written as = (X n2; ; Xn;T ) and Vn;T = (e n2; Vn3; ; VnT ), B n;t Y n;t = B n;t X n;t + V n;t, 9 In obtaining (5) where FnW nu nt = WnU nt, we have relied on Wn being row-normalized so that F nw nu nt = Fn Wn(FnF n + n lnl n )Unt = W n U nt because Wnln = ln and F n ln =. If Wn is not row-normalized, we will not have an SAR representation in the disturbances after the data transformation by F n. 2

14 where the variance of V n;t is 2 T I n with n = n and T = T ( ) of dimension (T ) (T ): T () = B (8) C A Denote s = (; ) and s = ( ; s; 2 ). At true parameters, s = ( ; ) and s = ( ; s; 2 ). The log likelihood function (when the disturbances are normal; otherwise, a quasi-log likelihood) of s is ln L nt ( s ) = n (T ) 2 +n ln jq T ()j ln(2 2 ) + (T ) ln jsn()j n 2 ln j T ()j (9) 2 2 V n;t ( s )( T () I n )V n;t ( s ), where Vn;T ( s) = B n;t ( s)(yn;t X n;t ) and jq T ()j =. Alternatively, by de ning L T ;T = B C as the rst di erence transformation matrix of dimension (T A )T with L n ;T ;T = L T ;T I n, we have the relation Vn;T ( s) = B n;t ( s)l n ;T ;T (YnT X nt ) with Y nt = (Y n; ; Y nt ). This implies that V n;t ( s )( T () I n )V n;t ( s ) = (Y nt X nt ) (J T () S n ()S n())(y nt X nt ) where J T () L T ;T Q T () T ()Q T ()L T ;T (2) is a non-negative de nite matrix of dimension T T, which has an explicit expression in (36) in Appendix A and it is uniformly bounded in row and column sums (for short, UB) in absolute value. 2 The J T () matrix eliminates the individual e ects (by L T ;T ) and serial correlation (by Q T ()) in the disturbances. When =, we have J T () = L T ;T T L T ;T = I T l T lt =T as the usual deviation from time mean Here we assume that the process has achieved stationarity so that e n (; 2 =( 2 )In). If the process has started at t = m, then the rst element of T () shall be + ( )( 2(m+2) )=( + ). We say a (sequence of n n) matrix P n is P uniformly bounded in row and column sums if sup n kp nk < and sup n kp nk <, where kp nk sup n in j= jp P ij;nj is the row sum norm and kp nk = sup n jn i= jp ij;n j is the column sum norm. 2 The T () is not necessarily UB if T goes to in nity. 3

15 operation, where l T is a T vector of ones. Thus, the log likelihood (9) can be rewritten as ln L nt ( s ) = n (T ) ln (T (Y nt ) ln js n()j X nt ) (J T () S n ()S n())(y nt n 2 ln j T ()j (2) X nt ). Using js n()j = ( ) js n()j with S n () = I n W n (see Lee and Yu, 2a), Y nt = F ny nt and F n F n = J n, (2) can also be written in terms of original variables as ln L nt ( s ) = (n )(T ) ln 2 2 n + (T ) ln js n ()j (T ) ln( ) ln j T ()j (Y nt X nt ) (J T () Sn()J n S n ())(Y nt X nt ), (22) with its score (39) and information matrix (4) in Appendix A. The evaluation of this log likelihood function involves the determinant of S n (), which would be the same as that for a conventional SAR model. The evaluation of the determinant j T ()j has a simple expression in Hsiao et al. (22). The separability of the space and time lters has the simpli cation in the separate evaluation of the spatial transformation S n () from J T () in the likelihood function in (2) or (22). In the above analysis, to eliminate individual e ects, we suggest the use of the rst di erence rather than the deviation from time mean. This is so because serial correlation in time series can be better dealt with in a recursive fashion. For a classical static panel model Y nt = X nt + c n + V nt, we can use either the group mean transformation J T = I T T l T l T transformation, as the variance matrix of the transformed disturbances 2 J T can rely on the eigenvector matrix transformation F T from J T. Using F T F T = I T or the rst time di erence. For the group mean for each unit is singular, we, the regression equation [Y n ; ; Y nt ]F T = P k j= [X n;j; ; X nt;j ]F T j + [V n ; ; V nt ]F T would have i.i.d. disturbances and the corresponding estimate of is the LSDV estimate and estimate of 2 is unbiased. For the rst time di erence equation Y nt = X nt + V nt, the variance matrix of the transformed disturbances would be 2 L T ;T L T ;T for each unit. By applying GLS to the rst di erenced equation (with T periods) and using L T ;T (L T ;T L T ;T ) L T ;T = J T, the estimate of is also the LSDV and the estimate of 2 would be also unbiased. Therefore, the group mean transformation and the rst time di erence will yield the same estimates for and 2 under the static model. When we specify serial correlation in the disturbances, the variance matrix of the group mean transformation is hard to track with the multiplication of F T Q T. On the contrary, the rst time di erence involves the multiplication of L T and ;T and Q T, which can be analyzed explicitly. Hsiao et al. (22) provide guidelines of such an analysis, and the current paper extends those results to the model with spatial lag and spatial time lag models. 4

16 3.2 Asymptotic Properties of QMLE For our asymptotic analysis, we make the following assumptions. Assumption. The disturbances fv it g, i = ; 2; ; n and t = 2; 3; ; T; are i:i:d: across i and t with zero mean, variance 2 and E jv it j 4+ < for some >. Also, the process in U nt has achieved the stationarity or has started a long time ago. Assumption 2. W n has properties that () W n is row-normalized and UB, and (2) W n is diagonalizable, i.e., W n = n $ n n where n is the eigenvector matrix, $ n is the diagonal eigenvalue matrix, and all the eigenvalues are real. Assumption 3. S n () is invertible for all 2, where is compact and the true parameter is in the interior of. Also, S n () is UB, uniformly in 2. Assumption 4. X nt are nonstochastic with sup n;t nt P T t= P n i= jx it;lj 2+ < for l = ; ; k for some >, where x it;l is the (i; l) element of X nt. Assumption 5. n tends to in nity. Assumption provides regularity assumptions for v it. If there were unknown heteroskedasticity in v it, the MLE would not be consistent. Methods such as the generalized method of moments (GMM) in Lin and Lee (2) and that in Kelejian and Prucha (2a) would be designed for that situation. The latter part of Assumption implies that e n (; 2 2 I n ) for the separable case. The rst part of Assumption 2 on UB is originated by Kelejian and Prucha (998, 2) and is also used in Lee (24, 27). When W n is row-normalized, the parameter can be considered as the average spatial e ect. Consequently, all eigenvalues of W n are less than one in absolute value. Kelejian and Prucha (2a) specify that the parameter space for an arbitrary W n can be ( $ n;max ; $ n;max ). When W n is row-normalized so that its largest eigenvalue is, the parameter space for can then be ( ; ). The uniform boundedness of W n is a condition that limits the spatial correlation to a manageable degree. The second part of Assumption 5 is convenient for the decomposition in (8) and will be required for the analysis of unstable models in Section 5. The practical speci cation of Ord (975) is constructed by the row-normalization of a symmetric matrix. Ord (975) showed that such a row-normalized matrix is diagonalizable and all of its eigenvalues are real. We consider a slightly generalized case in which W n is row-normalized and diagonalizable with all real eigenvalues. Assumption 3 guarantees that the reduced form of the disturbance in (4) is well de ned. Also, compactness of parameter spaces is a convenient condition for theoretical analysis on nonlinear functions. The UB property of Sn () is a condition to limit the spatial correlation to a manageable degree. When exogenous variables X nt are included in the model, it is convenient to assume that their elements are uniformly bounded or, more generally, with bounded moments as in Assumption 4. Assumption 5 allows 5

17 two cases: (i) T! as n! ; (ii) T can remain xed as n!. We also make assumptions in order to establish asymptotic properties of the QMLE. Assumption 6. (58) and (59) in Appendix D hold so that the and ( s; 2 ) are identi able. Assumption 7. The limit of the information matrix (4) in Appendix A is nonsingular. Assumption 6 speci es identi cation conditions for the parameters, where (58) represents the possible identi cation of and (59) states the identi cation of ( s; 2 ) through the disturbances. Assumption 7 is for the nonsingularity of the limit of the information matrix of the model. Theorem Under Assumptions, 2(), 3 7, the QML estimate ^ s;nt of s from maximizing (9) is consistent and asymptotically normal: p (n )(T )(^s;nt s )! d N(; lim n! nt + nt nt nt ), where nt in (4) is the information matrix, and nt in (4) is related to the third and fourth moments of v it ( nt = under normality). 4 Stable Models: Nonseparable Space-Time Filter The separable space-time lter speci cation rules out some interesting spatial and time dynamics. In terms of estimation, because the separable spatial-time lter might provide a misspeci ed model, it can cause estimation bias for the disturbances parameters and loss of e ciency for the regression coe cients, if the space-time lter for the DGP is in fact nonseparable. In this section, we will investigate the estimation of the regression panel model with a nonseparable space-time lter. In the asymptotic analysis of UB properties of relevant matrices, we extend the inverse of a generalized rst di erence matrix to the block matrix form (Hsiao et al. 22). 4. The Model and The Likelihood Function For DGP in (), after the F n transformation from J n and the rst di erence, the estimation equation is Y nt = X nt + U nt, t = 2; 3; ; T, (23) U nt = W nu nt + U n;t + W nu n;t + V nt. Here, the may not be equal to. We note that even though Y nt = Y nt Y n;t is observable for t = 2; 3; ; T, U n in the disturbance process of Y n2 might not be observed. Denote = (; ; ) and = ( ; ; 2 ). At the true parameters, = ( ; ; ) and = ( ; ; 2 ). Denote R n() = I n + W n and R n = R n( ). From (23), S nu nt = R nu n;t + V nt for all t. 6

18 With backward substitution, denoting A n() = Sn ()Rn() so that A n = Sn Rn, we have SnU n2 = P j= Aj n Vn;2 j. By denoting U n;t = (Un2; ; UnT ) and Sn Rn Sn B n;t = Rn, (24) B C. A Rn Sn we have B n;t U n;t = ((S nu n2) ; V n3; ; V nt ). For the variance matrix of B n;t U n;t, as Var[ P j= Aj n V n;2 j ] = 2 K n where K n = K n( ) with 3 K n() [I n + P j= Aj n ()(A n() I n )(A n() I n ) A j n ()]; (25) we have Var(B n;t U n;t ) = 2 n;t where n;t = n;t ( ) with n;t 2 () 6 4 Kn() I n I n 2I n I n I n 2I n I n I n I n 2I n 3. (26) 7 5 The log likelihood function for (23) is ln L nt () = n (T ) ln(2 2 ) + (T ) ln js 2 n()j 2 ln n;t () (27) 2 2 (Y nt X nt ) J nt ()(YnT X nt ), where J nt () = L n ;T ;T B n;t () n;t ()B n;t ()L n ;T ;T has an explicit form in (45) in Appendix B. 4 We see from (2) that for the separable space-time lter, J nt () is reduced to J T () S n ()S n(), the evaluation of which will involve separate evaluations of J T () with dimension T and S n() with dimension (n ). For the nonseparable case, as seen from Appendix B, the determinant of n;t is j(t )K n (T 2)I n j, and its inverse can be derived in terms of (n )(n ) matrices from (42) in Appendix B, so that we can avoid the computation of an (n )(T )(n )(T ) 3 Assuming that the process starts at t = m (so that S nu n; m = V n; m and S nu nt = R nu n;t + V nt for t > m), we have SnU n2 = V n2 +P m+ j= Aj n Vn;2 j +Am+2 n Vn; m instead. Under such a situation, Var[V n2 +P m+ j= Aj n Vn;2 j + A m+2 n Vn; m ] = 2 K n where K n [I n + P m+ j= Aj n (A n I n )(A n I n ) A j n ]. 4 From Appendix B, its counterpart J nt () without the F n transformation is UB; however, the counterpart of n;t () alone is not necessarily UB. 7

19 matrix when both n and T are large. Using js n()j = ( ) js n()j, Y nt = F ny nt and F n F n = J n, as is derived in Appendix B, (27) can be written as ln L nt () = (n )(T ) ln(2 2 ) + (T ) ln js n ()j (T ) ln( ) (28) 2 2 ln ji n + (T )(K n () I n )J n j 2 2 (Y nt X nt ) H nt ()(Y nt X nt ), where we have used n;t () = jin +(T )(K n () I n )J n j from (46) in Appendix B, with K n () de ned similar to K n() such that K n () [I n + P j= Aj n()(a n () I n )(A n () I n ) A j n ()]; (29) and H nt () = J nt L n;t ;T B n;t () n;t ()B n;t ()L n;t ;T J nt, (3) where J nt = I T J n, L n;t ;T = L T ;T I n, B n;t () has its counterpart without the F n transformation in (24), and n;t Appendix B. 4.2 Asymptotic Properties of QMLE () has its counterpart in (26). Its score (47) and information matrix (5) are in For our analysis of the asymptotic properties of estimators, we make the following assumptions. Assumption 7. The limit of the information matrix (5) in Appendix B is nonsingular. Assumption 8. P h= abs(ah n) is UB, where [abs(a n )] ij = ja n;ij j. Assumption 9. H nt () is UB. Assumption 7 is for the nonsingularity of the limit of the information matrix for the nonseparable model. Assumption 8 is the absolute summability condition and row/column sum boundedness condition for disturbances, which will play an important role for asymptotic properties of QML estimator. In order to justify the absolute summability of A n, a su cient condition is jja n jj < for any matrix norm (see Horn and Johnson (985), Corollary 5.6.6) that satis es jja n jj = jjabs (A n ) jj. When jja n jj <, P h= Ah n exists and can be de ned as (I n A n ). The UB property of H nt () in Assumption 9 is convenient for the model after the J n transformation. Under initial condition restrictions, 5 the UB of H nt () is satis ed. Theorem 2 Under Assumptions, 2(), 3-6,7 and 8-9, the QMLE ^ nt consistent and asymptotically normal: of from maximizing (27) is p (n )(T )(^nt )! d N(; lim n! nt + nt nt nt ); 5 The initial condition can be The inverse of the variance matrix of ( ) =2 V nt+(a n I n)(v n;t +A nv n;t 2 +A 2 nv n;t 3 + ) is UB for = ; ; and T 2 T. See the supplement le for details on how this restriction can imply the UB of H nt (). 8