Prediction in a Generalized Spatial Panel Data Model with Serial Correlation

Transcription

1 Prediction in a Generalized Spatial Panel Data Model with Serial Correlation Badi H. Baltagi, Long Liu February 4, 05 Abstract This paper considers the generalized spatial panel data model with serial correlation proposed by Lee and Yu 0 which encompasses a lot of the spatial panel data models considered in the literature, and derives the best linear unbiased predictor BLUP for that model. This in turn provides valuable BLUP for several spatial panel models as special cases. Key Words: Prediction; Panel Data; Fixed Eff ects; Random Eff ects; Serial Correlation; Spatial Error Correlation. Introduction Panel data has been used in forecasting gasoline demand across OECD countries, see Baltagi and Griffi n 997; Residential electricity and natural-gas demand using a panel of American states, see Maddala, Trost, Li and Joutz 997; World carbon dioxide emissions, see Schmalensee, Stoker and Judson 998; Growth rates of OECD countries, see Hoogstrate, Palm and Pfann 000; Cigarette sales using a panel of American states, see Baltagi and Li 004; The impact of uncertainty on U.K. investment authorizations using a panel of U.K. industries, see Driver, Imai, Temple and Urga 004; Sale of state lottery tickets using panel data on postal ZIP codes, see Frees and Miller 004; Exchange rate determination using industrialized countries quarterly panel data, see Rapach and Wohar 004; Migration to Germany from 8 source countries over the period , see Brucker and Siliverstovs 006; Short-term forecasts of employment in a panel of 36 West German regional labor markets observed over the period , see Longhi and Nijkamp 007; Annual growth rates of real gross regional product for a panel of Chinese regions, see Girardin and Kholodilin 00, to mention a few. See Baltagi 03 for a summary of selected empirical panel data forecasting applications. Address correspondence to: Badi H. Baltagi, Center for Policy Research, 46 Eggers Hall, Syracuse University, Syracuse, NY ; bbaltagi@maxwell.syr.edu. Long Liu: Department of Economics, College of Business, University of Texas at San Antonio, One UTSA Circle, TX ; long.liu@utsa.edu.

2 Wansbeek and Kapteyn 978, Lee and Griffi ths 979, and Taub 979 were among the first contributions in econometrics to the problem of prediction in an error component panel data model. Baltagi and Li 99 extended this prediction to the case of an error component panel model with serial correlation in the remainder disturbance term. While Baltagi and Li 004, 006 extended it to the case of spatial autocorrelation in the remainder disturbance term, and Baltagi, Bresson and Pirotte 0 carried out an extensive Monte Carlo study comparing forecasts in a spatial panel data model. See Baltagi 03 for a recent survey in the Handbook of Forecasting. This paper considers the generalized spatial panel data model with serial correlation proposed by Lee and Yu 0 which encompasses a lot of the spatial panel data models considered in the literature, and derives the best linear unbiased predictor BLUP for that model. This in turn provides valuable BLUP for several spatial panel models as special cases. Section gives a brief description of the Lee and Yu 0 generalized spatial panel data regression model with serial correlation, while Section 3 derives the best linear unbiased predictor BLUP for that model. The Lee and Yu 0 model encompasses a lot of the spatial and panel regression models used in empirical economics. The BLUP for these special cases are shown to follow easily from our BLUP derivation for the generalized model. The Model Lee and Yu 0 considered the following generalized spatial panel data regression model with serial correlation, spatial autocorrelation and random effects: y it = x itβ + u it, i =,..., N; t =..., T, where y it is the observation on the ith region for the tth time period, x it denotes the k vector of observations on the nonstochastic regressors and u it is the regression disturbance. In vector form, the disturbance vector of Equation is assumed to have random region effects, spatially autocorrelated residual disturbances and a first-order autoregressive remainder disturbance term: u t = u + u t, with u = λ W u + I N + δ M µ 3 u t = λ W u t + I N + δ M ν t 4

3 and ν t = ρν t + e t, 5 where u t = u t,..., u tn and ε t, ν t and e t are similarly defined. η = η,..., η N denote the vector of random region effects and µ = µ,..., µ N are assumed to be IIN 0, σ µ. λ and λ are the scalar spatial autoregressive coeffi cients with λ <, λ <, δ and δ are the scalar spatial moving average coeffi cients with δ <, δ <, while ρ is the time-wise serial correlation coeffi cient satisfying ρ <. Following Baltagi, Bresson and Pirotte 0, we define B = I N λ W, B = I N λ W, D = I N δ M and D = I N δ M. Equations 3 and 4 can be rewritten as: where A = D B and A = D B. Following Lee and Yu 0, we employ the following assumptions: u = A µ, 6 u t = A ν t, 7 Assumption W, W, M and M are nonstochastic spatial weights matrices with zero diagonal elements. Assumption The disturbances e it, i =,,..., n and t =, 3,..., T, are i.i.d. across i and t with zero mean, variance σ e, and E e it 4+κ < for some κ > 0; also, they are independent with ν t 0, σ e/ ρ I N. Assumption 3 B, B, D and D are invertible for all λ Λ, λ Λ, δ, δ and ρ P, where Λ, Λ,, are compact intervals and P is a compact subset in,. Furthermore, λ, λ, δ, δ and ρ are, respectively, in the interiors of Λ, Λ,, and P. Assumption 4 W, W, M and M are uniformly bounded in both row and column sums in absolute value for short, UB. Also, B, B, D and D are UB, uniformly in λ Λ, λ Λ, δ and δ. Assumption 5 N is large, whereas T is finite. Assumption 6 µ. 0, σ µi N is independent of ρ ν, e,..., e T Both of them are i.i.d. and independent of X. As pointed out by Lee and Yu 0, this model nests various spatial panel models in the literature including the following : See Lee and Yu 00,05 for nice surveys of spatial panel data models. 3

4 . When λ = 0 and δ = δ = 0, the model reduces to the random effects spatial autoregressive RE-SAR model with serial correlation in the remainder disturbances considered by Baltagi, Song, Jung and Koh When λ = λ = 0 and δ = δ = 0, the model reduces to the random effects panel data model with AR remainder error term and no spatial correlation considered by Baltagi and Li When λ = 0, δ = δ = 0 and ρ = 0, the model reduces to the random effects spatial autoregressive RE-SAR model with no serial correlation considered by Anselin When λ = λ = 0, δ = 0 and ρ = 0, the model reduces to the random effects spatial moving average RE-SMA model with no serial correlation described by Anselin, Le Gallo and Jayet When λ = λ, δ = δ = 0, ρ = 0 and W = W, the model reduces to the spatial autoregressive random effects SAR-RE model with no serial correlation considered by Kapoor, Kelejian and Prucha When λ = λ = 0, δ = δ, ρ = 0 and M = M, the model reduces to the spatial moving average random effects SMA-RE model with no serial correlation considered by Fingleton When δ = δ = 0 and ρ = 0, the model reduces to the generalized random effects spatial autoregressive model proposed by Baltagi, Egger and Pfaffermayr When λ = λ = 0, δ = δ = 0 and ρ = 0, the model reduces to the familiar random effects RE panel data model with no spatial effects and no serial correlation. The model in Equation can be rewritten in matrix notation as y = Xβ + u, 8 where y is of dimension NT, X is NT k, β is k and u is NT. X is assumed to be of full column rank and its elements are assumed to be bounded in absolute value. The disturbance term can be written in vector form as u = ι T A µ + IT A ν, 9 where v = v,..., v T and u is similarly defined. ι T is a vector of ones of dimension T. I T is an identity matrix of dimension T and denotes the Kronecker product. 4

5 Under the random effects model, Lee and Yu 0 showed that the variance covariance matrix of u can be written as where J T Ω = E uu = σ µj T A A + σ e V A A, 0 is a matrix of ones of dimension T, and E vv = σ ev I N, where V is the familiar AR variance-covariance matrix of dimension T, i.e., ρ ρ ρ T ρ ρ ρ T V = ρ ρ ρ ρ T ρ T ρ T ρ T 3 One can easily verify that V = C C, where ρ ρ ρ 0 0 C = ρ ρ is the Prais-Winsten transformation matrix as in Baltagi and Li 99. From Equation 9, the transformed spatial panel data regression disturbances are given by where Cι T u = C I N u = Cι T A µ + C A v = ρ ι T A µ + C A v, 3 = ρ ι T, where ι T =, ι T and = + ρ / ρ. Therefore, the variance covariance matrix of the Prais Winsten-transformed spatial panel data model is given by since C A where J T Ω = E u u = ρ σ µι T ι T A A + σ e I T A A 4 E vv C A. = σ e I T A A Replace ι T ι T by its idempotent counterpart d J T, = ι T ι T /d and d = ι T ι T = + T. Replace I T by E T + J T, where E T = I T J T, and collect like terms, see Baltagi and Li 99, we get Ω = E u u = J ] T d ρ σ µ A A + σ e A A + σ eet A A. 5 5

6 Hence we have Ω = J T Z + σ e ET A A. 6 ] where Z = d ρ σ µ A A + σ e A A. Therefore, Ω = C I N Ω C I N = C I N J T Z + σ e E T A A C IN. 7 One can easily verify that Equation 7 is equivalent to the inverse of the variance covariance matrix given by Lee and Yu 0 Ω = V d ρ ι T ι T V Z + σ e V ] d ρ V ι T ι T V A A 8 using and d ρ V ι T ι T V = d ρ C Cι T ι T C C = d C ι T ι T C = C J T C V d ρ V ι T ι T V = C C C J T C = C E T C. Assumption 7 Elements of the N k matrix of regressors X are nonstochastic and bounded, uniformly in N and T. Also, under the asymptotic setting in Assumption 5, the limit of NT T t= X Ω X exists and is nonsingular. Assumption 8 lim N,T NT ln Ω + NT ln Ω φ + p NT φ ] 0 for φ φ 0, where p NT φ = NT tr Ω φ Ω, φ = λ, λ, δ, δ, ρ, σ µ, σ e and φ0 denotes the true value of φ. Under Assumptions -8, Lee and Yu 0 establishes consistency and asymptotic normality of the quasimaximum likelihood estimator. They provided Matlab programs for these estimation methods. See also Millo 03 for R programs performing maximum likelihood estimation of panel data models with random eff ects, a spatially lagged dependent variable and spatially and serially correlated errors. In this paper we are interested in prediction. This is taken up in the next section. 3 BLUP Goldberger 96 showed that, for a known Ω, the best linear unbiased predictor BLUP for the ith individual s periods ahead y i,t +s is given by ŷ i,t +s = x i,t +sˆβ GLS + w Ω û GLS, 9 6

7 where w = E uu i,t +s is the covariance between the future disturbance u i,t +s and the sample disturbances u. ˆβ GLS is the GLS estimator of β from Equation 8 based on the true Ω. Also, û GLS = y x ˆβ GLS denotes the corresponding GLS residual vector. From Equation 9, u i,t +s can be rewritten as u i,t +s = η i +ε i,t +s = l i A µ + l i A v T +s, where l i as the ith row of I N and v T +s is the N vector of disturbances for the T + sth time period. Focusing on the last term of Equation 9, which we will call the Goldberger BLUP term, we get w Ω û GLS = E u i,t +s u Ω û GLS = E l ia µu Ω û GLS + E l ia v T +su Ω û GLS. 0 Consider the first term in Equation 0. Define Z = A A Z. Using Equation 3, The first term in Equation 0 can be expressed as: E l ia µu Ω û GLS = E l ia µu C I N Ω C I N û GLS { = E l ia µ ρ ι T A ] µ + C A } v Ω û GLS = l ia E µµ ρ ι T A Ω û GLS ] = ρ σ µ ι T l i A A J T Z + σ e ET A ] A û GLS ] = ρ σ µ ι T l i A A Z û GLS = ρ σ µ ι T l iz û GLS = ρ σ µ z ik û k + k= t= û kt ], where z ik is the i, kth elements of Z and û it is the itth elements of û GLS = C I N û GLS. This uses the following results: ι T J T = ι T, ι T E T = 0 and µ and v t are independent. Consider the second term in Equation 0. Notice that E l ia v T +su { = E l ia v T +s ιt A µ + IT A ν ] } = l ia E v T +sν I T A σ = l ia e ρ ρs ρ T, ρ T,, ρ, ] IT I N A σ e ρ = ρ ρs T, ρ T,, ρ, ] l i A A 7

8 since µ and v t are independent, and Ω in Equation 8 can be rewritten as Ω = V d ρ ι T ι T V Z ] + σ e V d ρ V ι T ι T V A A = σ e V A A + V d ρ ι T ι T V Z σ e A A = σ e V A A I T N + d ρ ι T ι T C σ ] ez I N, 3 where Z = A A Z. Alao, ι T V = ι T C C = ρ ι T C. Hence the second term in Equation 0 can be written as: = E l ia v T +su Ω û GLS σ e ρ ρs I T N + ρ T, ρ T,, ρ, l i d ρ ι T ι T C σ ez I N ] û GLS = ρ s 0, 0,, 0, l i] { û GLS + A A ] σ e V A A d ρ = ρ s 0, 0,, 0, l i] û GLS + d ρ = ρ s û i,t + σ eρ s z d ik û k + ρ k= ρ s t= ιt ι T σ ez I N ] û GLS } ι T σ el iz l i] û GLS ] û kt ρ s d ρ û i + t= û it, 4 where z ik is the i, kth element of Z and û GLS = C I N û GLS. This uses the following results: ρ ρ T, ρ T,, ρ, is the last row of V and 0, 0,, 0, ι T =. Combining Equations and 4, one gets the following Goldberger BLUP term: w Ω û GLS = ρ σ µ z ik û k + k= t= û kt ] ] +ρ s û i,t + σ eρ s z d ik û k + û ρ s kt ρ d û i + û it ρ k= t= t= = ρ s û i,t + ρ σ µz ik + ρs σ ] e d ρ z ik û k + û kt k= t= ρ s d û i + û it. 5 ρ t= Special case : When λ = 0 and δ = δ = 0, the model reduces to the random effects spatial autoregressive RE-SAR model with serial correlation considered by Baltagi, Song, Jung and Koh 007. In this case, we have A = I N, A = B, Z = d ρ σ µi N + σ e B B ], Z = Z and Z = 8

9 B B Z. The Goldberger BLUP term given in Equation 5 reduces to w Ω û GLS = ρ s û i,t + ρ s k= d ρ ρ σ µz ik + ρs σ e û i + t= û it d ρ g ik û k + where z ik and g ik are the i, kth elements of Z and B B Z, respectively. defined as the kth element of g i = b i B Z or g i = Z B b i, where b i t= û kt ], 6 Equivalently, g ik can be as the ith row of B. This is Goldberger s BLUP extra term derived by Song and Jung 00 for the random effects error component model with SAR correlation and serial correlation in the remainder disturbances. Special case : When λ = λ = 0 and δ = δ = 0, the model reduces to the random effects panel data model with AR remainder error term and no spatial correlation considered by Baltagi and Li 99. This model is special case, but with no spatial correlation. In this case, we have A = A = I N, ] Z = Z = Z = d ρ σ µi N + σ ei N = σ I N, where σ = d ρ σ µ + σ e. Substituting these terms into Equation 6, the Goldberger BLUP term given in Equation 5 reduces to ] ρ σ w Ω û GLS = ρ s µ ρ s σ e û i,t + σ l ik + k= d ρ σ l ik û k + û kt t= ρ s d û i + û it ρ t= ρ σ = ρ s µ ρ s σ e û i,t + σ + d ρ σ û i + û it t= ρ s d û i + û it ρ t= = ρ s û i,t + ρs ρ σ µ σ û i + û it, 7 where l ik is the i, kth elements of I N. When s =, it further reduces to Goldberger s BLUP extra term derived by Baltagi and Li 99 for the random effects panel data model with AR remainder error term and no spatial correlation. Special case 3: When λ = 0, δ = δ = 0 and ρ = 0, the model reduces to the random effects spatial autoregressive RE-SAR model with no serial correlation considered by Anselin 988. This is special case, but with no serial correlation. Note that ρ = 0 implies that =, d = T, û GLS = û GLS and Z = C T σ µi N + σ e B B ]. Substituting these into Equation 6, the Goldberger BLUP term t= 9

10 given in Equation 5 reduces to w Ω û GLS = σ µc ik k= T t= û kt = T σ µ c ik ū k., 8 where c ik is the i, kth elements of C and ū k. = T T t= ûkt. This is Goldberger s BLUP extra term derived by Baltagi and Li 004, 006 for the random effects error component model with SAR correlation in the remainder disturbances. Special case 4: When λ = λ = 0, δ = 0 and ρ = 0, the model reduces to the random effects spatial moving average RE-SMA model with no serial correlation described by Anselin, Le Gallo and Jayet 008. In this case, wee have A = I N, A = D. Note that ρ = 0 implies that =, d = T, û GLS = û GLS and Z = C T σ µi N + σ ed D ], Z = Z and Z = D D Z. Substituting these into Equation 6, the Goldberger BLUP term given in Equation 5 reduces to w Ω û GLS = σ µc ik k= T t= û kt = T σ µ k= c ik ū k., 9 where c ik is the i, kth elements of C and ū k. = T T t= ûkt. This is Goldberger s BLUP extra term derived by Baltagi and Li 004, 006 for the RE-SMA model with no serial correlation in the remainder disturbances. Special case 5: When λ = λ, δ = δ = 0, W = W and ρ = 0, the model reduces to the spatial autoregressive random effects SAR-RE model with no serial correlation considered by Kapoor, Kelejian and Prucha 007. In this case, we have A = A = B. Note that ρ = 0 implies that =, d = T, ] û GLS = û GLS and Z = T σ µ B B + σ e B B = σ B B, where σ = T σ µ + σ e and Z = Z = σ I N. Substituting these results into Equation 6, the Goldberger BLUP term given in Equation 5 reduces to w Ω û GLS = σ µ σ l ik k= T t= û kt k= = T σ µ σ ū i., 30 where l ik is the i, kth elements of I N and ū i. = T T t= ûit. This is equivalent to σ µ ι σ T l i û GLS, where l i as the ith row of I N. This is Goldberger s BLUP extra term derived by Baltagi, Bresson and Pirotte 0 for the SAR-RE model with no serial correlation in the remainder disturbances. Special case 6: When λ = λ = 0, δ = δ, M = M and ρ = 0, the model reduces to the spatial moving average random effects SMA-RE model with no serial correlation considered by Fingleton 008. In this case, we have A = A = D I N δ M. Note that ρ = 0 implies that =, d = T, û GLS = û GLS and Z = T σ µd D + σ ed D ] = σ D D, where σ = T σ µ + σ e and 0

11 Z = Z = σ I N. Substituting these results into Equation 6, the Goldberger BLUP term given in Equation 5 reduces to w Ω û GLS = σ µ σ l ik k= T t= û kt = T σ µ σ ū i., 3 where l ik is the i, kth elements of I N and ū i. = T T t= ûit. This is again equivalent to σ µ ι σ T l i û GLS, where l i as the ith row of I N. This is Goldberger s BLUP extra term derived by Baltagi, Bresson and Pirotte 0 for the SMA-RE model with no serial correlation in the remainder disturbances and it is the same as the one for SAR-RE model with no serial correlation in the remainder disturbances considered in special case 5. Note, however, that the feasible predictor will be based on different estimates of the residuals and variance components once the model is estimated by maximum likelihood or Generalized Moments. Special case 7: When δ = δ = 0 and ρ = 0, the model reduces to the generalized random effects spatial autoregressive model proposed by Baltagi, Egger and Pfaffermayr 03. In this case, we have A = B and A = B. Note that ρ = 0 implies that =, d = T, û GLS = û GLS and ] ] Z = T σ µ B B + σ e B B and Z = B B Z = T σ µi N + σ e B B B B C 3. Substituting these results into Equation 6, the Goldberger BLUP term given in Equation 5 reduces to w Ω û GLS = σ µc 3ik k= T t= û kt = T σ µ where c 3ik is the i, kth elements of C 3 and ū k. = T T t= ûkt. c 3ik ū k., 3 Special case 8: When λ = λ = 0, δ = δ = 0 and ρ = 0, the model reduces to the familiar random effects model without spatial or serial autocorrelation. In this case, we have A = A = I N. Note that ρ = 0 implies that =, d = T, û GLS = û GLS and Z = Z = Z = T σ µi N + σ ei N ] = σ I N, where σ = T σ µ + σ e. Substituting these results into Equation 6, the Goldberger BLUP term given in Equation 5 reduces to w Ω û GLS = σ µ σ l ik k= T t= û kt k= = T σ µ σ ū i., 33 where l ik is the i, kth elements of I N and ū i. = T T t= ûit. This is again equivalent to σ µ ι σ T l i û GLS, where l i as the ith row of I N. This is Goldberger s BLUP extra term derived by Wansbeek and Kapteyn 978, Lee and Griffi ths 979, and Taub 979 for the random effects error component model and it is the same as the one for SAR or SMA correlation in the remainder disturbances in special cases 5 and 6 but with different estimates of the residuals and variance components once the model is estimated by maximum likelihood or Generalized Moments.

12 4 Conclusion This paper derives Goldberger s 96 best linear unbiased predictor BLUP for the generalized spatial panel data model with serial correlation proposed by Lee and Yu 0. Since the latter model encompasses a lot of the spatial panel data models considered in the literature, this in turn provides valuable BLUP for several spatial panel models as special cases. Extensions of this BLUP should be applied to dynamic spatial panel models, see Baltagi, Fingleton and Pirotte 04, and to panel data models with a spatial lag, as well as higher order autoregressive and moving average processes, see Baltagi and Liu 03a, 03b. REFERENCES Anselin, L., 988, Spatial Econometrics: Methods and Models. Kluwer Academic Publishers, Dordrecht. Anselin, L., J. Le Gallo and H. Jayet, 008, Spatial panel econometrics. Ch. 9 in L. Mátyás and P. Sevestre, eds., The Econometrics of Panel Data: Fundamentals and Recent Developments in Theory and Practice, Springer-Verlag, Berlin, Baltagi, B.H., 008, Forecasting with panel data, Journal of Forecasting 7, Baltagi, B.H., 03, Panel Data Forecasting, Chapter 8 in the Handbook of Economic Forecasting, Volume B, edited by Graham Elliott and Allan Timmermann, North Holland, Amsterdam, Baltagi, B.H., Bresson, G. and Pirotte,A., 0, Forecasting with spatial panel data, Computational Statistics and Data Analysis 56, Baltagi, B.H., P. Egger and M. Pfaffermayr, 03, A generalized spatial panel data model with random effects, Econometric Reviews 3, Baltagi, B.H., B. Fingleton and A. Pirotte, 04, Estimating and forecasting with a dynamic spatial panel data model, Oxford Bulletin of Economics and Statistics 76, -38. Baltagi, B.H. and J.M. Griffi n, 997, Pooled estimators vs. their heterogeneous counterparts in the context of dynamic demand for gasoline, Journal of Econometrics 77, Baltagi, B.H. and D. Li, 004, Prediction in the panel data model with spatial correlation, Chapter 3 in L. Anselin, R.J.G.M. Florax and S.J. Rey, eds., Advances in Spatial Econometrics: Methodology, Tools and Applications, Springer, Berlin, Baltagi, B.H. and D. Li, 006, Prediction in the panel data model with spatial correlation: The case of liquor, Spatial Economic Analysis, Baltagi, B.H. and Q. Li, 99, Prediction in the one-way error component model with serial correlation, Journal of Forecasting, Baltagi, B.H. and L. Liu, 03a, Estimation and prediction in the random effects model with ARp remainder disturbances, International Journal of Forecasting 9,

13 Baltagi, B.H. and L. Liu, 03b, Prediction in the random effects model with MAq remainder disturbances, Journal of Forecasting 3, Baltagi, B.H., Song, S.H., Jung, B.C. and W. Koh, 007. Testing for serial correlation, spatial autocorrelation and random effects using panel data, Journal of Econometrics 40, 5-5. Brucker, H. and B. Siliverstovs, 006, On the estimation and forecasting of international migration: how relevant is heterogeneity across countries, Empirical Economics 3, Driver, C., K. Imai, P. Temple and A. Urga, 004, The effect of uncertainty on UK investment authorisation: homogeneous vs. heterogeneous estimators, Empirical Economics 9, 5-8. Fingleton, B., 008, A generalized method of moments estimators for a spatial panel model with an endogenous spatial lag and spatial moving average errors, Spatial Economic Analysis 3, Fingleton, B., 009, Prediction using panel data regression with spatial random effects, International Regional Science Review 3, Frees, E.W. and T.W. Miller, 004, Sales forecasting using longitudinal data models. International Journal of Forecasting 0, Girardin, E. and K. A. Kholodilin, 00, How helpful are spatial effects in forecasting the growth of Chinese provinces?, Journal of Forecasting, forthcoming. Goldberger, A.S., 96, Best linear unbiased prediction in the generalized linear regression model, Journal of the American Statistical Association 57, Hoogstrate, A. J., F. C. Palm, and G. A. Pfann, 000, Pooling in dynamic panel-data models: An application to forecasting GDP growth rates, Journal of Business and Economic Statistics 8, Kapoor, M., H.H. Kelejian and I.R.Prucha, 007, Panel data models with spatially correlated error components. Journal of Econometrics 40, Lee, L.F. and W.E. Griffi ths, 979, The prior likelihood and best linear unbiased prediction in stochastic coeffi cient linear models, working paper, Department of Economics, University of Minnesota. Lee, L. F. and J. Yu. 00, Some recent developments in spatial panel data models. Regional Science and Urban Economics 40, Lee, L. F. and J. Yu, 0, Spatial panels: Random components versus fixed effects, International Economic Review 53, Lee, L. F. and J. Yu, 05, Spatial panel data models, Chapter in the Oxford Handbook of Panel Data, Badi H. Baltagi, editor, Oxford University Press, Oxford, Longhi S. and P. Nijkamp, 007, Forecasting regional labor market developments under spatial heterogeneity and spatial correlation, International Regional Science Review 30,

14 Maddala, G.S., R.P. Trost, H. Li and F. Joutz, 997, Estimation of short-run and long-run elasticities of energy demand from panel data using shrinkage estimators, Journal of Business and Economic Statistics 5, Millo, G., 03, Maximum likelihood estimation of spatially and serially correlated panels with random effects, Computational Statistics and Data Analysis, forthcoming. Rapach, D.E. and M.E. Wohar, 004, Testing the monetary model of exchange rate determination: a closer look at panels, Journal of International Money and Finance 3, Schmalensee, R., T.M. Stoker and R.A. Judson, 998, World carbon dioxide emissions: , Review of Economics and Statistics 80, 5 7. Song, S.H. and B.C. Jung, 00, BLUP in the panel regression model with spatially and serially correlated error components. Statistical Papers 43, Taub, A.J., 979, Prediction in the context of the variance-components model, Journal of Econometrics 0, Wansbeek, T.J. and A. Kapteyn, 978, The separation of individual variation and systematic change in the analysis of panel data, Annales de l INSEE 30-3,