Random effects and Spatial Autocorrelations with Equal Weights

Transcription

1 Syracuse University SURFACE Center for Policy Research Maxwell School of Citizenship and Public Affairs 2006 Random effects and Spatial Autocorrelations with Equal Weights Badi H. Baltagi Syracuse University. Center for Policy Research, Follow this and additional works at: Part of the Mathematics Commons Recommended Citation Baltagi, Badi H., "Random effects and Spatial Autocorrelations with Equal Weights" (2006). Center for Policy Research This Working Paper is brought to you for free and open access by the Maxwell School of Citizenship and Public Affairs at SURFACE. It has been accepted for inclusion in Center for Policy Research by an authorized administrator of SURFACE. For more information, please contact

2 ISSN: Center for Policy Research Working Paper No. 89 RANDOM EFFECTS AND SPATIAL AUTOCORRELATION WITH EQUAL WEIGHTS Badi H. Baltagi Center for Policy Research Maxwell School of Citizenship and Public Affairs Syracuse University 426 Eggers Hall Syracuse, New York (35) Fax (35) December 2006 $5.00 Up-to-date information about CPR s research projects and other activities is available from our World Wide Web site at www-cpr.maxwell.syr.edu. All recent working papers and Policy Briefs can be read and/or printed from there as well.

3 CENTER FOR POLICY RESEARCH Fall 2006 Timothy Smeeding, Director Professor of Economics & Public Administration Associate Directors Margaret Austin Associate Director Budget and Administration Douglas Wolf Professor of Public Administration Associate Director, Aging Studies Program John Yinger Professor of Economics and Public Administration Associate Director, Metropolitan Studies Program SENIOR RESEARCH ASSOCIATES Badi Baltagi... Economics Pablo Beramendi...Political Science Dan Black... Economics Lloyd Blanchard... Public Administration William Duncombe... Public Administration Gary Engelhardt...Economics Deborah Freund...Public Administration Madonna Harrington Meyer...Sociology Christine Himes...Sociology William C. Horrace...Economics Duke Kao...Economics Eric Kingson... Social Work Thomas Kniesner...Economics Jeffrey Kubik...Economics Andrew London...Sociology Len Lopoo...Public Administration Jerry Miner...Economics Jan Ondrich...Economics John Palmer...Public Administration Lori Ploutz-Snyder...Exercise Science David Popp...Public Administration Grant Reeher... Political Science Christopher Rohlfs...Economics Stuart Rosenthal...Economics Ross Rubenstein...Public Administration Margaret Usdansky...Sociology Michael Wasylenko...Economics Janet Wilmoth...Sociology GRADUATE ASSOCIATES Javier Baez...Economics Sonali Ballal...Public Administration Jesse Bricker...Economics Maria Brown...Social Science Mike Eriksen...Economics Qu Feng...Economics Katie Fitzpatrick...Economics Alexandre Genest...Public Administration Julie Anna Golebiewski...Economics Nadia Greenhalgh-Stanley...Economics Tamara Hafner...Public Administration Yue Hu...Economics Joseph Marchand... Economics Neelakshi Medhi... Social Science Larry Miller... Public Administration Wendy Parker... Sociology Emily Pas... Economics Shawn Rohlin... Economics Cynthia Searcy... Public Administration Jacqueline Smith... Sociology Jeff Thompson... Economics Coady Wing... Public Administration Daniel Yanulavich... Public Administration Ryan Yeung... Public Administration STAFF Kelly Bogart...Administrative Secretary Martha Bonney.....Publications/Events Coordinator Karen Cimilluca...Administrative Secretary Kitty Nasto...Administrative Secretary Candi Patterson...Computer Consultant Mary Santy.... Administrative Secretary Tammy Tanner Librarian/Office Coordinator

4 Abstract: This note considers a panel data regression model with spatial autoregressive disturbances and random effects where the weight matrix is normalized and has equal elements. This is motivated by Kelejian, et al. (2005) who argue that such a weighting matrix, having blocks of equal elements, might be considered when units are equally distant within certain neighborhoods but unrelated between neighborhoods. We derive a simple weighted least squares transformation that obtains GLS on this model as a simple OLS. For the special case of a spatial panel model with no random effects, we obtain two sufficient conditions where GLS on this model is equivalent to OLS. Finally, we show that these results, for the equal weight matrix, hold whether we use the spatial autoregressive specification, the spatial moving average specification, the spatial error components specification or the Kapoor, et al. (2005) alternative to modeling panel data with spatially correlated error components. Keywords: Panel data, Spatial Error Correlation, Equal Weights, Error Components. JEL classification: C23, C2

5 Random E ects and Spatial Autocorrelation With Equal Weights by Badi H. Baltagi Department of Economics and Center For Policy Research, Syracuse University, 426 Eggers Hall, Syracuse, New York , USA (35) bbaltagi@maxwell.syr.edu Keywords: Panel data, Spatial Error Correlation, Equal Weights, Error Components. JEL classi cation: C23, C2 ABSTRACT This note considers a panel data regression model with spatial autoregressive disturbances and random e ects where the weight matrix is normalized and has equal elements. This is motivated by Kelejian, et al. (2005) who argue that such a weighting matrix, having blocks of equal elements, might be considered when units are equally distant within certain neighborhoods but unrelated between neighborhoods. We derive a simple weighted least squares transformation that obtains GLS on this model as a simple OLS. For the special case of a spatial panel model with no random e ects, we obtain two su cient conditions where GLS on this model is equivalent to OLS. Finally, we show that these results, for the equal weight matrix, hold whether we use the spatial autoregressive speci cation, the spatial moving average speci cation, the spatial error components speci cation or the Kapoor, et al. (2005) alternative to modelling panel data with spatially correlated error components. Introduction Spatial models deal with correlation across spatial units usually in a cross-section setting. However, Anselin (988) also considered spatial panel data models which allow for random e ects across these units, see also Baltagi, Song and Koh (2003) and Kapoor, Kelejian and I would like to thank Paolo Paruolo, Tom Kniesner and an anonymous referee for helpful comments and suggestions.

6 Prucha (2005) for two recent studies on testing and estimation in these models and Case (99) and Holtz-Eakin (994) for two empirical applications. This note focuses on a panel data regression model with spatial autoregressive disturbances and random e ects where the weight matrix is normalized and has equal elements. This is motivated by Kelejian and Prucha (2002) and Kelejian, et al. (2005) who use this weight matrix in the context of a spatially lagged dependent variable model. Kelejian, et al. (2005) argue that such a weighting matrix having blocks of equal elements might be considered when units are equally distant within certain neighborhoods but unrelated between neighborhoods. Section 2 introduces a panel regression model with spatial autoregressive (SAR) disturbances and random e ects. We show that for the equal weight matrix case, one can derive a simple weighted least squares transformation that obtains GLS on this model as a simple OLS. For the special case of a spatial panel model with no random e ects, we obtain two su cient conditions where GLS on this model is equivalent to OLS. Section 3 extends these results to other type of spatial error speci cations, including the spatial moving average (SMA) and spatial error components (SEC) cases as well as an alternative panel data regression model with spatially correlated error components suggested by Kapoor, et al. (2005). 2 The Model Consider the following panel data regression model y ti = + X 0 ti + u ti ; i = ; ::; N; t = ; ; T; () where y ti is the observation on the ith region for the tth time period, X ti denotes the kx vector of observations on the nonstochastic regressors and u ti is the regression disturbance. is a scalar and is a kx vector of slope parameters. In vector form, the disturbance vector of () is assumed to have random region e ects and spatially autoregressive (SAR) remainder disturbances: u t = + t ; (2) with t = W N t + t ; (3) where u 0 t = (u t ; :::; u tn ) and t and t are similarly de ned. 0 = ( ; 2 ; ::; N ) denote the vector of random region e ects which are assumed to be IID(0; 2 ): is the scalar spatial autoregressive coe cient with jj <, while ti IID(0; 2 ): We assume that and are independent. W N is an N N weighting matrix with zero elements across the diagonal, and equal elements (=(N )) o the diagonal. In other words, the disturbance for each unit is related to an average of the (N ) disturbances of the remaining units. Such a weighting matrix was recently considered by Kelejian and Prucha (2002), Lee (2002) and Kelejian, et al. (2005) and would naturally arise if all units are neighbors to each other and there is no other reasonable or observable measure of distance between them. Kelejian and Prucha (2002) consider the case of a spatially lagged dependent variable model with a row normalized weighting matrix with equal elements. They show that OLS and 2SLS are

7 inconsistent unless panel data is available. Kelejian, et al. (2005) give exact small sample results that corroborate the earlier asymptotic ndings. In addition, they demonstrate that for the spatial panel data case with xed e ects across time, OLS and 2SLS are both inconsistent. Lee (2002), on the other hand, shows that OLS can be consistent in an economic spatial environment where each unit can be in uenced aggregately by a signi cant portion of units in the population. One can rewrite (3) as t = (I N W N ) t = B N t (4) where B N = I N W N. The model () can be rewritten in matrix notation as y = NT + X + u = Z + u (5) where y is of dimension NT, NT is a vector of ones of dimension NT; X is NT k, u is NT and Z = ( NT ; X). 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 = ( T I N ) + (I T B N ) (6) where 0 = ( 0 ; 0 2 ; ::; 0 T ) and u is similarly de ned. T is a vector of ones of dimension T, I T is an identity matrix of dimension T and denotes the Kronecker product. Under these assumptions, the variance-covariance matrix of u can be written as = 2 (J T I N ) + 2 (I T (B 0 NB N ) ) (7) Kelejian and Prucha (2002) showed that W N can be written as W N = J N (N ) I N (N ) (8) where J N is a matrix of ones, and I N is an identity matrix, both of dimension N: De ning J N = J N =N and E N = I N JN ; and replacing J N and I N by their equivalent terms NJ N and (E N + J N ) and collecting like terms, see Wansbeek and Kapteyn (982), one gets W N = J N E N (N ) (9) with B N = (I N W N ) = (N + ) E N + ( (N ) ) J N Therefore B N = (I N W N ) = Note that W N is symmetric and (B 0 NB N ) = (N ) (N + ) E N + ( ) J N (0) (N )2 (N + ) 2 E N + ( ) 2 J N = c E N + c 2JN () 2

8 where c = (N )2 (N +) 2 and c 2 = ( ) 2 : Replacing J T and I T by their equivalent terms T J T and (E T + J T ) in (7), one gets = T 2 ( J T (E N + J N )) + 2 ((E T + J T ) (c E N + c 2 JN )) (2) Collecting like terms, we obtain the spectral decomposition of ; = (T 2 +c 2 )( J T E N )+(T 2 +c 2 2 )( J T J N )+c 2 (E T E N )+c 2 2 (E T J N ) (3) Hence = (T 2 + c 2 ) ( J T E N ) + (T 2 + c 2 2 ) ( J T J N ) + c 2 (E T E N ) + c 2 2 (E T J N ) (4) and =2 = q ( J T E N ) + (T 2 + c 2 ) q (T 2 + c 2 2 ) + p (E T E N ) + p (E T J N ) c 2 c2 2 ( J T J N ) (5) Note that if = 0; so that there is no spatial autocorrelation, then c = c 2 = and in (3) reduces to the familiar variance-covariance matrix of the random e ects panel data model given by = (T )( J T I N )+ 2 (E T I N ); see Baltagi (2005). Premultiplying p the regression model in (5) by c =2 from (5), one gets a similar transformation to the one suggested by Fuller and Battese (973) for the random e ects panel data model. In fact y p = c =2 y will have typical elements yti = (y ti y t: 2 y :i + 3 y :: ) where y t: denotes the sample average over regions, y :i denotes the sample average over time, and y :: denotes the average over the entire sample. The 0 s can be easily obtained from the corresponding (c ; c 2 ; 2 ; 2 ) parameters. For example, is the coe cient corresponding to (I T J N ) which can be veri ed to be ( c 2 ); 2 is the coe cient corresponding to r ( J c T I N ) which can be veri ed to be ( 2 (T 2 +c 2 ) ); and 3 is the coe cient corresponding to ( J T J q r r N ) which can be veri ed to be ( c c 2 c 2 (T 2 +c 2 ) + c 2 (T 2 +c 2 2)): If = 0; then = 3 = 0 and 2 = p ; which reduces y T ti to (y ti 2 y :i ): This is the familiar Fuller and Battese (973) random e ects transformation that allows us to obtain GLS as weighted least squares. If 2 = 0; i.e., the case of no random e ects, the variance-covariance matrix in (7) reduces to q c = 2 (I T (B 0 NB N ) ) (6) 3

9 which from () reduces to Hence and = 2 (I T (c E N + c 2 JN )) (7) = c 2 (I T E N ) + c 2 2 (I T J N ) (8) =2 = p c (I T E N ) + p c2 (I T J N ) (9) so that the typical element of y = p c =2 y is y ti = (y ti y t: ) where = ( Note that as N! ; q c c 2! ( ); and! : q c c 2 ): A special case of this model is the cross-section spatial regression model with (T = ) and equal weight matrix given by (8). In this case, one can show that OLS is equivalent to GLS as long as there is a constant in the regression. To prove this, note that the model in (5) becomes y N = N + X N + u N = Z N + u N (20) where Z N = ( N ; X N ) and = (; ) 0 : Here y N is a vector of observations on the dependent variable and N is a vector of ones, both of dimension N: X N is an N K matrix of observations on the K explanatory variables. The disturbance vector is assumed to follow a spatial autoregressive process u N = W N u N + N (2) where W N is an N N equal weighting matrix given in (8). In fact, one can prove that the Zyskind (967) necessary and su cient condition for OLS to be equivalent to GLS on (20) is satis ed. This calls for P Z = P Z, where Z = Z N = ( N ; X N ) is the matrix of regressors in (20) and = E(u N u 0 N ) = 2 is the variance-covariance matrix of the disturbances given in (2). It is straightforward to show that = c E N + c 2 JN and P Z = (N )2 (N + ) 2 (P Z J N ) + ( ) 2 J N (22) since P Z N = N ; P Z JN = J N and P Z E N = P Z JN : Similarly, P Z = (N )2 (N + ) 2 (P Z J N ) + ( ) 2 J N (23) 4

10 Hence, P Z = P Z. In fact, another necessary and su cient condition for OLS to be equivalent to GLS, which relies on ; is given by Milliken and Albohali (984) and this condition calls for Z 0 (I N P Z ) = 0: Here = (N + )2 (N ) 2 E N + ( ) 2 JN (24) and the fact that J N (I N (I N P Z ) = = P Z ) = 0, we get (N + )2 (N ) 2 E N (I N P Z ) + ( ) 2 JN (I N P Z ) (25) (N + )2 (N ) 2 (I N P Z ) Hence Z 0 (I N P Z ) = (N + )2 (N ) 2 Z 0 (I N P Z ) = 0 (26) since Z 0 P Z = Z 0 : Note that var(b OLS ) = var(b GLS ) = 2 (Z 0 Z) (27) = 2 ( ) 2 (Z 0 JN (N + )2 Z) + (N ) 2 (Z 0 E N Z) and this, in general, is not equal to the usual formula for var(b OLS ) computed by regression packages, i.e., 2 (Z 0 Z), unless = 0 which is the case of no spatial correlation. For the spatial panel regression with equal weights, this result is not necessarily true as long as 2 > 0: Two special cases where OLS on (5) is the same as GLS, i.e., b OLS = (Z 0 Z) Z 0 y = b GLS = (Z 0 Z) Z 0 y are the following: (i) the trivial case where 2 = 0 and = 0; In fact, when = 0; c = c 2 = ; = 0 and yti = y ti and GLS reduces to OLS; Also, (ii) when the matrix of regressors X is invariant across time, i.e., when X = T X N ; with X N being the N k matrix of exogenous regressors that is invariant across time. To show this, we prove that the Zyskind (967) necessary and su cient condition for OLS to be equivalent to GLS on (5) is satis ed. This calls for P Z = P Z, where Z = T Z N = T ( N ; X N ) is the matrix of regressors in (5) and is the variance-covariance matrix of the disturbances given in (7). It is straightforward to 5

11 show that P Z = J T P ZN and that P ZN N = N ; P ZN JN = J N and P ZN E N = P ZN JN : Hence Similarly, P Z = 2 ( J T P ZN )(I T (c E N + c 2 JN )) = 2 ( J T (c (P ZN JN ) + c 2 JN )) (28) P Z = 2 (I T (c E N + c 2 JN ))( J T P ZN ) = 2 ( J T (c (P ZN JN ) + c 2 JN )) (29) Hence, P Z = P Z. In fact, another necessary and su cient condition for OLS to be equivalent to GLS, which relies on ; is given by Milliken and Albohali (984) and this condition calls for Z 0 (I T N P Z ) = 0: Using (8), and the fact that J N (I N P ZN ) = 0, we get Hence (I T N P Z ) = ( c 2 (I T E N ) + c 2 2 (I T J N ))(I T N P Z ) (30) = ( c 2 (I T E N ) + c 2 2 (I T J N )) ( c 2 ( J T (P ZN JN )) + c 2 2 ( J T J N )) = c 2 ((I T E N ) ( J T (P ZN JN ))) + c 2 2 (E T J N ) Z 0 (I T N P Z ) = = = c 2 f( 0 T ZN)((I 0 T E N ) ( J T (P ZN JN )))g (3) c 2 f( 0 T ZNE 0 N ) ( 0 T ZN(P 0 ZN JN ) c 2 f( 0 T ZNE 0 N ) ( 0 T (ZN 0 ZN 0 J N ))g = 0 since 0 T J T = 0 T and Z0 N P Z N = Z 0 N : An anonymous referee pointed out that an alternative derivation of these results can be obtained using lemma 2. of Magnus (982). This lemma was derived in the context of a multivariate error component model, but when applied to the model considered in this paper, one common theme is the following: If W N = A N + 2 (I N A N ) with 6= 2 ; and A N is a symmetric idempotent matrix, then all powers of W N (also negative powers), its eigenvalues, and determinant can all be easily calculated. Here, we have I N W N = A N + 2 (I N A N ); with = ; and 2 = 2 : Assuming that 6= and 2 6= ; we get (I N W N ) = A N + (I N A N ): Hence 2 6

12 = (I N W N ) 2 = A 2 N + (I 2 N A N ): Applying the Milliken and Albohali (984) 2 necessary and su cient condition which calls for Z 0 (I N P Z ) = 0; we get: Z 0 (I N P Z ) = ( 2 2 2)Z 0 A N (I N P Z ) = 0;which is, Z 0 A N (I N P Z ) = 0: In the special case, where A N = J N ; we see that OLS is equivalent to GLS if and only if Z 0 N = 0 or (I N P Z ) N = 0; that is, if and only if the regressors are measured in deviations from the mean or the regression contains a constant. 3 Extensions So far, we have considered the spatial autoregressive (SAR) speci cation for the disturbances. An alternative speci cation is the spatial moving average (SMA) speci cation. Anselin (2003) classi es the spatial covariance structure induced by SAR as global, and that by SMA as local. In this case, the model given by () and (2) is the same, but the disturbances in (3) become t = W N t + t = (I N + W N ) t (32) In vector form, the panel disturbances in (6) become u = ( T I N ) + (I T (I N + W N )) (33) with variance covariance matrix = 2 (J T I N ) + 2 (I T (I N + W N ) 2 ) (34) since W N given by (9) is symmetric. In fact, (I N + W N ) 2 = (N )2 (N ) 2 E N + ( + ) 2 JN = d E N + d 2 JN (35) where d = (N )2 (N ) 2 and d 2 = ( + ) 2 : Replacing J T and I T by their equivalent terms T J T and (E T + J T ) in (34), and collecting like terms, we obtain = (T 2 + d 2 )( J T E N ) + (T 2 + d 2 2 )( J T J N ) (36) +d 2 (E T E N ) + d 2 2 (E T J N ) Hence, we get the same results for the SMA speci cation as for the SAR speci cation when the weigh matrix is equal. The only di erence between (36) and (3) is d and d 2 rather than c and c 2 : Similarly, and =2 are the same as (4) and (5) with d and d 2 replacing c and c 2 : The same holds true for the Fuller and Battese type transformation described below (5). Note that if = 0; so that there is no spatial autocorrelation, then d = d 2 = and in (36) reduces to the familiar variance-covariance matrix of the random e ects panel data model. If 2 = 0; i.e., the case of no random e ects, the variance-covariance matrix in (34) reduces to 7

13 = 2 (I T (d E N + d 2 JN )) (37) Hence, and =2 are the same as in (8) and (9) with d and d 2 replacing c and c 2 : The Fuller and Battese transformation given below (9) is now y p = d =2 q y with typical element yti = (y ti y t: ) where = ( d d 2 ): Note that as N! ; q d d 2! (+) ; and! (+) : Kelejian and Robinson (995) considered an alternative spatial error components (SEC) speci cation that di ers from the SAR and SMA speci cation. In this case, the model given by () and (2) is the same, but the disturbances in (3) become t = W N t + t (38) where t is an (N ) vector of spillover error components. The two component vectors t and t are assumed to consist of iid terms with respective variances 2 and 2 and are uncorrelated. In vector form, the panel disturbances in (6) become u = ( T I N ) + (I T W N ) + (39) with variance covariance matrix = 2 (J T I N ) + (I T ( 2 I N W N 2 )) (40) since W N given by (9) is symmetric. In fact, ( 2 I N W N 2 ) = ( where b = ( (N ) 2 )E N + ( ) J N = b E N + b 2 JN (4) (N ) 2 ) and b 2 = ( ): The rest of the derivations are the same as above with b and b 2 replacing c and c 2 : Note that as N! ; q b b 2! q ; and! q : For the cross-section (T = ) spatial regression model with SMA or SEC disturbances and an equal weight matrix, one can easily show that OLS is equivalent to GLS as long as there is a constant in the model, the proof is left to the reader. For the spatial panel regression with equal weights, this result is not necessarily true as long as 2 > 0: Two cases where OLS is the same as GLS are once again: (i) the trivial case where 2 = 0 and = 0 or (ii) when the matrix of regressors X is invariant across time. The proofs for the SMA and SEC cases are the same as that for the SAR case and are left for the reader. The key for these results is the fact that () for SAR, (35) for SMA and (4) for SEC are all linear combinations of E N and J N : These matrices are idempotent, orthogonal to each other and sum to I N : Alternatively, one can apply lemma 2. of Magnus (982) to obtain the same results. Next, we consider an alternative panel data model with spatially correlated error components suggested by Kapoor, Kelejian, and Prucha (2005). The regression model is the same as (), but the spatial error components structure given by (2) and (3) becomes: 8

14 u = (I T W N )u + (42) with = ( T I N ) + (43) where and are the same as before, i.e., IID(0; 2 ) and IID(0; 2 ) independent of each other andand among themselves. Performing the spatial Cochrane-Orcutt type transformation on (5) one gets: with y () = ( ) NT + X () + u () = Z () + u () (44) u () = (I T (I N W N ))u = (45) and y (); X () and Z () de ned similarly. The variance-covariance matrix = E( 0 ) = (T )( J T I N ) + 2 (E T I N ); is the usual panel data error components random e ects variance-covariance matrix. GLS on (44) can be obtained as OLS on the Fuller-Battese transformed equation (44), i.e., after premultiplying it by =2 = (E T I N ) + ( J q T I N ); where = T : In this case, y () = =2 y () = ((E T I N ) + ( J T I N ))(I T (I N W N ))y (46) For the equal weight matrix given in (8), (I N W N ) is given below (9) and can be rewritten as (I N W N ) = a E N + a 2JN (N +) ; where a = (N ) and a 2 = ( ): Substituting this expression in (46) yields: y () = [(E T (a E N + a 2JN )) + ( J T (a E N + a 2JN ))]y (47) = a (E T E N )y + a 2 (E T J N )y + a ( J T E N )y + a 2 ( J T J N )y once again, we see that for the Kapoor, Kelejian, and Prucha (2005) speci cation, with an equal weight matrix, GLS on (44) can be written as a weighted combination similar to that below (5), i.e., yti = a (y ti y t: 2 y :i + 3 y :: ) where y t: denotes the sample average over regions, y :i denotes the sample average over time, and y :: denotes the average over the entire sample. The 0 s can be easily obtained from the corresponding (a ; a 2 ; ; ) parameters. For example, is the coe cient corresponding to (I T J N ) which can be a veri ed to be ( 2 a ); 2 is the coe cient corresponding to ( J T I N ) which can be veri ed to be ( ); and 3 is the coe cient corresponding to ( J T J N ) which can be veri ed a to be ( 2 a + a 2 a ): If = 0; a = a 2 = ; then = 3 = 0 and 2 = ( ); which reduces yti to (y ti 2 y :i ): This is the familiar Fuller and Battese (973) random e ects transformation. 9

15 If 2 = 0; i.e., the case of no random e ects, then = ; 2 = 3 = 0 and = ( which reduces yti to a (y ti y t: ). In fact, y () from (2.39) reduces to a 2 a ); y () = [I T (a E N + a 2 JN )]y = a (I T E N )y + a 2 (I T J N )y (48) = a y + (a 2 a )(I T J N )y and note that a = (a a 2 ): Therefore, yti is a weighted combination of y ti and y t: : For this model, OLS is the same as GLS for the trivial case where 2 = 0 and = 0: The more interesting case where OLS on (44) is equivalent to GLS is the case where the matrix of regressors X is invariant across time. The proof is along the same lines as above and is left for the reader. In fact, one can easily show that when X = T X N ; with X N being the N k matrix of exogenous regressors that is invariant across time, both the Zyskind (967) and the Milliken and Albohali (984) necessary and su cient conditions are satis ed for this model. 4 Conclusion For spatial panel models with a weight matrix that is row normalized and has equal weights within a block but otherwise uncorrelated across blocks, we showed that GLS can be obtained as a simple weighted least squares transformation involving the means of the observations over time and over units. This is similar to the Fuller and Battese (973) transformation for the random e ects model in panel data with no spatial correlation. In addition, we showed that OLS is equivalent to GLS if there are no random or spatial e ects, or if the regressors vary across units but are invariant over time. We also showed that these results for the equal weight matrix hold whether we use the spatial autoregressive speci cation, the spatial moving average speci cation, the spatial error components speci cation, or the alternative panel data regression model with spatially correlated error components suggested by Kapoor, et al. (2005). 5 REFERENCES Anselin, L. (988). Spatial Econometrics: Methods and Models (Kluwer Academic Publishers, Dordrecht). Anselin, L. (2003). Spatial externalities, spatial multipliers and spatial econometrics, International Regional Science Review 26, Baltagi, B.H. (2005). Econometric Analysis of Panel Data (Wiley, Chichester). Baltagi, B.H., S.H. Song & W. Koh (2003). Testing panel data regression models with spatial error correlation. Journal of Econometrics 7, Case, A.C. (99). Spatial patterns in household demand. Econometrica 59,

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