Testing for Heteroskedasticity and Spatial Correlation in a Random Effects Panel Data Model

Transcription

1 Syracuse Uniersity SURFAC Center for Policy Research Maxwell School of Citizenship and Public Affairs 8 Testing for Heteroskedasticity and Spatial Correlation in a Random ffects Panel Data Model Badi H. Baltagi Syracuse Uniersity. Center for Policy Research, bbaltagi@maxwell.syr.edu Seuck Heun Song Jae Hyeok Kwon Follow this and additional works at: Part of the conometrics Commons Recommended Citation Baltagi, Badi H.; Song, Seuck Heun; and Kwon, Jae Hyeok, "Testing for Heteroskedasticity and Spatial Correlation in a Random ffects Panel Data Model" 8. 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 SURFAC. It has been accepted for inclusion in Center for Policy Research by an authorized administrator of SURFAC. For more information, please contact surface@syr.edu.

2 ISSN: Center for Policy Research Working Paper No. 8 Testing for Heteroskedasticity and Spatial Correlation in a Random ffects Panel Data Model Badi H. Baltagi, Seuck Heun Song, and Jae Hyeok Kwon Center for Policy Research Maxwell School of Citizenship and Public Affairs Syracuse Uniersity 46 ggers Hall Syracuse, New York Fax ctrpol@syr.edu July 8 $5. Up-to-date information about CPR s research projects and other actiities is aailable 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 CNTR FOR POLICY RSARCH Summer 8 Douglas Wolf, Interim Director Gerald B. Cramer Professor of Aging Studies, Professor of Public Administration Associate Directors Margaret Austin Associate Director Budget and Administration John Yinger Trustee Professor conomics and Public Administration Associate Director, Metropolitan Studies Program SNIOR RSARCH ASSOCIATS Badi Baltagi... conomics Kalena Cortes ducation Thomas Dennison... Public Administration William Duncombe... Public Administration Gary ngelhardt... conomics Deborah Freund... Public Administration Madonna Harrington Meyer... Sociology Christine Himes... Sociology William C. Horrace... conomics Duke Kao... conomics ric Kingson... Social Work Thomas Kniesner... conomics Jeffrey Kubik... conomics Andrew London... Sociology Len Lopoo... Public Administration Amy Lutz Sociology Jerry Miner... conomics Jan Ondrich... conomics John Palmer... Public Administration Lori Ploutz-Snyder... xercise Science Daid Popp... Public Administration Christopher Rohlfs... conomics Stuart Rosenthal... conomics Ross Rubenstein... Public Administration Perry Singleton conomics Margaret Usdansky... Sociology Michael Wasylenko... conomics Janet Wilmoth... Sociology GRADUAT ASSOCIATS Sonali Ballal... Public Administration Jesse Bricker... conomics Maria Brown... Social Science Il Hwan Chung Public Administration Mike riksen... conomics Qu Feng... conomics Katie Fitzpatrick... conomics Alexandre Genest... Public Administration Julie Anna Golebiewski...conomics Nadia Greenhalgh-Stanley... conomics Becky Lafrancois... conomics Larry Miller... Public Administration Phuong Nguyen... Public Administration Wendy Parker... Sociology Kerri Raissian... Public Administration Shawn Rohlin... conomics Carrie Roseamelia... Sociology Jeff Thompson... conomics Coady Wing... Public Administration Ryan Yeung... Public Administration STAFF Kelly Bogart Administratie Secretary Martha Bonney Publications/ents Coordinator Karen Cimilluca.... Administratie Secretary Roseann DiMarzo Receptionist/Office Coordinator Kitty Nasto..... Administratie Secretary Candi Patterson...Computer Consultant Mary Santy.... Administratie Secretary

4 Abstract A panel data regression model with heteroskedastic as well as spatially correlated disturbancesis considered, and a joint LM test for homoskedasticity and no spatial correlation is deried. In addition, a conditional LM test for no spatial correlation gien heteroskedasticity, as well as a conditional LM test for homoskedasticity gien spatial correlation, are also deried. These LM tests are compared with marginal LM tests that ignore heteroskedasticity in testing for spatial correlation, or spatial correlation in testing for homoskedasticity. Monte Carlo results show that these LM tests as well as their LR counterparts perform well een for small N and T. Howeer, misleading inference can occur when using marginal rather than joint or conditional LM tests when spatial correlation or heteroskedasticity is present. JL Code: C and C3 Keywords: Panel data; Heteroskedasticity; Spatial Correlation; Lagrange Multiplier tests; Random ffects.

5 Corresponding author. Department of conomics and Center for Policy Research, Syracuse Uniersity, Syracuse, NY 344, USA. Tel.: ; fax: mail address: Introduction The standard error component panel data model assumes that the disturbances hae homoskedastic ariances and no spatial correlation, see Hsiao 3 and Baltagi 5. These may be restrictie assumptions for a lot of panel data applications. For example, the cross-sectional units may be arying in size and as a result may exhibit heteroskedasticity. Also, for trade flows across a panel of countries, there may be spatial effects affecting this trade depending on the distance between these countries. The standard error components model has been extended to take into account spatial correlation by Anselin 988, Baltagi, Song and Koh 3, and Kapoor, Kelejian and Prucha 7, to mention a few. This model has also been generalized to take into account heteroskedasticity by Mazodier and Trognon 978, Baltagi and Griffin 988, Li and Stengos 994, Lejeune 996, Holly and Gardiol, Roy and Baltagi, Bresson and Pirotte 6 to mention a few. For a reiew of these papers, see Baltagi 5. Howeer, these strands of literature are almost separate in the panel data error components literature. When one deals with heteroskedasticity, spatial correlation is ignored, and when one deals with spatial correlation, heteroskedasticity is ignored. LM tests for spatial models are sureyed in Anselin 988, and Anselin and Bera 998, to mention a few. For a joint test of the absence of spatial correlation and random effects in a panel data model, see Baltagi, Song and Koh 3. Howeer, these tests ignore the heteroskedasticity in the disturbances. On the other hand, Holly and Gardiol deried an LM statistic which tests for homoskedasticity of the disturbances in the context of a one-way random effects panel data model. Howeer, this LM test ignores the spatial correlation in the disturbances. This paper extends the Holly and Gardiol model to allow for spatial correlation in the remainder disturbances. It deries a joint LM test for homoskedasticity and no spatial correlation. The

6 restricted model is the standard random effects error component model. It also deries a conditional LM test for no spatial correlation gien heteroskedasticity, as well as, conditional LM test for homoskedasticity gien spatial correlation. The paper then contrasts these LM tests with marginal LM tests that ignore heteroskedasticity in testing for spatial correlation, or spatial correlation in testing for homoskedasticity, again in the context of a random effects panel data model. These LM tests are computationally simple. Monte Carlo results show that misleading inference can occur when using marginal rather than joint or conditional LM tests when spatial correlation or heteroskedasticity is present. It is important to note that this paper does not consider alternatie forms of spatial lag dependence. It also does not allow for endogeneity of the regressors and requires the normality asssumption to derie the LM tests. The rest of the paper is organized as follows: Section reiews the general heteroskedastic one-way error component model with spatially autocorrelated residual disturbances. Section 3 deries the joint and conditional LM tests described aboe. Section 4 performs Monte Carlo simulations comparing the size and power of these LM tests along with their LR counterparts. Section 5 concludes. The Model Consider the following panel data regression model y ti = X tiβ + u ti, i =,,N;t =, T,. where y ti is the obseration on the ith region for the tth time period. X ti denotes the k ector of obserations on the non-stochastic regressors and u ti is the regression disturbance. In ector form, the disturbance ector of. is assumed to hae random region effects as well as spatially autocorrelated residual disturbances, see Anselin 988: u t = µ + ε t,. with ε t = λwε t + t,.3 3

7 where u = u t,...,u tn, ε t = ε t,...,ε tn and µ = µ,...,µ N are assumed independent and normally distributed according to: µ i N, µi, µi = µhf i θ, i =,...,N..4 h. is an arbitrary non-indexed strictly positie twice continuously differentiable function satisfying h. >, h =, h, where h denote the first deriatie of h. ealuated at zero. f i is a p ector of strictly exogenous regressors which determine the heteroskedasticity of the indiidual specific effects. θ isp ector of parameters, and λ is the scalar spatial autoregressie coefficient with λ <. W is a known N N spatial weight matrix whose diagonal elements are zero. W also satisfies the condition that I N λw is nonsingular for all λ <. t = t,..., tn, where ti is i.i.d. oer i and t and is assumed to be N,. The ti process is also independent of the µ i process. One can rewrite.3 as ε t = I N λw t = B t,.5 where B = I N λw and I N is an identity matrix of dimension N. The model. can be rewritten in matrix notation as y = Xβ + u,.6 where y is now 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 asymptotically bounded in absolute alue. quation. can be written in ector form as: u = ι T I N µ + I T B,.7 where = ν,...,ν T, ι T is a ector of ones of dimension T, I N is an identity matrix of dimension T and denotes the kronecker product. Under these assumptions, the ariance coariance matrix of u can be written as Ω = uu = ι T I N diag µ hf i θι T I N + I T B B = µ J T diag hfθ + I T B B,.8 where J T is a matrix of ones of dimension T and F is N p matrix of regressors that determine the heteroskedasticity. diaghf θ denotes a diagonal N N matrix with its ith diagonal element 4

8 being the ith element of the N ector hfθ. Note that the computational difficulty in dealing with this Ω is only hampered by the inersion of B B. Smirno 5 has designed an algorithm for computing the information matrix without storing B B. He deeloped a sparse ersion of the conjugate gradient method for which he reports good numerical stability and modest computational requirements. 3 LM Tests 3. Joint LM Test In this subsection, we derie the joint LM test for testing for no heteroskedasticity and no spatial correlation in a random effects panel data model. The null hypothesis is gien by H a o : θ = = θ p = and λ = µ >, >. The log-likelihood function under normality of the disturbances is gien by where ϕ = Lβ,ϕ = constant log Ω u Ω u, 3., µ, λ,θ,...,θ p. The information matrix is block-diagonal between β and ϕ. Since H a inoles only ϕ, the part of the information due to β is ignored, see Breusch and Pagan 98. In order to obtain the joint LM statistic, we need Dϕ = L/ ϕ and the information matrix Jϕ = L/ ϕ ϕ ealuated at the restricted ML estimator ϕ. Under the null hypothesis H a, the ariance-coariance matrix reduces to Ω = µ J T I N + I T I N = JT I N + T I N, 3. where T = I T J T with J T = J T /T and = T µ +. It is the familiar form of the random effects error component model with no spatial correlation or heteroskedasticity, see Baltagi 5. Its inerse is gien by Ω = JT I N + T I N

9 The score D ϕ, and information matrix J a ϕ,under the null hypothesis H a are deried in Appendix. The resulting LM statistic for H a is gien by: LM λθ = D ϕ where b = trw + W W, Dλ = L = λ H a J a ϕ D ϕ = D λ T b + 4 S F F F F S, 3.4 ũ T W + W ũ + ũ 4 J T W + W ũ, 3.5 with ũ = y X β ML denoting the restricted maximum likelihood residuals under H a, i.e., under a random effects panel data model with no spatial correlation or heteroskedasticity. S is an N ector with typical element S i = ũ i J T ũ i and F = I N J N F. Here, ũ i = ũ i,.., ũ it, i.e., the residuals are re-ordered according to time for each indiidual. F = I N J N F, and and are the restricted maximum likelihood estimates of and gien by =ũ J T I Nũ N =ũ T I N ũ NT, respectiely. Under the null hypothesis H a, LM λθ should be asymptotically distributed as χ p+. Although we do not explicitly derie the asymptotic distribution of our test statistics, they are likely to hold under a similar set of primitie assumptions deeloped by Kelejian and Prucha. Also, note that for large N, tricks for computing the tracew can be approximated as in Barry and Pace 999 to any desired accuracy using an ON algorithm. and 3. Conditional LM Tests The joint LM test deried in the preious section is useful especially when one does not reject the null hypothesis H a. Howeer, if the null hypotheses is rejected, one can not infer whether the presence of heteroskedasticity, or the presence of spatial correlation, or both factors caused this rejection. Alternatiely, one can derie two conditional LM tests. The first one tests for the absence of spatial correlation assuming that heteroskedasticity of the indiidual effects might be present. The second one tests for homoskedasticity assuming that spatial correlation might be present. All in the context of a random effects panel data model. For the first conditional LM test, the null hypothesis is gien by H b : λ = assuming at least one element of θ is not zero. Under the null hypothesis H b 6, the ariance-coariance matrix in.8

10 reduces to Ω = µj T diag hfθ + I T I N. 3.6 Replacing J T by T J T and I T by T + J T, where T = I T J T, and J T = J T /T, see Wansbeek and Kapteyn 98, one gets: Ω = where g i = T µhf i θ +. This also implies that JT diag g i + T I N, 3.7 Ω = J T diag + gi T I N. 3.8 Appendix deries the score D ϕ and information matrix J b ϕ under H b. The resulting conditional LM statistic for H b is gien by where b = trw + W W, LM λ θ = D ϕ J b D ϕ = D λ 4 d+t b, 3.9 W d = tr diag + W W diag + W, 3. gi gi D λ = ũ T W + Wũ + ũ J T diag W gi + Wdiag ũ, 3. gi with ũ = y X β ML denoting the restricted maximum likelihood residuals under H b, i.e., under a random effects panel data model with no spatial correlation but possible heteroskedasticity, g i = T µ hf i θ +, where θ, µ and are the restricted ML of θ, µ and, under Hb. Under the null hypothesis H b, LM λ θ should be asymptotically distributed as χ. For the second conditional LM test, the null hypothesis is gien by: H c : θ = assuming λ may not be zero. Under the null hypothesis H c, the ariance-coariance matrix in.8 reduces to Ω = µ J T I N + I T B B. 3. Replacing J T by T J T and I T by T + J T, where T = I T J T, and J T = J T /T, see Wansbeek and Kapteyn 98, one gets: Ω = JT TφI N + B B + T B B, 3.3 7

11 where φ = µ/. This also implies that where Z = Ω = JT Z + T B B, 3.4 TφI N + B B. Appendix 3 deries the score D ϕ and information matrix J c ϕ under H c, i.e., under a random effects panel data model with no heteroskedasticity but possible spatial correlation. The resulting conditional LM statistic for H c is gien by LM θ λ = D ϕ J c ϕ D ϕ, 3.5 and did not hae a simple expression as LM λ θ. Under the null hypothesis H c, the LM statistic should be asymptotically distributed as χ p. 4 Monte Carlo Results The experimental design for the Monte Carlo simulations is based on the format extensiely used in earlier studies in the spatial regression model by Anselin and Rey 99 and Anselin and Florax 995, and in the heteroskedastic panel data model by Roy. The model is set as follows: y it = α + x it β + u it i =,,N, t =,,T, where α = 5 and β =.5. x it is generated by a similar method to that of Nerloe 97. In fact, x it =.t +.5x i,t + z it, where z it is uniformly distributed oer the interal,. The initial alues x i are chosen as 5 + z i. For the disturbances, u it = µ i + ε it, ε it = λ N j= w ij ε it + ν it with ν it IIN,ν and µ i N,µi where, µ i = µ i x i. = µ + θ x i., or µ i = µ i x i. = µ expθ x i., 4. and x i. denoting the indiidual mean of x it. Denoting the expected ariance of µ i by µ i and following Roy, we fix µ i +ν =. We let ν take the alues 4 and 6. For each fixed alue 8

12 of ν, θ is assigned alues,, and 3 with θ = denoting the homoskedastic indiidual specific error. For a fixed alue of ν, we obtain the alues of µ i and using 4., we get the alues for µ for each θ alue considered. Then we obtain the alues of µ i for each µ under the four different θ alues considered. The matrix W is a rook or queen type weight matrix, and the rows of this matrix are standardized so that they sum to one. The spatial autocorrelation factor λ is aried oer the positie range from to.9 by increments of.. Two alues for N = 5 and 49, and three alues for T = 5,7 and are chosen. For each experiment, replications are performed. We chose small alues of N and T to demonstrate that the size of these tests work well een in small samples. Of course for larger samples, the computational difficulty increases, but the asymptotics should een be better behaed. Table gies the empirical size of the joint LM and LR tests for the null hypothesis H a : θ = λ = at the 5% significance leel when N = 5, 49 and T = 5, 7, for both the queen and rook weight matrices. Additionally, we ran a limited set of experiments for N = and T = 7 with a Rook design to show that the power improes as N gets large. These results are not shown here to sae space but are aailable upon request from the authors. Table gies the empirical size of the conditional LM and LR tests for the null hypothesis H b : λ = gien θ and Tables 3a, 3b gie the empirical size of the conditional LM and LR tests for the null hypothesis H c : θ = gien λ. As we can see from these Tables, the empirical size is not significantly different from 5% for a Bernoulli with replications and probability of success of.5. Table 4 gies the empirical size of the marginal LM and LR tests for the null hypothesis H d : λ = gien θ = and Tables 5a, 5b, gie the empirical size of the marginal LM and LR tests for the null hypothesis H e : θ = gien λ =. As we can see from the Tables, the empirical size of the LM and LR tests are seerely undersized for ν = 6, and high alues of λ, and this empirical size is significantly different from 5% for a Bernoulli with replications and probability of success of.5. The power of these tests for arious experiments are not shown here to sae space and are aailable as Tables A-A6 upon request from the authors. Figure plots the empirical power for the joint LM and LR test for the null hypothesis H a : θ = λ =, for N = 5 and 49, and T =. This is done for arious alues of λ, when ν takes the alues 4 and 6 and θ =,,,3. This is done for a Rook weight matrix and an exponential form of heteroskedasticity. The power of the joint 9

13 Table : stimated size of joint LM and LR tests for testing H a : λ = θ =. Quadratic heteroskedasticity xponential heteroskedasticity N=5 N=49 N=5 N=49 T W θ λ LM LR LM LR LM LR LM LR 5 Rook Queen Rook Queen Rook Queen LM and LR tests climbs to one quickly as soon as λ exceeds.3 for arious alues of θ. Also, this power increases with θ for < λ <.3. The power also increases with N. Figure repeats these power plots for the case of quadratic heteroskedasticity. Again, the same phenomenon is obsered for this alternatie form of heteroskedascity. Figure 3 plots the empirical power for the marginal and conditional LM and LR tests for the null hypothesis H b : λ = gien θ, for N = 5 and 49, and T =. This is done for arious alues of λ, when ν take the alue 6 and θ = 3. This is done for a Rook weight matrix and a quadratic form of heteroskedasticity. As clear from Figure 3, the conditional LM and LR tests perform better than their marginal counterparts for θ. Howeer, the power of all these tests is close to one for λ >.3. Figure 4 plots the empirical power for the marginal and conditional LM and LR tests for the null hypothesis H c : θ = gien λ, for N = 5 and 49, and T =. This is done for arious alues of θ, when ν take the alue 6 and λ =.9. This is done for a Rook weight matrix and an exponential form of heteroskedasticity. As clear from Figure 4, the conditional LM and LR tests perform better than their marginal counterparts for λ, and this power is increasing with θ. These figures gie a flaour of the power performance of these tests for a subset of the experiments. Of course, more plots can be gien, but we refrain from doing so because of space limitation.

14 Table : stimated size of conditional LM and LR tests for testing H b : λ = gien θ. Quadratic heteroskedasticity xponential heteroskedasticity N=5 N=49 N=5 N=49 T W θ λ LM LR LM LR LM LR LM LR 5 Rook Queen Rook Queen Rook Queen

15 Table 3a : stimated size of conditional LM and LR tests for testing H c : θ = gien λ when weight matrix is ROOK. Quadratic heteroskedasticity xponential heteroskedasticity N=5 N=49 N=5 N=49 T W θ λ LM LR LM LR LM LR LM LR 5 Rook Rook Rook Rook Rook Rook

16 Table 3b : stimated size of conditional LM and LR tests for testing H c : θ = gien λ when weight matrix is QUN. Quadratic heteroskedasticity xponential heteroskedasticity N=5 N=49 N=5 N=49 T W θ λ LM LR LM LR LM LR LM LR 5 Queen Queen Queen Queen Queen Queen

17 Table 4 : stimated size of marginal LM and LR tests for testing H d : λ = assuming θ =. Quadratic heteroskedasticity xponential heteroskedasticity N=5 N=49 N=5 N=49 T W θ λ LM LR LM LR LM LR LM LR 5 Rook Queen Rook Queen Rook Queen

18 Table 5a : stimated size of marginal LM and LR tests for testing H e : θ = assuming λ = when weight matrix is ROOK. Quadratic heteroskedasticity xponential heteroskedasticity N=5 N=49 N=5 N=49 T W θ λ LM LR LM LR LM LR LM LR 5 Rook Rook Rook Rook Rook Rook

19 Table 5b : stimated size of marginal LM and LR tests for testing H e : θ = assuming λ = when weight matrix is QUN. Quadratic heteroskedasticity xponential heteroskedasticity N=5 N=49 N=5 N=49 T W θ λ LM LR LM LR LM LR LM LR 5 Queen Queen Queen Queen Queen Queen

20 Figure. Frequency of rejections for H a : λ = and θ = N=5, 49 T=, xponential heteroskedasticity, weight matrix is ROOK. = 4, θ = = 4, θ = = 4, θ = = 4, θ = 3 = 6, θ = = 6, θ = = 6, θ = = 6, θ = 3 7

21 Figure. Frequency of rejections for H a : λ = and θ = N=5, 49 T=, Quadratic heteroskedasticity, weight matrix is ROOK. = 4, θ = = 4, θ = = 4, θ = = 4, θ = 3 = 6, θ = = 6, θ = = 6, θ = = 6, θ = 3 8

22 Figure 3. Frequency of rejections for H b : λ = assuming θ and for Hd : λ = assuming θ = T=, = 6, Quadratic heteroskedasticity, weight matrix is ROOK. N=5, θ = 3 N=49, θ = 3 Figure 4. Frequency of rejections for H c : θ = assuming λ and for He : θ = assuming λ = T=, = 6, xponential heteroskedasticity, weight matrix is ROOK. N=5, λ =.9 N=49, λ =.9 5 Conclusion This paper considered a panel data regression model with heteroskedasticity in the indiidual effects and spatial correlation in the remainder error. Testing for heteroskedasticity ignoring the spatial correlation as well as testing for spatial correlation ignoring heteroskedasticity is shown to lead to misleading results. The paper deried joint and conditional LM tests for heteroskedasticity and spatial correlation and studied their performance using Monte Carlo experiments. This paper generalized the Holly and Gardiol paper by allowing for spatial correlation in the remainder disturbances. The paper does not consider alternatie forms of spatial lag dependence and this should be the subject of future research. In addition, it is important to point out that the asymptotic 9

23 distribution of our test statistics were not explicitly deried in the paper but that they are likely to hold under a similar set of primitie assumptions deeloped by Kelejian and Prucha. The results in the paper should be tempered by the fact that the N = 5,49 used in our Monte Carlo experiments may be small for a typical micro panel. Larger N = did improe the performance of these tests whose critical alues were based on their large sample distributions. Although the computational difficulty increases with N, one can use computational tricks gien by Barry and Pace 999 and Smirno 5. Acknowledgement This work was supported by KOSFR We would like to thank the guest editor Manfred M. Fischer and two anonymous referees for helpful comments and suggestions. We dedicate this paper in memory of our colleague and co-author Seuck Heun Song who passed away March, 8. Appendices Appendix This appendix deries the joint LM test for testing for no heteroskedasticity and no spatial correlation in a random effects panel data model, i.e., Ho a : θ = = θ p = and λ = µ >, >. The ariance-coariance matrix of the disturbances in the unrestricted model is gien by.8 and can be written as Ω = µ J T diag hfθ + I T B B, A. where J T is a matrix of ones of dimension T and F is N p matrix of regressors that determine the heteroskedasticity. diaghf θ denotes a diagonal N N matrix with its ith diagonal element being the ith element of the N ector hfθ. The log-likelihood function under normality of the disturbances is gien by Lβ,ϕ = constant log Ω u Ω u, A.

24 where ϕ = Since H a, µ, λ,θ,...,θ p. The information matrix is block-diagonal between β and ϕ. inoles only ϕ, the part of the information due to β is ignored, see Breusch and Pagan 98. In order to obtain the joint LM statistic, we need Dθ = L/ θ and the information matrix Jθ = L/ θ θ ealuated at the restricted ML estimator ϕ. Under the null hypothesis H a, the ariance-coariance matrix reduces to Ω = µ I N J T + ǫ I N I T. It is the familiar form of the random effects error component model with no spatial correlation or heteroskedasticity, see Baltagi 5. Its inerse is gien by Ω = JT I N + T I N, A.3 where T = µ +, T = I T J T and J = J T /T. Hartley and Rao 967 or Hemmerle and Hartley 973 gie a general useful formula to derie the scores: Under the null hypothesis H a, we get L = ϕ r tr Ω Ω + u Ω Ω Ω u. A.4 ϕ r ϕ r Ω H = T a µ θ h J T diagf k, k Ω = λ H a IT W + W, Ω = I H a T I N Ω = J H a T I N, A.5 µ where h denote the first deriatie of h ealuated at zero, and F k is the kth column of F. This makes use of the fact that B B / λ = B B W B + B WB B, see Anselin 988, p.64. Substituting A.5 into A.4, we get L = H a tr JT I N + + u JT I N + = tr JT I N + T I N T I N I T I N T I N I T I N + u 4 JT I N + T I N u JT I N + 4 T I N u

25 = N NT u JT I N u + 4 u T I N u =, L = H a tr JT I N + µ + u JT I N + T I N = T tr T JT I N + 4 u JT I N u = TN + T 4 u JT I N u =, T I N J T I N J T I N JT I N + T I N u L = D λ = λ H a tr JT I N + JT I N + T I N + u T I N I T W + W I T W + W = tr JT W + W + T W + W + u T W + Wu + u 4 J T W + Wu = u T W + Wu + u 4 J T W + Wu, JT I N + T I N u L θ k H a = tr JT I N + + u = T µh = T µh JT I N + T I N JT I N + T I N T I N Tµ h J T diagf k Tµh J T diagf k tr JT diagf k + T µh 4 u JT diagf k u ι N F k + T µh 4 S F k for k =,..,p, u L θ H a θ = D = T µ h 4 S F T µ h ι NF

26 = T µh 4 S F T µh N S ι N ι NF = T µh 4 = T µ h 4 S F T µh 4 S JN F S I N J N F = T µ h 4 S F, A.6 This uses the fact that trw = trw =. From the second score equation, one gets that =ũ J T I Nũ N,where ũ = y X β ML denote the restricted maximum likelihood residuals under H a, i.e., under a random effects panel data model with no spatial correlation or heteroskedasticity. Substituting this in the first score equation, one gets =ũ T I N ũ NT. Note that is used to simplify the last score equation. S is an N ector with typical element S i = ũ i J T ũ i and F = I N J N F. Here, ũ i = ũ i,.., ũ it, i.e., the residuals are re-ordered according to time for each indiidual. Therefore, the score with respect to each element of ϕ, ealuated at the resticted ML is gien by D ϕ = = D λ ũ D θ T W + Wũ + ũ J 4 T W + Wũ. T µ h S F 4 For the information matrix, it is useful to use the formula gien by Harille977: J rs = L/ ϕ r ϕ s = Ω tr Ω Ω Ω, ϕ r ϕ s for r,s=,,..,p+3. Under the null hypothesis H a,we get: L H a = tr JT I N + T I N I T I N JT I N + T I N I T I N = tr JT 4 I N + 4 T I N = N T 4 + 4, 3

27 L µ H a = tr = T 4 JT I N + T I N J T I N JT I N + T I N tr NT JT I N =, 4 J T I N L λ H a = tr JT I N + T I N I T W + W JT I N + T I N I T W + W = 4 tr JT 4 W + W + T W + W 4 = 4 + T tr W + W W, L θ k θ l H a = tr = = JT I N + T I N Tµh JT diagf k JT I N + T I N Tµ h JT diagf l 4 tr J T diagf k diagf l T µ h T µ h 4 tr diagf k diagf l = Tµ h 4 F kf l, L θ θ L µ H a H a = Tµ h F F, = tr 4 JT I N + T I N I T I N JT I N + T I N J T I N = NT 4, L µ λ H a = tr JT I N + T I N J T I N JT I N + T I N 4 I T W + W =,

28 L µ θ k L µ θ H a H a = tr = T µ h 4 i NF k, = T µh 4 i N F, JT I N + T I N J T I N JT I N + T I N T µ h JT diagf k L λ L θ k L θ H a H a H a = tr = tr = T µh 4 i N F k, = T µ h 4 i NF, JT I N + T I N I T I N JT I N + T I N I =, T W + W JT I N + T I N I T I N JT I N + T I N Tµh JT diagf k L λ θ k H a = tr =. JT I N + T I N I T W + W JT I N + T I N Tµ h JT diagf k A.7 Therefore, the information matrix under the null hypothesis H a J a ϕ = N 4 + NT 4 T µh 4 T 4 NT 4 NT F T i µh N F i 4 N is gien by Tµ h i 4 N F T µh i 4 N F + T b T µ h 4 F F, 5

29 where b = trw + W W. Since D ϕ =,, D λ, D θ, and J a ϕ is a block diagonal matrix with respect to ϕ =, µ, θ and λ, we only need the lower p + p + block of the inerse of J a ϕ. Let J a ϕ = B B B B, where B = T b And using the partitioned inerse formula, we get J a ϕ λ,θ = B = B = B = T µ h 4 F F B B B B, with Tµh i 4 N F, T µh i 4 N F B B = N B B B = B B B B = T µ h 4 N T T F T i µ h N F i 4 N NT 4 NT 4 µ h F i N N 4, NT 4 +, T 4 Tµ h, and F JN F T b 4 Tµ h 4 F F,, where F = I N J N F. The resulting LM statistic for H a is gien by LM λθ = D ϕ J a ϕ D ϕ 6

30 = = D λ, T µh 4 S F 4 4 D λ T b + 4 S F F F F S. + T b T µh 4 F F D λ T µ h F S 4 A.8 Under the null hypothesis, LM λθ should be asymptotically distributed as χ p+. Appendix This appendix deries the conditional LM test for testing no spatial correlation assuming there might be heteroskedasticity, i.e., H b : λ = assuming some elements of θ may not be zero. Under the null hypothesis H b, the ariance-coariance matrix in A. reduces to Ω = µ J T diag hfθ + I T I N. A.9 Replacing J T by T J T and I T by T + J T, where T = I T J T, and J T = J T /T, see Wansbeek and Kapteyn 98, one gets: Ω = where g i = Tµ hf i θ +. This also implies that JT diag g i + T I N, A. In this case: Using A.4, we get Ω = J T diag + g i T I N. A. Ω = J H b T diag hfθ, µ Ω = I H b T I N, Ω H = T b µ JT diag h Fθ diag F k, and θ k Ω = IT W + W. A. λ H b L = H b tr J T diag + gi T I N I T I N 7

31 + u J T diag + gi T I N I T I N J T diag + gi T I N u = N i= g i + NT + u J T diag gi + 4 T I N u =, L = H b tr µ J T diag + gi T I N J T diag hfθ + u JT diag gi + T I N J T diag hfθ u JT diag gi + T I N = T N hf i θ + T hf g i= i u J T diag i θ gi u =, J T diag + gi L = D λ = λ H b tr + u JT diag gi + T I N I T W + W JT diag gi + T I N = W tr J T diag + W + T W + W gi + u T W + Wu + = u T W + Wu + T I N I T W + W u u J T diag W + Wdiag u gi gi u J T diag W + Wdiag u, gi gi L H = b θ k tr + u J T diag + gi T I N Tµ JT diag h Fθ diag F k JT diag gi + T I N Tµ JT diag h Fθ JT diag gi = T µ tr J T diag + T µ u + gi diag T I N h Fθ diagf k J T diag diag h Fθ diagf k diag u gi gi diag F k = for k =,..,p. A.3 8 u

32 Therefore, the score with respect to each element of ϕ, ealuated at the resticted ML under H b is gien by D ϕ = D λ = ũ T W + Wũ + ũ J T diag gi W + Wdiag gi ũ, where ũ = y X β ML denote the restricted maximum likelihood residuals under H b, i.e., under a random effects panel data model with no spatial correlation but possible heteroskedasticity, g i = T µhf i θ +, where θ and are the restricted ML of θ and, under H b. Also, the elements of the information matrix for this model are gien by L H b = tr JT diag gi JT diag gi = tr J T diag = N g + i= i g i NT 4, + T I N I T I N + T I N I T I N + 4 T I N L µ H b = tr JT diag gi JT diag gi = T tr J T diag + T I N J T diag hfθ + T I N J T diag hfθ diag hfθ gi gi diag hfθ diag = T N hf i θ, g i= i L λ H b = tr = tr JT diag gi + JT diag gi + 4 JT diag gi + T W + W T I N I T W + W T I N I T W + W W + Wdiag gi W + W 9

33 = 4 W tr diag + W W diag + W + T b, g i g i L θ k θ l H b JT diag gi + h Fθ diag F k = tr T I N Tµ JT diag JT diag gi + T I N Tµ JT diag h Fθ Tµ J T diag gi diag h Fθ diag F k = tr diag gi diag h Fθ diag F l Tµ N h f i = θ f kif li g i= i Tµ = f k r diag i f l, diag F l L θ θ L µ H b H b Tµ = F diag ri F, = tr JT diag gi JT diag gi + T I N I T I N = T tr J T diag diag hfθ diag gi gi + T I N T J T diag hfθ = T diag tr diag hfθdiag gi gi = T N hf i θ g, i= i L λ L θ i H b H b = tr = tr = tr JT diag gi + T I N I T I N JT diag gi + T I N I T W + W J T diag W + W + gi 4 JT diag gi + T W + W =, T I N I T I N JT diag gi + T I N Tµ JT diag h Fθ diag F i 3

34 = T µ tr J T diag gi = T µ tr diag gi diag = T µ N h f i θ g f ik i= i diag h Fθ diag F i h Fθ diag F i L θ H b L µ λ H b L µ θ i L µ θ L λ θ i H b H b H b = T µ z F k, = T µ z F, = tr JT diag gi + T I N T J T diag hfθ JT diag gi + T I N I T W + W = T W tr J T diag diag hfθdiag + W =, gi gi = tr JT diag gi + T I N T J T diag hfθ JT diag gi + T I N Tµ JT diag h Fθ = T µ tr J T diag diag hfθ diag g i g i = T µ tr = T µ = T µ s F k, diag diag hfθdiag g i N hf i θh f i θ g f ki i= i g i diag diag F k diag h Fθ diag F k h Fθ diag F k = T µ s F, = tr JT diag gi + T I N I T W + W JT diag gi + T I N Tµ JT diag h Fθ = T µ W tr J T diag + W diag diag h Fθ diag F i g i g i = T µ tr diag g i W + W diag g i diag diag F i h Fθ diag F i =, A.4 where r,z and s are N ectors with typical elements: r i = h f i θ/g i, z i = h f i θ/g i and s i = hf i θh f i θ/g i, respectiely. Therefore, the information matrix under the null hypothesis 3

35 H b is gien by: N NT T N hf + g i= i 4 i θ Tµ g i= z F i T N hf i θ T N hf i θ T g i= i g i µ s F i= J b ϕ =, 4 d+ T b Tµ F T z µ s T F µ F diag ri F where d = tr diag gi W + Wdiag gi W + W. Since D ϕ =,, D λ,, and J b ϕ is a block diagonal matrix with respect to ϕ =, µ, θ and λ, the resulting conditional LM statistic for H b is gien by LM λ θ = D ϕ J b D ϕ = 4 D λ d+t b. Under the null hypothesis H b, LM λ θ should be asymptotically distributed as χ. Appendix 3 This appendix deries the conditional LM test for testing for homoskedasticity assuming there might be spatial correlation, i.e H c : θ = assuming λ may not be zero. Under the null hypothesis Hc, the ariance-coariance matrix in A. reduces to Ω = µ J T I N + I T B B. A.5 Replacing J T by T J T and I T by T + J T, where T = I T J T, and J T = J T /T, see Wansbeek and Kapteyn 98, one gets: Ω = JT TφI N + B B + T B B, A.6 where φ = µ/. This also implies that Ω = JT Z + T B B, A.7 3

36 where Z = TφI N + B B. In this case: Using A.4, we get L µ H c L H c L λ H c L θ k H c Ω = T JT I H c N, µ Ω = I H c T B B, = tr + u Ω H = T c µ θ h JT diag F k, and k Ω = λ H c I T B B W B + B W B B. A.8 JT Z + T B B T JT I N, JT Z + T B B T JT I N JT Z + T B B u = T tr Z + T 4 u JT Z u =, = tr JT Z + T B B I T B B + u = tr JT Z + T B B I T B B JT Z + T B B JT Z B B + T I N + 4 u JT Z B B Z + T B B u =, = tr JT Z + T B B I T B B W B + B W B B + u u JT Z + T B B I T B B W B + B W B B JT Z + T B B = tr JT ZB B W B + B W B B + T W B + B W B B + u JT ZB B W B + B W B B Z + T W B + B W u =, = D θ k = tr JT Z + T B B Tµh JT diag F k + u JT Z + T B B Tµ h JT diag F k JT Z + T B B 33 u u

37 = T µh tr JT Zdiag F k + T µh 4 u JT Zdiag F k Z u = T µh d F k + T µh 4 u D k u for k =,..,p, where d is an N ector with its typical element being the diagonal element of the matrix Z. D k = JT Zdiag F k Z. Therefore, the score with respect to each element of ϕ, ealuated at the resticted ML under H c is gien by D ϕ = = D θ : : D θ p T µh T µ h d F + T µh u D 4 u d F p + T µ h u D 4 p u where ũ = y X β ML denote the restricted maximum likelihood residuals under H c, i.e., under a random effects panel data model with no heteroskedasticity but possible spatial correlation. Also, the elements of the information matrix for this model using A.3 are gien by L L µ L λ H c H c H c = tr JT Z + T B B I T B B JT Z + T B B I T B B = 4 tr JT Z B B Z B B + T I N = tr Z B B Z B B NT +, 4 = tr 4 JT Z + T B B T JT I N JT Z + T B B T JT I N = T 4 tr JT Z = T 4 tr Z, = tr JT Z + T B B I T B B W B + B WB B JT Z + T B B I T B B W B + B WB B = tr ZB B W B + B W B B, 34

38 L θ k θ l L θ θ L µ L λ L θ k L θ H c H c H c H c H c H c T W + tr B + B W B B, = tr JT Z + T B B Tµh JT diag F k JT Z + T B B Tµh JT diag F l = = = = T µh 4 T µh 4 T µ h 4 T µh = tr = T 4 = tr = 4 tr JT Zdiag F k Zdiag F l tr Zdiag F k Zdiag F l F k Z Z F l, F Z Z F, JT Z + T B B T JT I N JT Z + T B B I T B B tr J T Z B B = T tr Z B B, 4 JT Z + T B B I T B B JT Z + T B B I T B B W B + B WB B tr Z B B ZB B W B + B W B B T W + tr B + B W B B, = tr JT Z + T B B I T B B JT Z + T B B Tµ h JT diag F k = T µ h 4 tr JT Z B B Z diag Fk = T µh 4 tr = T µ h 4 d F k, = T µ h 4 d F, Z B B Z diag Fk 35

39 L µ λ L µ θ k µ θ L L λ θ k L λ θ H c H c H c H c H c = tr JT Z + T B B T JT I N JT Z + T B B I T B B W B + B WB B = T tr JT Z B B W B + B W B B = T tr Z B B W B + B W B B, = tr JT Z + T B B T JT I N JT Z + T B B Tµ h JT diag F k = T µ h 4 tr JT Z diag F k = T µh 4 tr = T µ h = tr 4 d 3 F, Z diag F k = T µh d 3 F k, 4 JT Z + T B B I T B B W B + B WB B JT Z + T B B Tµh JT diag F k = T µh tr ZB B W B + B W B B Zdiag F k = T µ h d 4 F k, = T µh d 4F, A.9 where stands for the Hadamard product, i.e., an element-by-element multiplication and d, d 3, d 4 are N ectors corresponding to the diagonal elements of matrices: ZB B Z, Z and ZB B W B + B W B B Z respectiely. Therefore, the information matrix under the null hypothesis H c is gien by: 36

40 J c ϕ = with b NT Tb 4 b 3 + T b 4 T µ h 4 Tb 4 T b 5 4 Tb 6 F d T µ h 4 b 3 + T b 4 Tb 6 b 7 + T b 8 F d 3 T µ h Tµ h d 4 F T µ h d 4 3 F Tµh d 4 F F Tµ d h 4 F Z Z F 4 b b b 3 = tr Z B B, = tr Z B B, = tr Z B B ZB B W B + B W B B, b 4 W = tr B + B W B B, b 5 = tr Z, b 6 = tr Z B B W B + B W B B, = tr ZB B W B + B W B B, and b 7 b 8 = tr W B + B W B B. A. The resulting conditional LM statistic for H c is gien by LM θ λ = D ϕ J c ϕ D ϕ. Under the null hypothesis, the LM statistic should be asymptotically distributed as χ p. References Anselin, L., 988, Spatial conometrics: Methods and Models Kluwer Academic Publishers, Dordrecht. 37

41 Anselin, L.,, Rao s score tests in spatial econometrics. Journal of Statistical Planning and Inference 97, Anselin, L. and A.K. Bera, 998, Spatial dependence in linear regression models with an introduction to spatial econometrics. In A. Ullah and D..A. Giles, eds., Handbook of Applied conomic Statistics, Marcel Dekker, New York. Anselin, L and S. Rey, 99, Properties of tests for spatial dependence in linear regression models. Geographical Analysis 3, -3. Anselin, L. and R. Florax, 995, Small sample properties of tests for spatial dependence in regression models: Some further results. In L. Anselin and R. Florax, eds., New Directions in Spatial conometrics, Springer-Verlag, Berlin, pp Baltagi, B.H., 5, conometrics Analysis of Panel Data Wiley, Chichester. Baltagi, B.H., G. Bresson and A. Pirotte, 6, Joint LM Test for Homoskedasticity in a One-Way rror Component Model, Journal of conometrics 34, Baltagi, B.H. and J.M Griffin, 988, A generalized error component model with heteroscedastic disturbances, International conomic Reiew 9, Baltagi, B.H., S.H. Song and W. Koh, 3, Testing panel data regression models with spatial error correlation, Journal of conometrics 7, 3-5. Barry, R., and R. Kelley Pace, 999, A Monte Carlo estimator of the log determinant of large sparse matrices, Linear Algebra and its Applications 89, Breusch, T.S. and A.R. Pagan, 979, A simple test for heteroskedasticity and random coefficient ariation, conometrica 47, Breusch, T.S. and A.R. Pagan, 98, The Lagrange Multiplier test and its application to model specification in econometrics, Reiew of conomic Studies 47, Hartley, H.O. and J.N.K. Rao, 967, Maximum likelihood estimation for the mixed analysis of ariance model. Biometrika 54,

42 Harille, D.A., 977, Maximum likelihood approaches to ariance component estimation and to related problems. Journal of the American Statistical Association 7, Hemmerle, W.J. and H.O. Hartley, 973, Computing maximum likelihood estimates for the mixed A.O.V. model using the W-transformation. Technometrics 5, Holly, A. and L. Gardiol,, A score test for indiidual heteroscedasticity in a one-way error components model, Chapter in J. Krishnakumar and. Ronchetti, eds., Panel Data conometrics: Future Directions North-Holland, Amsterdam, 99. Hsiao, C., 3, Analysis of Panel Data Cambridge Uniersity Press, Cambridge. Kapoor, M., H.H. Kelejian and I.R. Prucha, 7, Panel data models with spatially correlated error components, Journal of conometrics 4, Kelejian, H.H., Prucha, I.R.,. On the asymptotic distribution of the Moran I test with applications. Journal of conometrics 4, Lejeune B., 996, A full heteroscedastic one-way error components model for incomplete panel: maximum likelihood estimation and Lagrange multiplier testing, COR discussion paper 966, Uniersite Catholique de Louain, 8. Li, Q. and T. Stengos, 994, Adaptie estimation in the panel data error component model with heteroskedasticity of unknown form, International conomic Reiew 35, 98-. Magnus, J.R., 98, Multiariate error components analysis of linear and nonlinear regression models by maximum likelihood. Journal of conometrics 9, Mazodier, P. and A. Trognon, 978, Heteroskedasticity and stratification in error components models, Annales de l INS 3 3, Nerloe, M., 97, Further eidence on the estimation of dynamic economic relations from a time-series of cross-sections, conometrica 39, Roy, N.,, Is adaptie estimation useful for panel models with heteroscedasticity in the indiidual specific error component? Some Monte Carlo eidence, conometric Reiews,,