Moran's I test of spatial panel data model — Based on bootstrap method

inaiteolveboos artsp M1. Introductionometrics can test spatial effectsal heterandstenceacoreand spatialablesinbehavior.researchers more opportunities to extend model possibilities as com-pared to cross-sectional data models. The use of panel data also resultsconsidering the individual heterogeneity and the cross-sectional di-a good property ofabove problems. TheyEconomic Modelling 41 (2014) 914Contents lists available at ScienceDirectEconomic M/ldependence. Therefore, Morans I test has been most commonly used.As we all know, the panel data contains both the characteristicsofcross-sectional dataandtime seriesdata.Hence,itcan providericherare introduced into spatial cross-sectional data models by Lin et al.(2011). But spatial panel data models are more complicated andwould affect the bootstrap sampling. Therefore, the bootstrap methodsI test of spatial dependence is assumed to be a non-alternative hypothe-sis model, and, it can test spatial lag dependence and test spatial errorand low power. Bootstrap methods, which havenite samples, are an effective way to solve themany methods to test the existence of spatial dependence in spatialcross-sectional data models, but rarely are there methods to test the ex-istence of spatial dependence in spatial panel data models. The commonmethods to test the existence of spatial dependence in spatial cross-sectionaldatamodels,includeMoransI,LM,LR,Raosscore,etc.Moranssumptions cant satisfy strictly. For example, in real economic manage-ment analysis, data is non-normally distributed, or its sample size islimited for empirical discussion. Evidences (Lin et al., 2009; Yang,2011) showed that, in nite samples, the Morans I test referring to as-ymptotic critical values may suffer from the problems of size distortioninformation for regression analysis. The pane Corresponding author at: School of Economics, She518055, China. Tel.: +86E-mail address: (R. Zhang)./10.1016/j.econmod.2014.04.0220264-9993/ 2014 Elsevier B.V. All rights reserved.And the latter,rm, studies the inuencevior. Currently, there aretions that the error term is normally distributed or large samplesize. However, the real economy is a complex system which is affectedbydiversityfactors.Inalargenumberofempiricalstudies,theaboveas-whose spatial dependence exists in the error teof the error shock on neighboring region behamodels. And, to be precise, spatial econincluding spatial dependence and spatitial econometrics can get more reasonablethan classic econometrics. To test the exiinaspatialeconometricmodelhasbeenmodels includea spatiallagmodel (SLM)Theformerstudieshowdependentvariior affect other parts of the overall systemogeneity. Therefore, spa-realistic conclusionsof spatial dependenceissue.Spatialeconometricerrormodel(SEM).thevicinityofthebehav-mensions to test whether the spatial dependence exists. Thus, therearemuchbroaderprospectsofpracticalapplicationinactualresearches.However, panel data information is more plentiful than cross-sectionaldata.Hence,testingfor spatial dependence of spatialpanel data modelsis a difcult problem in spatial econometrics. Furthermore, the tradi-tionalmethodsoftestingspatialdependencedependedontheassump-Spatial econometric models have been extepastthirtyyears.Theyhavefewerassumptionstnsively studied in thehanclassiceconometricin a greater availability of degrees of freedom, and increases efciencyin the estimation. Spatial panel data models lead into spatial effects,Morans I test of spatial panel data model Tongxian Rena, Zhihe Longa,RenguiZhanga,b, QingqaSchool of Economics and Commerce, South China University of Technology, Guangzhou, ChinbSchool of Economics, Shenzhen Polytechnic, Shenzhen, Chinaabstractarticle infoArticle history:Accepted 15 April 2014Available online 15 May 2014Keywords:Bootstrap methodsMorans I testSpatial panel dataMonte Carlo simulationUndertheconditionofthenpanel data models isanunresmethod is used to constructCarlo simulation experimentpower. The experiment resulcould be rectied in bootstrajournal homepage: www.ell data models offer thenzhen Polytechnic, ShenzhenBased on bootstrap methodg Chenasampleortheunknowndistributederrorterm,testingforspatialdependenceind probleminspatial econometrics.In this paper, a fastdouble bootstrap(FDB)tstrap Morans I tests for Morans I test in spatial panel data models, and Montee used to prove the effectiveness from two aspects including size distortion andshow that, in asymptotic Morans I test, there is serious size distortion, whichorans I test. 2014 Elsevier B.V. All rights reserved.odellingocate/ecmodcant be applied to the spatial panel models directly. To our knowledge,under the conditions of nite samples, unknown distributed orheteroscedastic error terms, testing for spatial dependence in paneldata models is an unresolved problem in empirical studies of spatialeconometrics at present.10 T. Ren et al. / Economic Modelling 41 (2014) 914In this paper, we try to apply fast double bootstrap (FDB) methodto Morans I test in spatial panel data models, and then, Monte Carlosimulation experiments are used to prove the effectiveness of the teststatistics from size distortion and power.The rest of the paper proceeds as follows. Spatial panel modelsand Morans I test statistics are discussed in Section 2. The applicationsof the bootstrap methods are presented in Section 3.InSection 4,wereport results of Monte Carlo simulation to show size distortion andpower performance of bootstrap Morans I test. And, we will compareresults of bootstrap Morans I test with asymptotic test. Section 5contains some concluding remarks.2. Spatial panel data models and Morans I test statisticElhorst(2003,2010)extendedspatialcross-sectionaldatamodelstospatial panel models. And, they proposed spatial panel model estima-tion methods. Like the spatial cross-sectional data models, spatialpanel data models can be divided into spatial panel data lag modelsand spatial panel data error models. The expressions of two modelsare as follows:SLM : yt Wyt Xt tSEM : yt Xt t;t Wt vtt 1;2; TC261whereytisaN1vectorofadependentvariableforperiodt,isspatiallag dependence coefcient, Xtis a N K matrix of non-stochastics forperiod t, W is spatial weights matrix, is K 1 parameter vector, isspatialerrordependencecoefcient,tisaN1vectoroftheregressionerror term for period t, and vtis a N 1 vector of the remain error forperiodt.Itisassumedthat|b1,|b1,tN(0,2IN), vtN(0,v2IN).Spatial effects will be tested in spatial panel models before the spa-tial panel data models are established. At present, the most commonlyused method to test spatial dependence is Morans I test which wasproposed by Moran (1948). The test statistics developed for the cross-section were extended to the panel data model by Arbia (2005).Itsexpression is as follows:I e0WNTee0e2where I represents Morans I test statistic of a spatial panel data model,WNT=ITW isthespatialweightsmatrix,istheKroneckerproduct,and e is the residual.Anselin (1988) demonstrated when the error term does not obeythe classic distribution or heteroscedasticity, Morans I test would belapse.Thebootstrapmethodisaneffectivewaytosolveaboveproblems(Lin et al, 2009; Yang, 2011).3. Morans I test methodsBootstrapping is a method that performs inference using pseudo-datasets created by sampling from observed data (Efron, 1979). It isessentially a Monte Carlo simulation procedure. It does not need to as-sume that the error terms are independent and normally distributedand a parametric estimate of the variance of the estimate. And, it neednot provide an observational data distribution form. Therefore, thebootstrap methods are applied to non-classical error term conditions.Different bootstrap methods have been developed for differenttypes of regression. Such as residuals bootstrap (Efron, 1979), wildbootstrap (Beran, 1988), block bootstrap (Efron, 1979), pairs bootstrap(Freedman, 1981). The residuals bootstrap is used in cross-section andpanel data models (Efron, 1979). The block bootstrap is used in timeseriesmodels.Thewildbootstrapisusedtodealwithpaneldatamodelsand heteroscedasticity (Davidson and MacKinnon, 2007). The pairsbootstrap is used to dynamic model or the heteroscedastic modelwhich error term is unknown distributed.Among the bootstrap methods applied in the panel data model,Chang(2003)appliedbootstrapmethodtounitroottestsfordependentpaneldatamodel.Andtheyfoundthatbootstraptestsperformbetterinnite samples than in an asymptotic test. Cerrato and Sarantis (2007)used bootstrap methods to deal with cross-sectional dependence inpanel unit root tests of real exchange rates, their results showed thatthe statistic has good power and no size distortion for moderate andlarge samples.Godfrey(2009)suggestedthatthewildbootstrapproce-dure is well-behaved in nite samples under heteroscedasticity andmatch theperformancelessofrobusttests underclassical assumptions.However, it does not mean that bootstrap tests always perform well innite samples. Thus, Beran (1988) proposed the double bootstrap(DB). But DB tends to be very computationally demanding. Davidsonand MacKinnon (2007) developed a fast double bootstrap (FDB) basedon the double bootstrap. The FDB requires no more than about twiceas much computation as the general bootstrap, making it feasiblewhile the double bootstrap is not. This paper is the rst one which ap-plies the FDB to test the spatial dependence of the spatial panel datamodels. The most useful way to perform a bootstrap test is to calculatePvalueofbootstrap.And,MonteCarlosimulationexperimentsareusedto prove more effective than asymptotic test of the test statistics fromtwo aspects including size distortion and power.The simplest and most informative method to perform a bootstraptest is to calculate a bootstrap P value. We obtain a bootstrap P valueof the bootstrap test statistics which are more accurate than the actualtest statistic. When this P value is below the signicant value, we rejectthenullhypothesis.TheP valueofbootstrapiscalculatedbythefollow-ing expression:PC3 2 C3 min1BXBj1I C3j C16C17;1BXBj1I C3jN C16C1701A3where P* is the bootstrap P value, I() is a denote function, B representsthen bootstrap test frequency, denotes the test statistic of bootstrap.The double bootstrap which is proposed by Beran (1988) can beused to calculate P values. It should be more accurate than the generalbootstrapin theory. Therst stepin thedouble bootstrap is to generateB1rst-level bootstrap samples which are used to compute bootstraptest statistics j(j =1,2, B1). The second step is to get a second-level bootstrap DGP (data generating process) for each rst-level boot-strap sample indexed by j. Each second-level bootstrap DGP is used togenerate B2bootstrap samples that are used to calculate test statisticsjl(l =1,2, B2). In practice, the double bootstrap is very costly interms of computation. For each of B1bootstrap samples, B2+1teststatistics should be computed. Thus, the total amount of test statisticswill reach up to 1 + B1+ B1B2. The double bootstrap is costly becausewe need to generate B2s-level bootstrap samples for every rst-levelbootstrap sample. It is necessary because the distribution of the jlmay be dependent on statistic j.Davidson and MacKinnon (2007) Proposed the assumption that thejdistribution isindependentofjl.Itiscalledthefastdoublebootstraptest. And, the calculation of bootstrap test will be greatly reduced. Forthe FDB test, only one second-level bootstrap statistic j,iscomputedalong with each j. The expression of FDB P value was given byDavidson and MacKinnon (2006):pC3C3F1BXBj1I C3C3jNQC3C3B1pC3C0C1C16C174where p* is thebootstrap P value,pFC3C3is FDB P value, B is bootstraptesttimes, QB(1 p) represents the (1 p) quantile of the j.AhlgrenandAntell(2008)suggestedthattheFDBproducesafurtherimprovement in cases where the performance of the asymptotic test isunsatisfactory.Inthispaper,wecombinetheFDBmethodandotherbootstrapsam-pling methods which include wild bootstrap and cross-sectional boot-strap residuals to construct spatial panel data model Morans I teststatistic. The research of bootstrap Morans I test is as following:(1) Based on sample data (y, X), ignoring the spatial dependence,the rst step is to estimate the model (1) using OLS method.We can get a consistent parameter estimator and the residualvector yX.TheasymptoticMoransIstatisticcanbecalcu-lated by Eq. (2), and then, the P values of asymptotic test can beobtained.(2) Scalechanges.Itmeansthatthecenteroftheresidualvectortisas following:etNN1rC3t2q 1NXNC3t2q264375; k 1;2; N: 5(6) Repeat steps (2)(5) B times, the rst-level test statistic jandTable 2Heteroscedastic error term.FDB SamplingtimesNominal signicancesize 0SizeRook QueenAsy Bp Asy Bp199 0.05 0.096 0.048 0.089 0.053299 0.05 0.096 0.045 0.089 0.051399 0.05 0.096 0.050 0.089 0.050499 0.05 0.096 0.051 0.089 0.049599 0.05 0.096 0.054 0.089 0.051699 0.05 0.096 0.055 0.089 0.050799 0.05 0.096 0.052 0.089 0.051899 0.05 0.096 0.048 0.089 0.051999 0.05 0.096 0.050 0.089 0.048Table 3Normally distributed error term and samples size.Sample size(N,T)SizeRook QueenAsy Bp Asy Bp(16,10) 0.098 0.051 0.086 0.050(25,10) 0.106 0.067 0.105 0.062(36,10) 0.097 0.052 0.093 0.056(49,10) 0.096 0.050 0.089 0.050(81,10) 0.128 0.056 0.107 0.052(49,10) 0.096 0.050 0.089 0.050(49,20) 0.119 0.047 0.106 0.046(49,30) 0.149 0.055 0.137 0.059(49,40) 0.149 0.064 0.136 0.062(49,50) 0.155 0.064 0.143 0.05611T. Ren et al. / Economic Modelling 41 (2014) 9141hkl1 1hl(3) The third step is to sample vectore N times with replacement,and get a bootstrap sample e, which is a repeating randompermutations of vectore. Under the condition of e e,thestandard residuals bootstrap method is used to solve the prob-lem that the error term does not produce heteroscedasticity.When ekekC1 vC3k; k 1;2; N, the wild bootstrap methodsare used to solve the problem that the models produceheteroscedasticity. There are two forms of vk: ekekC1 vC3k; k 1;2; N:vk1; p 1.21; p 1.28:7where vkindicates symmetry wild bootstrap and asymmetricwild bootstrap.(4) Based on the residuals e, X and, we can get a new dependentvariable y1 X e. Then the new parameter estimator *andresidual vector eC3are obtained through the OLS method. Finally,the rst-level test statistic jof bootstrapMorans I, is calculatedby Eq. (2).(5) Based on the equation y2 X1 e2, the new parameters esti-mator2and residual vector e2 y2X2could be calculated.The second-level bootstrap sample Morans I test statistic isalso calculated by Eq. (2).Table 1Normally distributed error term.FDB SamplingtimesNominal signicancesize 0SizeRook QueenAsy Bp Asy Bp199 0.05 0.039 0.047 0.039 0.047299 0.05 0.039 0.044 0.039 0.049399 0.05 0.039 0.050 0.039 0.050499 0.05 0.039 0.046 0.039 0.051599 0.05 0.039 0.054 0.039 0.053699 0.05 0.039 0.048 0.039 0.048799 0.05 0.039 0.043 0.039 0.050899 0.05 0.039 0.044 0.039 0.049999 0.05 0.039 0.045 0.039 0.047the second-level test statistic j(j =1,2, B) are calculated.(7) TheP values of rst-level bootstraparecomputedbyEq.(3),andthe P values of FDB are calculated by Eq. (4).Following the above steps, we use Monte Carlo simulation to studythe validity of bootstrap Morans I test of spatial panel data modelfrom size distortion and power. When the nominal signicance level0isgiven,thesizedistortionreferstothedifferencebetweenbootstraptestleveland0.Thebootstraptestlevelisaprobabilityofthebootstraptest rejecting the null hypothesis. The formula of size distortion can beexpressed as:Sizedistortion 1MXMi1IpC3C3FIib00C08where 0is the nominal signicant level, usually 0=0.05,M areMonte Carlo experimental samples, I() is an indicator function, thevalue of I is 1 when pF(Ii) b 0,otherwise0.The power of bootstrap Morans I test of spatial panel data modelis the probability that bootstrap test rejects the null hypothesis whenspatial dependence exists. The formula of power can be expressed as:Power 1MXMi1IpC3C3FIib0C0C1: 94. Monte Carlo resultsWe apply Monte Carlo simulation to study whether FDB Morans Itest is more effective than the asymptotic test. The Monte Carloexperiments are carried out based on the following data generatingprocess:yt Wyt Xt t;t Wut vt: 10This study takes 5000 Monte Carlo experimental samples andbootstrap samples B = 199,299,999.Monte Carlo simulation experimental parameters are as follows:(1) Spatial weight matrix W is a rook matrix and queen matrix.(2) The error term t(t = 1, 2) includes standard normal error 1and heteroscedasticity error 2.(3) Dataisgeneratedbytheprocess(10),where=5,and=0.5.(4) When = = 0, we study the size distortion of Morans I test;when 0or 0, and , 0.9, 0.9, we study theThe FDB method is applied to spatial panel data model of MoransI test through Monte Carlo simulations when the error terms arenormally distributed and heteroscedastic. The steps are as follows:(1) X and are generated by Monte Carlo experiments, then thedependent variable y is obtained by y = X + . It means thatsamples (y, X) are calculated.(2) According to bootstrap Morans I test steps, the P value of FDBMorans I test pFand asymptotic Morans test piare obtained.(3) Repeating step (1) and step (2) M times, we can calculate theP value of FDB Morans I pFiC3and asymptotic Morans I pi.(4) AccordingtoEqs.(8)and(9), sizedistortionandpoweroftheas-ymptoticMoransItestarecalculated.SodoesFDBMoransItest.According to the above Monte Carlo simulation steps, we use Gauss10.0 software programming for Monte Carlo simulations.4.1. Size distortion of FDB Morans I testWe study how bootstrap simulation times, spatial weightmatrices and sample size, affect the size of FDB Morans I test.