Bootstrap LM Tests for Spatial Dependence in Panel Data Models with Fixed Effects
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摘要
This paper applies bootstrap methods to LM tests(including LM-lag test and LM-error test) for spatial dependence in panel data models with fixed effects, and removes fixed effects based on orthogonal transformation method proposed by Lee and Yu(2010). The consistencies of LM tests and their bootstrap versions are proved, and then some asymptotic refinements of bootstrap LM tests are obtained. It shows that the convergence rate of bootstrap LM tests is O((N T)~(-2)) and that of fast double bootstrap LM tests is O((NT)~(-5/2)). Extensive Monte Carlo experiments suggest that,compared to aysmptotic LM tests, the size of bootstrap LM tests gets closer to the nominal level of signifiance, and the power of bootstrap LM tests is higher, especially in the cases with small spatial correlation. Moreover, when the error is not normal or with heteroskedastic, asymptotic LM tests suffer from severe size distortion, but the size of bootstrap LM tests is close to the nominal significance level.Bootstrap LM tests are superior to aysmptotic LM tests in terms of size and power.
This paper applies bootstrap methods to LM tests(including LM-lag test and LM-error test) for spatial dependence in panel data models with fixed effects, and removes fixed effects based on orthogonal transformation method proposed by Lee and Yu(2010). The consistencies of LM tests and their bootstrap versions are proved, and then some asymptotic refinements of bootstrap LM tests are obtained. It shows that the convergence rate of bootstrap LM tests is O((N T)~(-2)) and that of fast double bootstrap LM tests is O((NT)~(-5/2)). Extensive Monte Carlo experiments suggest that,compared to aysmptotic LM tests, the size of bootstrap LM tests gets closer to the nominal level of signifiance, and the power of bootstrap LM tests is higher, especially in the cases with small spatial correlation. Moreover, when the error is not normal or with heteroskedastic, asymptotic LM tests suffer from severe size distortion, but the size of bootstrap LM tests is close to the nominal significance level.Bootstrap LM tests are superior to aysmptotic LM tests in terms of size and power.
This paper applies bootstrap methods to LM tests(including LM-lag test and LM-error test) for spatial dependence in panel data models with fixed effects, and removes fixed effects based on orthogonal transformation method proposed by Lee and Yu(2010). The consistencies of LM tests and their bootstrap versions are proved, and then some asymptotic refinements of bootstrap LM tests are obtained. It shows that the convergence rate of bootstrap LM tests is O((N T)~(-2)) and that of fast double bootstrap LM tests is O((NT)~(-5/2)). Extensive Monte Carlo experiments suggest that,compared to aysmptotic LM tests, the size of bootstrap LM tests gets closer to the nominal level of signifiance, and the power of bootstrap LM tests is higher, especially in the cases with small spatial correlation. Moreover, when the error is not normal or with heteroskedastic, asymptotic LM tests suffer from severe size distortion, but the size of bootstrap LM tests is close to the nominal significance level.Bootstrap LM tests are superior to aysmptotic LM tests in terms of size and power.
引文
[1]Horowitz J L.The bootstrap.Eds.by Heckman J J and Leamer E.Handbook of Econometrics,Elsevier Science B V,2001.
[2]Beran R.Prepivoting test statistics:A bootstrap view of asymptotic refinements.Journal of the American Statistical Association,1988,83:687-697.
[3]Hall P.The bootstrap and edgeworth expansion.Springer-Verlag,1992.
[4]Horowitz J L.Bootstrap-based critical values for the information matrix test.Journal of Econometrics,1994,61:395-411.
[5]Horowitz J L.Bootstrap methods in econometrics:Theory and numericalperformance.Eds.by Kreps D M,Wallis K F.Advances in Economics and Econometrics:Theory and Applications:Seventh World Congress.Cambridge University Press,Cambridge,1997.
[6]Hall P,Horowitz J L.Bootstrap critical values for tests based on generalizedmethodsof moments estimators.Econometrica,1996,4:891-916.
[7]Cameron A C,Trivedi P K.Microeconometrics methods and applications.Cambridge University Press,2005.
[8]Godfrey L.Bootstrap tests for regression models.Palgrave Macmillan,2009.
[9]Davidson,Mac Kinnon J G.The size distortion of bootstrap tests.Econometric Theory,1999,15:361-376.
[10]Davidson R,Mac Kinnon J G.The power of bootstrap and asymptotic tests.Journal of Econometrics,2006,133:421-441.
[11]Davidson R,Mac Kinnon J G.Fast double bootstrap tests of nonnested linear regression models.Econometric Reviews,2002,21:419-429.
[12]Lin K P,Long Z H,Ou B L.The size and power of bootstrap tests for spatialdependence in a linear regression model.Computational Economics,2011,38:153-171.
[13]Long Z H,Ou B L,Lin K P.Tests for patial dependence applying bootstrap methods:Mathematics derivation and simulation analysis.Beijing:Science Press,2012,(in Chinese).
[14]Ou B L,Long Z H,Lin K P.Size distortion of bootstrap Moran diagnostic testing for spatial autoregressive models.Systems Engineering,2009,27(8):69-73(in Chinese).
[15]Anselin L.Spatial econometrics:Methods and models.Kluwar Academic Publishing,1998.
[16]Anselin L.Some robust approaches to testing and estimation in spatial econometrics.Regional Science Urban Economics,1990,20:141-163.
[17]Burridge P,Fingleton B.Bootstrap inference in spatial econometrics:The J-test.Spatial Economic Analysis,2010,5:93-119.
[18]Fingleton B,Palombi S.Bootstrap J-test for panel data models with spatially dependent error components,a spatial lag and additional endogenous variables.Spatial Economic Analysis,2015,11(1):1-20.
[19]Su L J,Yang Z L.Asymptotics and bootstrap for transformed panel data regressions.Working paper,2009.
[20]Yang Z L.LM tests of spatial dependence based on bootstrap critical values.Journal of Econometrics,2015,185:33-59.
[21]Monchuk D C,Hayes D J,Miranowski J A,et al.Inference basedon alternative bootstrapping methods in spatial models with an application tocounty income growth in the United States.Journal of Regional Science,2011,51:880-896.
[22]Anselin L,Bera A K,Florax R,et al.Simple diagnostic tests for spatial dependence.Regional Science Urban Economics,1996,26:77-104.
[23]Kelejian H H,Robinson D P.A suggested test for spatial autocorrelation and/or heteroskedasticity and corresponding Monte Carlo results.Regional Science and Urban Economics,1998,28(4):389-417.
[24]Debarsy N,Ertur C.Testing for spatial autocorrelation in a fixed effects panel data model.Regional Science and Urban Economics,2010,40:453-470.
[25]Long Z H,Li W L,Chen Q Q.Bootstrap LM-error test of spatial correlation in fixed effects model.The Journal of Quantitative and Technical Economics,2015,8:149-160(in Chinese).
[26]Ren T X,Long Z H,Chen Q Q.LM-Error test of spatial panel data model based on bootstrap method.Statistical Research,2015,32(5):91-96(in Chinese).
[27]Jin F,Lee L F.On the bootstrap for Moran’s I test for spatial dependence.Journal of Econometrics,2015,184:295-314.
[28]Elhorst J P.Spatial panel data models.Eds.by Fischer M M,Getis A.Handbook of Applied Spatial Analysis,Springer:Berlin Heidelberg,New York,2010.
[29]Anselin L,et al.Spatial panel econometrics.Eds.by Matyas L,Sevestre P.The Econometrics of panel data,Springer-Verlag,2008.
[30]Burridge P.On the Cliff-Ord test for spatial autocorrelation.Journal of the Royal Statistical Society B,1980,42:107-108.
[31]Lee L F,Yu J H.Estimation of spatial autoregressive panel data models with fixed effects.Journal of Econometrics,2010,154:165-185.
[32]Ahlgren N,Antell J.Bootstrap and fast double bootstrap tests for cointegration rank with financial time series.Computational Statistics and Data Analysis,2008,52:4754-4767.
[33]Arasan J,Adam M B.Double bootstrap confidence interval estimates with censored and truncated data.Journal of Modem Applied Statistical Methods,2014,13:399-419.
[34]Davidson R,Mac Kinnon J G.Improving the reliability of bootstrap tests with the fast double bootstrap.Computational Statistics and Data Analysis,2007,51:3259-3281.
[35]Davidson R,Mac Kinnon J G.Diagnostics for the bootstrap and fast double bootstrap.Econometric Reviews,2017,36:6-9.
[36]Kelejian H H,Prucha I R.On the asymptotic distribution of the Moran I teststatistic with applications.Journal of Econometrics,2001,104:219-257.
[37]Lee L F,Yu J H.Estimation of spatial panels.Foundations and Trends in Econometrics,2011,4(1-2):1-164.
[38]Lee L F,Yu J H.Spatial panels:Random components versus fixed effects.International Economic Review,2012,53:1369-1412.
[39]Mammen E.When does bootstrap work?Asymptotic results and simulations.Springer-Verlag,1992.
[40]Kelejian H H,Prucha I R.A generalized moments estimator for the autoregressive parameter in a spatial model.Internaltional Economic Review,1999,40(2):509-533.
1 Similarly,Jin and Lee[27]proved the consistency of bootstrap Moran’s I tests in cross sectional data.
2 The expression of is Pj,1(x)similar to Equation(2.24)and Equation(2.25)in Page 48(Hall,1992)[3],and no repeat here.
3 Even the original test statistic is not asymptotically pivotal,fast double bootstrap tests also can obtain asymptotic refinements.
[2]Beran R.Prepivoting test statistics:A bootstrap view of asymptotic refinements.Journal of the American Statistical Association,1988,83:687-697.
[3]Hall P.The bootstrap and edgeworth expansion.Springer-Verlag,1992.
[4]Horowitz J L.Bootstrap-based critical values for the information matrix test.Journal of Econometrics,1994,61:395-411.
[5]Horowitz J L.Bootstrap methods in econometrics:Theory and numericalperformance.Eds.by Kreps D M,Wallis K F.Advances in Economics and Econometrics:Theory and Applications:Seventh World Congress.Cambridge University Press,Cambridge,1997.
[6]Hall P,Horowitz J L.Bootstrap critical values for tests based on generalizedmethodsof moments estimators.Econometrica,1996,4:891-916.
[7]Cameron A C,Trivedi P K.Microeconometrics methods and applications.Cambridge University Press,2005.
[8]Godfrey L.Bootstrap tests for regression models.Palgrave Macmillan,2009.
[9]Davidson,Mac Kinnon J G.The size distortion of bootstrap tests.Econometric Theory,1999,15:361-376.
[10]Davidson R,Mac Kinnon J G.The power of bootstrap and asymptotic tests.Journal of Econometrics,2006,133:421-441.
[11]Davidson R,Mac Kinnon J G.Fast double bootstrap tests of nonnested linear regression models.Econometric Reviews,2002,21:419-429.
[12]Lin K P,Long Z H,Ou B L.The size and power of bootstrap tests for spatialdependence in a linear regression model.Computational Economics,2011,38:153-171.
[13]Long Z H,Ou B L,Lin K P.Tests for patial dependence applying bootstrap methods:Mathematics derivation and simulation analysis.Beijing:Science Press,2012,(in Chinese).
[14]Ou B L,Long Z H,Lin K P.Size distortion of bootstrap Moran diagnostic testing for spatial autoregressive models.Systems Engineering,2009,27(8):69-73(in Chinese).
[15]Anselin L.Spatial econometrics:Methods and models.Kluwar Academic Publishing,1998.
[16]Anselin L.Some robust approaches to testing and estimation in spatial econometrics.Regional Science Urban Economics,1990,20:141-163.
[17]Burridge P,Fingleton B.Bootstrap inference in spatial econometrics:The J-test.Spatial Economic Analysis,2010,5:93-119.
[18]Fingleton B,Palombi S.Bootstrap J-test for panel data models with spatially dependent error components,a spatial lag and additional endogenous variables.Spatial Economic Analysis,2015,11(1):1-20.
[19]Su L J,Yang Z L.Asymptotics and bootstrap for transformed panel data regressions.Working paper,2009.
[20]Yang Z L.LM tests of spatial dependence based on bootstrap critical values.Journal of Econometrics,2015,185:33-59.
[21]Monchuk D C,Hayes D J,Miranowski J A,et al.Inference basedon alternative bootstrapping methods in spatial models with an application tocounty income growth in the United States.Journal of Regional Science,2011,51:880-896.
[22]Anselin L,Bera A K,Florax R,et al.Simple diagnostic tests for spatial dependence.Regional Science Urban Economics,1996,26:77-104.
[23]Kelejian H H,Robinson D P.A suggested test for spatial autocorrelation and/or heteroskedasticity and corresponding Monte Carlo results.Regional Science and Urban Economics,1998,28(4):389-417.
[24]Debarsy N,Ertur C.Testing for spatial autocorrelation in a fixed effects panel data model.Regional Science and Urban Economics,2010,40:453-470.
[25]Long Z H,Li W L,Chen Q Q.Bootstrap LM-error test of spatial correlation in fixed effects model.The Journal of Quantitative and Technical Economics,2015,8:149-160(in Chinese).
[26]Ren T X,Long Z H,Chen Q Q.LM-Error test of spatial panel data model based on bootstrap method.Statistical Research,2015,32(5):91-96(in Chinese).
[27]Jin F,Lee L F.On the bootstrap for Moran’s I test for spatial dependence.Journal of Econometrics,2015,184:295-314.
[28]Elhorst J P.Spatial panel data models.Eds.by Fischer M M,Getis A.Handbook of Applied Spatial Analysis,Springer:Berlin Heidelberg,New York,2010.
[29]Anselin L,et al.Spatial panel econometrics.Eds.by Matyas L,Sevestre P.The Econometrics of panel data,Springer-Verlag,2008.
[30]Burridge P.On the Cliff-Ord test for spatial autocorrelation.Journal of the Royal Statistical Society B,1980,42:107-108.
[31]Lee L F,Yu J H.Estimation of spatial autoregressive panel data models with fixed effects.Journal of Econometrics,2010,154:165-185.
[32]Ahlgren N,Antell J.Bootstrap and fast double bootstrap tests for cointegration rank with financial time series.Computational Statistics and Data Analysis,2008,52:4754-4767.
[33]Arasan J,Adam M B.Double bootstrap confidence interval estimates with censored and truncated data.Journal of Modem Applied Statistical Methods,2014,13:399-419.
[34]Davidson R,Mac Kinnon J G.Improving the reliability of bootstrap tests with the fast double bootstrap.Computational Statistics and Data Analysis,2007,51:3259-3281.
[35]Davidson R,Mac Kinnon J G.Diagnostics for the bootstrap and fast double bootstrap.Econometric Reviews,2017,36:6-9.
[36]Kelejian H H,Prucha I R.On the asymptotic distribution of the Moran I teststatistic with applications.Journal of Econometrics,2001,104:219-257.
[37]Lee L F,Yu J H.Estimation of spatial panels.Foundations and Trends in Econometrics,2011,4(1-2):1-164.
[38]Lee L F,Yu J H.Spatial panels:Random components versus fixed effects.International Economic Review,2012,53:1369-1412.
[39]Mammen E.When does bootstrap work?Asymptotic results and simulations.Springer-Verlag,1992.
[40]Kelejian H H,Prucha I R.A generalized moments estimator for the autoregressive parameter in a spatial model.Internaltional Economic Review,1999,40(2):509-533.
1 Similarly,Jin and Lee[27]proved the consistency of bootstrap Moran’s I tests in cross sectional data.
2 The expression of is Pj,1(x)similar to Equation(2.24)and Equation(2.25)in Page 48(Hall,1992)[3],and no repeat here.
3 Even the original test statistic is not asymptotically pivotal,fast double bootstrap tests also can obtain asymptotic refinements.