Empirical Economics

, Volume 55, Issue 1, pp 85–111 | Cite as

Testing spatial regression models under nonregular conditions

  • Sheena Yu-Hsien Kao
  • Anil K. Bera


In time series context, estimation and testing issues with autoregressive and moving average (ARMA) models are well understood. Similar issues in the context of spatial ARMA models for the disturbance of the regression, however, remain largely unexplored. In this paper, we discuss the problems of testing no spatial dependence in the disturbances against the alternative of spatial ARMA process incorporating the possible presence of spatial dependence in the dependent variable. The problems of conducting such a test are twofold. First, under the null hypothesis, the nuisance parameter is not identified, resulting in a singular information matrix (IM), which is a nonregular case in statistical inference. To take account of singular IM, we follow Davies (Biometrika 64(2):247–254, 1977; Biometrika 74(1):33–43, 1987) and propose a test procedure based on the supremum of the Rao score test statistic. Second, the possible presence of spatial lag dependence will have adverse effect on the performance of the test. Using the general test procedure of Bera and Yoon (Econom Theory 9:649–658, 1993) under local misspecification, we avoid the explicit estimation of the spatial autoregressive parameter. Thus our suggested tests are entirely based on ordinary least squares estimation. Tests suggested here can be viewed as a generalization of Anselin et al. (Reg Sci Urban Econ 26:77–104, 1996). We conduct Monte Carlo simulations to investigate the finite sample properties of the proposed tests. Simulation results show that our tests have good finite sample properties both in terms of size and power, compared to other tests in the literature. We also illustrate the applications of our tests through several data sets.


Spatial dependence Spatial ARMA Rao’s score test Davies test Nuisance parameter 

JEL Classification

C01 C12 C31 


  1. Anselin L (1988) Spatial econometrics: methods and models. Kluwer Academic Publishers, Dordrecht, The NetherlandsCrossRefGoogle Scholar
  2. Anselin L (2003) Spatial externalities, spatial multipliers, and spatial econometrics. International Regional Science Review 26(2):153–166CrossRefGoogle Scholar
  3. Anselin L, Bera AK, Florax R, Yoon MJ (1996) Simple diagnostic tests for spatial dependence. Regional Science and Urban Economics 26:77–104CrossRefGoogle Scholar
  4. Baltagi B, Liu L (2011) An improved generalized moments estimator for a spatial moving average error model. Econ Lett 113:282–284CrossRefGoogle Scholar
  5. Behrens K, Ertur C, Koch W (2012) ‘Dual’ gravity: Using spatial econometrics to control for multilateral resistance. J Appl Econom 27:773–794CrossRefGoogle Scholar
  6. Bera AK, Yoon MJ (1993) Specification testing with locally misspecified alternatives. Econom Theory 9:649–658CrossRefGoogle Scholar
  7. Davidson R, MacKinnon JG (1987) Implicit alternatives and the local power of test statistics. Econometrica 55:1305–1329CrossRefGoogle Scholar
  8. Davies RB (1977) Hypothesis testing when a nuisance parameter is present only under the alternative. Biometrika 64(2):247–254CrossRefGoogle Scholar
  9. Davies RB (1987) Hypothesis testing when a nuisance parameter is present only under the alternative. Biometrika 74(1):33–43Google Scholar
  10. Fingleton B (2008) A generalized method of moments estimator for a spatial model with moving average errors, with application to real estate prices. Empiri Econ 34:35–57CrossRefGoogle Scholar
  11. Florax RJGM (1992) The University: A Regional Booster? : Economic Impacts of Academic Knowledge Infrastructure, Aldershot, Hants, England and Brookfield. Vt, USAGoogle Scholar
  12. Harrison D, Rubinfeld DL (1978) Hedonic housing prices and the demand for clean air. J Environ Econ Manag 5:81–102CrossRefGoogle Scholar
  13. Kelejian HH, Prucha IR (2001) On the asymptotic distribution of the Moran I test statistic with applications. J Econom 104(2):219–257CrossRefGoogle Scholar
  14. Lam C, Souza PCL (2013) Regularization for spatial panel time series using adaptive Lasso. Mimeo, New YorkGoogle Scholar
  15. Pace RK, Gilley OW (1997) Using the spatial configuration of the data to improve estimation. J Real Estate Financ Econ 14:333–340CrossRefGoogle Scholar
  16. Poskitt DS, Tremayne AR (1980) Testing the specification of a fitted autoregressive-moving average model. Biometrika 67(2):359–363CrossRefGoogle Scholar
  17. Saikkonen P (1989) Asymptotic relative efficiency of the classical tests under misspecification. J Econom 42:351–369CrossRefGoogle Scholar
  18. Sen M, Bera AK, Kao YH (2012) A Hausman test for spatial regression model. In: Baltagi BH, CR Hill, Newey WK, White HL (eds) , vol 29. Advances in econometrics. Essays in honor of Jerry Hausman. Emerald Group Publishing Limited, Bingley, pp 547–559Google Scholar
  19. Sharpe K (1978) Some properties of the crossings process generated by a stationary chi-squared process. Adv Appl Probab 10:373–391CrossRefGoogle Scholar
  20. Silvey SD (1959) The lagrange multiplier test. Ann Math Stat 30:389–407CrossRefGoogle Scholar
  21. Yao Q, Brockwell PJ (2005) Gaussian maximum likelihood estimation for ARMA models I: time series. J Time Ser Anal 27(6):857–875CrossRefGoogle Scholar
  22. Yao Q, Brockwell PJ (2006) Gaussian maximum likelihood estimation for ARMA models II: spatial processes. Bernoulli 12(3):403–429CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of EconomicsUniversity of Illinois at Urbana-ChampaignChampaignUSA

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