Maximum Likelihood Estimation

  • R. Kelley Pace
Reference work entry


Maximum likelihood estimation has been the standard method employed for estimating spatial econometric models. This chapter introduces these methods, examines the specific case of a spatial error model, and provides an example based on a large data set. In addition, the chapter sets forth various solutions to the computational difficulties that arise for large data sets.


Ordinary Little Square Information Matrix Spatial Weight Matrix Spatial Error Model Observe Information Matrix 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



I would like to thank Mark Mclean, James LeSage, and Shuang Zhu for their very helpful comments.


  1. Anselin L (1988) Spatial econometrics: methods and models. Kluwer, DordrechtCrossRefGoogle Scholar
  2. Barry R, Pace RK (1999) LA Monte Carlo estimator of the log determinant of large sparse matrices, linear algebra and its applications 289(1–3):41–54CrossRefGoogle Scholar
  3. Beron KJ, Vijverberg WPM (2004) Probit in a spatial context: a Monte Carlo analysis. In: Anselin L, Florax RJGM, Rey SJ (eds) Advances in spatial econometrics: methodology, tools and applications. Springer, Berlin/Heidelberg/New York, pp 169–195CrossRefGoogle Scholar
  4. Bivand R (2010) Computing the Jacobian in spatial models: an applied survey (17 Aug 2010). NHH Department of Economics Discussion Paper No. 20/2010. Available at SSRN: or
  5. Burridge P (2012) A research agenda on general-to-specific spatial model search. Invest Reg 21:71–90Google Scholar
  6. Chen J, Jennrich R (1996) The signed root deviance profile and confidence intervals in maximum likelihood analysis. J Am Stat Assoc 91(435):993–998CrossRefGoogle Scholar
  7. Cramer JS (1986) Econometric applications of maximum likelihood methods. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  8. Davidson R, MacKinnon J (2004) Econometric theory and methods. Oxford University Press, New YorkGoogle Scholar
  9. George A, Liu J (1981) Computer solution of large sparse positive definite systems. Prentice-Hall, Englewood CliffsGoogle Scholar
  10. Geweke J (1991) Efficient simulation from the multivariate normal and student-t distributions subject to linear constraints. In: Computer science and statistics: proceedings of the twenty-third symposium on the interface. American Statistical Association, Alexandria, pp 571–578Google Scholar
  11. Griffith D (1989) Advanced spatial statistics. Kluwer, DordrechtGoogle Scholar
  12. Griffith D (2004) Faster maximum likelihood estimation of very large spatial autoregressive models: an extension of the Smirnov-Anselin result. J Stat Comput Simul 74(12):855–866CrossRefGoogle Scholar
  13. Haining R (1990) Spatial data analysis in the social and environmental sciences. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  14. Hajivassiliou V, McFadden D (1990) The method of simulated scores for the estimation of LDV models with an application to external debt crises. Cowles Foundation Discussion Paper 967, Yale UniversityGoogle Scholar
  15. Keane M (1994) A computationally practical simulation estimator for panel data. Econometrica 62(1):95–116CrossRefGoogle Scholar
  16. LeSage JP, Pace RK (2007) A matrix exponential spatial specification. J Econ 140(1):190–214CrossRefGoogle Scholar
  17. LeSage J, Pace RK (2009) Introduction to spatial econometrics. Taylor and Francis/CRC, Boca RatonCrossRefGoogle Scholar
  18. Mardia KV, Marshall RJ (1984) Maximum likelihood estimation of models for residual covariance in spatial regression. Biometrika 71(1):135–146CrossRefGoogle Scholar
  19. Martin RJ (1993) Approximations to the determinant term in Gaussian maximum likelihood estimation of some spatial models. Commun Stat Theory Methods 22(1):189–205CrossRefGoogle Scholar
  20. Ord JK (1975) Estimation methods for models of spatial interaction. J Am Stat Assoc 70(1):120–126CrossRefGoogle Scholar
  21. Pace RK, Barry RP (1997) Quick computation of spatial autoregressive estimators. Geogr Anal 29(3):232–246CrossRefGoogle Scholar
  22. Pace RK, LeSage JP (2002) Semiparametric maximum likelihood estimates of spatial dependence. Geogr Anal 34(1):76–90Google Scholar
  23. Pace RK, LeSage JP (2004) Chebyshev approximation of log-determinants of spatial weight matrices. Comput Stat Data Anal 45(1):179–196CrossRefGoogle Scholar
  24. Pace RK, LeSage J (2009) A sampling approach to estimating the log determinant used in spatial likelihood problems. J Geogr Syst 11(3):209–225CrossRefGoogle Scholar
  25. Pace RK, LeSage J (2011) Fast simulated maximum likelihood estimation of the spatial probit model capable of handling large samples. Available at SSRN: or
  26. Pace RK, Zhu S (2012) Separable spatial modelling of spillovers and dependence. J Geogr Syst 14(1):75–90CrossRefGoogle Scholar
  27. Pace RK, Zou D (2000) Closed-form maximum likelihood estimates of nearest neighbor spatial dependence. Geogr Anal 32(2):154–172CrossRefGoogle Scholar
  28. Phinikettos I, Gandy A (2011) Fast computation of high-dimensional multivariate normal probabilities. Comput Stat Data Anal 55(4):1521–1529CrossRefGoogle Scholar
  29. Smirnov O, Anselin L (2001) Fast maximum likelihood estimation of very large spatial autoregressive models: a characteristic polynomial approach. Comput Stat Data Anal 35(8):301–319CrossRefGoogle Scholar
  30. Walde J, Larch M, Tappeiner G (2008) Performance contest between MLE and GMM for huge spatial autoregressive models. J Stat Comput Simul 78(2):151–166CrossRefGoogle Scholar
  31. Zhang Y, Leithead WE, Leithead DJ (2007) Approximate implementation of logarithm of matrix determinant in Gaussian processes. J Stat Comput Simul 77(4):329–348CrossRefGoogle Scholar
  32. Zhu S, Pace RK Spatially interdependent mortgage decisions. J Real Estate Fin Econ (forthcoming)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Department of Finance, E.J. Ourso College of Business AdministrationLouisiana State UniversityBaton RougeUSA

Personalised recommendations