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Properties of the maximum likelihood estimator in spatial autoregressive models

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  • Grant Hillier

    () (Institute for Fiscal Studies and University of Southampton)

  • Federico Martellosio

    (Institute for Fiscal Studies)

Abstract

The (quasi-) maximum likelihood estimator (MLE) for the autoregressive parameter in a spatial autoregressive model cannot in general be written explicitly in terms of the data. The only known properties of the estimator have hitherto been its first-order asymptotic properties (Lee, 2004, Econometrica), derived under specific assumptions on the evolution of the spatial weights matrix involved. In this paper we show that the exact cumulative distribution function of the estimator can, under mild assumptions, be written down explicitly. A number of immediate consequences of the main result are discussed, and several examples of theoretical and practical interest are analysed in detail. The examples are of interest in their own right, but also serve to illustrate some unexpected features of the distribution of the MLE. In particular, we show that the distribution of the MLE may not be supported on the entire parameter space, and may be nonanalytic at some points in its support. Supplementary material relating to this working paper can be viewed here

Suggested Citation

  • Grant Hillier & Federico Martellosio, 2013. "Properties of the maximum likelihood estimator in spatial autoregressive models," CeMMAP working papers CWP44/13, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
  • Handle: RePEc:ifs:cemmap:44/13
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    File URL: http://www.cemmap.ac.uk/wps/cwp441313.pdf
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    1. Besley, Timothy & Case, Anne, 1995. "Incumbent Behavior: Vote-Seeking, Tax-Setting, and Yardstick Competition," American Economic Review, American Economic Association, vol. 85(1), pages 25-45, March.
    2. Göran Therborn & K.C. Ho, 2009. "Introduction," City, Taylor & Francis Journals, vol. 13(1), pages 53-62, March.
    3. Bao, Yong, 2013. "Finite-Sample Bias Of The Qmle In Spatial Autoregressive Models," Econometric Theory, Cambridge University Press, vol. 29(01), pages 68-88, February.
    4. Hillier, Grant & Kan, Raymond & Wang, Xiaolu, 2009. "Computationally Efficient Recursions For Top-Order Invariant Polynomials With Applications," Econometric Theory, Cambridge University Press, vol. 25(01), pages 211-242, February.
    5. Forchini, G., 2002. "The Exact Cumulative Distribution Function Of A Ratio Of Quadratic Forms In Normal Variables, With Application To The Ar(1) Model," Econometric Theory, Cambridge University Press, vol. 18(04), pages 823-852, August.
    6. Audretsch, David B & Feldman, Maryann P, 1996. "R&D Spillovers and the Geography of Innovation and Production," American Economic Review, American Economic Association, vol. 86(3), pages 630-640, June.
    7. Yang, Zhenlin, 2015. "A general method for third-order bias and variance corrections on a nonlinear estimator," Journal of Econometrics, Elsevier, vol. 186(1), pages 178-200.
    8. Giorgio Topa, 2001. "Social Interactions, Local Spillovers and Unemployment," Review of Economic Studies, Oxford University Press, vol. 68(2), pages 261-295.
    9. Martellosio, Federico, 2011. "Efficiency of the OLS estimator in the vicinity of a spatial unit root," Statistics & Probability Letters, Elsevier, vol. 81(8), pages 1285-1291, August.
    10. Case, Anne C, 1991. "Spatial Patterns in Household Demand," Econometrica, Econometric Society, vol. 59(4), pages 953-965, July.
    11. Joris Pinkse & Margaret E. Slade & Craig Brett, 2002. "Spatial Price Competition: A Semiparametric Approach," Econometrica, Econometric Society, vol. 70(3), pages 1111-1153, May.
    12. Giovanni Forchini, "undated". "The Distribution of a Ratio of Quadratic Forms in Noncentral Normal Variables," Discussion Papers 01/12, Department of Economics, University of York.
    13. Lung-Fei Lee, 2004. "Asymptotic Distributions of Quasi-Maximum Likelihood Estimators for Spatial Autoregressive Models," Econometrica, Econometric Society, vol. 72(6), pages 1899-1925, November.
    14. Hillier, G.H., 1999. "The density of a quadratic form in a vector uniformly distributed on the n-sphere," Discussion Paper Series In Economics And Econometrics 9902, Economics Division, School of Social Sciences, University of Southampton.
    15. J. Barkley Rosser, 2009. "Introduction," Chapters,in: Handbook of Research on Complexity, chapter 1 Edward Elgar Publishing.
    16. Harry H. Kelejian & Ingmar R. Prucha & Yevgeny Yuzefovich, 2006. "Estimation Problems In Models With Spatial Weighting Matrices Which Have Blocks Of Equal Elements," Journal of Regional Science, Wiley Blackwell, vol. 46(3), pages 507-515.
    17. Hillier, Grant, 2001. "THE DENSITY OF A QUADRATIC FORM IN A VECTOR UNIFORMLY DISTRIBUTED ON THE n-SPHERE," Econometric Theory, Cambridge University Press, vol. 17(01), pages 1-28, February.
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    Cited by:

    1. repec:eee:econom:v:202:y:2018:i:1:p:92-107 is not listed on IDEAS
    2. Blasques, Francisco & Koopman, Siem Jan & Lucas, Andre & Schaumburg, Julia, 2016. "Spillover dynamics for systemic risk measurement using spatial financial time series models," Journal of Econometrics, Elsevier, vol. 195(2), pages 211-223.
    3. Gupta, A, 2015. "Estimation of Spatial Autoregressions with Stochastic Weight Matrices," Economics Discussion Papers 15617, University of Essex, Department of Economics.
    4. Gupta, Abhimanyu & Robinson, Peter M., 2018. "Pseudo maximum likelihood estimation of spatial autoregressive models with increasing dimension," Journal of Econometrics, Elsevier, vol. 202(1), pages 92-107.
    5. repec:esx:essedp:772 is not listed on IDEAS

    More about this item

    Keywords

    spatial autoregression; maximum likelihood estimation; group interaction; networks; complete bipartite graph;

    JEL classification:

    • C12 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Hypothesis Testing: General
    • C21 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Cross-Sectional Models; Spatial Models; Treatment Effect Models

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