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Properties of the maximum likelihood estimator in spacial autoregressive models

  • Grant Hillier

    ()

    (Institute for Fiscal Studies and University of Southampton)

  • Federico Martellosio
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    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

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    File URL: http://www.cemmap.ac.uk/wps/cwp441313.pdf
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    Paper provided by Centre for Microdata Methods and Practice, Institute for Fiscal Studies in its series CeMMAP working papers with number CWP44/13.

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    Date of creation: Sep 2013
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    Handle: RePEc:ifs:cemmap:44/13
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    1. Case, Anne C, 1991. "Spatial Patterns in Household Demand," Econometrica, Econometric Society, vol. 59(4), pages 953-65, July.
    2. 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.
    3. 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.
    4. Joris Pinkse & Margaret E. Slade & Craig Brett, 2002. "Spatial Price Competition: A Semiparametric Approach," Econometrica, Econometric Society, vol. 70(3), pages 1111-1153, May.
    5. 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.
    6. J. Barkley Rosser, 2009. "Introduction," Chapters, in: Handbook of Research on Complexity, chapter 1 Edward Elgar.
    7. 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.
    8. 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.
    9. 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.
    10. G�ran Therborn & K.C. Ho, 2009. "Introduction," City, Taylor & Francis Journals, vol. 13(1), pages 53-62, March.
    11. 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.
    12. Topa, Giorgio, 1997. "Social Interactions, Local Spillovers and Unemployment," Working Papers 97-17, C.V. Starr Center for Applied Economics, New York University.
    13. 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 630-40, June.
    14. Giovanni Forchini, . "The Exact Cumulative Distribution Function of a Ratio of Quadratic Forms in Normal Variables with Application to the AR(1) Model," Discussion Papers 01/02, Department of Economics, University of York.
    15. Giovanni Forchini, . "The Distribution of a Ratio of Quadratic Forms in Noncentral Normal Variables," Discussion Papers 01/12, Department of Economics, University of York.
    16. 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.
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