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Exact Properties of the Maximum Likelihood Estimator in Spatial Autoregressive Models

Author

Listed:
  • Grant Hillier

    (CeMMAP and University of Southampton)

  • Federico Martellosio

    (University of Surrey)

Abstract

The (quasi-) maximum likelihood estimator (QMLE) 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 in terms of that of a particular quadratic form. A number of immediate consequences of this result are discussed, and some examples are analyzed 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.

Suggested Citation

  • Grant Hillier & Federico Martellosio, 2016. "Exact Properties of the Maximum Likelihood Estimator in Spatial Autoregressive Models," School of Economics Discussion Papers 0716, School of Economics, University of Surrey.
    • Handle: RePEc:sur:surrec:0716
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    File URL: https://repec.som.surrey.ac.uk/2016/DP07-16.pdf
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    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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