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Modified whittle estimation of multilateral spatial models

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  • Peter Robinson

    () (Institute for Fiscal Studies and London School of Economics)

  • J. Vidal Sanz Vidal Sanz

    (Institute for Fiscal Studies)

Abstract

We consider the estimation of parametric models for stationary spatial or spatio-temporal data on a d-dimensional lattice, for d = 2. The achievement of asymptotic efficiency under Gaussianity, and asymptotic normality more generally, with standard convergence rate, faces two obstacles. One is the 'edge effect', which worsens with increasing d. The other is the difficulty of computing a continuous-frequency form of Whittle estimate or a time domain Gaussian maximum likelihood estimate, especially in case of multilateral models, due mainly to the Jacobian term. An extension of the discrete-frequency Whittle estimate from the time series literature deals conveniently with the latter problem, but when subjected to a standard device for avoiding the edge effect has disastrous asymptotic performance, along with finite sample numerical drawbacks, the objective function lacking a minimum-distance interpretation and losing any global convexity properties. We overcome these problems by first optimizing a standard, guaranteed non-negative, discrete-frequency, Whittle function, without edge-effect correction, providing an estimate with a slow convergence rate, then improving this by a sequence of computationally convenient approximate Newton iterations using a modified, almost-unbiased periodogram, the desired asymptotic properties being achieved after finitely many steps. A Monte Carlo study of finite sample behaviour is included. The asymptotic regime allows increase in both directions, unlike the usual random fields formulation, with the central limit theorem established after re-ordering as a triangular array. When the data are non-Gaussian, the asymptotic variances of all parameter estimates are likely to be affected, and we provide a consistent, non-negative definite, estimate of the asymptotic variance matrix.

Suggested Citation

  • Peter Robinson & J. Vidal Sanz Vidal Sanz, 2003. "Modified whittle estimation of multilateral spatial models," CeMMAP working papers CWP18/03, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
  • Handle: RePEc:ifs:cemmap:18/03
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    File URL: http://cemmap.ifs.org.uk/wps/cwp0318.pdf
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    References listed on IDEAS

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    1. Robinson, Peter M, 1988. "The Stochastic Difference between Econometric Statistics," Econometrica, Econometric Society, vol. 56(3), pages 531-548, May.
    2. Hannan, E. J. & Dunsmuir, W. T. M. & Deistler, M., 1980. "Estimation of vector ARMAX models," Journal of Multivariate Analysis, Elsevier, vol. 10(3), pages 275-295, September.
    3. Korezlioglu, Hayri & Loubaton, Philippe, 1986. "Spectral factorization of wide sense stationary processes on 2," Journal of Multivariate Analysis, Elsevier, vol. 19(1), pages 24-47, June.
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    Cited by:

    1. Conley, Timothy G. & Molinari, Francesca, 2007. "Spatial correlation robust inference with errors in location or distance," Journal of Econometrics, Elsevier, vol. 140(1), pages 76-96, September.

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