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Modified Whittle Estimation of Multilateral Models on a Lattice

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  • Peter M Robinson
  • J Vidal Sanz

Abstract

In 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 possible difficulty of computing a continuous-frequency form of Whittle estimate or a time domain Gaussian maximum likelihood estimate, due mainly to the Jacobian term. This is especially a problem in "multilateral" models, which are naturally expressed in terms of lagged values in both directions for one or more of the d dimensions. An extension of the discrete-frequency Whittle estimate from the time series literature deals conveniently with the computational 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. The asymptotic regime allows increase in both directions of all d dimensions, with the central limit theorem established after re-ordering as a triangular array. However our work offers something new for "unilateral" models also. When the data are non-Gaussian, asymptotic variances of all parameter estimates may be affected, and we propose consistent, non-negative definite estimates of the asymptotic variance matrix.

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Bibliographic Info

Paper provided by Suntory and Toyota International Centres for Economics and Related Disciplines, LSE in its series STICERD - Econometrics Paper Series with number /2005/492.

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Date of creation: Jun 2005
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Handle: RePEc:cep:stiecm:/2005/492

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Web page: http://sticerd.lse.ac.uk/_new/publications/default.asp

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Keywords: spatial data; multilateral modelling; Whittle estimation; edge effect; consistent variance estimation;

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  1. Heyde, C. C. & Gay, R., 1993. "Smoothed periodogram asymptotics and estimation for processes and fields with possible long-range dependence," Stochastic Processes and their Applications, Elsevier, vol. 45(1), pages 169-182, March.
  2. Robinson, Peter M, 1988. "The Stochastic Difference between Econometric Statistics," Econometrica, Econometric Society, vol. 56(3), pages 531-48, May.
  3. Ma, Chunsheng, 2004. "Spatial autoregression and related spatio-temporal models," Journal of Multivariate Analysis, Elsevier, vol. 88(1), pages 152-162, January.
  4. 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.
  5. Tran, L. T. & Yakowitz, S., 1993. "Nearest Neighbor Estimators for Random Fields," Journal of Multivariate Analysis, Elsevier, vol. 44(1), pages 23-46, January.
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Citations

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Cited by:
  1. Jose Vidal-Sanz, 2009. "Automatic spectral density estimation for random fields on a lattice via bootstrap," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer, vol. 18(1), pages 96-114, May.
  2. Peter Robinson, 2008. "Developments in the analysis of spatial data," LSE Research Online Documents on Economics 25473, London School of Economics and Political Science, LSE Library.
  3. Peter M. Robinson, 2006. "Nonparametric spectrum estimation for spatial data," LSE Research Online Documents on Economics 4543, London School of Economics and Political Science, LSE Library.
  4. Peter Robinson, 2008. "Correlation testing in time series, spatial and cross-sectional data," LSE Research Online Documents on Economics 25470, London School of Economics and Political Science, LSE Library.
  5. Peter M Robinson, 2006. "Nonparametric Spectrum Estimation for SpatialData," STICERD - Econometrics Paper Series /2006/498, Suntory and Toyota International Centres for Economics and Related Disciplines, LSE.
  6. Beran, Jan & Ghosh, Sucharita & Schell, Dieter, 2009. "On least squares estimation for long-memory lattice processes," Journal of Multivariate Analysis, Elsevier, vol. 100(10), pages 2178-2194, November.
  7. Peter M Robinson, 2011. "Inference on Power Law Spatial Trends (Running Title: Power Law Trends)," STICERD - Econometrics Paper Series /2011/556, Suntory and Toyota International Centres for Economics and Related Disciplines, LSE.
  8. Sándor Baran & Gyula Pap, 2011. "Asymptotic inference for a one-dimensional simultaneous autoregressive model," Metrika, Springer, vol. 74(1), pages 55-66, July.
  9. Javier Hidalgo & Myung Hwan Seo, 2013. "Specification For Lattice Processes," STICERD - Econometrics Paper Series /2013/562, Suntory and Toyota International Centres for Economics and Related Disciplines, LSE.
  10. Robinson, P.M., 2008. "Correlation testing in time series, spatial and cross-sectional data," Journal of Econometrics, Elsevier, vol. 147(1), pages 5-16, November.
  11. Hidalgo, Javier, 2009. "Goodness of fit for lattice processes," Journal of Econometrics, Elsevier, vol. 151(2), pages 113-128, August.
  12. Peter Robinson, 2007. "Correlation testing in time series, spatial and cross-sectional data," CeMMAP working papers CWP01/07, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.

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