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One-step estimation of spatial dependence parameters: Properties and extensions of the APLE statistic

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  • Li, Hongfei
  • Calder, Catherine A.
  • Cressie, Noel

Abstract

We consider one-step estimation of parameters that represent the strength of spatial dependence in a geostatistical or lattice spatial model. While the maximum likelihood estimators (MLE) of spatial dependence parameters are known to have various desirable properties, they do not have closed-form expressions. Therefore, we consider a one-step alternative to maximum likelihood estimation based on solving an approximate (i.e., one-step) profile likelihood estimating equation. The resulting approximate profile likelihood estimator (APLE) has a closed-form representation, making it a suitable alternative to the widely used Moran’s I statistic. Since the finite-sample and asymptotic properties of one-step estimators of covariance-function parameters have not been studied rigorously, we explore these properties for the APLE of the spatial dependence parameter in the simultaneous autoregressive (SAR) model. Motivated by the APLE statistic’s closed from, we develop exploratory spatial data analysis tools that capture regions of local clustering or the extent to which the strength of spatial dependence varies across space. We illustrate these exploratory tools using both simulated data and observed crime rates in Columbus, OH.

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  • Li, Hongfei & Calder, Catherine A. & Cressie, Noel, 2012. "One-step estimation of spatial dependence parameters: Properties and extensions of the APLE statistic," Journal of Multivariate Analysis, Elsevier, vol. 105(1), pages 68-84.
    • Handle: RePEc:eee:jmvana:v:105:y:2012:i:1:p:68-84
    DOI: 10.1016/j.jmva.2011.08.006
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    References listed on IDEAS

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    1. H. Kelejian, Harry & Prucha, Ingmar R., 2001. "On the asymptotic distribution of the Moran I test statistic with applications," Journal of Econometrics, Elsevier, vol. 104(2), pages 219-257, September.
    2. Genton, Mark G. & Ruiz-Gazen, Anne, 2009. "Visualizing Influential Observations in Dependent Data," TSE Working Papers 09-051, Toulouse School of Economics (TSE).
    3. 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.
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    1. Suesse, Thomas, 2018. "Marginal maximum likelihood estimation of SAR models with missing data," Computational Statistics & Data Analysis, Elsevier, vol. 120(C), pages 98-110.
    2. Thomas Suesse, 2018. "Estimation of spatial autoregressive models with measurement error for large data sets," Computational Statistics, Springer, vol. 33(4), pages 1627-1648, December.
    3. Jin, Fei & Lee, Lung-fei, 2012. "Approximated likelihood and root estimators for spatial interaction in spatial autoregressive models," Regional Science and Urban Economics, Elsevier, vol. 42(3), pages 446-458.
    4. Kirillov, Andrew, 2021. "A study on spatial autocorrelation: Case of Russian regional inflation," Applied Econometrics, Russian Presidential Academy of National Economy and Public Administration (RANEPA), vol. 64, pages 5-22.
    5. Roger Bivand & Giovanni Millo & Gianfranco Piras, 2021. "A Review of Software for Spatial Econometrics in R," Mathematics, MDPI, vol. 9(11), pages 1-40, June.

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