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Some useful details about the Moran coefficient, the Geary ratio, and the join count indices of spatial autocorrelation

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  • Daniel A. Griffith

    (University of Texas at Dallas)

  • Yongwan Chun

    (University of Texas at Dallas)

Abstract

Popular spatial autocorrelation (SA) indices employed in spatial econometrics include the Moran Coefficient (MC), the Geary Ratio, (GR) and the join count statistics (JCS). Properties of these first two quantities rely on spatial weights matrix definitions [e.g., binary 0–1 (rook or queen adjacencies), nearest neighbors, inverse inter-point distance, row standardized], which may cause confusion about output from different software packages; to date, JCS calculations have been using only binary 0–1 definitions. The MC and GR expected values for linear regression residuals also merit closer examination; although the mean and other details of the sampling distribution for the MC are well-known, at least the details of those for the GR are not. The (MC + GR) sum furnishes a potential diagnostic for georeferenced data normality, one that warrants much further explication and scrutiny. The Moran scatterplot is a widely used graphic tool for visualizing SA; this paper formally introduces its Geary scatterplot counterpart (first appearing informally in 2019), together with some comparisons of the two. Meanwhile, established relationships between the JCS and the MC and the GR need additional inspection, too, especially in terms of their sampling variances. Preliminary analyses summarized in this paper also address derived asymptotic properties as well as links with the single spatial autoregressive parameter of the simultaneous autoregressive (SAR; spatial error) and autoregressive response (AR; spatial lag) model specifications. This paper describes selected little-known features of these standard SA indices, furthering a better understanding of, and a more complete set of details about, them. Results from a myriad of empirical spatial economics landscapes [e.g., Puerto Rico, Jiangsu Province, Texas, Houston (Harris County), and the Dallas-Fort Worth (DFW) metroplex] and a variety of planar surface partitionings (including the square and hexagonal tessellations, and randomly generated graphs) illustrate highlighted theoretical and conceptual traits. These include a corroboration of the contention in the literature that the MC more closely aligns with spatial autoregression, and the GR more closely aligns with geostatistics.

Suggested Citation

  • Daniel A. Griffith & Yongwan Chun, 2022. "Some useful details about the Moran coefficient, the Geary ratio, and the join count indices of spatial autocorrelation," Journal of Spatial Econometrics, Springer, vol. 3(1), pages 1-30, December.
    • Handle: RePEc:spr:jospat:v:3:y:2022:i:1:d:10.1007_s43071-022-00031-w
    DOI: 10.1007/s43071-022-00031-w
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    References listed on IDEAS

    as
    1. Giuseppe Arbia, 2006. "Spatial Econometrics," Advances in Spatial Science, Springer, number 978-3-540-32305-1, december.
    2. Daniel Griffith & Jean Paelinck, 2007. "An equation by any other name is still the same: on spatial econometrics and spatial statistics," The Annals of Regional Science, Springer;Western Regional Science Association, vol. 41(1), pages 209-227, March.
    3. 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.
    4. Qing Luo & Daniel A. Griffith & Huayi Wu, 2019. "Spatial autocorrelation for massive spatial data: verification of efficiency and statistical power asymptotics," Journal of Geographical Systems, Springer, vol. 21(2), pages 237-269, June.
    5. Yee Leung & Chang-Lin Mei & Wen-Xiu Zhang, 2000. "Testing for Spatial Autocorrelation among the Residuals of the Geographically Weighted Regression," Environment and Planning A, , vol. 32(5), pages 871-890, May.
    6. Daniel A. Griffith, 2019. "Negative Spatial Autocorrelation: One of the Most Neglected Concepts in Spatial Statistics," Stats, MDPI, vol. 2(3), pages 1-28, August.
    7. M Tiefelsdorf & D A Griffith & B Boots, 1999. "A Variance-Stabilizing Coding Scheme for Spatial Link Matrices," Environment and Planning A, , vol. 31(1), pages 165-180, January.
    8. Daniel A. Griffith, 2003. "Spatial Autocorrelation and Spatial Filtering," Advances in Spatial Science, Springer, number 978-3-540-24806-4, december.
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    More about this item

    Keywords

    Geary ratio; Join count statistics; Moran coefficient; Moran scatterplot; Spatial autocorrelation;
    All these keywords.

    JEL classification:

    • C18 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Methodolical Issues: General
    • C21 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Cross-Sectional Models; Spatial Models; Treatment Effect Models
    • C52 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Model Evaluation, Validation, and Selection

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