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

Daniel A. Griffith and Yongwan Chun ()
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Daniel A. Griffith: University of Texas at Dallas
Yongwan Chun: University of Texas at Dallas

Journal of Spatial Econometrics, 2022, vol. 3, issue 1, 1-30

Abstract: 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.

Keywords: Geary ratio; Join count statistics; Moran coefficient; Moran scatterplot; Spatial autocorrelation (search for similar items in EconPapers)
JEL-codes: C18 C21 C52 (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1007/s43071-022-00031-w

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