IDEAS home Printed from https://ideas.repec.org/a/spr/jospat/v3y2022i1d10.1007_s43071-022-00031-w.html
   My bibliography  Save this article

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

Author

Listed:
  • 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
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s43071-022-00031-w
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s43071-022-00031-w?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Giuseppe Arbia, 2006. "Spatial Econometrics," Advances in Spatial Science, Springer, number 978-3-540-32305-1, February.
    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. Daniel A. Griffith, 2003. "Spatial Autocorrelation and Spatial Filtering," Advances in Spatial Science, Springer, number 978-3-540-24806-4, February.
    5. 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.
    6. 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.
    7. 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.
    8. 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.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Christoph Grimpe & Roberto Patuelli, 2011. "Regional knowledge production in nanomaterials: a spatial filtering approach," The Annals of Regional Science, Springer;Western Regional Science Association, vol. 46(3), pages 519-541, June.
    2. Yongwan Chun, 2008. "Modeling network autocorrelation within migration flows by eigenvector spatial filtering," Journal of Geographical Systems, Springer, vol. 10(4), pages 317-344, December.
    3. Roberto Patuelli & Daniel A. Griffith & Michael Tiefelsdorf & Peter Nijkamp, 2011. "Spatial Filtering and Eigenvector Stability: Space-Time Models for German Unemployment Data," International Regional Science Review, , vol. 34(2), pages 253-280, April.
    4. Matías Mayor & Roberto Patuelli, 2012. "Short-Run Regional Forecasts: Spatial Models through Varying Cross-Sectional and Temporal Dimensions," Advances in Spatial Science, in: Esteban Fernández Vázquez & Fernando Rubiera Morollón (ed.), Defining the Spatial Scale in Modern Regional Analysis, edition 127, chapter 0, pages 173-192, Springer.
    5. Michael Tiefelsdorf & Daniel A Griffith, 2007. "Semiparametric Filtering of Spatial Autocorrelation: The Eigenvector Approach," Environment and Planning A, , vol. 39(5), pages 1193-1221, May.
    6. Roberto Patuelli & Andrea Vaona & Christoph Grimpe, 2010. "The German East‐West Divide In Knowledge Production: An Application To Nanomaterial Patenting," Tijdschrift voor Economische en Sociale Geografie, Royal Dutch Geographical Society KNAG, vol. 101(5), pages 568-582, December.
    7. Daniel A. Griffith & Yongwan Chun & Monghyeon Lee, 2020. "Deeper Spatial Statistical Insights into Small Geographic Area Data Uncertainty," IJERPH, MDPI, vol. 18(1), pages 1-16, December.
    8. Luc Anselin, 2010. "Thirty years of spatial econometrics," Papers in Regional Science, Wiley Blackwell, vol. 89(1), pages 3-25, March.
    9. Roberto Patuelli & Daniel A. Griffith & Michael Tiefelsdorf & Peter Nijkamp, 2006. "The Use of Spatial Filtering Techniques: The Spatial and Space-time Structure of German Unemployment Data," Tinbergen Institute Discussion Papers 06-049/3, Tinbergen Institute.
    10. Yong Bao & Xiaotian Liu & Lihong Yang, 2020. "Indirect Inference Estimation of Spatial Autoregressions," Econometrics, MDPI, vol. 8(3), pages 1-26, September.
    11. Tiziano Arduini & Eleonora Patacchini & Edoardo Rainone, 2014. "Identification and Estimation of Outcome Response with Heterogeneous Treatment Externalities," EIEF Working Papers Series 1407, Einaudi Institute for Economics and Finance (EIEF), revised Sep 2014.
    12. Baltagi, Badi H. & Liu, Long, 2008. "Testing for random effects and spatial lag dependence in panel data models," Statistics & Probability Letters, Elsevier, vol. 78(18), pages 3304-3306, December.
    13. Tse-Chuan Yang & Stephen A Matthews, 2015. "Death by Segregation: Does the Dimension of Racial Segregation Matter?," PLOS ONE, Public Library of Science, vol. 10(9), pages 1-26, September.
    14. Marco Helbich & Wolfgang Brunauer & Eric Vaz & Peter Nijkamp, 2014. "Spatial Heterogeneity in Hedonic House Price Models: The Case of Austria," Urban Studies, Urban Studies Journal Limited, vol. 51(2), pages 390-411, February.
    15. Guido M. Kuersteiner & Ingmar R. Prucha, 2020. "Dynamic Spatial Panel Models: Networks, Common Shocks, and Sequential Exogeneity," Econometrica, Econometric Society, vol. 88(5), pages 2109-2146, September.
    16. Daniel A. Griffith, 2023. "Understanding Spatial Autocorrelation: An Everyday Metaphor and Additional New Interpretations," Geographies, MDPI, vol. 3(3), pages 1-20, August.
    17. Badi H. Baltagi & Zhenlin Yang, 2013. "Standardized LM tests for spatial error dependence in linear or panel regressions," Econometrics Journal, Royal Economic Society, vol. 16(1), pages 103-134, February.
    18. Daniel A. Griffith & Manfred M. Fischer, 2016. "Constrained Variants of the Gravity Model and Spatial Dependence: Model Specification and Estimation Issues," Advances in Spatial Science, in: Roberto Patuelli & Giuseppe Arbia (ed.), Spatial Econometric Interaction Modelling, chapter 0, pages 37-66, Springer.
    19. Giuliano Guerra & Roberto Patuelli, 2014. "The influence of role models on immigrant self-employment: a spatial analysis for Switzerland," International Journal of Manpower, Emerald Group Publishing Limited, vol. 35(1/2), pages 187-215, May.
    20. Guowei Cui & Vasilis Sarafidis & Takashi Yamagata, 2020. "IV Estimation of Spatial Dynamic Panels with Interactive Effects: Large Sample Theory and an Application on Bank Attitude," Monash Econometrics and Business Statistics Working Papers 11/20, Monash University, Department of Econometrics and Business Statistics.

    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

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:jospat:v:3:y:2022:i:1:d:10.1007_s43071-022-00031-w. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.