The Exact Distribution of Moran's I

Abstract

In analogy to the exact distribution of the Durbin—Watson d statistic for serial autocorrelation of regression residuals, the exact small sample distribution of Moran's I statistic (or alternatively Geary's c) can be derived. Use of algebraic results by Koerts and Abrahamse and theoretical results by Imhof, allows the authors to determine by numerical integration the exact distribution function of Moran's I for normally distributed variables. For the case in which the explanatory variables have been neglected, an upper and a lower bound can be given within which the exact distribution of Moran's I for regression residuals will lie. Furthermore, the proposed methodology is flexible enough to investigate related topics such as the characteristics of the exact distribution for distinct spatial structures as well as their different specifications, the exact power function under different spatial autocorrelation levels, and the distribution of Moran's I for nonnormal random variables.