18. A Test for the Herfindahl Index

Regional scientists have adopted two rather different empirical views of agglomeration. One view, favored by economists, examines the distribution of some property across a class of entities (agents, regions). The other view, favored by geographers, examines the spatial correlation of that property based on the arrangement of those entities. Adopting the first approach, this paper develops new statistical properties for the Herfindahl–Hirschman concentration index. The methodology is based on the standard occupancy problem in physics, where r particles (points) are distributed across n cells (quadrats). Both r and n are random variables so the Herfindahl index itself is considered a random variable H(r, n). Including all the equi-probable states of this random variable, expected values for both the mean and standard deviation are specified. A test compares the Herfindahl score for a specific state (sample) to the score based on all possible states, allowing inferences to be made about whether the observed score conforms to a random process. A few applications follow and then a short discussion concludes the chapter.