Geographical Analysis

An International Journal of Theoretical Geography

Volume 36 Number 2 April 2004

Special Issue on

Methodological Developments in Spatial Econometrics and Statistics Edited by James P. LeSage, R. Kelley Pace, and Michael Tiefelsdorf

Introduction

Methodological Developments in Spatial Econometrics and Statistics James I? LeSage, R. Kelley Pace, and Michael Tiefelsdorf

Articles

Constructing the Spatial Weights Matrix Using a Local Statistic Arthur Getis andJared Aldstadt

Does Econometric Methodology Matter? An Analysis of Public Policy Using Spatial Econometric Techniques

Donald]. Lacombe

A Spatial Mixture Model of Innovation Diffusion Tony E. Smith and Sangyoung Song

Spatial Analysis of Employment and Population Density: The Case of the Agglomeration of Dijon 1999

Catherine Baumont, Cem Ertur, andJulie Le Gallo

Optimal Sampling Design for Variables with Varying Spatial Importance Peter A. Rogerson, Eric Delmelle, Rajan Batta, Mohan Akella,

Alan Blatt, and Glenn Wilson

Annual Index

87

90

105

119

146

177

195

Copyright 2004 The Ohio State University

88 / Geographical Analysis

The second paper by Lacombe examines a methodological approach to accommo- dating spatial effects used in the econometrics literature, known as border matching. Economists are often concerned with the magnitude and significance of economic policies that vary across states, countries, or regions. Ordinary regression methods have been applied to selected samples that contain only observations along both sides of a border. Lacombe compares estimates and inferences from this methodological approach to an extension of conventional spatial autoregressive regression models based on two mutually exclusive weight matrices. One weight matrix captures the spatial autoregressive influence of border counties in adjacent states while a second weight matrix measures the influence of contiguous counties within the state. He ar- gues that the regression-based border-matching method omits a large number of sample observations and fails to account for spatial autoregressive influences from bordering regions within the state. This can lead to an overestimate of the impact arising from changes in policy regime across states.

Another area of interest in the papers presented at the November 2002 conference was application of Bayesian methods to spatial statistical problems. The paper by Tony Smith and Sangyoung Song entitled “A Spatial Mixture Model of Innovation Diffusion” demonstrates an interesting case where Bayesian methods hold an advan- tage over conventional maximum-likelihood estimation and inference when sample sizes are small. They examine diffusion of new products or technical innovation in a spatial context, arguing that the event-based adoption process can be modeled as the outcome of two factors. One involves modeling the likelihood of a binary decision to adopt, which depends on spatial contacts or interaction between individuals and pre- vious adopters, while the other reflects individual characteristics. Since both interac- tion and individual characteristics are georeferenced, the spatial diffusion process is modeled as a probabilistic spatial mixture of these two forces. They show that stan- dard maximum-likelihood estimates behave poorly in the small samples typically en- countered with work involving early adoption of new technology or products. Two alternative approaches to estimation are provided: one based on the EM algorithm and the second that relies on Bayesian smoothing that arises from use of prior distri- butions. They demonstrate that both alternative estimation approaches result in more sensible estimates and inferences for the case of small sample sizes.

A number of conference papers also stressed the importance of robust estimates and inferences in the face of spatial heterogeneity and outliers. The paper by Cather- ine Baumont, Cem Ertur, and Julie Le Gallo entitled “Spatial Analysis of Employ- ment and Population Density: The Case of Agglomeration of Dijon, 1999” represents one such work. They examine population and employment distributions in the con- text of monocentric versus polycentric models. Given recent trends in decentraliza- tion that give rise to a number of employment subcenters, estimation and inference regarding these distributions could be quite sensitive to aberrant observations or out- liers. The very nature of an employment or population subcenter suggests that en- clave effects might result in a handful of influential observations exerting undue influence on dlstributional estimates based on small samples. They apply conven- tional subcenter identification methods to the metropolitan region centered on Dijon and provide an interesting comparison of a host of spatial modeling methods for esti- mating the population density including maximum likelihood, generalized method of moments, and heteroskedastic/robust Bayesian estimation techniques. For the Dijon region the authors find results consistent with a monocentric distribution, but never- theless conclude that aberrant observations or outliers can exert an impact on esti- mates and inferences regarding the distribution of economic activity.

Another area of interest in current spatial statistics work is that of local versus global modeling of spatial phenomena. There is intuitive appeal to the idea that rela- tions among variables change over the spatial sample of observations, limiting the

J a m s I? LeSage, R. Kelley Pace and Michael Tiefelsdorf / 89

usefulness of a single average set of estimates based on the entire sample. The paper by Peter A. Rogerson, Eric Delmelle, Rajan Batta, Mohan Akella, Alan Blatt, and Glenn Wilson entitled “Optimal Sampling Design for Variables with Varying Spatial Importance” represents work in this area. In an application involving signals received by cellular phone towers, the authors tackle the issue of infill spatial sampling where the information content provided by a given sample from certain points in space is of limited usefulness. For this application, the authors argue that sampling locations where the probability of a successful cellular phone call is near zero or one provides information of limited use. In contrast, the information content in cases where the probability of call success is bounded away from these two extremes is of greater rel- evance. For the problem under investigation, the data-generating spatial sampling design becomes a constrained optimization problem that aims at gathering observa- tions from those locations for which high-precision estimates can be achieved.

The papers in this volume as well as many of those presented at the November, 2002, conference deal with three tasks confronting practitioners using spatial regres- sion methods in applied work. First, there is the question of what type of weight ma- trix adequately reflects the relations among the spatial objects of the problem at hand. Second is the issue of which model specification is appropriate. Third is the usual problem encountered in regression regarding which explanatory variables are appropriate.

Research that tackles these issues separately may encounter problems since the weight matrix exerts an influence on both the model specification and appropriate ex- planatory variables. Similarly, the model specification and explanatory variables can exert an influence on the appropriate spatial connectivity structure. Most practition- ers have encountered situations whereby changing explanatory variables leads to sub- stantial changes in spatial dependence and explanatory variable parameter estimates. A unified approach addressing all three of these issues is needed to understand the spatial dimensions of the underlying data-generating process.