Applications of the Expansion Method

Emilio Casetti, John Paul Jones III, Emilio Casetti, John Paul Jones III

• Publisher: Routledge
• Year: 1991
• Print ISBN: 9780415034944
• eBook ISBN: 9781134966240
• Edition: 1
• Topic: Physical Sciences
• Subtopic: Geography
• Index: Geography

Bringing together researchers with an interest in the expanion method, this book examines the theoretical implications of the paradigm, contributes methodological advances and offers a variety of applications in substantive areas.

1
AN INTRODUCTION TO THE EXPANSION METHOD AND TO ITS APPLICATIONS

Emilio Casetti and John Paul Jones, III

In this introduction we present an overview of the applications of the expansion methodology appearing in this book. First, however, it is useful to outline what the expansion method is, and why you, the reader, might, or should be, interested in it.

Often, the processes of scientific inquiry identify critical variables and ‘important relationships’ among them. These relationships are likely to reflect and incorporate theoretical presuppositions, and are eventually formalized into mathematical models and estimated. Production functions, demand functions, the rank-size rule, and spatial interaction models are examples of such ‘important’ or ‘special status’ relationships.

Relationships such as these play a central role in the contemporary social sciences. Disciplines such as economics, psychology, or political science grew by carving from a common matrix certain ‘proprietary’ clusters of important relationships. The standing of individual scientific disciplines tends to be related to a major degree to their success in identifying, theorizing, modeling, and estimating such distinctive relationships.

There is no question that the abstraction of simple important relations from complex contexts can provide very significant additions to knowledge. Nevertheless, many limitations and shortcomings of the social sciences and their models can be traced to the same processes of abstraction that are also responsible for these advances. Simple and elegant models can yield important insights into naturam rerum (the nature of things), but they are in all likelihood inadequate for understanding complex realities and for intervening to change them. There is a need to reintroduce the complexities of the real world into simple theoretically grounded mathematical models without destroying these models in the process. In fact, the simple models and the simplified important relationships prevalent in the contemporary social sciences should be regarded as early steps in the growth of knowledge, rather than the end point and culmination of it. The expansion methodology combines a technique and a research philosophy that is especially well suited to bring together simple models and complex realities.

The expansion method is both a technique for creating or modifying mathematical models and a research paradigm. As a technique, it consists of the following well-defined operational steps: (a) an ‘initial model’ is specified; the model is made of variables and/or random variables and at least some of its parameters are in letter form; (b) at least some of the letter parameters in the initial model are redefined by ‘expansion equations’ into functions of variables and/or random variables; in many cases these are substantively significant indices representing a context; (c) the expanded parameters are replaced into the initial model to create a ‘terminal model’; and (d) the expansions can be iterated, since the terminal model produced by one expansion can become the initial model of a subsequent one.

Suppose that we take as initial model an important relationship with strong theoretical grounding, and that the expansion equations model the contextual variation of this relationship. Then the terminal model obtained from the two will encompass in the same entity both the model and its contextual drift. Thus, the identity of the initial model is preserved, but at the same time the initial model is rendered capable of addressing complex contextual realities that were previously not part of it.

The expansion methodology is also a research philosophy which carries within itself the suggestion that important theoretically grounded relationships should be regarded as building blocks of more complex theoretical structures encompassing both them and their contexts or environments. Specifically, these higher structures should reflect both the theory behind the initial model and the theory about the nexus between the initial model and its contexts.

Clearly, the expansion paradigm has major implications as regards estimation. The theoretically grounded relationships from individual disciplines tend to be investigated and estimated under the implicit presupposition that they possess some form of quasi universal validity (i.e. invariance). Certainly, in most cases, they are presumed to be invariant over the data sets from which they are estimated.

In contrast, the expansion methodology suggests that presuppositions of invariance are almost always unwarranted. Instead, the variation of relationships across contexts should be presumed, investigated, tested for, and theorized. The ‘invariance’ or ‘universal validity’ of a relation should be a conclusion arising from an extensive, protracted, and unsuccessful search for contextual variation, rather than a presupposition.

Furthermore, the contextual variation of relationships should not be regarded as a nuisance or an aberration, as is currently the case. On the contrary, the theoretical and empirical investigation of the variation of important relations across contexts should be regarded as the obvious second phase of any scientific effort that has brought these relationships into focus. In this next phase, potentially relevant contexts and environments should be focused upon to determine whether a relationship drifts across them, and to theorize why we should expect such drift to occur.

The word paradigm has diverse meanings. However, it is often used to denote an intertwined cluster of research questions and operational approaches/ techniques to obtain answers to these questions. In this sense, the expansion methodology is a paradigm, since it suggests that researchers ask questions about the contextual variation of relations while at the same time it provides the operational routines to model this variation and to test for its occurrence.

Research involving mathematical models involves diverse activities and is carried out within diverse schools of thought. To exemplify, let us consider some cases. One class of model-oriented research aims at determining the optimum states or optimum time paths of systems by techniques such as mathematical programming, optimum control, and others. Other activities are concerned with extracting the implications of models. Examples include research on systems of equations (as in input-output studies), the solving of differential equations, the execution of simulations (as in the System Dynamics tradition), and the investigation of the qualitative properties of dynamic systems.

Other types of model-oriented research are concerned with the estimation of a model’s parameters using empirical data. Estimation work is carried out by practitioners and theoreticians such as engineers, econometricians, statisticians, geographers, and physicists, to name a few, all of whom are very different in their objectives, concerns, and preferences. Their approaches may differ in the extent to which a researcher is committed to a specific model or, alternatively, is willing to consider variants or alternatives to it; in the emphasis on substantive modeling vis-à-vis the specification of error terms; in the manner and extent to which prior information is brought to bear upon the estimation process; and so on.

The vast diversity of mathematical modeling is placed into focus here in order to make the point that the expansion methodology can be applied to model-oriented research of any kind and within an open-ended spectrum of research approaches. For instance, it can be used to construct and modify very abstract models within a frame of reference encompassing the qualitative study of differential equations in which no estimation is contemplated, or to construct or modify models within, say, an econometric perspective. Indeed, the expansion method has a far greater potential, both methodologically and substantively, than is represented by the diversity of papers in this volume—most of which have been written by scholars with primarily substantive interests. These papers, introduced in the paragraphs that follow, reflect their authors’ perceptions of the expansion method and correspond to their diverse substantive and methodological preferences.

Casetti’s paper is a reprint of a 1986 statement on the expansion method that appeared in the IEEE Transactions on Systems, Man, and Cybernetics. The paper provides a guide to diverse applications of the expansion method. It also introduces ‘dual expansions’, a methodology which enables the researcher to investigate the duality between model and context using the expansion method. Casetti shows that when a model is expanded with respect to contextual variables, an implicit second model becomes defined in which the primal context becomes the dual model and the primal model becomes the dual context. The paper illustrates the model-context duality in an empirical study of economic development and population growth.

The next contribution, by Jones, discusses the paradigmatic aspects of the expansion methodology. He explores the implications of the expansion method for ‘open’ research, for altering research trajectories, for testing alternative theoretical frameworks, and for micro and macro level analyses. Jones then uses the expansion method to undercut the distinction between regional and systematic geography. The paper ends by drawing some parallels between the expansion method and analyses employing scientific realism.

Kodras’ paper focuses upon the spatial variation of the relationship between participation in welfare programs and welfare benefits. Traditionally, debates over welfare have been dominated by opposing theories reflecting liberal and conservative perspectives. The liberal theory views welfare provision as a policy response to social needs, with the corollary that greater welfare participation is the counterpart of greater social need. The conservatives’ work-disincentive theory formalizes the notion that high welfare benefits discourage participation in the labor force and encourage participation in welfare programs.

The research in this area has been largely aimed at determining the validity of these theories, or at most their comparative ability to explain reality. Kodras, instead, investigates spatial variation in the explanatory power of these theories. Her paper is concerned with participation in the Aid to Families with Dependent Children program. In a capsule, Kodras expands an initial model relating program participation and benefits into varimax rotated factors extracted from a number of relevant contextual variables. Her conclusions are that ‘each position in the welfare debate is more valid in some places than in others because the programs have different impacts in different contexts’, and that the spatial pattern of welfare provision is characterized by a mismatch between welfare services and welfare needs.

Foster, Gorr, and Wimberly address the comparative merits and the complementarities of drift analyses versus expansion approaches in the study of parametric variation. Drift analyses involve multiple estimations of an initial model, for instance at different points/regions in geographic space, or for different points/intervals in time. In this paper several specifications of moving window regressions are estimated and their results are contrasted with those produced by expansions. The initial model is a functional relation between the growth rates of physicians, the dependent variable, versus density of physicians and population growth, the independent variables. State level data from the American Medical Association master files are employed. The empirical analyses in the Foster et al. paper are suggested by the literature on the locational behavior of physicians, and are designed to estimate the effects of federal programs intended to end physician shortages and maldistribution.

Using random utility theory, Ellis and Odland specify a model of destination choice functionally similar to an originspecific gravity model. This model is expanded first to allow for the distinction between urban and rural destinations and then in order to examine the effects of age and gender on destination choices. All the variables in the model are categorical or categorized. The ages of migrants are categorized into age classes, and the migration distances are categorized into distance bands. The Ellis and Odland formulation yields a 240- cell contingency table; is characterized by a binomial sampling error; involves heteroskedasticity; and requires generalized least squares for its estimation. The sampling zeros in the contingency table are removed using pseudo-Bayesian estimates. The empirical analysis presented is based on an Ecuador data set with about 78,000 observations. This paper brings together concepts and formalisms from ANOVA, categorical data analysis, and the expansion method.

Krakover discusses four approaches to the investigation of metropolitan decentralization, of which two are applications of the expansion method. The latter approaches involve expanding into time polynomials, respectively, the parameters of a polynomial in distance from the CBD and the parameters of a trend surface. Population growth is the dependent variable in both formulations. The methods discussed in the paper are demonstrated and contrasted by a case study for the urban region of Tel Aviv.

The paper by Krakover and Morrill investigates a cluster of hypotheses concerning the dynamics of urban centralization and decentralization. Namely, they hypothesize (a) that the third Kondratieff cycle (1896–1933) coincided with metropolitan centralization; (b) that the fourth Kondratieff cycle (1933–72) coincided with metropolitan deconcentration; and that during both cycles, (c) periods of prosperity were characterized by the growth of more central counties of metropolitan areas and by decline in less central counties, while (d) recessionary years tended to exhibit inner county decline and outer county growth. These hypotheses are tested using county data for the metropolitan areas of Philadelphia, Chicago, and Atlanta. The analyses are based on a model obtained by expanding into time the parameters of an initial formulation relating population growth to distance.

Both Danta’s and Fan’s contributions employ expanded rank-size models. Danta analyzes the temporal and structural changes in the Hungarian urban hierarchy between 1870 and 1986. He starts from the classical formulation relating the sizes of urban centers to their rank, and argues that the parameter associated with the rank variable is a measure of hierarchical concentration. In the conventional unexpanded formulation this measure refers to one point in time, and to the entire system. By expanding the parameters of the rank-size model with respect to time and rank, Danta generates a terminal model that can portray agglomerative and deglomerative tendencies over time, and at various levels of an urban hierarchy. His empirical analyses estimate the temporal shifts of agglomerative and deglomerative trends of the Hungarian urban system, and test the effectiveness of that country’s policies aimed at reducing urban primacy.

Fan extends the potential use of the rank-size functions to the study of inequalities. She argues that the slope parameter of a log transformed rank-size relationship is a systemic measure of inequality in any system. Fan proposes using expanded rank-size relationships to investigate the change of inequality of a system across any context, as well as within any system. This approach is applied to investigate the dynamics of development inequalities for thirty-eight countries between 1913 and 1980.

A crucial suggestion arising from the expansion method paradigm is the stability of social science ‘laws’. Theoretically grounded empirical regularities with a law or quasi-law status are usually estimated under an implicit presupposition of invariance. As Pandit’s paper demonstrates, the drift of such laws across contexts is likely. She investigates the contextual drift of the law-like country level relationship between labor shares in agriculture, manufacturing, and services on one hand, and gross national product per capita on the other. Pandit’s starting point is a classical study by Chenery and Syrquin in which these relationships are estimated under an implicit assumption of invariance. Using a virtually identical data set, she is able to show that the relationships display a statistically significant drift over time and across space, meaning that they are not invariant and are thus not laws.

Visser shows that ‘expansions’ are required to estimate agricultural production functions from areal data in a manner that is consistent with location theory. His argument runs as follows. Location theory tells us that under competitive market equilibrium, agricultural types and intensities are distributed in space so as to maximize rents. For a single type of agriculture the spatial pattern thus produced is one of intensities decreasing with distance from markets. For multiple agricultural types, at each point in space, only one agricultural type will have an optimum intensity that maximizes rent. However, decreasing intensities with distance from market will still prevail for each agricultural type.

These propositions imply that when the parameters of an aggregate agricultural production function are estimated from areal data, they should be expanded into indices of the strength of various agricultural types in order to come to grips with the fact that each areal aggregate includes a mix of agricultural types. A successful empirical analysis concludes Visser’s paper.

The remote sensing application of the expansion methodology presented by Miles, Stow, and Jones is an effort that opens up a wide vista of similar applications. In remote sensing, measurements of phenomena taken at a distance, for instance from satellites, need to be functionally related to measurements taken at the surface of the earth. These relationships provide the basis for securing high resolution, inexpensive, and reliable information on earth surface phenomena. Miles et al. argue that initial models relating satellite measurements to surface measurements can be usefully expanded in terms of substantively relevant variables for the purpose of improving our ability to make accurate inferences about earth phenomena from space. In their application, trend surface expansions of a model relating satellites and surface measurements of several estuary water properties were tested, with results ranging from encouraging to very good.

The book ends with three papers centering upon theoretical and methodological themes. Sonis employs a generalization of the expansion methodology to link geographic diffusion theory, economic utility theory, and ecological competition theory. His is an example of expansion method applications that are not directly oriented toward estimation. Anselin, on the other hand, addresses estimation themes from a spatial econometrics point of view. His paper focuses upon the issues that arise when the error terms are spatially autocorrelated and/or heteroskedastic (possibly because of stochastic expansion equations). The classes of spatial estimation issues including the ones addressed by Anselin are attracting a growing interest in geography and regional science. In the final paper Hanham discusses the expansions into ‘splines’, thus integrating the expansion methodology with a class of techniques that has been diffusing from engineering into the social sciences. His paper includes an application of spline expansions to regional unemployment response functions.

In closing this introduction, we would like to express the hope that the readers of this book will experiment with expansion method techniques and themes in their own research.

2
THE DUAL EXPANSION METHOD: AN APPLICATION FOR EVALUATING THE EFFECTS OF POPULATION GROWTH ON DEVELOPMENT

Emilio Casetti

The progress of scientific research involves a recursive process in which two logically distinct phases can be identified. One phase is concerned with formulating ...

Table of contents

• COVER PAGE
• TITLE PAGE
• COPYRIGHT PAGE
• FIGURES
• TABLES
• CONTRIBUTORS
• ACKNOWLEDGMENTS
• 1: AN INTRODUCTION TO THE EXPANSION METHOD AND TO ITS APPLICATIONS
• 2: THE DUAL EXPANSION METHOD: AN APPLICATION FOR EVALUATING THE EFFECTS OF POPULATION GROWTH ON DEVELOPMENT
• 3: PARADIGMATIC DIMENSIONS OF THE EXPANSION METHOD
• 4: A CONTEXTUAL EXPANSION OF THE WELFARE MODEL
• 5: A COMPARISON OF DRIFT ANALYSES AND THE EXPANSION METHOD: THE EVALUATION OF FEDERAL POLICIES ON THE SUPPLY OF PHYSICIANS
• 6: PERSONAL CHARACTERISTICS IN MODELS OF MIGRATION DECISIONS: AN ANALYSIS OF DESTINATION CHOICE IN ECUADOR
• 7: ALTERNATIVE APPROACHES TO THE STUDY OF METROPOLITAN DECENTRALIZATION
• 8: LONG-WAVE SPATIAL AND ECONOMIC RELATIONSHIPS IN URBAN DEVELOPMENT
• 9: AN INVESTIGATION INTO THE DYNAMICS OF DEVELOPMENT INEQUALITIES VIA EXPANDED RANK-SIZE FUNCTIONS
• 10: IDENTIFYING HIERARCHICAL DEVELOPMENT TRENDS IN THE HUNGARIAN URBAN SYSTEM USING THE EXPANSION METHOD
• 11: AN EXPLORATION OF THE RELATIONSHIP BETWEEN SECTORAL LABOR SHARES AND ECONOMIC DEVELOPMENT
• 12: PRODUCTION FUNCTION ESTIMATION AND THE SPATIAL STRUCTURE OF AGRICULTURE
• 13: INCORPORATING THE EXPANSION METHOD INTO REMOTE SENSING-BASED WATER QUALITY ANALYSES
• 14: INNOVATION DIFFUSION THEORY AND THE EXPANSION METHOD
• 15: SPATIAL DEPENDENCE AND SPATIAL HETEROGENEITY: MODEL SPECIFICATION ISSUES IN THE SPATIAL EXPANSION PARADIGM
• 16: GENERATING VARYING PARAMETER MODELS USING CUBIC SPLINE FUNCTIONS