SPATIAL ECONOMETRICS: METHODS AND MODELS

Transcription

1 SPATIAL ECONOMETRICS: METHODS AND MODELS

2 STUDIES IN OPERATIONAL REGIONAL SCIENCE Folmer, H., Regional Economic Policy ISBN Brouwer, F., Integrated Environmental Modelling: Design and Tools ISBN X. Toyomane, N., Multiregional Input-Output Models in Long-Run Simulation ISBN X.

3 Spatial Econometrics: Methods and Models by Luc AnseIin Departments of Geography and Economics, University of California, Santa Barbara SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.

4 Library of Congress Cataloging in Publication Data Ansel in, Luc, Spatial econometrics methods and models I by Luc Anselin. p. cm. -- (Studies in operational regional science) Includes index. ISBN ISBN (ebook) DOI / Space in economlcs--econometric models. 2. Regional economics- -Econometric models. I. Title. 11. Serles. HB199.A '.028--dc CIP ISBN printed on acid free paper ts All Rights Reserved 1988 by Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 1988 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical including photocopying, recording or by any information storage and retrieval system, without written permissian fram the copyright awner

5 to E.T.

6 TABLE OF CONTENTS List of Tables List of Figures Preface xiii xiv xv CHAPTER 1: INTRODUCTION 1 PART I: FOUNDATIONS FOR THE ECONOMETRIC ANALYSIS OF SP ATIAL PROCESSES 5 CHAPTER Z: THE SCOPE OF SPATIAL ECONOMETRICS 2.1. Spatial Econometrics and Regional Science Spatial Econometrics and Standard Econometrics Spatial Econometrics and Spatial Statistics 2.2. Spatial Effects Spatial Dependence Spatial Heterogeneity CHAPTER S: THE FORMAL EXPRESSION OF SPATIAL EFFECTS 3.1. The Formal Expression of Connectivity in Space Neighbors in Space Spatial Contiguity Matrices General Spatial Weight Matrices Spatial Lag Operators Circularity and Redundancy in Spatial Lag Operators 3.2. Problems with the Concept of Space in the Formal Expression of Spatial Effects The Modifiable Areal Unit Problem Properties of Spatial Weight Matrices CHAPTER 4: A TYPOLOGY 01 SPATIAL ECONOMETRIC MODELS 4.1. A General Classification of Models 4.2. A Taxonomy of Spatial Linear Regression Models for Cross-Section Data 4.3. A Taxonomy of Spatial Linear Regression Models for Space-Time Data CHAPTER 5: SPATIAL STOCHASTIC PRO CES SES: TERMINOLOGY AND GENERAL PROPERTIES 5.1. Properties of Spatial Stochastic Processes Definitions Stationarity and Isotropy Locally Covariant Random Fields Mixing Sequences 5.2. Asymptotic Approaches to Spatial Processes Asymptotic Convergence Laws of Large Numbers Central Limit Theorems for Spatial Processes

7 viii PART II: A Closer Look at the Relevance of Statistical Approaches to Data in Space and Space-Time Spatial Sampie or Spatial Population AsymptoticB in Space and Space-Time ESTIMATION AND HYPOTHESIS TESTING CHAPTER 6: Appendix 6.A.: Appendix 6.B.: THE MAXIMUM LIKELmOOD APPROACH TO SPATIAL PROCESS MODELS Limitations of Ordinary Least Squares Estimation in Spatial Process Models OLS in the Presence of a Spatially Lagged Dependent Variable OLS in the Presence of Spatial Residual Autocorrelation Maximum Likelihood Estimation Properties of ML Estimators in the Presence of Spatially Lagged Dependent Variables Properties of the ML Estimator for the Linear Regression Model with Spatial Effects in the Error Terms The Likelihood Function and Jacobian for the General Model The First Order Conditions for ML Estimates in the General Model The Asymptotic Variance Matrix for the General Model Hypothesis Tests Based on the Maximum Likelihood Principle General Principles Wald Tests in Spatial Process Models Likelihood Ratio Tests in Spatial Process Models Lagrange Multiplier Tests in Spatial Process Models Finite Sampie Considerations Some Useful Results on Matrix Calculus Derivation of the Elements of the Information Matrix for the General Model CHAPTER 7: ALTERNATIVE APPROACHES TO INFERENCE IN SPATIAL PROCESS MODELS 7.1. Instrumental Variables Estimation in Spatial Process Models General Principles Implementation Issues in Models with Spatially Lagged Dependent Variables Implementation Issues in Models with Spatially Lagged Dependent Variables and Spatially Autoregressive Error Terms 7.2. Bayesian Approaches to the Estimation of Spatial Process Models A Bayesian Approach to Inference in a First Order Autoregressive Model A Bayesian Approach to Inference in the Linear Model with Spatially Autoregressive Error Terms 7.3. Robust Approaches to Spatial Process Models Bootstrapping in Regression Models: General Principles Bootstrapping in Models with Spatial Dependence CHAPTER 8: SPATIAL DEPENDENCE IN REGRESSION ERROR TERMS 8.1. Tests for Spatial Dependence in Regression Error Terms

8 ix Tests for Residual Spatial Autocorrelation Based on the Moran I Statistic Tests for Residual Spatial Autocorrelation Based on Maximum Likelihood Estimation Testing for Residual Spatial Autocorrelation in the Presence of Spatially Lagged Dependent Variables Testing for Residual Spatial Autocorrelation in the Presence of Heteroskedasticity Estimation in the Presence of Spatially Dependent Error Terms OLS, GLS, EGLS and ML Iterative Procedures Robustness Issues Testing for Spatial Dependence in the Presence of Other Misspecifications of Unknown Form A Heteroskedasticity-Robust Test for Residual Spatial Dependence Robust Error Covariance Matrix Estimates CHAPTER 9: SP ATIAL HETEROGENEITY General Aspects of Spatial Heterogeneity Testing for Heterogeneity in the Presence of Spatial Dependence Testing for Heteroskedasticity in the Presence of Spatial Dependence Testing for Structural Stability in the Presence of Spatial Dependence Spatial Expansion of Parameters General Principle of the Expansion Method Spatial Expansion and Heteroskedasticity Spatial Expansion and Spatial Autocorrelation Other Forms of Spatial Heterogeneity Random Coefficient Variation Error Component Models for Cross Section Data Spatially Switching Regressions Spatial Adaptive Filtering 133 CHAPTER 10: MODELS IN SP ACE AND TIME SUR and Spatial SUR General Formulation Spatial SUR with Spatial Error Autocorrelation Spatial SUR with Spatially Lagged Dependent Variables Testing for Spatial Autocorrelation in Spatial SUR Models Nested Spatial Effects Error Component Models in Space-Time General Formulation Spatial Autocorrelation in Error Component Models A Lagrange Multiplier Test for Spatial Autocorrelation in Error Component Models Simultaneous Models in Space-Time 156 Appendix 10.A.: First Order Conditions for the ML Estimator in the Spatial SUR Model with Spatially Autocorrelated Errors 157 Appendix 10.B.: Information Matrix for the ML Estimator in the Spatial SUR Model with Spatially Autocorrelated Errors 159

9 x Appendix 10.C.: Appendix 10.D.: Appendix 10.E.: ML Estimator in the Two-Equation Spatial SUR Model with Spatially Autocorrelated Errors The LM Test for Spatial Error Autocorrelation in the Two-Equation Spatial SUR Model Derivation of the LM Test for Spatial Autocorrelation in a 2ECM CHAPTER 11: PROBLEM AREAS IN ESTIMATION AND TESTING POR SPATIAL PROCESS MODELS Pre-Testing General Framework Pre-Test Estimators in Spatial Process Models The Boundary Value Problem The Problem and its Correction A Spatial Econometric Approach to Edge Effects The Specification of the Spatial Weight Matrix The Importance of Sam pie Size for Spatial Process Estimators CHAPTER 12: OPERATIONAL ISSUES AND EMPIRICAL APPLICATIONS Operational Issues Maximum Likelihood Estimation The Jacobian The Direct Search Approach to Optimization Other Optimization Methods Software Availability The Analysis of Cross-Sectional Data A Spatial Model of Determinants of Neighborhood Crime OLS Regression with Diagnostics for Spatial Effects ML Estimation of a Mixed Regressive Spatial Autoregressive Model Spatial Dependence in the Error Term A Spatial Durbin Model Diagnostics for Spatial Effects in the Spatial Expansion Method Testing for Spatial Error Dependence in a Heteroskedastic Model Testing for Structural Stability in the Presence of Spatial Error Dependence Joint Spatial Dependence and Heteroskedasticity in the Error Term The Analysis of Space-Time Data A Spatial Phillips-Curve OLS Regression with Diagnostics for Spatial Effects IV Estimation of a Mixed Regressive Spatial Autoregressive Model SUR with Diagnostics for Spatial Effects Error Components with Diagnostics for Spatial Effects Estimation of SUR with Spatial Dependence in the Error Terms 214 Appendix 12.A.: A Bisection Search for the Coefficient in a Pure Spatial Autoregressive Model 216

10 xi PART ill: MODEL VALIDATION 221 CHAPTER 13: MODEL VALIDATION AND SPECIlICATION TESTS IN SPATIAL ECONOMETRIC MODELS General Issues of Model Validation in Spatial Analysis Specification Testing in Spatial Econometrics Determination of Spatial Lag Length: Tests on Common Factors General Principle of the Spatial Common Factor Approach Evaluation and Practical Implications Determination of Spatial Structure: Non-Nested Tests General Principle of Tests on Non-Nested Hypotheses Non-Nested Tests on the Spatial Weight Matrix Based on Maximum Likelihood Estimation Non-Nested Tests on the Spatial Weight Matrix Based on Instrumental Variable Estimation Evaluation and Practical Implications CHAPTER 14: MODEL SELECTION IN SPATIAL ECONOMETRIC MODELS Measures of Fit in Spatial Models Information Based Criteria for Model Discrimination Bayesian Approaches to Model Selection Heuristics in Spatial Model Selection Practical Implications of Model Validation in Spatial Econometrics CHAPTER 15: REFERENCES INDEX CONCLUSIONS

11 LIST OF TABLES 3.1. Binary Contiguity Matrix for Gridded Data Second Power of a Binary Contiguity Matrix Third Power of a Binary Contiguity Matrix Determinants of Crime, Columbus, Ohio Neighborhoods First Order Contiguity, Columbus, Ohio Neighborhoods Lagrange Multiplier Diagn08tics for Spatial Effects in OLS Regression Estimates in the Mixed Regressive Spatial Autoregressive Model Bisection Search for the Spatial Autoregressive Parameter ML Estimation of the Model with Spatially Dependent Error Terms Iterations in the ML Estimation Estimation of the Spatial Durbin Model OLS Estimation of the Spatially Expanded Model Heteroskedastic-Robust Inference in the Spatially Expanded Model Lagrange Multiplier Diagnostics for Spatial Effects in the Spatially Expanded Model ML Estimates for Random Coefficient Models Estimation of the Unrestricted Model with Spatial Error Dependence Iterations of the Error Variance Parameters in the ML Estimation of the Model with Joint Spatial Dependence and Heteroskedasticity ML Estimation in the Model with Joint Spatial Dependence and Heteroskedasticity Observations for South-Western Ohio Counties Observations for South - Western Ohio Counties First Order Contiguity for South-Western Ohio Counties Lagrange Multiplier Diagnostics for Spatial Effects ML Estimation of the Spatial Phillips-Curve IV Estimation of the Spatial Phillips-Curve SLS Estimation of the Spatial Phillips-Curve SUR Estimation of the Phillips-Curve Pooled OLS Estimation of the Phillips-Curve ML Estimation of the Two Error Components Model Iterations for the Spatial Coefficients in the ML Estimation of the Spatial SUR Model ML Estimates for the SUR Model with Spatial Error Autocorrelation Estimation of a Spatial Autoregressive Model of Housing Values ML-Based Non-Nested Tests of Model i (Row) Against Model j (Column) Instrumental Variable Estimates for Models of Aggregate Housing Value Non-Nested Tests Based on an IV-Analogue to the Likelihood Ratio Non-Nested Tests Ba.sed on Instrumental Variable Estimation in an Augmented Regression Measures of Fit for Spatial Models of Neighborhood Crime Information Based Measures of Fit for Spatial Models of Neighborhood Crime 247 xiii

12 LIST OF FIGURES Spatial Dependence and Aggregation Contiguity on a Regular Lattice Spatial Lags on a Regular Lattice Neighborhoods in Columbus, Ohio Counties in South-Western Ohio xiv

13 PREFACE Spatial econometrics deals with spatial dependence and spatial heterogeneity, critical aspects of the data used by regional scientists. These characteristics may cause standard econometric techniques to become inappropriate. In this book, I combine several recent research results to construct a comprehensive approach to the incorporation of spatial effects in econometrics. My primary focus is to demonstrate how these spatial effects can be considered as special cases of general frameworks in standard econometrics, and to outline how they necessitate a separate set of methods and techniques, encompassed within the field of spatial econometrics. My viewpoint differs from that taken in the discussion of spatial autocorrelation in spatial statistics - e.g., most recently by Cliff and Ord (1981) and Upton and Fingleton (1985) - in that I am mostly concerned with the relevance of spatial effects on model specification, estimation and other inference, in what I cail a model-driven approach, as opposed to a data-driven approach in spatial statistics. I attempt to combine a rigorous econometric perspective with a comprehensive treatment of methodological issues in spatial analysis. Although I started working on these issues almost ten years ago, as part of my doctoral dissertation at CorneIl, most of the material in this book is much more recent. Some of it appeared earlier in various forms in a number of journal articles. However, much of the book consists of so far unpublished methods and findings, and the bulk of the empirical examples are new. By combining a fairly technical treatment of tests and estimators with an extensive set of illustrations and practical considerations, I hope I have achieved a mix which is of interest both to researchers in regional science and spatial analysis as weil as to applied econometricians. The breadth of the topics considered should also make the book appropriate as a graduate level text in spatial statistics, econometrics and regional analysis. In the development of the research behind this book, and in the writing of previous versions of various parts of the manuscript, I have benefitted greatly from the comments, suggestions and encouragement from a number of people, and especiaily from Walter Isard, Art Getis, Reg GoIledge, Geoff Hewings and Peter Nijkamp. Much of the research was supported by grants SES and SES from the National Science Foundation, and part of the empirical work sterns from agrant by the Ohio Board of Regents Urban University Research Program and a University of California at Santa Barbara Senate Research Grant. The Regional Science Association granted permission to use part of the materials from my article on "Specification Tests on the Structure of Interaction in Spatial Econometric Models," which appeared in volume 54, 1984, of the Papers, Regional Science A88ociation. xv

14 xvi Susan Kennedy and Serge Rey read the manuscript aud provided comments from a graduate student perspective. The graphics were produced by David Lawson and Serge Rey, and other general research assistance was provided at various points in time by Ayse Can, Mitchell Glasser, Steve Mikusa and Serge Rey. The book is dedieated to Emily, who served as general editor and graphie illustrator, but who also put up with my one-track mind during the past months. Santa Barbara, January 1988