Raymond J.G.M. Florax

Simple diagnostic tests for spatial dependence

In this paper we propose simple diagnostic tests, based on ordinary least-squares (OLS) residuals, for spatial error autocorrelation in the presence of a spatially lagged dependent variable and for spatial lag dependence in the presence of spatial error autocorrelation, applying the modified Lagrange multiplier (LM) test developed by Bera and Yoon (Econometric Theory, 1993, 9, 649-658). Our new tests may be viewed as computationally simple and robust alternatives to some existing procedures in spatial econometrics. We provide empirical illustrations to demonstrate the usefulness of the proposed tests. The finite sample size and power performance of the tests are also investigated through a Monte Carlo study. The results indicate that the adjusted LM tests have good finite sample properties. In addition, they prove to be more suitable for the identification of the source of dependence (lag or error) than their unadjusted counterparts.

New directions in spatial econometrics.

1 New Directions in Spatial Econometrics: Introduction.- 1.1 Introduction.- 1.2 Spatial Effects in Regression Models.- 1.2.1 Specification of Spatial Dependence.- 1.2.2 Spatial Data and Model Transformations.- 1.3 Spatial Effects in Limited Dependent Variable Models.- 1.4 Heterogeneity and Dependence in Space-Time Models.- 1.5 Future Directions.- References.- I-A: Spatial Effects in Linear Regression Models Specification of Spatial Dependence.- 2 Small Sample Properties of Tests for Spatial Dependence in Regression Models: Some Further Results.- 2.1 Introduction.- 2.2 Tests for Spatial Dependence.- 2.2.1 Null and Alternative Hypotheses.- 2.2.2 Tests for Spatial Error Dependence.- 2.2.3 Tests for Spatial Lag Dependence.- 2.3 Experimental Design.- 2.4 Results of Monte Carlo Experiments.- 2.4.1 Empirical Size of the Tests.- 2.4.2 Power of Tests Against First Order Spatial Error Dependence.- 2.4.3 Power of Tests Against Spatial Autoregressive Lag Dependence.- 2.4.4 Power of Tests Against Second Order Spatial Error Dependence.- 2.4.5 Power of Tests Against a SARMA (1,1) Process.- 2.5 Conclusions.- Acknowledgements.- References.- Appendix 1: Tables.- 3 Spatial Correlation: A Suggested Alternative to the Autoregressive Model.- 3.1 Introduction.- 3.2 The Spatial AR Model of Autocorrelation.- 3.3 The Singularity of (I - pM).- 3.3.1 Theoretical Issues.- 3.3.2 Independent Corroborative Evidence.- 3.4 The Parameter Space.- 3.5 A Suggested Variation of the Spatial AR Model.- 3.5.1 The Suggested Model.- 3.5.2 Some Limiting Correlations.- 3.5.3 A Generalization.- 3.6 Suggestions for Further Work.- Acknowledgements.- References.- Appendix 1: Spatial Weighting Matrices.- 4 Spatial Autoregressive Error Components in Travel Flow Models: An Application to Aggregate Mode Choice.- 4.1 Introduction.- 4.2 The First-Order Spatially Autoregressive Error Components Formulation.- 4.3 Estimation Issues.- 4.4 Empirical Example.- 4.4.1 An Illustration Based on Synthetic Data.- 4.5 Conclusions.- References.- I-B: Spatial Effects in Linear Regression Models Spatial Data and Model Transformations.- 5 The Impacts of Misspecified Spatial Interaction in Linear Regression Models.- 5.1 Introduction.- 5.2 Aggregation and the Identification of Spatial Interaction.- 5.3 Experimental Design.- 5.3.1 Sample Size.- 5.3.2 Spatial Interaction Structures.- 5.3.3 Spatial Models and Parameter Space.- 5.3.4 Test Statistics and Estimators.- 5.3.5 Forms of Misspecification.- 5.4 Empirical Results.- 5.4.1 Size of Tests Under the Null.- 5.4.2 Power of Tests.- 5.4.3 Misspecification Effects on the Power of Tests for Spatial Dependence.- 5.4.4 Sensitivity of Parameter Estimation to Specification of Weight Matrix.- 5.4.5 Impact of Misspecification of Weight Matrix on Estimation.- 5.5 General Inferences References.- 6 Computation of Box-Cox Transform Parameters: A New Method and its Application to Spatial Econometrics.- 6.1 Introduction.- 6.2 The Elasticity Method: Further Elaboration.- 6.2.1 Linearization Bias.- 6.2.2 Discretization Bias.- 6.2.3 Specification Bias.- 6.3 The One Exogenous Variable Test.- 6.4 An Application to Spatial Econometrics.- 6.5 The Multiple Exogenous Variable Computation.- 6.6 Conclusions.- References.- 7 Data Problems in Spatial Econometric Modeling.- 7.1 Introduction.- 7.2 Data for Spatial Econometric Analysis.- 7.3 Data Problems in Spatial Econometrics.- 7.4 Methodologies for Handling Data Problems.- 7.4.1 Influential Cases in the Standard Regression Model.- 7.4.2 Influential Cases in a Spatial Regression Model.- 7.4.3 An Example.- 7.5 Implementing Methodologies.- References.- 8 Spatial Filtering in a Regression Framework: Examples Using Data on Urban Crime, Regional Inequality, and Government Expenditures.- 8.1 Introduction.- 8.2 Rationale for a Spatial Filter.- 8.3 The Gi Statistic.- 8.4 The Filtering Procedure.- 8.5 Filtering Variables: Three Examples.- 8.5.1 Example 1: Urban Crime.- 8.5.2 Example 2: Regional Inequality.- 8.5.3 Example 3: Government Expenditures.- >8.6 Conclusions.- >Acknowledgments.- References.- II: Spatial Effects in Limited Dependent Variable Models.- 9 Spatial Effects in Probit Models: A Monte Carlo Investigation.- 9.1 Introduction.- 9.2 Sources of Heteroscedasticity.- 9.3 Heteroscedastic Probit.- 9.4 Monte Carlo Design.- 9.5 Tests.- 9.6 Monte Carlo Results.- 9.7 Conclusions.- References.- Appendix 1: Monte Carlo Results.- Appendix 2: Heteroscedastic Probit Computer Programs.- Appendix 3: Monte Carlo Computer Programs.- 10 Estimating Logit Models with Spatial Dependence.- 10.1 Introduction.- 10.1.1 Model.- 10.2 Simulation Example.- 10.3 Conclusions.- >References.- Appendix 1: Gauss Program for Finding ML Estimates.- Appendix 2: Gauss Program to Estimate Asymptotic Variances of ML Estimates.- 11 Utility Variability within Aggregate Spatial Units and its Relevance to Discrete Models of Destination Choice.- 11.1 Introduction.- 11.2 Theoretical Background.- 11.3 Estimation of the Maximum Utility Model.- 11.4 Model Specifications and Simulations.- 11.4.1 Specification Issues.- 11.4.2 Description of Simulation Method.- 11.4.3 Results.- 11.5 Conclusions.- Acknowledgement.- References.- III: Heterogeneity and Dependence in Space-Time Models.- 12 The General Linear Model and Spatial Autoregressive Models.- 12.1 Introduction.- 12.2 The GLM.- 12.3 Data Preprocessing.- 12.3.1 Analysis of the 1964 Benchmark Data.- 12.3.2 Evaluation of Missing USDA Values Estimation.- >12.4 Implementation of the Spatial Statistical GLM.- 12.4.1 Preliminary Spatial Analysis of Milk Yields: AR Trend Surface GLMs.- 12.4.2 AR GLM Models for the Repeated Measures Case.- 12.4.3 A Spatially Adjusted Canonical Correlation Analysis of the Milk Production Data.- 12.5 Conclusions.- >References.- >Appendix 1: SAS Computer Code to Compute the Popular Spatial Autocorrelation Indices.- Appendix 2: SAS Code for Estimating Missing Values in the 1969 Data Set.- Appendix 3: SAS Code for 1969 USDA Data Analysis.- 13 Econometric Models and Spatial Parametric Instability: Relevant Concepts and an Instability Index.- 13.1 Introduction.- 13.2 The Expansion Method.- 13.3 Parametric Instability.- 13.3.1 Example.- 13.4 Conclusions.- 13.4.1 Instability Measures: Scope.- 13.4.2 Instability Measures: Significance.- References.- 14 Bayesian Hierarchical Forecasts for Dynamic Systems: Case Study on Backcasting School District Income Tax Revenues.- 14.1 Introduction.- 14.2 Literature Review.- 14.3 The C-MSKF Model: Time Series Prediction with Spatial Adjustments.- 14.3.1 Multi-State Kaiman Filter.- 14.3.2 Spatial Adjustment via Hierarchical Random Effects Model.- 14.3.3 CIHM Method.- 14.3.4 C-MSKF.- 14.4 Case Study and Observational Setting.- 14.4.1 Data.- 14.4.2 Treatments.- 14.5 Results.- >14.6 Conclusions.- >References.- Appendix 1: Poolbayes Program.- 15 A Multiprocess Mixture Model to Estimate Space-Time Dimensions of Weekly Pricing of Certificates of Deposit.- 15.1 Introduction.- 15.2 A Dynamic Targeting Model of CD Rate-Setting Behavior.- 15.2.1 The Model.- 15.2.2 The Decision Rule.- 15.3 The Spatial Econometric Model.- 15.3.1 Spatial Time-Varying Parameters.- 15.3.2 Parameter Estimation.- 15.3.3 Testing Hypotheses with the Model.- 15.4 Implementing the Model.- 15.4.1 The Data.- 15.4.2 Prior Information.- 15.4.3 Empirical Results.- 15.5 Conclusions.- Acknowledgements.- References.- Appendix 1: FORTRAN Program for the Spatial Mixture.- Author Index.- Contributors.

Price and Income Elasticities of Residential Water Demand: A Meta-Analysis

This article presents a meta-analysis of variations in price and income elasticities of residential water demand. Meta-analysis constitutes an adequate tool to synthesize research results by means of an analysis of the variation in empirical estimates reported in the literature. We link the variation in estimated elasticities to differences in theoretical microeconomic choice approaches, differences in spatial and temporal dynamics, as well as differences in research design of the underlying studies. The occurrence of increasing or decreasing block rate systems turns out to be important. With respect to price elasticities, the use of the discrete-continuous choice approach is relevant in explaining observed differences. (JEL H31, Q25)

Small sample properties of tests for spatial dependence in regression models : some further results.

It has now been more than two decades since Cliff and Ord (1972) and Hordijk (1974) applied the principle of Moran’s Itest for spatial autocorrelation to the residuals of regression models for cross-sectional data. To date, Moran’sIstatistic is still the most widely applied diagnostic for spatial dependence in regression models [e.g., Johnston (1984), King (1987), Case (1991)]. However, in spite of the well known consequences of ignoring spatial dependence for inference and estimation [for a review, see Anselin (1988a)], testing for this type of misspecification remains rare in applied empirical work, as illustrated in the surveys of Anselin and Griffith (1988) and Anselin and Hudak (1992). In part, this may be due to the rather complex expressions for the moments of Moran’s I, and the difficulties encountered in implementing them in econometric Software [for detailed discussion, see Cliff and Ord (1981), Anselin (1992), Tiefelsdorf and Boots (1994)]. Recently, a number of alternatives to Moran’s Ihave been developed, such as the tests of Burridge (1980) and Anselin (1988b, 1994), which are based on the Lagrange Multiplier (LM) principle, and the robust tests of Bera and Yoon (1992) and Kelejian and Robinson (1992). These tests are all asymptotic and distributed as X 2variates. Since they do not require the computation of specific moments of the statistic, they are easy to implement and straightforward to interpret. However, they are all large sample tests and evidence on their finite sample properties is still limited.

THE VALUE OF STATISTICAL LIFE IN ROAD SAFETY: A META-ANALYSIS

Costs of accidents make up an important part of the total external cost of traffic. A substantial proportion of accident costs is related to fatal accidents. In the evaluation of fatal accident costs the availability of an estimate of the economic value of a statistical life is pivotal. We present an overview of the empirical literature on the value of statistical life in road safety (VOSL), and use meta-analysis to determine variables that explain the variation in VOSL estimates reported in the literature. We show that the magnitude of VOSL estimates depends on the value assessment approach (particularly, stated versus revealed preference), and for contingent valuation studies also on the type of payment vehicle and elicitation format. We explain that VOSL estimates cannot simply be averaged over studies. The magnitude of VOSL is intrinsically linked to the initial level of the risk of being caught up in a fatal traffic accident and to the risk decline implied by the research set-up.

Specification and estimation of spatial linear regression models: Monte Carlo evaluation of pre-test estimators

Abstract Spatially correlated residuals lead to various serious problems in applied spatial research. In this paper several conventional specification and estimation procedures for models with spatially dependent residuals are compared with alternative procedures. The essence of the latter is a search procedure for spatially lagged variables. By incorporating the omitted spatially lagged variables into the model spatially dependent residuals may be remedied, in particular if the spatial dependence is substantive. The efficacy of the conventional and alternative procedures in small samples will be investigated by means of Monte Carlo techniques for an irregular lattice structure.

A Meta-Analysis on the Relationship between Income Inequality and Economic Growth

In recent years there is a growing interest in determining the impact of inequality on economic growth. Theoretical papers as well as empirical applications have, however, produced controversial results. Although there is a considerable part of the literature that considers inequality detrimental to growth, more recent studies have challenged this result and found a positive effect of inequality on growth. In this paper, we provide a contribution to the empirical puzzle by using meta-analysis to systematically describe, identify and analyse the variation in outcomes of empirical studies. We find that estimation methods, data quality and sample coverage systematically affect the results. The results point out that it will be particularly useful to increasingly focus research on determining the impact of income inequality on economic growth using single-country data at the regional level, or a relatively homogeneous set of countries with adequate controls for country-wide differences in economic, social and institutional characteristics.

Determinants of the Regional Demand for Higher Education in The Netherlands: A Gravity Model Approach

Sá C., Florax R. J. G. M. and Rietveld P. (2004) Determinants of the regional demand for higher education in The Netherlands: a gravity model approach, Reg. Studies 38, 375–392. Studies on the determinants of the demand for higher education typically emphasize the relevance of socio-economic factors, but leave the spatial dimensions of the prospective students’ university choices largely unexplored. In this study, we investigate the determinants of university entrance for Dutch high school graduates in 2000, and pay particular attention to the attractiveness of the university, both in terms of its accessibility and the educational quality of its programme. We combine cross-section data on the region of origin of the high school graduate and the university destination region for first-year students with regional and university characteristics in a production- constrained gravity model. The main finding of the study is that the behaviour of prospective students is governed by a distance deterrence effect and a downward rent effect, but a positive impact results from regional/urban amenities rather than from the educational quality of the university programmes.

The Performance of Diagnostics for Spatial Dependence in Linear Regression Models: A Meta-Analysis of Simulation Studies

One of the reasons for A.D. Cliff and J.K. Ord’s 1973 book “Spatial Autocorrelation” achieving the status of a seminal work on spatial statistics and econometrics lies in their careful and lucid treatment of the autocorrelation problem in spatial data series. Cliff and Ord present test statistics for univariate spatial series of categorical (nominal and ordinal) and continuous (interval or ratio scale) data. They extend the use of autocorrelation statistics, specifically Moran’s I (Moran, 1948), to the analysis of regression residuals (see also Cliff and Ord, 1972). The detection of spatial autocorrelation among regression residuals implies either a nonlinear relationship between the dependent and independent variables, the omission of one or more spatially correlated regressors, or the appropriateness of an autoregressive error structure. Ignoring the presence of spatial autocorrelation among the population errors causes ordinary least squares (OLS) to be a biased variance estimator and an inefficient regression coefficient estimator. Anselin (1988b) shows that erroneously omitting the spatially lagged dependent variable from the set of explanatory variables causes the OLS estimator to be biased and inconsistent. Cliff and Ord (1981, p. 197) therefore urge the applied researcher to always apply “some check for autocorrelation,” and take remedial action when necessary.

Spatial Econometric Data Analysis: Moving Beyond Traditional Models:

This article appraises recent advances in the spatial econometric literature. It serves as the introduction to a collection of new papers on spatial econometric data analysis brought together in this special issue, dealing specifically with new extensions to the spatial econometric modeling perspective. Although the initial development of the field of spatial econometrics has been rather slow, the Dixit-Stiglitz revolution and the emergence of the New Economy Geography have been instrumental in uplifting the significance and the use of spatial data analysis techniques. Concurrent developments in other social sciences parallel this situation in economics. The upsurge in spatial econometrics is, among other things, driven by the recognition that traditional spatial econometric models are insufficient to capture modern theoretical developments. Therefore, this issue brings together a collection of articles on space-time and discrete choice modeling, spatial nonstationarity, and the methodology and empirics of regional economic growth models.