Spatial Weight Matrices Research Articles

Division of Weight Matrix Based on Pair-Distances and Resulting Nonlinear Spatial Dependency in Spatial Autoregressive Models

In spatial autoregressive models, spatial autocorrelations in the dependent (or omitted) variable are modeled. Dependency is measured under known spatial structures, typically represented as a spatial weight matrix (W). For ordinal spatial autoregressive models, a unique W exists, and the strength of influence of the variable (or autocorrelation) is expressed through this matrix. Elements of W are obtained as a monotonically decreasing function of distance between points within the solution space. Since the coefficients of W are a measure of the autocorrelation, the model is a linear function of the dependency. In actual situations, for example, locational or tax competition, highly complex interdependency may exist. Simultaneously, the model may exhibit negative autocorrelation at short distances, caused by substitution effects, and positive autocorrelation at long distances, caused by complementary effects. In this paper, we discuss realistic models by dividing spatial terms. The resulting nonlinear dependency function is represented through the coefficients of a pair of spatial weight matrices. As an example, we formulate land price models for all prefectures in Japan. From comparisons between the conventional ordinal one-weight-matrix model and the proposed two-weight-matrices model, we established that the latter may produce lower information criterion values. Our results show that division of spatial terms is important for extending the variability of spatial autoregressive models.

Spatial and Temporal Dependence in House Price Prediction

This paper incorporates spatial and temporal dependence among housing transactions in predicting future house prices. We employ the spatiotemporal autoregressive model and structure the spatial and temporal weighting matrices as in Pace et al. (1998). We control for the time variation of both the attribute prices and the spatial and temporal dependence parameters through performing the analysis on an annual basis. Spatial heterogeneity is accounted for using experience-based definition of submarkets by real estate professionals. Using a comprehensive housing transaction data set from the Dutch Randstad region, we show that integrating the spatial and temporal dependence within the hedonic modeling improves the prediction power as compared to traditional hedonic model that neglects these effects.

Gender, Space, and the Location Changes of Jobs and People: A Spatial Simultaneous Equations Analysis

This article summarizes a spatial econometric analysis of local population and employment growth in the Netherlands, with specific reference to impacts of gender and space. The simultaneous equations model used distinguishes between population- and gender-specific employment groups, and includes autoregressive and cross-regressive spatial lags to detect relations both within and among these groups. Spatial weights matrices reflecting different bands of travel times are used to calculate the spatial lags and to gauge the spatial nature of these relations. The empirical results show that although population–employment interaction is more localized for women's employment, no gender difference exists in the direction of interaction. Employment growth for both men and women is more influenced by population growth than vice versa. The interaction within employment groups is even more important than population growth. Women's, and especially men's, local employment growth mostly benefits from the same employment growth in neighboring locations. Finally, interaction between these groups is practically absent, although men's employment growth may have a negative impact on women's employment growth within small geographic areas. In summary, the results confirm the crucial roles of gender and space, and offer important insights into possible relations within and among subgroups of jobs and people.

Spatial interaction models for biomass consumption in the United States

Five alternative spatial interaction patterns of biomass consumption in the United States in 2005 are compared using the spatial autoregressive model. The influences of geographical locations, biomass price and income on biomass consumption are translated into the spatial weight matrices of spatial autoregressive model. The results indicate that not only the geographical locations but also both the biomass price and the income significantly affect spatial interaction among biomass consumption in the United States. The results also show that spatial interaction among biomass consumption in the United States becomes weaker with the farther neighbor states. Spatial interaction among biomass consumption incurred by the income becomes stronger than that incurred by the biomass price. When the influences of both the biomass price and the income are combined together into the hybrid spatial autoregressive model, spatial interaction among biomass consumption is the strongest.

On model specification and parameter space definitions in higher order spatial econometric models

Higher-order spatial econometric models that include more than one weights matrix have seen increasing use in the spatial econometrics literature. There are two distinct issues related to the specification of these extended models. The first issue is what form the higher-order spatial econometric model takes, i.e. higher-order polynomials in the spatial weights matrices vs. higher-order spatial autoregressive processes. The second issue relates to the parameter space in such models and how this can affect the choice of model specification, estimation, and inference. We outline a procedure that is simple both mathematically and computationally for finding the stationary region for spatial econometric models with up to K weights matrices for higher-order spatial autoregressive processes. We also compare and contrast this approach with the parameter space for models that incorporate higher-order polynomials in the spatial weights matrices. Regardless of the model utilized in empirical practice, ignoring the relevant parameter region can lead to incorrect inferences regarding both the nature of the spatial autocorrelation process and the effects of changes in covariates on the dependent variable.

A Novel Method to Incorporate the Spatial Location of the Lung Dose Distribution into Predictive Radiation Pneumonitis Modeling

Studies have proposed that patients who receive radiation therapy to the base of the lung are more susceptible to radiation pneumonitis than patients who receive therapy to the apex of the lung. The primary purpose of the present study was to develop a novel method to incorporate the lung dose spatial information into a predictive radiation pneumonitis model. A secondary goal was to apply the method to a 547 lung cancer patient database to determine whether including the spatial information could improve the fit of our model. The three-dimensional dose distribution of each patient was mapped onto one common coordinate system. The boundaries of the coordinate system were defined by the extreme points of each individual patient lung. Once all dose distributions were mapped onto the common coordinate system, the spatial information was incorporated into a Lyman-Kutcher-Burman predictive radiation pneumonitis model. Specifically, the lung dose voxels were weighted using a user-defined spatial weighting matrix. We investigated spatial weighting matrices that linearly scaled each dose voxel according to the following orientations: superior-inferior, anterior-posterior, medial-lateral, left-right, and radial. The model parameters were fit to our patient cohort with the endpoint of severe radiation pneumonitis. The spatial dose model was compared against a conventional dose-volume model to determine whether adding a spatial component improved the fit of the model. Of the 547 patients analyzed, 111 (20.3%) experienced severe radiation pneumonitis. Adding in a spatial parameter did not significantly increase the accuracy of the model for any of the weighting schemes. A novel method was developed to investigate the relationship between the location of the deposited lung dose and pneumonitis rate. The method was applied to a patient database, and we found that for our patient cohort, the spatial location does not influence the risk of pneumonitis.

Modeling and Prediction of Tree Height–Diameter Relationships Using Spatial Autoregressive Models

Three spatial autoregressive models were applied to model the tree height-diameter relationship, including spatial lag model (SLM), spatial error model (SEM), and spatial Durbin model (SDM), with ordinary least squares (OLS) as a benchmark. Five spatial weight matrices were used to evaluate the impacts of different weighting schemes on model fitting. It was evident that different schemes of spatial weight matrix strongly affected the model fitting and parameter estimation in these spatial autoregressive models. We found that the variogram or geostatistical weight matrix was superior to other spatial weight matrices such as contiguity, inverse distances, and local G i* statistics. Further, the spatial autoregressive models were used to predict tree heights at unsampled locations. The results showed that the three spatial autoregressive models overperformed the OLS model not only in model fitting and reducing spatial dependence, but also in model predictions. In general, SDM and SEM performed significantly better than SLM, whereas SDM was slightly better than SEM in each aspect of model fitting and prediction. However, if the complexity of the model structure is a concern, SEM with a geostatistical weight matrix is a reasonable choice over SDM because SEM offers the model coefficient estimates close to those of the OLS model, which makes the interpretation and understanding of the model much easier. FOR .S CI. 57(3):252-264.

Economic Structure and Regional Disparity in China: Beyond the Kuznets Transition

The literature on regional change in post-reform China suggests a consistent pattern of increasing regional disparity during 1990s. This article explains disparity through the lens of industrial transition as reflected in the three major economic sectors, agriculture, manufacturing, and services from 1995 to 2004. Increased economic output in China has been accompanied by dramatic changes in employment structure at both the national and regional level. Changes at the provincial level have been driven by national trends and changes of industry mix as well as regional characteristics. The importance of industry mix and regional competitive advantage varies across sectors and has different impacts on employment and output. This analysis indicates that employment loss in agriculture in the western and central regions has not been made up by the increases in other sectors. The differential rate of transition in economic structure toward the secondary and tertiary sectors has contributed to the widening gap between the coastal areas and the other parts of the country. Our exploratory spatial analysis with regard to the extended shift—share components indicates significant spatial autocorrelation both at the sectoral and provincial level. The necessity of integrating spillover effects into any policy intervention is demonstrated. Then sensitivity analysis is carried out to examine how the selection of two spatial weight matrices will influence the decomposition results.

A dynamic spatial panel data approach to the German wage curve

A wage curve is a decreasing function of wages on the regional unemployment rate. Most empirical studies on the wage curve ignore possible spatial interaction effects between the regions which are the primary units of research. This paper reconsiders the western German wage curve with a special focus on the geography of labour markets. Spillovers between regions are taken into account. The paper tests whether the unemployment rate in the larger surrounding region also affects wages. In addition, agglomeration effects and effects of local monopsony are assessed. The main database is a random sample of 974,179 employees observed over the period 1980–2004 and covering 326 NUTS3 units (districts). This rich data set is used to estimate a dynamic wage curve according to the two-step approach of Bell et al. (2002). In the first step one controls for individual heterogeneity and in the second step one allows for spatial effects of unemployment across regions on wages. We check the sensitivity of this wage elasticity to various spatial weight matrices as well as allowing for the endogeneity of unemployment. We also estimate the wage elasticity for various population groups.

Bayesians in Space: Using Bayesian Methods to Inform Choice of Spatial Weights Matrix in Hedonic Property Analyses

The choice of weights is a non-nested problem in most applied spatial econometric models. Despite numerous recent advances in spatial econometrics, the choice of spatial weights remains exogenously determined by the researcher in empirical applications. Bayesian techniques provide statistical evidence regarding the simultaneous choice of model specification and spatial weights matrices by using posterior probabilities. This paper demonstrate the Bayesian estimation approach in a spatial hedonic property model estimating the impacts of repeated wildfires on house prices in Southern California. We find that improper choice of spatial model and weights can result in up to 5 percent difference in estimated coefficients and in our case study up to a $15 Million difference in total benefits of reducing wildfires in Los Angeles County.

Forecasting with spatial panel data

Various forecasts using panel data with spatial error correlation are compared using Monte Carlo experiments. The true data generating process is assumed to be a simple error component regression model with spatial remainder disturbances of the autoregressive or moving average type. The best linear unbiased predictor is compared with other forecasts ignoring spatial correlation, or ignoring heterogeneity due to the individual effects. In addition, the root mean squared error performance of these forecasts is examined under misspecification of the spatial error process, various spatial weight matrices, and heterogeneous rather than homogeneous panel data models.

Seemingly unrelated regressions with spatial error components

This article considers various estimators using panel data seemingly unrelated regressions (SUR) with spatial error correlation. The true data generating process (DGP) is assumed to be SUR with spatial error of the autoregressive or moving average type. Moreover, the remainder term of the spatial process is assumed to follow an error component structure. Both maximum likelihood (ML) and generalized moments (GM) methods of estimation are used. Using Monte Carlo experiments, we check the performance of these estimators and their forecasts under misspecification of the spatial error process, various spatial weight matrices, and heterogeneous versus homogeneous panel data models.

Efficient Maximum Likelihood Estimation of Spatial Autoregressive Models with Normal But Heteroskedastic Disturbances

The likelihood functions for spatial autoregressive models with normal but heteroskedastic disturbances have been derived [Anselin (1988, ch.6)], but there is no implementation of maximum likelihood estimation for these likelihood functions in general cases with heteroskedastic disturbances. Therefore, less efficient IV-based methods must be applied if disturbance terms might be heteroskedastic. In the present paper, we develop a new computer program for maximum likelihood estimation of heteroskedastic models that also utilizes multiple spatial weight matrices and confirm the efficiency of the obtained estimator in cases with heteroskedastic disturbances through Monte Carlo simulations.

Economic Structure and Regional Disparity in China: Beyond the Kuznets Transition

The literature on regional change in post-reform China suggests a consistent pattern of increasing regional disparity during 1990s. This paper explains disparity through the lens of industrial transition as reflected in the three major economic sectors, agriculture, manufacturing, and services from 1995 to 2004. Increased economic output in China has been accompanied by dramatic changes in employment structure at both the national and regional level. Changes at the provincial level have been driven by national trends and changes of industry mix as well as regional characteristics. The importance of industry mix and regional competitive advantage varies across sectors and has different impacts on employment and output. This analysis indicates that employment loss in agriculture in the western and central regions has not been made up by the increases in other sectors. The differential rate of transition in economic structure toward the secondary and tertiary sectors has contributed to the widening gap between the coastal areas and the other parts of the country. Our exploratory spatial analysis with regard to the extended shift-share components indicates significant spatial autocorrelation both at the sectoral and provincial level. The necessity of integrating spillover effects into any policy intervention is demonstrated. Then sensitivity analysis is carried out to examine how the selection of two spatial weight matrices will influence the decomposition results.

Seemingly Unrelated Regressions with Spatial Error Components

This paper considers various estimators using panel data seemingly unrelated regressions (SUR) with spatial error correlation. The true data generating process is assumed to be SUR with spatial error of the autoregressive or moving average type. Moreover, the remainder term of the spatial process is assumed to follow an error component structure. Both maximum likelihood and generalized moments (GM) methods of estimation are used. Using Monte Carlo experiments, we check the performance of these estimators and their forecasts under misspecification of the spatial error process, various spatial weight matrices, and heterogeneous versus homogeneous panel data models.

A Dynamic Spatial Panel Data Approach to the German Wage Curve

A wage curve is a decreasing function of wages on the regional unemployment rate. Most empirical studies on the wage curve ignore possible spatial interaction effects between the regions which are the primary units of research. This paper reconsiders the western German wage curve with a special focus on the geography of labour markets. Spillovers between regions are taken into account. The paper tests whether the unemployment rate in the larger surrounding region also affects wages. In addition, agglomeration effects and effects of local monopsony are assessed. The main data base is a random sample of 974,179 employees observed over the period 1980-2004 and covering 326 NUTS3 units (districts). This rich data set is used to estimate a dynamic wage curve according to the two-step approach of Bell et al. (2002). In the first step one controls for individual heterogeneity and in the second step one allows for spatial effects of unemployment across regions on wages. We check the sensitivity of this wage elasticity to various spatial weight matrices as well as allowing for the endogeneity of unemployment. We also estimate the wage elasticity for various population groups.

Spatial Weights Matrices

The impetus for Cliff and Ord's landmark article, “The Problem of Spatial Autocorrelation,” was to avoid the problem of topological invariance in the spatial weights matrix. The problem and three points of view on it – theoretical, topological, and empirical – are discussed.

Service industry and cumulative growth in the regions of Europe

European regions have experienced a greater presence of service producers in their economy over the last few decades. Indeed, the manufacturing sector increasingly contracts out many activities to intermediate producer services. This is mostly because they are located close to each other and because services experience increasing returns to scale which reduce their marginal costs. In this paper, we propose to measure the extent to which productivity in services has converged across European regions. The model we use, originally developed by Verdoorn (1949), takes the increasing returns to scale explicitly into account. We apply spatial econometric techniques and control for border effects by introducing two different spatial weights matrices under the assumption that economic interactions decrease very substantially when a national border is passed. Furthermore, we take proper care of the presence of both types (spatial and non-spatial) of endogeneity by using spatial two stages least squares (Kelejian and Prucha 1998). Our conclusions bring new insights in the identification of regional productivity differentials.

Geographical distribution of crime in Italian provinces: a spatial econometric analysis

In the last years, the increasing level of criminality that has characterized the modern economies has drawn the attention of sociologists and economists in order to identify the causes leading to commit criminal offences. The aim of the pa- per is to investigate the causes of crime activity in 103 Italian provinces (NUTS3 regions) for the years 1999 and 2003. The Italian crime phenomenon is characterized by some stylized facts: high spatial and time variability of crime activities, and pres- ence of 'organized crime' (e.g. Mafia and Camorra) localized in some local territorial areas. Using exploratory spatial data analysis (ESDA), the paper firstly explores the spa- tial structure and distribution of four different typologies of crimes: murders, thefts, frauds and squeezes. ESDA allows us to detect some important geographical dimen- sions and to distinguish crucial micro- and macro- territorial aspects of offences. Further, on the basis of Becker-Ehrlich model, a spatial cross-sectional model - in- cluding deterrence, economic and socio-demographic variables - has been performed to investigate the determinants of Italian crime in 1999 and 2003 and its 'neighbour- ing' effects, measured in terms of 'geographical' and 'relational' proximities. The empirical results obtained by using different spatial weights matrices high- lighted that socio-economic variables have a relevant impact on crime activities, but their role changes enormously respect to crimes against person (murders) or against property (thefts, frauds and squeezes) and over time.

Forecasting with Spatial Panel Data

This paper compares various forecasts using panel data with spatial error correlation. The true data generating process is assumed to be a simple error component regression model with spatial remainder disturbances of the autoregressive or moving average type. The best linear unbiased predictor is compared with other forecasts ignoring spatial correlation, or ignoring heterogeneity due to the individual effects, using Monte Carlo experiments. In addition, we check the performance of these forecasts under misspecification of the spatial error process, various spatial weight matrices, and heterogeneous rather than homogeneous panel data models.