Properties of Tests for Spatial Dependence in Linear Regression Models

Based on a large number of Monte Carlo simulation experiments on a regular lattice, we compare the properties of Moran's I and Lagrange multiplier tests for spatial dependence, that is, for both spatial error autocorrelation and for a spatially lagged dependent variable. We consider both bias and power of the tests for six sample sizes, ranging from twenty-five to 225 observations, for different structures of the spatial weights matrix, for several underlying error distributions, for misspecified weights matrices, and for the situation where boundary effects are present. The results provide an indication of the sample sizes for which the asymptotic properties of the tests can be considered to hold. They also illustrate the power of the Lagrange multiplier tests to distinguish between substantive spatial dependence (spatial lag) and spatial dependence as a nuisance (error autocorrelation).     

Evaluating the Temporal and Spatial Heterogeneity of the European Convergence Process, 1980–1999*

Abstract. In this paper, we suggest a framework that allows testing simultaneously for temporal heterogeneity, spatial heterogeneity, and spatial autocorrelation in β‐convergence models. Based on a sample of 145 European regions over the 1980–1999 period, we estimate a Seemingly Unrelated Regression Model with spatial regimes and spatial autocorrelation for two sub‐periods: 1980–1989 and 1989–1999. The assumption of temporal independence between the two periods is rejected, and the estimation results point to the presence of spatial error autocorrelation in both sub‐periods and spatial instability in the second sub‐period, indicating the formation of a convergence club between the peripheral regions of the European Union.     

Using Spatial Filters and Exploratory Data Analysis to Enhance Regression Models of Spatial Data

Residual spatial autocorrelation is a situation frequently encountered in regression analysis of spatial data. The statistical problems arising due to this phenomenon are well‐understood. Original developments in the field of statistical analysis of spatial data were meant to detect spatial pattern, in order to assess whether corrective measures were required. An early development was the use of residual autocorrelation as an exploratory tool to improve regression analysis of spatial data. In this note, we propose the use of spatial filtering and exploratory data analysis as a way to identify omitted but potentially relevant independent variables. We use an example of blood donation patterns in Toronto, Canada, to demonstrate the proposed approach. In particular, we show how an initial filter used to rectify autocorrelation problems can be progressively replaced by substantive variables. In the present case, the variables so retrieved reveal the impact of urban form, travel habits, and demographic and socio‐economic attributes on donation rates. The approach is particularly appealing for model formulations that do not easily accommodate positive spatial autocorrelation, but should be of interest as well for the case of continuous variables in linear regression.     

A Geostatistical Linear Regression Model for Small Area Data. 一种适用于小区域数据的地统计线性回归模型

We present a new linear regression model for use with aggregated, small area data that are spatially autocorrelated. Because these data are aggregates of individual‐level data, we choose to model the spatial autocorrelation using a geostatistical model specified at the scale of the individual. The autocovariance of observed small area data is determined via the natural aggregation over the population. Unlike lattice‐based autoregressive approaches, the geostatistical approach is invariant to the scale of data aggregation. We establish that this geostatistical approach also is a valid autoregressive model; thus, we call this approach the geostatistical autoregressive (GAR) model. An asymptotically consistent and efficient maximum likelihood estimator is derived for the GAR model. Finite sample evidence from simulation experiments demonstrates the relative efficiency properties of the GAR model. Furthermore, while aggregation results in less efficient estimates than disaggregated data, the GAR model provides the most efficient estimates from the data that are available. These results suggest that the GAR model should be considered as part of a spatial analyst's toolbox when aggregated, small area data are analyzed. More important, we believe that the GAR model's attention to the individual‐level scale allows for a more flexible and theory‐informed specification than the existing autoregressive approaches based on an area‐level spatial weights matrix. Because many spatial process models, both in geography and in other disciplines, are specified at the individual level, we hope that the GAR covariance specification will provide a vehicle for a better informed and more interdisciplinary use of spatial regression models with area‐aggregated data.     

A Comparison of Spatially Varying Regression Coefficient Estimates Using Geographically Weighted and Spatial‐Filter‐Based Techniques

Geographically weighted regression (GWR) is a technique that explores spatial nonstationarity in data‐generating processes by allowing regression coefficients to vary spatially. It is a widely applied technique across domains because it is intuitive and conforms to the well‐understood framework of regression. An alternative method to GWR that has been suggested is spatial filtering, which it has been argued provides a superior alternative to GWR by producing spatially varying regression coefficients that are not correlated with each other and which display less spatial autocorrelation. It is, therefore, worthwhile to examine these claims by comparing the output from both methods. We do this by using simulated data that represent two sets of spatially varying processes and examining how well both techniques replicate the known local parameter values. The article finds no support that spatial filtering produces local parameter estimates with superior properties. The results indicate that the original spatial filtering specification is prone to overfitting and is generally inferior to GWR, while an alternative specification that minimizes the mean square error (MSE) of coefficient estimates produces results that are similar to GWR. However, since we generally do not know the true coefficients, the MSE minimizing specification is impractical for applied research.     

A New Perspective about Moran's Coefficient: Spatial Autocorrelation as a Linear Regression Problem. Moran系数的新视角：空间自相关视为线性回归问题

The computation of Moran's index of spatial autocorrelation requires the definition of a spatial weighting matrix. The eigendecomposition of this doubly centered matrix (i.e., one that forces the sums of all rows and columns to equal zero) has interesting properties that have been exploited in various contexts: distribution properties of the Moran coefficient (MC), spatial filtering in linear models, generalized linear models, and multivariate analysis. In this article, this eigendecomposition is used to propose a new view of MC based on its interpretation in the simple context of linear regression. I use this interpretation to demonstrate the different properties of MC and also the inefficiency of this index in some situations involving simultaneous positive and negative spatial autocorrelation. I propose some new statistics and procedures for testing spatial autocorrelation, and conduct a simulation study to evaluate these new approaches.     

PERSISTENCE OF REGIONAL UNEMPLOYMENT: APPLICATION OF A SPATIAL FILTERING APPROACH TO LOCAL LABOR MARKETS IN GERMANY*

ABSTRACT The geographical distribution and persistence of regional/local unemployment rates in heterogeneous economies (such as Germany) have been, in recent years, the subject of various theoretical and empirical studies. Several researchers have shown an interest in analyzing the dynamic adjustment processes of unemployment and the average degree of dependence of the current unemployment rates or gross domestic product from the ones observed in the past. In this paper, we present a new econometric approach to the study of regional unemployment persistence, in order to account for spatial heterogeneity and/or spatial autocorrelation in both the levels and the dynamics of unemployment. First, we propose an econometric procedure suggesting the use of spatial filtering techniques as a substitute for fixed effects in a panel estimation framework. The spatial filter computed here is a proxy for spatially distributed region‐specific information (e.g., the endowment of natural resources, or the size of the “home market”) that is usually incorporated in the fixed effects coefficients. The advantages of our proposed procedure are that the spatial filter, by incorporating region‐specific information that generates spatial autocorrelation, frees up degrees of freedom, simultaneously corrects for time‐stable spatial autocorrelation in the residuals, and provides insights about the spatial patterns in regional adjustment processes. We present several experiments in order to investigate the spatial pattern of the heterogeneous autoregressive coefficients estimated for unemployment data for German NUTS‐3 regions. We find widely heterogeneous but generally high persistence in regional unemployment rates.     

Comparative Spatial Filtering in Regression Analysis

One approach to dealing with spatial autocorrelation in regression analysis involves the filtering of variables in order to separate spatial effects from the variables’ total effects. In this paper we compare two filtering approaches, both of which allow spatial statistical analysts to use conventional linear regression models. Getis’ filtering approach is based on the autocorrelation observed with the use of the G_i local statistic. Griffith's approach uses an eigenfunction decomposition based on the geographic connectivity matrix used to compute a Moran's I statistic. Economic data are used to compare the workings of the two approaches. A final comparison with an autoregressive model strengthens the conclusion that both techniques are effective filtering devices, and that they yield similar regression models. We do note, however, that each technique should be used in its appropriate context.     

Symptoms of Instability in Models of Spatial Dependence

In recent years, there has been a growing interest in the problems caused by the existence of instability in cross-sectional regressions. The results about local autocorrelation measures are part of this debate, as are the proposals concerning the concept of geographically weighted regressions. This article also deals with the problem of stability (or the lack thereof), but focusing the discussion on the supposition of constancy in the parameter of spatial dependence. In most cases, this assumption is treated, with the risks that this involves, as a maintained hypothesis, which should be ascertained before continuing with the modeling exercise. In the article, we present a simple heterogeneity test for this type of parameters, based on the Lagrange Multiplier principle. To illustrate its use, we take the distribution of per capita income among the European regions as our discussion case. According to our results, there are clear signs of structural breaks in the spatial distribution of this variable and the scale factor and the autocorrelation coefficient appear to be principal actors.     

"The Problem of Spatial Autocorrelation" and Local Spatial Statistics

This article examines the relationship between spatial dependency and spatial heterogeneity, two properties unique to spatial data. The property of spatial dependence has led to a large body of research into spatial autocorrelation and also, largely independently, into geostatistics. The property of spatial heterogeneity has led to a growing awareness of the limitation of global statistics and the value of local statistics and local statistical models. The article concludes with a discussion of how the two properties can be accommodated within the same modelling framework.     

A NEW SPATIAL MULTIPLE DISCRETE‐CONTINUOUS MODELING APPROACH TO LAND USE CHANGE ANALYSIS

This paper formulates a multiple discrete‐continuous probit (MDCP) land use model within a spatially explicit economic structural framework for land use change decisions. The spatial MDCP model is capable of predicting both the type and intensity of urban development patterns over large geographic areas, while also explicitly acknowledging geographic proximity‐based spatial dependencies in these patterns. At a methodological level, the paper focuses on specifying and estimating a spatial MDCP model that allows the dependent variable to exist in multiple discrete states with an intensity associated with each discrete state. The formulation also accommodates spatial dependencies, as well as spatial heterogeneity and heteroskedasticity, in the dependent variable, and should be applicable in a wide variety of fields where social and spatial dependencies between decision agents (or observation units) lead to spillover effects in multiple discrete‐continuous choices (or states). A simulation exercise is undertaken to evaluate the ability of the proposed maximum approximate composite marginal likelihood (MACML) approach to recover parameters from a cross‐sectional spatial MDCP model. The results show that the MACML approach does well in recovering parameters. An empirical demonstration of the approach is undertaken using the city of Austin parcel level land use data.     

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

The aim of this paper is to analyze the intraurban spatial distributions of population and employment in the agglomeration of Dijon (regional capital of Burgundy, France). We study whether this agglomeration has followed the general tendency of job decentralization observed in most urban areas or whether it is still characterized by a monocentric pattern. To that purpose, we use a sample of 136 observations at the communal and at the IRIS (infraurban statistical area) levels with 1999 census data and the employment database SIRENE (INSEE). First, we study the spatial pattern of total employment and employment density using exploratory spatial data analysis. Apart from the CBD, few IRIS are found to be statistically significant, a result contrasting with those found using standard methods of subcenter identification with employment cut‐offs. Next, in order to examine the spatial distribution of residential population density, we estimate and compare different specifications: exponential negative, spline‐exponential, and multicentric density functions. Moreover, spatial autocorrelation, spatial heterogeneity, and outliers are controlled for by using the appropriate maximum likelihood, generalized method of moments, and Bayesian spatial econometric techniques. Our results highlight again the monocentric character of the agglomeration of Dijon.     

Designs for Detecting Spatial Dependence

The aim of this article is to find optimal or nearly optimal designs for experiments to detect spatial dependence that might be in the data. The questions to be answered are: how to optimally select predictor values to detect the spatial structure (if it is existent) and how to avoid to spuriously detect spatial dependence if there is no such structure. The starting point of this analysis involves two different linear regression models: (1) an ordinary linear regression model with i.i.d. error terms—the nonspatial case and (2) a regression model with a spatially autocorrelated error term, a so-called simultaneous spatial autoregressive error model. The procedure can be divided into two main parts: The first is use of an exchange algorithm to find the optimal design for the respective data collection process; for its evaluation an artificial data set was generated and used. The second is estimation of the parameters of the regression model and calculation of Moran's I , which is used as an indicator for spatial dependence in the data set. The method is illustrated by applying it to a well-known case study in spatial analysis.     

中国城市人口集聚分布——基于空间效应的研究

城市人口集聚分布以及城市带的形成是一个空间现象,空间力量对其形成和演化具有重要影响。本文应用空间计量经济学与趋势面分析相结合方法,以617个县级以上城市1998年和2011年的城镇人口和经纬度坐标数据来研究中国城市人口空间集聚分布与趋势。研究发现,中国城市人口分布与其周围相邻城市的人口分布关系密切,并且其6阶空间自相关效应是最强的;同时地理位置对中国城市人口体系的空间分布和纵向形态形成具有显著影响。在空间自相关和空间位置相关的共同作用下,城市人口规模聚集区已在东部沿海地区形成。     

Prediction of Bike-sharing Trip Counts: Comparing Parametric Spatial Regression Models to a Geographically Weighted XGBoost Algorithm

Regression models are commonly applied in the analysis of transportation data. This research aims at broadening the range of methods used for this task by modeling the spatial distribution of bike-sharing trips in Cologne, Germany, applying both parametric regression models and a modified machine learning approach while incorporating measures to account for spatial autocorrelation. Independent variables included in the models consist of land use types, elements of the transport system and sociodemographic characteristics. Out of several regression models with different underlying distributions, a Tweedie generalized additive model is chosen by its values for AIC, RMSE, and sMAPE to be compared to an XGBoost model. To consider spatial relationships, spatial splines are included in the Tweedie model, while the estimations of the XGBoost model are modified using a geographically weighted regression. Both methods entail certain advantages: while XGBoost leads to far better values regarding RMSE and sMAPE and therefore to a better model fit, the Tweedie model allows an easier interpretation of the influence of the independent variables including spatial effects.     

Application of LASSO to the Eigenvector Selection Problem in Eigenvector‐based Spatial Filtering

Eigenvector‐based spatial filtering is one of the often used approaches to model spatial autocorrelation among the observations or errors in a regression model. In this approach, a subset of eigenvectors extracted from a modified spatial weight matrix is added to the model as explanatory variables. The subset is typically specified via the selection procedure of the forward stepwise model, but it is disappointingly slow when the observations n take a large number. Hence, as a complement or alternative, the present article proposes the use of the least absolute shrinkage and selection operator (LASSO) to select the eigenvectors. The LASSO model selection procedure was applied to the well‐known Boston housing data set and simulation data set, and its performance was compared with the stepwise procedure. The obtained results suggest that the LASSO procedure is fairly fast compared with the stepwise procedure, and can select eigenvectors effectively even if the data set is relatively large (n = 10⁴), to which the forward stepwise procedure is not easy to apply.     

ART‐P‐MAP Neural Networks Modeling of Land‐Use Change: Accounting for Spatial Heterogeneity and Uncertainty

Spatial land‐use models over large geographic areas and at fine spatial resolutions face the challenges of spatial heterogeneity, model predictability, data quality, and of the ensuing uncertainty. We propose an improved neural network model, ART‐Probability‐Map (ART‐P‐MAP), tailored to address these issues in the context of spatial modeling of land‐use change. First, it adaptively forms its own network structure to account for spatial heterogeneity. Second, it explicitly infers posterior probabilities of land conversion that facilitates the quantification of prediction uncertainty. Extensive calibration under various test settings is conducted on the proposed model to optimize its utility in seeking useful information within a spatially heterogeneous environment. The calibration strategy involves building a bagging ensemble for training and stratified sampling with varying category proportions for experimentation. Through a temporal validation approach, we examine models’ performance within a systematic assessment framework consisting of global metrics and cell‐level uncertainty measurement. Compared with two baselines, ART‐P‐MAP achieves consistently good and stable performance across experiments and exhibits superior capability to handle the spatial heterogeneity and uncertainty involved in the land‐use change problem. Finally, we conclude that, as a general probabilistic regression model, ART‐P‐MAP is applicable to a broad range of land‐use change modeling approaches, which deserves future research.     

SPATIAL DEPENDENCE AND SPATIAL STRUCTURAL INSTABILITY IN APPLIED REGRESSION ANALYSIS*

The stability of regression coefficients over the observation set (“regional homogeneity”) is typically assessed by means of a Chow test or within a seemingly unrelated regression (SUR) framework. When spatial error autocorrelation is present in cross-sectional equations the traditional tests are no longer applicable. I evaluate this both in formal terms as well as empirically. I introduce a taxonomy of spatial effects in models for structural instability, and discuss its implication for testing. I compare the performance of traditional tests, robust approaches, maximum-likelihood procedures and pretest techniques by means of a series of simple Monte Carlo experiments.     

Exploring Heterogeneities with Geographically Weighted Quantile Regression: An Enhancement Based on the Bootstrap Approach

Geographically weighted quantile regression (GWQR) has been proposed as a spatial analytical technique to simultaneously explore two heterogeneities, one of spatial heterogeneity with respect to data relationships over space and one of response heterogeneity across different locations of the outcome distribution. However, one limitation of GWQR framework is that the existing inference procedures are established based on asymptotic approximation, which may suffer computation difficulties or yield incorrect estimates with finite samples. In this article, we suggest a bootstrap approach to address this limitation. Our bootstrap enhancement is first validated by a simulation experiment and then illustrated with an empirical U.S. mortality data. The results show that the bootstrap approach provides a practical alternative for inference in GWQR and enhances the utilization of GWQR.     

Local Spatial Autocorrelation Statistics: Distributional Issues and an Application

The statistics G_i(d) and G_i*(d), introduced in Getis and Ord (1992) for the study of local pattern in spatial data, are extended and their properties further explored. In particular, nonbinary weights are allowed and the statistics are related to Moran's autocorrelation statistic, I. The correlations between nearby values of the statistics are derived and verified by simulation. A Bonferroni criterion is used to approximate significance levels when testing extreme values from the set of statistics. An example of the use of the statistics is given using spatial-temporal data on the AIDS epidemic centering on San Francisco. Results indicate that in recent years the disease is intensifying in the counties surrounding the city.