ISSUES IN SPATIAL DATA ANALYSIS

ABSTRACT.  Misspecified functional forms tend to produce biased estimates and spatially correlated errors. Imposing less structure than standard spatial lag models while being more amenable to large datasets, nonparametric and semiparametric methods offer significant advantages for spatial modeling. Fixed effect estimators have significant advantages when spatial effects are constant within well-defined zones, but their flexibility can produce variable, inefficient estimates while failing to account adequately for smooth spatial trends. Though estimators that are designed to measure treatment effects can potentially control for unobserved variables while eliminating the need to specify a functional form, they may be biased if the variables are not constant within discrete zones.     

ESTIMATION AND HYPOTHESIS TESTING FOR NONPARAMETRIC HEDONIC HOUSE PRICE FUNCTIONS

ABSTRACT In contrast to the rigid structure of standard parametric hedonic analysis, nonparametric estimators control for misspecified spatial effects while using highly flexible functional forms. Despite these advantages, nonparametric procedures are still not used extensively for spatial data analysis due to perceived difficulties associated with estimation and hypothesis testing. We demonstrate that nonparametric estimation is feasible for large datasets with many independent variables, offering statistical tests of individual covariates and tests of model specification. We show that fixed parameterization of distance to the nearest rapid transit line is a misspecification and that pricing of access to this amenity varies across neighborhoods within Chicago.     

Geostatistical Smoothing of Areal Data: Mapping Employment Density with Factorial Kriging

This article summarizes area-to-point (ATP) factorial kriging that allows the smoothing of aggregate, areal data into a continuous spatial surface. Unlike some other smoothing methods, ATP factorial kriging does not suppose that all of the data within an area are located at a centroid or other arbitrary point. Also, unlike some other smoothing methods, factorial kriging allows the user to utilize an autocovariance function to control the smoothness of the output. This is beneficial because the covariance function is a physically meaningful statement of spatial relationship, which is not the case when other spatial kernel functions are used for smoothing. Given a known covariance function, factorial kriging gives the smooth surface that is best in terms of minimizing the expected mean squared prediction error. I present an application of the factorial kriging methodology for visualizing the structure of employment density in the Denver metropolitan area.     

Are Spells of Unemployment Longer in Nonmetropolitan Areas? Nonparametric and Semiparametric Evidence

I use nonparametric and semiparametric proportional hazard models to examine whether individuals resident in nonmetropolitan areas experience lower per period rates of exit from unemployment following job loss than metropolitan area residents. Results show that between 1989 and 1993 per period cumulative rates of exit from unemployment were slightly higher in nonmetropolitan areas, mainly due to nonmetropolitan–metropolitan differences in individual characteristics and local economic conditions. Employment density is found to have a positive association with rates of exit of unemployment in metropolitan areas but not in nonmetropolitan areas.     

Some Notes on Parametric Significance Tests for Geographically Weighted Regression

The technique of geographically weighted regression (GWR) is used to model spatial 'drift' in linear model coefficients. In this paper we extend the ideas of GWR in a number of ways. First, we introduce a set of analytically derived significance tests allowing a null hypothesis of no spatial parameter drift to be investigated. Second, we discuss 'mixed' GWR models where some parameters are fixed globally but others vary geographically. Again, models of this type may be assessed using significance tests. Finally, we consider a means of deciding the degree of parameter smoothing used in GWR based on the Mallows C_p statistic. To complete the paper, we analyze an example data set based on house prices in Kent in the U.K. using the techniques introduced.     

Gaussian Process Regression and Bayesian Model Averaging: An Alternative Approach to Modeling Spatial Phenomena

Gaussian Process Regression (GPR) is a nonparametric technique that is capable of yielding reliable out‐of‐sample predictions in the presence of highly nonlinear unknown relationships between dependent and explanatory variables. But in terms of identifying relevant explanatory variables, this method is far less explicit about questions of statistical significance. In contrast, more traditional spatial econometric models, such as spatial autoregressive models or spatial error models, place rather strong prior restrictions on the functional form of relationships, but allow direct inference with respect to explanatory variables. In this article, we attempt to combine the best of both techniques by augmenting GPR with a Bayesian Model Averaging (BMA) component that allows for the identification of statistically relevant explanatory variables while retaining the predictive performance of GPR. In particular, GPR‐BMA yields a posterior probability interpretation of model‐inclusion frequencies that provides a natural measure of the statistical relevance of each variable. Moreover, while such frequencies offer no direct information about the signs of local marginal effects, it is shown that partial derivatives based on the mean GPR predictions do provide such information. We illustrate the additional insights made possible by this approach by applying GPR‐BMA to a benchmark BMA data set involving potential determinants of cross‐country economic growth. It is shown that localized marginal effects based on partial derivatives of mean GPR predictions yield additional insights into comparative growth effects across countries.     

Exploring Spatial Process Dynamics Using Irregular Cellular Automaton Models

Cellular automaton models have enjoyed popularity in recent years as easily constructed models of many complex spatial processes, particularly in the natural sciences, and more recently in geography also. Most such models adopt a regular lattice (often a grid) as the basis for the spatial relations of adjacency that govern evolution of the model. A number of variations on the cellular automaton formalism have been introduced in geography but the impact of such variations on the likely behavior of the models has not been explored. This paper proposes a method for beginning to explore these issues and suggests that this is a new approach to the investigation of the relationships between spatial structure and dynamics of spatial processes. A framework for this exploration is suggested, and details of the required methods and measures are provided. In particular, a measure of spatial pattern—spatial information—based on entropy concepts is introduced. Initial results from investigation along the proposed lines are reported, which suggest that a distinction can he made between spatially robust and fragile processes. Some implications of this result and the methodology presented are briefly discussed.     

INTRODUCTION: WHITHER SPATIAL ECONOMETRICS?

ABSTRACT Spatial econometrics has become a mainstay for regional scientists aiming to estimate geographic spillovers in regional outcomes. Yet, many remain skeptical, especially urban economists who prefer natural experimental approaches. Their concerns revolve around identification and a general lack of a theoretical foundation in the estimation of spatial econometric models. This theme issue includes three papers from leading regional scientists to appraise the status of spatial econometrics. The outcome is sweeping proposals from (1) abandoning standard spatial econometrics because it cannot identify causality, (2) using nonparametric approaches, and (3) implementing more nuanced changes revolving around better theoretical and empirical modeling.     

Semiparametric Maximum Likelihood Estimates of Spatial Dependence

We semiparametrically model spatial dependence via a combination of simpler weight matrices (termed spatial basis matrices) and fit this model via maximum likelihood. Estimation of the model relies on the intuition that bounds to the log‐determinant term in the log‐likelihood can provide penalties to overfitting both the level and pattern of spatial dependence. By relying on symmetric and doubly stochastic spatial basis matrices that reflect different weight specifications assigned to neighboring observations, we are able to derive a mathematical expression for bounds on the log‐determinant term that appears in the likelihood function. These bounds can be conveniently calculated allowing us to solve for maximum likelihood estimates at the bounds using a simple optimization over two quadratic forms that involve small matrices. An intuitively pleasing aspect of our approach is that the objective function for the bounded log‐likelihoods contains one quadratic form equal to the sum‐of‐squared errors measuring the quality of fit, and another quadratic form reflecting a penalty to overfitting spatial dependence. We apply our semiparametric estimation method to a housing model using 57,647 U.S. census tracts.     

Employment Densities, Spatial Autocorrelation, and Subcenters in Large Metropolitan Areas

Employment density functions are estimated for 62 large metropolitan areas. Estimated gradients are statistically significant for distance from the nearest subcenter as well as for distance from the traditional central business district. Lagrange Multiplier (LM) tests imply significant spatial autocorrelation under highly restrictive ordinary least squares (OLS) specifications. The LM test statistics fall dramatically when the models are estimated using flexible parametric and nonparametric methods. The results serve as a warning that functional form misspecification causes spatial autocorrelation.     

Incorporating the effects of spatial structure in discrete choice models of migration

Discrete-choice theory and logit models are evaluated for their usefulness in analyzing migration patterns in a zonal system. The authors "argue that spatial effects and more specifically the relative location of zones are not taken into account in such analyses. We, therefore, introduce a measure of spatial structure and advocate its usage as a predictor of migration in such models. In an example of intrametropolitan migration in Toronto [Canada], we demonstrate that this variable is not only significant but also it improves the performance of all the other variables with the greatest impact on the distance between zones. In addition, inclusion of this variable improves the overall performance of the model in terms of residuals."     

An Anisotropic Model for Spatial Processes

One of the key assumptions in spatial econometric modeling is that the spatial process is isotropic, which means that direction is irrelevant in the specification of the spatial structure. On the one hand, this assumption largely reduces the complexity of the spatial models and facilitates estimation and interpretation; on the other hand, it appears rather restrictive and hard to justify in many empirical applications. In this article a very general anisotropic spatial model, which allows for a high level of flexibility in the spatial structure, is proposed. This new model can be estimated using maximum likelihood and its asymptotic properties are derived at length. When the model is applied to the well-known 1970 Boston housing prices data, it significantly outperforms the isotropic spatial lag model. It also provides interesting additional insights into the price determination process in the properties market. Finally, a Monte Carlo simulation study is used to confirm the optimal properties of the model.     

现代服务业集聚形成机理空间计量分析

在纳入空间效应前提下,构建现代服务业集聚形成机理空间面板计量模型,对我国28个省域相关数据实证研究表明:我国现代服务业集聚在省域之间有较强的空间依赖性和正的空间溢出效应。技术差异在时间维度上对现代服务业集聚促进作用显著,在空间维度上并不显著;交易费用与现代服务业集聚有显著的负相关性;知识溢出、规模经济、政府行为对现代服务业集聚促进作用显著。     

Constructing the Spatial Weights Matrix Using a Local Statistic

Spatial weights matrices are necessary elements in most regression models where a representation of spatial structure is needed. We construct a spatial weights matrix, W , based on the principle that spatial structure should be considered in a two‐part framework, those units that evoke a distance effect, and those that do not. Our two‐variable local statistics model (LSM) is based on the G_i* local statistic. The local statistic concept depends on the designation of a critical distance, d_c, defined as the distance beyond which no discernible increase in clustering of high or low values exists. In a series of simulation experiments LSM is compared to well‐known spatial weights matrix specifications—two different contiguity configurations, three different inverse distance formulations, and three semi‐variance models. The simulation experiments are carried out on a random spatial pattern and two types of spatial clustering patterns. The LSM performed best according to the Akaike Information Criterion, a spatial autoregressive coefficient evaluation, and Moran's I tests on residuals. The flexibility inherent in the LSM allows for its favorable performance when compared to the rigidity of the global models.     

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.     

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.     

A CONDITIONALLY PARAMETRIC PROBIT MODEL OF MICRODATA LAND USE IN CHICAGO

Spatial data sets pose challenges for discrete choice models because the data are unlikely to be independently and identically distributed. A conditionally parametric spatial probit model is amenable to very large data sets while imposing far less structure on the data than conventional parametric models. We illustrate the approach using data on 474,170 individual lots in the City of Chicago. The results suggest that simple functional forms are not appropriate for explaining the spatial variation in residential land use across the entire city.     

MODELING SPATIAL STRUCTURES THROUGH EQUILIBRIUM STATES FOR TRANSITION MATRICES*

ABSTRACT. In this paper we present a model for spatial interaction within a network of towns. This interaction is modeled through equilibrium states for certain Markov chains where, in particular, explicit formulas for these states are given. Our model exploits and intertwines ideas from gravity models, the competing destinations model and the intervening opportunities model. The central idea in the paper is to capture the effect of spatial structure in a framework where interaction is determined by the global network configuration.     

Modeling the Uncertainty of Slope and Aspect Estimates Derived from Spatial Databases

Estimates of slope and aspect are commonly made from digital elevation models (DEMs), and are subject to the uncertainty present in such models. We show that errors in slope and aspect depend on the spatial structure of DEM errors. We propose a general-purpose model of DEM errors in which a spatially auto-regressive random field is added as a disturbance term to elevations. In addition, we propose a general procedure for propagating such errors through GIS operations. In the absence of explicit information on the spatial structure of DEM errors, we demonstrate the potential utility of a worst-case analysis. A series of simulations are used to make general observations about the nature and severity of slope and aspect errors.     

Bayesian Model Averaging for Spatial Econometric Models

We extend the literature on Bayesian model comparison for ordinary least-squares regression models to include spatial autoregressive and spatial error models. Our focus is on comparing models that consist of different matrices of explanatory variables. A Markov Chain Monte Carlo model composition methodology labeled MC ³ by Madigan and York is developed for two types of spatial econometric models that are frequently used in the literature. The methodology deals with cases where the number of possible models based on different combinations of candidate explanatory variables is large enough such that calculation of posterior probabilities for all models is difficult or infeasible. Estimates and inferences are produced by averaging over models using the posterior model probabilities as weights, a procedure known as Bayesian model averaging. We illustrate the methods using a spatial econometric model of origin–destination population migration flows between the 48 U.S. states and the District of Columbia during the 1990–2000 period.