Closed‐Form Maximum Likelihood Estimates of Nearest Neighbor Spatial Dependence

Models of n² potential spatial dependencies among n observations spread irregularly over space seem unlikely to yield simple structure. However, the use of the nearest neighbor leads to a very parsimonious eigenstructure of the associated adjacency matrix which results in an extremely simple closed form for the log determinant. In turn, this leads to a closed‐form solution for the maximum likelihood estimates of the spatially autoregressive and mixed regressive spatially autoregressive models. With the closed‐form solution, one can find the neighbors and compute maximum likelihood estimates for 100,000 observations in under one minute. The model has theoretical, pedagogical, diagnostic, modeling, and methodological applications. For example, the model could serve as a more enlightened null hypothesis for geographic data than spatial independence.     

Estimation Bias in Spatial Models with Strongly Connected Weight Matrices

This article shows that, for both spatial lag and spatial error models with strongly connected weight matrices, maximum likelihood estimates of the spatial dependence parameter are necessarily biased downward . In addition, this bias is shown to be present in general Moran tests of spatial dependency. Thus, positive dependencies may often fail to be detected when weight matrices are strongly connected. The analysis begins with a detailed examination of downward bias for the extreme case of maximally connected weight matrices. Results for this case are then extended by continuity to a broader range of (appropriately defined) strongly connected matrices. Finally, a simulated numerical example is presented to illustrate some of the practical consequences of these biases.     

Model Uncertainty in Matrix Exponential Spatial Growth Regression Models

This article considers the most important aspects of model uncertainty for spatial regression models, namely, the appropriate spatial weight matrix to be employed and the appropriate explanatory variables. We focus on the spatial Durbin model (SDM) specification in this study that nests most models used in the regional growth literature, and develop a simple Bayesian model‐averaging approach that provides a unified and formal treatment of these aspects of model uncertainty for SDM growth models. The approach expands on previous work by reducing the computational costs through the use of Bayesian information criterion model weights and a matrix exponential specification of the SDM model. The spatial Durbin matrix exponential model has theoretical and computational advantages over the spatial autoregressive specification due to the ease of inversion, differentiation, and integration of the matrix exponential. In particular, the matrix exponential has a simple matrix determinant that vanishes for the case of a spatial weight matrix with a trace of zero. This allows for a larger domain of spatial growth regression models to be analyzed with this approach, including models based on different classes of spatial weight matrices. The working of the approach is illustrated for the case of 32 potential determinants and three classes of spatial weight matrices (contiguity‐based, k‐nearest neighbor, and distance‐based spatial weight matrices), using a data set of income per capita growth for 273 European regions.     

Likelihood Dominance Spatial Inference

Many users estimate spatial autoregressions to perform inference on regression parameters. However, as the sample size or the number of potential models rise, computational exigencies make exact computation of likelihood‐based inferences tedious or even impossible. To address this problem, we introduce a lower bound on the likelihood ratio test that can allow users to conduct conservative maximum likelihood inference while avoiding the computationally demanding task of computing exact maximum likelihood point estimates. This form of inference, known as likelihood dominance, performs almost as well as exact likelihood inference for the empirical examples examined. We illustrate the utility of the technique by performing likelihood‐based inference on parameters from a spatial autoregression involving 890,091 observations in less than a minute (given the spatial weight matrix)     

PANEL DATA MODELS WITH SPATIALLY DEPENDENT NESTED RANDOM EFFECTS

This paper focuses on panel data models combining spatial dependence with a nested (hierarchical) structure. We use a generalized moments estimator to estimate the spatial autoregressive parameter and the variance components of the disturbance process. A spatial counterpart of the Cochrane‐Orcutt transformation leads to a feasible generalized least squares procedure to estimate the regression parameters. Monte Carlo simulations show that our estimators perform well in terms of root mean square error compared to the maximum likelihood estimator. The approach is applied to English house price data for districts nested within counties.     

EVIDENCE OF POLITICAL YARDSTICK COMPETITION IN FRANCE USING A TWO‐REGIME SPATIAL DURBIN MODEL WITH FIXED EFFECTS*

ABSTRACT This research proposes a two‐regime spatial Durbin model with spatial and time‐period fixed effects to test for political yardstick competition and exclude any other explanation that might produce spatial interaction effects among the dependent variable, the independent variables, or the error term. The study also derives the maximum likelihood estimator and variance–covariance matrix of the parameters of this model. Data pertaining to welfare spending by 93 departments in France during 1992–2000 provide significant empirical evidence in support of political yardstick competition. Departments governed by a small political majority mimic neighboring expenditures on welfare to a greater extent than do departments governed by a large political majority.     

A Methodology for Neural Spatial Interaction Modeling

This paper attempts to develop a mathematically rigid and unified framework for neural spatial interaction modeling. Families of classical neural network models, but also less classical ones such as product unit neural network ones are considered for the cases of unconstrained and singly constrained spatial interaction flows. Current practice appears to suffer from least squares and normality assumptions that ignore the true integer nature of the flows and approximate a discrete‐valued process by an almost certainly misrepresentative continuous distribution. To overcome this deficiency we suggest a more suitable estimation approach, maximum likelihood estimation under more realistic distributional assumptions of Poisson processes, and utilize a global search procedure, called Alopex, to solve the maximum likelihood estimation problem. To identify the transition from underfitting to overfitting we split the data into training, internal validation, and test sets. The bootstrapping pairs approach with replacement is adopted to combine the purity of data splitting with the power of a resampling procedure to overcome the generally neglected issue of fixed data splitting and the problem of scarce data. In addition, the approach has power to provide a better statistical picture of the prediction variability. Finally, a benchmark comparison against the classical gravity models illustrates the superiority of both, the unconstrained and the origin constrained neural network model versions in terms of generalization performance measured by Kullback and Leibler's information criterion.     

Bayesian Point Process Modeling to Quantify Geographic Regions of Excess Stillbirth Risk

Motivated by the paucity of high‐quality stillbirth surveillance data and the spatial analysis of such data, we describe the pattern of stillbirth events as a step toward increased understanding of risk factors which can better guide future measures to mitigate these events. A challenge in such an analysis is that some mothers experience stillbirth events from independent pregnancies within a defined study period. To account for these dependencies, we parameterize our model to include a maternal contextual effect and broaden the appeal of Bayesian Poisson point process modeling to quantify excess stillbirth risk while considering a form of a log‐Gaussian Cox process. In the presence of extra unobserved spatial variation in risk, we demonstrate a pragmatic methodologic strategy to model the risk surface in relation to covariates and that there is a variance‐bias trade‐off associated with the use of a maternal contextual effect. We applied our strategy to the spatial distribution of stillbirth in Iowa during the years 2005–2011 using data obtained from an active, statewide public health surveillance program. We identified areas of excess risk for further investigation based on model components that captured important features of the data.     

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 Structural Equation Approach to Models with Spatial Dependence

We introduce the class of structural equation models (SEMs) and corresponding estimation procedures into a spatial dependence framework. SEM allows both latent and observed variables within one and the same (causal) model. Compared with models with observed variables only, this feature makes it possible to obtain a closer correspondence between theory and empirics, to explicitly account for measurement errors, and to reduce multicollinearity. We extend the standard SEM maximum likelihood estimator to allow for spatial dependence and propose easily accessible SEM software like LISREL 8 and Mx. We present an illustration based on Anselin's Columbus, OH, crime data set. Furthermore, we combine the spatial lag model with the latent multiple-indicators–multiple-causes model and discuss estimation of this latent spatial lag model. We present an illustration based on the Anselin crime data set again.     

SPATIAL ECONOMETRIC MODELING OF ORIGIN‐DESTINATION FLOWS*

ABSTRACT Standard spatial autoregressive models rely on spatial weight structures constructed to model dependence among n regions. Ways of parsimoniously modeling the connectivity among the sample of N=n² origin‐destination (OD) pairs that arise in a closed system of interregional flows has remained a stumbling block. We overcome this problem by proposing spatial weight structures that model dependence among the N OD pairs in a fashion consistent with standard spatial autoregressive models. This results in a family of spatial OD models introduced here that represent an extension of the spatial regression models described in Anselin (1988) .     

A Spatial Mixture Model of Innovation Diffusion

The diffusion of new product or technical innovation over space is here modeled as an event‐based process in which the likelihood of the next adopter being in region r is influenced by two factors: (i) the potential interactions of individuals in r with current adopters in neighboring regions, and (ii) all other attributes of individuals in r that may influence their adoption propensity. The first factor is characterized by a logit model reflecting the likelihood of adoption due to spatial contacts with previous adopters, and the second by a logit model reflecting the likelihood of adoption due to other intrinsic effects. The resulting spatial diffusion process is then assumed to be driven by a probabilistic mixture of the two. A number of formal properties of this model are analyzed, including its asymptotic behavior. But the main analytical focus is on statistical estimation of parameters. Here it is shown that standard maximum‐likelihood estimates require large sample sizes to achieve reasonable results. Two estimation approaches are developed which yield more sensible results for small sample sizes. These results are applied to a small data set involving the adoption of a new Internet grocery‐shopping service by consumers in the Philadelphia metropolitan area.     

Nonlinear Multivariate Spatial Modeling Using NLPCA and Pair‐Copulas

A novel geostatistical modeling approach is developed to model nonlinear multivariate spatial dependence using nonlinear principal component analysis (NLPCA) and pair‐copulas. In spatial studies, multivariate measurements are frequently collected at each location. The dependence between such measurements can be complex. In this article, a multivariate geostatistical model is developed that can capture both nonlinear spatial dependence across locations and nonlinear dependence between measurements at a particular location. Nonlinear multivariate dependence between spatial variables is removed using NLPCA. Subsequently, a pair‐copula based model is fitted to each transformed variable to model the univariate nonlinear spatial dependencies. NLPCA and pair‐copulas, within the proposed model, are compared with stepwise conditional transformation (SCT) and conventional kriging. The results show that, for the two case studies presented, the proposed model that utilizes NLPCA and pair‐copulas reproduces nonlinear multivariate structures and univariate distributions better than existing methods based on SCT and kriging.     

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.     

An Introduction to the Network Weight Matrix

This study introduces the network weight matrix as a replacement for the spatial weight matrix to measure the spatial dependence between links of a network. This matrix stems from the concepts of betweenness centrality and vulnerability in network science. The elements of the matrix are a function not simply of proximity, but of network topology, network structure, and demand configuration. The network weight matrix has distinctive characteristics, which are capable of reflecting spatial dependence between traffic links: (1) elements are allowed to have negative and positive values capturing the competitive and complementary nature of links, (2) diagonal elements are not fixed to zero, which takes the self‐dependence of a link upon itself into consideration, and (3) elements not only reflect the spatial dependence based on the network structure, but they acknowledge the demand configuration as well. We verify the network weight matrix by modeling traffic flows in a 3 × 3 grid test network with 9 nodes and 24 directed links connecting 72 origin‐destination (OD) pairs. Models encompassing the network weight matrix outperform both models without spatial components and models with the spatial weight matrix. The network weight matrix represents a more accurate and defensible spatial dependency between traffic links, and offers the potential to augment traffic flow prediction.     

A SPATIAL CLIFF‐ORD‐TYPE MODEL WITH HETEROSKEDASTIC INNOVATIONS: SMALL AND LARGE SAMPLE RESULTS*

ABSTRACT In this paper, we specify a linear Cliff‐and‐Ord‐type spatial model. The model allows for spatial lags in the dependent variable, the exogenous variables, and disturbances. The innovations in the disturbance process are assumed to be heteroskedastic with an unknown form. We formulate multistep GMM/IV‐type estimation procedures for the parameters of the model. We also give the limiting distributions for our suggested estimators and consistent estimators for their asymptotic variance‐covariance matrices. We conduct a Monte Carlo study to show that the derived large‐sample distribution provides a good approximation to the actual small‐sample distribution of our estimators.     

Spatial and temporal effects in Mexican direct elections for the chamber of deputies

By using data of the elections for the Chamber of Deputies of 1997 and 2000 in Mexico, we fit spatial autologistic models with temporal effects to test the significance of spatial and temporal effects on those elections. The binary variable of interest is the one that indicates a win of the National Action Party (PAN) or the alliance that it formed. By spatial effect, we refer to the fact that neighbouring constituencies present dependence on their electoral results. The temporal effect refers to the existence of dependence, for the same constituency, of the result of the election with the result of the previous election. The model that we used to test the significance of spatial and temporal effects is the spatial autologistic model with temporal effects for which estimation is complex and requires simulation techniques. By defining an urban constituency as one that contains at least one population center of 200,000 inhabitants or more, among our principal results, we find that, for the Mexican election of 2000, the spatial effect is significant only when neighbouring constituencies are both urban. For the election of 1997, the spatial effect is significant independent of the type of neighbouring constituencies. The temporal effect is significant on both elections.     

COMPUTATION OF MAXIMUM LIKELIHOOD ESTIMATES OF GRAVITY MODEL PARAMETERS*

ABSTRACT. This paper presents the derivations of several new algorithms for the computation of maximum likelihood estimates of the parameters of a very general form of the gravity model. The algorithms are then compared with previously available algorithms including GLIM and that given in Sen (1986). One of the new algorithms emerges as far superior in just about every way to its competitors. In particular, it is usually much more than an order of magnitude faster than the GLIM procedure and that given in Sen (1986). It is also not substantially affected by pitfalls such as multicollinearity and (unlike the GLIM procedure) is capable of comfortably handling large O-D matrices.     

Model‐based Cluster Analysis of Artefact Compositional Data

Cluster analysis is the most widely used multivariate technique in archaeometry, with the majority of applications being exploratory in nature. Model‐based methods of clustering have their advocates, but have seen little application to archaeometric data. The paper investigates two such methods. They have potential advantages over exploratory techniques, if successful. Mixture maximum‐likelihood worked well using low‐dimensional lead isotope data, but had problems coping with higher‐dimensional ceramic compositional data. For our most challenging example, classification maximum‐likelihood performed comparably with more standard methods, but we find no evidence to suggest that it should supplant these.     

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.