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Statistical methods of spatial analysis are often successful at either prediction or explanation, but not necessarily both. In a recent paper, Dearmon and Smith (2016) showed that by combining Gaussian Process Regression (GPR) with Bayesian Model Averaging (BMA), a modeling framework could be developed in which both needs are addressed. In particular, the smoothness properties of GPR together with the robustness of BMA allow local spatial analyses of individual variable effects that yield remarkably stable results. However, this GPR-BMA approach is not without its limitations. In particular, the standard (isotropic) covariance kernel of GPR treats all explanatory variables in a symmetric way that limits the analysis of their individual effects. Here we extend this approach by introducing a mixture of kernels (both isotropic and anisotropic) which allow different length scales for each variable. To do so in a computationally efficient manner, we also explore a number of Bayes-factor approximations that avoid the need for costly reversible-jump Monte Carlo methods.

To demonstrate the effectiveness of this Variable Length Scale (VLS) model in terms of both predictions and local marginal analyses, we employ selected simulations to compare VLS with Geographically Weighted Regression (GWR), which is currently the most popular method for such spatial modeling. In addition, we employ the classical Boston Housing data to compare VLS not only with GWR but also with other well-known spatial regression models that have been applied to this same data. Our main results are to show that VLS not only compares favorably with spatial regression at the aggregate level but is also far more accurate than GWR at the local level.

We propose a generic algorithm for numerically accurate likelihood evaluation of a broad class of spatial models characterized by a high-dimensional latent Gaussian process and non-Gaussian response variables. The class of models under consideration includes specifications for discrete choices, event counts and limited-dependent variables (truncation, censoring, and sample selection) among others. Our algorithm relies upon a novel implementation of efficient importance sampling (EIS) specifically designed to exploit typical sparsity of high-dimensional spatial precision (or covariance) matrices. It is numerically very accurate and computationally feasible even for very high-dimensional latent processes. Thus, maximum likelihood (ML) estimation of high-dimensional non-Gaussian spatial models, hitherto considered to be computationally prohibitive, becomes feasible. We illustrate our approach with ML estimation of a spatial probit for US presidential voting decisions and spatial count data models (Poisson and Negbin) for firm location choices.

We formulate generalized maximum entropy estimators for the general linear model and the censored regression model when there is first order spatial autoregression in the dependent variable. Monte Carlo experiments are provided to compare the performance of spatial entropy estimators relative to classical estimators. Finally, the estimators are applied to an illustrative model allocating agricultural disaster payments.

For this discussion, assume there are n sample observations of the dependent variable y at unique locations. In spatial samples, often each observation is uniquely associated with a particular location or region, so that observations and regions are equivalent. Spatial dependence arises when an observation at one location, say y i is dependent on “neighboring” observations y j, y j∈ϒi. We use ϒi to denote the set of observations that are “neighboring” to observation i, where some metric is used to define the set of observations that are spatially connected to observation i. For general definitions of the sets ϒi,i=1,…,n, typically at least one observation exhibits simultaneous dependence, so that an observation y j, also depends on y i. That is, the set ϒj contains the observation y i, creating simultaneous dependence among observations. This situation constitutes a difference between time series analysis and spatial analysis. In time series, temporal dependence relations could be such that a “one-period-behind relation” exists, ruling out simultaneous dependence among observations. The time series one-observation-behind relation could arise if spatial observations were located along a line and the dependence of each observation were strictly on the observation located to the left. However, this is not in general true of spatial samples, requiring construction of estimation and inference methods that accommodate the more plausible case of simultaneous dependence among observations.

The most used spatial regression models for binary-dependent variable consider a symmetric link function, such as the logistic or the probit models. When the dependent variable represents a rare event, a symmetric link function can underestimate the probability that the rare event occurs. Following Calabrese and Osmetti (2013), we suggest the quantile function of the generalized extreme value (GEV) distribution as link function in a spatial generalized linear model and we call this model the spatial GEV (SGEV) regression model. To estimate the parameters of such model, a modified version of the Gibbs sampling method of Wang and Dey (2010) is proposed. We analyze the performance of our model by Monte Carlo simulations and evaluate the prediction accuracy in empirical data on state failure.

Most spatial econometrics work focuses on spatial dependence in the regressand or disturbances. However, Lesage and Pace (2009) as well as Pace and LeSage2009 showed that the bias in β from applying OLS to a regressand generated from a spatial autoregressive process was exacerbated by spatial dependence in the regressor. Also, the marginal likelihood function or restricted maximum likelihood (REML) function includes a determinant term involving the regressors. Therefore, high dependence in the regressor may affect the likelihood through this term. In addition, Bowden and Turkington (1984) showed that regressor temporal autocorrelation had a non-monotonic effect on instrumental variable estimators.

We provide empirical evidence that many common economic variables used as regressors (e.g., income, race, and employment) exhibit high levels of spatial dependence. Based on this observation, we conduct a Monte Carlo study of maximum likelihood (ML), REML and two instrumental variable specifications for spatial autoregressive (SAR) and spatial Durbin models (SDM) in the presence of spatially correlated regressors.

Findings indicate that as spatial dependence in the regressor rises, REML outperforms ML and that performance of the instrumental variable methods suffer. The combination of correlated regressors and the SDM specification provides a challenging environment for instrumental variable techniques.

We also examine estimates of marginal effects and show that these behave better than estimates of the underlying model parameters used to construct marginal effects estimates. Suggestions for improving design of Monte Carlo experiments are provided.

This paper revisits the joint and conditional Lagrange multiplier tests derived by Debarsy and Ertur (2010) for a fixed effects spatial lag regression model with spatial autoregressive error, and derives these tests using artificial double length regressions (DLR). These DLR tests and their corresponding LM tests are compared using an empirical example and a Monte Carlo simulation.

From a theoretical point of view, a spatial econometric model can contain both a spatially lagged dependent variable (spatial lag) and a spatially autocorrelated error term (spatial error). However, such models are rarely used in practice. This is because (assuming a lattice model approach is used for both the spatial lag and spatial error) the model is difficult to estimate¹ unless the weight matrices are different for the spatial lag and the spatial error.

This research aims to empirically analyze the spatial bank branch network in Canada. The authors study the market structure (both industrial and geographic concentrations) via its own or adjacent postal areas. The empirical framework of this study considers branch density (the ratio of the total number of branches to area size) by employing a spatial two-way fixed effects model. The main finding of this study is that there are no effects associated with market structure, however, there are strong spatial within and nearby effects associated with the socioeconomic variables. In addition, the authors also study the effect of spatial competition from rival banks: they find that large banks and small banks tend to avoid markets dominated by their competitors.

Within spatial econometrics a whole family of different spatial specifications has been developed, with associated estimators and tests. This lead to issues of model comparison and model choice, measuring the relative merits of alternative specifications and then using appropriate criteria to choose the “best” model or relative model probabilities. Bayesian theory provides a comprehensive and coherent framework for such model choice, including both nested and non-nested models within the choice set. The paper reviews the potential application of this Bayesian theory to spatial econometric models, examining the conditions and assumptions under which application is possible. Problems of prior distributions are outlined, and Bayes factors and marginal likelihoods are derived for a particular subset of spatial econometric specifications. These are then applied to two well-known spatial data-sets to illustrate the methods. Future possibilities, and comparisons with other approaches to both Bayesian and non-Bayesian model choice are discussed.