BAYESIAN MODEL AVERAGING IN THE CONTEXT OF SPATIAL HEDONIC PRICING: AN APPLICATION TO FARMLAND VALUES

ABSTRACT Specification uncertainty arises in spatial hedonic pricing models because economic theory provides no guide in choosing the spatial weighting matrix and explanatory variables. Our objective in this paper is to investigate whether we can resolve uncertainty in the application of a spatial hedonic pricing model. We employ Bayesian Model Averaging in combination with Markov Chain, Monte Carlo Model Composition. The proposed methodology provides inclusion probabilities for explanatory variables and weighting matrices. These probabilities provide a clear indication of which explanatory variables and weighting matrices are most relevant, but they are case specific.     

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.     

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.     

Stochastic Efficiency of Bayesian Markov Chain Monte Carlo in Spatial Econometric Models: An Empirical Comparison of Exact Sampling Methods

Spatial econometric specifications pose unique computational challenges to Bayesian analysis, making it difficult to estimate models efficiently. In the literature, the main focus has been on extending Bayesian analysis to increasingly complex spatial models. The stochastic efficiency of commonly used Markov Chain Monte Carlo (MCMC) samplers has received less attention by comparison. Specifically, Bayesian methods to analyze effective sample size and samplers that provide large effective size have not been thoroughly considered in the literature. Thus, we compare three MCMC techniques: the familiar Metropolis‐within‐Gibbs sampling, Slice‐within‐Gibbs sampling, and Hamiltonian Monte Carlo. The latter two methods, while common in other domains, are not as widely encountered in Bayesian spatial econometrics. We assess these methods across four different scenarios in which we estimate the spatial autoregressive parameter in a mixed regressive, spatial autoregressive specification (or, spatial lag model). We find that off‐the‐shelf implementations of the newer high‐yield simulation techniques require significant adaptation to be viable. We further find that the effective sizes are often significantly smaller than nominal sizes. In addition, we find that stopping simulation early may understate posterior credible interval widths when effective sample size is small. More broadly, we suggest that sample information and stopping rules deserve more attention in both applied and basic Bayesian spatial econometric research.     

THE SLX MODEL

We provide a comprehensive overview of the strengths and weaknesses of different spatial econometric model specifications in terms of spillover effects. Based on this overview, we advocate taking the SLX model as point of departure in case a well‐founded theory indicating which model is most appropriate is lacking. In contrast to other spatial econometric models, the SLX model also allows for the spatial weights matrix W to be parameterized and the application of standard econometric techniques to test for endogenous explanatory variables. This starkly contrasts commonly used spatial econometric specification strategies and is a complement to the critique of spatial econometrics raised in a special theme issue of the Journal of Regional Science (Volume 52, Issue 2). To illustrate the pitfalls of the standard spatial econometrics approach and the benefits of our proposed alternative approach in an empirical setting, the Baltagi and Li (2004) cigarette demand model is estimated.     

Principal Component Regression in Spatial Lag Model: Teen Employment in the City of Rosario,Argentina

Teen employment is a very important socioeconomic phenomenon because of its consequences on human capital formation. We assess the relation between teen employment and poverty, education, and unemployment in the city of Rosario, using information from the 2010 Argentina Census disaggregated at census block level. We use two different spatial models: The spatial lag model (SLM) and a linear regression model with the spatial component filtered (filtering model, FM). Given the nature of the variables employed, multicollinearity is an issue. One of the techniques proposed in the literature to deal with multicollinearity problems is principal component regression (PCR). We develop an adaptation of such methodology to be used in the SLM. Both models are estimated using their traditional methodologies (instrumental variables for the SLM and OLS for the FM) and using PCR. Although results are similar between the two models, depending on the methodology used in the estimations they differ greatly. Under traditional methodologies estimations show high variability, instability, and contradictory outcomes, but under PCR, results behave according to the literature.     

The Determinants of Regional Freight Transport: A Spatial,Semiparametric Approach

In the context of modeling regional freight the four‐stage model is a popular choice. The first stage of the model, freight generation and attraction, however, suffers from three shortcomings: first of all, it does not take spatial dependencies among regions into account, thus potentially yielding biased estimates. Second, there is no clear consensus in the literature as to the choice of explanatory variables. Second, sectoral employment and gross value added are used to explain freight generation, whereas some recent publications emphasize the importance of variables which measure the amount of logistical activity in a region. Third, there is a lack of consensus regarding the functional form of the explanatory variables. Multiple recent studies emphasize nonlinear influences of selected variables. This article addresses these shortcomings by using a spatial variant of the classic freight generation and attraction models combined with a penalized spline framework to model the explanatory variables in a semiparametric fashion. Moreover, a Bayesian estimation approach is used, coupled with a penalized Normal inverse‐Gamma prior structure, to introduce uncertainty regarding the choice and functional form of explanatory variables. The performance of the model is assessed on a real‐world example of freight generation and attraction of 258 European NUTS‐2 level regions, covering 25 European countries.     

Human capital accumulation and long‐term income growth projections for European regions

We propose an econometric framework to construct projections for per capita income growth and human capital for European regions. Using Bayesian methods, our approach accounts for model uncertainty in terms of the choice of explanatory variables, the nature of spatial spillovers, as well as the potential endogeneity between output growth and human capital accumulation. This method allows us to assess the potential contribution of future educational attainment to economic growth and income convergence among European regions over the next decades. Our findings suggest that income convergence dynamics and human capital act as important drivers of income growth for the decades to come.     

DETERMINANTS OF KNOWLEDGE PRODUCTION AND THEIR EFFECTS ON REGIONAL ECONOMIC GROWTH

ABSTRACT We provide a Bayesian spatial Markov Chain Monte Carlo model composition (MC³) analysis of growth rates in European regional patenting activity. Based on theoretical models on innovation and growth, we identify a large set of candidate explanatory variables that characterize regional stocks of knowledge, including: human resources devoted to innovative activity, scientific and technical capabilities, public and private investments, government policies, as well as regional industry structure, and indicators of regional technology gaps that reflect distance from the technological frontier. Our analysis shows that accommodating spatial dependence and heterogeneity leads to different conclusions regarding factors important for technological transfer and knowledge spillovers.     

A Bayesian Approach to Modeling Binary Data: The Case of High‐Intensity Crime Areas

This paper reports the fitting of a number of Bayesian logistic models with spatially structured or/and unstructured random effects to binary data with the purpose of explaining the distribution of high‐intensity crime areas (HIAs) in the city of Sheffield, England. Bayesian approaches to spatial modeling are attracting considerable interest at the present time. This is because of the availability of rigorously tested software for fitting a certain class of spatial models. This paper considers issues associated with the specification, estimation, and validation, including sensitivity analysis, of spatial models using the WinBUGS software. It pays particular attention to the visualization of results. We discuss a map decomposition strategy and an approach that examines properties of the full posterior distribution. The Bayesian spatial model reported provides some interesting insights into the different factors underlying the existence of the three police‐defined HIAs in Sheffield.     

Spatial Autoregressive Models for Geographically Hierarchical Data Structures

This article discusses how standard spatial autoregressive models and their estimation can be extended to accommodate geographically hierarchical data structures. Whereas standard spatial econometric models normally operate at a single geographical scale, many geographical data sets are hierarchical in nature—for example, information about houses nested into data about the census tracts in which those houses are found. Here we outline four model specifications by combining different formulations of the spatial weight matrix W and of ways of modeling regional effects. These are (1) groupwise W and fixed regional effects; (2) groupwise W and random regional effects; (3) proximity‐based W and fixed regional effects; and (4) proximity‐based W and random regional effects. We discuss each of these model specifications and their associated estimation methods, giving particular attention to the fourth. We describe this as a hierarchical spatial autoregressive model. We view it as having the most potential to extend spatial econometrics to accommodate geographically hierarchical data structures and as offering the greatest coming together of spatial econometric and multilevel modeling approaches. Subsequently, we provide Bayesian Markov Chain Monte Carlo algorithms for implementing the model. We demonstrate its application using a two‐level land price data set where land parcels nest into districts in Beijing, China, finding significant spatial dependence at both the land parcel level and the district level.     

Vector Autoregression (Var) — An Approach to Dynamic analysis of Geographic Processes

Vector autoregression (VAR) is a widely used econometric technique for multivariate time series modelling. This paper shows that with several very attractive features, VAR may also provide a valuable tool for analysing the dynamics among geographic processes and for spatial autoregressive modelling. After a brief discussion of the VAR approach, a VAR model for the dynamics of the US population between 1910 and 1990 is estimated and interpreted to illustrate the techniques. The VAR makes it possible to view the interactions among the four variables used in the model (total population, birth rate, immigration and per capita GNP) more adequately. The paper then discusses recent developments in the VAR methodology such as Bayesian vector autoregression (BVAR), spatial prior for regional modelling and cointegration, as well as the limitations and problems that arise from the application of VARs.     

INTERPRETING SPATIAL ECONOMETRIC ORIGIN‐DESTINATION FLOW MODELS

Spatial interaction or gravity models have been used to model flows that take many forms, for example population migration, commodity flows, traffic flows, all of which reflect movements between origin and destination regions. We focus on how to interpret estimates from spatial autoregressive extensions to the conventional regression‐based gravity models that relax the assumption of independence between flows. These models proposed by LeSage and Pace ( 2008 , 2009 ) define spatial dependence involving flows between regions. We show how to calculate partial derivative expressions for these models that can be used to quantify these various types of effect that arise from changes in the characteristics/explanatory variables of the model.     

Bayesian Spatial Filtering for Hedonic Models: An Application for the Real Estate Market

This article presents a Bayesian method based on spatial filtering to estimate hedonic models for dwelling prices with geographically varying coefficients. A Bayesian Adaptive Sampling algorithm for variable selection is used, which makes it possible to select the most appropriate filters for each hedonic coefficient. This approach explores the model space more systematically and takes into account the uncertainty associated with model estimation and selection processes. The methodology is illustrated with an application for the real estate market in the Spanish city of Zaragoza and with simulated data. In addition, an exhaustive comparison study with a set of alternatives strategies used in the literature is carried out. Our results show that the proposed Bayesian procedures are competitive in terms of prediction; more accurate results are obtained in the estimation of the regression coefficients of the model, and the multicollinearity problems associated with the estimation of the regression coefficients are solved.     

TESTING THE SPATIAL SCALE AND THE DYNAMIC STRUCTURE IN REGIONAL MODELS (A CONTRIBUTION TO SPATIAL ECONOMETRIC SPECIFICATION ANALYSIS)*

This article addresses the problem of specification uncertainty in modeling spatial economic theories in stochastic form. It is ascertained that the traditional approach to spatial econometric modeling does not adequately deal with the type and extent of specification uncertainty commonly encountered in spatial economic analyses. Two alternative spatial econometric modeling procedures proposed in the literature are reviewed and shown to be suitable for analyzing systematically two sources of specification uncertainty, viz., the level of aggregation and the spatio-temporal dynamic structure in multiregional econometric models. The usefulness of one of these specification procedures is illustrated by the construction of a simple multiregional model for The Netherlands.     

A Spatial Prior for Bayesian Vector Autoregressive Models

In this paper we develop a Bayesian prior motivated by cross-sectional spatial autoregressive models for use in time-series vector autoregressive forecasting involving spatial variables. We compare forecast accuracy of the proposed spatial prior to that from a vector autoregressive model relying on the Minnesota prior and find a significant improvement. In addition to a spatially motivated prior variance as in LeSage and Pan (1995) we develop a set of prior means based on spatial contiguity. A Theil-Goldberger estimator may be used for the proposed model making it easy to implement.     

MODELING SPATIAL AUTOCORRELATION IN SPATIAL INTERACTION DATA: AN APPLICATION TO PATENT CITATION DATA IN THE EUROPEAN UNION*

ABSTRACT Spatial interaction models of the gravity type are widely used to model origin–destination flows. They draw attention to three types of variables to explain variation in spatial interactions across geographic space: variables that characterize an origin region of a flow, variables that characterize a destination region of a flow, and finally variables that measure the separation between origin and destination regions. This paper outlines and compares two approaches, the spatial econometric and the eigenfunction‐based spatial filtering approach, to deal with the issue of spatial autocorrelation among flow residuals. An example using patent citation data that capture knowledge flows across 112 European regions serves to illustrate the application and the comparison of the two approaches.     

Duration Modeling of Spatial Point Patterns

This paper attempts to further the research by Odland and Ellis (1992) in applying event history methodology to the analysis of spatial point patterns (that is, event patterns). Its empirical focus is the event pattern derived from the adoption of an agricultural innovation, the Harvestore, in southern Ontario, Canada, from 1963 to 1986. Event history analysis involves the use of discrete-state, continuous-time stochastic models to investigate a temporal longitudinal record on discrete variables. Event history models are usually concerned with durations of time between events and the effects of intertemporal time dependencies on future event occurrences. As such, they are often referred to as duration models. Many of the methods used in event history analysis allow the use of other nonnegative interval measurements in place of standard temporal intervals to investigate a series of events. In particular, spatial intervals (or durations) of distances between events may also be accommodated by event history models. Our analysis extends the previous research of Odland and Ellis (1992) by using a wider range of parametric models to explore duration dependence, investigating the role of spatial censoring, and using a more extensive set of explanatory variables. In addition, simulation experiments and graphical tests are used to evaluate the empirical event pattern against one generated from Complete Spatial Randomness. Results indicate that the event pattern formed by the Harvestore adopter farms is clustered (that is, is described by positive duration dependency), the sales agent is a significant factor in the distribution of adopters, and that contrasting results are obtained from the analysis using censored data versus uncensored data.     

Calibration and Test of a Discrete Choice Model with Endogenous Choice Sets

Conventional discrete choice models assume implicitly that the choice set is independent of the decisionmaker's preferences conditional on the explanatory variables of the models. This assumption is implausible in many choice situations where the decisionmaker selects his or her choice set. This paper estimates and tests a discrete choice model with endogenous choice sets based on Horowitz' theoretical work. To calibrate the model, a new probability simulator is introduced and a sequential estimation procedure is developed. The model and calibration methods are tested in an empirical application as well as Monte Carlo simulations. The empirical results are used to test the theory of endogenous choice sets and to examine the differences between the new model and a conventional choice model in parameter estimates and predicted choice probabilities. The empirical results strongly suggest that ignoring the endogeneity of choice sets in choice modeling can have serious consequences in applications.     

The Spatial Association of Demographic and Population Health Characteristics with COVID-19 Prevalence Across Districts in India

In less-developed countries, the lack of granular data limits the researcher's ability to study the spatial interaction of different factors on the COVID-19 pandemic. This study designs a novel database to examine the spatial effects of demographic and population health factors on COVID-19 prevalence across 640 districts in India. The goal is to provide a robust understanding of how spatial associations and the interconnections between places influence disease spread. In addition to the linear Ordinary Least Square regression model, three spatial regression models—Spatial Lag Model, Spatial Error Model, and Geographically Weighted Regression are employed to study and compare the variables explanatory power in shaping geographic variations in the COVID-19 prevalence. We found that the local GWR model is more robust and effective at predicting spatial relationships. The findings indicate that among the demographic factors, a high share of the population living in slums is positively associated with a higher incidence of COVID-19 across districts. The spatial variations in COVID-19 deaths were explained by obesity and high blood sugar, indicating a strong association between pre-existing health conditions and COVID-19 fatalities. The study brings forth the critical factors that expose the poor and vulnerable populations to severe public health risks and highlight the application of geographical analysis vis-a-vis spatial regression models to help explain those associations.