Bayesian Estimation of Limited Dependent Variable Spatial Autoregressive Models

A Gibbs sampling (Markov chain Monte Carlo) method for estimating spatial autoregressive limited dependent variable models is presented. The method can accommodate data sets containing spatial outliers and general forms of non‐constant variance. It is argued that there are several advantages to the method proposed here relative to that proposed and illustrated in McMillen (1992) for spatial probit models.     

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.     

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.     

Crime Risk Maps: A Multivariate Spatial Analysis of Crime Data

In crime analyses, maps showing the degree of risk help police departments to make decisions on operational matters, such as where to patrol or how to deploy police officers. This study statistically models spatial crime data for multiple crime types in order to produce joint crime risk maps. To effectively model and map the spatial crime data, we consider two important characteristics of crime occurrences: the spatial dependence between sites, and the dependence between multiple crime types. We reflect both characteristics in the model simultaneously using a generalized multivariate conditional autoregressive model. As a real‐data application, we examine the number of incidents of vehicle theft, larceny, and burglary in 83 census tracts of San Francisco in 2010. Then, we employ a Bayesian approach using a Markov chain Monte Carlo method to estimate the model parameters. Based on the results, we detect the crime hotspots, thus demonstrating the advantage of using a multivariate spatial analysis for crime data.     

BAYSIAN INFERENCE FOR ORDERED RESPONSE DATA WITH A DYNAMIC SPATIAL‐ORDERED PROBIT MODEL

ABSTRACT Many databases involve ordered discrete responses in a temporal and spatial context, including, for example, land development intensity levels, vehicle ownership, and pavement conditions. An appreciation of such behaviors requires rigorous statistical methods, recognizing spatial effects and dynamic processes. This study develops a dynamic spatial‐ordered probit (DSOP) model in order to capture patterns of spatial and temporal autocorrelation in ordered categorical response data. This model is estimated in a Bayesian framework using Gibbs sampling and data augmentation, in order to generate all autocorrelated latent variables. It incorporates spatial effects in an ordered probit model by allowing for interregional spatial interactions and heteroskedasticity, along with random effects across regions or any clusters of observational units. The model assumes an autoregressive, AR(1), process across latent response values, thereby recognizing time‐series dynamics in panel data sets. The model code and estimation approach is tested on simulated data sets, in order to reproduce known parameter values and provide insights into estimation performance, yielding much more accurate estimates than standard, nonspatial techniques. The proposed and tested DSOP model is felt to be a significant contribution to the field of spatial econometrics, where binary applications (for discrete response data) have been seen as the cutting edge. The Bayesian framework and Gibbs sampling techniques used here permit such complexity, in world of two‐dimensional autocorrelation.     

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.     

Bayesian Estimation of Regional Production for CGE Modeling

Abstract Computable general equilibrium (CGE) models are often criticized for using restrictive functional forms and relying on external sources for parameter values in their calibration. CGE modelers argue that in many instances reliable econometric estimates of important model parameters are unavailable because they must be estimated using small numbers of time‐series observations. To address these criticisms, this paper uses a Bayesian approach to estimate the parameters of a translog production function in a regional computable general equilibrium model. Using priors from more reliable national estimates, and parameter restrictions required by neoclassical production theory, estimation is done by Markov chain Monte Carlo simulation. A stylized regional CGE model is then used to contrast policy responses of a Cobb‐Douglas specification with those from the estimated translog equation.     

IS SPATIAL BOOTSTRAPPING A PANACEA FOR VALID INFERENCE?

Bootstrapping methods have so far been rarely used to evaluate spatial datasets. Based on an extensive Monte Carlo study we find that also for spatial, cross‐sectional data, the wild bootstrap test proposed by Davidson and Flachaire ( 2008 ) based on restricted residuals clearly outperforms asymptotic as well as competing bootstrap tests, like the pairs bootstrap.     

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.     

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.     

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.     

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.     

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).     

Aggregation of Regional Economic Time Series with Different Spatial Correlation Structures. 具有不同空间相关结构的区域经济时间序列集成

In this article, we compare the relative efficiency of different forecasting methods of space‐time series when variables are spatially and temporally correlated. We consider two cases: (1) univariate forecasting (i.e., a space‐time series aggregated into a single time series) and (2) the more general instance of multivariate forecasting (i.e., a space‐time series aggregated into a coarser spatial partition). We extend the results in the literature by including the consideration of larger datasets and the treatment of edge effects and of negative spatial correlation. We first introduce a statistical framework based on the space‐time autoregressive class of random field models, which constitutes the basis of our simulation study, and we present the various alternative forecasting methods considered in the simulation. We then present the results of a Monte Carlo study related to univariate forecasting. In order to allow a comparison with the findings of Giacomini and Granger (2004), we consider the same forecasting strategies and the same combinations of the parameter values used there, but with a larger parametric set. Finally, we extend our analysis to the case of multivariate forecasting. The outcomes obtained provide operational suggestions about how to choose between alternative forecasting methods in empirical circumstances.     

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.     

Bayesian Analysis of Knowledge Spillovers in European Regions*

Abstract. This paper estimates the effects of knowledge spillovers on patent growth rates across 335 European regions over the 1989–1999 period. We propose a dynamic model based on an innovation production function. A Bayesian approach is used to take into account area‐specific innovation and spatial spillovers. The estimation of the model proceeds via Markov Chain Monte Carlo simulation. The results show significant positive and negative spatial effects on innovative activity. The model allows for a rich spatial specification, which we illustrate by incorporating transport proximity measured by transportation time between regions to augment the typical spatial proximity measure of connectivity between regions. Doing this produces more pronounced spatial spillovers that exhibit a more polarized spatial pattern than a model relying on spatial proximity alone.     

Weighted‐Average Least Squares Applied to Spatial Econometric Models: A Monte Carlo Investigation

Recently, model averaging techniques have been employed widely in empirical investigations as an alternative to the conventional model selection procedure, a procedure criticized because it disregards a major component of uncertainty, namely, uncertainty regarding the model itself, and, thus, it leads to the underestimation of uncertainty regarding the quantities of interest. Bayesian model averaging (BMA) is one of the most popular model averaging techniques. Some studies indicate that BMA has cumbersome aspects. One of the major practical issues of using BMA is its substantial computational burden, which obstructs the process of obtaining exact estimates. A simulation method, such as Markov chain Monte Carlo (MCMC), is required to resolve this problem. Weighted‐average least squares (WALS) estimation has been proposed as an alternative to BMA. The computational burden of WALS estimation is negligible; therefore, it does not require the MCMC method. Furthermore, WALS estimation has theoretical advantages over BMA estimation. This article presents two contributions to the WALS literature. First, it applies WALS to spatial lag/error models in order to consider spatial dependence. Second, it extends WALS in order to consider explicitly the problem of multicollinearity by employing the technique of principal component regression. The small sample properties of the estimators of the proposed models are examined using Monte Carlo experiments; the results of these experiments suggest that the standard WALS may produce biased estimates when the underlying data‐generating process is a spatial lag process. Results also indicate that when the correlation among the regressors is high, the standard WALS estimators may suffer from large variances and root mean squared errors. Both of these problems are significantly mitigated by using the proposed models. Las técnicas de promediado de modelos (model averaging) vienen siendo empleadas con creciente frecuencia en las investigaciones empíricas como una alternativa a los procedimientos convencionales de selección de modelos estadísticos. Dichos procedimientos convencionales han sido criticados por no tomar en cuenta un componente clave de la incertidumbre: la incertidumbre del modelo en sí, y por lo tanto, conducen a la subestimación de la incertidumbre en la cuantificación de las valores estimados. El promediado bayesiano de modelos (Bayesian Model Averaging‐BMA) es una de las técnicas de promediado más usadas. Algunos estudios indican que BMA tiene aspectos engorrosos: uno de los principales aspectos prácticos a considerar en su uso es su pesada carga computacional, la cual obstruye el proceso de obtención de estimaciones exactas. Esta limitación hace necesario el uso de métodos de simulación, como el de la cadena de Markov de Monte Carlo (Markov Chain Monte Carlo‐MCMC). La estimación de mínimos cuadrados usando un promediado ponderado (Weighted‐Average Least Squares‐WALS) ha sido propuesta como alternativa a BMA. La carga computacional de la estimación WALS es mínima y por lo tanto no requiere del uso de MCMC. Más aun, la estimación WALS posee ventajas teóricas sobre BMA. Este artículo presenta dos contribuciones a la literatura especializada de WALS. En primer lugar, aplica WALS a modelos espaciales tipo lag/error con el fin de incorporar la dependencia espacial. En segundo lugar, modifica el método WALS, a fin de considerar explícitamente el problema de la multicolinealidad entre variables mediante el empleo de la técnica de regresión de componentes principales (Principal Component Regression‐PCR). Luego los autores utilizan experimentos Monte Carlo para examinar las propiedades de tipo “muestra pequeña” (small simple) de los estimadores de los modelos propuestos. Los resultados de los experimentos sugieren que el método WALS estándar puede producir estimaciones sesgadas cuando el proceso generador de datos subyacente (Data Generating Process‐DGP) es un proceso de retardo espacial (Spatial Lag Process‐SLP). Los resultados también indican que cuando la correlación entre las variables es alta, los estimadores estándar de WALS pueden padecer de varianzas y errores cuadráticos medios (root mean squared errors‐RMSEs) atípicamente grandes. Ambos problemas son mitigados significativamente mediante el uso de los modelos propuestos en el presente artículo. 近来,模型平均技术作为与传统模型选择流程可替换的方法,在经验调查中得到广泛应用。传统的模型选择流程忽视了模型本身的不确定性,进而低估了感兴趣样本数量的不确定性而受到批评。贝叶斯模型平均技术(BMA)是最为流行的模型平均技术之一。但已有研究表明,BMA在某些方面较为繁琐复杂,一个最主要的问题是其巨大的计算负荷阻碍模型了精确估计的过程,因此需要利用马尔可夫‐蒙特卡洛(MCMC) 之类的模拟方法进行解决。加权平均最小二乘(WALS)估计可作为BMA的可替换方法,其优点在于计算负荷可以忽略不计,因此不需要采用MCMC方法解决计算负荷问题。此外,WALS估计相比于BMA估计在理论上有一定的优势。本文针对WALS的贡献有两点：将WALS应用于空间滞后/空间误差模型以考虑空间依赖性,并利用主成分回归(PCR)拓展WALS以明确考虑多重共线性问题。本文利用蒙特卡洛实验对所提模型估计的小样本特征进行测试,结果显示当潜在数据生成过程(DGP)是一个空间滞后过程时,标准WALS可能产生有偏估计;此外,当回归量的相关性较高时,标准WALS估计量可能有较大的方差和根均方差(RMSEs).而本文提出的加权平均最小二乘估计模型能很好地缓解这两个问题。     

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.     

Decision Making and Uncertainty: Bayesian Analysis of Potential Flood Heights

This paper provides a case study of a method to estimate the value of additional information, before its acquisition, to aid decision making in the face of uncertainty. The approach employs conditional simulation in a Monte Carlo framework to conduct a Bayesian assessment of the value of information in an explicitly spatial setting. This paper demonstrates the procedure as applied by a hypothetical decision maker concerned with coastal flood control where flood damage is dependent on the spatial distribution of elevation. A set of known survey points provides the decision maker with limited knowledge of elevation. The method explored in the paper allows the decision maker to ascertain the potential value of additional survey information in terms of its ability to reduce uncertainty about flood damage.