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.     

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.     

ACCOUNTING FOR LOCAL SPATIAL HETEROGENEITIES IN HOUSING MARKET STUDIES

We adopt a novel method to deal with omitted spatial heterogeneities in hedonic house price analysis. A Gaussian variant of the conditional autoregressive (CAR) model is used to study the impact of spatial effects. In a general linear modeling framework, we include zone‐specific random effects that are allowed to interact spatially with neighboring zones. The results demonstrate that this estimator accounts for missing spatial information, producing more reliable results on estimated spatially related coefficients. The CAR model is benchmarked against a fixed effects model. Socioeconomic neighborhood characteristics are found to have only modest impact on spatial variation in housing prices.     

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.     

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.     

Beyond Moran's I: Testing for Spatial Dependence Based on the Spatial Autoregressive Model

The statistic known as Moran's I is widely used to test for the presence of spatial dependence in observations taken on a lattice. Under the null hypothesis that the data are independent and identically distributed normal random variates, the distribution of Moran's I is known, and hypothesis tests based on this statistic have been shown in the literature to have various optimality properties. Given its simplicity, Moran's I is also frequently used outside of the formal hypothesis-testing setting in exploratory analyses of spatially referenced data; however, its limitations are not very well understood. To illustrate these limitations, we show that, for data generated according to the spatial autoregressive (SAR) model, Moran's I is only a good estimator of the SAR model's spatial-dependence parameter when the parameter is close to 0. In this research, we develop an alternative closed-form measure of spatial autocorrelation, which we call APLE , because it is an approximate profile-likelihood estimator (APLE) of the SAR model's spatial-dependence parameter. We show that APLE can be used as a test statistic for, and an estimator of, the strength of spatial autocorrelation. We include both theoretical and simulation-based motivations (including comparison with the maximum-likelihood estimator), for using APLE as an estimator. In conjunction, we propose the APLE scatterplot, an exploratory graphical tool that is analogous to the Moran scatterplot, and we demonstrate that the APLE scatterplot is a better visual tool for assessing the strength of spatial autocorrelation in the data than the Moran scatterplot. In addition, Monte Carlo tests based on both APLE and Moran's I are introduced and compared. Finally, we include an analysis of the well-known Mercer and Hall wheat-yield data to illustrate the difference between APLE and Moran's I when they are used in exploratory spatial data analysis.     

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.     

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.     

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) .     

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.     

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.     

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.     

Effects of Vaccination and the Spatio-Temporal Diffusion of Covid-19 Incidence in Turkey

This study assesses the spatio-temporal impact of vaccination efforts on Covid-19 incidence growth in Turkey. Incorporating geographical features of SARS-CoV-2 transmission, we adopt a spatial Susceptible–Infected–Recovered (SIR) model that serves as a guide of our empirical specification. Using provincial weekly panel data, we estimate a dynamic spatial autoregressive (SAR) model to elucidate the short- and the long-run impact of vaccination on Covid-19 incidence growth after controlling for temporal and spatio-temporal diffusion, testing capacity, social distancing behavior and unobserved space-varying confounders. Results show that vaccination growth reduces Covid-19 incidence growth rate directly and indirectly by creating a positive externality over space. The significant association between vaccination and Covid-19 incidence is robust to a host of spatial weight matrix specifications. Conspicuous spatial and temporal diffusion effects of Covid-19 incidence growth were found across all specifications: the former being a severer threat to the containment of the pandemic than the latter.     

乡村振兴导向下乡村发展能力的多维测度及时空交互特征

乡村振兴要求走乡村内生性发展道路,客观认知乡村发展能力的时空分异格局和交互关系对乡村振兴战略实施具有重要意义。本文从农业产业发展、农村基础建设和农民社会生活三个维度构建测度指标体系,并引入面板向量自回归模型识别影响乡村发展能力的敏感因素,以此测度乡村发展能力指数,进而采用探索性时空数据分析方法揭示乡村发展能力的时空交互特征。结果表明：(1)我国中部地区多维乡村发展能力提升,但呈明显的空间异质性,不同维度间差异化显著;(2)中部地区多维乡村发展能力局域空间格局呈现较强的动态性并整体表现出较强的空间依赖关系;(3)邻接地级市多维乡村发展能力的时空交互网络以正向关联为主,局部空间呈差异化的竞合态势。     

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.     

A SPATIAL AND TEMPORAL AUTOREGRESSIVE LOCAL ESTIMATION FOR THE PARIS HOUSING MARKET

ABSTRACT This original study examines the potential of a spatiotemporal autoregressive Local (LSTAR) approach in modeling transaction prices for the housing market in inner Paris. We use a data set from the Paris Region notary office (Chambre des notaires d’Île‐de‐France) which consists of approximately 250,000 transactions units between the first quarter of 1990 and the end of 2005. We use the exact X–Y coordinates and transaction date to spatially and temporally sort each transaction. We first choose to use the STAR approach proposed by Pace et al., 1998 . This method incorporates a spatiotemporal filtering process into the conventional hedonic function and attempts to correct for spatial and temporal correlative effects. We find significant estimates of spatial dependence effects. Moreover, using an original methodology, we find evidence of a strong presence of both spatial and temporal heterogeneity in the model. It suggests that spatial and temporal drifts in households socio‐economic profiles and local housing market structure effects are certainly major determinants of the price level for the Paris Housing Market.     

Comparative Spatial Filtering in Regression Analysis

One approach to dealing with spatial autocorrelation in regression analysis involves the filtering of variables in order to separate spatial effects from the variables’ total effects. In this paper we compare two filtering approaches, both of which allow spatial statistical analysts to use conventional linear regression models. Getis’ filtering approach is based on the autocorrelation observed with the use of the G_i local statistic. Griffith's approach uses an eigenfunction decomposition based on the geographic connectivity matrix used to compute a Moran's I statistic. Economic data are used to compare the workings of the two approaches. A final comparison with an autoregressive model strengthens the conclusion that both techniques are effective filtering devices, and that they yield similar regression models. We do note, however, that each technique should be used in its appropriate context.     

TESTING CRITERIA FOR DETERMINING LEADING REGIONS IN WAGE TRANSMISSION MODELS*

In this study, we use a vector autoregressive modeling framework to test the zero restrictions implied by alternative criteria for ranking regions into hierarchies in the wage-transmission debate. This approach allows formal statistical tests to be carried out for competing criteria suggested for determining leading and following regions. The ability of the modeling technique proposed here to produce a set of nested hypothesis tests of alternative criteria is in stark contrast to the historical literature in this debate. Researchers have traditionally proposed a criterion for ranking regions into a hierarchy, and then argued for the merits of their particular criterion by demonstrating that an econometric model of wage formation produces statistically significant coefficient estimates when their criterion is used to rank the regions of their data sample. We apply the methods proposed here to a sample of eight midwestern cities in the U.S. in order to test the following criteria: Beaumont (1983), Hart and MacKay (1977), Reed and Hutchinson (1976), unemployment rate, and earnings level. The test is for consistency with the Granger-causal structure of wage interactions inherent in the wage diffusion idea. We argue that the technique set forth here is a real step forward that should allow a resolution of this particular debate. The proposed procedures might also be applied to empirically test other regional science hypotheses concerning for example, intercity and interindustry causal structures.     

A Spatial Price Determination Model for Multi‐Market Heterogeneously Distributed Resources: An Application to Forestry Markets in Sweden

The spatial dimension is a key paradigm in price determination, as attested by recent studies in the literature that highlighted the differential in market behavior between spatial and non‐spatial pricing settings. In this paper, we develop a model of spatial pricing for multi‐market heterogeneously distributed resources, with an application to the Swedish forestry sector. The focus of the model is to estimate the impact of spatial interaction on the demand for resources in terms of resource allocation, competition, and pricing. In its core, the pricing mechanism relies on a supply–demand framework. Using disaggregated data at the gridcell level for forest feedstock supply and harvesting costs in Sweden, we construct regional supply curves for each gridcell assuming a maximum transportation distance to delimit the potential market. Demand nodes are exogenously determined and are adjusted using a distance‐decay model to assess demand pressure across locations. We apply the model empirically to assess the impact on forest feedstock prices of a 20 TWh increase in biofuel production.     

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.