Journal of Econometrics Spatial models have a long history in the regional science and geography literature. Somewhat more recently, spatial models have also become an important tool in economics for the analysis of spatially dependent data. The purpose of this volume is to bring together a variety of studies that relate both to the further theoretical development of such models, and their application to various economic issues.

Volume 140, Issue 1 (September, 2007)

1-4 Baltagi, Badi H. & Kelejian, Harry H. & Prucha, Ingmar R.   Analysis of spatially dependent data
  Overview
5-51 Baltagi, Badi H. & Heun Song, Seuck & Cheol Jung, Byoung & Koh, Won   Testing for serial correlation, spatial autocorrelation and random effects using panel data
  Baltagi, Song, Jung, and Koh consider a spatial panel data model which involves time series autocorrelation, as well as spatial dependence between spatial units at each point in time. The model also allows for heterogeneity of spatial units via random effects. They derive several Lagrangian multiplier tests for this model including a joint test for serial correlation, spatial correlation, and random effects. The joint LM test derived in this paper encompass those derived in earlier studies by Anselin and Bera (1998) and by Baltagi et al. (2003). It is shown that the earlier LM tests are marginal LM tests that ignore either serial correlation over time or spatial error correlation. Testing for any one of these ignoring the other two is shown to lead to misleading results. The paper also derives conditional LM and LR tests that do not ignore these correlations and contrast them with their marginal LM and LR counterparts. The small sample performance of these tests is investigated using Monte Carlo experiments.
52-75 Brock, William A. & Durlauf, Steven N.   Identification of binary choice models with social interactions
  Spatial models do not only relate to spatial dependencies and interactions across geographic space, but may be viewed more generally as models for cross sectional dependence and interactions. Lee (2004) relates the class of spatial Cliff–Ord models to certain social interaction models. The paper by Brock and Durlauf in this volume aims at developing an analysis of the identification problem for social interactions in binary choice models using individual level data. The paper explores the identifiability of model parameters without assuming that the distribution of random payoff terms is known, and extends the analysis of Manski (1988) and (Brock and Durlauf, 2001a) and (Brock and Durlauf, 2001b). Among other things, the paper also analyzes the identification of social interactions in the presence of unobserved group effects.
76-96  Conley, Timothy G. & Molinari, Francesca   Spatial correlation robust inference with errors in location or distance
 

A key ingredient in spatial model is the choice of metric space and locations for the observed agents. However, in many cases the agents’ locations may not be known with certainty by the econometrician. Conley and Molinari investigate the consequences of measurement errors in locations/distances upon inference. Their approach is to use smoothed periodogram covariance matrix estimators that are consistent in the presence of bounded, potentially endogenous location errors. They present Monte Carlo results that relate to the impact of location errors upon the precision of estimators of an asymptotic variance. They also give Monte Carlo results which relate to two new specification tests for parametric estimators of that asymptotic variance. Among other things, these results relate to comparisons of parametric and nonparametric estimators of that asymptotic variance. They find that nonparametric estimators are quite robust to location errors, as are method of moments estimators. On the other hand, MLE estimators perform poorly.

97-130 Kapoor, Mudit & Kelejian, Harry H. & Prucha, Ingmar R.    Panel data models with spatially correlated error components
  The paper by Kapoor, Kelejian, and Prucha considers a panel data model where the disturbances have an error component structure. The innovations are modeled as a first order spatially autoregressive process and are thus allowed to be spatially correlated. The model blends specifications typically considered in the spatial literature with those considered in the error components literature. The paper introduces generalizations of the generalized moments (GM) estimators suggested in Kelejian and Prucha (1999) for estimating the spatial autoregressive parameter and the variance components of the disturbance process, and discusses alternative weighting schemes for the moments. Kapoor, Kelejian, and Prucha then use those GM estimators to define a feasible generalized least squares procedure for the regression parameters, and give formal large sample results for the proposed estimators. The estimators remain computationally feasible even in large samples.
131-154 Kelejian, Harry H. & Prucha, Ingmar R.   HAC estimation in a spatial framework
  Kelejian and Prucha suggest a nonparametric heteroscedasticity and autocorrelation consistent (HAC) estimator for an asymptotic variance covariance (VC) matrix which would naturally arise in a spatial framework in which an instrumental variable (IV) procedure is used to estimate the model parameters. They formally demonstrate consistency of their estimator under a set of relatively simple assumptions that covers, among others, the important and widely used class of Cliff–Ord type spatial models. The specification of the HAC estimator allows for more than one measure of distance, each of which may be measured with error, and only assumes that one of the distance measures considered by the researcher corresponds to the true one. The authors also derive the asymptotic distribution of an IV estimator for the parameters of a general spatial model and demonstrate that a consistent estimator of the VC matrix involved can be based on the suggested HAC procedure.
155-189 Lee, Lung-fei   The method of elimination and substitution in the GMM estimation of mixed regressive, spatial autoregressive models
  The paper by Lee considers the estimation of a mixed regressive spatial autoregressive model, which is, in the terminology of Anselin, also often referred to as a spatial autoregressive model with autoregressive disturbances, for short SARAR(1,1). The paper introduces a computationally simple generalized method of moments (GMM) for the estimation of this model. This method is based on the method of elimination and substitution in linear algebra, and the modified GMM procedure can substantially reduce the computational burden. The approach reduces the joint estimation of the entire unknown parameter vector into the estimation of separate components. The paper shows that for the mixed regressive spatial autoregressive model, the nonlinear estimation is reduced to the estimation of the (single) spatial effect parameter. The paper furthermore identifies situations under which the modified GMM estimator can be as efficient as the joint GMM estimator. Among other situations, this will be the case if the disturbances have zero third moments, e.g., if the disturbances are distributed Gaussian.
190-214 LeSage, James P. & Kelley Pace, R.   A matrix exponential spatial specification
  Among the most widely used spatial models are variations on the one put forth by (Cliff and Ord, 1973) and (Cliff and Ord, 1981). For these models maximum likelihood estimation is tedious and even forbidding in large samples because of certain computational issues. LeSage and Pace suggest an alternative model which is based on a matrix exponential specification. Their model has certain features that simplify the computational burdens involved in estimation, as well as certain theoretical analyses, including Bayesian analysis. LeSage and Pace also give results which suggest that empirical results based on their model are quite similar to those which would be based on Cliff and Ord-type models.
215-225 Pinkse, Joris & Shen, Lihong & Slade, Margaret   A central limit theorem for endogenous locations and complex spatial interactions
  The paper by Pinkse, Shen, and Slade introduces a new central limit theorem (CLT) for spatial processes. The CLT is derived under a very general set of assumptions, that plausibly covers many economic applications in which location is endogenous. The notion of weak dependence employed by the theorem extends that of Douchan and Louhichi (1999). The theorem is derived using Bernstein blocking techniques.
226-259 Sain, Stephan R. & Cressie, Noel   A spatial model for multivariate lattice data
  The paper by Sain and Cressie develops a random Markov field model for multivariate lattice data. More specifically the paper extends the univariate conditional autoregressive (CAR) model to what the authors call a canonical multivariate conditional autoregressive (CAMCAR) model. The model allows for covariates as well as general forms of spatial dependence between the variables observed at respective spatial locations. The paper then develops a hierarchical model that incorporates the CAMCAR model. Using Bayesian inference and Markov chain Monte Carlo methods the paper examines the racial distribution of residents of southern Louisiana in relation to the location of sites listed with the U.S. Environmental Protection Agency's Toxic Release Inventory.
260-281 Baltagi, Badi H. & Egger, Peter & Pfaffermayr, Michael     Estimating models of complex FDI: Are there third-country effects?
  The paper by Baltagi, Egger, and Pfaffermayr considers the recent general equilibrium theory of trade and multinationals which emphasizes the importance of third countries and the complex integration strategies of multinationals. They test this theory empirically by considering not only bilateral determinants, but also spatially weighted third-country determinants of FDI. Since the dependency among host markets is particularly related to multinationals’ trade between them, they use trade costs (distances) as spatial weights. Using a panel of manufacturing and non-manufacturing industries and a large group of host countries observed over the 1989–1999 period, they estimate a bilateral three-factor knowledge-capital model that allows for spatial correlation in the independent variables and the error term. They find that third-country effects are significant, lending support to the existence of various modes of complex FDI. The linkage between host countries seems to be positively related to goods traded by multinationals and declines with bilateral trade costs among the host countries.
282-303 Conley, Timothy G. & Topa, Giorgio   Estimating dynamic local interactions models
  Conley and Topa present empirical methods that are useful in calibrating, estimating, and evaluating a class of local interactions models in which agents’ transitions are affected by their neighbors’ states. They apply it to urban unemployment and social networks in job search using publicly available cross-section and retrospective data. In their model, each agent is directly connected to a small number of others and their employment status influences his/her employment transitions. Most links in their model are local, but some span an entire metropolitan area. They also present methods to investigate how well the individual spell distributions implied by their estimated model match those from retrospective Current Population Survey data. Finally, they present descriptive methods to illustrate the implications of the model estimates by computing Impulse Response Functions, both in time and in space, to local employment shocks. This allows them to study, among other things, how far a given shock in one area can propagate to nearby areas.
304-332  Keller, Wolfgang & Shiue, Carol H   The origin of spatial interaction
  The Keller–Shiue paper is an application of spatial techniques to the analysis of trade patterns relating to Chinese rice prices. Specifically, they use spatial methods to explain rice price differences in Chinese prefectural markets for the years 1742–1795. Their analysis is based on a data set which is carefully constructed. Among other things, variables explicitly relating to geographic features such as rivers and coastal waterways are considered. Their modeling efforts involve spatial lags in the dependent variable, as well as in the error term. They also consider time autocorrelation involving a spatial lag of the dependent variable. Various tests of model specification are considered. They show that explicit modeling of spatial features is key to understanding the expansion of interregional trade as well as the rise and fall of trading hubs.