A Biparametric Approach to Spatial Autocorrelation

Abstract

In spatial econometric models, autocorrelation among error terms is usually incorporated by means of the so-called contiguity matrix W, determining the interdependence between the spatial observations on the dependent variable. In this paper, the analysis is generalized by introducing two contiguity matrices, related to two autocorrelation parameters. This may be useful when dealing with variables representing flows between regions, where both the origin and the destination regions have a different impact on the autocorrelation scheme. It is shown analytically and illustrated empirically that the presence of such autocorrelation can be tested with the likelihood-ratio test, whereas the parameters can be estimated by the maximum-likelihood approach.1 Introduction In test procedures for autocorrelation among spatial data, the alternative hypothesis is mostly restricted to the first-order spatial Markov scheme. Thereby the spatial units are linked by a so-called contiguity matrix, which contains weights indicating the degree of interaction between the spatial units with respect to the phenomenon that is studied, for example, income in or flows between regions. Estimation and testing methods based on such a scheme were studied among others by Cliff and Ord (1973), Hepple (1976), andBrandsma andKetellapper (1979). Alternative approaches based on less prior information concerning interactional weights were proposed by Arora and Brown (1977) and Kooyman (1976).These latter two argued that ambiguity in specifying the contiguity matrix W on a priori considerations prevents successful application of this concept of autocorrelation. In fact, Arora and Brown suggested the use of econometric techniques like joint generalized least squares in case the errors of the model display autocorrelation, whereas Kooyman constructed a test statistic for autocorrelation by maximizing the Moran statistic as a function of its weight coefficients under certain constraints. Both papers start from the assumption that only very vague knowledge about possible alternative hypotheses of autocorrelation is available. However, we are of the opinion that testing and estimation based on a specification of W which is theoretically well founded may still be appropriate since it gives the analyst insight into how to respecify his model in order to incorporate the factors that caused the autocorrelation explicitly (for example, the introduction of spatially lagged variables when W represents the Markov scheme-cf Haining, 1978b); the approaches useful for exploratory rather than explanatory analysis. In addition, it is virtually impossible to estimate all elements of W from the data, unless W is postulated to depend only on a few parameters, for example, by means of a distance function. More on this approach follows in section 4.A more general approach to the problem of spatial dependence was followed by Haining (1978b), who (1) defined two-dimensional moving-average schemes, (2) discussed a specification technique based on spatial autocorrelation coeffic...

Multivariate Modeling with Interdependent Network Data

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Modèle d’explication de flux à composantes d’erreurs spatialement corrélées

“…La formulation complète du modèle est désignée par l'acronyme EC-SAR(I). Brandsma et Ketellapper (1979) (BK) et Bolduc, Dagenais et Gaudry (1989) (BDG) constituent deux tentatives intéressantes pour résoudre le problème de corrélation spatiale dans les erreurs de flux. L'approche BK est fondamentalement correcte mais ignore un groupe de facteurs quelquefois importants pour expliquer la corrélation.…”

Modelling Spatial Autocorrelation in Spatial Interaction Data

“…Spatial interaction models are said to be misspecified if the residuals are spatially autocorrelated, violating the independence assumption. This problem has been largely neglected so far, with very few exceptions (see, for example, Brandsma and Ketellapper, 1979, Griffith and Jones, 1980, Baxter, 1987, Bolduc, Laferiere and Santarossa, 1992, 1995, Fischer, Reismann and Scherngell, 2006a, LeSage and Pace, 2007. This neglect may be because spatial interaction models are more complex than models for the geographic distribution of attribute data, with each region being associated with several values as an origin as well as a destination so that specification of the autocorrelation structure is less obvious.…”