Symptoms of Instability in Models of Spatial Dependence

Abstract

In recent years, there has been a growing interest in the problems caused by the existence of instability in cross-sectional regressions. The results about local autocorrelation measures are part of this debate, as are the proposals concerning the concept of geographically weighted regressions. This article also deals with the problem of stability (or the lack thereof), but focusing the discussion on the supposition of constancy in the parameter of spatial dependence. In most cases, this assumption is treated, with the risks that this involves, as a maintained hypothesis, which should be ascertained before continuing with the modeling exercise. In the article, we present a simple heterogeneity test for this type of parameters, based on the Lagrange Multiplier principle. To illustrate its use, we take the distribution of per capita income among the European regions as our discussion case. According to our results, there are clear signs of structural breaks in the spatial distribution of this variable and the scale factor and the autocorrelation coefficient appear to be principal actors. IntroductionThere is a quite extended consensus that heterogeneity and dependence are the fundamental characteristics of spatial data (Griffith 2003). The literature has focused mainly on the treatment of spatial dependence, with extremely relevant results. The question of heterogeneity has not received so much attention, which could be attributed to a desire to limit the complexity and uncertainty involved in the specification of any work that models spatial data. Nevertheless, there is widespread evidence that spatial behavioural relations tend to be unstable, as shown by Haining (2003, chapter 4) and previously by Cressie (1991) and Upton and Fingleton (1985), among others. The complexity appears in the different dimensions associated with this broad topic, which can hide problems of global instability in the specification (relatively little studied in the specialized literature), of nonstability in the parameters of the equation or of lack of constancy of the mechanism of spatial interaction that operates in the model.Our article focuses on the last aspect and, specifically, on the problem of ''(1) testing for the presence of spatial heterogeneity by means of appropriate diagnostics'' as Anselin (1990, p. 186) indicates. To this end, we develop a heterogeneity test for the parameter of spatial dependence, based on the principle of the Lagrange Multiplier, which is relatively simple to obtain and can be easily adapted to different situations of interest. Continuing with the reference of Anselin (1990, p. 186), we believe that the issues associated with ''(2) implementing alternative estimation techniques when needed'' are well dealt with in the articles of Brunsdon, Fotheringham, and Charlton (1998b) and Pace and Lesage (2004), which we adopt as our guidelines.To illustrate the applicability of our proposal, we take the case of the distribution of per capita income on the European continent as the element of discussion. It i...

Further Topics in Spatial Econometrics

“…The literature in this field is vast, rapidly-growing and still in the early stages of development in many respects (for a review, see Fotheringham et al, 2002;McMillen and McDonald, 2004;Pace and LeSage, 2004;Wheeler and Paez, 2010). An obvious extension of the basic model presented in section 4.4.2 concerns the use of spatial econometric models belonging to the SARAR family in place of the simple OLS local regression (see Brunsdon et al, 1998;Paez et al, 2002;Pace and LeSage, 2004;and Mur et al, 2008). However, this extension encounters severe computational challenges when operating with large sample size.…”

Testing the hypothesis of stability in spatial econometric models

“…Mur et al. (2008) wonder whether the large differences observed between the European regions can also affect the capacity of the regions to relate to their immediate neighbours.…”

“…(2002a, 2002b). Recently, Mur et al. (2008) try to bring the two topics even closer by developing several tests of instability in the coefficient of spatial autocorrelation.…”

“…The last two papers may be seen as a natural extension of the locally weighted regressions (LWR) algorithm, introduced by the seminal papers of Cleveland (1979) and Cleveland and Devlin (1988), and subsequently developed by the works of, among others, McMillen (1996), McMillen and McDonald (1997), Brunsdon et al (1998a), Leung et al (2000Leung et al ( , 2003 or Paez et al (2002aPaez et al ( , 2002b. Recently, Mur et al (2008) try to bring the two topics even closer by developing several tests of instability in the coefficient of spatial autocorrelation. The present paper insists along the same line of combining heterogeneity and spatial dependence, although adopting a broader perspective than the previous works.…”

A Simulation Study on Specifying a Regression Model for Spatial Data: Choosing between Autocorrelation and Heterogeneity Effects

“…Unfortunately, all such model specification issues are almost always difficult to address with any objectively, and commonly involve analytical impasses and confounders (Anselin , 204; Pace and LeSage , 31; Mur, López, and Angelo ). Analytical issues are particularly pertinent for spatial data, as their collection are rarely part of a statistically designed experiment—that by definition should negate confounders (e.g., Stefanova, Smith, and Cullis ).…”