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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. [ABSTRACT FROM AUTHOR]