Ecological inference and spatial heterogeneity: an entropy‐based distributionally weighted regression approach

. In this article we compare two competing approaches to ecological modelling using test data. The first approach is based on the “traditional” method of Ordinary Least Squares (OLS), assuming constancy of parameters across disaggregated spatial units (spatial homogeneity). The second (new) approach is based on the method of Generalised Cross‐Entropy (GCE), assuming varying parameters (spatial heterogeneity). The latter approach is designated as entropy‐based “distributionally weighted regression” (DWR). The two approaches are tested in a real‐world application, using data on per‐capita GDP for the 17 regions and some covariates for the 50 provinces of Spain. Specifically, the performances of the two approaches are assessed by examining their capability in tracking the actual per‐capita GDP data for the provinces (while treating them as if they were not observed by the econometrician), and in showing evidence of spatial heterogeneity. Our findings indicate that the GCE varying‐parameter approach outperforms the OLS approach in terms of predictive power. Specifically, we find that the GCE predictions make efficient use of the lower‐level information that is available. In addition, it is shown that entropy‐based DWR has some potential as a useful technique for investigating spatially heterogeneous relationships at the lower level of analysis that might otherwise be overlooked.