A spatial error model with continuous random effects and an application to growth convergence

Abstract

We propose a spatial error model with continuous random effects based on Matérn covariance functions and apply this model for the analysis of income convergence processes ($\beta$-convergence). The use of a model with continuous random effects permits a clearer visualization and interpretation of the spatial dependency patterns, avoids the problems of defining neighborhoods in spatial econometrics models, and allows projecting the spatial effects for every possible location in the continuous space, circumventing the existing aggregations in discrete lattice representations. We apply this model approach to analyze the economic growth of Brazilian municipalities between 1991 and 2010 using unconditional and conditional formulations and a spatiotemporal model of convergence. The results indicate that the estimated spatial random effects are consistent with the existence of income convergence clubs for Brazilian municipalities in this period.

Appendix

This appendix briefly presents a robustness analysis using a SPDE-Matérn model specification relying on an alternative triangulation, based on a convex grid generated by the geographic centroid of each municipality. This specification can be thought of as the most basic triangulation that preserves the existing information in the spatial distribution of municipalities, with a unique focal point for each observed region. Figure 15 shows this triangulation, constructed using only nodes generated by the geographic centroid of each municipality.

Fig. 15

Triangulated Delauney mesh: Brazilian municipalities using centroids

The posterior distribution of the estimated parameters is shown in Table 8. We can note that the results are quite similar to those observed with the use of finer triangulation (see Table 2), showing that the results are robust to the choice of an alternative mesh. Figure 16 shows the posterior mean for spatial random effects estimated based on this alternative triangulation. In general, the results are quite similar to those observed in the original formulation. The model captures the existing precision in the distribution of points in space, with more precise confidence intervals for the regions with the highest density of points, and intervals more open in regions with lower density of points, especially in the northern region of the country. Note that this result is due to the interpolation mechanism used in the representation of the solution of the partial differential stochastic equation used in the representation of the spatial random field (see Eq. (7)), which is also used in determining the posterior distribution of spatial random effects.

Table 8 Estimation results: $\beta$-convergence for Brazilian municipalities, SPDE-Matérn model with alternative triangulation (1991–2010)

Fig. 16

Posterior mean of the spatial random effects: SPDE-Matérn model with an alternative triangulation

This result also indicates that continuous representation of space can properly handle the presence of an irregular lattice structure, a major problem in model specifications with areal data, as discussed in Wall (2004). The posterior distribution of spatial random effects captures the amount of data available for density regions on the map, avoiding the problems associated with the use of spatial weights where this information is not used optimally.