Spatial regimes in regional European growth: an iterated spatially weighted regression approach

Abstract

This paper proposes a new technique based on an iterated spatially weighted regression procedure to verify the presence of economic growth heterogeneities in EU regions in the period 1981–2009. The approach extends a procedure originally proposed in the field of image analysis based on the assumption of local homogeneity of the signal. The presence of the heterogeneity is a criterion to divide the sample of observations (i.e. regions) into smaller homogeneous groups. Our results highlight the presence in the EU regions of five different clubs with different growth paths within each subgroup. Spatial dependence is also considered in the definition of the economic convergence model.

Appendix

1.1 Main steps

• 1. Initialization for each region i calculate initial estimates ${\hat{\mathbf{b}}}^{0} (i)$ with standard GWR, where $g_{ij}^{0} = K_{d} (d_{ij} ) = e^{{ - \gamma d_{ij} }}$ and compute ${\mathbf{\hat{\sigma }}}_{\varepsilon }^{2}$;

• 2. Main step: computation of the weights $\dot{g}_{ij}^{l}$ at each iteration l compute the statistics $T_{ij}^{l} = \left( {{\hat{\mathbf{b}}}^{l - 1} (i) - {\hat{\mathbf{b}}}^{l - 1} (j)} \right)\hat{\Sigma }^{ - 1} \left( {{\hat{\mathbf{b}}}^{l - 1} (i) - {\hat{\mathbf{b}}}^{l - 1} (j)} \right)^{t}$, $K_{d} (d_{ij}^{l} )$, $K_{T} (T_{ij}^{l} )$ and determine $g_{ij}^{l} = K_{d} (d_{ij}^{l} ) \cdot K_{T} (T_{ij}^{l} ) = \left( {e^{{ - \gamma d_{ij} /l}} } \right) \cdot \left( {e^{{ - \tau T_{ij}^{l} }} } \right)$. Apply the convex combination $\dot{g}_{ij}^{l} = (1 - \eta )g_{ij}^{l} + \eta \dot{g}_{ij}^{l - 1}$ to determine the new weights that represent the elements of G ^l;

• 3. Stopping if $\hbox{max} \left| {\dot{g}_{ij}^{l - 1} - \dot{g}_{ij}^{l} } \right| < \omega ,\quad \forall i,j$ with ω a fixed small value, the procedure is stopped. The final weight matrix is G* with generic element given by $g_{ij}^{*}$. Otherwise continue;

• 4. Estimation of the new model use $\bf {\text{G}}^{ * } (i)$, diagonal matrix with the elements of main diagonal constituted by the i-th row of G*, to estimate the parameters of the model.