1280 Spatial Analysis of Economic Convergence
wherexi,tis per capita income for a given country. Canova and Marcet (1995, p. 9)
argue that this alleviates potential problems of income cross-correlations among
countries since recessions and expansions which affect the whole aggregate of
regions would be factored out. This is not spatial autocorrelation, however, but
rather a shift in the mean of the series over time.
In the same spirit, a number of approaches to regional conditioning have been
suggested as ways to mitigate the impact of spatial autocorrelation in income series.
Quah (1996b) and Arbiaet al. (2003) rely on the following transformation:
̃xi,t=xi,t
∑
jwi,jxj,t, (27.30)where a region’s income is now expressed relative to its geographical neighbors
rather than the national average. Applying this transformation to European data
results in the removal of the bimodal characteristic of the income distribution.
Fischer and Stumpner (2008) apply the spatial filtering approach of Getis and
Ord (1992) to European data using the following filter:
x ̃i,t=xi,t[
1
n− 1 Wi]Gi(δ)
, (27.31)whereWi=
∑
jwi,j(δ)andwi,j(δ)is the(i,j)th element of the binary spatial weights
matrix such thatwi,j(δ)=1 if a region’siandjare separated by a distance of less
than the critical distance bandδ, andwi,j(δ)=0 otherwise.Gi(δ)is the local statistic
defined as:
Gi(δ)=∑
j∑wi,j(δ)xj,t
jxj,t. (27.32)
Comparing the unfiltered,f(x), and the spatially filtered,f(x ̃), distributions, Fis-
cher and Stumpner (2008) find that the latter displays much less dispersion than
the former, yet over the 1995–2003 period, the level of dispersion in the filtered
series increases by 15%, while the unfiltered series experiences an actual decline
in dispersion of 3.3%. Because the unfiltered series is highly spatially autocorre-
lated, the overall finding ofσ-convergence is attributed to the role of the spatial
dependence.
While the spatial filtering and regional conditioning approaches provide avenues
to explore the impact of spatial dependence on the evolution of regional income
densities, several questions remain. First, as Fischer and Stumpner (2008) note,
there is currently a lack of a formal inferential framework to test hypotheses about
distribution dynamics in the presence of spatial effects. Second, the filters applied
in (27.30) and (27.32) keepwi,jfixed over time. In other words, the spatial structure
is specified as time-invariant. Finally, the interpretation of just what the identified
spatial components represent is not at all clear.
27.3.4 Space-time kernels
Rather than filtering out the spatial component, kernels that explicitly incorporate
space can be developed. Figure 27.7 contains examples of how this can be done.