Sergio J. Rey and Julie Le Gallo 1269also been adopted to explore the question of regional heterogeneity. Bickenbach
and Bode (2003) do this by estimating transition matrices for the groups of states in
each of the Bureau of Economic Analysis (BEA) regions of the US. They also suggest
formal tests of regional homogeneity and find strong evidence that the transitional
dynamics are not homogeneous across the US space economy.
While the spatial Markov approach offers an interesting extension of the classic
Markov chain to the geographical context, there are a number of methodological
issues associated with the approach that require further investigation. The first issue
surrounds the data-intensive requirements needed to estimate a spatial Markov
matrix. Rather than having to estimatek^2 transition probabilities as in the classic
approach, when one is conditioning onkclasses for the spatial lag, the number of
probabilities to estimate grows tok^3. This can lead to many zero, or small count,
observations for specific types of transitions, which in turn means a loss of precision
in the estimation of those probabilities.
One solution to this degrees of freedom problem is to combine the thin count
cells with other cells and estimate the transition probabilities for the new aggre-
gated cells. This could be done in several ways. One approach would be to keep the
number of classes for the spatial lag fixed, and then collapse the same group of cells
within each of thekconditional transition matrices. This would result inkcondi-
tional matrices of orderr×c, withr≤kandc≤k. An alternative approach would
be to keep the number of income classes fixed atkbut reduce the number of classes
for the spatial lag, essentially aggregating together cells across the conditional tran-
sition matrices. In this case one would havel≤kconditional transition matrices
of orderk×k. Of course, a third option would be to aggregate both the number
of classes for the income variable as well as the spatial lag. The choice between
these alternatives is not inconsequential, as the first approach would trade a loss
in resolution of the income class transitions for maintaining detail in the effect of
spatial context (that is, the spatial lag). In the second approach, more detail on
the class transitions is gained at the expense of a coarser view of spatial context
effects. To date, however, the relative merits of these different approaches remain
unexplored.
A final issue with the spatial Markov approach is that it focuses only on the
transitions of individual economies within the income distribution subject to the
level of income in the surrounding economies in the initial period. It does not also
treat the transition of the spatial lag, but uses that as the conditioning variable.
Joint consideration of the transitions of the spatial lag and the income of an econ-
omy would require estimatingk^4 transition probabilities (assumingkclasses for
incomes and the spatial lag). Clearly this would exacerbate the degrees-of-freedom
problem.
27.3.1.4 Spatial rank mobility
Rank mobility measures attempt to address the problems with discretization to
provide a more comprehensive picture of regional income mobility (Boyle and
McCarthy, 1997; Webberet al., 2005). One classic measure is Kendall’s rank