1270 Spatial Analysis of Economic Convergence
correlation statistic:
τ=
nc−nd
(n^2 −n)/ 2, (27.17)wherencis the number of concordant pairs of economies over a given interval.
Concordant pairs are those that maintain the same relative pair-wise ranking in
two periods.ndis the number of discordant pairs, where a discordant pair are
two economies whose relative ranks are reversed over the period. If all pairs are
concordant in a given period thennc=(n^2 −n)/2,nd=0 andτ=1. Conversely,
if all pairs are discordant, thenτ=−1.
To introduce a geographical dimension to this rank mobility measure, Rey
(2004b) suggested a decomposition of the concordance measure as:
nc=nc,r+nc,o, (27.18)wherenc,ris the number of concordant pairs in which the economies involved were
geographical neighbors, whilenc,oincludes those discordant pairs that involve
non-neighboring economies. Together with a similar spatial decomposition of the
discordant pairs, spatial versions of the rank correlation statistic can be defined as:
τr=nc,r−nd,r
(n^2 r−nr)/ 2, (27.19)for the neighboring pairs, and:
τo=nc,o−nd,o
(n^2 o−no)/ 2, (27.20)for the non-neighboring pairs of economies. The traditional measure of rank
mobility can be related to these spatial versions:
τ=ψτr+( 1 −ψ)τo, (27.21)where:
ψ=
ωr
ω, (27.22)withωrbeing the number of neighboring pairs of economies, andω=(n^2 −n)/ 2
being the number of all pairs.
Figure 27.1 summarizes the results of a spatial rank mobility analysis of state
incomes. The right-hand figure shows the definition of the neighborhood sets, here
based on BEA regions. The left-hand figure shows the z-value for theτtest statistic
standardized against its expected value based on spatial permutations of the income
values. The statistic is significantly negative in the earlier periods, indicating that
the amount of intraregional concordance was below what would be expected if
there was no spatial structure to income growth. In other words, the differences in