1266 Spatial Analysis of Economic Convergence
whereα,β, andσ
2 are as defined in equation (27.1). If 0<β<1 the difference
equation is stable andβ-convergence gives rise to the steady-state variance:
σ∗^2 =
σ
2
1 −( 1 −β)^2. (27.16)
The cross-sectional variance in incomes will fall with increases inβ(in the stable
range) but rise with the initialσ
2.
There are a number of limitations ofσ-convergence for studying the dynamics
of the distribution. First, it focuses on only the second moment of the distribution
and is thus silent on other moments, such as skewness and kurtosis, which can be
important from a substantive perspective. Second, the sample variance would also
mask any multimodality or twin-peakedness in the distribution, which tends to
be a common finding at the international scale. In addition to being silent on the
morphology of the distribution, measures ofσ-convergence provide no insight on
the degree of intradistributional mixing and mobility.
While these criticisms are generally well known, there are also several lesser
known problems with the application ofσ-convergence to spatially referenced
data. The first is a spatial identification problem. This arises from the sample vari-
ance being what is known as a “whole map” statistic. More specifically, given a
map (spatial distribution) ofnincomesyi,t:i=1, 2,...,n, with sample variance
σˆt^2 , there aren!spatial permutations of the map that would have the same sample
variance.
The second difficulty withσ-convergence in a spatial context relates to the i.i.d.
assumption on. As is the case for the confirmatory econometric modeling of
convergence, the presence of spatial dependence in the error term of the model
complicates the analysis. The impact of spatial dependence on the interpretation
ofσ-convergence has been examined by Egger and Pfaffermayr (2006) and Rey and
Dev (2006), who show that, in addition to theβcoefficient and the initial variance
level, the value of the sample variance will also reflect the level and structure of the
spatial dependence. More specifically, if the underlying data generating process is
the spatial lag specification, then the sample variance for the income levels will
be sensitive to the value of the spatial lag parameter and the specific structure of
the spatial weights matrix. These additional sources of dynamics complicate the
interpretation of the sample variance as a measure ofσ-convergence.
27.3.1.2 Markov chain models
Quah (1993a, 1996a) has adopted a discrete Markov chain approach to study the
evolution of income distributions. Using international data for 1962–84, Quah
discretizes the income distribution in each period intokclasses, and the proba-
bility of an economy transitioning between each pair of classes is estimated from
the income series for the country economies. An application of this approach to
the lower 48 state incomes for the US over the period 1929–2000 is reported in
Table 27.1. The income values are standardized to the national value each year and
the class cut-offs are taken from the quintiles of the relative distribution for the
first year in the sample. Thus, for a state that had an income level below 66% of