1098 The Econometrics of Convergence
23.4 σ-convergence
An alternative statistical convergence concept focuses on the cross-section dis-
persion of log per capita output across countries, and whether it is increasing or
shrinking. This has a natural connection to debates on whether inequality across
countries is widening or diminishing. Discussion of this form of convergence often
emphasizes the cross-section variance of log incomes, but the variance is not a
sufficient statistic for the overall dispersion, and so can mask interesting forms
of cross-section inequality. Although most empirical studies use the log variance,
other measures of inequality can easily be used, such as the Gini coefficient or the
Atkinson (1970) class of measures. These will sometimes be preferable to the log
variance on axiomatic grounds.
A reduction in the dispersion of log income is interpreted as convergence because,
as withβ-convergence, it suggests that contemporary income differences are tran-
sitory. Lettingσlog^2 y,tdenote the variance acrossiof logyi,t,σ-convergence occurs
betweentandt+Tif:
σlog^2 y,t−σlog^2 y,t+T>0. (23.13)Barro and Sala-i-Martin (2004, Ch. 11) report declines in the variance of the log-
arithm of income for the US states between 1880 and 2000, for the Japanese
prefectures between 1930 and 1990, and for regions within five European countries
between 1950 and 1990. In contrast, the variance of log income per capita in the
world as a whole increased between 1960 and 2000.
These results are consistent with the outcomes of unconditionalβ-convergence
tests, but there is no necessary relationship betweenβ- andσ-convergence. It is
easy to see howσ-divergence can occur even when, by an economic measure, con-
vergence is present. For example, if all countries start from the same position but
shocks are present, then divergence will occur, regardless of the speed of mean
reversion. A more subtle point is that, if output changes for all countries obey
yi,t−yi,t− 1 =βyi,t− 1 +εi,t, thenβ<0 is compatible with a constant cross-sectional
variance, which in this example will equal the variance ofyi,t. The alternative, but
mistaken, idea that mean reversion in a time series must imply a falling variance is
known as Galton’s fallacy. The mistake here is to ignore the role of ongoing shocks
in sustaining the variance. The relevance of these types of arguments to under-
standing the relationship between convergence concepts in the growth literature
was identified by Friedman (1992) and Quah (1993a).
Several authors have recently proposed tests forσ-convergence that employ
regression specifications. Following Friedman (1992), Cannon and Duck (2000)
argue that a possible test forσ-convergence could use regressions of the form:
γi=T−^1 (logyi,t+T−logyi,t)=α+πlogyi,t+T+εi. (23.14)As the probability limit of the OLS estimator ofπisT−^1
(
1 −σlogyi,t,logyi,t+T/σlog^2 yi,t+T
)
, a negative estimate ofπ impliesσlogyi,t,logyi,t+T >σlog^2 yi,t+T. This