Steven Durlauf, Paul Johnson and Jonathan Temple 1099inequality in turn impliesσlog^2 y,t>σlog^2 y,t+T, as otherwise(σlogyi,t,logyi,t+T)^2 <
σlog^2 yi,t+Tσlog^2 yi,t, the condition for positive definiteness of the variance-covariance
matrix of logyi,tand logyi,t+T, would be violated. Hence a test that rejects the null
hypothesisπ=0 in favor of the alternativeπ<0 is evidence ofσ-convergence.
Their application of the test finds convergence among the US states and the
European regions but not among the countries of the world.
Egger and Pfaffermayr (2007) suggest a test of conditionalσ-convergence that, in
the growth context, would involve a test of the hypothesis thatπT= 1 −σu^2 T/σlog^2 yi,t
against the alternative thatπT< 1 −σu^2 T/σlog^2 yi,t, whereπTis the coefficient on
logyi,tin the cross-section regression logyi,t+T=πTlogyi,t+Zi+uiT, where,
as above,Ziis a vector of country-ispecific, time-invariant control variables. This
test generalizes the unconditionalσ-convergence test of Carree and Klomp (1997),
which those authors use to provide evidence ofσ-convergence within the OECD
countries. Egger and Pfaffermayr (2007) apply their test to a large dataset on the
size of European manufacturing firms; its power for the samples typically used in
growth applications is not yet clear.
Bliss (1999a, 1999b) points out that the interpretation of tests ofσ-convergence
can be problematic in the presence of non-stationarities; an evolving distribution
for the data makes it difficult to think about the distributions of test statistics under
the null hypothesis. Further difficulties arise when unit roots are present.
23.5 Convergence and the cross-country distribution of per
capita income
Bothβ- andσ-convergence are directly motivated by the law of motion of the
neoclassical growth model. A distinct approach to convergence was pioneered by
Quah (1993a, 1993b, 1996a, 1996b, 1996c, 1997). He focuses on “distribution
dynamics”: the evolution of the entire cross-country distribution of income per
capita. We will question whether analyses of this kind can speak directly to the
convergence hypothesis, but the approach has helped to establish some stylized
facts that could be important in assessing the empirical salience of different growth
theories.
One strand of this literature takes snapshots of the distribution of income per
capita at points in time. For example, Bianchi (1997) tests for multimodality in
kernel density estimates of the cross-country distribution of per capita income. He
finds evidence of bimodality in densities estimated using Penn World Table data on
gross domestic product (GDP) per capita for 1970, 1980 and 1989.^19 He also notes
a tendency for the two modes to become more distant from each other over time,
supporting the view that the cross-country distribution of per capita income has
become increasingly polarized. He finds very little mobility within the distribution;
most of the countries nearer to either the upper or lower mode in 1970 are still there
in 1989. Henderson, Parmeter and Russell (2007) confirm Bianchi’s findings using
a longer span of data and more advanced statistical tests.