1108 The Econometrics of Convergence
and hence convergence requires output for a pair of countries to behave similarly in
the long run. This is a strong requirement since it does not permit logyi,t−logyj,t
to be stochastic in the long run, as would be the case if the two countries have
different short-run shock processes. Hallet al.point out the lacuna in the definition
(23.19) in the case of long-run deviations whose current direction is not predictable.
If, for example, logyi,t−logyj,tobeys a random walk with current value zero,
then definition (23.19) would hold despite the fact that future output deviations
between countriesiandjcould be large. This problem is avoided by the modified
definition:
∀r≥0, lim
t→∞E(logyi,t−logyj,t|ρi,r,ρj,r)=0. (23.22)This modification has no bearing on the relationship between long-run unfore-
castability of differences and either theory or the various convergence tests we
have described.
Theβ-convergence concept represents the idea of the long-run irrelevance of
initial output per capita by decomposing the growth rate ofyi,tinto two parts,
technological progress and “catching up.” In the Solow model a positive rate of
conditional convergence,λ, implies that final output is ultimately independent
of initial conditions, so that the initial gap between output and its steady-state
value ceases to play a role in long-run growth. As we have discussed, a testable
implication of a positiveλis a negativeβ.
In contrast, the literature that focuses on nonlinearities in the growth process
seeks to identify sub-groups of countries that obey a common growth model dis-
tinct from that obeyed by the countries in other groups. Such behavior is consistent
with initial conditions that are relevant even in the long run, as it is consis-
tent with a global growth model with multiple steady-states in which economies
with similar initial conditions tend to converge to one another. The convergence
approaches that study the behavior of the entire cross-country distribution of out-
put per capita look for evidence of the long-run importance of initial conditions in
the form of accumulation points in the distribution, as evinced by intervals with
large quantities of probability mass, between which there is little mobility. This
behavior is consistent with a dynamic growth process that exhibits multiple basins
of attraction and so produces convergence clubs.
The convergence definitions given in (23.19) and (23.20) above make no dis-
tinction between the long-run effects of initial conditions and those of structural
heterogeneity, and so fail to distinguish conditional convergence from the exis-
tence of convergence clubs. This is a serious deficiency for those seeking to use the
outcomes of convergence tests to adjudicate between competing theories of eco-
nomic growth. Long-run effects of cross-country differences in preferences are, for
example, consistent with both the neoclassical and endogenous or “new” classes
of growth theories. However, neoclassical theories are inconsistent with long-run
effects of cross-country differences in initial human and physical capital stocks,
whereas this would not be the case for all endogenous growth models. In other
words, the finding of a long-run role for initial conditions constitutes evidence in