Steven Durlauf, Paul Johnson and Jonathan Temple 1107economic outcomes. As argued in Durlaufet al.(2005), the empirical task is then to
determine the role of initial conditions in explaining cross-country differences in
per capita output. This task is complicated by the role of structural heterogeneity
in also explaining those differences; so the empirical literature must disentan-
gle the effects of initial conditions and structural heterogeneity. There are three
possibilities: (i) unconditional convergence (to a common long-run level) occurs
if differences in per capita incomes are temporary; (ii) conditional convergence
occurs if permanent differences reflect only cross-country structural heterogeneity;
and (iii) club convergence occurs if initial conditions determine, to some extent at
least, long-run outcomes, with countries with similar initial conditions exhibiting
similar long-run outcomes.^30
To formalize these ideas, we associate with economyiinitial conditionsρi,0and
say that these initial conditions do not matter in the long run if:
lim
t→∞μ(logyi,t|ρi,0)does not depend onρi,0, (23.17)whereμ(·)is a probability measure.^31 Letting‖‖denote a metric for computing the
distance between probability measures, we say that countriesiandjconverge if:
lim
t→∞‖μ(logyi,t|ρi,0)−μ(logyj,t|ρj,0)‖=0, (23.18)which implies convergence in average income levels in the sense that:
lim
t→∞E(logyi,t−logyj,t|ρi,0,ρj,0)=0. (23.19)This definition can be modified to require that the limiting expected difference
between logyi,tand logyj,tis bounded if the equality of steady-state growth rates is
of interest. This definition is the one that underlies all of the time series approaches
to convergence: the differences between Harvey and Carvalho (2002) and Phillips
and Sul (2006) and the earlier time series tests of Bernard and Durlauf (1995) and
others reflect differences in how this long-run forecast similarity is calculated. This
is also consistent with the economic notion of convergence that appears in the
neoclassical growth model. Our criticism of some of the cross-section and panel
approaches to convergence partially derives from their failure to evaluate this
condition fully.
Bernard and Durlauf (1996) suggest a definition of partial convergence that
requires contemporaneous income differences be expected to diminish, that is:
E(logyi,t−logyj,t|ρi,0,ρj,0)<logyi,0−logyj,0, (23.20)for logyi,0>logyj,0. Hall, Robertson and Wickens (1997) suggest a definition that
requires the variance of output differences to diminish to zero, that is:
lim
t→∞E((logyi,t−logyj,t)^2 |ρi,0,ρj,0)=0, (23.21)