1102 The Econometrics of Convergence
estimates imply that the ergodic distributions of both GDP per capita and the pro-
ductivity residual are bimodal, while those of the accumulable factors of production
are not.^25 Consistent with the emphasis of Klenow and Rodriguez-Clare (1997) and
Hall and Jones (1999) on productivity differences, he concludes that the proximate
cause of the “twin peaks” in the distribution of GDP per capita is bimodality in the
distribution of productivity, and accordingly advocates more research on models
that emphasize traps in TFP.
Fitting a Markov chain to a continuous variable like GDP per capita requires a
discretization of the state space. This is problematic, as it can easily alter the prob-
abilistic properties of the data (Quah, 1996c, 1997, 2001; Bulli, 2001). Reichlin
(1999) showed that the inferred dynamic behavior and the long-run implica-
tions of that behavior can depend on the discretization scheme that is used. To
address this problem, Quah (1996c, 1997) proposed a continuous state space ver-
sion of the approach that avoids the problems caused by discretization. If the
cross-country income distribution at timethas a density function,ft(x), and if the
process describing the evolution of the distribution is time-invariant and first-order
Markov, the density at timet+τ,τ>0, will be given byft+τ(x)=
∫∞
0 gτ(x|z)ft(z)dz,
wheregτ(x|z)is theτ-period-ahead density ofxconditional onz. The function
gτ(x|z)is the continuous analog of the transition matrix. The implied ergodic (long-
run) density function,f∞(x), if it exists, solvesf∞(x)=
∫∞
0 gτ(x|z)f∞(z)dz. Quah
(1996c, 1997) uses kernel density methods to estimate variousgτ(x|z)using cross-
country data on output per capita, and finds a general concentration of the mass
near points wherex=z, that is, along the “main diagonal,” as well as a tendency
for peaks in the plot near the ends of the main diagonal and a trough in the mid-
dle. These features imply a lack of mobility within the cross-country distribution
of income per capita and a tendency for mass to accumulate in the tails of the
long-run distribution.^26 The estimated ergodic densities in Bulli (2001), Johnson
(2005) and Fiaschi and Romanelli (2008) are also bimodal and hence support this
conclusion. While Quah (2001) observes that there is not yet a theory of inference
for these methods, Fiaschi and Romanelli (2008) propose a bootstrap procedure
for computation of confidence intervals for the ergodic density, and their results
suggest that the bimodality is statistically significant.
These methods have been important in establishing stylized facts concerning
the cross-country distribution of per capita output, but there have been relatively
few attempts to explore the implications for the empirical relevance of alternative
growth theories. Quah (1996c) finds that conditioning on measures of physical
and human capital accumulation similar to those used by Mankiwet al.(1992),
and a dummy variable for the African continent, has little effect on the estimated
dynamics of the distribution. This suggests that the heterogeneity revealed by
the distributional approaches is, at least in part, due to the existence of conver-
gence clubs rather than heterogeneity in steady-state determinants.^27 Azariadis and
Stachurski (2003) derive the form of thegτ(x|z)function implied by a stochastic
version of the model in Azariadis and Drazen (1990). They estimate the model’s
parameters and compute forward projections of the sequence of cross-country