Paul Johnson, Steven Durlauf and Jonathan Temple 1141estimate models of the form:
si,t=ai+ρisi,t− 1 +εi,t. (24.26)The long-run forecast ofsi,tis given by 1 a−iρ
i
, with 1−ρibeing the rate of con-
vergence towards that value. Restricting the parametersaiandρito be constant
acrossigives a standardβ-convergence test and yields an estimated annual rate of
convergence of approximately 2%, similar to other findings in the literature. But
allowing for heterogeneity in these parameters produces a “substantial” and sta-
tistically significant dispersion of the implied long-runsi,tforecasts. The positive
correlation of those forecasts with the initial values ofsi,timplies a dependence of
long-run outcomes on initial conditions that contradicts the convergence hypoth-
esis. For the country-level data, differences in initial conditions explain almost
half the cross-sectional variation in long-run forecasts. This shows how key find-
ings can be sensitive to the treatment of parameter heterogeneity, and we return
to this issue when we discuss panel data models in section 24.5.
24.4.3 Nonlinearity and multiple regimes
We now discuss research that has attempted to disentangle the roles of hetero-
geneous structural characteristics and initial conditions in determining growth
outcomes. These papers employ a variety of statistical methods, but there is con-
siderable agreement in their findings. Many of them indicate the existence of
convergence clubs even after accounting for the role of structural characteristics.
We have discussed some of this work in our companion chapter on convergence
(Chapter 23 in this volume), and here we concentrate on the wider implications
of multiple regimes for the statistical methods that should be adopted.
One of the first contributions to this literature was Durlauf and Johnson
(1995). They used classification and regression tree (CART) methods to search for
nonlinearities.^13 More specifically, the CART procedure identifies sub-groups of
countries that obey a common linear growth model based on the Solow variables.
These sub-groups are identified by initial income and literacy; a typical sub-groupl
is defined by countries whose initial income lies within the intervalθl,y≤yi,0<θ ̄l,y
and whose literacy rateLilies in the intervalθl,L≤Li<θ ̄l,L. The number of sub-
groups and the boundaries for the variable intervals that define them are chosen
by an algorithm that trades off model complexity (the number of sub-groups) and
goodness of fit. Because the procedure uses rules to sequentially split the data into
finer and finer sub-groups, it organizes the data into a tree structure, where the
branches of the tree ultimately divide the sample into groups of countries that
follow distinct regimes.
Durlauf and Johnson (1995) also test the null hypothesis of a common growth
regime against the alternative hypothesis of a growth process with multiple
regimes. Taking Mankiwet al.(1992) as their starting point, and using income
per capita and the literacy rate as possible threshold variables, Durlauf and John-
son reject the single regime model. This finding has been confirmed in subsequent
research by Papageorgiou and Masanjala (2004). They estimate a version of the