Steven Durlauf, Paul Johnson and Jonathan Temple 109123.2.2 Cross-country regressions andβ-convergence
In order to translate this economic notion of convergence into a statistical model,
it is necessary to address the unobservable variablesyiE,∞andAi,0. One possibility
is to assume that (i) all countries have a common steady-state so thatyiE,∞is a
constant,y∞E; and (ii) logAi,0=logA+εiwhere,εiis a country-specific shock.^7
SubsumingyE∞, logAandginto the constant term, unconditionalβ-convergence
holds ifβ<0 in the regression:
γi=α+βyi,0+εi. (23.6)When using a sample of countries, one rarely finds unconditionalβ-convergence,
unless the sample is restricted to similar entities such as the Organization for
Economic Cooperation and Development (OECD) member countries. There is
rarely a strong correlation between initial income and subsequent growth in large,
heterogeneous samples, such as those found in the Penn World Table.^8
Relative to equation (23.6), heterogeneity in countries naturally suggests that
yEi,∞is not, in fact, constant. In the augmented Solow model estimated by Mankiw
et al.(1992), for example,yEi,∞is shown to depend on the rates of physical and
human capital accumulation and the rate of population growth. More generally,
lettingZibe the set of variables that determine a country’s steady-state income
level, conditionalβ-convergence is said to hold ifβ<0 in the regression:
γi=α+βyi,0+Zi+εi. (23.7)While many differences exist in the choice of controls, most studies include several
determinants inspired by some version of the Solow growth model. Unlike uncon-
ditionalβ-convergence, evidence of conditionalβ-convergence has been found in
many contexts. For the cross-country case, the finding is generally attributed to
Barro (1991), Barro and Sala-i-Martin (1992) and Mankiwet al.(1992).
The Mankiwet al.(1992) analysis is of particular interest, as it is based on a
regression suggested by the dynamics of the Solow growth model. Their findings
have been widely interpreted as evidence in favor of diminishing returns to capital
(the source ofβ<0 in the Solow model) and as evidence against some endogenous
growth models. The analysis especially calls into question models which emphasize
increasing returns in capital accumulation (either human or physical) as a source of
perpetual growth. However, some endogenous growth models are consistent with
β-convergence, and therefore some caution is needed in drawing inferences about
the nature of the growth process from the results ofβ-convergence tests.^9
Given an estimate ofβ, an estimate ofλ, the implied rate of convergence, can be
obtained fromβ=−t−^1 ( 1 −e−λt). In cross-section studies, this typically yields an
estimated convergence rate of about 2% per year; in other words, over the course
of a year, countries will close just 2% of the gap between their current position and
the balanced growth path.^10 This result is found using datasets on a wide variety
of economic entities and from time periods more than 100 years apart.^11 While