The general question of convergence, understood as the tendency of differences between countries to disappear over time, is of long-standing interest to social scientists. In the 1950s and early 1960s, many analysts discussed whether capitalist and socialist economies would converge over time, in the sense that market institutions would begin to shape socialist economies just as government regulation and a range of social welfare policies grew in capitalist ones.

In modern economic parlance, convergence usually refers specifically to issues related to the persistence or transience of differences in per capita output between economic units, be they countries, regions or states. Most research has focused on convergence across countries, since the large contemporaneous differences between countries generally dwarf intra-country differences. In the context of economic growth, the convergence hypothesis arguably represents the most commonly studied aspect of growth, although the effort to identify growth determinants is arguably the main area of contemporary growth research.

In this overview of convergence, our primary emphasis will be on the development of precise statistical definitions of convergence. This reflects an important virtue of the current literature, namely, the introduction of statistical methods to adjudicate whether convergence is present. At the same time, there is no single definition of convergence in the literature, which is one reason why empirical evidence on convergence is indecisive. Our discussion focuses on convergence across countries, which has dominated empirical studies, although there is reference to studies that focus on other units.

β-Convergence

The primary definition of convergence used in the modern growth literature is based on the relationship between initial income and subsequent growth. The basic idea is that two countries exhibit convergence if the one with lower initial income grows faster than the other. The local (relative to steady state) dynamics of the neoclassical growth model in both its Solow and Cass–Koopmans variants imply that lower-income economies will grow faster than higher-income ones.

As a statistical question, this notion of convergence can be operationalized in the context of a cross-country regression. Let gi denote real per capita growth of country i across some fixed time interval and yi,0 denote the initial per capita income for country i. Then, unconditional β-convergence is said to hold if, in the regression

$$ {g}_i=k+\log {y}_{i,0}\beta +{\varepsilon}_i,\beta <0. $$
(1)

For cross-country regression analysis, one typically does not find unconditional β-convergence unless the sample is restricted to very similar countries, for example, members of the OECD. This finding is in some ways not surprising, since unconditional β-convergence is typically not a prediction of the existing body of growth theories. The reason for this is that growth theories universally imply that growth is determined by factors other than initial income. While different theories may propose different factors, they collectively imply that (1) is misspecified. As a result, most empirical work focuses on conditional β-convergence. Conditional β-convergence holds if β < 0 for the regression

$$ {g}_i=k+\log {y}_{i,0}\beta +{Z}_i\gamma +{\varepsilon}_i $$
(2)

where Zi is a set of those growth determinants that are assumed to affect growth in addition to a country’s initial income. While many differences exist in the choice of controls, it is nearly universal to include those determinants predicted by the Solow growth model, that is, population growth and human and physical capital accumulation rates.

Unlike unconditional β-convergence, evidence of conditional β-convergence has been found in many contexts. For the cross-country case, the basic finding is generally attributed to Barro (1991), Barro and Sala-i-Martin (1992) and Mankiw et al. (1992). The Mankiw, Romer and Weil analysis is of particular interest as it is based on a regression suggested by the dynamics of the Solow growth model. Hence, their findings have been widely interpreted as evidence in favour of decreasing returns to scale in capital (the source of β < 0 in the Solow model), and therefore as evidence against the Lucas–Romer endogenous growth approach, which emphasizes increasing returns in capital accumulation (either human or physical) as a source of perpetual growth.

From the perspective of the neoclassical growth model, the term – β also measures the rate at which an economy’s convergence towards its steady-state growth rate, that is, the growth rate determined exclusively by the exogenous rate of technical change. The many findings in the cross-country literature are often summarized by the claim countries converge towards their steady-state growth rates at a rate of about two per cent per year, although individual studies produce different results. The convergence rate has received inadequate attention in the sense that a finding of convergence may have little consequence for questions such as policy interventions if it is sufficiently slow.

As is clear from (2), any claims about conditional convergence necessarily depend on the choice of control variables Zi. This is a serious concern given the lack of consensus in growth economics on which growth determinants are empirically important. Doppelhofer et al. (2004) and Fernandez et al. (2001) use model averaging methods to show that the cross-country findings that have appeared for conditional β-convergence are robust to the choice of controls. A number of additional statistical issues such as the role of measurement error and endogeneity of regressors are surveyed and evaluated in Durlauf et al. (2005).

The assumption in cross-section growth regressions that the unobserved growth terms εi are uncorrelated with log yi,0 rules out the possibility that there are country-specific differences in output levels; if such effects were present, they would imply a link between the two. For this reason, a number of researchers have investigated convergence using panel data. This leads to models of the form

$$ {g}_{i,t}={c}_i+\log {y}_{i,t-1}\beta +{Z}_{i,t}\gamma +{\varepsilon}_{i,t} $$
(3)

where growth is now measured between t – 1 and t. This approach not only can handle fixed effects, but can allow for instrumental variables to be used to address endogeneity issues. Panel analyses have been conducted by Caselli et al. (1996), Islam (1995) and Lee et al. (1997). These studies have generally found convergence with rather higher rates than appear in the cross-section studies; for example, Caselli et al. (1996) report annual convergence rate estimates of ten per cent.

As discussed in Durlauf and Quah (1999) and Durlauf et al. (2005), panel data approaches to convergence suffer from the problem that, once country specific effects are allowed, it becomes more difficult to interpret results in terms of the underlying economics. The problem is that, once one allows for fixed effects, then the question of convergence is changed, at least if the goal is to understand whether initial conditions matter; simply put, the country-specific effects are themselves a form of initial conditions. When studies such as Lee et al. (1997) allow for rich forms of parameter heterogeneity across countries, β-convergence become equivalent to the question of whether there is some mean reversion in a country’s output process, not whether certain types of contemporaneous inequalities diminish. This does not diminish the interest of these studies as statistical analyses, but means their economic import can be unclear.

σ-Convergence and the Cross-Section Distribution of Income

A second common statistical measure of convergence focuses on the whether or not the cross-section variance of per capita output across countries is or is not shrinking. A reduction in this variance is interpreted as convergence. Letting \( {\sigma}_{\log\ y,t}^2 \) denote the variance across i of log yit, σ-convergence occurs between t and t + T if

$$ {\sigma}_{\log\ y,t}^2-{\sigma}_{\log\ y,t+T}^2>0. $$
(4)

There is no necessary relationship between β- and σ-convergence. For example, if the first difference of output in each country obey log yi,t– log yi,t–1 = β log yi,t–1 + εi,t, then β < 0 is compatible with a constant cross-sectional variance (which in this example will equal the variance of log yi,t). The incorrect idea that mean reversion in time series implies that its variance is declining is known as Galton’s fallacy; its relevance to understanding the relationship between convergence concepts in the growth literature was identified by Friedman (1992) and Quah (1993a). While it is possible to construct a cross-section regression to test for σ-convergence (cf. Cannon and Duck 2000), they do not test β-convergence per se.

Work on β-convergence has led to general interest in the evolution of the crosscountry income distribution. Quah (1993b, 1996) has been very influential in his modelling of a stochastic process for the distribution itself, with the conclusion that it is converging towards a bimodal steady-state distribution. Other studies of the evolution of the cross-section distribution include Anderson (2004) who uses nonparametric density methods to identify increasing polarization between rich and poor economies across time. Increasing divergence between OECD and non-OECD economies is shown in Maasoumi et al. (2007), working with residuals from linear growth regressions.

One difficulty with convergence approaches that emphasize changes in the shape of the cross-section distribution is that they may fail to address the original question of the persistence of contemporaneous inequality. The reason for this is that it is possible, because of movements in relative position within the distribution, for the cross-section distribution to flatten out while at the same time differences at one point in time are reversed; similarly, the cross-section distribution can become less diffuse while gaps between rich and poor widen. That being said, an examination of the locations of individual countries in various distribution studies typically indicates that the increasing polarization of the world income distribution is mirrored by increasing gaps between rich and poor. A useful extension of this type of research would be to employ the dynamics of individual countries to provide additional information on how the cross-section distribution evolves.

Time Series Approaches to Convergence

An alternative approach to convergence is focused on direct evaluation of the persistence of transitivity of per capita output differences between economies. This approach originates in Bernard and Durlauf (1995), who equate convergence with the statement that

$$ {\mathrm{lim}}_{T\Rightarrow \infty }E\left(\log {y}_{i,t+T}-\log {y}_{j,t+T}|{F}_t\right)=0 $$
(5)

where Ft denotes the history of the two output series up to time t. They find that convergence does not hold for OECD economies, although there is some cointegration in the individual output series. Hobijn and Franses (2000) find similar results for a large international data-set. Evans (1996) employs a clever analysis of the evolution of the cross-section variance to evaluate the presence of a common trend in OECD output, and finds one is present; his analysis allows for different deterministic trends in output and so in this sense is compatible with Bernard and Durlauf (1995).

The relationship between cross-section and time series convergence tests is complicated. Bernard and Durlauf (1996) argue that the two classes of tests are based on different assumptions about the data under study. Cross-section tests assume that countries are in transition to a steady state, so that the data for a given country at time t is drawn from a different stochastic process from the data at some future t + T. In contrast, time series tests assume that the underlying stochastic processes are time-invariant parameters, that is, that countries have transited to an invariant output process. They further indicate how convergence under a cross-section test can in fact imply a failure of convergence under a time series test, because of these different assumptions. For these reasons, time series tests of convergence seem appropriate for economies that are at similar stages and advanced stages of development.

From Statistics to Economics

The various concepts of convergence we have described are all purely statistical definitions. The economic questions that motivated these definitions are not, however, equivalent to these questions, so it is important to consider convergence as an economic concept in order to assess what is learned in the statistical studies. As argued in Durlauf et al. (2005), the economic questions that underlie convergence study revolve around the respective roles of initial conditions versus structural heterogeneity in explaining differences in per capita output levels or growth rates. It is the permanent effect of initial conditions, not structural features that matters for convergence. If we define initial conditions as ρi0 and the structural characteristics as θi,0, convergence can be defined via

$$ {\mathrm{lim}}_{t\to \infty }E\left(\log \, {y}_{i,t}{-} \log \, {y}_{j,t}|{\rho}_{i,0},{\theta}_{i,0},{\theta}_{j,0}\right)=0\, \mathrm{if}\, {\theta}_{i,0}={\theta}_{j,0}. $$
(6)

The gap between the definition (6) and the statistical tests that have been employed is evident when one considers whether the statistical tests can differentiate between economically interesting growth models, some of which fulfil (6) and others of which do not. One such contrast is between the Solow growth model and the Azariadis and Drazen (1990) model of threshold externalities, in which countries will converge to one of several possible steady states, with initial conditions determining which one emerges. By definition (6), the Solow model produces convergence whereas the Azariadis–Drazen model does not. However, as shown by Bernard and Durlauf (1996) it is possible for data from the Azariadis–Drazen model to produce estimates that are consistent with a finding of β-convergence.

There is in fact a range of empirical findings of growth nonlinearities that are inconsistent with convergence in the sense of (6). Durlauf and Johnson (1995) is an early study of this type, which explicitly estimated a version of the Azariadis–Drazen model in which the Solow model, under the assumption of a Cobb–Douglas aggregate production function, is a special case. Durlauf and Johnson rejected the Solow model specification and found multiple growth regimes indexed by initial conditions. Their findings are consistent with the presence of convergence clubs in which different groups of countries are associated with one of several possible steady states. These results are confirmed by Papageorgiou and Masanjala (2004) using a CES production function specification.

The Durlauf and Johnson analysis uses a particular classification procedure, known as a regression tree, to identify groups of countries obeying a common linear model. Other statistical approaches have also identified convergence clubs. For example, Bloom et al. (2003) use mixture distribution methods to model countries as associated with one of two possible output processes, and conclude that individual countries may be classified into high-output manufacturing- and service-based economies and low-output agriculture-based economies. Canova (2004) uses Bayesian methods to identify convergence clubs for European regions.

As discussed in Durlauf and Johnson (1995) and Durlauf et al. (2005), studies of nonlinearity also suffer from identification problems with respect to questions of convergence. One problem is that a given data-set cannot fully uncover the full nature of growth nonlinearities without strong additional assumptions. As a result, it becomes difficult to extrapolate those relationships between predetermined variables and growth to infer steady-state behaviour. Durlauf and Johnson give an example of a data pattern that is compatible with both a single steady and multiple steady states. A second problem concerns the interpretation of the conditioning variables in these exercises. Suppose one finds, as do Durlauf and Johnson, that high- and low-literacy economies are associated with different aggregate production functions. One interpretation of this finding is that the literacy rate proxies for unobserved fixed factors, for example culture, so that these two sets of economies will never obey a common production function, and so will never exhibit convergence in the sense of (6). Alternatively, the aggregate production function could structurally depend on the literacy rate, so that, as literacy increases, the aggregate production functions of currently low-literacy economies will converge to those of the high-literacy ones. Data analyses of the type that have appeared cannot distinguish between these possibilities.

Conclusions

While the empirical convergence literature contains many interesting findings and has helped identify a number of important generalizations about cross-country growth behaviour, it has yet to reach any sort of consensus on the deep economic questions for which the statistical analyses were designed. The fundamentally nonlinear nature of endogenous growth theories renders the conventional crosssection convergence tests inadequate as ways to discriminate between the main classes of theories. Evidence of convergence clubs may simply be evidence of deep nonlinearities in the transitional dynamics towards a unique steady state. Crosssection and time series approaches to convergence not only yield different results but are predicated on different views of the nature of transitory versus steady-state behaviour of economies, differences that themselves have yet to be tested.

None of this is to say that convergence is an empirically meaningless question. Rather, progress requires continued attention to the appropriate statistical definition of convergence and the use of statistical procedures consistent with the definition. Further, it seems important to move beyond current ways of assessing convergence both in terms of better use of economic theory and by a broader view of appropriate data sources. Graham and Temple (2006) illustrate the potential for empirical analyses of convergence that employ well-delineated structural models. The research programme developed in Acemoglu, Johnson and Robinson (2001, 2002) provides a perspective on the micro-foundations of country-specific heterogeneity that speaks directly to the convergence question and which shows the power of empirical analysis based on careful attention to economic history. For these reasons, research on convergence should continue to be productive and important.

See Also