The phenomenon of modern economic growth is fairly new. It started less than two centuries ago, but it changed our lives significantly. One of the main changes is that income gaps between countries have greatly increased. One of the main questions that concern economists who study economic growth is whether these gaps are still growing, or countries are instead converging to the same level of income. This question is empirical, but it has important theoretical implications, as our main growth theories predict convergence between countries. This question is clearly also important from a policy point of view. If the poor countries will converge to the global frontier anyway, there is no need to provide them with extensive assistance. But if there is no convergence and even divergence, then such countries need help badly.

# Alternative methods of measuring convergence

The most common method of measuring convergence or divergence of output per capita across countries has been through growth regressions. This method was developed by Barro (1991) and by Mankiw et al. (1992). In this method, the average rate of growth of output per capita over a long period is regressed over the initial level of output per capita and some additional control variables.

- If countries converge, the coefficient of initial output should be negative, meaning a country that begins poorer will grow faster while a rich country will grow more slowly.

The size of this negative coefficient measures the rate of convergence. The control variables used in such regressions are viewed as explanatory variables for economic growth, and they usually include variables that measure education, political stability, institutions, and similar variables.

- Growth regressions have indeed shown that there is convergence, namely that the coefficient of initial output on the rate of growth is negative.

As for the size of the coefficient – the rate of convergence – results are mixed and vary between 0.02 (Barro and Sala-i-Martin 1992) and 0.14 (Caselli et al. 1996).

Other studies of convergence, which have followed an alternative path – a direct test of how the distribution of output across countries evolves over time – reached very different conclusions.

- These alternative empirical studies found that the levels of output per capita in countries tend to diverge over time.

See Bernard and Durlauf (1995) and Quah (1996), and the exhaustive survey of growth empirics by Durlauf et al. (2005).

This discrepancy between growth regressions and between distributional studies of convergence is one of the main issues dealt with in our recent paper (Battisti et al. 2013). We show how to reconcile the two types of studies and their conflicting results.

# Our approach

Our main point of departure is the benchmark model of growth regressions as described in Durlauf et al. (2005). The basic assumption of this model is that for each country the ratio of output per capita (or output per worker) *y* to total factor productivity *A*, that is, *y*/*A*, converges to a long-run value* yE* at a rate *b*. Note that early growth regressions examined economic growth only at short time horizons, due to lack of data, so the changes in *A* were small and were usually ignored. Later on, more data were accumulated and growth regressions turned from cross-section regressions to panel estimations, but these studies still did not specify the exact dynamics of productivity *A* for the estimation (see Durlauf et al. 2005, for a full description of these developments). Our study suggests adding a specification of the dynamics of productivity – one that is simple, intuitive, and can be tested empirically.

We assume in the paper that a country’s productivity follows the global frontier of technology *F*, but may follow it only partially. We assume that a country may adopt a share *d* of all new technologies every year, where *d* is a country-specific parameter that can vary from 0 to 1. If it is 0, the country has no technical change, while if *d* is 1, the country follows the global frontier fully.^{1}

Adding this specification of productivity to the benchmark growth regression model leads to the following insights:

- Each country converges to a long-run growth path which is determined by
*A*. - The rate of convergence to the long-run path is
*b*, and it is equal to the convergence coefficient measured by all growth regressions. - However, the long-run growth path itself can diverge from the global frontier
*F*if the country’s*d*is less than 1.

In that case, the country becomes poorer, and poorer relative to the countries at the frontier.

# Estimation

Embedding our productivity specification into the benchmark model yields an extended growth regression that combines convergence to the long-run growth path with the dynamics of this path itself. This allows us to empirically test this extended model, and estimate for each country the convergence coefficient b and also the coefficient d that measures global divergence.

We use two methods of estimation.

- Panel cointegration with estimation of heterogeneous coefficients for each country.

Here *d* is the cointegrating coefficient, and *b* is the error-correction coefficient.

- Estimation by differences.

We find that the two methods of estimation yield similar results, but we present here the results of the panel cointegration only.

Our empirical test of the extended growth regression model uses a panel of 140 countries over the years 1950–2008, in which we use GDP per capita as our main measure of economic development. The data are in PPP-adjusted units, 1990 Geary–Khamis dollars, taken from the Groningen database. The global frontier is represented by US GDP per capita, both because the US leads global growth and also because its rate of growth is very stable over time.

Interestingly, we do not need any additional control variables in our dynamic estimation. We use them in a later stage of the paper, not reported here, in which we estimate in a cross-country regression the effects of the usual explanatory variables on the parameters *d* and *a*. Hence, the dynamic estimation that finds these parameters for each country does not require the use of control variables. This is another important result of the paper, since the use of control variables in the dynamic estimation of growth regressions has been criticised as ad hoc and rather arbitrary.

# Results

Our results show that countries both converge and diverge. Each country converges to its long-run growth path, and the rate of convergence is 0.04, which is quite similar to the convergence coefficient found in many growth regressions. But the long-run growth paths of countries diverge from the global frontier, as the average estimated *d* is 0.7 and it is significantly lower than 1. Hence, our study succeeds in reconciling the two strands of the literature: growth regressions that point at convergence, and distribution dynamics that point at divergence (for details, see Battisti et al. 2013).

# Summary

In this column we show how we reconcile the results of growth regressions, which indicate convergence, with the results of distribution dynamics tests, which indicate divergence.

- We find that almost all countries converge to their own steady-state paths. However, the steady-state paths of most countries diverge from the global frontier.

We also try to test how the common explanatory variables of growth regressions affect the coefficient *d*, that is, the divergence of a country. This means that our method enables us to test separately the effect of various empirical variables on the long-run and on the short-run of economic growth.

In addition to the reconciliation, our work offers a methodological extension to growth regressions. We hope it will stimulate research by other economists who will use our approach in order to improve our understanding of economic growth across countries. Cross-country growth differentials are, after all, one of the main puzzles of our times.

*Authors’ note: The views in this paper are those of the authors and do not necessarily represent their affiliated institutions.*

# References

Barro, R J (1991), “Economic growth in a cross section of countries”, *Quarterly Journal of Economics* 106(2), pp. 407–443.

Barro, R J and X Sala-i-Martin (1992), “Convergence”, *Journal of Political Economy* 100(2), pp. 223–251.

Battisti, M, G Di Vaio, and J Zeira (2013), “Global Divergence in Growth Regressions [1]”, CEPR Discussion Paper 9687.

Caselli, F, G Esquivel, and F Lefort (1996), “Reopening the convergence debate: a new look at cross-country growth empirics”,* Journal of Economic Growth* 1, pp. 363–389.

Durlauf, S N, P Johnson, and J Temple (2005), “Growth Econometrics”, in P Aghion and S N Durlauf (eds), *Handbook of Economic Growth*, Amsterdam: North Holland.

Mankiw, N G, D Romer, and D N Weil (1992), “A Contribution to the Empirics of Economic Growth”, *Quarterly Journal of Economics* 107, pp. 408–437.

Quah, D T