Convergence and soccer: testing for convergence.

Szymanski, Stefan

According to the convergence hypothesis, the growth of a nation's GDP should be negatively correlated with its historical level of GDP; low income nations should be owing faster than high income nations, and the variance of national incomes should fall over time. In recent years, there has been considerable debate about whether we do in fact observe convergence in GDP, and results are mixed. This paper examines a variant of the convergence debate by examining convergence in national team soccer results. Soccer is the most popular sport in world, and almost every nation on the planet has a national team that regularly plays in international competition. This paper examines the results of national soccer teams between 1950 and 2010 and finds that, whether measured by the percentage of games won or by goal difference (goals scored minus goals conceded), there is significant evidence of convergence. This paper then speculates about why it might be so much easier to find evidence of convergence in national team soccer results than for GDP.

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The Convergence Debate

The convergence hypothesis is a straightforward consequence of neoclassical growth theory. The essential insight of economic growth theory is that increasing output (measured by GDP) requires more capital (equipment, machinery, infrastructure). The productivity of each worker increases when they have more capital to use, but this process cannot continue without limit. Adding more capital raises productivity, but at an ever decreasing rate (in the jargon of economics, the marginal product of capital is positive but diminishing).

From this insight follows two very simple observations. If we compare two economies operating with two different levels of accumulated capital, then we will see that 1) the economy with more capital will have a higher income per capita, and 2) a unit of investment in the economy with the lower level of capital will generate more growth than a unit of investment in the economy with the higher level of capital. This is the basis for convergence. If capital is mobile, then returns will be greatest in the low income economy, the rate of investment will be higher, and thus its rate of growth will also be higher, and so will tend to catch up with the high income economy over time. Eventually, income levels will reach equality, referred to in economics as the steady state.

Of course, in addition to mobile capital, this model requires some additional assumptions in order to work. In particular, one must assume that the technology embedded in all capital is available everywhere, so that the low income economy can adopt the technology that it currently lacks. This includes concepts such as "know-how", which in practice tends to be jealously guarded by the governments of wealthy economies and thus difficult for low income economies to acquire.

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Evidence in support of the convergence hypothesis is overall mixed. Most research focuses on tests of convergence across countries, although some have also tested for convergence across regions, such as the individual states of the USA. Anecdotally, the USA was the wealthiest nation at the end of the Second World War. Since then, standards of living in many other regions have tended to catch up, first in western Europe, then Japan, Korea, and more recently China. However, some regions, notably sub-Saharan Africa, showed limited evidence of convergence from the era of independence (roughly between 1950 and 1970) to the millennium.

The literature also distinguishes between conditional and unconditional convergence. Unconditional convergence implies that convergence is observed regardless of variations in specific national characteristics (e.g. climate, political institutions, education levels), while conditional convergence implies that these factors play a role and must also be accounted for if convergence is to be identified. In the literature, evidence is generally only found to support conditional convergence. However, a study by Dani Ro-drik, published in the Quarterly journal of Economics in 2012, examined the productivity of manufacturing plants across the world and is one of the few studies to find unconditional convergence. He argues that convergence is more likely to be observed in sectors where international trade ensures competition and the adoption of best practices (like manufacturing) than in service sectors which tend to be closed to international trade. It seems reasonable to conjecture that international sports, where representatives of different nations compete against each other intensively, might be another sector in which unconditional convergenee might be observed.

International Soccer

The first national soccer federation was formed in England in 1863. A second federation was formed in Scotland, and the two nations played the first international representative game in 1872 (in fact, the game was played on November 30, 1872, while the Scottish Football Association was not created until March 13, 1873). This pattern has been repeated for several nations, but the overwhelming majority of national team games are played under the auspices of a national federation recognized by FIFA, the federation of national soccer federations. As the game spread around the world, national federations proliferated, and so did the number of international games. For more than a century, the number of national federations and national soccer teams has approximately equaled the number of nations on the planet. By 1950, there were around 50 nations playing international soccer and around 250 international games played each year.

As Figure 1 shows, since 1950, the number of nations has grown to be more than 200 following decolonization and the collapse of the Soviet Union. FIFA now has 209 member federations, although not all of these play every year. The UN actually has fewer members (only 193) partly because some countries have managed to negotiate representation for regional teams (the United Kingdom is allowed four national teams England, Scotland, Wales and Northern Ireland) and also because some dependencies have been permitted to field national teams (e.g. Guadeloupe is a Caribbean island that is part of France but has its own soccer team). The number of international games played has grown to around 1500, reflecting easier international transport, improved broadcasting technology, and the growing appetite for the game.

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When comparing the economic performance of different nations, there are usually substantial statistical challenges. Methods of collecting statistics differ substantially across different countries, and reliability is often an issue. Soccer results, by contrast, can be measured with a high degree of confidence. Although there have been more than 30,000 international games, the precise result is seldom disputed (as distinct from disputing whether the result was fair or deserved). The game played is the same for every nation and changes little from year to year, so comparisons across years are relatively unproblematic.

Convergence and Soccer

One of the first acts of a new nation is to create a national soccer team and build a national soccer stadium. However, new nations face a number of disadvantages in playing international soccer, namely that the players and coaches lack experience. However, soccer nations appear to learn over time. One way to identify this process is to examine the performance of teams from different continental federations. There are six continental federations: UEFA (Europe), CONMEBOL (South America), Asia (AFC), Africa (CAF), North and Central America and the Caribbean (CONCACAF), and Oceania (OFC). Nations from UEFA and CONMEBOL had a significant head start in competition, with most nations having established teams before the Second World War. By contrast, colonialism meant that there were few recognized independent nations in Asia and Africa, and the independence of nations from these continents was generally achieved between 1950 and 1970. Figure 2 shows the cumulative win percentage (treating ties as half a win) of national teams from these two federations against European and South American teams since 1960.

From Figure 2, it is apparent that teams from Asia have improved their performance significantly over the past half century. Note that in the early years, there are relatively few games. In the case of African nations, performance seems to actually decline in the early years, but has been improving since the 1970s (these early results are dominated by the games played by Egypt, which played several games against relatively weak European opponents such as Malta, tending to give an overly favorable view of the team's performance). The great Brazilian player Pele famously predicted that an African nation would win the World Cup by 2000--a prediction which clearly did not come true. Moreover, the performance of African nations in international competition seems to have stagnated in recent years. Nonetheless, these results still suggest that over the last four or five decades, the emerging nations of Asia and Africa appear to have improved relative to the established powers of Europe and South America--consistent with convergence.

Formal testing for convergence usually relies on a simple statistical model. If there is convergence, we should observe that changes in performance ("growth") are inversely related to historical levels of performance. In the context of soccer results, this should mean that teams with low win percentages in the past should increase their win percentage, while teams with high win percentages in the past should stagnate and decline. An alternative measure of performance is goal difference. Winning teams have a positive goal difference on average and losing teams a negative goal difference, but even a losing team can be getting better if the absolute size of the goal difference is diminishing. To estimate this effect, the average performance of national teams across seven eight-year cycles is calculated, starting from the years from 1955-1962 up until 20032010 (these cycles coincide with two FIFA World Cups). Eight year cycles ensure that enough games are captured in each measure of performance to provide a reasonable estimate.

Figure 3 charts the relationship between the level (on the horizontal axis) and the change(on the vertical axis) of win percentage and of goal difference for each country.

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The solid lines in each chart represent the regression line, the most accurate summary of the relationship between the level and the change in the variable. The fact that in both charts the regression line slopes downward from left to right implies that there is indeed convergence in international soccer results for both win percentage and goal difference. The further to the left on the chart (the worse results were in the past) then the higher up the chart the country will be (the better the improvement in results).

Further analysis shows that these patterns hold up when the data is broken down into subperiods, when only games played between teams from different federations are considered, and when only competitive games (e.g. in the World Cup) are included and "friendly" games are excluded. Recall that this data suggests that there is unconditional convergence-the weaker teams appear to be catching up regardless of the underlying conditions of die nations concerned.

These results seem to lend strong support to the notion of convergence in international soccer results. However, this approach has been strongly challenged by several researchers e.g. Quah (1996). They argue that results of this type may simply be examples of Galton's fallacy, or regression to the mean. Galton, a biologist and statistician of the 19th century, noted that tall fathers tended to have shorter sons and short fathers tended to have taller sons. The fallacy is to think that this result necessarily implies convergence--it can be the case that the tallest fathers have shorter sons, but their sons in turn may still be tall--while the sons of short fathers may be taller, but their sons in turn may turn out to be short. There might be no tendency for the two groups (generally tall and generally short) to converge. If two variables are correlated, and one is regressed on another, the regression will uncover this correlation, but this does not imply that the dispersion of the variables is diminishing over time. A more direct test, therefore, is simply to examine dispersion over time and see if it is decreasing. Figure 4 shows the dispersion of winning percentages across nations for six world cup cycles.

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If we plot the win percentages of different countries at any point in time they will form a bell curve, with most nations centered around the average, but with a few teams significantly above and some significantly below average. Inspection of the distributions, which are all drawn to the same scale, shows that over time the bell shapes are getting both "taller" and "thinner"- implying that the dispersion of win percentages is indeed falling over time, consistent with convergence. These results are confirmed by statistical tests, and similar results are obtained for goal difference.

Implications and Conclusion

This paper has shown that there is evidence of unconditional convergence in the results of international soccer teams over the last six decades. This contrasts with the evidence of convergence in national income, where unconditional convergence is not observed. It would, of course, be foolish to expect that the results of competition in soccer would operate in the same way as the growth of GDP, but nonetheless one can speculate that there are factors which might facilitate convergence in soccer competition which are absent in many other economic spheres.

Soccer is played in a highly competitive international arena, and so teams are able to learn from the performance of their rivals. Generally one might expect to see convergence in sectors which produce internationally traded goods (Rodrik advances this argument) but less so in sectors which do not. Although national team play is popular, club soccer played in domestic leagues is even more popular, and the top players are internationally mobile. This means that athletes from all countries can learn to play at the highest level and, often, repatriate some of the skills they have learned. Soccer players are also better able to appropriate the returns from their investment. Appropriability, meaning the capacity of those who invest to obtain their return, is often a problem in developing countries because of bureaucracy and corruption (others appropriate the investment returns, thus diminishing the incentive to invest in the first place). This can amount to a form of theft, and investors may be in a very weak position to prevent this from happening. Since soccer players usually carry high status at home and abroad, the appropriability problem tends to be less severe.

From a different perspective, FIFA, the world governing body, long ago adopted an explicit policy of allocating more places in competitions to teams from weaker federations than would be justified purely on sporting merit, a policy which has helped to raise the standards and expectations of teams in Africa, Asia, and Central/North America. While the developed nations have sometimes offered privileged access to their markets in order to help developing countries, domestic lobbies have as often imposed limits on access.

"Convergence is more likely to be observed in sectors where international trade ensures competition and adoption of best practices..."

STEFAN SZYMANSKI is one of the world's leading and most influential sports economists. Between 2008-2012 he was a Professor of Economics at the Cass Business School of the City University of London. Currently, he serves as a Co-Director of the Michigan Centre for Sport Management at the University of Michigan.