Lack of global convergence and the formation of multiple welfare clubs across countries: An unsupervised machine learning approach

Mendez, Carlos Article Lack of global convergence and the formation of multiple welfare clubs across countries: An unsupervised machine learning approach Economies Provided in Cooperation with: MDPI – Multidisciplinary Digital Publishing Institute, Basel Suggested Citation: Mendez, Carlos (2019) : Lack of global convergence and the formation of multiple welfare clubs across countries: An unsupervised machine learning approach, Economies, ISSN 2227-7099, MDPI, Basel, Vol. 7, Iss. 3, pp. 1-17, https://doi.org/10.3390/economies7030074 This Version is available at: https://hdl.handle.net/10419/257006 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/ economies Article Lack of Global Convergence and the Formation of Multiple Welfare Clubs across Countries: An Unsupervised Machine Learning Approach Carlos Mendez Graduate School of International Development, Nagoya University, Aichi 464-8601, Japan; [email protected] Received: 6 June 2019; Accepted: 8 July 2019; Published: 17 July 2019   Abstract: The cross-country convergence hypothesis is one of the central topics of long-run macroeconomics. This paper revisits this hypothesis in a context beyond GDP. It uses a novel welfare index that incorporates measures of consumption, leisure, life expectancy, and inequality. Based on a sample of 128 countries over the 1980–2007 period, the lack of global sigma and beta convergence is first documented. Next, the paper incorporates some recent developments from the unsupervised machine learning literature to evaluate the existence of local convergence. In particular, the application of a distribution-based clustering algorithm suggests the formation of three local convergence clubs. Under this classification, beta convergence is recovered for each club. However, only the core members of the richest club appear to be reducing their welfare differences in a way that is consistent with the strong notion of sigma convergence. Overall, these results re-emphasize the finding that beta convergence is necessary, but not sufficient for sigma convergence, even within convergence clubs and in a context beyond GDP. Keywords: convergence; welfare; machine learning; non-parametric distribution; distribution dynamics; distribution-based clustering 1. Introduction In the study of long-run macroeconomics, a central topic is the empirical testing of the convergence hypothesis across countries (Johnson and Papageorgiou 2018). Differences across countries are conceptually defined in terms of living standards or national welfare. However, from an empirical and operational standpoint, GDP per-capita has been used in most of the literature as the key proxy variable for measuring living standards. Although the use of GDP per-capita is sometimes useful and informative, 1 economist are well aware that it is an incomplete measure of national welfare (Fleurbaey 2009;Fleurbaey and Blanchet 2013,Stiglitz et al. 2010,2019). In this context, a series of different alternatives has been proposed by economists and non-economists alike (Becker et al. 2005;Cordoba and Verdier 2008;Fleurbaey and Gaulier 2009). One of the most recent attempts to go beyond GDP is the work of Jones and Klenow (2016). Using a rigorous expected utility framework, these authors combined measures of national consumption (private and public), leisure, life expectancy, and inequality to construct a theoretically-appealing new welfare index. Given the comprehensive coverage of this index, it can be used as an alternative proxy variable for the study of the convergence hypothesis. Machine learning is a fast-growing field of research that focuses on the development of prediction, classification, and clustering algorithms (Athey 2018;Athey and Imbens 2019). It is commonly divided 1 GDP per capita is a useful variable in the sense that it correlates with other human development variables such as educational attainment, life expectancy, and even subjective happiness. Economies 2019,7, 74; doi:10.3390/economies7030074 www.mdpi.com/journal/economies Economies 2019,7, 74 2 of 17 into two sub-fields: supervised and unsupervised machine learning. The latter focuses on finding clusters of observations using covariates, and it is the main approach used in the current paper. Traditional clustering frameworks such K-means and hierarchical clustering derive their results based on some criteria of distance. In contrast, distribution-based clustering frameworks derive their results by defining clusters as regions of high-probability density separated by low-probability regions (Hartigan 1975;Kaufman and Rousseeuw 2010). This paper revisits the cross-country convergence hypothesis in the context beyond GDP suggested by Jones and Klenow (2016). From a methodological standpoint, this paper first applies the classical sigma and beta convergence tests of Barro and Sala-i Martin (1992). Then, its fundamental contribution relies on the implementation of the distributional convergence approach of Quah (1993, 1996b,1997). Furthermore, by taking advantage of some recent developments in the unsupervised machine learning literature, it extends the distributional convergence framework by integrating the distribution-based clustering algorithm of Azzalini and Menardi (2014a). This extension allows for a more detailed identification and characterization of convergence clubs. Based on a sample of 128 countries over the 1980–2007 period, this paper finds that welfare differences across countries are characterized by a lack of both sigma and beta convergence. Moreover, limited country mobility suggests the possible existence of convergence clubs. The distributional convergence approach builds on these findings and suggests that the cross-country welfare distribution appears to be characterized by three convergence clubs or clusters. Interestingly, under this classification, the beta convergence is recovered for each club. However, only the core members of the richest club appear to be converging in a way that is consistent with the strong notion of sigma convergence. Overall, these results re-emphasize a central finding in the long-run macroeconomics literature: beta convergence is necessary, but not sufficient for sigma convergence, even within convergence clubs and in a context beyond GDP. The rest of the paper is organized as follows: Section 2describes the data and the different convergence frameworks. Section 3introduces some stylized convergence facts about welfare differences across countries. Section 4presents the convergence clubs’ results. Section 5discusses the results and outlines some suggestions for further research. Finally, Section 6offers some concluding remarks. 2. Data and Methods 2.1. Beyond GDP Data The work in Jones and Klenow (2016) proposed a new summary statistic that aims to quantify the level of welfare of people in a country. This statistic incorporates measures of consumption (private and public), leisure, life expectancy, and inequality. Internationally-comparable measures of consumption are taken from the Penn World Table 8.0. This database also provides the necessary inputs to compute a measure of leisure time from the available series of hours worked, employment, and population data. Life expectancy data are from the World Bank’s HNPStatsdatabase. Finally, inequality data are from the UNI-WIDERWorld Income Inequality database 3.0.2 To aggregate these measures, in a theoretically-consistent way, the work in Jones and Klenow (2016) calibrated an expected utility function and evaluated the consumption-equivalent level of the four variables. Using this statistic, they found that, on average, welfare is highly correlated with GDP per capita. However, they also reported that there are often large deviations from GDP, in particular in developing countries. For instance, in terms of the relative welfare differences across regions, Western Europe appears closer to the United States; fast-growing Asia has not caught up as much; and most countries in Latin America and Africa are lagging further behind. The work in Jones and Klenow 2For further details on the measurement and calculations of the variables, see the Appendix of Jones and Klenow (2016). Economies 2019,7, 74 3 of 17 (2016) constructed this welfare statistic for 152 countries in the year 2007. They also calculated the average growth rate for the 1980–2007 period using a sub-sample of 128 countries 3 . Given the welfare level data for the final year (2007) and the average growth rate (1980–2007 period), it is possible to recover cross-sectional data for the initial year (1980). Thus, based on the availability of the data, this paper uses a sample of 128 countries for two time periods: 1980 and 2007.4 2.2. Sigma and Beta Convergence In the empirical literature of economic growth, two concepts of convergence are typically discussed. On the one hand, the concept of sigma ( σ ) convergence describes the evolution of the cross-sectional dispersion of a variable. From this perspective, convergence occurs when the cross-sectional dispersion declines over time, so the level of the variable under study becomes increasingly more similar across countries (Barro and Sala-i Martin 1992;Baumol 1986;Dowrick and Nguyen 1989). Typically, sigma convergence is measured by the coefficient of variation or by the standard deviation of the logarithm of the variable under study. In the context of the current paper, the latter indicator is adopted, and the measurement of sigma convergence is implemented as follows: σt=v u u t 1 N−1 N ∑ i=1log(yi)−log(y)2, (1) where σt is the cross-country dispersion in national welfare, N is the number of countries, log(yi) is the natural logarithm of the welfare level of country i , and log(y) is the sample average of the logarithm of welfare. On the other hand, the concept of unconditional beta ( β ) convergence describes the inverse relationship between the initial level of a variable and its average growth rate. From this perspective, if such an inverse relationship exists, it means that, on average, poor countries tend to grow faster than the rich ones, so over time, poor countries tend to catch up with the level of the rich ones (Barro and Sala-i Martin 1992;Baumol 1986;Sala-i Martin 1996). Typically, beta convergence is measured by the estimation of an Ordinary Least Squares (OLS) regression, in which the growth rate of a variable inversely depends on its initial level. In the context of the current paper, the measurement of beta convergence is implemented as follows: 1 Tlog yt y0=c−1−e−βt tlog(y0), (2) where the left side represents the average rate of welfare growth, which shows an inverse relationship to its initial level in log terms, log(y0) , β represents the speed of convergence to the steady-state, and c represents unobserved parameters, such as steady-state values. These two measures of convergence are related 5 . Keeping other variables constant, when poor countries tend to grow faster than rich ones (that is, beta convergence), then the cross-country dispersion declines over time (that is, sigma convergence). In other words, beta convergence is one determinant of sigma convergence. However, the effect of beta convergence can be offset by other variables and shocks that increase the dispersion at any point in time. As pointed out by Quah (1993) and Sala-i Martin (1996), beta convergence is necessary, but not sufficient to achieve sigma convergence. 3The database can be accessed at https://web.stanford.edu/~chadj/BeyondGDP500.xls. 4 Although one could use the latest versions of the Penn World Table, UNI-WIDER, and the World Bank databases to extend the analysis beyond 2007, the purpose of this paper is to be directly comparable with the paper of Jones and Klenow (2016). Thus, the reference period for comparison is still the 1980–2007 period. Further research, beyond the scope of this paper, could extend the period of analysis and evaluate the persistence of the convergence clusters. 5See Sala-i Martin (1996) for further details. Economies 2019,7, 74 4 of 17 2.3. Distributional Convergence and Convergence Clubs The distributional convergence approach aims to capture the evolution of the entire cross-country distribution of a variable. From this perspective, convergence occurs when the shape of the cross-sectional distribution tends to one density peak (one mode) over time. In this framework, the emergence of multiple peaks or modes is usually associated with the existence of convergence clubs (Galor 1996;Magrini 2009;Quah 1996a 1997). Typically, distributional convergence is measured by the shape and the number of modes of a stochastic kernel distribution and its corresponding ergodic distribution. In the current paper, only the former is adopted, and the measurement of distributional convergence6is implemented as follows: 1. The variable under study (that is, national welfare) is expressed relative to a benchmark economy, which in the literature is usually the United States. The purpose of this normalization is to abstract from systematic forces that might simultaneously affect all countries. 2. To facilitate comparison and visualization, the natural logarithm of the relative variable is applied. The log of a relative variable can be interpreted as the proportional difference between a country and the benchmark country (i.e., the convergence frontier). 3. The stochastic kernel is a conditional distribution that is calculated as follows: G(yt+s|yt) = ft+s,t(yt+s,yt) ft(yt), (3) where ft(·) is the univariate kernel distribution of relative welfare in the initial year, t , and ft+s,t(·) is the (inter-temporal) bivariate kernel distribution between the years. 4. The bivariate kernel distribution is estimated as follows: ft+s,t(yt+s,yt) = 1 nht+sht n ∑ i=1 Kt+syt+s−yt+s,i ht+sKtyt−yi ht, (4) where yt+s and yt denote the relative welfare of each country at time t+s and t , respectively, Kt+s and Kt denote kernel functions, and ht+s and ht denote the smoothing parameters of yt+s and yt , respectively. Following the convention of the literature, the kernel functions adopt a Gaussian form, and the smoothing parameters are selected based on the minimization of the Asymptotic Mean Integrated Square Error (Magrini 2009). The stochastic kernel is a tree-dimensional object that is commonly represented by a surface plot or a contour plot. If most countries are concentrated around the main diagonal of these graphs, then there is evidence of distributional persistence over time. Global distributional convergence is found when most of the countries are located around zero in the the (t+s)-axis and parallel to the t-axis. Finally, the distribution-based clustering algorithm developed by Azzalini and Menardi (2014a) is applied with the previously-described kernel functions and smoothing parameters. The implementation of this new clustering framework is useful for two purposes. First, it helps us identify the location of each country within the distribution. Second, it helps us allocate each country to its nearest convergence club. 3. Some Stylized Facts 3.1. Lack of Sigma and Beta Convergence The classical analysis of convergence shows that, similar to patterns of GDP per capita documented by Barro and Sala-i Martin (1992), welfare differences across countries are characterized by a lack 6 For a more comprehensive and recent presentation of the distributional convergence approach, see Dal Bianco (2016), Durlauf et al. (2005), Epstein et al. (2003), or Mendez (2018) Economies 2019,7, 74 5 of 17 of both sigma and beta convergence. Figures 1and 2summarize this finding. Figure 1shows that, although the distribution has shifted to the right, the cross-country dispersion of welfare has increased between 1980 and 2007. As such, this result highlights the lack of sigma convergence in the context of welfare differences across countries. Figure 1. Lack of sigma convergence in welfare across countries. Figure 2. Lack of beta convergence in welfare across countries. More specifically, the standard deviation of the cross-country welfare distribution in the year 1980 was 1.22. By the year 2007, this dispersion increased to 1.41. This higher degree of cross-country inequality in welfare is also observable in the increasing distance between the third quartile and the first quartile of the boxplot presented in Figure 1. Figure 2shows that, on average, welfare-poor countries are not growing faster than welfare-rich countries. As such, this result indicates a lack of beta convergence. Note also that, if anything, the positive (but not significant) slope of the regression line would suggest that welfare-rich countries Economies 2019,7, 74 6 of 17 are growing faster than welfare-poor countries, and thus contributing to the increasing dispersion reported in Figure 1. Finally, it also worth noting that the triangular shape of the scatter plot is highly similar to that reported in the studies that document a lack of beta convergence in income (Barro and Sala-i Martin 1992;Sala-i Martin 1996). 3.2. Limited Forward and Backward Mobility Figure 3highlights the fact that there is a limited degree of forward and backward mobility across countries. The measurement of country mobility here is relative to that experienced by the frontier, which in this case is the United States. Thus, the lack of relative mobility does not imply the lack of progress in absolute terms. In Figure 3, most countries are located around the 45-degree line. Indeed, the relation between the initial and the final level of welfare is summarized by a linear regression in which the coefficient of the slope is statistically equal to one. The 95% confidence interval for this slope coefficient is between 0.97 and 1.14. Moreover, the R-squared of this regression highlights that 83 percent of the cross-country welfare variation of the year 2007 is explained by the welfare variation of the year 1980. Figure 3. Limited forward and backward mobility in welfare across countries. Another interesting mobility fact is that 76 out the 128 countries in the sample (that is, almost 60 percent of the total sample) are moving backwards. In other words, relative to its initial position in 1980, 60 percent of the countries ended up with lower relative welfare in 2007. This backward mobility is more pronounced in the poorest countries in the sample. Finally, both the limited degree country mobility and the overall lack of cross-country convergence suggest the possible existence of local convergence clubs. The results associated with this hypothesis are presented in the next section. 4. Results 4.1. Transitional Dynamics via the Stochastic Kernel Distribution Figure 4shows the 3D surface of the stochastic kernel distribution. The transitional dynamics between 1980 and 2007 were characterized by three density modes along the main diagonal. In this Economies 2019,7, 74 7 of 17 convergence framework, the lack of country mobility and the emergence of multiple basins of attraction (that is, multiple density modes) in the main diagonal are typically interpreted as suggestive evidence of convergence clubs. In this case, given the existence of three modes, then three different convergence clubs are likely to be present. Interestingly, these tree convergence clubs in welfare are qualitatively similar to the three convergence clubs in income reported in Pittau et al. (2016). Although the number of clubs is the same, the methodology of Pittau et al. (2016) is based on a finite Gaussian mixture model, which is a semi-parametric distributional alternative to the non-parametric distribution framework implemented in the current paper. Figure 4. Stochastic kernel distribution and convergence clubs (3D surface plot). Figure 5shows the contour plot representation stochastic kernel overlapped with the scatter plot of the countries of Figure 3. It provides a rough overview of the country composition of each convergence club. From the figure, it is clear that some countries were located at the center-core of each club, while the position of other countries raises some doubts about their club membership. In particular, there seemed to be a considerable number of countries between the bottom club and the middle club. For this sub-sample, some countries were moving forward and transitioning towards a superior club. In contrast, other countries were moving backward and transitioning towards an inferior club. To help clarify the membership of these kind of countries, the novel distribution-based clustering algorithm of Azzalini and Menardi (2014a) is implemented in the next subsection. Economies 2019,7, 74 8 of 17 Figure 5. Stochastic kernel distribution and convergence clubs (contour plot). 4.2. Core Clusters and Classification of Countries In a series of papers, Azzalini and Torelli (2007) and Azzalini and Menardi (2014a,2014b) used the modes of a non-parametric distribution as a criterion for identifying clusters. High-density observations group themselves in what Azzalini and Menardi call “cores clusters”. Low-density observations, on the other hand, could be allocated to a proximate cluster based on a Delaunay triangulation. Figure 6shows the results of this approach in the context of the mobility scatter plot of Figure 3. Consistent with the stochastic kernel of Figures 4and 5, three clusters or clubs were identified. The main advantage of this clustering framework is the endogenous identification of core club members (those with a “1”, “2”, or “3” prefix). In addition, low-density observations (i.e., those with a “0” prefix) were also identified and allocated to their more proximate core club. 7 In this low-density context, countries such as the Maldives or South Korea are interesting cases to be studied. Although both countries experienced forward mobility (catching up behavior), their progress has not been classified as that of a “core” member of the immediately superior club. In these cases, the clustering classification was still informative in the sense that it was also clear that these kinds of countries did not belong to their immediately inferior “core” club either. Finally, the bottom right of Figure 6includes a cluster tree. It is meant to provide a measure of the robustness of the clubs to different density thresholds. Although for a considerably large set of density thresholds, three clubs were identifiable, it is also possible that the countries of Club 1 and Club 2 could be converging to a similar steady state. Even in this case, the cross-country distribution of welfare is likely to be characterized by more than one convergence club. 7See Appendix Afor a list of countries their respective clubs. Economies 2019,7, 74 15 of 17 Table A1. Cont. ID Name ISO Code Clubs Core Club Relative Welfare in 1980 Relative Welfare in 2007 54 Mauritius MUS Middle Middle 8.3 16 55 Iran IRN Middle NA 2.65 16 56 Venezuela VEN Middle NA 11.85 15.3 57 Malaysia MYS Middle Middle 7.45 15.1 58 Panama PAN Middle Middle 8.58 14.3 59 Maldives MDV Low NA 0.98 13.5 60 Jamaica JAM Middle Middle 6.01 13.1 61 Brazil BRA Middle Middle 4.46 11.5 62 Tunisia TUN Middle Middle 4.84 11.2 63 Fiji FJI Middle Middle 5.54 10.9 64 Thailand THA Middle NA 3.99 10.9 65 Jordan JOR Middle NA 9.75 10.8 66 Peru PER Middle NA 4.12 9.9 67 Ecuador ECU Middle Middle 5.36 9.2 68 Colombia COL Middle NA 6.92 9.1 69 Egypt EGY Low NA 1.56 8.9 70 Suriname SUR Middle NA 8.89 8.7 71 Syria SYR Middle NA 6.17 8.2 72 Sri Lanka LKA Middle NA 3.06 7.8 73 Cape Verde CAF Low NA 1.58 7.7 74 Guatemala GTM Middle NA 3.62 7.3 75 Honduras HND Middle NA 3.87 7.2 76 Gabon GAB Middle NA 6.58 6.6 77 China CHN Low NA 1.86 6.6 78 Mongolia MNG Low NA 2.04 6.3 79 Paraguay PRY Middle NA 4.17 5.9 80 Bhutan BTN Low NA 0.93 5.9 81 Indonesia IDN Low NA 2.1 5.7 82 Iraq IRQ Low NA 1.45 5.3 83 Morocco MAR Low Low 3.39 5.2 84 Philippines PHL Low NA 4.08 4.9 85 Bolivia BOL Low NA 1.52 4.8 86 South Africa ZAF Low NA 4.38 4.5 87 Pakistan PAK Low Low 2.68 4.4 88 Botswana BWA Low Low 1.97 4.3 89 Namibia NAM Low Low 3.15 4.1 90 Vietnam VNM Low NA 1.25 4 91 India IND Low Low 1.61 3.9 92 Sudan SDN Low Low 1.62 3.8 93 Sao Tome/Princi STP Low Low 3.54 3.7 94 Ghana GHA Low Low 2.1 3.3 95 Djibouti DJI Low Low 3.48 3.2 96 Swaziland SWZ Low NA 4.67 3.1 97 Zimbabwe ZWE Low Low 3.44 3.1 98 Lao LAO Low NA 1.02 3 99 Mauritania MRT Low Low 2.26 2.9 100 Cambodia CMR Low NA 0.66 2.7 101 Bangladesh BGD Low Low 2.08 2.5 102 Senegal SEN Low Low 2 2.4 103 Comoros COM Low Low 1.77 2.3 104 Nigeria NGA Low Low 1.49 2.3 105 Cameroon CAN Low Low 2.44 2.2 106 Lesotho LSO Low Low 2.28 2.2 107 Cote d’Ivoire CIV Low Low 2.81 2 108 Congo COD Low NA 3.29 1.9 109 Kenya KEN Low Low 2.72 1.9 110 Benin BEN Low Low 1.46 1.9 Economies 2019,7, 74 16 of 17 Table A1. 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New York: The New Press. c 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).