# Lack of Global Convergence and the Formation of Multiple Welfare Clubs across Countries: An Unsupervised Machine Learning Approach

## Abstract

**:**

## 1. Introduction

## 2. Data and Methods

#### 2.1. Beyond GDP Data

#### 2.2. Sigma and Beta Convergence

#### 2.3. Distributional Convergence and Convergence Clubs

- 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.
- 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).
- The stochastic kernel is a conditional distribution that is calculated as follows:$$G({y}_{t+s}\mid {y}_{t})=\frac{{f}_{t+s,t}({y}_{t+s},{y}_{t})}{{f}_{t}\left({y}_{t}\right)},$$
- The bivariate kernel distribution is estimated as follows:$${f}_{t+s,t}({y}_{t+s},{y}_{t})=\frac{1}{n{h}_{t+s}{h}_{t}}\sum _{i=1}^{n}{K}_{t+s}\left(\right)open="("\; close=")">\frac{{y}_{t+s}-{y}_{t+s,i}}{{h}_{t+s}},$$

## 3. Some Stylized Facts

#### 3.1. Lack of Sigma and Beta Convergence

#### 3.2. Limited Forward and Backward Mobility

## 4. Results

#### 4.1. Transitional Dynamics via the Stochastic Kernel Distribution

#### 4.2. Core Clusters and Classification of Countries

#### 4.3. Sigma and Beta Convergence within Clubs

## 5. Discussion

## 6. Concluding Remarks

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

ID | Name | ISO Code | Clubs | Core Club | Relative Welfare in 1980 | Relative Welfare in 2007 |
---|---|---|---|---|---|---|

1 | Luxembourg | LUX | High | NA | 52.98 | 125 |

2 | Iceland | ISL | High | High | 44.59 | 111.7 |

3 | United States | USA | High | High | 43.74 | 100 |

4 | Sweden | SWE | High | High | 44.3 | 91.2 |

5 | France | FRA | High | High | 37.72 | 91.1 |

6 | Australia | AUS | High | High | 35.83 | 90.7 |

7 | United Kingdom | GBR | High | High | 34.97 | 90.4 |

8 | Switzerland | CHE | High | High | 44.6 | 87.1 |

9 | The Netherlands | NLD | High | High | 42.88 | 86.2 |

10 | Austria | AUT | High | High | 32.06 | 85.5 |

11 | Belgium | BEL | High | High | 42.61 | 83 |

12 | Cyprus | CYP | High | High | 18.07 | 83 |

13 | Japan | JPN | High | High | 28.8 | 82.6 |

14 | Canada | CPV | High | High | 43.5 | 82.3 |

15 | Norway | NOR | High | High | 43.15 | 81 |

16 | Italy | ITA | High | High | 32.21 | 78 |

17 | Spain | ESP | High | High | 27.26 | 77.6 |

18 | Germany | DEU | High | High | 33.63 | 77.3 |

19 | Denmark | DNK | High | High | 42.91 | 75.8 |

20 | Finland | FIN | High | High | 27.75 | 74.4 |

21 | New Zealand | NZL | High | High | 28.05 | 71 |

22 | Greece | GRC | High | High | 22.62 | 70.5 |

23 | Ireland | IRL | High | High | 23.58 | 69.6 |

24 | Israel | ISR | High | High | 30.08 | 63.4 |

25 | Malta | MLT | High | High | 19.27 | 61.8 |

26 | Hong Kong | HKG | High | High | 22.35 | 59 |

27 | Singapore | SGP | High | NA | 11.82 | 56.7 |

28 | Qatar | QAT | High | NA | 49.23 | 52.1 |

29 | Barbados | BRB | High | High | 20.24 | 50.7 |

30 | Portugal | PRT | High | High | 16.14 | 50.7 |

31 | South Korea | KOR | Middle | NA | 5.56 | 45.3 |

32 | Macao | MCO | High | NA | 16.28 | 44.9 |

33 | Kuwait | KWT | High | High | 22.13 | 43 |

34 | Hungary | HUN | Middle | NA | 12.43 | 34.2 |

35 | Poland | POL | Middle | NA | 8.7 | 31.5 |

36 | Bahamas | BHS | Middle | NA | 12.13 | 31.1 |

37 | Bahrain | BHR | High | NA | 26.16 | 23.6 |

38 | Costa Rica | CRI | Middle | NA | 13.83 | 22.8 |

39 | Mexico | MEX | Middle | NA | 12.07 | 22.6 |

40 | Oman | OMN | Middle | Middle | 7.23 | 22.6 |

41 | St. Vincent | VCT | Middle | Middle | 6.66 | 22.5 |

42 | Turkey | TUR | Middle | NA | 4.65 | 22.3 |

43 | Argentina | ARG | Middle | NA | 5.78 | 21.8 |

44 | Saint Lucia | LCA | Middle | NA | 20.29 | 20.9 |

45 | Bulgaria | BGR | Middle | Middle | 9.13 | 20.6 |

46 | Chile | CHL | Middle | Middle | 9.88 | 19.7 |

47 | Saudi Arabia | SAU | Middle | Middle | 8.16 | 19.6 |

48 | Trinidad/Tobago | TTO | High | NA | 31.63 | 19 |

49 | Lebanon | LBN | Middle | Middle | 7.16 | 19 |

50 | Belize | BLZ | Middle | Middle | 11.16 | 18.5 |

51 | Uruguay | URY | Middle | NA | 12.7 | 18 |

52 | Albania | ALB | Middle | Middle | 6.42 | 16.8 |

53 | Dominican Rep. | DOM | Middle | Middle | 7.03 | 16.2 |

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 |

111 | Angola | AGO | Low | Low | 1.45 | 1.9 |

112 | Chad | TCD | Low | Low | 1.12 | 1.9 |

113 | Zambia | ZMB | Low | Low | 2.53 | 1.8 |

114 | Togo | TGO | Low | Low | 1.48 | 1.8 |

115 | Nepal | NPL | Low | NA | 1.04 | 1.8 |

116 | Uganda | UGA | Low | NA | 0.87 | 1.7 |

117 | Tanzania | TZA | Low | Low | 2.4 | 1.6 |

118 | Rwanda | RWA | Low | Low | 1.57 | 1.6 |

119 | Madagascar | MDG | Low | Low | 1.51 | 1.6 |

120 | Guinea | GIN | Low | Low | 2.36 | 1.5 |

121 | Burkina Faso | BFA | Low | NA | 0.97 | 1.5 |

122 | Mali | MLI | Low | NA | 0.77 | 1.5 |

123 | Sierra Leone | SLE | Low | Low | 2.21 | 1.4 |

124 | Liberia | LBR | Low | Low | 2.18 | 1.3 |

125 | Ethiopia | ETH | Low | NA | 1.2 | 1.3 |

126 | C. Afr. Republic | KHM | Low | NA | 1.54 | 1.2 |

127 | Niger | NER | Low | NA | 1.33 | 1.2 |

128 | Malawi | MWI | Low | NA | 1.26 | 1.2 |

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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. |

2 | For further details on the measurement and calculations of the variables, see the Appendix of Jones and Klenow (2016). |

3 | The 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. |

5 | See Sala-i Martin (1996) for further details. |

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) |

7 | See Appendix A for a list of countries their respective clubs. |

8 | Although the frameworks are similar, they are not identical. The work in Battisti and Parmeter (2012) used a semi-parametric density-based clustering framework, while this paper uses a non-parametric density-based clustering framework. |

Standard Deviation | Standard Deviation | Dispersion Ratio | ANOVA Test | |
---|---|---|---|---|

Log of Relative Welfare 1980 | Log of Relative Welfare 2007 | 1980/2007 | p-Value | |

Total (128 members) | 1.22 | 1.41 | 0.86 | 0.10 |

Club 1 (All 54 members) | 0.46 | 0.58 | 0.79 | 0.09 |

Club 2 (All 40 members) | 0.45 | 0.49 | 0.92 | 0.62 |

Club 3 (All 34 members) | 0.37 | 0.39 | 0.96 | 0.80 |

Club 1 (Core 30 members) | 0.31 | 0.39 | 0.80 | 0.23 |

Club 2 (Core 17 members) | 0.25 | 0.27 | 0.92 | 0.75 |

Club 3 (Core 28 members) | 0.32 | 0.21 | 1.47 | 0.05 |

Club 1 | |||
---|---|---|---|

Variable | Coefficient | t-Statistic | p-Value |

constant | 0.0337 | 6.9442 | 0.0000 |

log(y0)/T | −0.8252 | −4.76 | 0.0000 |

R2 | 0.3 | ||

Speed of convergence ($\beta $) | 6.46% | ||

Half-life (periods) | 11 | ||

Club 2 | |||

Variable | Coefficient | t-Statistic | p-Value |

constant | 0.0599 | 5.4013 | 0.0000 |

log(y0)/T | −0.4388 | −2.9187 | 0.0059 |

R2 | 0.18 | ||

Speed of convergence ($\beta $) | 2.14% | ||

Half-life (periods) | 32 | ||

Club 3 | |||

Variable | Coefficient | t-Statistic | p-Value |

constant | 0.0966 | 4.6296 | 0.0001 |

log(y0)/T | −0.5231 | −3.1832 | 0.0032 |

R2 | 0.24 | ||

Speed of convergence ($\beta $) | 2.74% | ||

Half-life (periods) | 25 |

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## Share and Cite

**MDPI and ACS Style**

Mendez, C.
Lack of Global Convergence and the Formation of Multiple Welfare Clubs across Countries: An Unsupervised Machine Learning Approach. *Economies* **2019**, *7*, 74.
https://doi.org/10.3390/economies7030074

**AMA Style**

Mendez C.
Lack of Global Convergence and the Formation of Multiple Welfare Clubs across Countries: An Unsupervised Machine Learning Approach. *Economies*. 2019; 7(3):74.
https://doi.org/10.3390/economies7030074

**Chicago/Turabian Style**

Mendez, Carlos.
2019. "Lack of Global Convergence and the Formation of Multiple Welfare Clubs across Countries: An Unsupervised Machine Learning Approach" *Economies* 7, no. 3: 74.
https://doi.org/10.3390/economies7030074