Convergence and Paths of Economic Growth

4.1 Cluster Analysis

In Figure 1 Panel A, we plot the percentage of variance explained by varying numbers of clusters. As the cluster characteristics of the data remain broadly consistent across years, we present the results each 20 years, 1960, 1980, 2000, and 2014. When the number of clusters exceeds three, the increases in percentage of explained variance slows down rapidly, indicating that additional clusters yield rapidly diminishing returns. In addition, when cluster number is three, the percentage of explained variance surpasses 80 % for all years. Above supports three cluster is the optimal choice. Because this pattern holds at all sampled time points, we conclude that three distinct clusters represent the most natural and robust categorization. The constancy of the number of clusters over time supports the hypothesis that steady-state attractors constrain long-run growth paths. The world economy may follow three distinct long-term growth convergence paths, each corresponding to different per capita income levels.

Figure 1:

Elbow method and K-means clustering result. Notes: The results of 162 economies of the Elbow Method in 1960, 1980, 2000, and 2014 are shown in Panel A. The vertical coordinate is % Explained Variation = (Total Sum of Squares – WCSS)/Total Sum of Squares * 100, and the horizontal coordinate is the number of clusters k. Panel B presents the K-means clustering results for economies that remained in the same cluster over the sample period. The horizontal axis represents the log of GDP per capita, and the vertical axis shows the growth rate of the log of GDP per capita. Economies in each cluster are depicted in distinct colors, and the center of each cluster is also indicated. The two dashed lines in each coordinate system mark the boundaries that separate the clusters. Since the characteristics of each year are similar, we only present 1960, 1980, 2000, and 2014 here.

Figure 1 Panel B shows the K-means clustering results of countries that consistently remain in the same cluster across all assessment years, adopting three clusters following the elbow method recommendation. We observe that both the cluster boundaries and centroids shift over time, consistent with our growth convergence path framework and our hypothesis. Then, we define distinct clusters as high-, middle-, and low-income groups based on the per capita income level of each cluster. There are 25, 19, and 22 economies consistently remain on the high-, middle-, and low-income cluster across all assessment years. A detailed list of economies sees in Appendix Table A1.

4.2 Convergence Test

Table 1 Panel A and B present the results of convergence test (equation (3)) using 5-year interval data for each cluster. All β _j and α _j,t are statistically significant at the 1 % level confirming the robustness of convergence dynamics. Panel A presents the results for the economies that remained in the same cluster throughout the entire sample period (1960–2019), excluding economies experienced path transitions. The β _j for the high-, middle-, and low-income clusters are all negative and significant at the 1 % level. These results confirm the presence of robust absolute β-convergence within each income group, consistent with our theoretical framework based on Lyapunov stability. The adjusted R ² values (0.38 for high-income, 0.23 for middle-income, and 0.11 for low-income) further suggest stronger convergence dynamics and better model fit among high-income countries, reflecting greater stability and predictability in their growth trajectories. The results of the 1-year and 10-year intervals data are similar to these of the 5-year interval (see Appendix Table A2 and A3).

Panel B includes almost all economies by only excluded observations from periods immediately surrounding path transitions, as transition period elimination. This allows for a more inclusive test while still mitigating the distortion caused by structural changes. The β _j remain negative and highly significant (−0.141, −0.122, and −0.069), though slightly smaller in magnitude than in Panel A. This decline is expected due to the inclusion of residual noise from economies approaching or recovering from transitions. Importantly, the adjusted R ² values also decrease, highlighting how transitional dynamics introduce additional heterogeneity that weakens the explanatory power of the model. The consistency of the β _j estimates and their statistical significance across both stringent and relaxed samples confirms the correctness of our specification in equation (3), under which the convergence behavior of each economy is governed by its steady-state attractor. These findings demonstrate that once transition phases are excluded, convergence within each growth path becomes distinct and statistically robust, underscoring the necessity of removing transition observations before testing convergence.

To validate our panel model, which only includes year fixed effects, we conduct F-tests for both country and the year fixed effects in each income level path. The F-test values of country fixed effects for high-, middle-, and low-income paths are 0.80, 0.36, and 0.73, respectively, with p-values above 0.10. Conversely, the p-values for the year fixed effects are all below 0.01, highlighting the substantial impact of the temporal factors, including technological advancements and economic shocks, in line with prior findings (Mankiw et al. 1992; Barro 2012; Ito 2017). F-tests confirm that once paths are identified, country fixed effects lose significance while year effects remain, implying that cross-country heterogeneity is largely captured by path classification. The results of the 1-year and 10-year intervals data are same.

Building on earlier studies, such as Kremer et al. (2021) and Acemoglu and Molina (2021), which emphasize the importance of properly accounting for fixed effects in growth models, we find that once the three paths are clearly identified, only year fixed effects remain significant. This result suggests that distinguishing the paths adequately captures the cross-country heterogeneity previously explained by country fixed effects, thereby reducing the risk of bias in our estimations. Overall, the heterogeneity documented in many convergence studies may come from imprecise path classification and outlier noise. By delineating distinct convergence paths, we effectively address these concerns, reinforcing the utility of multi-path frameworks in explaining persistent differences across global economies.

In Table 1 Panel C, Panel D, and Panel E, using the same cross-sectional test method commonly employed in traditional conditional convergence studies, stable β-convergence exists within each income group in all sample periods. This consistency in both panel and cross-sectional results strengthens confidence in our findings, indicating that the three convergence paths are stable and well-supported by the empirical evidence.

Our study provides a strong and robust evidence that there are three growth convergence paths globally. Our conclusion that three clusters provide the most natural categorization is consistent with historical income classifications used by agencies such as the United Nations and the World Bank. This suggests that steady-state growth paths, segmented by income levels, substantially constrain the economic growth of economies, reaffirming core principles of the Solow-Swan model and the broader convergence framework. Empirically, the classification based on per-capita income effectively captures underlying structural and technological heterogeneity. Once economies are grouped by path, the explanatory power of growth regressions markedly improves, confirming that the observed dynamics are consistent with the theoretical attractor interpretation. Meanwhile, we provide insights into club convergence in a more general framework.

4.3 Correlate Variable Analysis

The above tests provide an insight into club convergence. Based on the three growth convergence paths, whether the understanding of conditional convergence will be refreshed. In Table 2, we examine the impact of correlated variables on cross-sectional data from 1970 to 2019, 1970 to 1995, and 1995 to 2019. The results associated with the coefficient significance show that on the basis that the three paths we previously obtained converge significantly, the evidence on the influence of certain variables is weak. In the early literature, population growth, investment rate, human capital, and institutional quality factors (such as political rights and rule of law) are conditional variables that significantly affect convergence (Barro 1991; Barro and Sala-i-Martin 1992; Mankiw et al. 1992; Kremer et al. 2021). Only in the 1970–2019 period, Solow fundamental variables are significant in the high-income sample, but after splitting the samples, only log(I/GDP) is significant in the 1970–1995 sample, and only log(n + g + δ) is significant in the 1995–2019 sample. Also, only log(n + g + δ) is significant in the 1995–2019 middle-income sample, and GS is significant in the low-income 1970–1995 sample. The others are all not significant at 10 %. Furthermore, the effects of short period correlated variables such as Rule of Law, Political Right, and Civil liberties also appear weak (see Appendix Table A4). Under the three convergence paths framework, the evidence for conditional convergence is weak and unconvincing.

A key finding from our analysis is that the evidence of conditional convergence is highly sensitive to sample periods and different income levels’ growth paths. Once path heterogeneity is accounted for, most coefficients lose significance, especially for middle- and low-income groups. While earlier studies (e.g., Mankiw et al. 1992) find conditional convergence, results vary by period. For example, convergence appears in 1960–1985 and 2005–2015 but not in 1985–1995 (Kremer et al. 2021).

A natural speculation following the identification of three growth convergence paths is that the evidence of conditional convergence observed in previous studies may be caused by path heterogeneity. To explore this, we systematically evaluate the distribution of correlated variables within each identified convergence path. using box plots. As expected, key variables frequently associated with long-run growth, such as investment rates, population growth, human capital, and institutional quality, exhibit substantial differences across the identified paths. Figure 2 presents box plots covering economies of each cluster. These plots confirm that the variables diverge substantially among the three paths. Specifically, the distribution is closely linked to differences in income levels, with mean values showing a pronounced monotonic pattern. For instance, looking at the quartiles for variables like H and CL, there is almost no overlap in their distributions across the different paths.

Figure 2:

Boxplot of correlate variables. Notes: In Figure 2, we draw boxplots with the average value of correlated variables, X ‾ i , j , t , △ t , according to the path in each coordinate.

By integrating boxplot analysis with conditional convergence tests, our findings reveal systematic monotonic variation in key determinants across identified convergence paths, underscoring the critical importance of explicitly accounting for structural heterogeneity when evaluating convergence hypotheses. Previous studies (e.g., Mankiw et al. 1992; Kremer et al. 2021) generally found evidence of conditional convergence but did not differentiate among distinct development paths. Correlated variables highlighted in earlier literature, such as population growth, investment rate, human capital, and institutional quality factors (e.g., political rights and rule of law), demonstrate clear monotonicity across development paths, as illustrated by boxplots. Such monotonic patterns imply that differences in these correlated variables might drive or moderate convergence outcomes previously attributed to intrinsic country-specific effects and conditional convergence.

Furthermore, our findings indicate that after differentiating growth paths, the significance of correlated variables largely disappears, especially within middle- and low-income trajectories. Additionally, the conditional convergence evidence documented among high-income paths (e.g., OECD countries) appears sensitive to the chosen sample periods, occurring sporadically rather than consistently across different time frames. Consequently, the limited and inconsistent evidence for conditional convergence suggests that past findings may primarily reflect unrecognized heterogeneity arising from distinct developmental trajectories. In contrast, the three absolute convergence paths we identified previously exhibit statistically significant convergence consistently across all sample periods, highlighting a clear distinction from the unstable and period-sensitive nature of conditional convergence observed in prior studies.

Our analysis shows that the correlated variables that are often used to classify economic clubs are strongly tied to income levels, such that countries with similar variable profiles tend to follow the same growth path. Therefore, the three absolute convergence paths identified can also be viewed through the lens of conditional or club convergence, since countries on each path share comparable conditions. The substantial differences between paths in these variables underscore the role specific conditions can play in shaping convergence, though the precise impact on each path remains unclear. Our results reconcile the conditional and club-convergence perspectives: economies on the same path share similar structural conditions, and convergence within groups is primarily absolute. In this way, our work presents a fresh perspective on the confluence of conditional and club convergence.

4.4 Growth Paths and Transition Countries

In Figure 3, we present a visualization of representative economies that potentially experienced path transition. These transitions include transitions from original growth paths to higher paths, descents to lower paths, and oscillations between initial and alternative paths. The two red dashed lines represent the income level boundary of economies on the middle-income path and the high-income path, and the other one represents the income level boundary of economies on the low-income path and middle-income path. Such details illuminate the intricate dynamics of global development process.

Figure 3:

The economic growth path of transition countries. Notes: In Figure 3, The horizontal coordinates are year, and the vertical coordinates are log GDP per capita. The two red dashed lines represent the income level boundary of economies on the middle-income path and the high-income path, and the other one represents the income level boundary of economies on the low-income path and middle-income path. Different colors represent different clusters. In the same coordinate system, different countries are represented by points with different colors and shapes.

Figure 3. Panel A and Panel B illustrate economies, such as China, Singapore, Malta, and the Republic of Korea, that have moved from their initial economic paths to higher ones. Figure 3. Panel C shows examples like Barbados and South Africa, where economies have shifted from their original paths to lower-income paths. Figure 3. Panel D highlights cases in which economies initially drop to a lower path but later revert to their original trajectory. Figure 3. Panel E depicts economies that briefly transition to a new path and then return to their original one, a pattern often associated with the notion of middle-income and poverty traps. Finally, Figure 3. Panel F presents economies whose growth remains on a single path with no transitions.

The analysis of economies potentially experienced transitions reveals a critical factor that may have distorted earlier convergence assessments. Many growth miracles arise from a leap from one path to another, with the rapid expansion of these economies and transformations in their economic structures as mentioned in the previous study (Rodrik 2012). These observations highlight the dynamic nature of economic growth and the importance of incorporating path transitions in analyzing economic convergence.

Our findings also carry policy implications. Identifying three stable convergence paths offers a practical framework for stage-specific strategies. Institutions such as the World Bank can use this classification to monitor countries’ positions and identify windows for potential transitions. For middle- and low-income economies, targeted interventions (especially in education, human-capital development, and institutional quality (governance, rule of law)) may facilitate upward movement to higher paths. The framework thus serves as both an empirical clarification and a diagnostic tool for policy design.

4.5 Policy Experience and Stage-Specific Interventions

The three-path framework can also be interpreted as a diagnostic device that separates economies that remain close to a given steady-state attractor from those that manage discrete upward transitions or suffer downward collapses. The experience of successful transition economies suggests that path shifts are rarely the outcome of smooth extrapolation of existing trends. Rather, they tend to be associated with deliberate and sometimes discontinuous policy reforms that are implemented on top of a reasonably stable domestic political and social foundation.

Korea illustrates how such preconditions can be combined with an aggressive, export-led industrialization strategy to engineer a sustained upward transition. From the early 1960s, Korean governments used directed credit, temporary protection, and export targets to promote manufacturing, while investing heavily in education and basic infrastructure. Empirical evidence shows that subsequent trade liberalization and exposure to external competition were associated with higher productivity growth in Korean manufacturing, underscoring the importance of an outward-oriented policy mix (Lee 1996). In our classification, Korea moves from the low-income path in the early 1960s to the middle- and then high-income paths as these policies take effect and the industrial structure upgrades. Singapore followed a different but complementary route, combining a strongly interventionist developmental state with radical openness to trade and foreign capital. The government relied on state-linked enterprises, targeted fiscal incentives, and strategic land-use and infrastructure policies to attract multinationals, while maintaining tight macroeconomic management and low political uncertainty (Huff 1995). This pattern helps explain how a very small economy was able to jump directly from the low-income path to the high-income path in our data.

China and Egypt provide modern examples of upward transitions from low- to middle-income paths under broadly similar conditions. In both cases, a period of relative political stability and basic improvement in living standards preceded major economic reforms. Only after domestic settled down and basic food security, education, and health provision had improved did governments implement packages of market-oriented reforms: opening to trade and foreign investment, reforming systems, strengthening legal and regulatory frameworks, and reducing discretionary state intervention in production. In China, the reform and opening process (from rural collectivization and township-and-village enterprises to coastal opening and WTO accession) combined gradual market liberalization with strong state direction of investment and experimentation. Existing work shows that this policy package underpins China’s escape from the low-income path and subsequent movement into the middle-income path in the late twentieth century (Fang et al. 2022, 2024). These reforms, combined with large inflows of foreign capital and technology, allowed China and Egypt to shift away from inward-looking, state-dominated models toward more open, investment- and export-driven growth, which in our classification corresponds to a move from the low-income path to the middle-income path.

Viewed alongside Korea and Singapore, these cases point to a common pattern in successful path transitions. First, all four economies undertook major reforms only after achieving a domestic stability, no ongoing war, a consolidated central government, and a basic social floor for the population. Second, the reforms that triggered transitions were not incremental policy tweaks but broad, outward-oriented packages that combined opening to trade and FDI with institutional changes, greater reliance on rules rather than discretion, clearer property-rights protection, and stronger legal and bureaucratic capacity. Third, these reforms were accompanied by sustained efforts to raise human capital and expand the tradable sector. These elements map naturally into the variables in our empirical analysis (governance, openness, and human-capital differences between paths) suggesting that the structural gaps we document are precisely those that policy must close to make an upward transition feasible.

Failed or incomplete transitions in our data (e.g., Zimbabwe, Cameroon, and Djibouti) highlight how fragile upward shifts can be when reforms rest on a weak institutional and social base. When major liberalization or adjustment is launched before basic political stability, social cohesion, and administrative capacity are in place, reforms tend to trigger distributional conflict, undermine state legitimacy, and interrupt convergence. Wars, civil conflict, and large commodity shocks can similarly destroy the fiscal and productive foundations needed to remain on a higher path, especially when governments respond with ad hoc measures rather than institution-building and diversification.

Based on our framework, above experiences suggest clear stage-specific priorities for policymakers. For economies still on the low-income path, the precondition for any successful transition is to secure basic political stability and a minimal social floor while building administrative capacity, rather than rushing into broad liberalization. For middle-income economies, the priority shifts to closing the structural gaps we document (e.g., governance, openness, and human capital) through outward-oriented reforms that deepen integration into global markets, strengthen legal and bureaucratic capacity, and support upgrading in the tradable sector. Even for economies that have already moved onto the high-income path, the negative cases in our data warn that inadequate institutions, unmanaged shocks, or poorly sequenced reforms can undo earlier gains, underscoring that sustaining an upward path is as much a policy challenge as achieving the transition itself.