The study examines six BRICS member countries: Brazil, Russia, India, China, the South African Republic, and Indonesia. These countries were selected because data was available for analysis. If sufficient data become available on the countries that joined BRICS in 2024 — Iran, Egypt, Ethiopia, and the UAE — the study may be expanded to include them.

Data on control countries were added to the dataset to identify their homogeneity. Researchers have previously used a similar technique (Artis & Zhang, 1998, 2001, 2002; Ozer & Özcan, 2007, 2008a, 2008b; Sarlin, 2011), in which control countries such as Canada and Japan — and to some extent the United States, which are not members of the European Monetary Union (EMU) — were used to calibrate clustering models. In this study, rather than using control countries to calibrate the level of fuzziness in the clustering, we use them to challenge the distribution of the BRICS countries, allowing BRICS economies to potentially cluster with control economies instead of with each other. In other words, the study examines whether BRICS economies are more homogeneous in relation to each other than to external economies. Thus, control countries are chosen to represent not only non-BRICS economies but also different regions and economic characteristics. In line with previous studies, the United States, Japan and Canada — all of which are developed market economies — are included. To broaden the scope, two emerging or developing market economies were added: Colombia and Turkey, which has applied for BRICS membership. Spain — a developed market country and a member of the EMU — was included to represent the EMU.

This study aims to apply FCM clustering models with different numbers of clusters to observe whether all BRICS countries, or specific subsets of them, consistently form a single cluster. A key focus is to examine whether this grouping persists as the number of clusters increases. The two-cluster model is the most informative scenario and serves as a baseline test of homogeneity. If all the BRICS countries are grouped together in one of the two clusters, this would suggest a significant level of internal similarity and potential distinction from the control group. However, increasing the number of clusters also yields important information. It enables the assessment of whether the BRICS grouping remains intact or begins to fragment, which helps to gauge the robustness of their homogeneity under more granular classification. The primary aim is to determine whether these countries — or a subset of them — consistently form a coherent group, regardless of which non-BRICS economies may also be included in the same cluster. At the same time, it would be useful to find out whether the BRICS countries cluster distinctly from the rest or tend to align with certain external economies.

An important advantage of the FCM method is its ability to provide not only categorical cluster assignments but also quantitative estimates of each country’s degree of association with each cluster, as expressed through membership grades. The algorithm is run for 1000 iterations. For the purposes of this analysis, each country is assigned to the cluster in which it has the highest membership grade, even if that grade is below 0.5.

It is important that in this study a cluster does not represent an actual currency area, but rather a group of countries that exhibit homogeneity with respect to OCA criteria. Including countries within the same cluster means that they are at a similar level of readiness for forming a currency union, whether high or low. This does not imply that these countries are necessarily prepared to establish such a union. In fact, a cluster may consist of countries that have a low level of alignment with OCA standards, indicating limited feasibility for currency integration despite their similarity.

In this context, the concept of a central or “reference country” becomes particularly important. In earlier studies on the EMU, most OCA-related indicators were calculated relative to Germany, which served as the economic benchmark due to its size and level of development. Similarly, within the BRICS framework, China as the largest and most economically advanced member might naturally assume a comparable role. However, when a clustering model is fitted using China as the sole reference country, the results reflect the homogeneity of the other BRICS economies in relation to China but exclude China itself from the comparison. Our objectives, however, involve answering the question: is China homogeneous with the other BRICS countries? This is why the reference country is rotated sequentially, with factors being calculated for each of the other countries — including China — in relation to the new reference country. In this paper, the term “reference country” does not imply a central role in terms of monetary leadership or anchor status; it simply denotes the country relative to which indicators are computed. This approach allows us to assess whether China’s level of readiness for a currency area is comparable to that of other economies. Although cluster distribution shifts with a change in the reference country, the tendency shows whether homogeneity is sustained.

In their paper ‘The Endogeneity of the Optimum Currency Area Criteria’ (1996), Frankel and Rose argue that international trade and the correlation of business cycles between countries are endogenous. This implies that economic integration, particularly a monetary union, can enhance the synchronization of business cycles and trade between countries. They conclude that assessing the ability of economies to form an OCA based solely on historical data may be misleading, as these indicators may change once a currency union has been formed. The economies that could not form an OCA according to the criteria prior to unification converge upon joining the currency union. As a result, the area formed can be considered optimum.

Without disputing the endogeneity of criteria or inappropriateness of analyzing historical data in this article, we note that the results of our study are still useful for understanding the initial state of economies before unification. Even if we assume that economies would begin to converge after the decision to form a monetary union and the adoption of appropriate measures, it is useful to imagine the path that participating countries would need to follow before forming an optimum area.

Mundell (1961) established the basis of the theory of OCAs, proposing that an OCA was founded on the mobility of production factors within the currency area, subject to flexible prices and wages. MacKinnon (1963) observed that the benefits from unification within a currency area increased with the degree of economic openness. Several other criteria were introduced later, including diversification of production and exports, fiscal integration (Kenen, 1981/1969). In addition to these, financial and political integration and the similarity of inflation rates were added to the conditions for an OCA, and later, when the theory of optimum areas was re-evaluated, researchers singled out the similarity of shocks as an important criterion (Mongelli, 2002). The theory gained renewed popularity in the late 1980s and 1990s amid discussions about a common European currency, a topic that Paul Krugman addressed in 1990. As Mongelli (2002) notes, empirical studies in the field of OCA began to appear during this period, examining EMU data and employing econometric and other mathematical methods.

In a series of papers, Artis and Zhang (1998, 2001, 2002) employed the FCM method to address questions concerning the EMU countries’ classification as either “core” or “peripheral” within the monetary union. For FCM clustering, they used quantitative variables reflecting a number of the main conditions of the OCA. Almost all of these factors reflected the proximity of each country to Germany. The authors designated Germany as the ‘central’ country, grouping countries based on how close they were to it according to a number of features. These features included synchronization of the business cycles of each country and Germany; volatility of the real exchange rate of each country’s currency and the German mark (in later calculations and other papers, the euro was used); synchronization of the real interest rates of each country and Germany; openness to trade with Germany; and employment protection legislation ratings calculated independently for each country.

There were some changes in the calculation and use of criteria in the studies of other authors who used the same methodology. For example, Ozer and Özcan (2007) and Sarlin (2011) omitted the rating of labour legislation. Boreiko (2002) omitted both the rating of labour legislation and the synchronization of real interest rates. The indicator of trade openness was calculated relative to the EMU countries (Boreiko, 2002) or EU25 (Ozer & Ozkan, 2007; Sarlin, 2011) rather than Germany. Ozer and Özcan (2007) used different filters to identify the seasonal components necessary for calculating business cycle similarity, and also slightly transformed the correlation coefficients.

The present study focuses on factors that correspond to the key criteria for an optimum currency area (OCA) as identified in the literature. These criteria can be interpreted and measured using mathematical and statistical methods. The symmetry of economic shocks is proxied by the synchronization of business cycles. The similarity of monetary policies, which reflects the cost of relinquishing independent national policy, is assessed through similarities in exchange and interest rates. Inflation convergence is measured by the absolute differences in national inflation rates. Labor market flexibility and labor mobility are approximated using the labor market regulations index. The export product concentration index captures export diversification. Further studies could expand the set of factors. The OCA criteria and the methodology for calculating them in the current study are described in more detail below.

1. The synchronization of the business cycles of the country in question and the reference country was calculated using the correlation of the seasonal components of the monthly intra-annual data on industrial production. Seasonal components are extracted from monthly data on industrial production in absolute values ​​(thousands of US dollars, the base year is 2005) by means of the Hodrick-Prescott filter (H-P filter). Artis and Zhang (2001) point out that this factor is included to assess the symmetry of shocks of two economies. Data for this study are taken from the Global Economic Monitor database, World Bank Databank (World Bank, n.d.-a, GEM Data). Application of the Hodrick-Prescott filter was made by means of the “hpfilter” function of the “statsmodels” package version 0.14.0 for Python version 3.9.7. The documentation recommends a lambda value of 129,600 for monthly data, referencing a detailed study of this issue by Ravn and Uhlig (2001).

2. The volatility of the real exchange rate between the currency of the researched country and that of the reference country is calculated as the standard deviation of the real exchange rate index, expressed in reverse quotation (i.e., units of the reference currency per unit of the country’s currency). While some authors calculate this indicator using the standard deviation of the logarithmic differences in real exchange rates, this study adopts the methodology used by the Central Bank of the Russian Federation (Central Bank of the Russian Federation, n.d.). The nominal cross-rates between the researched currencies and the reference currency were derived from the official average monthly exchange rates against the US dollar, sourced from the Global Economic Monitor (World Bank, n.d.-a, GEM Data). The real exchange rate indices were computed using monthly year-on-year consumer price indices from the same source. The economic rationale for including the real exchange rate in the dataset is grounded in the assumption that countries with similar monetary policies — affecting both nominal and real exchange rates — will experience smaller bilateral exchange rate fluctuations. Accordingly, the lower the volatility of a country’s exchange rate relative to that of the reference country, the lower the expected cost of transitioning to a shared currency (Artis & Zhang, 2001).

3. The synchronization of the real interest rates of the countries under consideration and the reference country is calculated as the correlation of their respective seasonal components. These components are calculated as the difference between the short-term interest rates and the year-on-year consumer price index. The economic rationale also rests on the assumption that if two countries pursue relatively similar monetary policies, their intra-annual interest rate movements will exhibit comparable patterns. The greater the similarity in these movements, the closer the alignment of their monetary policies — and consequently, the lower the expected cost of transitioning to a shared currency (Artis & Zhang, 2001). Data on consumer price indices are sourced from the Global Economic Monitor (World Bank, n.d.-a, GEM Data), and data on short-term rates — from the OECD database (OECD, n.d., Data). Minor gaps in the time series for Russia and the United States are addressed using linear interpolation. In cases where short-term interest rate data are unavailable — specifically for Brazil and Turkey — substitute data on “instant rates” are used.

4. Trade openness (also trade integration in other studies) is calculated using the formula

$\frac{(x_i^{BRICS}+m_i^{BRICS})}{(x_i+m_i)}$, (1)

where x_i and m_i are the total exports and total imports of country i (under consideration) for the period under consideration, $x_i^{BRICS}$ and $m_i^{BRICS}$ are the total exports and total imports of country i to and from the BRICS group of countries. It is important to emphasize that, in this case, the BRICS group comprises ten of its members. Although five of the countries included in the analysis were not BRICS members during the study period, this research assumes that if the BRICS or control countries maintained a trade relationship with them, those interactions should be included in the analysis. Trade integration — or trade openness, as it is sometimes referred to in the literature — has been identified as a key criterion for optimum currency areas (OCAs) by both McKinnon (1963) and Krugman (1990). Data are sourced from the World Bank’s WITS database (World Bank, n.d.-c, WITS Data by countries).

5. Inflation convergence is calculated as the average difference between a country’s inflation rate and the inflation rate in a reference country over a given period. Annual inflation data are sourced from the World Development Indicators database, World Bank (World Bank, n.d.-b, World Development Indicators Data). The convergence of inflation rates with those in the reference country demonstrates the extent to which the two countries are synchronized in their efforts to achieve their targeted inflation rates. Quah (2016) in his analysis calculates this parameter slightly differently, not simply as the difference in inflation rates, but as the modulus of the difference. This approach is logically consistent: if the primary concern is how close the inflation rates are to those of the reference country, disregarding the direction of deviation by using the absolute value may seem appropriate. However, this method can produce misleading results. For example, a country with inflation significantly lower than the reference rate could end up in the same cluster as a country with significantly higher inflation, simply because the magnitude of the deviation is similar. In such cases, the actual difference in inflation between these countries may be substantial. We argue that it is important to consider the direction of inflation rate deviation relative to the reference country. For countries with relatively low inflation, abandoning an effective inflation-targeting regime may result in higher economic costs and so face greater political opposition than in countries with higher inflation. This asymmetry is analytically significant and should be reflected in the evaluation. Accordingly, this study measures inflation convergence as the average difference in inflation rates rather than using the absolute value.

6. The labor market regulations index, a composite taken from Fraser Institute data as part of the Index of Economic Freedom, reflects the flexibility of the labor market, measured from 0 to 10; the closer the index to 10, the less regulated the labor market is (Fraser Institute, 2023, Data by countries). Artis and Zhang (2001) use an analog of this ranking from the OECD as a proxy for labor market flexibility. They also mention that, according to some studies, it is not labor mobility itself that is important, but rather the difference between inter-regional labor mobility within a country and international labor mobility within a region. It should be noted that, while they were able to argue that low labor mobility between European countries is similar to low inter-regional labor mobility within these countries, we cannot claim the same for the BRICS countries. Even if the assumption of low labor mobility within the BRICS countries is correct, there are strong reasons to believe that it will still be significantly higher than intra-country labor mobility within the BRICS group. This group brings together countries whose populations speak languages that differ greatly from one another and often belong to different language families. These language differences will pose a significant obstacle to labor mobility, even when legal regulations are not taken into account. However, it is precisely this final factor that is addressed by the Fraser Institute’s labor market regulations index. The general similarity of the institutions affecting labor market flexibility can help evaluate the extent to which the labor markets of the countries studied could converge, at least on this parameter.

7. The export product concentration index measures a country’s export concentration, which is the opposite of diversification and can therefore be used to determine how similar countries’ export diversification is. This factor was not included in the original series of papers by Artis and Zhang, Ozer and Özcan, and Sarlin. However, it was used by Nguyen (2007) and Quah (2016), who refer to Kenen (1981/1969), who introduced the criterion. Although Quah calculated the export diversification index as the sum of squared shares of export product types, we use the export product concentration index as defined by UNCTAD (UNCTAD, n.d.-a, Data). It is calculated as

$H_j=\frac{\sqrt{\sum _{i=1}^N{\left(\frac{X_{i,j}}{X_j}\right)}^2}-\sqrt{\frac{1}{N}}}{1-\sqrt{\frac{1}{N}}}$, (2)

where H_j is the product concentration index of exports for country j, X_i,j is the value of exports of product i by country j, X_j is the total value of exports of country j, and N is the number of products exported at the three-digit level of the SITC Revision 3 (UNCTAD, 2019). We use the UNCTAD concentration index rather than the UNCTAD export diversification index because the latter does not represent the variety of exports, but the absolute deviation of a country’s trade structure from the global trade structure. (UNCTAD, n.d.-b, Indicator explanation).

All the calculated factors are standardized before the FCM algorithm is fitted. The factors based on the OCA criteria calculated in reference to China before standard scaling are presented in Table 1. Factors before scaling in reference to other BRICS economies can be found in Appendices 1–5.

The Fuzzy C-Means (FCM) clustering method is a clustering method in which each element (in our case, countries) of a multidimensional space (whose dimensions are factors; in our case, the seven OCA criteria) is assigned a number between zero and one for each of the initially specified clusters. This is known as the membership grade, or the degree to which a country belongs to a cluster. Hereafter, the membership grade will be defined as the degree to which the j-th country belongs to the i-th cluster, and will be designated as u_ij. In summary, the membership values for each element in clusters are all equal to one.

The FCM algorithm can be described as follows: each element (country) is a vector x_j = (x_j1, ... x_jk) in a k-dimensional space, where k is the number of factors (in our case, seven). The distance between elements (vectors), for example, x_a and x_b, is calculated as the Euclidean (in this study):

$d(x_a,x_b)=\parallel x_a-x_b\parallel =\sqrt{\sum _{p=1}^k{(x_{ap}-x_{bp})}^2}=\sqrt{{(x_a-x_b)}^T(x_a-x_b)}$ (3.1)

Like many clustering algorithms, FCM begins with a certain or random initial disposition — in this case, a membership matrix U (of size c × N, c is the number of clusters, N is the number of elements, i.e. countries) and cluster centers v_i , i = 1, ... c, where v_i is the vector of the center of the i-th cluster (or the i-th centroid). The number of clusters c, as in the case of the classical k-means method, is set a priori and does not change during the course of the algorithm. What changes — by being specified — are the cluster centers and the membership grades of elements in clusters. This occurs due to the minimization of the objective function, which in our case of Euclidean distances and norms appears as in formula (3.2) (Bezdek et al., 1984):

$J_m(U,v)=\sum _{j=1}^N\sum _{i=1}^c{\left(u_{ij}\right)}^m{\parallel x_j-v_i\parallel }^2$, (3.2)

where u_ij is the grade of membership of the j-th element to the i-th cluster, u_ij Î [0,1], U = [u_ij] is the membership matrix c × N, c is the number of clusters, N is the number of elements (countries), x_j is the vector of the j-th element (country), v_i is the vector of the i-th centroid, m is a so-called “weighting exponent” (Bezdek et al., 1984), or a level of fuzziness (Ozer & Ozkan, 2007, 2008a, 2008b), the measure of the cluster borders imprecision, m Î [1,+¥).

Local minimization of the objective function J_m (U, v) for m > 1 occurs by passing it $(\hat{U},\hat{v})$, which are calculated using the formulas:

${\hat{v}}_i=\frac{\sum _{j=1}^N{\left(u_{ij}\right)}^m{x}_j}{\sum _{j=1}^N{\left(u_{ij}\right)}^m},1\le i\le c$ (3.3)

$\widehat{u_{ij}}={\left(\sum _{p=1}^c{\left(\frac{\parallel x_j-{\hat{v}}_i\parallel }{\parallel x_j-{\hat{v}}_p\parallel }\right)}^{2/m-1}\right)}^{-1},1\le j\le N,1\le i\le c$ (3.4)

At the beginning of the algorithm, we determine the number of clusters c, the value m, and the method for calculating the distance between elements (in this study, Euclidean). We select some specific or random initial matrix U ^(q) q = 0, 1, … ITERMAX and centroids. Then we move the centroids to new points v ^{(q + 1)}, calculated by formula (3.3), recalculate the membership values in the matrix U ^{(q + 1)} by formula (3.4), and check whether the objective function (3.2) has decreased. If it has decreased, we shift the clusters again; if not, or if the number of iterations that we have planned for the algorithm has ended, we stop the algorithm and examine the membership matrix.

The measure m varies from 1 to infinity. It can be adjusted, but the default value is 2 (Ozer & Ozkan, 2008a). The question of the optimal number of clusters also arises for researchers who use the clustering algorithm. Ozer and Özcan (2007, 2008a) selected control countries that were not part of the EMU in order to adjust the number of clusters and the value of m, ensuring that they would fall into a distinct cluster. This is not an option for this study as we are not sure that the BRICS countries are homogeneous enough to form a currency area. Therefore, including or excluding the BRICS countries from the group of control countries will not indicate the optimal number of clusters or correctly selected coefficient m. Control countries are still useful for our study as they help us track the behavior of the BRICS countries when the number of clusters changes and determine whether these countries are homogeneous in terms of OCA criteria when the number of clusters increases. The development of a methodology to determine the optimal value of m is a topic for separate research. In this study, this value is set to 2.

The study using the C-means fuzzy clustering method was performed using the skfuzzy package version 0.4.2 for Python version 3.9.7. The random state was 42.