Equity markets’ clustering and the global financial crisis

The opinions and statements in this article are the sole responsibility of the authors, and do not represent neither those of Banco de la República nor The Bank of Korea.

Abstract

The effect of the Global Financial Crisis (GFC) has been substantial across markets and countries worldwide. We examine how the GFC has changed the way equity markets group together based on the similarity of stock indices’ daily returns. Our examination is based on agglomerative clustering methods, which yield a hierarchical structure that represents how stock markets relate to each other based on their cross-section similarity. Main results show that both hierarchical structures, before and after the GFC, are readily interpretable, and indicate that geographical factors dominate the hierarchy. The main features of equity markets’ hierarchical structure agree with most stylized facts reported in related literature. The most noticeable change after the GFC is an increase in (geographical) clustering. However, the increase in clusters’ compactness and the decrease in clusters’ separateness point out that world equity markets became more interconnected after the GFC. Some changes in the hierarchy that do not conform to geographical clustering are explained by well-known idiosyncratic features or shocks.

Keywords:

• Clustering
• Unsupervised learning
• Stock market
• Connectedness

JEL Classification:

• C38
• L22
• G15

Acknowledgements

We are thankful to Peter Sarlin and the two reviewers for their contributive comments and suggestions. We are grateful to Hernando Vargas, Pamela Cardozo, Clara Machado, Alejandro Reveiz, Esteban Gómez, Sandra Benítez, Sebastián Rojas, Daniel Osorio, Fredy Gamboa, José Fernando Moreno, Andrés Ortiz, Brian Lesmes and Diego Roa for their comments to an early version of the article. Any remaining errors are the authors’ own.

Notes

The opinions and statements in this article are the sole responsibility of the authors, and do not represent neither those of Banco de la República nor The Bank of Korea.

¹ For instance, interconnectedness is one of the five factors commonly used to assess systemic importance, as suggested by BCBS (2013). Furthermore, non-substitutability is another systemic importance factor quite related to interconnectedness. Different higher loss absorbency requirements (i.e. an additional buffer in the form of common equity) will be imposed based on systemic importance to further reduce the probability of failure of systemically financial institutions.

² There are other methods beyond the three reported here, such as planar maximally filtered graphs (see Tumminello et al. 2005) or clique percolation (see IMF 2012). An exhaustive revision of related methods is not intended in our article.

³ Mantegna (1999) is credited for first studying the hierarchical structure of financial data (i.e. the US stock market) by means of minimal spanning trees. Afterwards, other markets have been studied by means of minimal spanning trees, such as foreign exchange markets (see Mizuno et al. 2006, Naylor et al. 2007), and credit default swaps (see Marsh and Stevens 2003, León et al. 2014).

⁴ Studying the hierarchical structure of financial data by means of asset graphs is less common than by minimal spanning trees. To the best of our knowledge, Onnela et al. (2003) introduces asset graphs for examining the US stock market.

⁵ As in Martínez and Martínez (2008), exploratory data analysis is the philosophy that data should first be explored without assumptions for the purpose of discovering what they can tell us about the phenomena we are investigating; it is a collection of techniques for revealing information about the data, and methods for visualizing them, to see what they can tell us about the underlying process that generated it.

⁶ Other—more complex- clustering methods are available (e.g. fuzzy clustering, model-based clustering, spectral clustering). These other methods are described in Martínez and Martínez (2008), Kolaczyk (2009) and Martínez et al. (2011).

⁷ Divisive clustering methods exist as well. Unlike agglomerative ones (i.e. bottom-up), divisive starts with a single group containing all observations and successively split the groups until there are m groups with one observation per group (i.e. top-down). As reported by Martínez and Martínez (2008), divisive methods are less common.

⁸ Euclidean distance is the most often used for continuous data because of its simplicity and interpretability as a physical distance. However, other measures of distance exist as well (see Martínez and Martínez 2008, Everitt et al. 2011), including some transformations of the correlation coefficient.

⁹ Using correlation not only requires making an assumption about the normal distribution of returns, but also may be misleading due to the positive bias in estimated correlation coefficients introduced by volatility (see Forbes and Rigobon 2002). Hence, distances based on correlation may be biased downwards with market volatility, and comparisons between different periods (with different volatilities) may be misleading.

¹⁰ This is evident in Diagram 2. As long as the closest (farthest) elements in each cluster are preserved, single (complete) linkage method would yield the same distance between clusters irrespective of the organization of the remaining elements. On the other hand, changes in the organization of the remaining elements in average and centroid linkage methods affect the distance between clusters to some extent.

¹¹ To the best of our knowledge, there are no empirical studies that examine which linkage method is better for our case (i.e. financial time series). However, consistent with results obtained with the Calinski and Harabasz index, unrelated empirical studies tend to favour Ward’s linkage method (see Milligan and Cooper 1987, Ferreira and Hitchcock 2009, Everitt et al. 2011, Hossen et al. 2015). Dendrograms obtained with other methods (available upon request) are not easily interpretable as they do not produce clear clusters. However, visual inspection reveals that they do not contradict the results attained with Ward’s method.

¹² Marginal changes to the size of the samples have no material effect on the resulting hierarchy, or on the conclusions.

¹³ Several methods have been used to deal with the time zone problem (see Olbrys 2013). Besides the rolling average two-day returns (Forbes and Rigobon 2002), some of them switch to another frequency (e.g. weekly or monthly data), or take a certain hour in a leader market to register the quotes of all stock markets in the sample, whereas others use specific data-matching procedures based on opening and closing prices. Lagging indices based on the second eigenvector of the distance matrix are also possible (see Sandoval 2013).

¹⁴ For instance, it is feasible that results for Panamá and Costa Rica are driven by their features as small open Central American economies with representative services sectors (e.g. tourism, financial, transport). Moreover, the lack of other stock indices from small open Central American countries may also affect the results. In the case of Venezuela, it is reasonable to conjecture that government’s particular economic stance and investors’ risk perception may be affecting the results. For instance, Conti and Gibert (2012) report that the Venezuelan stock market is small and closed, in which government policies and the presence of public funds in the listed companies give a special connotation to this stock market.

¹⁵ It has been documented that spillovers from China’s stock market volatility have been significant for other Asian economies during 2015 (see Guimaraes-Filho and Hong 2016). Thus, it is arguable that China has increasingly approximated its regional cluster after the GFC.

¹⁶ The integration process amid MILA is of a virtual nature; there are no corporate changes (e.g. merge or acquisition), but an integration based on technological tools and regulatory standardization. The first phase of this integration process (including Chile, Colombia and Peru) started on 8 September 2009, but it was only until the end of May 2011 that MILA formally started operations (see Mellado and Escobari 2015). On December 2014, the entry of Mexico became official.