AN APPLICATION OF MINIMAL SPANNING TREES AND HIERARCHICAL TREES TO THE STUDY OF LATIN AMERICAN EXCHANGE RATES

. This paper analyzes a group of nine Latin American currencies with the aim of identifying clusters of exchange rates with similar co-movements. In this work the study of currency relationships is formulated as a network problem, where each currency is represented as a node and the relationship between each pair of currencies as a link. The paper combines two methods, Symbolic Time Series Analysis (STSA) and a clustering method based on the Minimal Spanning Tree (MST), from which we obtain a Hierarchical Tree (HT). Symbolic Time Series Analysis consists in the transformation of a given time series into a symbolic sequence with the aim of identifying patterns in the set of data. The Minimal Spanning Tree condenses the core information on the global structure of the network and its main advantage is that it greatly simpliﬁes comparisons by dramatically reducing the number of elements that must be compared. We identify two main clusters in the currency network, as well as speciﬁc currencies that function as transmission channels between clusters. Using data regarding the degree of ﬁnancial liberalization, as well as the distinction between inﬂation targeting (IT) and non-IT countries, the analysis suggests that the obtained taxonomy is economically relevant.


1.
Introduction. Throughout recent years, research on exchange rates has focused on the analysis of co-movements, particularly in East Asia around the Renminbi or in other regions with the US dollar or the Euro as the main anchors [20,21,22,23,26,45]. However, these studies have bypassed the analysis of co-movemets among Latin American currencies. In this work we address this issue through a non-parametric approach proposed by [4,6] and [40]. Following this methodology, our aim is to identify clusters of co-movements among Latin American currencies. Using symbolic sequences, we introduce a notion of distance between the dynamical paths of different exchange rates. Then, a hierarchical tree is constructed with the distances between time series. This hierarchical tree corresponds to groups of exchange rates sharing similar performance.
The paper combines two methods, Symbolic Time Series Analysis and a clustering algorithm based on the Minimal Spanning Tree (MST), from which we obtain a Hierarchical Tree (HT). Symbolic Time Series Analysis consists in the transformation of a given time series into a symbolic sequence with the aim of identifying patterns in the set of data. On the other hand, given a network of elements, the MST extracts its most relevant information and arranges it as a graph, whereas the HT utilizes the MST as an input to determine if hubs exist inside the analyzed network.
The clustering process used to form the MST is known in cluster analysis as the single-linkage clustering method, also known as the nearest-neighbor technique. This method is the simplest of an important group of clustering methods known as agglomerative hierarchical clustering method [30]. One of the main advantages of the minimal spanning tree is that it greatly simplifies comparisons by dramatically reducing the number of elements that must be compared directly. This is because spanning trees provide the underlying structure of the analyzed network [27]. It is worth mentioning that, given the fact that the MST selects the strongest couplings among network nodes, it condenses the core information on the global structure of the network. Thus, the overall characteristics of the network can be reproduced in the MST [34].
This combination of Symbolic Time Series and clustering methodologies has been applied by Brida et al [4,6,7]. In this work we utilize a notion of distance based on symbolic methods. In a previous paper, [10], we develop a mathematical model to show how symbolic dynamics can be used to describe the evolution of the exchange rates. In that work we describe the dynamic of exchange rates as a sequence of two regimes in which the state space is divided by a certain threshold. We apply those ideas in the present work, where the symbolized time series data are grouped into different clusters. The study considers the exchange rates of nine Latin American countries: Argentina, Brazil, Chile, Colombia, Honduras, Mexico, Peru, Uruguay and Venezuela and utilizes daily data, from 16 January 2007 through 29 December 2017.
One point to consider is whether the identified clusters are related to the degree of financial openness and whether they can also be associated with the presence or absence of inflation targeting (IT) regimes. The reason for this is that the degree of openness of the capital account is closely linked to the volatility of the financial flows [1], and these in turn influence the dynamics of the exchange rate. It could be expected that countries with similar levels of financial liberalization would show similar levels of volatility in their capital flows and that, consequently, the movements of their exchange rates would show similarity. Financial openness intensifies linkages that, in a context of growing capital mobility, may increase the risk of cross-border contagion [47].
As the financial ties are strengthened, exchange rate stability might be necessary for investors seeking to promote their financial activities by exporting short-and long-term capital flows [31]. Moreover, the level of financial openness can moderate the risks associated to exchange rate fluctuations by providing access to hedging instruments [16]. Following [2] and [15], it might be expected, ceteris paribus, that central banks in countries with a similar degree of financial liberalization would have similar margins of maneuver to intervene in the exchange market.
On the other hand, it is also relevant to ask whether the clusters maintain a relationship with the presence or absence of IT regimes. The reason is the following. Inside an IT regime, monetary policy aims at keeping the inflation rate at a certain level. In order to do so, central banks use the interest rate as the main instrument of monetary policy. In this regime, central banks do not intervene in the face of appreciations of the exchange rate, since this helps to contain the inflation rate. This creates a bias towards an appreciated exchange rate [29].
Thus, under inflation targeting regimes, central banks in Latin America would be more prone to let their currency appreciate against the USD in order to reach price stability. As a result, this monetary policy framework should augment currency co-movements during episodes of appreciation [31].
Alternatively, when facing depreciations, central banks increase their interest rate to prevent the exchange rate from affecting inflation through the pass-through effect. This increase might modify the interest rate differentials, which, in turn, might affect exchange rate correlation dynamics in inflation-targeting countries [36].
In non-IT regimes, particularly with the presence of an anchor, central banks buy or sell currencies in order to maintain the exchange rate at a certain predetermined level or margin. Thus, the distinction between IT and non-IT would imply different types of response to fluctuations in the exchange rate.
The remainder of this paper is organized as follows. In section 2 the literature referring to the analysis of currency markets from a network theory and complex systems perspective is reviewed. In particular, we review how the Minimal Spanning Trees (MST), Hierarchical Trees (HT) and Symbolic Time Series Analysis (STSA) have been used to study relationships between currencies. In section 3, the methodology of this paper is presented. Section 4 introduces the data, whereas section 5 presents and discusses the empirical results. Finally, section 6 draws the conclusions.
2. Literature review. The Minimal Spanning Tree graphs were introduced by Kruskal in graph theory [33], but they started to be utilized in the realm of Econophysics due to the work of Mantegna [38], who utilized Minimal Spanning Trees (MST) to analyze stock market correlations. The clustering process used to form the MST is known in cluster analysis as the single-linkage clustering method, also known as the nearest-neighbor technique. One of the main advantages of the minimal spanning tree is that it greatly simplifies comparisons by reducing the number of elements that must be compared directly. Additionally, since MST selects the strongest relationships among network nodes, it condenses the core information on the global structure of the network. Thus, the general characteristics of the network can be reproduced in the MST [27,34].
The use of MST in currency markets is relatively recent [18,24,25,32,35,41,42], although generally these studies have bypassed the question of co-movements among Latin American currencies. A spanning tree links vertices (in our case currencies) in such a way that there is exactly one path between any pair of vertices. An edge connecting two vertices in a spanning tree denotes a relationship between those two currencies.
Currency markets are complex systems composed of many interacting elements that exhibit numerous forms of emergent collective dynamics [3,19]. These systems cannot be explained by studying the constituent parts in isolation, but by considering the interrelation with the other elements of this system. Following the ideas of [38,39,41], the study of currency relationships can be formulated as a network problem, where each currency would be represented as a node (also known as vertex), and the relationship between each pair of currencies as a link (also known as edge). Thus, currency markets would be a complex network of mutually interacting nodes and each link would represent the exchange rate of two currencies. Currencies are expressed in terms of a given currency that is called the base currency. This approach is consistent with the paradigm of irregularity, where irregularity is the general case, whereas regularity is the rare property of time series [4].
On this regard, [41] develop a network analysis of the foreign exchange market (FX). They show that clustering techniques, particularly the minimal spanning trees (MST), provide a meaningful representation of the global FX dynamics. They analyze hourly data in a period of two years, 1993-1994, and find that there are links (i.e. correlations between a pair of currencies) which last over the entire twoyear period. According to the authors, this result implies that a relatively robust structure in the FX markets exists.
In the same line, [37] study correlations of exchange rate volatility in the FX market. The authors point out that the currency market is a complex system, where each currency corresponds to an interacting element in the network. The paper introduces correlation-based network methods to build a minimal spanning tree to detect the correlation relations between elements within the FX network. The analysis shows that currency co-movements in the same region are highly correlated. Although the study focuses on the European and the East Asian currency modules in the FX network, the authors find that when the effect of the US Dollar and Euro is removed, the Mexican Peso has influence on the resulting subnetwork (i.e. without USD and EUR).
In [44], the authors apply the MST method to analyze correlations in a network composed by European and Asiatic currencies. The authors suggest that the construction of minimal spanning trees can reflect not only economic relations but also possible speculations in the currency market. Thus, currency networks would also imply information networks. On this regard, [19] employ a dynamical clustering to develop a network analysis of the FX market and suggest that currencies that are located on the edges of communities, i.e. currencies that might be clustered in more than one group or bloc of currencies, are key for information transfer in the FX market. Although the authors do not include Latin American currencies in the analysis, they find that the Mexican peso crisis led to important changes in the community relationships of the studied currencies.
In [34] the authors study the currency network based on daily exchange rates of 46 currencies in the interval from 1998 to 2008. One of the main findings is that during this period the USD was gradually losing its centrality in the network, whereas the Euro became slightly more central than in its early years. The MST shows that, when the currency base is the Dollar, the currencies of Peru, Mexico, Chile, Colombia and Brazil were part of the same cluster and that the Chilean peso occupied the central position in this cluster.
The daily FX rates of 44 currencies are analyzed in [46]. By examining three minimal spanning trees (MSTs) from the period of 2007 to 2012, the study confirms the USD and EUR as the predominant world currencies. Additionally, the results show that the Middle East cluster is very stable, while the Asian cluster and the Latin America cluster are not stable. On this regard, the outcome also shows that the Mexican Peso, Colombian Peso, Chilean Peso and Brazilian Real tended to be clustered, whereas the positions of the Peruvian Sol, Argentinian Peso and Venezuelan Bolivar in the currency network were more dispersed.
In [43] the minimal spanning tree and the ultrametric hierarchical tree are used to extract a topological map for a network of 44 currencies during the years 1995-2001. The resulting taxonomy is meaningful and shows, in the case of Latin America, that Brazil and Chile share relevant co-movements. In [40] the authors examine a set of 28 real exchange rates of developed and developing countries by means of the minimal spanning tree (MST) and the ultrametric distance. The study utilizes monthly data in the period 1992-2002 and identifies a hierarchical structure of the FX network, where the European Union, Asian group and Latin American countries emerge as three different real exchange rate areas. However, the study also shows that among the three big clusters, Latin America seems to be a more disconnected region, which would imply that it is not a homogeneous group, although Brazil plays a central position in the exchange rate dynamics of the region, particularly for Chile, Peru and Colombia, whereas Chile has diffused connections with other regions.
Additionally, [4] introduce a new method to describe dynamical patterns of the real exchange rate movements and to analyze contagion in currency crisis. The methodology combines the Minimal Spanning Tree with the methods of Symbolic Time Series Analysis. The authors find that the Spanning Tree and the associated Hierarchical Tree are useful as a theoretical description of the currency markets, showing the most narrowly connected countries and those who seem to be more distant. The analysis focuses mainly on clusters of countries in Europe and Asia, although they find that, in Latin America, Brazil and Chile could be generators of currency crises because they became centre of all the regional links in crises periods.
Regarding assets, [7] utilize this method to analyze the Euro Stoxx market during the period from 2002 to 2014. The paper shows that the combination of both, the Minimal Spanning Tree and the methodology of Symbolic Time Series Analysis, offers interesting insights into the understanding of the hierarchical organization of the Euro Stoxx market. In this article, the volume and prices of assets are used jointly to analyze the structure of the market, whereas the symbolization methods are applied to characterize the market behavior in normal and critical situations. Among their findings, the authors show that there is an increase in synchronization and connectivity in the Euro Stoxx 50 market during the global crisis. Additionally, during this period the network becomes a more centralized one. This result implies higher coupling between stocks during volatile regimes.
This methodology of cluster analysis that combines MST with Symbolic Time Series Analysis has also been utilized in other research areas. For instance, [9] study the regional convergence of subnational states in the Mercosur from 1961 to 2005 and finds evidence that supports the existence of convergence clubs of performance.
Finally, in [8] the authors employ this methodology to explore the relationship between economic growth and public debt. By using symbolic sequences, the paper introduces a measure of distance between the dynamical patterns of different countries. Then, these symbolic time series are utilized as input to construct a Minimal Spanning Tree and a Hierarchical Tree with the aim of detecting the existence of groups of countries sharing similar economic performance. The analysis identifies three groups (or clusters) of countries: high, mid and low indebted countries.
The next section explains the Minimal Spanning Tree and Hierarchical Tree methodology, as well as the Symbolic Time Series Analysis methods.

Methodology.
3.1. Minimal spanning tree and hierarchical clustering. In his seminal work [38], Mantegna proposed analyzing financial markets by using elements of graph theory. In his work he pointed out that stocks could be considered as the nodes of a graph, while the relationships between the stocks would be represented by the links of the graph. This way, the links correspond to the Pearson's correlations between the pairs of stock returns. From these correlations, Mantegna constructs the Minimal Spanning Tree (MST), which is the minimal graph that covers all nodes without loops. In other words, the minimal spanning tree links nodes (in this case stocks) in such a way that there is exactly one path between any pair of nodes. The first step of this methodology is the calculation of the returns, r i , of each of the stocks. Subsequently, the Pearson's correlations are computed between the n × n pairs of stock returns: Where r i is the average value of r i during the considered period. Then, the correlation matrix is built with the correlation coefficients ρ ij . By definition ρ ij takes values in the interval (-1,1), where -1 means completely anti-correlation, 1 complete correlation and 0 that the two variables are uncorrelated. This matrix is symmetrical, with ρ ii = 1 in this main diagonal. As it is well known, the Pearson correlation coefficient, (equation 1), does not fulfill the three axioms that define a Euclidean metric. For this reason, the correlation matrix is transformed into the distance matrix according que the following formula Which fulfills the three axioms of a Euclidean distance: Then, the distance matrix is used to determine the minimal spanning tree (MST) connecting the n vertices. In our case, each vertex corresponds to a currency. The MST is constructed following the Kruskals algorithm, which links all the vertices together in a single graph that minimizes the distances between the corresponding time series. Then, the distance matrix is used to determine the minimal spanning tree (MST) connecting the n vertices. In our case, each node corresponds to a currency. The MST is constructed following the Kruskal's algorithm, which links all the vertices together in a single graph that minimizes the distances between the corresponding time series.
The Kruskal method is implemented by following the next steps. First, the algorithm chooses a pair of currencies with the nearest distance and connects them with a line that is proportional to the distance. Then, the algorithm connects another pair of currencies with the second nearest distance. In the third step, the nearest pair that is not connected by the same tree is also linked. This step is repeated until all the currencies are connected in a single tree. The importance of the MST is that it provides a representation of the relationship between the currencies that tells us which connections are more relevant.
From the MST we obtain the subdominant ultrametric distance matrix X, which corresponds to the matrix whose elements are the subdominant ultrametric distance, d < (i, j). The subdominant ultrametric distance d < (i, j) between the vertices i and j is the maximum value of any Eucliedan distance d < (i, j) detected by moving one step from i to j through the shortest path connecting i and j in the MST. An ultrametric distance d < (i, j) must satisfy the first two properties of a metric distance, d < (i, j) = 0 ⇔ i = j and d < (i, j) = d < (j, i), while the usual triangular inequality d(i, j) ≤ d(i, k) + d(k, j) is replaced by a stronger inequality, called an ultrametric inequality We compute the ultrametric distance d < (i, j) between each pair of currencies and from it we construct the Hierarchical Cluster, which is represented as a tree known as dendogram. From this three we obtain taxonomic information of the co-movements between the currencies -i.e. we can group and classify the currencies according to the degree of synchronization of their co-movements.
It is worth observing that in the methodology described here, the Pearson correlation coefficient, (equation 1), is used as an input to obtain the distance between the time series of each of the currencies. However, it is also possible to use other alternative measures. In this paper we will follow the methodology introduced in [4,5,6,7,8] and [9], which combines the methodology of the MST with the Symbolic Analysis of Time Series (STSA). This way, we will transform the original series into a symbolic representation, (i.e symbolic time series), from which we will calculate the Euclidean distance between them. This Euclidean distance will allow us to construct the distance matrix, from which we will obtain the ultrametric distances needed to construct the MST and the corresponding HT.

Symbolic Time Series Analysis (STSA). Symbolic Time Series
Analysis is a methodology that allows the extraction of qualitative information from time series. Each value of the time series is transformed into a symbol, in such a way that the original time series is transformed into a symbolic time series. Based on the analysis of the symbolic time series, it is possible to obtain information referring to the dynamics of the original time series.
In order to transform the original series into a symbolic one, it is necessary to introduce a partition into the state spaces of the original series. Thus, the set of values taken by the original series is partitioned into a finite number of regions. Then, once the partition has been specified, each measurement of the original series is transformed into a symbol depending on which region of the partition the observation fell into. In this context, the set of symbols is known as alphabet. In order to give an example, figure 1 shows the time series of the variation of the exchange rate of the Mexican peso against the dollar. It is a daily series, starting on 07.05.2013 and ending on 05.17.2013. In the same figure, the horizontal line indicates the average value of the time series, µ = 0.0025. This µ value is used as the threshold value that separates the state space in two regions: deviations above the mean and deviations below the mean. In this example, the alphabet has two symbols, 0 and 1. Thus, when the original value is above the mean µ, the symbolic value is 1, whereas if the original value is below the mean µ, the symbolic sequence shows the value 0.
Thus, the original series is transformed into the symbolic sequence S = 100100101. These symbolic sequences are called words in the symbolic dynamics literature and represent paths or patterns -i.e. which regions of the partition are visited by the variable in each period.
In this paper we will use the average of each time series to make the corresponding partition. Consequently, the state space is divided into two regions: above and below the trend. If the value of the series is below its trend, the symbol used is 0, and 1 otherwise. Thus, the data is symbolized according to the following rule: Where ∆e i is the change in the exchange rate of currency i, µ i is the trend of ∆e i and s i is the corresponding symbolic value. Once the original series have been transformed into symbolic series, we can measure the distance between them. For this purpose we will use the most commonly used distance measure, the Euclidean distance. Given the method of symbolization we have used, the Euclidean distance shows the degree of co-movement, or synchronization, between the currencies. If the distance between a pair of currencies i and j is 0, then both currencies have a complete co-movement: when one is above its trend, the other one is also above its own trend, and the same happens when any of them is below their tendency. On the other hand, if the distance is maximal, then there is no co-movement, since in that case the currencies would be moving in opposite directions (when the currency i is above the trend, the currency j is below its own trend, and vice versa).
Given two symbolic sequences s i and s j , the Euclidean distance between s i and s j is the following function Where s i and s j are two symbolic sequences for currencies for countries i and j, respectively, and the difference s it -s jt takes value 1 if s it = s jt and 0 if s it = s jt . Next section presents the data and section 5 presents the empirical results of this paper.
4. Data. The study considers the exchange rates of nine Latin American countries: Argentina, Brazil, Chile, Colombia, Honduras, Mexico, Peru, Uruguay and Venezuela. We analyze daily data, from 16 January 2007 through 29 December 2017, which is obtained from the Pacific Exchange Rate Exchange Rate Service available online. To denote the currencies, we follow the ISO4217 standard by using three-letter codes (table 1). In this analysis, the exchange rate is defined as the value of one currency in terms of the US Dollar (USD). Table 1. Countries, currencies and three-letter codes.
First, we calculate the variation in the exchange rates: where e i,t is the daily exchange rate from country i, at day t. As we mentioned in section 3, for each currency we compute its trend value, µ i . Later, this value is utilized as the threshold value that will introduce a discrete partition into the state space. In our case, the state space consists of all the possible values that the changes in the exchange rates may adopt. We say that currencies show comovements when they occupy the same region at the same time. Given the fact that our set of currencies is integrated by nine countries, it follows that there are 36 (=9x8/2) possible co-movements. With the MST we will find the most relevant linkages and with the HT we will find the groups of currencies that tend to move together.

5.
Results. In this section we present the empirical results of this article. As mentioned in the previous section, in our study we analyze the relationships between nine currencies. In the context of this work, these relationships refer to the Euclidean distances that exist between the symbolic series corresponding to each currency. Our objective is to determine which currencies tend to follow similar dynamics. For this purpose, in this section we build the minimal spanning tree and the corresponding hierarchical tree for each period. The period analyzed covers from 2007 to 2017. This period will be divided into four subperiods. The reason for dividing the analysis into different periods is to see if the relationships between the currencies are stable over time, that is, we want to know if there are groups of countries that stay grouped in the same cluster during all periods. In that case, the results would imply that there are structural components that persist over time. Alternatively, with this analysis we could also observe if there are isolated currencies i.e. currencies that tend to be separated from the rest of the currencies. In addition, another interesting question lies in finding out if there are nodes that are located on the border between different groups i.e. if there are currencies that could be grouped in more than one cluster. These kind of currencies have a very relevant role because they are the transmission channel between different clusters and can spread contagion processes.
Additionally, the idea of dividing the analysis into subperiods would allow us to extract some insights referring to the effects of the financial crisis. Having said that, the analysis is divided into four time intervals. The first period analyzed will be year 2007. Our objective here is to have an image of the structure of the network before the financial crisis. A second period is 2008-2010: our aim is to determine the impact of the financial crisis on the currency network, particularly in the co-movements of the nodes. A third period is 2011-2014, here our objective is to analyze if the changes observed in the previous period are conjunctural or if, on the contrary, the network has assimilated the changes. Finally, in the fourth period, 2015-2017, we might observe the effects of the post-commodity boom.

Period: 2007.
In this part, we discuss the results corresponding to the year 2007. In the Hierarchical Tree (HT) we observe the following. During this period, at least two main clusters are clearly identified. In the first one, the Honduras lempira and the Venezuelan bolivar are located. The second cluster consists of the nodes corresponding to Mexico, Colombia, Brazil and Chile, whereas the currencies from Argentina, Peru and Uruguay were the most distant. In the case of Argentina, although distant, its most similar behavior was with the cluster integrated by Honduras and Venezuela. In the case of Peru, which also had a different dynamic than the rest, its closest proximity was to the cluster integrated by Mexico, Colombia, Brazil and Chile. The Uruguayan peso was the currency that had the most heterogeneous dynamics in this period. Regarding the Minimal Spanning Tree, we observe some interesting facts. First of all, the Brazilian real and the Chilean peso occupy the central positions in the network. Furthermore, both currencies are located on the border with other clusters. The case of Chile is significant, as in addition to being linked to the currencies of its own cluster, it was also related to two currencies outside its own cluster: the Uruguayan peso and the Peruvian sol. The role of the Brazilian real also stands out, since it was directly related to two members of its own cluster -Colombian peso and Chilean peso -but it was also linked to the Lempira currency.
The importance of Brazil and Chile in this period lies not only in the number of links, but also in the fact that they are in connection with more than one cluster, which implies that these currencies are the channels that transmit information between different groups of countries. We must also highlight the case of the Mexican peso which, although it did not occupy a central position in the Minimal Spanning Tree (MST), its distance to the Chilean node was the minimal distance during the period i.e. the strongest co-movement in this period was the one observed between the Chilean peso and the Mexican peso. So, thanks to its close connection with the Chilean currency, the Mexican peso could also affect or be affected by what happens in other clusters. Finally, the two least related currencies during the period were the Venezuelan bolivar and the Uruguayan peso.

5.2.
Period: 2008-2010. As in the previous case, in this period the Honduran lempiras and the Venezuelan bolivar also formed a cluster. Additionally, the Brazilian real, the Mexican peso, the Colombian peso, the Peruvian sol and the Chilean peso integrated a second cluster. Within this cluster, the closest relationship was between the Brazilian real and the Mexican peso, while the Chilean peso had the most heterogeneous behavior. On the other hand, both the Argentinian peso and the Uruguayan peso had the most isolated dynamics. In this period, the maximum distance was between the Uruguayan peso and the Venezuelan bolivar i.e. these two currencies had the lowest degree of co-movement within the network. In this period, the shortest distance was the one observed between the Honduran lempiras and the Venezuelan bolivar. Now let us analyze the Minimal Spanning Tree (MST). In this period, the Mexican peso occupied the central position, since it was the currency with the most direct links to other currencies; furthermore, it was also linked directly with currencies from other clusters, such as the Venezuelan bolivar. Similarly, the case of the Venezuelan bolivar is interesting, since it was also directly linked to the Argentinian peso. This implies that during this period, the Bolivar would be a transmission channel between the Mexican peso and the Argentinian peso. It is also noteworthy that the second shortest distance i.e. the second strongest co-movement-was the one observed between the Mexican peso and the Brazilian real. The results of this period, together with the results of the previous period, would be pointing to some structural relationships in the currency network. First, the results suggest that the co-movements between the Honduran lempiras and the Venezuelan bolivar would be one of the structural characteristics of this currency network. The same would be happening with the case of the Brazilian real and the Mexican peso. Similarly, it seems that the case of the Argentinian peso and the Uruguayan peso would also be a structural component of the network. Finally, in this period the two currencies with the most dissimilar behavior were again the Uruguayan peso and the Venezuelan bolivar, so this fact would also be pointing to another structural component of the currency network.

5.3.
Period: 2011-2014. In the analysis of the hierarchical tree of this period, we observe the following. Once again, two main clusters are identified. On one hand, we confirm that the relationship between the Honduran lempiras and the Venezuelan bolivar is a structural component of the currency network. The second major cluster identified, and which in fact is the one with the highest number of members, is the one integrated by the Peruvian sol, the Chilean peso, the Colombian peso, the Brazilian real and the Mexican peso. Within this cluster, there are three interesting dynamics, each one corresponding to a subcluster within this group of currencies. First, the Peruvian sol is the most isolated component within this cluster. Second, in this period the Chilean peso and the Colombian peso form a subcluster. Finally, the close relationship between the Brazilian real and the Mexican peso is confirmed once more. Again, Argentina and Uruguay had isolated dynamics. Concerning the Minimal Spanning Tree, we observe that both the Mexican peso and the Chilean peso were in the central positions and each one was linked directly with three other currencies. In the case of the Chilean peso, it was directly linked not only to its own cluster but also to a node (corresponding to Uruguay) external to its group of countries. Thus, in this period the Chilean peso would be a transmission channel between heterogeneous components of the network. In the case of the Mexican peso, although it is not directly linked to external nodes, it is directly linked to the Chilean peso. Thus, the Mexican peso, via the Chilean peso, would have an impact on the Uruguayan peso. As it might be expected, the maximum distance was between the Venezuelan bolivar and the Uruguayan peso. 5.4. Period: 20155.4. Period: -2017. In this period, as in the previous ones, we observed two main clusters. On one hand, the cluster composed by the Honduran lempiras and the Venezuelan bolivar, and a second cluster, integrated by the Brazilian real, the Peruvian sol, the Colombian peso, the Chilean peso and the Mexican peso. Once more, the Uruguayan peso and the Argentinian peso had more isolated dynamics. Regarding the biggest cluster, there are two relevant facts. Firstly, the Brazilian real was the currency with the most heterogeneous behavior within this cluster. Secondly, the Mexican peso and the Chilean peso showed greater homogeneity in their dynamics. Regarding the Minimal Spanning Tree, we observe, as in the previous period, that the Mexican peso and the Chilean peso occupied the central positions; both had three direct links with other currencies and were also directly linked to each other. However there is an important qualitative difference in comparison with the former period: this time the Mexican peso occupies the role held by the Chilean peso in the previous period. Thus, the Mexican peso, linked to elements of both inside and outside its own cluster, would be the main transmission channel within this network. Another interesting point confirmed through the analysis of the Minimal Spanning Tree (MST) is the close structural relationship between the Mexican peso and the Brazilian real, since, as in the two preceding periods, these two currencies were directly linked in the MST. Finally, as in the previous cases, the two currencies with the most heterogeneous behavior between them were the Venezuelan bolivar and the Uruguayan peso.
To sum up, the following can be concluded from this analysis. First of all, the stability of the cluster composed by the Honduran lempiras and the Venezuelan bolivar. The case of the Argentinian peso is also relevant because, although it tended to have a very heterogeneous behavior, its closest relationship was either with the Honduran lempiras or with the Venezuelan bolivar. A possible explanation behind this fact could be the impact that the US Dollar has on these currencies. Another important point of the analysis is the relationship between the Brazilian real, the Colombian peso, the Chilean peso, the Peruvian sol and the Mexican peso, which remained within the same cluster. As for the Peruvian sol, its dynamics were the most unstable and heterogeneous within this cluster. On the other hand, the Brazilian real, Chilean peso and Mexican peso showed important co-movements and, in all periods, at least one of them occupied a central position within the network and served as a transmission channel between different clusters. However, in the period 2008-2010, the Chilean peso decreased the pace of its co-movements with the rest of the currencies of this group, whereas at the same time the closest relationship was between the Brazilian real and the Mexican peso. A possible explanation would suggest that the impact of the crisis on the currencies was asymmetric and differentiated, affecting the Mexican peso and the Brazilian real in a more symmetric way. Lastly, the Uruguayan peso is confirmed as an isolated element of the network, with the most heterogeneous dynamics among the group of analyzed currencies. 5.5. Discussion. In this section, we argue that the obtained results exhibit a meaningful economic taxonomy. Firstly, it is observed that the clusters match the levels of financial liberalization. In figure 6, the capital account openness index of Chinn and Ito, [13,14], is shown for each of the countries considered. This index is the most commonly used measure of capital controls and focuses on regulatory restrictions on capital account transactions. The countries with the lowest level of financial liberalization are Argentina, Honduras and Venezuela, which is consistent with the analysis of clusters, since the co-movements of the currencies of these three countries have always been grouped. On the other hand, Brazil, Chile, Colombia and Mexico show an intermediate level of financial liberalization, and these countries were always part of the same cluster. Peru and Uruguay have the highest level of financial liberalization. This is particularly relevant in the case of Uruguay, since the Uruguayan peso was the currency that showed the most divergent behavior.
It is worth mentioning that in emerging developing countries, the fear of floating [11] is behind the interventions of central banks. Such ability depends, not only on the size of foreign reserves, but also on the degree of financial openness [2]. Thus, financial flows mean an important challenge for the central banks in the region, as these flows do not lead to an automatic adjustment. Financial flows can generate cycles of appreciation, depreciation or volatility that might result in problems of exchange rate misalignment and volatility, as it has happened in cases such as Brazil, Chile or Mexico [1]. In the case of African countries, [12] finds that capital account openness is strongly related to foreign exchange co-movement and suggests that this relationship might be driven by a signaling effect. Table 2. Monetary Policy Framework. Argentina maintains a de facto exchange rate anchor to the U.S. dollar. Uruguay has an inflation target regime with monetary aggregates control. Source: [28].
On the other hand, the observed results are also relevant as far as the distinction between IT and non-IT goes. Table 2 shows the monetary policy framework of the countries considered. Out of the sample analyzed, Brazil, Chile, Colombia, Mexico and Peru are IT countries. The co-movements of these currencies, as previously mentioned, were part of the same cluster. In line with this result, and for the cases of Brazil, Chile and Mexico, [17] finds evidence that, after controlling for the exchange rate regime, inflation targeting has tended to reduce conditional volatility in these countries. The author points out that this might be due to the fact that IT is a credible and predictable monetary framework that tends to reduce unexpected shocks.
Alternatively, Argentina, Honduras and Venezuela had the US dollar as anchor, which would explain the similarity of their co-movements. These countries have used the US dollar as anchor with the aim of achieving price stability. Uruguay, whose currency showed the most isolated dynamics, is the only country in the sample whose monetary policy framework is monetary aggregate target, a regime where the central bank uses its instruments to achieve a target growth rate for a monetary aggregate and the aggregated target becomes the nominal anchor. 6. Conclusions. In this paper, we analyzed the co-movements among nine Latin American exchange rates. The currencies analyzed were the Argentinian peso, the Brazilian real, the Chilean peso, the Colombian peso, the Honduran lempiras, the Mexican peso, the Peruvian sol, the Uruguayan peso and the Venezuelan bolivar. In the analysis, we used daily data for the period 2007-2017. The starting point of the study was to consider that this group of currencies is a complex system that can be conceptualized as a network. This network is made up of nine nodes (that is, nine currencies) and 36 links (i.e. the total number of possible co-movements for each currency).
The analysis combined elements of symbolic time series analysis with a clustering algorithm (minimal spanning tree and hierarchical tree). Among the results of our analysis, the following stands out. On one hand, the studied currency network has structural features that are stable over time, for example, the lempiras and the Venezuelan bolivar were always located in the same cluster and the Argentinian peso was always gravitating around this cluster. On the other hand, the Mexican peso, the Chilean peso, the Colombian peso, the Peruvian sol and the Brazilian real compose the largest cluster of the network. Within this cluster, the Peruvian sol was the one that tended to have the most heterogeneous behavior. In contrast, the Mexican peso, the Chilean peso and the Brazilian real tended to occupy central positions in the network. They were usually the ones with the most connections and were connected to more than one cluster. By being in contact with more than one cluster, these currencies function as channels of information flows between the clusters. Finally, using data regarding the degree of financial liberalization, as well as the distinction between inflation-targeting (IT) and non-IT countries, the analysis suggests that the obtained taxonomy is economically relevant. Future research might consider the introduction of other partitions and classification methods in order to detect different dynamics associated to volatile periods.