key: cord-148358-q30zlgwy
authors: Pang, Raymond Ka-Kay; Granados, Oscar; Chhajer, Harsh; Legara, Erika Fille
title: An analysis of network filtering methods to sovereign bond yields during COVID-19
date: 2020-09-28
journal: nan
DOI: nan
sha: 
doc_id: 148358
cord_uid: q30zlgwy

In this work, we investigate the impact of the COVID-19 pandemic on sovereign bond yields amongst European countries. We consider the temporal changes from financial correlations using network filtering methods. These methods consider a subset of links within the correlation matrix, which gives rise to a network structure. We use sovereign bond yield data from 17 European countries between the 2010 and 2020 period as an indicator of the economic health of countries. We find that the average correlation between sovereign bonds within the COVID-19 period decreases, from the peak observed in the 2019-2020 period, where this trend is also reflected in all network filtering methods. We also find variations between the movements of different network filtering methods under various network measures.

The novel coronavirus disease 2019 (COVID-19) epidemic caused by SARS-CoV-2 began in China in December 2019 and rapidly spread around the world. The confirmed cases increased in different cities of China, Japan, and South Korea in a few days of early January 2020, but spread globally with new cases in Iran, Spain, and Italy within the middle of February. We focus on sovereign bonds during the COVID-19 period to highlight the extent to which the pandemic has influenced the financial markets. In the last few years, bond yields across the Euro-zone were decreasing under a range of European Central Bank (ECB) interventions, and overall remained stable compared with the German Bund, a benchmark used for European sovereign bonds. These movements were disrupted during the COVID-19 pandemic, which has affected the future trajectory of bond yields from highly impacted countries, e.g., Spain and Italy. However, in the last months, the European central banks intervened in financial and monetary markets to consolidate stability through an adequate supply of liquidity countering the possible margin calls and the risks of different markets and payment systems. These interventions played a specific role in sovereign bonds because, on the one side, supported the stability of financial markets and, on the other side, supported the governments' financial stability and developed a global reference interest rate scheme. Understanding how correlations now differ and similarities observed in previous financial events are important in dealing with the future economic effects of COVID- 19. We consider an analysis of sovereign bonds by using network filtering methods, which is part of a growing literature within the area of econophysics [29, 44, 30, 28, 17] . The advantages in using filtering methods is the extraction of a network type structure from the financial correlations between sovereign bonds, which allows the properties of centrality and clustering to be considered. In consequence, the correlation-based networks and hierarchical clustering methodologies allow us to understand the nature of financial markets and some features of sovereign bonds. It is not clear which approach should be used in analyzing sovereign bond yields, and so within this paper, we implement various filtering methods to the sovereign bond yield data and compare the resulting structure of different networks.

Our analysis shows that over the last decade, the mean correlation peaks in October 2019 and then decreases during the 2020 period, when COVID-19 is most active in Europe. These dynamics are reflected across all network filtering methods and represent the wide impact of COVID-19 towards the spectrum of correlations, compared to previous financial events. We consider the network centrality of sovereign bonds within the COVID-19 period, which remains consistent with previous years. These trends are distinctive between filtering methods and stem from the nature of correlations towards economic factors e.g., positive correlations show a stable trend in the individual centrality, compared with the volatile trends for negative correlations, where central nodes within these networks are less integrated in the Euro-area. Although there is a change in the magnitude of correlations, the overall structure relative to the central node is maintained within the COVID-19 period.

Previous studies have used different methods to analyze historic correlations as random matrix theory to identify the distribution of eigenvalues concerning financial correlations [27, 39, 23] , the approaches from information theory in exploring the uncertainty within the financial system [20, 12] , multilayer network methods [1, 7, 46, 24, 18, 40] , and filtering methods. Several authors have used network filtering methods to explain financial structures [31, 37] , hierarchy and networks in financial markets [50] , relations between financial markets and real economy [34] , volatility [51] , interest rates [33] , stock markets [21, 52, 53, 2] , future markets [8] or topological dynamics [45] to list a few. Also, the comparison of filtering methods to market data has been used for financial instruments. Birch, et al [10] consider a comparison of filtering methods of the DAX30 stocks. Musmeci, et al [35] propose a multiplex visual network approach and consider data of multiple stock indexes. Kukreti, et al [26] use the S&P500 market data and incorporate entropy measures with a range of network filtering methods. Aste, et al [5] apply a comparison of network filtering methods on the US equity market data and assess the dynamics using network measures.

In order to evaluate the European sovereign bonds, based on filtering methods, this work is organized as follows. In Section 2, we describe the network filtering methods and present the data sets with some preliminary empirical analyses. We apply in Section 3 the filtering methods to sovereign bond yields and analyze the trend of financial correlations over the last decade and consider aspects of the network topology. We construct plots in Section 4 representing the COVID-19 period for all methods and analyze the clustering between countries. In Section 5, we discuss the results and future directions.

We introduce a range of network filtering methods and consider a framework as in [31] for sovereign bond yields. We define n ∈ N to be the number of sovereign bonds and bond yields Y i (t) of the ith sovereign bond at time-t, where i ∈ {1, ..., n}. The correlation coefficients r ij (t) ∈ [−1, 1] are defined using Pearson correlation as:

with · denoting the average of yield values. The notion of distance d ij ∈ [0, 2] considers the values of the entries r ij of the correlation matrix R ∈ [−1, 1] n×n , with d ij = 2(1 − r ij ). A distance of d ij = 0 represents perfectly positive correlations and d ij = 2 represents bonds with negative correlations. The network filtering methods are then applied to the distance matrix D ∈ [0, 2] n×n , where a subset of links (or edges) are chosen under each filtering method. The set of edges is indicated by {(i, j) ∈ E(t) : nodes i and j are connected} at time-t, defined for each filtering method.

We define the time frames of financial correlations as X for the set of observations, with n different columns and T rows. From the set of observations X, we consider windows of length 120, which is equal to six months of data values. We then displace δ windows by 10 data points, which is equal to two weeks of data values, and discard previous observations until all data points are used. By displacing the data in this way, we can examine a time series trend between each window X.

We verify the statistical reliability of correlations by using a non-parametric bootstrapping approach as in Efron [15] , which is used in Tumminello, et al [48, 49] . We randomly choose rows equal in number to the window length T , allowing repeated rows to be chosen. We compute the correlation matrix for this window X * m and repeat the procedure until m samples are generated, which is chosen at 10,000. The error between data points described in Efron [15] is equal to (1 − ρ 2 )/T , where highly positive and negative correlated values ρ have the smallest errors.

The minimum spanning tree (MST) method is a widely known approach which has been used within currency markets [22] , stocks markets [42, 43] and sovereign bond yields [13] . The MST from Table  1 considers the smallest edges and prioritizes connections of high correlation to form a connected and undirected tree network. This approach can be constructed from a greedy type algorithm e.g. Kruskal's and Prim's algorithm and satisfies the properties of subdominant ultrametric distance i.e, d ij ≤ max{d ik , d kj } ∀i, j, k ∈ {1, ..., n}.

A maximum spanning tree (MaST) constructs a connected and undirected tree network with n − 1 edges in maximizing the total edge weight. Analyses involving MaST have been used as comparisons to results seen within MST approaches [14, 19] . An MaST approach is informative for connections of perfectly anti-correlation between nodes, which are not observed within the MST.

A network formed from asset graphs (AG) considers positive correlations between nodes of a given threshold. Within the MST, some links of positive correlation are not considered in order to satisfy the properties of the tree network. All n − 1 highest correlations are considered in an AG, allowing for the formation of cliques not observed within a MST or MaST network. The use of AG has been considered in Onnela, et al [38] , which identifies clustering within stock market data. As the method only considers n − 1 links, some nodes within the AG may not be connected

Minimum Spanning Tree (MST) n − 1 [25] A connected and undirected network for n nodes which minimizes the total edge weight.

Maximum Spanning Tree (MaST) n − 1 [41] A connected and undirected network for n nodes which maximizes the total edge weight.

Asset Graph (AG) n − 1 [36] Choose the smallest n−1 edges from the distance matrix. Triangulated Maximal Filtering Graph (TMFG)

[32] A planar filtered graph under an assigned objective function. for the given threshold and therefore the connection of unconnected nodes is unknown, relative to connected components. The triangulated maximal filtering graph (TMFG) constructs a network of 3(n − 2) fixed edges for n nodes, similar to the planar maximal filtered graph (PMFG) [47] , which has been used to analyze US stock trends [35] . The algorithm initially chooses a clique of 4 nodes, where edges are then added sequentially, in order to optimize the objective function e.g., the total edge weight of the network, until all nodes are connected. This approach is non-greedy in choosing edges and incorporates the formation of cliques within the network structure. A TMFG is also an approximate solution to the weighted planar maximal graph problem, and is computationally faster than the PMFG. The resulting network includes more information about the correlation matrix compared with spanning tree approaches, while still maintaining a level of sparsity between nodes.

The European sovereign debt has evolved in the last ten years, with some situations affecting the convergence between bond yields. After the 2008 crisis, European countries experienced a financial stress situation starting in 2010 that affected bond yields, thus the investors saw an excessive amount of sovereign debt and demanded higher interest rates in low economic growth situations and high fiscal deficit levels. During 2010-2012, several European countries suffered downgrades in their bond ratings to junk status that affected investors' trust and fears of sovereign risk contagion resulting, in some cases, a differential of over 1,000 basis points in several sovereign bonds. After the introduction of austerity measures in GIIPS countries, the bond markets returned to normality in 2015.

The 2012 European debt crisis particularly revealed spillover effects between different sovereign bonds, which have been studied using various time series models e.g. VAR [11, 4] and GARCH [6] . The results showed that Portugal, Greece, and Ireland have a greater domestic effect, Italy and Spain contributed to the spillover effects to other European bond markets and a core group of ABFN (Austria, Belgium, France, and Netherlands) countries had a lower contribution to the spillover effects, with some of the least impacted countries residing outside of the Euro zone.

During the sovereign debt crisis, public indebtedness increased after Greece had to correct the public finance falsified data, and other countries created schemes to solve their public finance problems, especially, bank bailouts. In consequence, the average debt-to-GDP ratio across the Euro-zone countries rose from 72% in 2006 to 119.5% in 2014, as well as the increase in sovereign credit risk [3, 9] . After the Fiscal Compact Treaty went into effect at the start of 2013, which defined that fiscal principles had to be embedded in the national legislation of each country that signed the treaty, the yield of sovereign bonds started a correction, although some investors and institutions . Four of the listed countries are part of the G7 and G20 economic groups (Germany, France, Italy and the UK). We consider sovereign bond yields with a 10 year maturity between January 2010 and June 2020. This data is taken from the financial news platform 1 . In total, there are 2,491 data values for each country with an average of 240 data points within 1 year. Table 2 provides summary statistics of the 10Y bond yield data. The results show Greek yields to have the highest values across all statistical measures compared with other countries yields, particularly within the 2010-2011 (max yield of 39.9). In contrast, Swiss bond yields exhibit the smallest mean and variance, with a higher than average positive skewness compared with other sovereign bonds. Under the JB test for the normality of data distributions, all bond yield trends have a negligible p-value with non-Gaussian distributions. The left skewed yield distributions (except for Iceland), which represent an average decrease in yield values each year are high for GIIPS countries compared with the UK, France, and Germany, with flattening yield trends.

We compute the correlation matrix for each window X with a displacement of δ between windows, and consider the mean and variance for the correlation matrix. We define the mean correlation r(t) given the correlations r ij for n sovereign bonds 

From Figure 1 , we find that the mean correlation r(t) is highest at 0.95 in Oct 2019. This suggests that a COVID-19 impact was a continuation on the decrease of the mean correlation, and throughout the punitive lock down measures introduced by the majority of European countries in Feb-Mar 2020. The decreases in mean correlation are also observed within the in the 2012 period during the European debt crisis, in which several European countries received EU-IMF bailouts to cope with government debt and in 2016, under a combination of political events within the UK and the increased debt accumulation by Italian banks. The variance u(t) also follows a trend similar to the mean correlation, with the smallest variance of 0.002 in October 2019. Within 2020, the variance increases between sovereign bonds and reflects the differences between the correlations of low and high yield.

We consider the normalized network length L(t), which is introduced in Onnela, et al [36] as the normalized tree length. We define the measure as the normalized network length, as this measure is considered for AG and TMFG non-tree networks. The network length is a measure of the mean link weights on the subset of links E(t), which are present within the filtered network on the distance matrix at time-t 

The plots in Figure 2 represent the mean and variance of the network length. As each filtering method considers a subset of weighted links, the normalized length L(t) is monotonic between all methods and decreases with the increased proportion of positive correlated links within the network. We highlight the movements in the normalized network length during the COVID-19 period, which is reflected across all filtering methods. This movement is observed within 2016, but only towards a subset of correlations, in which the network length of the MaST and TMFG increases compared with the MST and AG. The relative difference between the normalized networks lengths is least evident in periods of low variance; this is observed in the 2019-2020 period, where the difference between all methods decreases.

We find the variance is highest within the TMFG and lowest with the AG approach. The increased inclusion of links with a higher reliability error in the TMFG increases the variance, particularly within the 2014-2017 period. The variance of the MST on average is higher compared with the MaST, but when considering only the highest correlated links in the AG, the variance decreases.

We define the degree centrality for the node of maximum degree C(t) at time-t. This measure considers the number of direct links

The mean occupation layer η(t) (MOL) introduced in Onnela, et al [36] is a measure of the centrality of the network, relative to the central node υ(t). We define lev i (t) as the level of the node, which is the distance of the node relative to υ(t), where the central node and nodes unconnected relative to the central node have a level value of 0

We use the betweenness centrality to define the central node υ(t) for the MOL. Introduced in Freeman [16] , the betweenness B(t) considers the number of shortest paths σ ij (k) between i and j which pass through the node k, relative to the total number of shortest paths σ ij between i and j,

Within the MST, the degree centrality ranges between 3 to 5 for Euro-zone countries. The trend within the MST remains stable, where the central node under degree centrality is associated with multiple sovereign bonds e.g., Netherlands 19%, Portugal 10% and Belgium 9% across all periods. The MaST has the highest variation, with a centralized network structure in some periods e.g., C(t) of 16, forming a star shaped network structure. This is usually associated with Greece, Iceland and Hungary, which are identified as the central node 55% of the time. The degree centrality on average is naturally highest with the TMFG, under a higher network density, where the central nodes are identified as Hungary and Romania sovereign bonds, similar to the MaST. The AG identifies the Netherlands and Belgium within the degree centrality, under a higher proportion of 25% and 13% compared with the MST. Within Figure 3 , the MOL on average is smallest for the AG, because of the 0 level values from unconnected nodes, in which an unconnected node is present within 94% of considered windows. We find that the nodes within the TMFG are closest within the network, where the central node is directly or indirectly connected for all nodes, with an average path length of 1.1 across all periods. Between the MST and MaST, the MOL is higher within the MaST, where nodes within the network have a higher degree centrality.

We analyze the temporal changes of sovereign bond yields between October 2019 and June 2020. The associated link weights on each filtering method for window X are the proportions in which the link appears within the correlation matrix, under the statistical reliability, across all samples m for the randomly sampled windows X * m . Under the MST, Austria has the highest degree centrality of 4. The network also exhibits clusters between southern European countries connected by Spain, and the UK towards Polish and German sovereign bond yields. Within the network, there is a connection between all ABFN countries, but countries within this group also facilitate the connecting component within GIIPS countries, where Belgium is connected with Spain and Irish sovereign bonds. The UK and eastern European countries remain on the periphery, with ABFN countries occupying the core of the network structure.

For the MaST in Figure 4 , there exists a high degree centrality for Polish sovereign bonds between western European countries e.g., France and Netherlands. This contrasts to the observed regional hub structure within the MST, with the existence of several sovereign bonds with high degree centrality in the network. The UK remains within the periphery of the MaST structure when considering anti-correlations, and shows UK bond yields fluctuate less with movements of other European bonds compared with previous years. This is also observed for sovereign bonds for other countries with non-Euro currencies such as Czech Republic, Hungary, and Iceland. We find nodes within the TMFG to have the highest degree in Iceland at 13 and Poland at 10. Although the MST is embedded within the TMFG network structure, a high resemblance is observed to links from the MaST, where 69% of links which are present within the MaST are common in both networks. There is also the associated degree centrality of the MaST, which is observed within the TMFG connected nodes. Under the TMFG, nodes have a higher degree connectivity when considering an increased number of links, this is the case for the UK, which has 9 links compared with other spanning tree approaches. The AG exhibits three connected components between western European countries, southern European countries and the UK with eastern European countries. These unconnected nodes within the AG are associated with non-Euro adopting countries, with the remaining countries connected in an individual component. By solely considering the most positive correlations, we observe the formation of 3-cliques between countries, which is prevalent within the western European group of 6 nodes.

The average statistical reliability is highest at 0.92 within the MaST and AG, 0.89 for the MST and 0.82 for the TMFG. Under the TMFG, the increased inclusion of links with a lower magnitude in correlations decreases the reliability in link values. Other filtering approaches which consider a smaller subset can still result in low reliability values between some nodes e.g. Austria and Romania at 0.51 in the MST, Germany and Netherlands at 0.47 in AG.

Under various constraints, we observe a commonality between sovereign bonds across network filtering methods. We find for tree networks, that Euro-area countries have a high degree centrality and countries with non-Euro currencies e.g. Czech Republic and the UK are predominately located within the periphery of the network. This is further observed within the AG, where cliques are formed between GIIPS and ABFN countries, which is distinctive during the COVID-19 period compared with previous years. The anti-correlations within the MaST inform the trends of the negative correlations between eastern European countries and other European countries. By considering the TMFG with an increased number of links for positive correlations, we find similarities with the MaST degree centrality.

As a response to the COVID-19 pandemic, most countries implemented various socio-economic policies and business restrictions almost simultaneously. An immediate consequence was an increase in yield rates for these nations. The resulting upward co-movement and upward movements in other yield rates explain the decrease in the mean correlation in bond dynamics, coinciding with the pandemic outbreak. Thus, understanding the dynamics of financial instruments in the Euro area is important to assess the increased economic strain from events seen in the last decade.

In this paper, we consider the movements of European sovereign bond yields for network filtering methods, where we particularly focus on the COVID-19 period. We find that the impact of COVID-Banks starts to drop off, the market dynamics could adjust to economic performance and not its financial performance. In other words, the resulting dynamics could explain an increase in mean correlation in bond dynamics coinciding with the economic dynamics after the pandemic and the increment in yield rates.

Although we consider the sovereign bond yields with a 10Y maturity as a benchmark, this research can be extended to sovereign bonds with different maturities (e.g., short term 1Y, 2Y or 5Y, and long term 20Y or 30Y) because these bonds could reveal interesting effects and confirm that sovereign bonds are a good indicator to identify the economic impact of COVID-19. As each sovereign bond has different yield and volatility trends, we considered using the zero-coupon curve to evaluate the full extent of COVID-19 on sovereign bonds.

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