Connectedness between G10 currencies: Searching for the causal structure

Bettendorf, Timo; Heinlein, Reinhold Article — Published Version Connectedness between G10 currencies: Searching for the causal structure International Journal of Finance & Economics Provided in Cooperation with: John Wiley & Sons Suggested Citation: Bettendorf, Timo; Heinlein, Reinhold (2022) : Connectedness between G10 currencies: Searching for the causal structure, International Journal of Finance & Economics, ISSN 1099-1158, John Wiley & Sons, Ltd., Chichester, UK, Vol. 28, Iss. 4, pp. 3938-3959, https://doi.org/10.1002/ijfe.2629 This Version is available at: https://hdl.handle.net/10419/288116 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. http://creativecommons.org/licenses/by/4.0/ RESEARCH ARTICLE Connectedness between G10 currencies: Searching for the causal structure Timo Bettendorf 1 | Reinhold Heinlein 2 1 DG Economics, Deutsche Bundesbank, Frankfurt am Main, Germany 2 Bristol Business School, University of the West of England, Bristol, UK Correspondence Timo Bettendorf, DG-Economics, Deutsche Bundesbank, Frankfurt am Main, Germany. Email: [email protected] Reinhold Heinlein, Bristol Business School, University of the West of England, Bristol, UK. Email: [email protected]c.uk Abstract This paper presents a new approach for modelling the connectedness between asset returns. We adapt the measure of Diebold and Yilmaz, which is based on the forecast error variance decomposition of a VAR model. However, their connectedness measure hinges on critical assumptions with regard to the variance–covariance matrix of the error terms. We propose to use a more agnostic empirical approach, based on a machine learning algorithm, to identify the contemporaneous structure. In a Monte Carlo study, we compare the different connectedness measures and discuss their advantages and disadvantages. In an empirical application we analyse the connectedness between the G10 currencies. Our results suggest that the US dollar as well as the Norwegian krone are the most independent currencies in our sample. By contrast, the Swiss franc and New Zealand dollar have a negligible impact on other currencies. Moreover, a cluster analysis suggests that the currencies can be divided into three groups, which we classify as: commodity currencies, European currencies and safe haven/carry trade financing currencies. KEYWORDS connectedness, exchange rates, graph theory, networks 1|INTRODUCTION Triggered by the seminal work of Diebold and Yilmaz (2009, 2014), the measurement of spillover effects and connectedness between asset returns has gained popularity in the economic literature. Their approach, which is based on the forecast error variance decomposition (FEVD) of a VAR model, hinges on critical assumptions with regard to a recursive ordering of the variables (e.g., Cholesky). In this paper, we propose an alternative and more agnostic approach to modelling the connectedness between asset returns, which is based on a causal search algorithm that imposes no a priori recursive ordering. We compare its properties with those of other identification measures using aMonteCarloexperimentandapplyittotheG10 currencies. i Given the new procedure, our first goal is to estimate the network structure between nine currencies vis-à-vis an appropriate numéraire currency (i.e., pound sterling). 1 We focus on the network structure and not the dynamics of a connectedness measure, because we aim to understand the relationships between currencies. Such estimates provide important information for policy makers and practitioners. The network indicates the extent to which a certain currency or group of currencies is affected by domestic and foreign shocks. In this sense, it Received: 13 July 2021 Revised: 1 March 2022 Accepted: 9 April 2022 DOI: 10.1002/ijfe.2629 This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited. © 2022 The Authors. International Journal of Finance & Economics published by John Wiley & Sons Ltd. 3938 Int J Fin Econ. 2023;28:3938–3959. wileyonlinelibrary.com/journal/ijfe helps to gain a better understanding of potential contagion. The second goal is to utilise spillover intensities in order to identify clusters which can be interpreted as currency blocs or groups of common influence factors such as target currencies for carry trades. Research in this area has created several extensions to the original work by Diebold and Yilmaz (2009), who estimated the return and volatility spillovers with respect to global equity markets. Diebold and Yilmaz (2014) studied the connectedness of financial institutions during the financial crisis period. In Diebold and Yilmaz (2015), they also estimated the connectedness between returns of other asset classes such as bilateral exchange rates, for instance. The approach is based on the idea that the network—or the spillover effects—between asset returns can be estimated given a FEVD of a vector autoregressive (VAR) model. Diebold and Yilmaz (2009) suggest orthogonalising the VAR model residuals with the help of a Cholesky decomposition, while pointing to the problem that the results of such a factorisation depend on the ordering of the variables, as zero restrictions are imposed on the upper triangular contemporaneous matrix without any theoretical or statistical motivation (i.e., preventing contemporaneous spillover effects between certain variables). Given the arbitrariness with respect to the ordering of the variables, the estimated model may not capture the spillover effects correctly. Instead of choosing one specific ordering, Klößner and Wagner (2014) propose considering all possible variable permutations. They replicate the paper by Diebold and Yilmaz (2009) and show that given different permutations, differences in spillover intensity can be large. Their approach, however, is not only computationally intensive, but also induces a high degree of model uncertainty. In other words, Klößner and Wagner (2014) average on many misspecified models and one correct model, which is unknown. Apart from Cholesky decompositions, the literature also employs generalised impulse response functions (see Pesaran & Shin, 1998) in order to obtain variance decompositions which are invariant to the ordering of the variables, see for example Diebold and Yilmaz (2012,2014). Greenwood-Nimmo et al. (2016), for instance, use generalised variance decompositions to study exchange rate return and volatility connectedness. In a rolling-window approach, Greenwood-Nimmo et al. (2017) apply the generalised approach in order to analyse the change in European debt connectedness distributions over time. This approach, however, has two shortcomings: First, the explained shares of forecast error variance do not sum to unity. In order to avoid re-scaling the shares, Lanne and Nyberg (2016) propose an alternative generalised FEVD which yields shares summing up to unity by construction. However, shocks are not orthogonalised. Second, and more importantly, the approach is unable to model contemporaneous causal linkages. The weaknesses of generalised variance decompositions have been pointed out by De Santis and Zimic (2018), who instead propose absolute magnitude restrictions to identify SVAR models. They show that generalised variance decompositions tend to overestimate connectedness. An alternative and more agnostic approach is identification with the help of causal search algorithms from the machine learning literature. Such an approach for structural VAR models has been suggested by Swanson and Granger (1997). Demiralp and Hoover (2003) introduced the causal search methods for identification. These algorithms use information from the reduced VAR residuals in order to uncover the contemporaneous causal structure. Applications can be seen in Heinlein and Krolzig (2012) and Demiralp et al. (2014). We follow this literature on empirical identification and systematically analyse in a Monte Carlo experiment as well as in an application on returns of G10 currencies how the identification strategy impacts on the measures of connectedness. To the best of our knowledge, the only papers using an empirical identification strategy in the connectedness literature are Scida (2018) and Yang et al. (2021), but they do not systematically study the impact of this approach or compare the empirical identification with other identification methods. The machine learning approach is very appealing because it does not require any prior assumptions with regard to the contemporaneous causal structure between the variables. On the contrary, we derive with our data-driven approach a causal ordering, which can be evaluated and discussed. Another important strand of literature in this context focuses on VAR model parameter reduction. With an increasing number of variables to be modelled, the number of coefficients to be estimated increases exponentially. This problem is often referred to as the curse of dimensionality. Demirer et al. (2018) use lasso-type dimension reduction methods combined with generalised variance decompositions in order to estimate the connectedness between 150 bank stocks. An even sparser approach is proposed by Barigozzi and Brownlees (2019), who use lasso-type reduction methods not only to shrink the VAR lag matrices but also to shrink the variance– covariance matrix. Our causal search algorithm delivers an over-identified model, reducing the number of coefficients to be estimated, and hence eases the issue of dimensionality. This paper contributes in several ways to the existing literature. First, we propose an alternative identification strategy which detects causal linkages. As Demiralp and Hoover (2003) show, the empirical procedure is very effective in detecting the true causal connections among BETTENDORF AND HEINLEIN 3939 different variables. Second, we analyse the performance of our algorithm with respect to the Diebold and Yilmaz (2014) measure of connectedness and show in a Monte Carlo experiment that our algorithm outperforms other approaches. 2 Third, we apply our algorithm to the G10 currencies and pay special attention to the choice of the numéraire currency. This choice is of particular importance because it can have strong effects on the estimates, as we will discuss later. Our results suggest that the US dollar as well as the Norwegian krone are the most independent currencies in our sample. By contrast, the Swiss franc and New Zealand dollar have a negligible impact on other currencies. Moreover, a cluster analysis suggests that the currencies can be divided into three groups, which can be identified as: commodity currencies, European currencies, and safe haven/ carry trade financing currencies. We show that following the Brexit referendum, the within cluster dispersion is very low. 2|METHODOLOGY For ytbeing a K1 vector of endogenous variables, we consider a SVAR(1) as follows: B0yt¼Byt1þwt,ð1Þ where Brefers to the KKcoefficient matrix of the lagged vector of endogenous variables. B0defines the KKcontemporaneous coefficient matrix. Uncorrelated structural shocks are denoted by wtNID 0,Σw ðÞ. Note that the off-diagonal entries of Σware 0. We follow the notation of Kilian and Lütkepohl (2017). For brevity, we work here with just one lag and no deterministic terms. For the estimation, however, a constant is included, and the lag order is chosen according to AIC. The reduced form of this model can be written as follows: yt¼Ayt1þut,ð2Þ with A¼B1 0Band ut¼B1 0wt. Traditionally, the contemporaneous matrix B0is uncovered with the help of restrictions motivated by economic theory. For a VAR model of exchange rate returns, economic theory does not provide a unique causal structure that can be imposed on the contemporaneous matrix. However, we achieve (over-)identification using a graph theoretical causal search algorithm which finds contemporaneous causality in the reduced form residuals ut. The correct contemporaneous effects are an important factor in the computation of the FEVD and consequently in the connectedness measure of Diebold and Yilmaz (2014). 2.1 |The PC causal search algorithm and its application to the identification of SVAR models The PC algorithm belongs to the literature on graphtheoretic analysis of causal structures, see Pearl (2000) and Spirtes et al. (2001). 3 A causal structure is represented by a graph with arrows from causes to caused variables. The algorithm uses the residual variance– covariance matrix of the reduced form model as an input to detect the causal structure of a system, a directed acyclical graph (DAG). The PC algorithm cannot necessarily determine the DAG uniquely, but only down to a Markov equivalence class of the DAG. All members of an equivalence class encode the same conditional independence information. By using the conditional independence information the algorithm can only determine the equivalence class, but not distinguish between members of a class. In this way, the algorithm finds some undirected edges. We will determine these undirected edges with the help of a bootstrap procedure, which we will explain in the following. To find the DAG, the algorithm performs an elimination stage and an orientation stage. The elimination stage starts with a graph where all the variables are linked to each other with an undirected link. Then, links are removed based on unconditional and conditional correlation tests, with a tuning parameter αfor Fisher's Z-statistic being used as a significance level. First, connections are removed between two variables, which are unconditionally uncorrelated. Then, connections are eliminated for variables which are uncorrelated conditional on other variables. Here, the correlation of a pair of variables is conditioned on every other variable individually, then on all possible pairs of variables, thereafter on all subsets of three variables and so on up to all possible subsets of conditioning. When there is no more link to be removed, the elimination stage is finished and the skeleton of the graph is identified. In the orientation stage, triples of linked variables A—B—Care analysed. Unshielded colliders (v-struc- tures) A!B Ccan be determined when A and C are independent when conditioned on possible sets of variables, but dependent when conditioned also on B. The algorithm searches for unshielded colliders and directs the edges accordingly. Finally, some more links might be oriented on the basis of logic. Some directions of links would lead to new unshielded colliders or to cyclicality, hence they need to be directed the other way around. 3940 BETTENDORF AND HEINLEIN Cyclicality, like A!B!C!A, is not permitted, hence bi-directional links are likewise not possible. Demiralp et al. (2008) show that a bootstrap procedure is successful in directing the undirected edges. The residuals of the reduced form VAR are drawn randomly with replacement, and new dataset are generated, to which the PC algorithm is applied. The undirected edges are finally directed in the direction, which is prominent more often in the bootstrap runs. Sampling errors or latent variables can lead to conflicting information about edge directions. In these cases, the algorithm returns a bi-directed edge. We decide on the bi-directed edges via our bootstrap procedure. Thus, the bootstrap procedure decides on the undirected edges (Markov equivalence class) and on the edges with conflicting information. If the final graph is a DAG, then it can be mapped in the contemporaneous matrix B 0 , and due to the acyclicality property of the DAG, the contemporaneous matrix can be written as an overidentified lower triangular matrix for some ordering of the variables. Hence the SVAR model is identified. If the final graph contains cyclicality, which might arise due to some conflicting information about certain v-structures, the order condition is fulfilled, but it will not be possible to writetheSVARmodelasanoveridentifiedrecursive form. 4 It is not clear from the onset which alpha value should be chosen in the PC algorithm. With increasing alpha values the algorithm becomes more liberal and so chooses fewer zero restrictions. Following our Monte Carlo simulation and Demiralp et al. (2014), we choose an alpha value of 10% in our application. 2.2 |A connectedness measure using forecast error variance decompositions We use the connectedness measure of Diebold and Yilmaz (2014). The approach is based on the computation of FEVD. 5 The stationary SVAR model in Equation (1) can be written in an MA representation as yt¼X ∞ i¼0 Φiuti¼X ∞ i¼0 Θiwti,ð3Þ where Φiare reduced-form impulse responses and Θiare the structural impulse responses with ΘiΦiB1 0. The matrixes Φican be retrieved recursively by computing Φ0¼IKand Φi¼B1 0B  i. We compute a FEVD dh jk ¼100X h1 i¼0 e0 jΘiek  2=X h1 i¼0X K k¼1 θ2 jk,i,ð4Þ where θ jk,i are the jkth element of Θiand e k is the kth column of I K . The measure dh jk is the proportion of the h-step forecast error variance of variable j, accounted for by innovations from variable k. We multiply the fractions by 100 to obtain percentages. Following Diebold and Yilmaz (2014), the pairwise directional connectedness from kto jis defined as Ch j k¼dh jk:ð5Þ In general Ch j k≠Ch k j, so there are K 2 -Kseparate pairwise directional connectedness measures. The measure of total connectedness can be defined as Ch¼1 KX K j,k¼1,j≠k dh jk:ð6Þ In the following sections, we will compare this measure with other measures of connectedness. One of these alternative measures is the generalised forecast error variance decomposition (GFEVD). For the computation of a GFEVD we follow Lanne and Nyberg (2016) 6 dh jk,g¼100 P h1 i¼0 e0 jΦiΣuekσ1=2 kk  2 P h1 i¼0P K k¼1 e0 jΦiΣuekσ1=2 kk  2,ð7Þ where σ kk are the diagonal entries of Σu. 2.3 |The algorithm We make use of the R software package ‘pcalg’by Kalisch et al. (2012). 7 Our proposed algorithm (see Algorithm 1) starts with the estimation of a reduced form VAR model where the lag order is determined by the Akaike information criterion (AIC). Then, we apply the PC algorithm (PC) to the reduced form residuals and test if the resulting graph is a DAG. If this is the case, we can proceed and determine the contemporaneous matrix (B0) in accordance with the obtained DAG. Otherwise, we bootstrap the reduced form VAR 10,000 times, apply the PC algorithm in each run, and collect the 10,000 suggested graphs. Note that it is important to draw BETTENDORF AND HEINLEIN 3941 vectors from the residuals in such a way that the correlation between the residuals is preserved. Afterwards, we modify the original graph in such a way that the undirected edges become directed according to the direction preferred by the bootstrap. Note that we consider the bootstrap only in order to decide on edges which were originally undirected or bi-directed. Having obtained a DAG, we may proceed with the specification of B0(line 10 of Algorithm 1). Finally, the connectedness measure— or spillover matrix—can be derived from the estimated structural VAR model where the shocks are orthogonalised by B0. 3|THE EFFECTIVENESS OF THE PC ALGORITHM IN THE CONNECTEDNESS APPROACH: AMONTECARLOSTUDY In this section, we evaluate the impact of different identification strategies on the measures of connectedness by performing a Monte Carlo experiment. We generate artificial data with the help of a known data generating process (DGP). Afterwards, we estimate the connectedness matrices for different identification strategies and benchmark them with the theoretical result of the known DGP. We compare the empirical identification with the generalised approach (see Lanne & Nyberg, 2016) and the average-of-all-Cholesky-orderings (see Klößner & Wagner, 2014) approach. The empirical identification is performed with two different algorithms, the PC algorithm and the greedy equivalence search (GES) algorithm of Chickering (2002). 8 Because the appropriate significance level alpha for the individual conditional independence tests of the PC algorithm is not clear, we use two conventional options: 5% and 10%. The artificial data are generated recursively according to the SVAR(1): yt¼B1 0Byt1þB1 0wt,ð8Þ with structural shocks wtNID 0,Σw ðÞand y0¼0.To eliminate dependence on the initial condition we discard the first 80% of the generated data in all cases. The lag matrix, B,isKKwith random uniform coefficients between 0.05 and 0.05. The residuals, wt, are drawn randomly from independent normal distributions with mean 0 and variance 1. For the contemporaneous matrix, B0, we generate random directed acyclic graphs (DAGs) with a fixed expected number of neighbours. We use random Erd} os-Rényi graphs for the DAGs, multiply the matrix entries by 1 and add an identity matrix. In this way we generate a sparse contemporaneous matrix with some negative off-diagonal entries between 0 and 1. 9 We perform this Monte Carlo study for N=100 datasets in each MC experiment. The categories are: two different ALGORITHM 1 1: procedure IDENTIFICATION 2: ut,A½ VAR data,p ¼AICðÞ 3: graph PC algorithm α,ut ðÞ 4: if graph is directed-acyclical-graph then 5: DAG graph 6: else 7: DAG Bootstrap(ut,A,α,graph) 8: end if 9: B0 DAG 10: connectedness FEVD SVAR B0,AðÞ½ 11: end procedure 12: function BOOTSTRAP(ut,A,α,p,graph) 13: for runs {1, 2,…, 10,000} do 14: artificial data Create artificial data ut,AðÞ 15: uBS t  VAR artificial data,pðÞ 16: bootstrap-graph runsðÞ PC algorithm α,uBS t  17: end for 18: return direct undirected edges ingraphaccording tobootstrapgraph 19: end function 3942 BETTENDORF AND HEINLEIN system dimensions (K=8/16), three different levels of sparsity (d=1/3/5) and two different sample lengths (T=250/2500). 10 The results are evaluated as follows. For each identification method, we compute four measures in terms of recovering the true connectedness matrix. For all four measures, we report the mean absolute error (MAE) of the estimated measure relative to the measure for the true connnectedness matrix. The first measure, C, is the MAE of the off-diagonal elements of the connectedness matrix: C¼1 NX N i¼1 1 K2KX K j,k¼1,j≠k jCh j k,iCh j k,ij,ð9Þ whereby the variables with a star are the true connectedness values. Cis an important measure, as it places a strong weight on the direction of the connectedness. The second measure, T, is the MAE of the total connectedness: T¼1 NX N i¼1 jCh iCh ij:ð10Þ Here, it is not so much the direction of the links that is evaluated, but rather whether the over-identifying zeros of the PC algorithm are appropriate. The third measure, S, is the MAE of the skewness of the distribution of the off-diagonal entries of the connectedness matrix: S¼1 NX N i¼1 jSkew Ch j k,i no j,k¼1…K,j≠kSkew Ch j k,i no j,k¼1…K,j≠kj: ð11Þ While the fourth measure, K, is the MAE of the kurtosis of the distribution of the off-diagonal entries of the connectedness matrix: TABLE 1 Monte Carlo simulation: Comparing connectedness measures for different identification strategies relative to the correct connectedness measures using 100 random DAGs dimension 8 d=1 T=250 T=2500 CT SK CTSK avgChol 1.849 3.648 1.798 14.836 1.487 0.558 1.576 13.606 GFEVD 2.823 12.233 1.523 11.965 2.233 7.126 1.260 10.400 PCalg 5% 1.644 2.594 0.455 4.245 1.031 0.534 0.228 2.166 PCalg 10% 1.664 2.824 0.452 4.334 1.045 0.548 0.245 2.300 GES 1.421 2.731 0.444 4.243 1.017 0.405 0.238 2.290 d=3 T=250 T=2500 CT SKCT SK avgChol 5.386 4.337 1.331 5.342 5.195 2.219 1.310 5.350 GFEVD 8.276 28.478 1.207 4.644 8.284 23.241 0.862 3.637 PCalg 5% 4.722 3.249 0.520 3.261 3.584 2.738 0.354 2.217 PCalg 10% 4.668 3.256 0.502 3.086 3.630 2.554 0.366 2.247 GES 5.221 3.983 0.411 2.449 4.143 1.811 0.363 2.208 d=5 T=250 T=2500 CTSKCTSK avgChol 9.695 6.007 1.559 3.596 9.643 5.091 1.564 3.586 GFEVD 11.552 30.634 1.113 3.098 13.508 28.869 0.858 2.642 PCalg 5% 9.592 8.656 0.754 3.730 9.157 6.858 0.576 2.561 PCalg 10% 9.728 7.621 0.671 3.217 9.227 6.807 0.579 2.634 GES 10.914 6.249 0.442 1.865 10.628 5.452 0.415 1.762 Note: 100 random Erd} os-Rényi graphs with eight nodes. d(1, 3, 5) corresponds to the expected number of neighbours per node, more precisely the expected sum of the in- and out-degree. Sample size 250/2500 observations. Cis the MAE of the off-diagonal entries of the connectedness matrix. Tis the MAE of the total connectedness. Sand Kare the MAEs of the skewness and kurtosis of the distribution of the off-diagonal entries of the connectedness matrix. BETTENDORF AND HEINLEIN 3943 K¼1 NX N i¼1 jKurt Ch j k,i no j,k¼1…K,j≠kKurt Ch j k,i no j,k¼1…K,j≠kj: ð12Þ These distributional measures aim to evaluate whether the extreme values in the connectedness matrices of the identification strategies are comparable to the theoretical connectedness. The results of the Monte Carlo experiments are displayed in Tables 1and 2. The PC algorithm performs, in general, better than the average-of-all-Cholesky-orderings approach and the generalised approach. While the other two identification strategies usually split the connectedness between j and k in such a way that Ch j k≈Ch k j, the causal search algorithm manages to find the true causal connectedness. Even when the PC algorithm might find incorrect directions for some links, overall the true causal structure is uncovered to a much higher degree, which can be seen in the lower MAE values of our measure C. The generalised approach overestimates the total connectedness strongly in all cases, which can be seen in the high MAE in the measure T. This result is in line with findings by De TABLE 2 Monte Carlo simulation: Comparing connectedness measures for different identification strategies relative to the correct connectedness measures using 100 random DAGs dimension 16 d=1 T=250 T=2500 CT SK CTSK avgChol 1.291 8.006 2.603 33.602 0.864 0.997 2.219 30.262 GFEVD 2.066 20.489 2.572 30.278 1.334 8.775 1.763 22.862 PCalg 5% 1.020 5.867 0.613 10.554 0.581 0.680 0.322 5.870 PCalg 10% 1.023 6.179 0.586 9.586 0.586 0.670 0.321 5.810 GES 0.995 6.202 0.602 9.510 0.497 0.641 0.251 4.384 d=3 T=250 T=2500 CT SK CT SK avgChol 3.021 7.009 1.791 12.483 2.762 2.477 1.752 12.510 GFEVD 4.871 36.769 2.112 13.264 4.468 26.757 1.460 9.546 PCalg 5% 2.309 2.932 0.505 5.531 1.729 1.896 0.494 5.094 PCalg 10% 2.276 3.332 0.515 5.633 1.671 1.796 0.453 4.607 GES 2.455 5.582 0.430 4.572 1.881 1.850 0.417 4.272 d=5 T=250 T=2500 CT SK CT SK avgChol 5.018 8.309 2.163 10.084 4.900 5.410 2.140 10.111 GFEVD 6.420 37.451 1.720 9.183 6.684 33.617 1.593 8.289 PCalg 5% 4.662 4.533 0.772 6.909 3.952 4.408 0.640 5.155 PCalg 10% 4.606 4.381 0.822 7.276 4.000 4.532 0.685 5.542 GES 5.262 7.762 0.648 4.871 4.854 5.255 0.511 3.802 Note: 100 random Erd} os-Rényi graphs with eight nodes. d(1, 3, 5) corresponds to the expected number of neighbours per node, more precisely the expected sum of the in- and out-degree. Sample size 250/2500 observations. Cis the MAE of the off-diagonal entries of the connectedness matrix. Tis the MAE of the total connectedness. Sand Kare the MAEs of the skewness and kurtosis of the distribution of the off-diagonal entries of the connectedness matrix. TABLE 3 Monte Carlo simulation: comparing connectedness measures for different identification strategies relative to the correct connectedness measures using 100 random DAGs. Specification similar to the application: dimension =9, d=3.8, T=2048 CT SK avgChol 6.042 3.342 1.332 4.750 GFEVD 9.371 28.402 0.803 3.249 PCalg 5% 4.863 4.033 0.467 2.725 PCalg 10% 4.875 3.661 0.484 2.831 GES 5.410 2.939 0.463 2.645 Note: 100 random Erd} os-Rényi graphs with nine nodes. d=3.8 corresponds to the expected number of neighbours per node, more precisely the expected sum of the in- and out-degree. Sample size 2048 observations. Cis the MAE of the off-diagonal entries of the connectedness matrix. Tis the MAE of the total connectedness. Sand Kare the MAEs of the skewness and kurtosis of the distribution of the off-diagonal entries of the connectedness matrix. 3944 BETTENDORF AND HEINLEIN Santis and Zimic (2018). The average-of-all-Cholesky- orderings approach usually overestimates the total connectedness slightly. The PC algorithm partly overestimates and partly underestimates the total connectedness, depending on the sparsity of the network, performing well for many DGPs, but poorly in a small number of cases. The GES algorithm tends to overestimate the total connectedness more often. The empirical approaches are superior in estimating connectedness for models with sparse contemporaneous matrices. In general there is no clear favourite between a PC algorithm with a 5% and 10% significance level. The GES algorithm performs equally well as the PC algorithm and is particularly strong for sparse contemporaneous matrices. To reinforce the relevance of the Monte Carlo simulation to our empirical application, we perform a simulation experiment where we mimic the settings of the application, see Table 3. In the application we have a dimension of 9, a level of sparsity of d=3.8 and 2048 observations. 11 In the Monte Carlo experiment, the causal search PC algorithm performs strongly, achieving low MAEs, especially in category C, which measures the direction of connectedness. 4|AN APPLICATION TO EXCHANGE RATE DATA The bilateral exchange rate can be interpreted as the relative price between two currencies. Here, it is defined as the foreign currency price of buying one unit of home currency (quantity quotation). A positive shock to the bilateral exchange rate in quantity quotation can thus be interpreted as a positive shock to the demand of the home currency or a negative shock to the demand of the foreign currency. These shocks can trigger movements in other exchange rates as well. The reasons behind the international effects are manifold. One could think of currency (basket) pegs or international substitution effects, for instance. The aim of this exercise is to uncover the network of spillover effects between exchange rate returns. 12 We apply the proposed algorithm to the G10 bilateral euro exchange rates and cluster the exchange rates in order to uncover potential currency blocs afterwards. All bilateral exchange rates are downloaded in daily frequency from the ECB statistical data warehouse (SDW) and correspond to the ECB reference rates, representing the 14:15 CET fixing. 13 The sample covers the period between January 2010 and December 2017. We start in 2010 in order to exclude potential effects arising from the 2008 financial crisis. As ECB reference rates are expressed in quantity quotation and quoted against the euro, we transform the rates in such a way that the pound sterling becomes the numéraire. All transformed series enter our model in log differences. The reasoning behind changing the numéraire currency is discussed in the following section. 4.1 |Choice of the numéraire currency The bilateral (or multilateral) nature of exchange rates poses a problem for researchers and practitioners. When regressing exchange rate returns on exchange rate returns, the correct choice of the numéraire currency (or basket) is crucial, because the numéraire can have substantial effects on the estimates. If both currencies were pegged to the numéraire, the regression coefficient would be zero, implying that despite the common peg no relationship would exist. This problem has been extensively discussed by the literature on currency baskets and blocks, for example by Frankel and Wei (2008), Frankel and Xie (2010), or Ohno (1999). The US dollar, the most heavily traded currency, appears to be a good choice as the numéraire currency. But the afore mentioned statistical problems arise if currencies, pegged to the dollar enter the model. Despite the peg, these exchange rates—expressed in US dollar— would appear to be unconnected. More importantly, however, the connectedness of the US dollar could not be estimated if it served as the numéraire currency. This would eliminate important information, as several studies have pointed out that the US dollar shares certain properties with other currencies, for example its status as a safe haven currency (see Hossfeld & MacDonald, 2015). Other studies such as Frankel and Wei (2008) and Ohno (1999) relied on the Swiss franc as the numéraire currency. The Swiss franc seemed to be an appealing choice, because its trading volume is high and the currency was independent at that time. The Swiss franc lost this property when the Swiss National Bank introduced a minimum rate vis- a-vis the euro on 6 September 2011. 14 When the numéraire currency is pegged to another currency in the sample, the exchange rate has no variance, which can be explained by other currencies. Apart from the numerical problems that arise from this, the series would not have any variance and should thus not be employed within the Diebold and Yilmaz (2014)approach. 15 Deutsche Bundesbank (2019) provides numerical examples with respect to this issue. The literature on basket weights proposes using a basket of different currencies as the numéraire. Frankel and Xie (2010) claim that monetary authorities are more likely to use a weighted average of currencies as a reference for possible interventions when the exchange rate BETTENDORF AND HEINLEIN 3945 obtained spillover matrix (see Table C1) shows a very similar pattern when compared to Table 9. 21 The total connectedness increases slightly because of missing overidentifying restrictions. 22 However, the relative importance of the shocks remains the same, therefore the results are qualitatively similar. For instance, shocks to the euro explain 8.1% (9.1% with over-identifying restrictions) of Swiss franc forecast error variance and only 0.3% (0.0% with over-identifying restrictions) of New Zealand forecast error variance, although the coefficient b 97 is now unrestricted. It is also surprising to see that the measure of total connectedness, C 10 =41.0 is very similar to the Cholesky application with random ordering (here, C 10 =41.1). This finding supports Diebold and Yilmaz (2014), who argue that the system-wide summary measure (C h ) is often robust to the Cholesky ordering. However, this does not change the fact that the result is due to a random ordering and thus (potentially) a result of misspecification. In summary, the PC algorithm provides us with welldefined directed edges, which enable us to unveil a directed network of exchange rates. This gives the PC algorithm a clear advantage over the other presented methods. However, the PC algorithm is computationally also the most intense procedure, particularly because of the time-consuming bootstrap. 23 4.4 |Cluster analysis In this section, we exploit the connectedness between exchange rates (displayed in Table 9) in order to divide the network into clusters (also known as communities or modules). A cluster is characterised by a high number of edges between nodes within the cluster, relative to the number of edges to nodes outside the cluster. In this sense, we visualise the previously estimated connectedness and identify groups of exchange rates with a relatively high intra-group connectedness. These groups can be interpreted as currency blocs. The currencies of a bloc are likely to move in tandem, which is important information for policy makers and the management of currency risk. Note that our definition of a currency bloc is more general than the definition by Fischer (2016), for instance. The quality of the partitioning of a whole network, which can consist of as many clusters as nodes, is thus often expressed by a measure, depending on the differences between the numbers of edges within clusters and the numbers of edges that would exist if it were a random network model. Hence, positive values indicate the existence of clusters. This measure is referred to as modularity.Fora detailed explanation, we refer the reader to Blondel et al. (2008), who propose a popular algorithm (hereafter: Louvain algorithm) which detects the best clustering by maximizing modularity. Initially, the algorithm assigns each node to a single cluster. In a second step, the algorithm moves nodes to new clusters if gains in modularity can be achieved until no additional gain can be achieved (see Blondel et al., 2008). One drawback of the Louvain algorithm is that it is designedforundirectednetworks. Consequently, it is not feasible given the causal structure of our network. Dugué and Perez (2015)provideasolutionto this common problem. They modify the Louvain algorithm in such a way that it allows for directed modularity as defined by Leicht and Newman (2008). Using the Directed Louvain algorithm by Dugué and Perez (2015), we aim to partition the network in Table 9. TABLE 9 Connectedness: PC algorithm (α=0.1) AUD CAD CHF EUR NOK NZD SEK USD JPY IN AUD 53.0 16.8 0.1 0.0 24.1 0.0 0.0 5.9 0.1 47.0 CAD 0.1 61.6 0.1 0.1 16.8 0.0 0.0 21.2 0.1 38.4 CHF 0.1 0.0 61.9 9.1 8.4 0.0 3.5 8.9 8.0 38.1 EUR 0.5 0.3 0.1 39.6 36.1 0.0 13.8 6.9 2.7 60.4 NOK 0.4 0.1 0.0 0.0 98.6 0.0 0.2 0.2 0.5 1.4 NZD 24.3 11.2 0.4 0.0 14.5 42.3 0.0 6.6 0.6 57.7 SEK 1.6 0.6 0.0 0.0 52.0 0.0 45.1 0.3 0.5 54.9 USD 0.2 0.1 0.0 0.1 0.2 0.0 0.0 99.3 0.0 0.7 JPY 0.0 0.0 0.0 0.0 0.3 0.0 0.1 41.2 58.4 41.6 OUT 27.1 29.2 0.7 9.4 152.4 0.1 17.6 91.1 12.5 C 10 =37.8 Note: The table shows the (10 periods ahead) forecast error variance decomposition of the SVAR model, which is identified by the PC algorithm. The column ‘IN’corresponds to the row sum of the non-diagonal variance shares (i.e., the total share of variance which is explained by [international] shocks). The column ‘OUT’corresponds to the column sum of the non-diagonal variance shares (i.e., the total share of variance which is explained by the corresponding column variable). C 10 refers to the measure of total connectedness (see section 2.2). 3952 BETTENDORF AND HEINLEIN We observe in Figure 2that the algorithm classifies the G10 currencies into three different clusters. The first cluster contains the AUD, CAD and NZD, which are often referred to as commodity currencies. Another common property is that investments in these countries provide the investor with a relatively high yield. The NOK is often also referred to as a commodity currency, but it is part of the second cluster. In addition to the NOK, this cluster also contains the EUR as well as the SEK and thus European currencies only. The third cluster contains the FIGURE 2 Partition according to the directed Louvain algorithm. This figure shows the exchange rates clustered according to the spillover matrix of the PC algorithm (Table 9). Cluster 1 (dark grey): AUD, CAD, NZD; Cluster 2 (grey): EUR, NOK, SEK; Cluster 3 (light grey): CHF, USD, JPY. Causation propagates clockwise. FIGURE 3 Exchange rate movements following the Brexit referendum. This figure shows pound sterling exchange rate movements following the Brexit referendum (pound sterling in quantity quotation). Exchange rates are marked according to their corresponding cluster. Cluster 1 (solid): AUD, CAD, NZD; Cluster 2 (dashed): EUR, NOK, SEK; Cluster 3 (dotted): CHF, USD, JPY. Source: ECB. BETTENDORF AND HEINLEIN 3953 CHF, USD and JPY. These currencies are often referred to as safe haven and/or carry funding currencies (see Ferreira Filipe & Suominen, 2013; Hossfeld & MacDonald, 2015). Thus, the latter group has the tendency to appreciate in times of financial stress, either because investors are seeking a safe haven for their investments or due to the unwinding of carry trades. 4.5 |Empirical assessment of the clustering: Brexit referendum In this section, we assess the quality of the partition suggested by the Directed Louvain algorithm. To this end we normalise all exchange rates before the referendum on the UK's membership of the EU, the result of which surprised many market participants, and discuss their movements during the trading days following the referendum (see Figure 3). The Brexit referendum is an appealing example, as it is a shock to the numéraire, which affects all other currencies. Following the referendum, the pound sterling depreciated against all currencies in our sample. Figure 3shows the movements of the pound sterling exchange rates against all currencies in quantity quotation (cross rates of ECB reference rates). In order to simplify the interpretation, exchange rates have been normalised to 100 on the day of the referendum (23 June 2016 is day 0). 24 The similarity of movements within clusters is striking. Currencies within the first (solid lines) and second (dashed lines) cluster, in particular, move closely in tandem. Only currencies in the third cluster (dotted lines) display a slightly larger dispersion. Nor is it surprising that the pound sterling depreciates strongly against the third cluster, which reflects safe haven and carry funding currencies. These are supposed to appreciate in times of financial stress. Additionally, it is expected that the European currencies appreciate the least of the three clusters, because the uncertainty surrounding Brexit means uncertainty for the European Monetary Union. The Swiss franc, which has been found by our procedure to belong to the cluster of safe havens, moves in the case of the Brexit experiment more closely in line with European currencies. Overall, we observe that exchange rate movements follow a very similar pattern, but we also see that the dispersion within clusters is strikingly low. 5|CONCLUSIONS The literature on connectedness between exchange rates has so far ignored a potential causal structure. Research along the lines of Diebold and Yilmaz (2014) is based on a FEVD in a VAR framework. The difficulty in this context is the identification of the variance–covariance matrix in order to orthogonalise the shocks. We show that a Cholesky decomposition, which is frequently used, can lead to arbitrary results, as the outcome depends heavily on the ordering of the variables. A generalised FEVD is independent of the ordering of the variables, but it is unable to detect causality between the shocks. The same applies when all possible orderings of variables are considered (see Klößner & Wagner, 2014). We address this problem by employing a causal search algorithm from the machine learning literature, which is able to find causality in contemporaneous data. This approach is then applied to the G10 currencies, whereby nine currencies are modelled vis- a-vis the pound sterling as the numéraire currency. Our results suggest that the US dollar and the Norwegian krone are the most independent currencies in our sample. Shocks to these currencies affect a large set of other currencies. We also observe that connectedness between commodity currencies and those that are often referred to as safe haven and/or carry funding currencies is particularly high. Using a clustering algorithm, we identify three currency clusters which confirm the previous findings. The first cluster contains commodity currencies such as the AUD, CAD and NZD. The second cluster comprises the European currencies EUR, NOK and SEK. Finally, the third cluster contains the CHF, USD and JPY—currencies, which are often referred to as safe haven or carry funding currencies. They have the tendency to appreciate in times of financial stress. In an additional exercise, we evaluate the movements of currencies with respect to their clusters following the Brexit referendum. We observe that the dispersion of exchange rate movements within clusters is indeed relatively low, particularly for the first and second clusters. The third cluster shows the strongest appreciation against the pound sterling following the referendum. The Swiss franc, however, appears to move more closely in line with other European currencies (second cluster). Overall, these estimates provide important information for policy makers and practitioners, as they shed light on potential co-movements between certain exchange rates. ACKNOWLEDGEMENTS The authors are grateful to Joscha Beckmann, Geert Bekaert, Christoph Fischer, Ulrich Grosch, Kevin D. Hoover, M. Hashem Pesaran and an unknown referee, as well as participants of the Bundesbank research seminar, the ICMAIF 2019, Rethymno, 25th CEF conference, Ottawa, the 6th IAAE conference, Nicosia, and the Swiss Society of Economics and Statistics conference 2021, Zürich, for helpful comments and suggestions. The views 3954 BETTENDORF AND HEINLEIN expressed in this paper are those of the authors and do not necessarily coincide with the views of the Deutsche Bundesbank or the Eurosystem. DATA AVAILABILITY STATEMENT The data that support the findings of this study are openly available in the ECB Statistical Data Warehouse (SDW) at https://urldefense.com/v3/__https://sdw.ecb. europa.eu/ ORCID Timo Bettendorf https://orcid.org/0000-0002-6375-055X ENDNOTES i Note that the G10 currencies refer to the 10 most heavily traded currencies and not to the Group of Ten countries. 1 In order to measure spillover effects using variance decompositions, the numéraire currency should neither be an anchor currency nor be pegged to another currency. Returns of fixed exchange rates have no volatility, implying that variance decompositions would be meaningless. Moreover, we interpret an exchange rate as an asset price and focus in our empirical analysis on exchange rate changes. Within this strand of literature, exchange rate changes (log differences) are referred to as exchange rate returns. 2 Previous studies have focussed on the ability of the PC algorithm, a causal search algorithm which will be explained in the following sections, to detect the correct causal structure. We, however, focus on the ability of different approaches to detect the correct degree of network connectedness. 3 PC stands for the initials of its inventors, Peter Spirtes and Clark Glymour. 4 On these grounds, we estimate our SVAR models equation by equation using OLS. 5 For an introduction to FEVDs, see Lütkepohl (2005). 6 Chan-Lau (2017) studies the advantages of the Lanne and Nyberg (2016) approach in a connectedness application. Note that this is not the same GFEVD approach as in Diebold and Yılmaz (2014). Nevertheless, it is also subject to the shortcoming that it fails in modelling contemporaneous causality. 7 The chosen settings are: (conservative =TRUE, solve. confl =TRUE, u2pd =c(‘relaxed’)). By choosing the conservative rule instead of the retry option, the algorithm produces a fully order-independent output; see Colombo and Maathuis (2014). 8 Note that the PC algorithm is a constraint-based approach, while the GES algorithm is a score-based method. Constraint-based approaches work with conditional independence tests. Scorebased approaches assign scores to particular graph structures based on the data fit, for example using scoring metrics like the BIC score, which we use here. 9 Given positive correlations between variables in applications to most markets, it is a reasonable assumption to focus on negative entries in the contemporaneous matrix. 10 In contrast to our application, we perform 100 bootstrap runs in the Monte Carlo experiment. 11 We identified 17 links in our application. Hence, the sum of in- and out-degree is 34. Thirty-four divided by 9 is 3.8. 12 As our focus is on the identification of the contemporaneous causal structure and its impact on the connectedness between returns, we are not interested in dynamic total connectedness, which is a standard procedure in this literature. 13 Note that all exchange rates are fixed at the same time. Hence, trading times do not overlap. 14 The minimum rate was abandoned on 13 January 2015, which caused the Swiss franc to appreciate strongly against several major currencies. 15 Note that even if the numéraire is an independent currency, hard pegs among other currencies in the sample cause collinearity. 16 When performing Ljung-Box tests on the residuals of the VAR (1), the null hypothesis of no autocorrelation cannot be rejected in each of the equations under the 5% significance level. Hence, a lag length of one in the VAR is adequate to model the multivariate dynamics of the system. 17 Bi-directed edges have not been found in our application. Where a bi-directed edge is detected in the bootstrap of the application, these edges are displayed under ‘bi-directed’in Table 5. Overall, bi-directed edges are not prominent in our application, no bi-directed edge has been found on the original data, and only one edge showed a relevant occurrence of bi-directed outcomes in the bootstrap. The absence of bi-directed edges indicates that the system of exchange rates is contemporaneously self-contained. Latent variables, which might affect several exchange rates, seem to enter only with a lag. 18 Note that due to the over-identifying restrictions, this ordering is not unique. For example, the Japanese yen could also be ordered behind the Norwegian krone and the following other currencies, but it needs to be before the euro. Hence, some other orderings would also be consistent with the output of the causal search algorithm. However, the computed connectedness measures in the following are not influenced by our choice of a recursive ordering. 19 We observe that the sum of variance shares is close to unity, but not exactly unity. 20 The minimum exchange rate was introduced by the Swiss National Bank on 6 September 2011 and abandoned on 15 January 2015. It served as a key monetary policy instrument. 21 Note that in Table C1 the currencies follow the recursive ordering. 22 Given the Cholesky decomposition, we no longer have exact zero entries in the lower diagonal of the contemporaneous matrix. The non-zero coefficients translate into minor changes in the variance shares. 23 However, we found that working with a small number of bootstrap runs usually leads to the same result as with a large number of runs. For example, for our application we ran 100 experiments with bootstraps with 100 runs each and found that 86 experiments lead to the same contemporaneous matrix as a bootstrap with 10,000 runs, while in 14 experiments one of the edges is found to be directed differently. 24 Note that the fixing of ECB reference rates takes place at 14:15 CET—before the results of the referendum were published. BETTENDORF AND HEINLEIN 3955 REFERENCES Aloosh, A., & Bekaert, G. (2021). Currency factors. Management Science 0(0). https://doi.org/10.1287/mnsc.2021.4023 Barigozzi, M., & Brownlees, C. (2019). Nets: Network estimation for time series. Journal of Applied Econometrics,34(3), 347–364. Blondel, V., Guillaume, J.-L., Lambiotte, R., & Lefebvre, E. (2008). Fast unfolding of communities in large networks. Journal of Statistical Mechanics: Theory and Experiment,P10008,1–12. Chan-Lau, M. J. A. (2017). Variance decomposition networks: Potential pitfalls and a simple solution. Working paper 17/107. International Monetary Fund. Chickering, D. M. (2002). Optimal structure identification with greedy search. Journal of Machine Learning Research,3(Nov), 507–554. Colombo, D., & Maathuis, M. H. (2014). Order-independent constraint-based causal structure learning. The Journal of Machine Learning Research,15(1), 3741–3782. De Santis, R. A., & Zimic, S. (2018). Spillovers among sovereign debt markets: Identification through absolute magnitude restrictions. Journal of Applied Econometrics,33(5), 727–747. Demiralp, S., & Hoover, K. D. (2003). Searching for the causal structure of a vector autoregression. Oxford Bulletin of Economics and Statistics,65, 745–767. Demiralp, S., Hoover, K. D., & Perez, S. J. (2008). A bootstrap method for identifying and evaluating a structural vector autoregression. Oxford Bulletin of Economics and Statistics,70(4), 509–533. Demiralp, S., Hoover, K. D., & Perez, S. J. (2014). Still puzzling: Evaluating the price puzzle in an empirically identified structural vector autoregression. Empirical Economics,46(2), 701–731. Demirer, M., Diebold, F. X., Liu, L., & Yilmaz, K. (2018). Estimating global bank network connectedness. Journal of Applied Econometrics,33(1), 1–15. Deutsche Bundesbank. (2019). Parallels in the exchange rate movements of major currencies. Monthly Report,July 2019,19–37. Diebold, F. X., & Yilmaz, K. (2009). Measuring financial asset return and volatility spillovers, with application to global equity markets. The Economic Journal,119(534), 158–171. Diebold, F. X., & Yilmaz, K. (2012). Better to give than to receive: Predictive directional measurement of volatility spillovers. International Journal of Forecasting,28(1), 57–66. Diebold, F. X., & Yilmaz, K. (2014). On the network topology of variance decompositions: Measuring the connectedness of financial firms. Journal of Econometrics,182(1), 119–134. Diebold, F. X., & Yilmaz, K. (2015). Financial and macroeconomic connectedness: A network approach to measurement and monitoring. Oxford University Press. Dugué, N., & Perez, A. (2015). Directed louvain: Maximizing modularity in directed networks (PhD thesis). Université d'Orléans. Ferreira Filipe, S., & Suominen, M. (2013). Currency carry trades and funding risk. AFA 2014 Philadelphia Meetings Paper. Fischer, C. (2016). Determining global currency bloc equilibria: An empirical strategy based on estimates of anchor currency choice. Journal of International Money and Finance,64, 214–238. Frankel, F., & Wei, S.-J. (2008). Estimation of de facto exchange rate regimes: Synthesis of the techniques for inferring flexibility and basket weights. IMF Staff Papers,55(3), 384–416. Frankel, J., & Xie, D. (2010). Estimation of de facto flexibility parameter and basket weights in evolving exchange rate regimes. American Economic Review,100, 568–572. Greenwood-Nimmo, M., Nguyen, V. H., & Rafferty, B. (2016). Risk and return spillovers among the G10 currencies. Journal of Financial Markets,31,43–62. Greenwood-Nimmo, M., Nguyen, V. H., & Shin, Y. (2017). What's mine is yours: Sovereign risk transmission during the european debt crisis. Working Paper 17/17, Melbourne Institute. Heinlein, R., & Krolzig, H.-M. (2012). Effects of monetary policy on theUSDollar/UKpoundexchangerate.Istherea‘delayed overshooting puzzle’?Review of International Economics,20, 443–467. Hossfeld, O., & MacDonald, R. (2015). Carry funding and safe haven currencies: A threshold regression approach. Journal of International Money and Finance,59, 185–202. Huynh, T. L. D., Nasir, M. A., & Nguyen, D. K. (2020). Spillovers and connectedness in foreign exchange markets: The role of trade policy uncertainty. The Quarterly Review of Economics and Finance.https://doi.org/10.1016/j.qref.2020.09.001 IMF. (2016). Annual report on exchange arrangements and exchange restrictions 2016. Technical Report, International Monetary Fund. Kalisch, M., Mächler, M., Colombo, D., Maathuis, M. H., & Bühlmann, P. (2012). Causal inference using graphical models with the R package pcalg. Journal of Statistical Software,47(11), 1–26. Kilian, L., & Lütkepohl, H. (2017). Structural vector autoregressive analysis. Cambridge University Press. Klößner, S., & Wagner, S. (2014). Exploring all VAR orderings for calculating spillovers? Yes, we can!—A note on Diebold and Yilmaz (2009). Journal of Applied Econometrics,29(1), 172–179. Kühl, M. (2018). Excess comovements between the euro/US dollar and pound sterling/US dollar exchange rates. Applied Economics,50(34–35), 3664–3685. Lanne, M., & Nyberg, H. (2016). Generalized forecast error variance decomposition for linear and nonlinear multivariate models. Oxford Bulletin of Economics and Statistics,78(4), 595–603. Le, T., Martin, F., & Nguyen, D. (2018). Dynamic connectedness of global currencies: A conditional Granger-causality approach. Working Paper hal-01806733. Leicht, E. A., & Newman, M. E. (2008). Community structure in directed networks. Physical Review Letters,100(11), 118703. Lütkepohl, H. (2005). New introduction to multiple time series analysis. Springer. Ohno, K. (1999). Exchange rate management in developing asia. Working paper 1. Asian Development Bank Institute. Pearl, J. (2000). Causality: Models, reasoning, and inference. Cambridge University Press. Pesaran, H. M., & Shin, Y. (1998). Generalized impulse response analysis in linear multivariate models. Economics Letters,58(1), 17–29. Scida, D. (2018). Structural VAR and financial networks: A minimum distance approach to spatial modeling. SSRN.https://doi. org/10.2139/ssrn.2860866 Spirtes, P., Glymour, C., & Scheines, R. (2001). Causation, prediction, and search (2nd ed.). MIT Press. Swanson, N. R., & Granger, C. W. (1997). Impulse response functions based on a causal approach to residual orthogonalization in vector autoregressions. Journal of the American Statistical Association,92(437), 357–367. 3956 BETTENDORF AND HEINLEIN Wan, Y., & He, S. (2021). Dynamic connectedness of currencies in G7 countries: A Bayesian time-varying approach. Finance Research Letters,41, 101896. Wen, T., & Wang, G.-J. (2020). Volatility connectedness in global foreign exchange markets. Journal of Multinational Financial Management,54, 100617. Yang, J., Tong, M., & Yu, Z. (2021). Housing market spillovers through the lens of transaction volume: A new spillover index approach. Journal of Empirical Finance,64, 351–378. How to cite this article: Bettendorf, T., & Heinlein, R. (2023). Connectedness between G10 currencies: Searching for the causal structure. International Journal of Finance & Economics, 28(4), 3938–3959. https://doi.org/10.1002/ijfe.2629 BETTENDORF AND HEINLEIN 3957 APPENDIX A: ESTIMATED MATRICES A which can be inverted and scaled by the standard deviations of the residuals to. APPENDIX B: MAXIMUM AND MINIMUM CONNECTEDNESS B TABLE B1 Maximum connectedness considering all permutations of recursive orderings AUD CAD CHF EUR NOK NZD SEK USD JPY AUD 99.8 44.0 9.9 18.9 28.4 58.1 26.0 11.8 9.5 CAD 43.8 99.1 11.0 17.7 24.7 31.9 20.2 28.4 12.7 CHF 9.9 11.0 99.5 36.5 15.9 11.2 18.8 15.4 23.0 EUR 18.8 17.7 36.5 99.3 40.6 18.7 55.7 18.3 20.2 NOK 28.5 24.5 15.8 40.3 98.6 22.6 52.5 5.6 5.2 NZD 58.1 32.0 11.2 18.8 22.6 99.7 20.7 9.5 10.8 SEK 25.9 20.1 18.9 55.6 52.7 20.6 99.0 9.0 8.5 USD 12.0 28.4 15.4 18.2 5.7 9.6 9.1 99.4 41.1 JPY 9.4 12.7 23.0 19.9 4.9 10.7 8.1 41.2 99.6 Note: The table shows (10 periods ahead) forecast error variance decomposition values of SVAR models which are identified by a Cholesky decomposition. The entries of the matrix show the maximum entries which can be achieved with a Cholesky decomposition approach considering all possible orderings. 3958 BETTENDORF AND HEINLEIN APPENDIX C: ROBUSTNESS ANALYSIS C TABLE C1 Connectedness: Cholesky (ordering according to PC algorithm) USD JPY NOK CAD AUD SEK EUR CHF NZD IN USD 99.4 0.0 0.2 0.1 0.2 0.0 0.1 0.0 0.0 0.6 JPY 41.2 58.4 0.3 0.0 0.0 0.1 0.0 0.0 0.0 41.6 NOK 5.6 1.2 92.5 0.1 0.4 0.2 0.0 0.0 0.0 7.5 CAD 28.4 0.1 14.6 56.5 0.1 0.0 0.1 0.1 0.0 43.5 AUD 11.8 1.3 20.9 16.2 49.6 0.0 0.0 0.1 0.0 50.4 SEK 9.0 1.9 44.4 0.3 1.1 43.2 0.0 0.0 0.0 56.8 EUR 18.3 5.2 28.6 0.2 0.1 12.3 35.1 0.1 0.0 64.9 CHF 15.4 8.9 8.5 0.0 0.0 2.0 8.1 57.2 0.0 42.8 NZD 9.5 3.0 16.1 10.6 21.0 0.0 0.3 0.1 39.4 60.6 OUT 139.2 21.6 133.6 27.5 22.9 14.7 8.6 0.4 0.1 C 10 =41.0 Note: The table shows the (10 periods ahead) forecast error variance decomposition of the SVAR model which is identified by a Cholesky decomposition determined by the PC algorithm. The column ‘IN’corresponds to the row sum of the non-diagonal variance shares (i.e., the total share of variance which is explained by [international] shocks). The column ‘OUT’corresponds to the column sum of the non-diagonal variance shares (i.e., the total share of variance which is explained by the corresponding column variable). C 10 refers to the measure of total connectedness (see section 2.2). TABLE B2 Minimum connectedness considering all permutations of recursive orderings AUD CAD CHF EUR NOK NZD SEK USD JPY AUD 32.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.1 CAD 0.0 42.3 0.0 0.0 0.1 0.0 0.0 0.0 0.1 CHF 0.0 0.0 57.1 0.0 0.0 0.0 0.1 0.0 0.0 EUR 0.0 0.0 0.0 30.7 0.0 0.0 0.0 0.0 0.2 NOK 0.3 0.0 0.0 0.0 38.9 0.0 0.0 0.0 0.2 NZD 0.0 0.0 0.0 0.0 0.0 39.4 0.0 0.0 0.1 SEK 0.0 0.0 0.0 0.0 0.0 0.0 32.1 0.0 0.3 USD 0.0 0.0 0.0 0.0 0.1 0.0 0.0 45.6 0.0 JPY 0.0 0.0 0.0 0.0 0.2 0.0 0.0 0.0 50.3 Note: The table shows (10 periods ahead) forecast error variance decomposition values of SVAR models which are identified by a Cholesky decomposition. The entries of the matrix show the minimum entries which can be achieved with a Cholesky decomposition approach considering all possible orderings. BETTENDORF AND HEINLEIN 3959