Structural Estimation of Time-Varying Spillovers: an Application to International Credit Risk Transmission

We propose a novel approach to quantify spillovers on financial markets based on a structural version of the Diebold-Yilmaz framework. Key to our approach is a SVARGARCH model that is statistically identified by heteroskedasticity, economically identified by maximum shock contribution and that allows for time-varying forecast error variance decompositions. We analyze credit risk spillovers between EZ sovereign and bank CDS. Methodologically, we find the model to better match economic narratives compared with common spillover approaches and to be more reactive than models relying on rolling window estimations. We find, on average, spillovers to explain 37% of the variation in our sample, amid a strong variation of the latter over time.


NON-TECHNICAL SUMMARY
Assessing spillovers between financial assets is a difficult exercise. When a shock occurs in one market and then spreads to others, prices of those markets are affected in a quasicontemporary manner. It is difficult then, ex post, to identify the source of the shock and thus to distinguish correlation from causality in the movement of financial time series.
Moreover, the magnitude of the transmission of such shocks is not stable over the observed period. Therefore, a good spillover model should not only succeed in identifying the shocks in question, but also take into account the time-varying effects they have on other markets. Several papers in the literature have proposed to solve this problem by estimating models on rolling windows. However, this methodology has downsides: in rolling windows new observations have little weight compared to past observations, so that such model lack in responsiveness.
In this paper, we propose a novel model to quantify spillovers based on the work of Diebold and Yilmaz (2009) and Lütkepohl and Milunovich (2016). We estimate this model on sovereign and bank Credit Default Swaps (CDS) in the Eurozone. More specifically, we use this methodology to assess the national and international propagation of credit risk shocks and to analyze the extent of the sovereign-bank nexus across countries.
Our main results are methodological. By comparing our estimates with those of other models used in the literature, we observe a superior performance of our methodology with respect to the two issues mentioned: the identification of shocks and the reactivity to new events. Concerning the identification of shocks, we compare the capacity of the models to clearly distinguish between bank and sovereign shocks. For example, during the Italian political crisis of May 2018, Italian bank and sovereign CDS spreads increased significantly at the same time, thus reinforcing the likelihood for any spillover model to mistake this sovereign shock for a bank shock. As shown in the graph below, for this particular event the SVAR-GARCH model we propose is the only one correctly identifying the shock. Analysing the performance more generally on a large list of bank and sovereign events, the SVAR-GARCH compares favourable to competing models. Concerning the reactivity of spillover estimates, we compare the different methodologies by performing Granger causality tests. We find that the estimates produced by the SVAR-GARCH model are more reactive, especially compared to models estimated on rolling windows.
We also present economic results that further support our identification strategy: The spillovers the model produces retrace well the Eurozone crisis; for example by underlining the importance of Irish shocks at the beginning of the crisis, followed by a rise of Italian and Spanish shocks. Moreover, we find that the spillover estimates are positively associated with channels of credit risk transmission that the theoretical and empirical literature suggests.
All in all, the model we propose appears well suited for estimating spillovers between CDS markets, combining an attractive identification approach with time variation in the spillover estimates, while contributing to the active literature on methodologies for spillover estimations. Moreover, to the extent that our model imposes relatively few restrictions, it lends itself to be a useful tool for the analysis of spillover dynamics on a broad set of financial markets, instruments and variables.
Banque de France WP798 iii

Spillover indices from Italian shocks (sovereign and bank)
Note: The upper and middle parts of the graph represent spillover indices from Italian sovereign and bank shocks (i.e. how much the latter affect the variances of other variables). The lower part of the graph represents the difference between Italian sovereign and bank spillovers. The vertical red bar highlights the period of May 2018, when Italy was rattled by political turmoil. The models represented correspond to the following references: our paper (SVAR-GARCH), Diebold and Yilmaz (2009, VAR Cholesky), Diebold and Yilmaz (2012, VAR GIRF), Fengler and Herwartz (2018, DCC Fengler), and a contagion model built on Engle (2002, DCC Cholesky).

Introduction
Assessing financial spillovers between different markets can be highly challenging. To evaluate how a specific shock propagated from one market to another requires first to identify this shock. Yet, this task may cause significant difficulties as asset prices contemporaneously affect each other and thus co-move significantly. As for numerous asset classes, this problem applies to spillovers of credit risk; itself a topic of substantial interest for both researchers and policy makers due to their pivotal role in the European debt crisis (Coeuré (2018)).
A recent example for credit risk contagion 1 that has lead to asset price comovement is the political turmoil in Italy and heightened fear of a referendum on the Euro membership in May 2018 2 . The event, which can be interpreted as a sovereign Italian shock, led to a considerable surge in the Italian sovereign CDS spreads, a proxy for credit risk.
This shock then propagated to Italian bank CDS as well as to CDS of other Euro Area sovereigns and banking sectors, for example in Spain (Figure 1 presents daily CDS of the Euro Area sovereigns and banking sector, the red bar indicates the peak of the political turmoil). As can be seen from Figure 1 the daily CDS series increased simultaneously, therefore a mere visual analysis of Figure 1 cannot help to identify the source of the upsurge: was it a sovereign or a bank shock, and originating from which country? This example highlights the two main needed features of an econometric model with the aim to capture credit risk spillovers. First, the estimated model should be able to handle endogeneity and strong asset co-movements. Second, spillover estimates need to reflect the time variation in financial spillovers as these latter are unlikely to stay constant over time.
The main contribution of this paper is to combine an attractive identification approach 1 Here, we use the terms spillover and contagion interchangeably. Section 2 differentiates more clearly between the concepts.

Figure 1: Sovereign and bank CDS spreads for Italy and Spain
On the graph are represented the CDS spreads from the Italian and Spanish sovereign and banking sectors (dashed lines). The red bar indicates the peak of the Italian political turmoil. The sources of the data and the underlying methodology can be found in Section 4.
for a set of endogenous variables with time variation in the estimates of the spillovers. To do so, we rely on a SVAR-GARCH approach combined with the framework of Diebold and Yilmaz (2009). As an application, we estimate the model on a sample of 16 banking sector and sovereign CDS series in the Eurozone (EZ), including the CDS series presented in Figure 1. First, we show that economic identification of the shocks is feasible in this framework, even in a 16-variable system. Second, we test and validate that the economic channels behind the estimates fit economic theories on financial contagion.
The seminal work by Diebold and Yilmaz (2009), as well as a large number of subsequent papers (Alter and Beyer (2014); Claeys and Vašíček (2014); De Santis and Zimic 2 Electronic copy available at: https://ssrn.com/abstract=3827913 (2018); Demirer et al. (2018)), propose to base spillover estimates on the off-diagonal entries of forecast error variance decompositions (FEVDs) of rolling window structural vector autoregressions (SVARs). While the approach allows for the construction of mutual consistent spillovers, the literature faces the econometric challenge of identification (De Santis and Zimic (2018)). Earlier papers rely on short-run zero restrictions for the coefficients of the SVAR. However this assumption is unlikely to hold with very reactive financial data (see Alter and Beyer (2014)). Later papers sidestep any structural identification by using reduced form shocks in the form of Generalized FEVD analysis (GFEVD, see Pesaran and Shin (1998)). Yet, reduced form shocks have no economic interpretation and cannot be used for quantifying causal relationships of the data (Kilian and Lütkepohl (2017)). Other standard identification approaches are not appealing either: sign restrictions (Fry and Pagan (2011) We propose a novel approach on handling such econometric modeling choices by exploiting a SVAR-GARCH model that is statistically identified by the heteroskedasticity in the data (Lütkepohl and Milunovich (2016)). We show that this modelization is also attractive as it yields time-varying FEVDs based on the conditional variances of estimated structural errors. To the best of our knowledge, we are the first to exploit the timevarying properties of the conditional variances for Diebold-Yilmaz spillover estimates in a SVAR-GARCH setting. Moreover, we show that it is feasible to achieve economic identification between structural shocks and financial market variables in a nontrivial 3 Electronic copy available at: https://ssrn.com/abstract=3827913 one-to-one relationship, even in a system of 16 variables. We label shocks with a maximum contribution to the forecast error variance of a variable as a shock of precisely that variable (following the 3 to 4-variable identification of Grosse Steffen and Podstawski (2016) and Dungey et al. (2010)). Due to the GARCH component in our estimation, spillover estimates are up-to-date (as in Fengler and Herwartz (2018)) and not drown in a moving window average (as in Diebold and Yilmaz (2009).
We present both methodological as well as economic results. First on the methodological side, we show that the identification of the SVAR-GARCH model yields shock estimates that fit known economic and market events, thus supporting the initial maximum contribution identification. We manage to match major shocks to credit risk to 117 news events, either for bank or for sovereign CDS. In a second step, building either on this list of events or on the lists used in Candelon et al. (2011) and Alexandre et al. (2016), we compare the economic match of the spillovers implied by the SVAR-GARCH to a wide range of different DY-models. We find that the SVAR-GARCH outperforms, on this measure, identification schemes used in Fengler and Herwartz (2018), Diebold and Yilmaz (2009) or Diebold and Yilmaz (2012). Third, we show that the SVAR-GARCH yields more up-to-date spillover estimates compared to traditional moving window estimates as it Granger causes the latter.
Economically, we find cross-section results that corroborate our identification strategy as spillover estimates match (i) the economic narratives of the EZ debt crisis and (ii) economic contagion channels proposed by the theoretical and empirical literatures. For example, we find that during the European debt crisis, spillovers from periphery countries increased markedly, while elevated spillovers from core countries are more centered around the 2008/09 financial crisis. As for the underlying economic channels, we find international credit risk spillovers between sovereigns to be higher when the two countries have stronger ties in trade and portfolio investments, in line with the business 4 Electronic copy available at: https://ssrn.com/abstract=3827913 cycle network literature (Foerster et al. (2011)). We also find international credit risk spillovers between banking systems to be higher when they exhibit more similar portfo- lios; yet we find spillovers not to be significantly associated with bank cross-holdings (as suggested in Brunetti et al. (2019)). Concerning the national sovereign-bank nexuses, we find that (i) a lower capital ratio and higher debt to GDP ratio increase domestic bank to sovereign spillovers in both low and high debt countries; while (ii) reliance of the non bank sector on domestic bank funding is significantly associated with domestic bank to sovereign spillovers only in low debt countries. In turn, we find domestic sovereign to bank spillovers to be higher for countries with a stronger bank exposure to domestic government debt. Moreover, we find that in high debt countries domestic sovereign to bank spillover are stronger when the domestic banking sector shows higher non-performing loan ratios and disposes of a lower share of liquid assets to short term liabilities.
Overall, we find credit risk in the Euro Area to be less integrated than suggested by estimates based on the more standard Diebold-Yilmaz style VAR models. We estimate that, on average, credit risk spillovers explain about 37% of the total variation in our sample. Yet, we show that the importance of spillover fluctuates distinctively, peaking at 61%.

Estimating Contagion in the Literature
Throughout this paper, we define spillovers as the degree to which exogenous shocks to one CDS market drive the variation of CDS spreads in other markets, in line with the FEVD-analysis of Diebold and Yilmaz (2009 sign and not only the magnitude of the impact) while they label FEVD-estimates as "connectedness" and the coefficient estimates of their SVAR purged from the size-effect of the shocks as "contagion". Similarly, Claeys and Vašíček (2014) and Dungey et al. (2015) term contagion as significant changes in the propagation mechanism, not the propagation mechanism itself. Diebold and Yilmaz (2009 propose in a set of papers a prominent approach to quantify time-varying spillovers on financial markets. The model is widely reused in the literature (e.g. Claeys and Vašíček (2014), Alter and Beyer (2014), Fengler and Gisler (2015), Diebold et al. (2018), Hale and Lopez (2018), Greenwood-Nimmo et al. (2017) or Greenwood-Nimmo et al. (2019)). Yet the Diebold-Yilmaz approach relies on orthogonalized SVARs and the identification of the latter is challenging.
Three different streams of the contagion-literature do offer attractive identification strategies. First, De Santis and Zimic (2018) and De Santis and Zimic (2019) apply a methodology close to ours. They gauge the interconnectedness among sovereign debt markets or between medium-term interest rates with a Diebold-Yilmaz approach based on a SVAR that is identified by "magnitude restrictions", that is by imposing that a shock stemming from one country impacts the most its own country. Second, Ando et al. (2018) add numerous exogenous variables to their vector autoregressions with the aim to purge their variables from common factors. Once this filtering is done, they obtain (quasi) orthogonal shocks. Finally, several papers focusing on financial spillovers (Ehrmann et al. (2011), Dungey et al. (2015, Ehrmann and Fratzscher (2017), Fratzscher and Rieth (2019)) apply the idea of Rigobon (2003) and rely on the identification by heteroskedasticity. The authors use the variations in the variance-covariance matrix of the reduced form shocks to identify the structural shocks.
However, the time variation in the first two streams of the literature comes from a rolling window estimation. These papers use relatively long window length in order to have a 6 Electronic copy available at: https://ssrn.com/abstract=3827913 sufficient accuracy in their parameter estimates. Nevertheless, with this feature, their models will lack responsiveness as past observations mitigate the effect of new ones.
The third stream of the literature focuses on specific sub-periods (e.g. Ehrmann and Fratzscher (2017) or Dungey et al. (2015)) and do not provide a continuous estimation of their spillover indices.
In contrast, a recent literature has exploited MGARCH models that are capable of generating up-to-date spillovers (Fengler and Herwartz (2018), Strohsal and Weber (2015)).
However, these models lack attractive identification approaches for structural analysis 3 .
The same drawback applies to variation of the approach using time-varying VARs as in Geraci and Gnabo (2018) or in Korobilis and Yilmaz (2018).

Measuring spillovers
We follow the key idea of Diebold and Yilmaz (2009 and base a set of mutual consistent spillover measures, from pairwise to system wise, on FEVDs. Table 1 depicts a FEVD which is amended with an additional bottom row that captures the off-diagonal column sums, an additional column on the right that captures the off-diagonal row sums and a bottom right element that captures the grand average of either off-diagonal column or row sums.
The FEVD is populated by elements d H ij , which give the proportion of the H step forecast error variance of variable y j that is driven by an orthogonal shock to y i . Following  Diebold and Yilmaz (2009 we define d H ij as a pairwise directed spillover from i to j: The pairwise spillovers allow to construct more aggregated spillover indices. For example, the off-diagonal column sums indicate to which degree the H step forecast error variation of variable y j is driven by other variables in the system. Diebold and Yilmaz (2009) define therefore inward spillovers as: Vice versa, the off-diagonal row sums indicate to what degree variable y j drives the variation of all other variables in the system. Outward spillovers are therefore defined as: 8 Electronic copy available at: https://ssrn.com/abstract=3827913 Total spillovers in the system are finally defined as average of inward or outward spillovers.
As underlined above, Diebold and Yilmaz (2009

Description of the Model
For the development of a structural version of the Diebold-Yilmaz index, we rely on a SVAR model with a GARCH error structure and an identification by heteroskedasticity, similar in spirit to Normandin and Phaneuf (2004). We choose the model for the following reasons: first, a GARCH error structure appears a natural choice given that first differences of CDS, alike many other financial variables, show clustering of volatility over time and is therefore well approximated by GARCH processes. Second, the model has the property of time-varying conditional volatility of the errors, given the GARCH structure of the model. This property is crucial for the identification of structural shocks (Rigobon (2003)). Third, still relying on this property, we can construct time-varying FEVDs. This last feature allows us to estimate the model over the whole period, thus enabling more responsiveness compared to a time-varying FEVD based on a rolling estimation.

SVAR identification through heteroskedasticity
We base the empirical model on a structural vector autoregression of order p, that allows 9 Electronic copy available at: https://ssrn.com/abstract=3827913 our variables to be determined simultaneously.
where Y t is a vector containing the endogenous variables of interest, typically sovereign and bank sector CDS time series. The matrices B i contain the contemporaneous and lagged effects of the endogenous variables. ǫ t denote structural errors with zero mean and an unconditional diagonal variance covariance matrix λ ǫ . As the SVAR cannot be estimated directly, we first estimate a reduced form VAR: where the reduced form shocks µ t have zero mean and a non-diagonal variance covariance matrix Σ µ . The structural errors ǫ t are then defined through µ t and the contemporaneous interaction matrix B 0 : The well known VAR identification problem arises as we try to obtain estimates for the contemporaneous interaction matrix B 0 from the relationship Σ µ = B −1 0 λ ǫ B −1′ 0 . Yet without further restrictions B 0 is not identified since Σ µ provides only N (N +1) 2 equations for N 2 unknowns if we normalize λ ǫ = I.
The SVAR-GARCH model we are using relies on Rigobon (2003) identification scheme that exploits the general heteroskedasticity in financial data. Suppose that the variances (or conditional variances) of µ t vary over time -implying that the structural error variance does too -while B 0 is constant 4 . This feature implies that there is more than one volatility regime in the data, defined by a different reduced form variance-covariance 4 In Annex A.6 we relax this assumption.

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Electronic copy available at: https://ssrn.com/abstract=3827913 matrix Σ µ (m). If there are M different volatility regimes, then we have: where λ m are the diagonal matrices of the structural shocks (λ 1 is normalized to I).
Lanne and Saikkonen (2007) show that B 0 is locally uniquely determined if ∀(k, l) there is sufficient heterogeneity in the volatility changes.

SVAR-GARCH
Conditional heteroskedasticity can be modeled in different ways (see Lütkepohl and Netšunajev (2017a)). We rely on the methodology first proposed by Normandin and Phaneuf (2004) and assume that it is driven by GARCH processes. Similar models have been applied in Bouakez and Normandin (2010), Lütkepohl and Milunovich (2016) and Lütkepohl and Netšunajev (2017a).
We assume that the structural shocks are orthogonal and that their variances follow a univariate GARCH(1,1) process: where γ k > 0, g k ≥ 0, γ k + g k < 1, 1 k N so that the GARCH(1,1) processes are non-trivial.
Then, we can express the reduced form shocks as:

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Electronic copy available at: https://ssrn.com/abstract=3827913 where: is a (N x N) diagonal matrix with the univariate GARCH processes on the diagonal.
Therefore, the distribution of µ t conditional on past information has mean zero and a covariance matrix: Rigobon (2003) shows that for full (local) statistical identification, 2 different volatility regimes is enough. With a SVAR-GARCH we have T (number of observations) different volatility "regimes". In this study, using daily CDS data between 2008 and 2019, this translates into more then 2800 regimes. We estimate the parameters of the SVAR-GARCH model by Maximum Likelihood as in Lütkepohl and Milunovich (2016).

Forecasts for FEVD
Estimates for time-varying conditional variance-covariance matrices allow us to construct FEVDs for each time period, i.e. for each day. Note that for the computation of FEVDs in each period t, one cannot take the actual estimated structural varianceŝ λ t|t−1 . Instead, we need to compute, by definition of the FEVD, in-sample forecasts for the structural variances λ * t+h|t conditional on the information set in t, as in Fengler and Herwartz (2018). Contrary to the approach in the latter, our matrix B 0 is constant over time, so that the only change between a classic SVAR-FEVD and our approach is the computation of future structural variances.
We have with Equation 10: 12 Electronic copy available at: https://ssrn.com/abstract=3827913 Taking conditional expectation at time t, with h ≥ 2: Using the law of iterated expectations, we get: That is: We thus obtain λ * t+h|t for each h as this matrix is diagonal and is only composed of the different σ 2 k,t+h|t . To build the FEVDs, we then first compute the MSPE. The Θ i matrices come from the Moving Average (MA) representation of the SVAR as detailed in Kilian and Lütkepohl (2017): With the structural variances estimated, we get: We can then evaluate the contribution of shock j to MSPE of y kt with the usual MSPEformula, the only difference with a classic SVAR is that variances of structural shocks 13 Electronic copy available at: https://ssrn.com/abstract=3827913 are no longer normalized to 1. With θ kj,h the kj th element of Θ h : With: We get: Eventually the time-varying FEVDs enable to build the time-varying spillover indices, as explained in Section 3.1.

Data and filtering for common shocks 4.1 Data
We focus on credit risk of major EZ sovereigns and banks. We attempt to strike a balance between a sufficiently high coverage of important CDS markets and the limited number of variables our empirical approach allows. As a result, we limit the sample to 9 countries (Greece, Ireland, Italy, Portugal, Spain, Germany, France, Belgium and Netherlands). For each country we include two variables in the sample, sovereign credit risk and credit risk in the banking sector, except for Ireland and Greece where we lack banking credit risk series due to data constraints 5 . This leaves us with 16 variables all together.
As standard in this literature (see Greenwood-Nimmo et  14 Electronic copy available at: https://ssrn.com/abstract=3827913 risk using CDS spreads on senior unsecured debt, modified-modified restructuring, mid spread and a maturity of 5 years 6 . We retrieve CDS spreads for non-US sovereigns and US banks denominated in USD while CDS spreads for the US sovereign and European banks are denominated in EUR. Our sample covers daily data between January 2008 and March 2019, covering the GFC, European debt crisis and several sovereign and banking turbulence such as the Italian political turmoil of May 2018. We construct country banking variables as an unweighted average of bank CDS from that country as in Greenwood-Nimmo et al. (2017). In the selection of banks, we follow Alter and Beyer (2014)

Filtering for common shocks
The literature agrees that global and regional variables may exert a common influence on credit spreads (Longstaff et al. (2011)). Ignoring such common shocks that have a simultaneous effect on different variables in an econometric analysis may result in an overestimation of contagion patterns. We would falsely attribute common shocks to the propagation of idiosyncratic shocks. We therefore follow Alter and Beyer (2014) in including the following set of pan-European credit risk factors, including (i) the Itraxx Europe index (which comprises investment grade rated European entities, reflecting the overall credit performance of the European real economy), (ii) the Itraxx Crossover index (which comprises below investment grade rated European entities, reflecting the lowerend credit performance of the European real economy), and (iii) the spread between the ditions equivalent to the TED spread). Moreover, we control for the Eurostoxx 50 (the European stock market index), the VIX index (as a proxy for investors' risk aversion) and US and UK sovereign and banking CDS series (to account for foreign shocks).
We account for common shocks in a two-step approach. First, we regress each CDS series individually on a vector of common factors and then we run the SVAR-GARCH model specified in Section 3.2 on the obtained residuals, as in Dungey et al. (2010) That is, in a first step, we filter first differences bank and sovereign CDS series by the following OLS regression: where ∆z jt represents the first difference of a CDS series j in the sample, α j is a constant and ∆X t is a vector of common factors in first differences. y jt contains the residuals of the regression and serves as input data for the SVAR-GARCH. Annex A.6 reports robustness checks using a smaller set of exogenous variables.

Results
In this section we present the results for the SVAR-GARCH model outlined above. We estimate the model with 2 lags as indicated by the information criteria from a simple VAR estimated on the same dataset. Moreover, in line with Diebold and Yilmaz (2009, we choose a forecast horizon for the FEVD of 10 days. In Section 5.1 we present the results of our identification approach, that is the labeling of structural shocks, as well as comparisons of timeliness and of identification performances between competing models. In Section 5.2 we present the economic results of our application.

Statistical and economic identification
Statistical identification is achieved when the number of univariate GARCH components underlying the GARCH structure are larger or equal to N-1. That means that for full local identification we may have at most one series that is not well approximated by a GARCH process in order to have sufficient heteroskedasticity in the structural shocks.
We follow the identification test proposed by Lanne and Saikkonen (2007) and reject fewer than N-1 GARCH processes in our sample (see Annex A.1).
However, full local identification implies only statistical identification up to sign changes and ordering. To make the orthogonal shocks economic meaningful we need to label them, ideally in such a way that each orthogonal shock corresponds to a different variable.
In line with Grosse Steffen and Podstawski (2016) This table represents the average over time of the FEVDs obtained with the SVAR-GARCH. We can see that the originating shocks (in line) impact the most their own variables (in column).
number of historical events. In the spirit of Antolín-Díaz and Rubio-Ramírez (2018), we compare major shocks with historical economic and market events 8 . We define major shocks as those shocks that are higher than 6 times their own standard deviations. Of the 79 shocks that meet this criteria, we are able to match 62 events (covering 78% of major shocks) 9 . On Figure 2 we present the time series of the estimated structural shocks (in black) along with the timing of the matched events (red vertical lines). Figure 2 shows also isolated events of spillovers that fall short of the threshold for major events. Again, we are able to match a large amount of such shocks to economic and financial events, extending the list of events to 117 items. The identified events are typically rating downgrades or political shocks (for sovereigns) or bank stress episodes (for banking sectors). Annex A.3 reports the exhaustive list of events. This exercise suggests that our identification strategy based on major shock contribution is further supported by the event-analysis on structural shocks of Figure 2, something which is rarely performed in the SVAR literature.

Figure 2: Structural Shocks and Events
On the different graphs above are represented the estimated structural shocks of the model (ǫ t ) as well as identified historical events for each variable represented in vertical red lines. The list of events used is available in Annex A.3.

Total Spillover Comparison
How reactive is our model to new events? To assess its timeliness, we compare total spillovers (S H in Equation 4) from our SVAR-GARCH model with total spillover estimates from other Diebold-Yilmaz approaches of the literature. More precisely, we estimate S H for the following 4 models 10 : Model 1 A SVAR estimated on a rolling window and identified by Cholesky decomposition (as in Diebold and Yilmaz (2009), labeled here VAR Cholesky); Model 2 A SVAR estimated on a rolling window and identified by GIRF/GFEVD (as in Diebold and Yilmaz (2012), labeled here VAR GIRF); Model 3 A DCC-GARCH identified by Cholesky decomposition and estimated over the entire sample, labeled here DCC Cholesky. More precisely, we estimate a DCC GARCH as a reduced form VAR, that is: following the notations of Engle (2002). We then switch to the structural form with a Cholesky decomposition at each period t: Model 4 Similarly to Model 3, we estimate a VAR-GARCH based on a DCC-GARCH, but with the identification of Fengler and Herwartz (2018) where the columns of Γ contain the eigenvectors of H and Λ 1/2 is diagonal with the positive square roots of the eigenvalues on its diagonal.
12 As De Santis and Zimic (2018) show, when contemporaneous interaction effects between variables are not equal to 0, the estimated standard errors of structural shocks obtained with GIRF are biased upwards, equally biasing upwards spillovers estimates based on FEVDs. The 0 restrictions the Cholesky identification introduces are likely to be at odds with the data generating process. In a numerical exercise De Santis and Zimic (2018) show that also this DY-model is likely to misspecify estimated spillovers.
identification imposes restrictions that are likely to be at odds with the data generating process. SVAR models relying on this identification are thus susceptible to over-or underestimate total spillovers. We come back more formally on this point in Annex A.4.
Second, we see across all modelizations the 2010-2012 the "financial fragmentation" of the EZ. Indeed, each total spillover index is U-shaped: declining during those years, before increasing again afterwards. Ehrmann and Fratzscher (2017) and De Santis and Zimic (2018), who find similar shapes of total spillovers, argue that the latter decreased over 2010-2012 since shocks from peripheric countries had a decreasing impact on core countries.
Third, intuitively, indices relying on a rolling window estimation should be less responsive to new events compared to the SVAR-GARCH. However, there is no clear distinction a priori in responsiveness between the different models with a GARCH-component. This intuition is confirmed by a Granger causality analysis between the different S H given in Table 3. Indeed S H from the SVAR-GARCH does Granger cause S H indices from the rolling window estimated models (VAR GIRF and VAR Cholesky), but not the indices stemming from a DCC-GARCH (DCC Cholesky and DCC Fengler). When we reverse the perspective, SVAR-GARCH is only Granger caused by DCC Fengler and not by VAR Cholesky or VAR GIRF. In that sense, S H index estimated by DCC Fengler appears to be the most responsive to new events. However, as we show in the next section, the underlying pairwise spillovers estimated by DCC Fengler and DCC Cholesky are at odds with economic narratives. So that, contrary to the SVAR GARCH, these models fulfill the second condition of a good contagion model (responsiveness) but not the first one (good identification of the events).

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Electronic copy available at: https://ssrn.com/abstract=3827913 The different lines represent the Total Spillover indices S H built from the five different models outlined above. The rolling window models are estimated on a 100-day period, as standard in this strand of literature. For readability we show 10 day moving averages of the indices.

Spillover Comparison and Narrative Events
To evaluate the performance of our identification strategy compared to other models, we analyze how the different spillovers evolve along well-known narrative events.
To showcase our approach, we focus on the May 2018 political turmoil in Italy. At that time, the formation of a Eurosceptic coalition brought about a sharp increase in Italian

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Electronic copy available at: https://ssrn.com/abstract=3827913  Figure 4 shows that only the spillover estimates from the SVAR-GARCH do so during this period (highlighted in red), while other methodologies' spillovers remain subdued.

Figure 4: Outward Spillovers from Italian sovereign and bank shocks
The two upper parts of the graph represent the Outward Spillover Index (S H j→• ) from, respectively, the Italian sovereign and the Italian banks, built from the five different models outlined above. The bottom part of the graph represents the "net" spillovers (outward sovereign spillovers minus outward bank spillovers). The periods highlighted in red represent the May 2018 Italian political turmoil.
between sovereign and bank CDS series from the same country, there is a high risk that a model confuses bank shocks with their corresponding sovereign shocks. Accordingly, at the time of a sovereign event, outward spillovers from the country's banking sector should remain flat or decrease. Therefore, for a sovereign event to be correctly identified, not only the sovereign spillovers should increase, they should also increase by more than the corresponding bank spillovers. On the middle and lower parts of Figure 4 we display the outward spillovers from Italian banks as well as the difference between sovereign and bank outward spillovers ("net" spillovers). While most of the models exhibit flat 24 Electronic copy available at: https://ssrn.com/abstract=3827913 or negative net spillovers, only the SVAR-GARCH manages well to identify this specific event on this measure.
To evaluate on a more systematic basis the identification strategies of the different models, we replicate the analysis of Figure 4 over the set of our identified events available in Annex A.3. We estimate that a sovereign (bank) event is well identified if, 5 days around the day of the event, the spillover estimate stemming from the sovereign (banking sector) increases more than the spillover estimate from the banking sector (sovereign) in the same country. We evaluate the identification performance of the models on different  Table 4 suggests that the SVAR-GARCH outperforms, on every set of events, the other models in terms of identification. Note also that the competing models barely exceed the 50% threshold of identification, meaning that they tend to confuse more sovereign events with banking events than a random selection 14 .

Economic results
Figure 3 on total spillover indices shows that we estimate credit risk to be less integrated than other models would suggest. According to our S H estimates, on average about 37% of the variation in the filtered CDS rates can be explained by spillovers. Yet, we find substantial variation in this magnitude over time. To investigate the sources of heightened spillovers, this section analyses first the time-variation of both bank and sovereign spillovers from the EZ countries, and then the economic channels behind the spillovers we estimate. As our estimates match both the narrative of spillovers in the EZ debt crisis (Section 5.2.1) and the theoretical channels of credit risk spillovers (Section 5.2.2), we interpret these economic results as a further validation of our identification strategy.

Group pairwise spillovers
In this section, we analyse credit risk spillovers in terms of (i) timing, (ii) magnitude and (iii) origin. Given, that we estimate spillovers between 16 CDS series, presenting the resulting 240 pairwise spillovers is not feasible. We focus therefore on pairwise spillovers from different sets of countries/banking sectors. In the "Peripheric" group are included the high-debt countries at the time of the EZ debt crisis: Italy, Spain, Portugal, Greece, Belgium and Ireland 15 . The "Core" group, on the reverse, is constituted by Germany, France and the Netherlands. The "Peripheric banks" and "Core banks" include the corresponding banking sectors. However, as indicated in Section 4, due to data-constraints the group "Peripheric banks" does not include Greek and Irish banking sectors. Figure 5 presents estimates of group pairwise spillovers for each variable sets defined above. In line with Section 3.1 we define the group pairwise spillover from group G 1 to group G 2 as the average outward spillovers from G1 restricted to the variables of G 2 .
More formally we have: With N G 1 and N G 2 the number of variables in G 1 and G 2 16 . Each line represents by how much shocks from a variable set drive the variation of other variable sets on average. The analysis of time-varying spillovers here differs from the presentation of snapshots spillovers around narrative events in Section 5.1.3 (as we focus here on a much broader time period) and also from the presentation of the shocks in Section 15 Note that we include Belgium in the Periphery-group as the country exhibited high public debt/GDP ratio. However the results are very similar if we define Belgium as a Core country.
16 Contrary to Equations 2 and 3, we divide here the index by the number of pairwise directed spillovers considered. Likewise the different indices of Figure 5 are expressed in the same unit, that is: by how much, on average, a single variable of G1 has an impact on a single different variable of G2.

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Electronic copy available at: https://ssrn.com/abstract=3827913 5.1.1. This is because spillover estimates in our DY-framework are not only functions of the time-varying variances of structural shocks (λ t ), but also of the interaction matrix (B 0 ) and of the VAR coefficients (A i ). Therefore a large structural shock does not necessarily translate into a large spillover if it is associated with low coefficients in the corresponding matrices or if the magnitude of the shock is low relatively to other shocks' variances. Figure 5 can be read in two ways, either from a shock to perspective by rows, or from a shock from perspective by columns. In the following, we take the shock to perspective. Figure Figure 5 suggests that periphery sovereign shocks affect strongest CDS rates in other sovereign periphery countries followed by periphery banking sectors. Yet, core sovereigns and banks were also significantly affected periphery sovereign shocks.
We also find sizable spillovers from the periphery banking sector to other blocks in the EZ. For example, Figure 5 shows elevated spillovers at the beginning of 2013, when investors worried about the health of the Italian banking sector (due to high NPL ratios amid excessive reliance on debt), as well as at the beginning of 2016, when again concerns 28 Electronic copy available at: https://ssrn.com/abstract=3827913 about NPLs and the lack of credibility in the Italian banking sector heightened. We also find increased spillovers around dates between 2011 and mid-2012 when the Spanish banking sector signaled problems.
While we find spillovers from periphery EZ countries to increase with the beginning of the Euro debt crisis, we find spillovers from core EZ countries to be stronger during 2008/09 financial crisis. As such, we estimate strong sovereign core spillovers in January 2009 when the Dutch government announced plans to provide a backup facility to cover the risks of the ING's securitised mortgage portfolio. Moreover, Figure 5 shows increased sovereign core spillovers around dates that coincide with a downgrade of France by SP as well as the second round presidential election stand-off between Emmanuel Macron and Marine Le Pen. Finally, we find strong core bank spillovers, for example around the dates when ING received 10bn EUR from the Dutch government or when BNP entered a liquidity crunch when the bank was no longer able to borrow in USD. Overall, compared to their periphery counterparts, we find sudden increases of spillovers from core countries to be less frequent.

What economic channels explain spillovers?
While in the previous section we discussed the sources and time-variation of outward and total spillovers, this section focuses on the economic channels underlying the pairwise spillovers we estimate. More specifically, given a shock to a sovereign or banking sector in our sample, we vet whether the resulting pairwise spillovers match the economic channels proposed by the theoretic and empirical literature as an additional test for our identification strategy. We focus here on four different types of spillovers: (i) international sovereign to sovereign spillovers, (ii) international bank to bank spillovers, (iii) national bank to sovereign spillovers and finally (iv) national sovereign to bank spillovers.

International spillovers
First, we address the following question: given a sovereign shock in country i, what factors are the international spillovers to the sovereign risk in country j associated with?
We follow broadly the regression approach by De Santis and Zimic (2018) and regress the credit risk spillover of sovereign i on sovereign j in quarter t on a set of regressors that can be divided into two main groups: distance and exposure. We estimate: The results, reported in Table 5 suggest that similarity in business cycles cannot explain spillovers in sovereign risk. Instead, we find that similar credit risk in terms of similar debt to GDP ratios as well as both stronger trade and portfolio exposure are significantly related to higher sovereign risk spillovers. This finding supports the business cycle network literature (such as Foerster et al. (2011)) which models contagion channels through exactly those two exposure variables 18 . 17 We multiply difference variables by -1, such that the indicators increase in similarity. 18 As the use of generated dependent variables in the regression can induce heteroskedasticity (see De

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Electronic copy available at: https://ssrn.com/abstract=3827913 The distance variables include credit risk distances which we estimate by the squared difference between country i and country j's banking sector's non-performing loans and capital ratios in period t 19 . In terms of exposures we test for two economic channels that are frequently used to model financial institution linkages: cross asset holdings and similarities in portfolios across banking sectors (see Giudici et al. (2020), Brunetti et al. (2019, Greenwood et al. (2015)). We construct bank sector portfolios from BIS Santis and Zimic (2018)), we report White heteroskedasticity-consistent standard errors. 19 Here again, we multiply difference variables by -1, such that the indicators increase in similarity.

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Electronic copy available at: https://ssrn.com/abstract=3827913 Consolidated Banking Statistics data, following Greenwood et al. (2015), and calculate squared differences of those portfolios for each time period t. Cross asset holdings between banking systems are measured as the share of banks claims of country j vis-à-vis country i.
The results, shown in Table 6, suggest that cross-asset holdings are not significantly linked to the bank to bank spillovers. We do find however, that portfolio similarities are significantly associated with bank to bank spillovers. Both these findings are in line with the literature (Brunetti et al. (2019)). Similarly to the sovereign regressions, risk distances have some explicative power: we find that international bank spillovers are significantly associated with similar capital ratios for pairs of banking systems. However, similar NPL ratios turn out not to be of statistical significance.  (Podstawski and Velinov (2018)).
We focus first on the economic transmission channels of domestic spillovers from banks to sovereigns. First, one reason for higher spillovers may simply be a more vulnerable economy. We include in the regression measures of debt to GDP ratios, current account and GDP growth as predictor variables. Second, bank risk may also affect domestic sovereign risk through the "bailout channel", that is explicit or implicit public guarantees, in case of distress of the banking sector (Alter and Schüler (2012)). To proxy this effect, we add as a predictor the capital ratio of the banking sector. Intuitively, the bailout channel should be significant if domestic banks are undercapitalized and in potential need of public support. Third, another potential channel of spillovers is that when a banking sector is in distress, it can trigger fire sales of the government bonds it holds, increasing in turn the credit risk of the sovereign issuer. Fourth, distress for banks may affect their lending activity and therefore impact sovereign risk through a slowdown in economic growth (Podstawski and Velinov (2018)). For the third and fourth channels, we therefore include two exposure variables in the regression set: the share of domestic government bonds and the share of domestic non-bank assets that the banking sector holds. Denoting v k s the vulnerability variable k for sector s, Equation 27 restates the OLS regressions we estimate: Note that for the vulnerability variables, we use dummies instead of continuous variables contrary to Equations 25 and 26 20 . We define high and low realisations of the variables with regards to their overall sample mean 21 . Since the sample is split according to debt levels, using the mean debt/GDP as threshold for the construction of a debt dummy does not yield much variation in the high debt subsamples. We therefore use the subsample mean for the high debt country group, and the overall sample mean for the low debt country group 22 .
We find in Table 7 that low capital ratios and high debt to GDP ratios are significantly associated with stronger domestic spillovers from banks to sovereigns. This suggests that the "bailout channel" may indeed be important in explaining the sovereign-bank nexus (Fratzscher and Rieth (2019)), as is higher sovereign indebtedness. Moreover, we find that neither the capital account nor GDP growth is significantly associated with the spillovers we estimate. While the vulnerability variables yield similar results concerning the significance of the indicators across country groups, the results for the exposure variables differ. For high debt countries, both the dependence of the domestic non-bank corporate sector and government on domestic bank lending are not significant. In contrast, we find for low debt countries that higher non-bank exposure to domestic lending is significantly associated with higher domestic bank to sovereign spillover, suggesting that reduced lending activity in the case of a banking shock may indeed feed through the corporate sector into sovereign risk (see Pagano (2018)). As for high debt countries, we also find non-significant effects of sovereign debt exposure to domestic bank for low 20 The underlying reason for using continuous variables for Equations 25 and 26 is that investors on the CDS markets may pass the shock of one sovereign (bank) to the price of another sovereign (bank) CDS if they judge them as similar. However for bank-to-sovereign or sovereign-to-bank regressions we cannot rely on such similarity metrics as the giving and the receiving variables are of different types. Therefore for Equations 27 and 28 we consider that investors pass the shock of a bank (sovereign) to a sovereign (bank) CDS if they judge the receiving variable as not resilient enough. This kind of reasoning is discrete, therefore we turn to dummy variable so as to illustrate the threshold that investors may consider.
21 Defined by 15.2% for the capital ratio, 0.5% for the capital account and 0.3% for real GDP growth. 22 We use the overall sample mean (86.5 %) for the low debt group, and not the subsample mean (70%) as crossing the latter is unlikely to appear as warning signal for investors. Indeed, Germany has crossed this threshold between 2009 and 2016 while keeping its status as safe heaven. The subsample mean is at 101.7 % for the high debt group debt countries.
Capital -0.91 * * * -0.91 * * * -1.01 * * -5.01 * * * -5.07 * * * -3.88 * * * (0 Finally, we investigate the determinants of domestic credit risk spillovers from a country's sovereign to its banking sector (see Equation 28). We test the following hypotheses: First, are domestic spillovers to banks stronger if the banking sector is more vulnerable? We proxy here bank vulnerability with capital ratio, liquidity (measured by liquid assets to short term liabilities) and NPL ratios. Second, are spillovers stronger if the domestic banking sector holds more domestic government debt, expressed in % of total assets (the "balance sheet channel" as described in Angeloni andWolff (2012) andBuch et al. (2016))? Third, are spillovers stronger if the domestic banking sector holds more assets of domestic non-financial firms, expressed in % of total assets (the "real economy channel" 23 )? Here again, we express vulnerability variables in terms of dummies, where 23 The underlying rationale for this hypothesis is that a sovereign shock can feed into the real sector

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Electronic copy available at: https://ssrn.com/abstract=3827913 the thresholds between high and low realisations are set to sample averages 24 . We estimate:ω the "real economy channel" seems to matter only for low debt countries (positive and significant coefficient for NFC exposures). Concerning the role of bank vulnerability, we find mixed results across country groups: For high debt countries, both higher NPL and lower liquidity ratios are significantly associated with higher domestic sovereign to bank spillovers in contrast to the capital ratio, which we find not to be significantly linked to the latter. For low debt countries, we find both NPL and capital ratios not to be significantly linked to spillovers, while we find lower liquidity ratios to be significantly associated with higher spillover only in one out of three regressions.

Conclusion
We propose a novel approach of the popular Diebold-Yilmaz framework by exploiting a SVAR-GARCH model that is statistically identified by the heteroskedasticity in the data. We show that this identification approach is attractive as it yields time-varying FEVDs based on the conditional variances of estimated structural errors. Moreover, we show that it is feasible to achieve economic identification between structural shocks and financial market variables in a nontrivial bijective relationship, even in a system of 16 variables. We show the advantages of this methodological contribution by comparing the results with other common identification approaches used in the time-varying spillover literature. Overall, the identification scheme is supported by the fact that the results outperform other models in terms of timeliness and narrative fit. Additionally, we show that the obtained pairwise spillovers match theoretical contagion channels.
This study has some limitations that could be addressed in future search. First, our identification approach relies on a constant B 0 matrix over the full sample period 25 . In principal, this constraint can be relaxed by estimating the model on shorter subsamples, for example defined on dates for which the researcher expects a structural break in interdependencies. While in Annex A.6 we allow for a single change in the B 0 matrix, we leave a more profound analysis of this avenue for future research. Second, by imposing fewer constraints than previous models, the SVAR-GARCH could be applied to investigate contagion issues on time series that have been less considered in the literature, notably market liquidity data.

A.1 Test for identification and estimated coefficients
We rely on the original test proposed by Lanne and Saikkonen (2007) to test for the identification of B 0 . The recursive test applied here gives strong evidence for full identification of B 0 , see Table 9. For a more thorough description of the test, see Lütkepohl and Milunovich (2016). Note that the result of the test can be explained despite the reported low power of this latter, because (i) of the size of our dataset (ii) our 16 GARCH processes have a high persistence (γ k + g k close to 0.9 ∀k) which tends to increase the power of the test.

A.4 IRF assumptions
An additional advantage of our econometric framework is that it imposes less restrictions on the impulse response functions (IRFs) compared to other contagion-models. More specifically, Cholesky-identified SVARs, as used in VAR Cholesky and DCC Cholesky, postulate a recursive structure of the IRFs. Generalized impulse response functions, as used in VAR GIRF, impose that the IRF of a one-unit shock i on variable j has the same initial impact as an IRF from shock j to variable i. Eventually the same criticism applies also to the orthogonalization in Fengler and Herwartz (2018) as used for the model DCC Fengler -see demonstrations below. On the reverse, the SVAR-GARCH framework does not impose such a structure (Lütkepohl and Netšunajev (2017b)). Therefore the leveldifferences observed on Figure 3 may come from the overly strong assumptions of the competing models, over-or underestimating the spillovers. is equivalent to its moving average representation: Which can be rewritten in its structural form: Generally speaking, the IRF of a vector shock δ = (δ 1 , ..., δ n ) on Y t is defined, at horizon h and with Ω t−1 the information set at t, as: Due to the orthogonality of the structural shocks, one uses δ = (0, ..., 0, δ j , 0, ..., 0) in order to consider the impact of a single shock. In that case we get, with e j a vertical vector full of zeros apart for its j th element that is equal to 1: In our SVAR-GARCH setting, B 0 is identified by heteroskedasticity and by economic identification (with Σ ǫ and Σ µ evolving over time and being equal to, respectively, λ t|t−1 and Σ µ,t|t−1 , Equation 13). This identification strategy does not impose any structure on the IRFs. Conversely, in Diebold and Yilmaz (2009) . Although convenient, this orthogonalization imposes a recursive structure in the Data Generating Process as B 0 is then lower-triangular.
Identification by GIRF works differently since, instead of considering structural shocks, the GIRF looks at reduced form shocks. Using the notation Σ µ = (σ ij ) i,j∈ 1,n 2 , a one standard deviation shock j and the same remaining notations, the GIRF is defined as: GIRF (h, σ jj e j , Ω t−1 ) = E(Y t+h |µ t = σ jj e j , Ω t−1 ) − E(Y t+h |Ω t−1 ) (A.32) If one assumes that µ t ∼ N (0, Σ µ ), then we can write (see Pesaran and Shin (1998)): E(µ t |µ jt = σ jj ) = [(σ 1j , ..., σ mj ) ′ σ −1 jj ]σ jj = Σ µ e j (A.33) So that the impact of a one standard deviation j shock on variable i at horizon 0 is 53 Electronic copy available at: https://ssrn.com/abstract=3827913 (with Equation A.29): GIRF i (0, σ jj e j , Ω t−1 ) = e ′ i Φ 0 Σ µ e j (A.34) As Φ 0 = I we get: GIRF i (0, σ jj e j , Ω t−1 ) = e ′ i Σ µ e j = σ ij = σ ij = GIRF j (0, σ ii e i , Ω t−1 ) (A.35) Similarly, the identification strategy of Fengler and Herwartz (2018) used in DCC Fengler yields also symmetric IRFs on impact. This is because the time-varying matrices B −1 0,t (and hence B 0,t ) are symmetric as, ∀t and knowing that Λ t is diagonal and therefore symmetric: To conclude, identification with GIRFs or the identification of Fengler and Herwartz (2018) impose a symmetric structure of impulse responses upon impact while identification by Cholesky assumes a recursive one. These assumptions may be controversial when it comes to financial data which tend to respond rapidly to shocks and where variables react asymmetrically to each other.

A.5 Data sources OLS regressions
• Similar Business Cycle : the quarterly squared difference between country i and country j's GDP growth (multiplied by (-1) so that a higher number indicates more similar tendencies) [this is similar to De Santis and Zimic (2018) Electronic copy available at: https://ssrn.com/abstract=3827913 • Similar D/GDP : Same approach for quarterly D/GDP ratios (multiplied by (-1) so that a higher number indicates more similar tendencies); Source: IMF • Trade exposure: Second, as exposed in Section 3.2, we assume a constant B 0 in our study. Some authors argued that the increase in CDS-correlation during the EZ debt crisis came mainly from changes in volatility and not in propagation mechanisms (Caporin et al. (2018)), but this point is disputed (De Santis and Zimic (2018) 27 Note that Ehrmann and Fratzscher (2017) use 3 smaller subperiods, with a "crisis period" for the Greek turmoil that starts in March 2010 and ends in March 2012, and a "post-crisis period" that starts in October 2012. However, to build the total spillover indices S H , we need the exact identification underlined in Section 5.1.1, i.e. being able to assign each shock to a single variable. This identification is not granted for any subsample, and is hard to achieve on short time intervals. Thus, on Figure A.2 we rely on a large "crisis period" so as to obtain this identification.

Figure A.2: Robustness graphs
The different lines represent the Total Spillover indices S H built from our main specification ("SVAR-GARCH") as well as from the other specifications outlined above. For the upper part of the graph, the different indices are named according to the exogenous variables included (Oil price, Macro news from Citibank, US and UK bank or sovereign CDS). For the bottom part of the graph, the different indices represent our main specification estimated on subsamples (before and after 01/10/2012 as outlined in Ehrmann and Fratzscher (2017)). For readability we show 10 day moving averages of the indices.