COVID-19 spreading in financial networks: A semiparametric matrix regression model
Billio Monica, Casarin Roberto, Costola Michele, Iacopini Matteo
CCOVID-19 spreading in financial networks: Asemiparametric matrix regression model ∗ Monica Billio † Roberto Casarin ‡ Michele Costola § Matteo Iacopini ¶ January 5, 2021
Abstract
Network models represent a useful tool to describe the complex set of financial relationshipsamong heterogeneous firms in the system. In this paper, we propose a new semiparametricmodel for temporal multilayer causal networks with both intra- and inter-layer connectivity.A Bayesian model with a hierarchical mixture prior distribution is assumed to captureheterogeneity in the response of the network edges to a set of risk factors including theEuropean COVID-19 cases. We measure the financial connectedness arising from theinteractions between two layers defined by stock returns and volatilities. In the empiricalanalysis, we study the topology of the network before and after the spreading of the COVID-19 disease.
Keywords : multilayer networks; financial markets; COVID-19;
JEL Classification : C11; C58; G10. ∗ We thank the participants at the 14th International Conference Computational and FinancialEconometrics (CFE 2020) for the provided comments. We also thank Maurizio La Mastra for theexcellent research assistant. This research used the SCSCF and the HPC multiprocessor cluster systemsprovided by the Venice Centre for Risk Analytics (VERA) at University Ca’ Foscari of Venice. MatteoIacopini acknowledges financial support from the Marie Sk(cid:32)lodowska-Curie Actions, European Union, SeventhFramework Program HORIZON 2020 under REA grant agreement n. 887220. † Ca’ Foscari University of Venice, Email: [email protected]. (corresponding author) ‡ Ca’ Foscari University of Venice, Email: [email protected]. § Ca’ Foscari University of Venice, Email: [email protected]. ¶ Vrije Universiteit Amsterdam, Email: [email protected]. a r X i v : . [ ec on . E M ] J a n Introduction
The recent outbreak of the COVID-19 disease has severely affected the economy and thefinancial markets due to the consequences of the lockdowns and travel limitations. Accordingto the International Monetary Fund (IMF, 2020), the world growth in 2020 is projected tohave a contraction of 4.4%. During February-March of 2020, the global financial market hassuffered multiple crashes having the largest drop of around 13% on 16 March 2020. To ensurethe financial stability and avoid the breakdown of the markets, central banks supported thefunctioning of the system with asset purchase programs. While the global financial crisis hasbeen originated by the vulnerabilities of the US mortgage market, which in turn has beenthe root of the European sovereign debt crisis, the COVID-19 pandemic has representedan (unprecedented) exogenous shock to the financial system that could not be reasonablyforeseen and hence, priced by the financial markets. The analysis the financial connectednessthrough the network modeling can provide interesting insights to policy makers on the effectof the COVID-19 on the financial system.The literature of financial connectedness and network modeling has rapidly increasedafter the recent financial crisis both in theoretical (i.e., Elliott et al., 2014; Acemoglu et al.,2015) and empirical analyses (i.e., Billio et al., 2012; Diebold and Yılmaz, 2014). Regardingthe econometric methods, the methodologies proposed for extracting unobserved networkshave been particularly flourishing, especially in the Bayesian framework. Ahelegbey et al.(2016a,b) combine a Bayesian graphical approach with vector autoregressive (VAR) modelswhere the contemporaneous and temporal causal structures of the model are represented bymeans of two distinct graphs. In a graphical model framework, Bianchi et al. (2019) propose aMarkov-switching graphical SUR model to investigate changes in systemic risk. The authorsshow that connectivity increases in during 1999-2003 and in 2008-2009. Using a time-varyingparameter vector autoregressive model (TVP-VAR), Geraci and Gnabo (2018) estimate adynamic Granger Network in the S&P 500 market and find a gradual decrease in networkconnectivity not detectable using a rolling window approach. Similarly, in the framework1f forecast variance error decomposition (Diebold and Yilmaz, 2009; Diebold and Yılmaz,2014). In high-dimensional multivariate time-series, Billio et al. (2019) propose a Bayesiannonparametric Lasso prior for VAR models. The causal networks are extracted throughclustering and shrinking effects, and well describe real-world networks features. Bernardiand Costola (2019) propose a shrinkage and selection methodology for network inferencethrough a regularised linear regression model with spike-and-slab prior on the parameters.The financial linkages are expressed in terms of inclusion probabilities resulting in a weighteddirected network. Referring to econometric models for networks, Billio et al. (2018b) proposea dynamic linear regression model for tensor-valued response variables and covariates witha parsimonious parametrization based on the parallel factor decomposition. Billio et al.(2018a) propose a Bayesian Markov-switching regression model for multidimensional arrays(tensors) of binary time series. The coefficient tensor can switch between multiple regimesin order to capture time-varying sparsity patterns of the network structure.In this paper we propose a new semiparametric model for temporal multilayer causalnetworks. A Bayesian model with mixture prior distribution is assumed in order to modelheterogeneity in the response of the network edges to a set of exogenous variables.Recently, the literature has focused on multilayer networks where different channelsof relationships, defined as layers, characterize a given set of nodes (system). Thestudy of connectedness involving the interdependence among these layers can improve themeasurement of the network topology. For instance, Wang et al. (2020) consider a multilayerGranger causality network composed by mean spillover, volatility spillover, and extremerisk spillover layers. The authors show that significant changes in connectivity on extremerisk and volatility spillover layers before a general financial turmoil. Casarin et al. (2020)propose a Bayesian graphical vector autoregressive model to extract multilayer network inthe international oil market and show that oil production network is a lagged driver forprices.In this paper, we follow this stream of literature and consider multilayer networks with2oth intra- and inter-layer connectivity. We measure the financial connectedness arisingfrom the interactions between two layers defined by assets returns and volatilities. Theseconnectivity effects are represented at four levels: (i) return linkages (returns causes returns);(ii) volatility linkages (volatility causes volatility); (iii) risk premium linkages (volatilitycauses returns); and (iv) leverage linkages (return causes volatility).To investigate the role of selected risk factors on the dynamic evolution of the multilayerfinancial network, we propose a novel semiparametric model for panels of matrix-valueddata. The use of matrix-valued statistical models in time series econometrics has becomeincreasingly popular over the last decades. In the seminal paper by Harrison and West(1999), matrix-valued distributions were exploited for representing state space models.More recently, Carvalho and West (2007); Wang and West (2009) used the matrix normaldistribution in Bayesian dynamic linear models while Carvalho et al. (2007) applied thehyper-inverse Wishart distribution in a Gaussian graphical model. Other applications ofmatrix-variate distributions include stochastic volatility (Uhlig, 1997; Gouri´eroux et al.,2009; Golosnoy et al., 2012), classification of longitudinal datasets (Viroli, 2011), modelsfor network data (Zhu et al., 2017, 2019), and factor models (Chen et al., 2019). Withrespect to the existing literature, our approach proposes two main contributions. First, weextend the matrix-valued linear model to panels of matrix-valued data. Second, we proposea hierarchical mixture prior to cope with overfitting and loss of efficiency in high-dimensionalsettings. This prior choice allows for a semiparametric model that grants higher flexibility ininvestigating the impact of covariates on matrix-valued response variables. Our model andinference are well suited for the analysis of multilayer temporal networks, where the intra-and inter-layer connectivity at each point in time is encoded by a cross section of adjacencymatrices.Finally, an original application to a European financial network among 412 firms basedin Germany, France, and Italy, shows that the proposed framework scales well in high-dimensions (i.e., hundreds of nodes) and can be successfully used to provide new insights on3hock transmission in financial markets. Inspired by the causality definition between returnand volatility (i.e., Bekaert and Wu, 2000), we label the intra-connectivity risk premiumaccording to the time-varying risk premium hypothesis (volatility causes returns) and theleverage according to the leverage hypothesis (return shocks lead to changes in volatilities).In the proposed model, each adjacency matrix related to the four levels of connectivity ismodelled as a function of a set of risk factors, including market returns, implied volatility,corporate credit risk, and the number of COVID-19 new confirmed European cases. In theempirical analysis, we study the topology of the European financial network before and afterthe spreading of the COVID-19 disease.Our findings highlight that COVID-19 is the most relevant factor in explaining theconnectivity of the European financial network at firm and sector level, in particularindustrial, real estate, and health care. The probabilities of volatility and risk premiumlinkages are the most positively affected by the COVID-19, which at the same time has anegative effect on leverage linkages. Moreover, we find evidence of a positive relationshipbetween firm centrality and the number of its linkages that are impacted by the COVID-19,across all layers except leverage.The remaining of the paper is structured as follows. Section 2 introduces a noveleconometric framework for matrix-valued panel data, then Section 3 presents the Bayesianinference procedure. Section 4 illustrates the empirical analysis and our major results.Finally, Section 5 concludes.
Let G t = ( G t , G t , G t , G t , E ) a two-layer temporal network (Boccaletti et al., 2014),where G ijt ⊂ E × E is the connectivity graph between layers i and j , E = { , . . . , N } is theset of nodes (firms). In the proposed framework, each node represents a firm, the two layersare given by firm stock return (layer 1) and volatility (layer 2). Four graphs encode the4onnectivity between and within layers. Here, we focus on causal financial network whereeach adjacency matrix is directed and hence, asymmetric. This implies that each element ofthe matrix indicates a causal relationship between two nodes. The connectivity is representedthrough the intra-layer adjacency matrices Y ,t and Y ,t and inter-layer adjacency matrices Y ,t and Y ,t .For the intra-connectivity graphs, we label return linkages the sub-network Y ,t (returnscauses returns) and volatility linkages the sub-network Y ,t (volatility causes volatility).Regarding the inter-connectivity graphs, we label the two graphs inspired by the causalitydefinition in asset pricing between return and volatility as discussed in Bekaert and Wu(2000). The label risk premium linkages for the sub-network Y ,t refers to the time-varyingrisk premium hypothesis (volatility causes return) while the label leverage linkages for thesub-network Y ,t is based on the leverage hypothesis (return causes volatilities).We propose the following matrix-variate linear model for studying the impact of a set of R covariates ( f ,t , . . . , f R,t ) on the linkages Y ,t = R (cid:88) r =1 B ,r f r,t + E ,t , E ,t ∼ MN n,n ( O, Σ , , Σ , ) Y ,t = R (cid:88) r =1 B ,r f r,t + E ,t , E ,t ∼ MN n,n ( O, Σ , , Σ , ) Y ,t = R (cid:88) r =1 B ,r f r,t + E ,t , E ,t ∼ MN n,n ( O, Σ , , Σ , ) Y ,t = R (cid:88) r =1 B ,r f r,t + E ,t , E ,t ∼ MN n,n ( O, Σ , , Σ , ) (1)where B lk,r are ( n × n ) matrices of coefficients and MN n,n ( O, Σ , Σ ) denotes the zero-meanmatrix normal distribution (see Gupta and Nagar, 1999, Ch.2, for further details) with twovariance/covariance matrices Σ and Σ . If the ( n × p ) random matrix X is distributed asa matrix normal with mean M and covariance matrices Σ , Σ , with Σ of size ( n × n ) and5 of size ( p × p ), written X ∼ MN n,p ( M, Σ , Σ ), then P ( X | M, Σ , Σ ) = (2 π ) − np/ | Σ | − p/ | Σ | n/ exp (cid:16) −
12 tr (cid:0) Σ − ( X − M ) (cid:48) Σ − ( X − M ) (cid:1)(cid:17) . (2)For identification purposes, we assume Σ lk, = I n and Σ lk, = diag( σ ,lk , . . . , σ n,lk ), for each l, k = 1 , As regards the prior assumption on the model parameters, we choose the followingindependent mixture of Normal distributions for the coefficients b ij,lk,r , for i, j = 1 , . . . , n (with i (cid:54) = j ), l, k = 1 ,
2, and r = 1 , . . . , Rb ij,lk,r | p lk , µ lk , γ lk ∼ p ,lk N ( b ij,lk,r | , γ ,lk ) + M b (cid:88) m =2 p m,lk N ( b ij,lk | µ m,lk , γ m,lk ) . (3)To solve the label switching problem, we impose the identification constraint µ ,lk < µ ,lk <. . . < µ M b ,lk . As regards the prior distribution for the variances ( σ ,lk , . . . , σ n,lk ) we assumethe following mixture of Inverse Gamma distributions σ i,lk,r | q lk , α lk , β lk ∼ M σ (cid:88) m =1 q m,lk IG ( σ i,lk | α m,lk , β m,lk ) , (4)and impose the identification constraint on the mean by assuming β ,lk / ( α ,lk − <β ,lk / ( α ,lk − < . . . < β M σ ,lk / ( α M σ ,lk − p ,lk , p ,lk , . . . , p M b ,lk ) ∼ D ir ( φ b , φ b , . . . , φ b ) (5)( q ,lk , q ,lk , . . . , q M σ ,lk ) ∼ D ir ( φ σ , φ σ , . . . , φ σ ) (6)6 ,lk = 0 , (7) γ ,lk ∼ G a ( a , b ) , (8) µ m,lk ∼ N (0 , s ) , m = 2 , . . . , M b (9) γ m,lk ∼ IG ( a , b ) , m = 2 , . . . , M b (10) α m,lk ∼ G a ( a , b ) , m = 1 , . . . , M σ (11) β m,lk ∼ G a ( a , b ) , m = 1 , . . . , M σ , (12)which is a standard choice in Bayesian mixture modeling (Fr¨uhwirth-Schnatter, 2006). Thefirst component of the mixture prior for coefficients b ij,lk,r is a Bayesian Lasso (Park andCasella, 2008) distribution, that corresponds to setting a = 1. This prior specificationstrategy overcomes overparametrization and overfitting issues by clustering coefficients intogroups and by shrinking the coefficients in the first group toward zero, thus improvingthe estimation efficiency in high-dimensions. The proposed hierarchical mixture priordistribution naturally induces a mixture model for the matrix-valued observations. Foreach matrix Y lk,t , with l, k = 1 , t = 1 , . . . , T , by integrating out the parameters in thelikelihood, one obtains the following marginal likelihood. P ( Y lk,t | µ lk , γ lk , α lk , β lk ) = M nσ (cid:88) m (cid:48) =1 M n b (cid:88) m =1 ˜ p m,lk ˜ q m (cid:48) ,lk P ( Y lk,t | ˜ θ bm,lk , ˜ θ σm (cid:48) ,lk ) , (13)where ˜ θ bm,lk = ( ˜ µ m,lk , ˜ γ m,lk ), ˜ θ σm,lk = ( ˜ α m (cid:48) ,lk , ˜ β m (cid:48) ,lk ), and P ( Y lk,t | ˜ θ bm,lk , ˜ θ σm (cid:48) ,lk ) == (cid:90) (cid:90) P ( Y lk,t | B lk, , . . . , B lk,R , σ lk ) P ( σ lk | ˜ θ σm (cid:48) ,lk ) P ( B lk, , . . . , B lk,R | ˜ θ bm,lk ) d B lk, · · · d B lk,R d σ lk . See the Appendix for a proof. We summarize our Bayesian semiparametric model in theDirected Acyclic Graph representation of Figure 1.7 a b a b a b a b φ b φ σ p lk q lk µ m,lk γ ,lk γ m,lk B lk,r α m,lk β m,lk σ i,lk Y lk,t f t Figure 1: Directed Acyclic Graph of the proposed Bayesian semiparametric modelfor multilayer networks. It exhibits the conditional independence structure of theobservation model for Y lk,t with covariates f t = ( f ,t , . . . , f R,t ) (cid:48) (grey circles), theparameters p lk = ( p ,lk , . . . , p M b ,lk ), q lk = ( q ,lk , . . . , q M σ ,lk ), B lk,r , σ i,lk , the component-specific parameters µ m,lk , γ m,lk , α m,lk , β m,lk (white solid circles) and the fixed hyperparameters s, a , b , a , b , a , b , a , b (white dashed circles). The directed arrows show the causaldependence structure of the model. Start by introducing two set of allocation variables D bij,lk,r , for i, j = 1 , . . . , n , l, k = 1 , r = 1 , . . . , R , and D σi,lk , for i = 1 , . . . , n , l, k = 1 ,
2. Denote the collection of all parameterswith θ = ( B lk, , . . . , B lk,R , σ ,lk , . . . , σ n,lk ) (cid:48) , and let Y , f be the collection of all observednetworks and risk factors, respectively. Letting σ lk = ( σ ,lk , . . . , σ n,lk ) (cid:48) , the likelihood of themodel in Eq. (1) is P ( Y , f | θ ) = T (cid:89) t =1 2 (cid:89) l =1 2 (cid:89) k =1 (2 π ) − n / | diag( σ lk ) | − n/ | I n | n/ · exp (cid:16) −
12 tr (cid:16) diag( σ lk ) − (cid:0) Y lk,t − R (cid:88) r =1 B lk,r f r,t (cid:1) (cid:48) I − n (cid:0) Y lk,t − R (cid:88) r =1 B lk,r f r,t (cid:1)(cid:17)(cid:17) = (2 π ) − n T (cid:89) l =1 2 (cid:89) k =1 | diag( σ lk ) | − nT/ · exp (cid:16) −
12 tr (cid:16) (cid:88) l =1 2 (cid:88) k =1 diag( σ lk ) − T (cid:88) t =1 (cid:0) Y lk,t − R (cid:88) r =1 B lk,r f r,t (cid:1) (cid:48) (cid:0) Y lk,t − R (cid:88) r =1 B lk,r f r,t (cid:1)(cid:17)(cid:17) . b ij,lk, , . . . , b ij,lk,R ) from the Normal distribution P ( b ij,lk,r |− ).2. Draw σ i,lk from Inverse Gamma distribution P ( σ i,lk |− ).3. Draw the allocations ( d b ,lk,r , . . . , d bn ,lk,r , d b ,lk,r , . . . , d bn ,lk,r , d b n,lk,r , . . . , d bnn,lk,r ) fromthe discrete distribution P ( d bij,lk,r |− ).4. Draw the allocations ( d σ ,lk , . . . , d σn,lk ) from the discrete distribution P ( d σi,lk |− ).5. Draw ( p ,lk , . . . , p M b ,lk ) from the Dirichlet distribution P ( p lk |− ).6. Draw ( q ,lk , . . . , q M σ ,lk ) from the Dirichlet distribution P ( q lk |− ).7. Draw the hyperparameters:a) µ m,lk , for m = 2 , . . . , M σ , from the Normal distribution P ( µ m,lk |− ).b) γ ,lk from the Generalized Inverse Gaussian distribution P ( γ ,lk |− ).c) γ m,lk , for m = 2 , . . . , M b , from the Inverse Gamma distribution P ( γ m,lk |− ).d) α m,lk , for m = 1 , . . . , M σ , from the distribution P ( α m,lk |− ).e) β m,lk , for m = 1 , . . . , M σ , from the Gamma distribution P ( β m,lk |− ). In this section, we first describe the firms’ dataset and the set of risk factors (source:Bloomberg and Eikon/Datastream), then we illustrate the network extraction procedureby means of Granger causality tests. Finally, we discuss the results of the proposed networkmodel. 9 .1 Data description
The European firms.
The dataset includes 412 European firms (176 German, 162 French,74 Italian) belonging to 11 GICS sectors: Financials (43 firms), Communication Services (38firms), Consumer Discretionary (61 firms), Consumer Staples (17 firms), Health Care (48firms), Energy (9 firms), Industrials (88 firms), Information Technology (45 firms), Materials(24 firms), Real Estate (22 firms), Utilities (12 firms), and not classified in a specific GICSsector (5 firms). The data sample ranges from the 4th of January 2016 to the 30th of September 2020, atweekly frequency (Friday-Friday), thus including the period before and after the outbreakof the COVID-19. The weekly logarithmic return for firm i , r i,t , is obtained from the totalreturns series, whereas the weekly volatility is computed using the estimator of the varianceproposed by Garman and Klass (1980):ˆ σ i,t = 0 . H i,t − L i,t ) − . C i,t − O i,t ) − . C i,t − O i,t )( H i,t + L i,t − O i,t ) − H i,t − O i,t )( L i,t − O i,t )] , (14)where H i,t is the weekly logarithmic high price, L i,t is the weekly logarithmic low price, O i,t is the weekly logarithmic opening price, and C i,t is the logarithmic closing price. The weeklyprices have been obtained by taking in a given week the maximum among the daily highprices (weekly High Price), the minimum among the daily low prices (weekly Low Price),the opening price of the first available day in a week (weekly Opening Price), and the closingprices of the last available day in a week (weekly Closing Price). The risk factors.
We consider the following risk factors: (i) the log-returns on the EuroSTOXX 50 index (SX5E), (ii) the implied volatility on the Euro STOXX 50 index (V2X),(iii) the Bloomberg Barclays EuroAgg Corporate Average OAS (LECPOAS) as a proxy for The list of the firms, the countries, and the information about their GICS sectors and industries areavailable upon request to the Authors. S X E V X L E C P O A S N C O V E U R Figure 2: The considered risk factors in the analysis: the returns on the Euro STOXX50 index (top-left), the implied volatility on the Euro STOXX 50 index (top-right), theBloomberg Barclays EuroAgg Corporate Average OAS (bottom-left) and the new EuropeanCOVID-19 cases (bottom-right) for the considered European firms over time.
We estimate the dynamic networks of European financial institutions using pairwise Granger-causality (e.g., see Billio et al., 2012). In this respect, we make use of a rolling windowapproach (104 observations, that is 2 years) and consider the intra- and inter-connectivity,namely, the return linkages, the volatility linkages, the risk premium linkages, and theleverage linkages: x i,t = m (cid:88) l =1 b l x i,t − l + m (cid:88) l =1 b l x j,t − l + ε it x j,t = m (cid:88) l =1 b l x i,t − l + m (cid:88) l =1 b l x j,t − l + ε jt (15)11 , t r e t u r n Y , t l e v e r ag e Y , t v o l a t ili t y Y , t r i s k p r e m i u m Figure 3: The network densities (solid blue line) at 1% level of statistical significance forthe return linkages (top-left), the volatility linkages (bottom-left), the leverage linkages (top-right), and the risk premium (bottom-right) for the considered European firms over time. r e t u r n , v o l a t ili t y l e v e r ag e , r i s k p r e m i u m Figure 4: The network densities at 1% level of statistical significance from the 21st ofFebruary 2020 to the 1st of May 2021 (same scale). The return linkages (solid line), thevolatility linkages (dashed line), the leverage linkages (dashed-dotted line) and the riskpremium (dotted line) for the considered European firms over time.where i, j = 1 , . . . , k and x = { r, ˆ σ } . Each entry ( i, j ) of the adjacency matrix Y lk,t associatedto layer lk , with l, k = 2, is defined as Y ij,lk,t = pval ( b ij ) for i (cid:54) = j . Therefore, theelement Y ij,lk,t represents the observed probability that the relationship between x i,t and x j,t is statistically different from zero. We estimate a total of 145 × Figure 3 provides the density of the four sub-networks over time at 1% level of statistical The estimation algorithm has been parallelized and implemented in MATLAB on two nodes at the HighPerformance Computing (HPC) cluster (VERA - Ca’ Foscari University). Each node has 2 CPUs Intel Xeonwith 20 cores 2.4 Ghz and 768 GB of RAM. . Despite sharing some similarities, the dynamics of the intra- and inter-connectivity linkages can provide different signals on shock propagation in the financialmarkets. This calls for a joint modeling of the four connectivity layers. On average, thedensity is higher for the volatility linkages (0 .
16) followed by the risk premium linkages(0 . . . The density of the network is defined as the total number of observed linkages over the total number ofpossible linkages. If the density is 0, no connections exist, while if density is 1, the network is fully connected. r e t u r n Y r e t u r n Y v o l a t ili t y Y v o l a t ili t y Figure 5: The intra-layer directed networks: returns linkages (top) and volatility linkages(bottom) on the 17th of January 2020 (left) and the 27th of March 2020 (right). Edgesare clockwise directed. Node size: proportional to the total degree averaged over timewithin each regime. Edge color according to the Industry of the source node: Financials(light green), Communication Services (violet), Consumer Discretionary (pink), ConsumerStaples (orange), Health Care (light blue), Energy (light orange), Industrials (olive green),Information Technology (dark orange), Materials (green), Real Estate (dark pink), Utilities(blue), and not classified in a specific GICS sector (dark green). For exposition purposes,we drop edges with p-value larger than 0.1%.14 r i s k p r e m i u m Y r i s k p r e m i u m Y l e v e r ag e Y l e v e r ag e Figure 6: The inter-layer directed networks: risk premium linkages (top) and leveragelinkages (bottom) on the 17th of January 2020 (left) and the 27th of March 2020 (right).Edges are clockwise directed. Node size: proportional to the total degree averaged over timewithin each regime. Edge color according to the Industry of the source node: Financials(light green), Communication Services (violet), Consumer Discretionary (pink), ConsumerStaples (orange), Health Care (light blue), Energy (light orange), Industrials (olive green),Information Technology (dark orange), Materials (green), Real Estate (dark pink), Utilities(blue), and not classified in a specific GICS sector (dark green). For exposition purposes,we drop edges with p-value larger than 0.1%.15etworks on the 17th of January 2020 (one week before the first European COVID-19 case)and the 27th of March 2020. As shown in the Figures, the connectivity increases in the intra-layer networks for the return linkages and especially, on the volatility linkages. Regarding theinter-layer directed networks, the connectivity of the risk premium linkages increase whileit decreases significantly in the leverage linkages. It is therefore interesting to apply theproposed network model to measure the impact of the COVID-19 and the other risk factors,on the intra- and inter-linkages of the considered European firms.
In this section, we apply the model and inference proposed in Sections 2-3 to estimate theimpact of the risk factors on the multilayer European financial network. Since our analysisis focused on investigating the role of COVID-19, we describe the effects of the other riskfactors (returns on the Euro STOXX 50 index, the implied volatility on the Euro STOXX 50index, and the Bloomberg Barclays EuroAgg Corporate Average OAS) on financial linkagesin the Appendix. Figure 7 reports the impact of COVID-19 on each linkage ( i, j ), from firm j to firm i , across all layers. In particular, we report only non-null effects, that is coefficientssuch that zero is not included in the corresponding 95% posterior high probability densityinterval (HPDI). The blue color indicates a positive impact on the probability of an edgefrom firm j to firm i , while the red color indicates a negative impact. Since our responsevariable is an increasing transformation of the p-value of Granger causality test performedon a pair ( i, j ) of time-series, a negative (positive) COVID-19 coefficient implies an increase(decrease) in the p-value, hence in the probability to observe a linkage from j to i . As shownin the Figure, the COVID-19 has a weaker effect on return linkages in terms of magnitudeand number of impacted linkages. Conversely, it increases the probability of volatility andrisk premium linkages, whereas, in most of the cases, it reduces the probability of leveragelinkages. Overall, among the selected risk factors, the COVID-19 has the greatest impacton the European financial networks. The other risk factors, such as the market returns,16he implied volatility, and the corporate credit risk, have some effect on the volatility andleverage linkages and very weak effect on return and risk premium linkages (see Figure 11in the Appendix). r e t u r n -5-4-3-2-1012345 r i s k p r e m i u m -5-4-3-2-1012345 v o l a t ili t y -5-4-3-2-1012345 l e v e r ag e -5-4-3-2-1012345 Figure 7: Impact of COVID-19 on financial linkages for intra-layer (left column) and inter-layer (right column) networks. In each plot, the coefficient in position ( i, j ) refers to theimpact of COVID-19 on the edge from firm j to firm i . Only non-null coefficients arereported: blue indicates positive impact on edge existence, red indicates negative impact onedge existence. A coefficient is considered null if its posterior HPDI contains zero.Figure 8 shows the net effect of COVID-19 on the linkages between sectors. In eachpanel, the block in position ( i, j ) refers to the number of linkages (net effect) from sector j to sector i impacted by COVID-19. The main empirical findings are the following: • there is evidence of a heterogeneous net impact on return and risk premium linkages, ofa substantial increase in volatility linkages, and of a decrease in the leverage linkages;17 the Industrial sector plays a pivotal role in the connectivity structure of the multilayernetwork. In return linkages, risk premium, and volatility linkages, there is an increasein the connectivity from and to the other sectors (except for Consumer Staples, Energy,and Utilities). The Industrial sector exhibits the largest increase in the connectivitylevel within sector (except in the leverage layer); • in the return linkages, the Financial sectors shows the largest decrease in theconnectivity to the other sectors (red squares in the column). Conversely, in the riskpremium and volatility linkages, there is an increase in the connectivity to the othersectors (except for Utilities); • utilities are the unique sector that exhibits a decrease in the connectivity from othersectors in the risk premium linkages, especially from the Industrial sector. A similarbehavior can be found also in the return linkages for the Financial, Real Estate, andUtilities sectors; • energy is the only sector that does not affect and is not unaffected by COVID-19 in allthe multilayer network (except for the incoming connectivity in the return linkages); • the Health Care and the information technology are the sectors that show the largestdecrease in the connectivity from other sectors in the leverage linkages.In conclusion, the results discussed above show that the COVID-19 has affected theconnectivity of the European financial network between the sectors. We further investigatethe relationship between the impact of COVID-19 on financial linkages and the centrality ofeach firm in the network.Centrality is measured either by in-/out-degree or betweenness centrality. Betweennesscentrality quantifies the number of times a node acts as a bridge along the shortest pathbetween two other nodes. Firms with large betweenness contribute to spreading contagionin the networks, thus requiring to be monitored for the stability of the financial system. We18 e t u r n C o m m un i c a t i on S e r v i c e s C on s u m e r D i sc r e t i ona r y C on s u m e r S t ap l e s E ne r g y F i nan c i a l s H ea l t h C a r e I ndu s t r i a l s I n f o r m a t i on T e c hno l og y M a t e r i a l s R ea l E s t a t e U t ili t i e s Communication ServicesConsumer DiscretionaryConsumer StaplesEnergyFinancialsHealth CareIndustrialsInformation TechnologyMaterialsReal EstateUtilities -80-70-60-50-40-30-20-1001020 r i s k p r e m i u m C o m m un i c a t i on S e r v i c e s C on s u m e r D i sc r e t i ona r y C on s u m e r S t ap l e s E ne r g y F i nan c i a l s H ea l t h C a r e I ndu s t r i a l s I n f o r m a t i on T e c hno l og y M a t e r i a l s R ea l E s t a t e U t ili t i e s Communication ServicesConsumer DiscretionaryConsumer StaplesEnergyFinancialsHealth CareIndustrialsInformation TechnologyMaterialsReal EstateUtilities -300-250-200-150-100-500 v o l a t ili t y C o m m un i c a t i on S e r v i c e s C on s u m e r D i sc r e t i ona r y C on s u m e r S t ap l e s E ne r g y F i nan c i a l s H ea l t h C a r e I ndu s t r i a l s I n f o r m a t i on T e c hno l og y M a t e r i a l s R ea l E s t a t e U t ili t i e s Communication ServicesConsumer DiscretionaryConsumer StaplesEnergyFinancialsHealth CareIndustrialsInformation TechnologyMaterialsReal EstateUtilities -1600-1400-1200-1000-800-600-400-200 l e v e r ag e C o m m un i c a t i on S e r v i c e s C on s u m e r D i sc r e t i ona r y C on s u m e r S t ap l e s E ne r g y F i nan c i a l s H ea l t h C a r e I ndu s t r i a l s I n f o r m a t i on T e c hno l og y M a t e r i a l s R ea l E s t a t e U t ili t i e s Communication ServicesConsumer DiscretionaryConsumer StaplesEnergyFinancialsHealth CareIndustrialsInformation TechnologyMaterialsReal EstateUtilities
Figure 8: Impact of COVID-19 on sector linkages for intra-layer (left column) and inter-layer(right column) networks. In each plot, the block in position ( i, j ) refers to the number oflinkages (net effect) from sector j to sector i impacted by COVID-19. Blue (red) indicatesan increase (decrease) of the number of linkages. return risk premium leverage volatility Figure 9: Number of edges impacted by COVID-19, across layers (columns), for the nodes’total degree. In each scatterplot: node total degree on the 27th March 2020 (horizontalaxis) versus the number of non-null coefficients. Only non-null coefficients are reported:blue indicates positive impact on edge existence, red indicates negative impact on edgeexistence. A coefficient is considered null if its posterior HPDI contains zero.19easure the impact by considering the sum of the negative (blue) and positive (red) nodecoefficients of a risk factor, that is:˜ b IN, + i,lk,r = n (cid:88) j =1 ˆ b ij,lk,r I (ˆ b ij,lk,r > , ˜ b IN, − i,lk,r = n (cid:88) j =1 ˆ b ij,lk,r I (ˆ b ij,lk,r < , ˜ b OUT, + i,lk,r = n (cid:88) j =1 ˆ b ji,lk,r I (ˆ b ji,lk,r > , ˜ b OUT, − i,lk,r = n (cid:88) j =1 ˆ b ji,lk,r I (ˆ b ji,lk,r < , ˜ b BT W, + i,lk,r = n (cid:88) j =1 ˆ b ij,lk,r I (ˆ b ij,lk,r >
0) + n (cid:88) j =1 ˆ b ji,lk,r I (ˆ b ji,lk,r > , ˜ b BT W, − i,lk,r = n (cid:88) j =1 ˆ b ij,lk,r I (ˆ b ij,lk,r <
0) + n (cid:88) j =1 ˆ b ji,lk,r I (ˆ b ji,lk,r < . (16)Figure 9 shows the number of linkages of each firm which are impacted by COVID-19 versusthe firms’ total degree. We find evidence, across the different inter- and inter-layer networks,of a positive relationship between firm centrality and the effect of COVID-19, except forleverage linkages. In particular, in the risk premium and volatility layer, almost half of thelinkage of each node have been impacted by the COVID-19.Figure 10 shows the node centrality on the 27th March 2020 versus the sum of thenegative (blue) and positive (red) node coefficients. In each plot, the triangles indicatethe firms with an increased betweenness centrality after the outbreak of the COVID-19.More specifically, we identify the firms which moved from the 1st tercile of the betweennesscentrality distribution on the 17th January 2020 to the 3rd tercile on the 27th March 2020.The negative impact of COVID-19 on edge existence (red color) uniformly affects thefirms with low and high centrality in the multilayer networks. This is also true for thepositive impact of COVID-19 (blue color) in the leverage linkages. The most interestingfindings concern the leverage and risk premium linkages. In the volatility linkages, there isa positive relationship between the COVID-19 coefficients and the IN degree (last columnin the first row) which indicates that the connectivity of firms with higher IN degree is lessaffected by COVID-19. Conversely, there is a negative relationship between the COVID-1920eturn risk premium leverage volatility I N d e g r ee O U T d e g r ee b e t w ee nn e ss Figure 10: Impact of the COVID-19 factors on financial linkages versus firm centrality,across layers (columns) and centrality measures (rows). In each scatterplot: node centralityon the 27th March 2020 (horizontal axis) versus the sum of the negative (blue) and positive(red) node coefficients of a given variable (vertical axis). Filled triangles indicate firms thatmoved from the 1st tercile of the betweenness centrality distribution on the 17th January2020 to the 3rd tercile on the 27th March 2020. Only non-null coefficients are reported: blueindicates positive impact on edge existence, red indicates negative impact on edge existence.A coefficient is considered null if its posterior HPDI contains zero.coefficients and the OUT degree (last column in the second row) which indicates that theconnectivity of firms with higher OUT degree is more affected by the COVID-19. Therefore,firms with higher OUT degree become more prone in transmitting volatility shocks to thesystem (last column, second row), with impact on other firms’ volatility and returns (secondcolumn). Similar conclusions can be drawn by considering the effect of COVID-19 on firmswith large betweenness centrality (last column, last row).21
Conclusion
In this paper, we have proposed a novel semiparametric framework for matrix-valued paneldata. The model has been applied to study multilayer temporal networks among Europeanfinancial firms (France, Germany, Italy). We measure the financial connectedness arisingfrom the interactions between two layers defined by assets returns and volatilities. Theconnectivity effects are represented at four levels: (i) return linkages; (ii) volatility linkages;(iii) risk premium linkages; and (iv) leverage linkages.We have investigated the impact of COVID-19 on the structure of the network, whichrepresents an unprecedented case since no previous disease outbreak had affected the realeconomy and the financial markets as the COVID-19 pandemic. There is evidence of theexplanatory power of COVID-19 for the connectivity of the European financial networkat firm and sector level (e.g., Industrial, Real Estate, and Health Care). The COVID-19has a heterogeneous effect across layers, increasing the probabilities of volatility and riskpremium linkages while decreasing the probability of leverage linkages. Finally, our resultsshow a positive relationship between firm centrality and the number of its linkages that areimpacted by the COVID-19.The network modeling of financial connectedness is a useful tool for policy-makers andother authorities in monitoring the financial system. Moreover, despite being motivated byand applied to a European financial network, the proposed econometric framework is generaland can be of interest for studying a wide spectrum of matrix-variate datasets emerging inseveral fields of data science.
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Proof of the results in the paper
A.1 Model properties
Proof of Eq. 13. P ( Y lk,t | µ lk , γ lk , α lk , β lk ) = (cid:90) (cid:90) P ( Y lk,t | B lk, , . . . , B lk,R , σ lk ) P ( σ lk | α lk , β lk , q lk ) d σ lk · P ( B lk, , . . . , B lk,R | µ lk , γ lk , p lk ) d B lk, . . . d B lk,R = (cid:90) (cid:90) P ( Y lk,t | B lk, , . . . , B lk,R , σ lk ) M nσ (cid:88) m (cid:48) =1 ˜ q m (cid:48) ,lk P ( σ lk | ˜ α m (cid:48) ,lk , ˜ β m (cid:48) ,lk ) d σ lk · M n b (cid:88) m =1 ˜ p m,lk P ( B lk, , . . . , B lk,R | ˜ µ m,lk , ˜ γ m,lk ) d B lk, . . . d B lk,R = M nσ (cid:88) m (cid:48) =1 M n b (cid:88) m =1 ˜ p m,lk ˜ q m (cid:48) ,lk P ( Y lk,t | ˜ α m (cid:48) ,lk , ˜ β m (cid:48) ,lk , ˜ µ m,lk , ˜ γ m,lk ) . Since: P ( σ lk | α lk , β lk , q lk ) = n (cid:89) i =1 P ( σ i,lk | α lk , β lk , q lk )= n (cid:89) i =1 M σ (cid:88) m =1 q m,lk P ( σ i,lk | α m,lk , β m,lk )= M σ (cid:88) i =1 . . . M σ (cid:88) i n =1 n (cid:89) u =1 q i u ,lk P ( σ u,lk | α i u ,lk , β i u ,lk ) . By relabeling the indices using the inverse lexicographic order: u = 1 + n (cid:88) (cid:96) =1 ( i (cid:96) − M (cid:96) − σ one obtains P ( σ lk | α lk , β lk , q lk ) = M nσ (cid:88) m (cid:48) =1 ˜ q m (cid:48) ,lk P ( σ lk | ˜ α m (cid:48) ,lk , ˜ β m (cid:48) ,lk ) , P ( σ lk | ˜ α m (cid:48) ,lk , ˜ β m (cid:48) ,lk ) = n (cid:89) i =1 P ( σ i,lk | α π ( i,m (cid:48) ) ,lk , β π ( i,m (cid:48) ) ,lk )˜ q m (cid:48) ,lk = n (cid:89) i =1 q π ( i,m (cid:48) ) ,lk with π mapping the pair of indices ( i, m (cid:48) ) (cid:55)→ m , and m ∈ [1 , M σ ]. A similar argument appliesto P ( B lk, , . . . , B lk,R | µ lk , γ lk , p lk ). A.2 Posterior distributions
In the following, we provide the derivation of the full conditional distribution using the Gibbssampler.The combination of the likelihood of the model in Eq. (1) and the mixture priors inEqs. (3)-(4) yields a high-dimensional integral with no closed-form solution. To addressthis issue, we follow the data-augmentation principle and introduce two sets of latentallocation variables in the set of observations, D blk,r = { d bij,lk,r for i, j = 1 , . . . , n } , for l, k = 1 , r = 1 , . . . , R , and D σlk = { d σi,lk for i = 1 , . . . , n } , for l, k = 1 ,
2. Denotewith α lk = ( α ,lk , . . . , α M σ ,lk ) (cid:48) , β lk = ( β ,lk , . . . , β M σ ,lk ) (cid:48) , µ lk = ( µ ,lk , . . . , µ M b ,lk ) (cid:48) , γ lk =( γ ,lk , . . . , γ M b ,lk ) (cid:48) . Thus, we obtain the data-augmented joint posterior distribution for theparameters of the likelihood, B lk,r and σ lk , the first stage hyper-parameters α lk , β lk , µ lk ,and γ lk , the latent allocation variables D blk,r , D σlk , and the mixing probabilities, p lk and q lk ,with l, k = 1 ,
2, and r = 1 , . . . , R , as: (cid:89) l =1 2 (cid:89) k =1 M σ (cid:89) m =1 P ( α m,lk ) P ( β m,lk ) P ( q m,lk ) M b (cid:89) m =1 P ( µ m,lk ) P ( γ m,lk ) P ( p m,lk ) · (cid:89) l =1 2 (cid:89) k =1 n (cid:89) i =1 P ( σ i,lk | α lk , β lk , d σi,lk ) P ( d σi,lk | q lk ) n (cid:89) j =1 R (cid:89) r =1 P ( b ij,lk,r | µ lk , γ lk , d bij,lk,r ) P ( d bij,lk,r | p lk ) · (cid:89) l =1 2 (cid:89) k =1 T (cid:89) t =1 P ( Y lk,t | f t , B lk, , . . . , B lk,R , σ lk ) 27 ull conditional distribution of b ij,lk,r The full conditional distributions of thecoefficients are given as follows. For each entry ( i, j ), define (cid:15) ij,lk,r,t = y ij,lk,t − (cid:80) r (cid:48) (cid:54) = r b ij,lk,r (cid:48) f r (cid:48) ,t ,thus obtaining P ( b ij,lk,r |− ) ∝ exp (cid:16) − ( b ij,lk,r − µ d bij,lk,r ) γ d bij,lk,r (cid:17) T (cid:89) t =1 exp (cid:16) ( (cid:15) ij,lk,r,t − b ij,lk,r f r,t ) σ i,lk (cid:17) ∝ N ( µ, γ )where γ = (cid:16) γ d ij,lk,r + T (cid:88) t =1 f r,t σ i,lk (cid:17) − µ = γ (cid:16) (cid:80) Tt =1 f r,t (cid:15) ij,lk,r,t σ i,lk + µ d ij,lk,r γ d ij,lk,r (cid:17) Full conditional distribution of σ i,lk The full conditional distributions of the noisevariances are given by P ( σ i,lk |− ) ∝ ( σ i,lk ) − α dσi,lk − exp (cid:16) − β d σi,lk σ i,lk (cid:17) · T (cid:89) t =1 ( σ i,lk ) − n/ exp (cid:16) −
12 tr (cid:0) diag( σ lk ) − ( Y lk − R (cid:88) r =1 B lk,r f r,t ) (cid:48) ( Y lk − R (cid:88) r =1 B lk,r f r,t ) (cid:1)(cid:17) ∝ IG (cid:16) α ˜ d i,lk ,lk + T n , β ˜ d i,lk ,lk + 12 T (cid:88) t =1 E ii,lk,t (cid:17) where E lk,t = ( Y lk,t − (cid:80) r B lk,r f r,t ) (cid:48) ( Y lk,t − (cid:80) r B lk,r f r,t ). Full conditional distribution of d bij,lk,r and d σi,lk The full conditional distributions ofthe allocation variables are given by P ( d bij,lk,r = m |− ) ∝ p m,lk N ( b ij,lk,r | µ m,lk , γ m,lk ) P ( d σi,lk = m |− ) ∝ q m,lk IG ( σ i,lk | α m,lk , β m,lk )28 ull conditional distribution of p m,lk and q m,lk The full conditional distribution of themixing probabilities for each mixture are P ( p lk |− ) ∝ D ir (cid:16) φ b + (cid:88) i,j,r I ( d bij,lk,r = 1) , . . . , φ b + (cid:88) i,j,r I ( d bij,lk,r = M b ) (cid:17) P ( q lk |− ) ∝ D ir (cid:16) φ σ + (cid:88) i I ( d σi,lk = 1) , . . . , φ σ + (cid:88) i I ( d σi,lk = M σ ) (cid:17) Full conditional distribution of µ m,lk For each m = 2 , . . . , M b , the posteriordistributions of the component-specific means are obtained as P ( µ m,lk |− ) ∝ exp (cid:16) − µ m,lk s (cid:17) (cid:89) { i,j,r : d bij,lk,r = m } exp (cid:16) − ( b ij,lk,r − µ m,lk ) γ m,lk (cid:17) ∝ N ( µ, s )where µ = s (cid:88) { i,j,r : d bij,lk,r = m } b ij,lk,r γ m,lk s = (cid:16) s + (cid:88) { i,j,r : d bij,lk,r = m } γ m,lk (cid:17) − Full conditional distribution of γ m,lk For m = 1, since µ ,lk = 0, the posteriordistributions of the component-specific variances are obtained as P ( γ ,lk |− ) ∝ ( γ ,lk ) a − exp (cid:16) − γ ,lk b (cid:17) (cid:89) { i,j,r : d bij,lk,r =1 } ( γ ,lk ) − / exp (cid:16) − b ij,lk,r γ ,lk (cid:17) ∝ GiG (cid:16) a − (cid:80) i,j,r I ( d bij,lk,r = 1)2 , b , (cid:88) { i,j,r : d bij,lk,r =1 } b ij,lk,r (cid:17) Instead, for each m = 2 , . . . , M b , the posterior distributions are P ( γ m,lk |− ) ∝ ( γ m,lk ) − a − exp (cid:16) − b γ m,lk (cid:17) (cid:89) { i,j,r : d bij,lk,r = m } ( γ m,lk ) − / exp (cid:16) − ( b ij,lk,r − µ m,lk ) γ m,lk (cid:17) IG (cid:16) a + (cid:80) i,j,r I ( d bij,lk,r = m )2 , b + (cid:88) { i,j,r : d bij,lk,r = m } ( b ij,lk,r − µ m,lk ) (cid:17) ∝ GiG (cid:16) − a − (cid:80) i,j,r I ( d bij,lk,r = m )2 , , b + (cid:88) { i,j,r : d bij,lk,r = m } ( b ij,lk,r − µ m,lk ) (cid:17) Full conditional distribution of α m,lk For m = 1 , . . . , M σ , the posterior distributions ofthe component-specific shapes are obtained as P ( α m,lk |− ) ∝ α a − m,lk exp (cid:16) − α m,lk b (cid:17)(cid:16) β α m,lk m,lk Γ( α m,lk ) (cid:17) { i : d σi,lk = m } (cid:16) (cid:89) { i : d σi,lk = m } σ i,lk (cid:17) − α m,lk we sample from this distribution using an adaptive RWMH with proposal. Full conditional distribution of β m,lk For m = 1 , . . . , M σ , the posterior distributions ofthe component-specific scales are given by P ( β m,lk |− ) ∝ β a − m,lk exp (cid:16) − β m,lk b (cid:17) (cid:89) { i : d σi,lk = m } β α m,lk m,lk exp (cid:16) − β m,lk σ i,lk (cid:17) ∝ G a (cid:16) a + α m,lk · { i : d σi,lk = m } , b + (cid:88) { i : d σi,lk = m } σ i,lk (cid:17) B Additional results r e t u r n -5-4-3-2-1012345 -5-4-3-2-1012345 -5-4-3-2-1012345 -5-4-3-2-1012345 r i s k p r e m i u m -5-4-3-2-1012345 -5-4-3-2-1012345 -5-4-3-2-1012345 -5-4-3-2-1012345 l e v e r ag e -5-4-3-2-1012345 -5-4-3-2-1012345 -5-4-3-2-1012345 -5-4-3-2-1012345 v o l a t ili t y -5-4-3-2-1012345 -5-4-3-2-1012345 -5-4-3-2-1012345 -5-4-3-2-1012345 Figure 11: Impact of risk factors (column) on financial linkages of the multilayer network.In each plot, the coefficient in position ( i, j ) refers to the impact of the risk factor in columnon the edge from institution j to institution ii