Measuring interconnectedness between financial institutions with Bayesian time-varying vector autoregressions

Transcription

1 Measuring interconnectedness between financial institutions with Bayesian time-varying vector autoregressions Marco Valerio Geraci, and Jean-Yves Gnabo ECARES, Université libre de Bruxelles CeReFiM, Université de Namur August 19, 216 Abstract We propose a market-based framework that exploits time-varying parameter vector autoregressions to estimate the dynamic network of financial spillovers. We apply it to all financials listed in the Standard & Poor s 5 index and uncover interesting features at the individual, sectorial and system wide level. This includes a gradual decrease in interconnectedness after the crisis, not observable using the classical rolling window approach, and more stable interconnectedness-based rankings that can be used for monitoring purposes. Keywords: financial interconnectedness, time-varying parameter, systemic risk JEL Classification: G1, G18, C32, C51, C58 This is a revised version of our ECARES WP No We acknowledge support of the Communauté française de Belgique through the Projet d Actions de Recherche Concertées 13/17-55 grant. We are grateful to the Editor Paul Malatesta and an anonymous referee, Matteo Barigozzi, Christiane Baumeister, Sophie Béreau, Oscar Bernal, Monica Billio, Alain de Combrugghe, Ruben Hipp, Michele Lenza, Matteo Luciani, Nicolas Scholtes, David Veredas, Sebastian Werner, and Kamil Yılmaz for helpful comments and suggestions. We also thank participants at the Conference on Financial Risk & Network Theory in Cambridge, the seminar participants at the Economics Department of the University of Fordham, ECARES (Université libre de Bruxelles), GESS (University of Mannheim), and naxys (Université de Namur) for their feedback and valuable discussion. All remaining errors are our own.

2 I. Introduction In this paper we develop a market-based statistical framework that uses Bayesian estimation of time-varying parameter vector autoregressions (TVP-VAR) to model the dynamic nature of connections between financial institutions. The framework allows connections to evolve gradually through time as opposed to the classical approach, which favours sudden, often unjustified, changes in interconnectedness. Paired with graph theory, our framework allows us to reconstruct a continuously evolving network of directed spillover effects. We use our framework to study the evolution of interconnectedness of the US financial sector over the past two decades. Market-based measures of interconnectedness apply statistical measures of association (e.g. correlation, Granger causality, tail dependence) to asset prices, in order to map and analyse the network of spillover effects between financial institutions (see for example Billio, Getmansky, Lo, and Pelizzon, 212, Barigozzi and Hallin, 215, Diebold and Yılmaz, 29, 214, Dungey, Luciani, and Veredas, 213, Hautsch, Schaumburg, and Schienle, 214, 215). These standard statistical measures presuppose that the inferred relationships are time-invariant over the sample used for the estimation. To retrieve a dynamic measure of interconnectedness, the usual approach has been to divide the original sample period into multiple subsamples and calculate these statistical measures over rolling windows of data. We argue that this is potentially unsuitable if the system studied is deemed to be timevarying. By relying on short subsamples, rolling windows lower the power of inference and induce dimensionality problems. Moreover, the rolling window approach is known to be susceptible to outliers because, in small subsamples, these have a larger impact on estimates (Zivot and Wang, 26). On the other hand, choosing longer windows will lead to estimates that are less reactive to change, biasing results towards time-invariant connections (Clark and McCracken, 29). Thus, the rolling window approach will always involve a trade-off between the precision and reactivity of estimates of interconnectedness. Our framework intuitively parallels the literature that uses temporal dependencies for measuring interconnectedness (Billio et al., 212, Barigozzi and Brownlees, 216, Barigozzi and Hallin, 215, Diebold and Yılmaz, 29, 214). We construct a TVP-VAR model (Cogley 1

3 and Sargent, 25, Primiceri, 25) with several additional features that can accommodate many of the properties of stock returns (e.g. heteroskedasticity, skewness, and excess kurtosis). The time-varying cross-coefficients of the TVP-VAR capture, in every time period, the strength of directed connections between financial institutions. We determine connections by evaluating the posterior distribution of the parameters using Bayesian inference. The framework is based on the whole sample of observed data and does not require establishing any window size. Our framework is extremely flexible and can also be applied to study interconnectedness based on contemporaneous dependencies, as is done by Acharya, Pedersen, Philippon, and Richardson (21), Adrian and Brunnermeier (216). This is possible thanks to a time-varying scale matrix, which we include in the model and which effectively captures the contemporaneous connections between returns. Thus, the framework contributes to the literature of both categories of interconnectedness measures: those based on temporal dependencies and those based on contemporaneous dependencies. 1 We apply our framework to analyse the US network of financial spillovers. To do this, we use monthly stock price data for all financial institutions listed from 199 to 214 on the Standard & Poor s 5 index, including firms that have since gone defunct. We proceed in two steps. First, we estimate the time-varying network of spillovers at an aggregate level between the four sectors comprising our data: banks, broker-dealers, insurance companies and real estate companies. In a second step, we estimate the network at a disaggregated level between the 2 most systemically important financial institutions, according to the Financial Stability Board and related studies (Diebold and Yılmaz, 214). Our analysis yields four main results. First, our emprical investigation reveals the limitations of the rolling window approach and the gains of adopting the time-varying parameter framework in the network context. Measures of connectivity and centrality computed under the rolling window approach are more volatile because of the approach s high sensitivity to extreme observations. This gives the impression that interconnectedness rose after the crisis, whereas the time-varying pa- 1 Our analysis of the US financial sector is based on temporal dependencies, since these can be readily interpreted as directed spillover effects. 2

4 rameter framework reveals a decrease, concurring with the credit freeze that occurred. Second, we examine interconnectedness between the four financial sectors included in our study and find that banks and insurance companies were the largest contributors to financial spillovers. The real estate sector, composed primarily of real estate investment trusts, was the most influenced by these spillovers. This is due primarily to the fact that the real estate companies in our sample are mostly real estate investment trusts, which were less leveraged than banks and less exposed to non-agency mortgage-backed securities. Third, we show that interconnectedness-based rankings, computed using the rolling window approach, are extremely volatile and unlikely to be useful for policy decisions. On the other hand, the time-varying parameter framework produces more stable rankings that could be appropriate for time-dependent monitoring. We find that our framework yields more stable rankings than other market-based measures (e.g. the marginal expected short fall (Acharya, Engle, and Richardson, 212, Brownlees and Engle, 216) and the market beta) but has more reactive rankings than measures based on book value data (e.g. leverage). Fourth, at the individual institution level, the empirical application identifies some interesting cases. American International Group, Goldman Sachs, and Merrill Lynch were found to be among the largest propagators of financial spillovers, highlighting their potential widespread influence. By contrast, Bear Stearns did not play a major role in the propagation of spillovers but rather was very receptive of incoming spillovers. This could explain why its collapse did not represent a systemic event. These results were confirmed in an out-of-sample exercise using only data up to the end of 27. In the next section, we will briefly review the recent literature on financial interconnectedness measures and how this relates to our modelling choices. The remainder of the paper is then structured as follows. In Section III, we go through the estimation framework, partly reviewing the Granger causality approach. In Section IV we discuss the empirical application and we draw our conclusions in Section V. 3

5 II. Literature Review The greatest difficulty faced by research into financial interconnectedness and stability has been associated with data availability. Little data regarding bank cross-exposures is available, primarily because of banks confidentiality concerns. For this reason, part of the literature has focused on simulating the data using network models of contagion (e.g. Allen and Gale (2), Gai and Kapadia (21), Nier, Yang, Yorulmazer, and Alentorn (27)) and agent-based models (e.g. Ha laj, Kochańska, and Kok (215), Giansante, Chiarella, Sordi, and Vercelli (212), Georg (213), Montagna and Kok (213)). Even if data on cross-exposures were available, it might not be sufficient to fully capture interconnectedness between banks. This is because banks can also be connected by off-balance sheet items, such as derivatives, or indirectly by common portfolio holdings (Langfield and Soramäki (214)). These instruments can potentially have contagion effects far more severe than defaults on bilateral credit exposures (Roukney, Bersini, Pirotte, Caldarelli, and Battiston (213)). Moreover, concentrating solely on interbank exposures confines the analysis to depositary institutions, while neglecting important players in the financial system, such as brokers, insurers and real estate companies who are not part of this market. In order to overcome such data limitations, a growing area of research with pioneering contributions from Acharya et al. (212, 21), Adrian and Brunnermeier (216), Billio et al. (212) and Diebold and Yılmaz (214), has turned to market data to recover measures of interconnectedness. 2 These market-based measures are based on the notion of forwardlooking markets and the observed phenomenon of comovement in stock prices during times of crisis. The idea is that stock prices include all available information up to the most recent point in time. Then, financial distress travelling from one financial institution to another should be reflected by comovement in stock prices. Market-based measures of interconnectedness can be sub-divided into two categories: those that make use of contemporaneous dependencies and those that exploit temporal dependencies. 2 Benoit, Colletaz, Hurlin, and Pérignon (213), Giglio, Kell, and Pruitt (213), Nucera, Schwaab, Koopman, and Lucas (216) analyse different market-based measures of interconnectedness and systemic risk. 4

6 The former category exploits simultaneous spillovers to gauge connections between financial institutions. Such measures include the Conditional value-at-risk (CoVaR) of Adrian and Brunnermeier (216), the Marginal expected shortfall (MES) of Acharya et al. (21), and the SRisk of Acharya et al. (212) and Brownlees and Engle (216), as well as related measures developed by Balla, Ergen, and Migueis (214), Dungey et al. (213), Hautsch et al. (215), and Peltonen, Piloiu, and Sarlin (215). Although conditional measures, as in the case of CoVaR, MES, and SRisk, allow to infer some sense of directedness of spillovers, in general measures that use contemporaneous dependencies cannot distinguish the influencing financial institution from the influenced. On the other hand, measures that exploit temporal dependencies use the notions of precedence and predictability to infer directed connections between financial institutions. These studies have implicitly assumed that spillovers are impounded in prices with some temporal order and thus can be captured using Granger causality (Billio et al., 212) and vector autoregressions (Barigozzi and Brownlees, 216, Barigozzi and Hallin, 215, Diebold and Yılmaz, 29, 214, Demirer, Diebold, Liu, and Yılmaz, 215). A major issue with the aforementioned studies is that the underlying connections that obtain among financial institutions plausibly vary over time, whereas the statistical measures adopted are designed for time-invariant connections. These studies examined the evolution of interconnectedness through time by applying the given statistical measures to rolling windows of observations. As mentioned in the introduction, this results in a trade-off whereby larger windows lead to greater precision but less reactivity of estimates. We build on studies pertaining to the second category by developing a flexible framework accounts for time-varying connections. Our framework does not require rolling windows and exploits the whole data sample. At the same time, we also contribute to the first category of studies by modelling contemporaneous dependencies using a time-varying scale matrix. We also contribute to the literature of Bayesian measures of interconnectedness. In this literature, Bernardi, Gayraud, and Petrella (215) develop a Bayesian approach for measuring CoVaR. They focus on quantile effects, whereas we parallel Billio et al. (212) and focus on average effects. Nonetheless, our framework could be extended with slightly different distributional assumptions (Yu and Moyeed, 21) to model quantile effects as in 5

7 the VAR for VaR framework of White, Kim, and Manganelli (215). Ciccarelli and Rebucci (27) also adopt Bayesian TVP-VARs to measure contagion in the South American FX market. They assume asset returns follow a multivariate t- distribution but with constant scale. Our approach is more flexible as we use a multivariate t-distribution with time-varying scale, which allows for heteroskedasticity and time-varying contemporaneous correlations between asset returns. Moreover, we include the possibility for returns to exhibit skewness by allowing for a so-called leverage effect between shocks to asset returns and shocks to volatilities. A related study by Adams, Füss, and Gropp (214) examines interconnectedness between four U.S. financial sectors (commercial banks, investment banks, hedge funds and insurance companies) by proposing a state-dependent sensitivity Value-at-Risk (SDSVaR) model. The SDSVaR model is comparable to our proposed framework as it allows for three states of connections between sectors, according to whether financial markets are in a volatile, normal or tranquil condition. However, our TVP-VAR is more flexible as it does not restrict the number of states and allows connections to vary freely through time. Finally, it is worth mentioning a study put forward by Blasques, Koopman, Lucas, and Schaumburg (214) who estimate the time-varying dependence between European sovereign credit spreads with a lagged spacial model. In their case, the (spacial) network used is timeinvariant and exogenously given by the BIS database. On the other hand, we propose a statistical framework for retrieving the underlying time-varying network from market data. III. Empirical methodology The classical approach for measuring interconnectedness by temporal dependencies uses Granger causality testing to infer connections between financial institutions (see e.g. Barigozzi and Brownlees, 216, Billio et al., 212). According to this definition of interconnectedness, financial institution i has a directed spillover to financial institutions j, denoted (i j), if the stock returns of i, denoted r i, cause (in the sense of Granger (1969)) the stock returns of j, denoted r j. As shown by Geweke (1982, 1984), the actual test is carried out by estimating a vector 6

8 autoregression (VAR) of the returns and testing the joint significance of the cross-coefficients linking r i to r j within the VAR. Then, the (absolute) value of the cross-coefficients can be interpreted as the strength of the directed relationship. Effectively, the classical approach relies on an in-sample test, which is conditional on all observations collected from the start of the sample up to period T. If the direction of causality were to change within this sample period, the VAR estimation and test inference might be affected. One possibility to account for such changes would be by performing the Granger causality test on a rolling window basis (as, for example, is done by Billio et al. (212), or, with different techniques, by Diebold and Yılmaz (214) and Hautsch et al. (214)). However, as highlighted in the introduction, this solution involves a trade-off between precision and reactivity of estimates when selecting the appropriate window size (Clark and McCracken, 29). Our framework does not require rolling windows and, as shown in the Monte Carlo study in Appendix D, it performs well, compared to the rolling window approach, both in terms of estimation (measured by the mean squared error of the estimated parameters) and in terms of inference (measured by Receiver-Operator Curves and precision-recall curves). A. The model We parallel the classical approach by allowing time-varying effects with the following time-varying vector autoregression (TVP-VAR): (1) R t = c t + B t R t 1 + u t X tθ t + u t, where R t [r 1t,..., r Nt ], the vector of stock returns. We stack the time-varying intercepts c t and matrix of time-varying coefficients B t in θ t so that equation (1) can be interpreted as the measurement equation of a state space model. 3 Now, we define a time-varying spillover at period t from i to j, denoted (i t j), if the 3 The framework can be extended to account for higher order lags. We used the BIC criterion to select the most appropriate lag specification. 7

9 (j, i) element of B t, denoted b (j i) t (j i) of b t represents the strength of the spillover from i to j., is different than zero. In particular, the (absolute) value We let the vector of disturbances u t have the following form: u t = λ t Σ t ε t, where ν/λ t χ 2 ν and ε t is a vector of standard normal errors. This means that u t is t-distributed with scale matrix Σ t Σ t Ξ t and degrees of freedom ν. The t-distribution allows for the returns in our VAR to exhibit fat tails for a wide range of kurtosis. Moreover, the time-varying scale matrix Ξ t can capture heteroskedastic volatility and time-varying correlations between errors, which can be interpreted as the contemporaneous spillovers occurring between the financial institutions in the network. For the present study, we concentrate on B t, which represents temporal spillovers and have the advantage of being interpretable as a directed network. Nonetheless, the framework is highly flexible and lends itself to the study of both temporal and contemporaneous spillovers. As is done in the macroeconomic literature (e.g. Cogley and Sargent (25), D Agostino, Gambetti, and Giannone (213), Primiceri (25)), we let parameters evolve according to a driftless random walk. The state equation of the model is thus, (2) θ t+1 = θ t + v t+1, v t N (, Q t ), where we assume that ε t and v s are independent at all t and s. 4 We allow the time-varying parameters to have heteroskedastic variance Q t. This allows for additional flexibility for B t and effectively means that the unconditional distribution of B t is fat-tailed (Jacquier, Polson, and Rossi, 1994). We assume that Q t is a diagonal matrix, with diagonal elements collected in q t and evolving as a geometric random walk, ln q t+1 = ln q t + ω t+1. Regarding the time-varying scale matrix of the measurement equation Ξ t, we assume it 4 This assumption is taken solely for reasons of convenience and could be easily relaxed. 8

10 takes the usual triangular form, Ξ t = A 1 t H t (A 1 t ), where H t is diagonal matrix containing the stochastic volatilities and A t is lower triangular with ones across its diagonal and the contemporaneous interactions as lower diagonal elements. Let h t denote the vector of diagonal elements of H t, and α t the lower diagonal elements of A t stacked by rows. Then we assume the following laws of motion for the time-varying parameters: ln h t = ln h t 1 + η t α t = α t 1 + τ t The vector of errors of the model [ε t, η t, ω t, τ t ] is jointly normal with mean zero and variance-covariance matrix V, defined as I Ω Ω Z V = η Z ω S where, ρ 1 σ 1 σ 1. ρ Ω = 2 σ σ , Z η = , and ρ n σ N σ N σ ω,1. σ Z ω = ω, σ ω,n (1+N) The variance-covariance matrix V and its submatrix Ω allow the error terms of the 9

11 measurement and volatility equations, namely ε t and η t, to be contemporaneously correlated row-by-row. In the case of stock returns this correlation is often negative and is known as the leverage effect (Nakajima and Omori, 212). This assumption offers increased flexibility as it allows for the possibility of having skewness in the errors of our VAR equation in (1). Finally, following Primiceri (25) and Baumeister and Benati (213), we adopt a blockdiagonal structure for S, which implies that the non-zero and non-unity elements of A t belonging to different rows evolve independently. The model is estimated by simulating the posterior distribution of the parameters as in the Bayesian tradition. An overview of the prior specification and the sampling algorithm are given, respectively, in Appendix A and Appendix B. B. Inferring the dynamic network In order to determine our dynamic network of spillover effects, we evaluate, for every pair of institutions i j, the following time-varying hypothesis: (j i) (3) H,t : b t =. That is, we draw a directed time-varying spillover (i t j) if there is sufficient evidence, given our data and prior beliefs, that hypothesis H,t is not true. Then, if a connection is found to exist, we weigh it according to the absolute value of the given cross coefficient b (j i) t. In this way, we recover a completely dynamic weighted network of temporal directed spillovers. We evaluate the hypothesis using Bayes factor, as in the Bayesian tradition. We find that this is an informationally more efficient approach than is recursive Granger causality testing by rolling windows. Bayes Factor gives the odds in favour of the restricted model, given by the null hypothesis given in equation (3), against the unrestricted model, given by equations (1) and (2). We follow Koop, Leon-Gonzalez, and Strachan (21) who show a convenient way of calculating Bayes factor by the Savage-Dickey density ratio (SDDR). We give further details on how this is done in Appendix C. 1

12 The estimated Bayes factor K t gives us the odds that the null hypothesis of no connection at time t holds. The implied probability is then just K t /(1 + K t ). This can be subjected to a threshold to make the decision whether to accept or reject the null hypothesis. Unlike classical frequentist testing, Bayes factor weighs evidence in favour of the restricted and unrestricted model equally. Effectively, the threshold is a filtering mechanism and a higher threshold leads to a denser network with more links. Here, we show results using a threshold of 8%. Similar results are obtained using different thresholds and are available upon request. IV. Interconnectedness of US financial institutions In order to analyse the time-varying interconnectedness of US financial institutions we conducted two separate empirical investigations using our framework. First, working at an aggregate level, we estimate the time-varying network of spillovers between four financial sectors: banks, broker-dealers, insurers, and real estate companies. Analogously to Adams et al. (214), we do this by computing indices from the stock price returns of 154 financial institutions listed on the Standard & Poor s 5 index (S&P 5). In the second investigation, we move to the disaggregated level and analyse interconnectedness between a set of 2 financial institutions that were identified as systemically important by the Financial Stability Board (FSB) and by other studies (e.g. by Diebold and Yılmaz (214)). The two empirical investigations can provide a clear view on the spread of spillovers and the dynamics of interconnectedness during the crisis, which can be particularly important for risk management and supervisory purposes. Several studies have performed similar exercises by adopting a rolling window approach (e.g. Billio et al. (212), Diebold and Yılmaz (214), Hautsch et al. (214)). On the other hand, we apply TVP-VARs to retrieve two completely dynamic networks (at the sectorial and firm level), exploiting the whole sample of data. 11

13 A. Data We selected financial institutions with Standard Industrial Classification (SIC) codes from 6 to 6799 that were components of the S&P 5 from January 1993 to December 214. For these companies we collected the monthly cum-dividend stock price from Thomson Reuters Eikon for the same time period. The monthly frequency makes it possible to reduce the amount of noise in the data. Data at the intra-daily or even daily frequency reveals a higher number of linkages, because stocks are more susceptible to market shocks that lead to a higher degree of co-movement. Initially the sample contained 182 firms but was reduced to 154 after constraining our analysis to stocks with at least 36 monthly observations. Table 1 shows the financial institutions included in the final sample. The sample can be subdivided into four sectors based on the SIC code of the companies. These were identified as banks (SIC codes 6 to 6199), broker/dealers (SIC codes 62 to 6299), insurers (SIC codes 63 to 6499) and real estate companies (SIC codes 65 and 6799). The final sample included 71 banks, 21 brokers/dealers, 4 insurers and 23 real estate companies. We define monthly stock returns for company i at month t as ( ) pi t + d i t r i t = log, p i t 1 where p i t is the stock price of company i at the end of month t and d i t are dividends paid that month. Finally, we netted the risk free rate from the stock returns, as is often done in asset pricing. For this we used the monthly return on the three-month US Treasury Bills. For our first investigation, we constructed four equally weighted indices from the stock returns of banks, brokers, insurers and real estate companies in our sample. We then inferred the dynamic network of spillovers effects between sectors using a conditional approach. That is, we inferred the dynamic network of spillovers effects between sectors by estimating a four-variable TVP-VAR with one lag. We then determined the existence of a connection, at a given period, between two sectors (conditional on all other sectors) by using Bayes factor, as explained in Section III. If a connection was found to exist, we captured 12

14 its strength using the absolute value of the given cross coefficient in the TVP-VAR at that time period. For our second investigation, we inferred the dynamic network of spillovers effects between 2 systemically important financial institutions. We considered all banks and insurers from the FSB s list of systemically important financial institutions and systemically important insurers. We also considered the financial institutions that were acquired or that went bankrupt during the crisis (as done by Diebold and Yılmaz, 214). Thus, the subsample included: American Express, American International Group (AIG), Bank of America, Bank of New York Mellon, Bear Stearns, Citigroup, Fannie Mae, Freddie Mac, Goldman Sachs, JP Morgan, Lehman Brothers, Merrill Lynch, MetLife, Morgan Stanley, PNC Group, Prudential, State Street, US Bancorp, Wachovia, and Wells Fargo. Since our sample is unbalanced, with several stock time series substantially shorter than the complete sample period, we adopted a pairwise approach. For each pair of financial institutions, we estimated a bivariate TVP-VAR (with one lag), taking into account the longest common time period of available data between any given pair. Again, we determined the existence of a connection at a given period by Bayes factor and the strength of the connection by the absolute value of the appropriate time-varying cross coefficient in the bivariate TVP-VAR. We compare the two dynamic networks obtained with the TVP-VAR framework to two corresponding networks obtained with conditional and pairwise Granger causality testing over rolling windows of data. For this, we used rolling windows of 36 months and 24 months with a critical value of 1% for the Granger causality tests. B. Financial network density and sectorial interconnectedness Here we analyse the network obtained from the first investigation, in which we estimate conditional time-varying interconnectedness between banks, brokers, insurers and real estate companies. To do so we make use of two widely applied network measures: density and degree centrality. The density is the average strength of a connection in the network at a given time period. It can be computed as 13

15 (4) Density t = 1 N(N 1) N t i=1 (j i) (i t j) b t, j i (j i) with i, j {Banks, Brokers, Insurers, Real Estate} and i j, where b t is the cross coefficient connecting i to j, in period t, in the TVP-VAR, and where, in this case, N = 4. time. It gives us a measure of how integrated the financial system is at a given moment in For comparison we also computed the density of the network estimated using the rolling windows of 36 months (as was used in Billio et al. (212)) and 24 months. The evolution of the density is given in Figure 1 together with some significant events for the US economy. The figure shows the density estimated using the time-varying parameter framework in bold solid. The density estimated using rolling windows of 36 months is depicted by the light dashed line with circle markers, whereas that estimated using rolling windows of 24 months is depicted with triangle markers. [ FIGURE 1 ABOUT HERE] The time-varying parameter framework density identifies two main events in terms of the interconnectedness of spillover effects across US financial sectors: the collapse of the Long-Term Capital Management (LTCM) fund in 1998 and the financial crisis of The fall and subsequent bailout of LTCM was a major event in terms of contagion because the fund was highly leveraged and was contractually connected with an extensive number of counterparties (McDonough, 1998). Concerns for spillovers effects bringing down financial markets lead the Federal Reserve Bank of New York to coordinate a consortium of banks to bailout LTCM. The financial crisis represented an even more important event in terms of interconnectedness. Our network density measure, based on the time-varying parameter framework shows that interconnectedness gradually grew by more than twofold between the beginning of 25 and October 28. After the financial crisis, the density gradually decreased to below pre-crisis levels. This could reflect two phenomena. First, after the default of Lehman Brothers, in October 28, there was an immediate market freeze generated by 14

16 the high uncertainty among market agents. This caused a drastic decrease in interconnectedness. However, the policies introduced to counter this sentiment, such as the Troubled Asset Relief Program and Dodd-Frank act could have slowed down the perceived drop in interconnectedness. Dungey et al. (213) found a similar decrease using a realized volatility-based measure of systemic risk. Looking at the density computed from the rolling window estimates, we notice that it is substantially more volatile than the time-varying parameter counterpart, exhibiting sudden short-lived peaks. The volatility of the network density found using the time-varying parameter framework was.6, whereas it was.13 using the rolling window approach with 36 months window size. Using a smaller window size of 24 months leads to more reactive density measure with an even higher volatility of.18. There are several discrepancies in the two density measures estimated using the rolling window approach. The density computed using the 24-month rolling window is far noisier because it relies on fewer observations and therefore is more susceptible to outliers. particular, after the financial crisis, the measure appears to jump strongly, whereas the same movement is not observable using 36 month rolling windows. We believe that this peak is generated by the crisis observations, relating to October 28, exiting the rolling windows and creating an artificial jump in interconnectedness. In order to study the evolution of interconnectedness between sectors we analysed the time-varying cross-coefficients of the four variable TVP-VAR. Figure 2 gives the posterior (j i) density mean of the off-diagonal elements of B t (from equation 1), i.e. the value of b t i j. The (absolute) value of the time-varying cross-coefficient represents the strength of the directed spillover effect from a given sector (indicated by the rows of the figure) to another sector (indicated by the columns of the figure). Moreover, the sign of the cross-coefficient can help us understand whether the spillover effect was positive or negative. As can be noticed from Figure 2, the coefficients evolve substantially through time, evidencing the usefulness of the time-varying parameter framework. In for [ FIGURE 2 ABOUT HERE ] Figure 2 shows the LTCM 1998 crisis and the Lehman default crisis with vertical dashed 15

17 lines. Prior to the two crisis events, some sectors appear to have had a stronger increase in interconnectedness compared to others. In particular, banks and insurance companies seem to have had the largest increase in outgoing connections during this period. Between these two sectors, it seems that banks were influencing insurers to a greater extent than insurers were influencing banks. The fourth row of Figure 2 shows that real estate companies were not a major contributor of spillovers during the financial crisis. Rather, the fourth column of Figure 2 shows that real estate companies were mostly receiving spillovers from other sectors, primarily banks and insurers. The role of the real estate sector was crucial during the financial crisis and, at a first glance, our results do not seem to corroborate with this. Two explanations can be put forward for our results regarding the real estate sector. First, it must be said that 21 of the 23 real estate companies in our sample are real estate investment trusts (REITs), which primarily own income-producing real estate and in some cases finance real estate. Only in specific cases do REITs own mortgages and generally, if they do, these will be through agency mortgage backed securities (MBSs), which are guaranteed by agencies of the US government. On the other hand, banks were directly hit by troubles in the real estate sector because of direct exposure to mortgages and non-agency or private label MBSs. [ TABLE 2 ABOUT HERE ] Second, banks were heavily leveraged prior and during the financial crisis, whereas REITs were far less leveraged. 5 Table 2 shows the average leverage ratios for the financial institutions in our sample. During the financial crisis, the publicly traded our sample of banks were more than six times more leveraged than our sample of real estate institutions. This could explain the low outwards interconnectedness found for the real estate sector using the time-varying parameter framework. Effectively, their low leverage and their less direct exposure to non-agency MBSs meant that these real estate companies were not strong risk propagators. 5 We are grateful to an anonymous referee for this insight regarding REITs. 16

18 C. Financial institution centrality In the second investigation, we worked at the disaggregated level and considered a smaller subsample of 2 financial institutions, which were the systemically most important financial institutions according to the FSB and other similar studies (e.g. Diebold and Yılmaz, 214). We constructed the dynamic network of spillover effects between the 2 institutions by applying our framework to their stock returns in a pairwise fashion. That is, we estimated many bivariate TVP-VARs for all possible pairs of the 2 financial institutions and inferred directed connections between pairs using Bayes Factor. This is similar to the pairwise approach adopted by Billio et al. (212) using classical Granger causality testing over rolling windows. In order assess the importance of each financial institution within the dynamic network, we analysed their degree centrality, which measures the weighted sum of the connections to and from a given financial institution. Since our network is directed, we can measure both the in-degree centrality as well as the out-degree centrality, respectively given by: (5) (6) In-Degree i,t = Out-Degree i,t = 1 (N t 1) 1 (N t 1) (i j) (j t i) b t, j i (j i) (i t j) b t, j i where i, j {1,..., 2} and N t is the number of financial institutions present at time t. Notice that, as for the density measure in equation (4), we chose to weight connections by the absolute value of the underlying cross-coefficient in the bivariate VAR. In-degree centrality measures the average strength of incoming connections to financial institution i. Effectively, it is an indicator of the extent to which other financial institutions influence the stock price of financial institution i. Therefore, it is a measure of vulnerability to stock spillovers. Out-degree centrality measures the strength of outgoing links from financial institution i. Thus, a financial institution with high out-degree is heavily influencing many of its neighbours, making it a propagator of spillovers. As suggested by Hautsch et al. (214), such 17

19 financial institutions should be monitored closely because they are highly interconnected. Figures 3 and Figure 4 depict, respectively, the in- and out-degree for the financial institutions studied. The measures computed using our time-varying parameter framework are shown in bold solid; whereas the light dashed lines depict those computed from the rolling window approach with window size of 36 months. [ FIGURE 3 AND 4 ABOUT HERE] By visually inspecting each chart, we can immediately notice that the two methodologies examined, the time-varying parameter framework and rolling window approach, produce very different pictures. Figure 3 shows that the financial institution to attain the highest level of in-degree was Bear Stearns, peaking at a value of 2.4, reached in March 28 (the peak is not shown in the figure for visibility reasons). This means that it was receiving financial spillovers from a growing number of firms, effectively increasing its fragility. Bear Stearns was the first large bank to collapse as a result of the subprime mortgage crisis of 27. The fact that it had many incoming connections and few outgoing connections (shown in Figure 4) may explain why the collapse of Bear Stearns was, at the time, not warranted as a systemic event even though it marked the prelude to the financial crisis. Another interesting aspect that is worth mentioning regarding the interconnectedness of Bear Stearns is that the data involved in calculating its in-degree did not contain the financial meltdown of September 28. This is because the sample time-series of Bears Stearns stock price was quoted only until March 28. Therefore, the high in-degree cannot be attributable to any noise caused by the many events that occurred during September-October 28. Other institutions with a high level of in-degree during the financial crisis were AIG, Fannie Mae, Freddie Mac, Lehman Brothers and Wachovia. It is interesting to notice that the in-degree Fannie Mae and Freddie Mac peaked at very different periods according to the framework used to compute the measure, i.e. the time-varying parameter framework or rolling window approach. Figure 4 shows the out-degree for the financial institutions examined. Here levels are more homogeneous between financial institutions compared to the in-degree levels showed in 18

20 Figure 3. However, we can notice that among the institutions with the highest out-degree were American International Group, Goldman Sachs and Merrill Lynch. This concurs with the findings of Hautsch et al. (214) and Dungey et al. (213) who also identified these banks as very central. We can notice Wachovia s out-degree increased through time until the governmentinduced forced sale to Wells Fargo, completed in December 28. Again, the out-degree calculated using the classical rolling window approach gives a measure that is more volatile and sporadic. It jumps suddenly in September 28, while the corresponding measure calculated with our time-varying parameters framework shows an increase that began long before this event. For Lehman Brothers, the level of out-degree found using our time-varying parameter framework is higher than that found using the rolling window approach. Nonetheless, it was not among the highest across the financial institutions analysed. The result is consistent with the policy decision taken by the government and Federal Reserve to not bailout Lehman Brothers during the crisis because it was considered as not systemic (Committee on Oversight and Government Reform (28)). Whether Lehman Brothers had a role in propagating financial spillovers is still a debated question and unfortunately, we do not have the counterfactual event to answer this question. Nonetheless, several considerations can be made. According to an asset and liability exposure analysis conducted by Scott (212), Lehman s bankruptcy was not particularly destabilizing for its direct counterparties. However, without doubt Lehman s bankruptcy triggered a sentiment of fear and uncertainty in markets. Fear was primarily due to the realization that government rescue of large financial institutions was no longer guaranteed, whereas uncertainty surrounded the extent of losses incurred by other institutions due to Lehman s default. This led to a freeze in the short-term funding market and effectively a liquidity crisis that affected financial institutions indiscriminately of their connections with Lehman Brothers. On the other hand, Lehman s reliance on short-term rather than long-term funding made it vulnerable to external shocks (Scott (212)). This is detectable in the growing in-degree of Lehman, which according to our time-varying framework had began since 26, as can be 19

21 seen in Figure 3. D. Stability of centrality-based rankings Using the approach developed by the Basel Committee on Banking Supervision (BCBS), the FSB ranks financial institutions according to their systemic importance and uses this ranking to determine their additional loss absorbency requirements. 6 In a similar spirit, we ranked the 2 systemically important financial institutions analysed in our second investigation according to their in- and out-degree. These two rankings give only a partial view on systemic risk, one based only on interconnectedness, whereas the rankings drawn by the FSB are also based other determinants of systemic risk such as size and leverage. Nonetheless, these rankings can help us identify, at every point in time, the most exposed institutions to spillovers effects and the most important propagators of spillover effects. As put forward by Daníelsson, James, Valenzuela, and Zer (215) and Dungey et al. (213), the usefulness of rankings for policy makers is severely limited if these are prone to frequent, drastic changes that lead to unmotivated excessive alarm. In an attempt to assess this for the rankings produced by the time-varying framework and the classical rolling window approach, we developed a series of stability indicators. Let Z in i t be the ordinal ranking of institution i at time t in terms of in-degree. Similarly, let Z out i t be the ordinal ranking of institution i at time t in terms of out-degree. We compute the quadratic ranking stability indicator as SI in Q = 1 T 1 T t=2 Nt i=1 (Z in i t Z in i t 1) 2 N t, SI out Q = 1 T 1 Similarly, we construct the absolute stability indicator as SI in A = T N t t=2 i=1 Zi in t Z in N t (T 1) i t 1, SI out A = T t=2 T N t t=2 i=1 Nt i=1 Z out i t (Z out i t Z out N t (T 1) Z out i t 1) 2 N t. i t 1 6 Additional loss absorbency requirements have phased in starting in January 216 with full implementation by January 219 (see Financial Stability Board (211)).. 2

22 The quadratic stability indicators, SI in Q and SIout Q, measures the average change in the ranking between adjacent time periods. The quadratic term used in the calculation causes large deviations in the rankings to have a much higher weight in the stability indicator compared to smaller deviations. On the other hand, for the absolute stability indicator, SI in A and SI out, the weight increases only linearly with the distance between ranking positions. A The above-mentioned stability indicators are calculated to grasp the suitability, from a policy perspective, of rankings based on degree-centrality. As additional indicators of stability, we also computed the average percentage of financial institutions that kept their position in the ranking between adjacent time periods and average changes in the composition of the top 5 and top 1 financial institutions in rankings. In order to have a reference for comparison, and in a similar spirit to Nucera et al. (216), we computed the stability indicators for four additional rankings based on the following measures of systemic risk: SRisk (Acharya et al., 212, Brownlees and Engle, 216), marginal expected shortfall (MES) (Acharya et al., 21), leverage ratio (Engle, Jondeau, and Rockinger, 215), and beta CAPM (Engle, 212). 7 Table 3 shows the stability indicators for rankings based on degree centrality measures found with our time-varying parameter framework and with the rolling window approach (top and bottom panel), as well as for rankings based on the four other systemic risk measures (middle panel). [TABLE 3 ABOUT HERE] The first four columns of Table 3 display the quadratic and absolute stability indicators. Both indicators, for both centrality measures (in- and out-degree), show that the rankings based on our time-varying parameter framework are far more stable than the rankings based on the classical Granger causality approach over rolling windows. In fact, according to both stability indicators, the rolling window approach appears to produce rankings that are more than twice as unstable than those produced by the time-varying parameter framework. The middle panel of Table 3 shows the stability of rankings obtained from other systemic risk measures. Notice that leverage provides the most stable ranking, according to the 7 We obtain the monthly time-series of systemic risk measures from the Vlab website, The data covers periods

23 absolute and quadratic stability indicators. This is because leverage is based on the book value of assets, which is generally observed at low frequencies. On the other hand, MES and Beta, which are based on higher frequency market data, provide the most unstable rankings. Comparing the middle panel to the first panel of Table 3, which shows the stability of the time-varying parameter and rolling window rankings for comparable periods of our sample, we see that the time-varying parameter framework produces rankings with similar stability to SRisk. The latter uses a combination of book value and market data, whereas the former uses exclusively market data. The successive two columns headed % Invariance, denote the average percentage of firms that kept their position in the ranking between adjacent time periods. Between the years 2 and 214, on any given month, on average, about 55% of firms kept the same position held in the previous month in the in- and out-degree rankings calculated with the time-varying parameter framework. Quite to the contrary, only about 37% kept the same position for centrality rankings calculated using the rolling window approach. The same difference was found when using the whole sample, from 1993 to 214 (shown in the bottom panel of Table 3). This confirms the higher stability of rankings found with the time-varying parameter framework. Looking at the stability of rankings produced by other systemic risk measures (shown in the middle panel of Table 3), the time-varying parameter approach yields more stable rankings than MES and market Beta, but appears to be similar, in terms of % Invariance, to SRisk and less stable than Leverage. The columns headed Top 5 and Top 1 show the average changes (in percentage terms) in the composition of, respectively, the top 5 and the top 1 financial institutions in the rankings. For example, for the rankings computed using the rolling window approach, we can expect, on average, one firm to change in the top 5 (19% for in-degree, 2.9% for out-degree, see bottom panel of Table 3) every month. On the other hand, for the rankings computed using the time-varying parameter framework, only between 7.3% and 9.3% of the top 5 changed on average, so substantially less than one firm per month. Similar magnitudes of results were obtained for the period (shown in the top panel of Table 3) and for the stability of the top 1 firms in the rankings. 22

24 All measures used to quantify stability indicate that the rolling window approach, with standard window size, provides less stable rankings compared to the time-varying parameter framework even though both approaches make use of the same data. The reasons for the higher stability offered by the time-varying parameter framework are to be found in the transition law imposed for time-varying connections. By allowing some degree of inertia between successive time periods, large exceptional observations have less influence on the estimated path of connections. On the other hand, with the rolling window approach, these observations have a larger weight in the estimation of connections. The high instability of the rankings found using the rolling window approach would make these rankings difficult to use for policy purposes. It would be hard to justify policy decisions based on a ranking that changes, on average, one component in its top 5 most interconnected institutions every month. On the other hand, the time-varying parameter framework offers a generally stable ranking while allowing some degree of flexibility that can be useful to motivate policy intervention. E. Out-of-sample interconnectedness In order to further illustrate the use of our framework for policy purposes, we conducted an out-of-sample exercise, which consists in estimating interconnectedness using a smaller data sample restricted up to the end of We then evaluate which institutions were the most interconnected, in terms of in-degree and out-degree, prior to the crisis at the end of December 27. Table 4 shows the top 5 financial institutions, in terms of in-degree and out-degree, found using our time-varying framework with the restricted data sample. [ TABLE 4 ABOUT HERE ] Concurrent with previous results, Bear Stearns was in the top 5 ranking in terms of indegree but was only ranked 18 th (not shown) in terms of out-degree. Thus, the result that Bear Stearns was not an important propagator of spillover effects, whereas it was susceptible to receiving spillover effects from other institutions, was also detectable in December 27 using a restricted data sample. 8 We are grateful towards an anonymous referee for suggesting this exercise. 23