Modeling systemic risk with Markov Switching Graphical SUR models

Abstract

We propose a Markov Switching Graphical Seemingly Unrelated Regression (MS-GSUR) model to investigate time-varying systemic risk based on a range of multi-factor asset pricing models. Methodologically, we develop a Markov Chain Monte Carlo (MCMC) scheme in which latent states are identified on the basis of a novel weighted eigenvector centrality measure. An empirical application to the constituents of the S&P100 index shows that cross-firm connectivity significantly increased over the period 1999–2003 and during the financial crisis in 2008–2009. Finally, we provide evidence that firm-level centrality does not correlate with market values and it is instead positively linked to realized financial losses.

Introduction

The financial crisis of 2008–2009 has shown that liquidity and valuation shocks may quickly propagate across the economic system and affect financial institutions operating in different markets, with different size and business structure, thus causing widespread losses and domino effects. Understanding the dynamics of cross-asset and cross-equity linkages is therefore of key importance to both systemic risk management purposes and to deal with contagion waves in times of crisis. Systemic risk shocks are conventionally referred to in the network literature as abrupt increases in the density of cross-firm connectivity (see, e.g. Billio et al., 2012, and references therein). Modeling firms’ connectedness through network analysis has been recently supported by a series of papers that have shown its in- and out-of-sample superior performance over traditional, correlation-based approaches.¹

We extend the existing literature in a number of ways. First, we propose a system-wide inferential scheme based on a Markov-switching graphical model that allows us to simultaneously consider all the possible linkages among firms through constraints on the regime-specific conditional dependence structure. Second, we propose an identification scheme for different regimes of cross-firm connectivity based on a novel weighted eigenvector centrality measure, which is related to both the number and the strength of the connections between firms. Third, we provide an asset pricing application based on otherwise standard multi-factor pricing models in which the exposures of the assets to the risk factors (their regression betas) are allowed to change according to the regimes in cross-firm connectivity. This allows us to develop a unified framework where systematic and systemic risks are not mutually exclusive, in the sense that firm-specific exposures to sources of systematic risk may directly depend on the level of aggregate network connectivity.² Finally, we provide a Metropolis-within-Gibbs sampling scheme which permits to jointly draw both the parameters of the factor pricing model, the latent states and the underlying regime-specific graphs.

Methodologically, we build upon the Gaussian Graphical model for multi-variate systems proposed by Whittaker (1990), Dawid and Lauritzen (1993), Lauritzen (1996), Carvalho et al. (2007b), Wang and West (2009), Wang (2010), Rodriguez et al. (2011), Wang et al. (2011) and Ahelegbey et al. (2016). In particular, Wang et al. (2011) developed a dynamic matrix-variate graphical model which allows to capture conditional dependencies under time-invariant graphs. We generalize and extend their framework by introducing Markov-switching dynamics in the graph structure within a Seemingly Unrelated Regression (SUR) model. More specifically, we propose a new Markov Switching Graphical SUR (MS-GSUR) which makes it possible to identify different regimes of network connectivity. The regime-switching identification problem is solved by exploiting the graph-theoretic properties of the state-specific conditional dependence structures of the error terms in the model. Thus, we provide a new weighted eigenvector centrality measure, which accounts not only for the number of adjacent nodes, but also for the weights of the edges and for the number of indirect connections between nodes, i.e. the number of walks between nodes. We formally show that our measure can be related to both (Bonacich, 1972) and communicability (see, e.g., Estrada and Hatano, 2008 and Estrada and Hatano, 2009), as well as to other measures used in the analysis of topological features of complex networks.

Our empirical application focuses on the cross-section of daily excess returns of the constituents of the S&P100 index over the period 1996–2014. The emphasis on stock returns is motivated by a widespread desire by policy makers and regulators to incorporate the most current information for the purpose of systemic risk measurement: stock prices of largely traded stocks reflect information more rapidly than other non-traded measures such as accounting variables. Informative cross-firm connectivity is estimated on the basis of the residual covariance structure of stock returns, conditioning on a set of tradable risk factors used in some of the most popular linear asset pricing models, namely the Capital Asset Pricing Model (CAPM), the three-factor model of Fama and French (1993), and Merton’s (1973) intertemporal CAPM (I-CAPM). The results are robust across model specifications.

Our main findings reveal that the dynamics of systemic risk can be captured by two regimes, in which a state of high connectedness characterizes the period 1999–2003 (marked by the passing of the Gramm–Leach–Bliley act, the inflating and bursting of the dot.com bubble, and the ensuing financial scandals) and subsequently, the great financial crisis of 2008–2009. We show that a few financial institutions turned out to heavily outweigh other firms in the network during these periods and that shocks to the Financial sector turned out to be the most systemically important. Finally, both a cross-sectional regression and rank-correlation analysis show that market capitalization does not significantly explain the relevance of a given firm within the network. However, firms which are more relevant within the network are more likely to suffer significant losses during periods of high systemic risk.