Identifying risk spillovers in financial markets is of great importance for assessing systemic risk and portfolio management. Granger causality in tail (or in risk) tests whether past extreme events of a time series help predicting future extreme events of another time series. The topology and connectedness of networks built with Granger causality in tail can be used to measure systemic risk and to identify risk transmitters. Here we introduce a novel test of Granger causality in tail which adopts the likelihood ratio statistic and is based on the multivariate generalization of a discrete autoregressive process for binary time series describing the sequence of extreme events of the underlying price dynamics. The proposed test has very good size and power in finite samples, especially for large sample size, allows inferring the correct time scale at which the causal interaction takes place, and it is flexible enough for multivariate extension when more than two time series are considered in order to decrease false detections as spurious effect of neglected variables. An extensive simulation study shows the performances of the proposed method with a large variety of data generating processes and it introduces also the comparison with the test of Granger causality in tail by Hong et al. (2009). We report both advantages and drawbacks of the different approaches, pointing out some crucial aspects related to the false detections of Granger causality for tail events. An empirical application to high frequency data of a portfolio of US stocks highlights the merits of our novel approach.

# Author Archives: Piero Mazzarisi

# P. Mazzarisi, S. Zaoli, C. Campajola, F. Lillo (2020). Tail Granger causalities and where to find them: Extreme risk spillovers vs spurious linkages,

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P. Mazzarisi, F.Lillo, S. Marmi (2019). When panic makes you blind: A chaotic route to systemic risk, * Journal of Economic Dynamics and Control *, 100, 176-199

We present an analytical model to study the role of expectation feedbacks and overlapping portfolios on systemic stability of financial systems. Building on Corsi et al. (2016), we model a set of financial institutions having Value-at-Risk capital requirements and investing in a portfolio of risky assets, whose prices evolve stochastically in time and are endogenously driven by the trading decisions of financial institutions. Assuming that they use adaptive expectations of risk, we show that the evolution of the system is described by a slow-fast random dynamical system, which can be studied analytically in some regimes. The model shows how the risk expectations play a central role in determining the systemic stability of the financial system and how wrong risk expectations may create panic-induced reduction or over-optimistic expansion of balance sheets. Specifically, when investors are myopic in estimating the risk, the fixed point equilibrium of the system breaks into leverage cycles and financial variables display a bifurcation cascade eventually leading to chaos. We discuss the role of financial policy and the effects of some market frictions, as the cost of diversification and financial transaction taxes, in determining the stability of the system in the presence of adaptive expectations of risk.

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P. Mazzarisi, P. Barucca, F. Lillo, D. Tantari (2020), * A dynamic network model with persistent links and node-specific latent variables, with an application to the interbank market * , European Journal of Operational Research, 281, 1, 50-65

We propose a dynamic network model where two mechanisms control the probability of a link between two nodes: (i) the existence or absence of this link in the past, and (ii) node-specific latent variables (dynamic fitnesses) describing the propensity of each node to create links. Assuming a Markov dynamics for both mechanisms, we propose an Expectation-Maximization algorithm for model estimation and inference of the latent variables. The estimated parameters and fitnesses can be used to forecast the presence of a link in the future. We apply our methodology to the e-MID interbank network for which the two linkage mechanisms are associated with two different trading behaviors in the process of network formation, namely preferential trading and trading driven by node-specific characteristics. The empirical results allow to recognize preferential lending in the interbank market and indicate how a method that does not account for time-varying network topologies tends to overestimate preferential linkage.

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P.Mazzarisi, P.Barucca, F.Lillo, D.Tantari, *A dynamic network model with persistent links and node-specific latent variables, with an application to the interbank market*

We propose a dynamic network model where two mechanisms control the probability of a link between two nodes: (i) the existence or absence of this link in the past, and (ii) node-specific latent variables (dynamic fitnesses) describing the propensity of each node to create links. Assuming a Markov dynamics for both mechanisms, we propose an Expectation-Maximization algorithm for model estimation and inference of the latent variables. The estimated parameters and fitnesses can be used to forecast the presence of a link in the future. We apply our methodology to the e-MID interbank network for which the two linkage mechanisms are associated with two different trading behaviors in the process of network formation, namely preferential trading and trading driven by node-specific characteristics. The empirical results allow to recognise preferential lending in the interbank market and indicate how a method that does not account for time-varying network topologies tends to overestimate preferential linkage.

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P.Barucca, P.Mazzarisi, F.Lillo, D.Tantari (2017), * Disentangling group and link persistence in dynamic stochastic block models*

We study the inference of a model of dynamic networks in which both communities and

links keep memory of previous network states. By considering maximum likelihood inference from

single snapshot observations of the network, we show that link persistence makes the inference of

communities harder, decreasing the detectability threshold, while community persistence tends to make

it easier. We analytically show that communities inferred from single network snapshot can share a

maximum overlap with the underlying communities of a specific previous instant in time. This leads

to time-lagged inference: the identification of past communities rather than present ones. Finally

we compute the time lag and propose a corrected algorithm, the Lagged Snapshot Dynamic (LSD)

algorithm, for community detection in dynamic networks. We analytically and numerically characterize

the detectability transitions of such algorithm as a function of the memory parameters of the model.