In this paper, we investigate the interconnections among and within the Energy, Agricultural, and Metal commodities, operating in a risk management framework with a twofold goal. First, we estimate the Value-at-Risk (VaR) employing GARCH and Markov-switching GARCH models with different error term distributions. The use of such models allows us to take into account well-known stylized facts shown in the time series of commodities as well as possible regime changes in their conditional variance dynamics. We rely on backtesting procedures to select the best model for each commodity. Second, we estimate the sparse Gaussian Graphical model of commodities exploiting the Graphical LASSO (GLASSO) methodology to detect the most relevant conditional dependence structure among and within the sectors. A novel feature of our framework is that GLASSO estimation is achieved exploring the precision matrix of the multivariate Gaussian distribution obtained using a Gaussian copula with marginals given by the residuals of the aforementioned selected models. We apply our approach to the sample of twenty-four series of commodity futures prices over the years 2005–2022. We find that Soybean Oil, Cotton, and Coffee represent the major sources of propagation of financial distress in commodity markets while Gold, Natural Gas UK, and Heating Oil are depicted as safe-haven commodities. The impact of Covid-19 is reflected in increased heterogeneity, as captured by the strongest relationships between commodities belonging to the same commodity sector and by weakened inter-sectorial connections. This finding suggests that connectedness does not always increase in response to crisis events.

In this paper we develop a linear expectile hidden Markov model for the analysis of cryptocurrency time series in a risk management framework. The methodology proposed allows to focus on extreme returns and describe their temporal evolution by introducing in the model time-dependent coefficients evolving according to a latent discrete homogeneous Markov chain. As it is often used in the expectile literature, estimation of the model parameters is based on the asymmetric normal distribution. Maximum likelihood estimates are obtained via an Expectation-Maximization algorithm using efficient M-step update formulas for all parameters. We evaluate the introduced method with both artificial data under several experimental settings and real data investigating the relationship between daily Bitcoin returns and major world market indices.

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