Multi-feature evaluation of financial contagion.

Financial contagion refers to the spread of market turmoils, for example from one country or index to another country or another index. It is standardly assessed by modelling the evolution of the correlation matrix, for example of returns, usually after removing univariate dynamics with the GARCH model. However, significant events like crises visible in one financial market are usually reflected in other financial markets/countries simultaneously in several dimensions, i.e., in general, entire distributions of returns in other markets are affected. These distributions are determined/described by their expected value, variance, skewness, kurtosis and other statistics that determine the shape of the distribution function of returns, which can be based on higher (mixed) moments. These descriptive statistics are not constant over time, and, moreover, they can interreact within the given market and among the markets over time. In this article we propose, and use for the daily values of five indexes (CAC40, DAX30, DJIA, FTSE250 and WIG20) over the time period 2006-2017, a new, simple and computationally inexpensive methodology to automatically extend contagion evaluation from the evolution of the correlation matrix to the evolution of multiple higher mixed moments as well. Specifically, the joint distribution of normalized variables for each pair of indexes is modeled as a polynomial with time evolving coefficients estimated using an exponential moving average. As we can obtain any arbitrary number of evolving mixed moments this way, its dimensionality reduction using PCA (principal component analysis) is also discussed, obtaining a lower number of dominating and relatively independent features, which can each be interpreted through a polynomial that describes the corresponding perturbation of joint distribution. We obtain features that describe the interrelations among stock markets in several dimensions and that provide information about the current stage of crisis and the strength of the contagion process.