Clustering of time series via non-parametric tail dependence estimation

We present a procedure for clustering time series according to their tail dependence behaviour as measured via a suitable copula-based tail coefficient, estimated in a non-parametric way. Simulation results about the proposed methodology together with an application to financial data are presented showing the usefulness of the proposed approach

Copula-based measures of tail dependence with applications

With the advent of globalization and the recent financial turmoil, the interest for the analysis of dependencies between financial time series has significantly increased. Risk measures such as value-at-risk are heavily affected by the joint extreme comovements of associated risk factors. This thesis suggests some copula-based statistical tools which can be useful in order to have more insights into the nature of the association between random variables in the tail of their distributions. Preliminarily, an overview of important definitions and properties in copula theory is given, and some known measures of tail dependence based on the notion of tail dependence coefficients and rank correlations are introduced. A first proposal consists of a graphical tool based on the so-called tail concentration function, in order to distinguish different families of copulas in a 2D configuration. This can be used as a copula selection tool in practical fitting problems, when one wants to choose one or more copulas to model the dependence structure in the data, highlighting the information contained in the tail. The thesis mainly deals with financial time series applications, where copula functions and the related concepts of tail copula and tail dependence coefficients are used to characterize the dependence structure of asset returns. Classical cluster analysis tools are revisited by introducing suitable copula-based tail dependence measures, which are exploited in the identification of similarities or dissimilarities between the variables of interest and, in particular, between financial time series. Such an approach is designed to investigate the joint behaviour of pairs of time series when they are taking on extremely low values. Either the asymptotic and the finite behaviour are assessed. The proposed methodology is based on a suitable copula-based time series model(GARCH-copula model), in order to model the marginal behaviour of each time series separately from the dependence pattern. Moreover, non-parametric estimation procedures are adopted for describing the pairwise dependencies, thus avoiding any model assumption. Simulation studies are conducted in order to check the performances of the proposed procedures and applications to financial data are presented showing their practical implementation. The information coming from the output of the introduced clustering techniques can be exploited for automatic portfolio selection procedures in order to hedge the risk of a portfolio, by taking into account the occurrence of joint losses. A two-stage portfolio diversification strategy is proposed and empirical analysis are provided. Results show how the suggested approach to the clustering of financial time series can be used by an investor to have more insights into the relationships among different assets in crisis periods. Moreover, the application to portfolio selection framework suggests a cautious usage of standard procedures that may not work when the markets are expected to experience periods of high volatility