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This article investigates the causality structure of financial time series.We concentrate on three main approaches to measuring causality: linear Grangercausality, kernel generalisations of Granger causality (based on ridgeregression and the Hilbert--Schmidt norm of the cross-covariance operator) andtransfer entropy, examining each method and comparing their theoreticalproperties, with special attention given to the ability to capture nonlinearcausality. We also present the theoretical benefits of applying non-symmetricalmeasures rather than symmetrical measures of dependence. We apply the measuresto a range of simulated and real data. The simulated data sets were generatedwith linear and several types of nonlinear dependence, using bivariate, as wellas multivariate settings. An application to real-world financial datahighlights the practical difficulties, as well as the potential of the methods.We use two real data sets: (1) U.S. inflation and one-month Libor; (2) S$\&$Pdata and exchange rates for the following currencies: AUDJPY, CADJPY, NZDJPY,AUDCHF, CADCHF, NZDCHF. Overall, we reach the conclusion that no single methodcan be recognised as the best in all circumstances, and each of the methods hasits domain of best applicability. We also highlight areas for improvement andfuture research.

Measures of Causality in Complex Datasets with application to financial data