Measures of the direction and strength of the interdependence among time series from multivariate systems are evaluated based on their statistical significance and discrimination ability. The best-known measures estimating direct causal effects, both linear and nonlinear, are considered, i.e., conditional Granger causality index (CGCI), partial Granger causality index (PGCI), partial directed coherence (PDC), partial transfer entropy (PTE), partial symbolic transfer entropy (PSTE) and partial mutual information on mixed embedding (PMIME). The performance of the multivariate coupling measures is assessed on stochastic and chaotic simulated uncoupled and coupled dynamical systems for different settings of embedding dimension and time series length. The CGCI, PGCI and PDC seem to outperform the other causality measures in the case of the linearly coupled systems, while the PGCI is the most effective one when latent and exogenous variables are present. The PMIME outweighs all others in the case of nonlinear simulation systems.

In the aim to explore the complex relationships between S&P500, VIX and volume we introduce a Granger causality test using the nonlinear statistic of Asymmetric Partial Transfer Entropy (APTE). Through a simulation exercise, it arises that the APTE offers precise information on the nature of the connectivity. Our empirical findings concretize the information flow that links volume, S&P500 and VIX, and merge the leverage effect and the asymmetric stock return-volume relationship into a unified framework of analysis. More specifically, when we condition on the tails, the detected causal channel provides empirical validation of the noise trading contribution to large swings in financial markets, because of the increase of trading volume and the subsequent worsening ability of market prices to adjust to new information.