Partial Symbolic Transfer Entropy

In this paper, we introduce the partial symbolic transfer entropy (PSTE), an extension of the symbolic transfer entropy that accounts only for the direct causal effects among the components of a multivariate system. It is an information theoretic measure, and as such does not suffer from model mis-specification bias. The PSTE is defined on the ranks of vectors that are formed from the reconstructed vectors, instead of the original time series values. The statistical significance of PSTE is assessed by randomization test making use of surrogate time series. The PSTE is evaluated on multivariate time series of different types of coupled and uncoupled systems and compared with conditional Granger causality index (CGCI). It is shown that the PSTE is not affected by the existence of outliers, it is directly applicable to time series that are non-stationary in mean and in variance, and it is also not affected by data filtering. As a real application, the causal effects among three economic indexes are investigated. Computations of PSTE and CGCI on both the initial returns and the VAR filtered returns, and only of PSTE on the original indexes, showed consistency of the PSTE in estimating the causal effect.