eGC - Matlab Tool for computing the extended Granger Causality

Granger causality (GC) is a very popular tool for assessing the presence of directional interactions between two time series of a multivariate data set [1] . The present toolbox implements the estimation of the GC measure, both in its traditional formulation [1] and in an extended formulation (the so-called extended GC (eGC)) which we recently proposed to account for the possibly confounding effects of zero-lag correlations among the observed time series [2].

The traditional formulation of the GC is based on a strictly causal vector autoregressive (VAR) model which describes only the time-lagged effects between the time series, but does not account for the instantaneous (i.e. not-lagged) effects. As a consequence of this incomplete description, the presence of significant instantaneous effects may lead the standard VAR models to provide wrong interpretations about causality [3]. The measure of extended GC proposed in our recent study [2] overcomes the limitations of GC by properly accounting for zero-lag effects in the linear regression schemes implemented by the VAR model. In [2] we also present a procedure for the practical estimation of eGC which addresses the problem of assigning instantaneous effects to the regression models. This procedure employs a two-step approach based on first estimating the existence of zero-lag correlations between two processes, and then orienting them along one of the two possible causal directions using pairwise measures of non-Gaussianity [4].



[1] Bressler S L and Seth A K 2011 Wiener-Granger Causality: A well established methodology. Neuroimage 58 323-9.

[2]  L Schiatti, G Nollo, G Rossato, L Faes, 'Extended Granger causality: a new tool to identify the structure of physiological networks', Phys. Meas. 2015; 36:827-843.

[3] Hyvarinen A, Zhang K, Shimizu S and Hoyer P O 2010 Estimation of a Structural Vector Autoregression Model Using Non-Gaussianity. J. Machine Learning Res. 11 1709-31.

[4] Hyvarinen A and Smith S M 2013 Pairwise Likelihood Ratios for Estimation of Non-Gaussian Structural Equation Models. J. Machine Learning Res. 14 111-52.


Zip file with all scripts and functions:

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The toolbox contains functions for computing the GC (gcMVAR.m; auxiliary function: LinReg.m, LinReg_Ftest.m, SetLag.m) and the eGC (egcMVAR.m; this function finds the directions of instantaneous effects exploiting the pairwise measures of non-Gaussianity proposed by Hyvarinen and Smith [4]: functions pwling.m, mentappr.m).
For testing the tool, we provide the script test_simulation.m that implements the simulation described in [2]; the scripts makes use of the functions CElin_analytic.m and CElin_analytic_0.m (to find exact theoretical values of GC and eGC for linear systmes with known coefficients), diag_coeff_rev.m (to move from strictly causal to extended VAR representation), and InstModelfilter.m and MVARfilter.m (to generate realizations of the simulation).