In the analysis of multivariate time series, the primary interest is in investigating inter-dependencies of the observed variables, which are often assumed to be representing possibly coupled systems. In the last two decades many different measures have been proposed to quantify directed coupling, or inter-dependence, most prominent being the Granger causality measures. The original concept of Granger causality (X Granger causes Y if the prediction of Y from its own past improves by including also the past of X) has been extended in various ways. To name some, nonlinear models were employed for fitting, the linear models were transferred to the frequency domain, and information measures were used instead of model fitting. Recently, much attention has been given to direct Granger causality measures, i.e. causal effects from one variable to another in the presence of other variables, and further to the formation of complex networks having as nodes the observed variables and links the Granger causality measure values.
I will review some of the most prominent Granger causality measures and stress their conceptual differences. I will discuss in more detail Granger causality measures based on information theory, where our group has made some contribution. An important issue with all measures is the spurious evidence of Granger causality. I will address this problem in terms of significance statistical testing, and assess the sensitivity and specificity of the measures on linear and nonlinear stochastic, as well as chaotic, simulated systems. Finally, applications of Granger causality to brain dynamics and finance will be discussed.