  Granger causality is a fundamental technique for causal inference in time
series data, commonly used in the social and biological sciences. Typical
operationalizations of Granger causality make a strong assumption that every
time point of the effect time series is influenced by a combination of other
time series with a fixed time delay. However, the assumption of the fixed time
delay does not hold in many applications, such as collective behavior,
financial markets, and many natural phenomena. To address this issue, we
develop variable-lag Granger causality, a generalization of Granger causality
that relaxes the assumption of the fixed time delay and allows causes to
influence effects with arbitrary time delays. In addition, we propose a method
for inferring variable-lag Granger causality relations. We demonstrate our
approach on an application for studying coordinated collective behavior and
show that it performs better than several existing methods in both simulated
and real-world datasets. Our approach can be applied in any domain of time
series analysis.
