  Transfer entropy, an information-theoretic measure of time-directed
information transfer between joint processes, has steadily gained popularity in
the analysis of complex stochastic dynamics in diverse fields, including the
neurosciences, ecology, climatology and econometrics. We show that for a broad
class of predictive models, the log-likelihood ratio test statistic for the
null hypothesis of zero transfer entropy is a consistent estimator for the
transfer entropy itself. For finite Markov chains, furthermore, no explicit
model is required. In the general case, an asymptotic chi-squared distribution
is established for the transfer entropy estimator. The result generalises the
equivalence in the Gaussian case of transfer entropy and Granger causality, a
statistical notion of causal influence based on prediction via vector
autoregression, and establishes a fundamental connection between directed
information transfer and causality in the Wiener-Granger sense.
