  Uncovering causal interdependencies from observational data is one of the
great challenges of nonlinear time series analysis. In this paper, we discuss
this topic with the help of information-theoretic concept known as R\'enyi
information measure. In particular, we tackle the directional information flow
between bivariate time series in terms of R\'enyi transfer entropy. We show
that by choosing R\'enyi $\alpha$ parameter appropriately we can control
information that is transferred only between selected parts of underlying
distributions. This, in turn, provides particularly potent tool for quantifying
causal interdependencies in time series, where the knowledge of "black swan"
events such as spikes or sudden jumps are of a key importance. In this
connection, we first prove that for Gaussian variables, Granger causality and
R\'enyi transfer entropy are entirely equivalent. Moreover, we also partially
extend this results to heavy-tailed $\alpha$-Gaussian variables. These results
allow to establish connection between autoregressive and R\'enyi entropy based
information-theoretic approaches to data-driven causal inference. To aid our
intuition we employ Leonenko et al. entropy estimator and analyze R\'enyi
information flow between bivariate time series generated from two
unidirectionally coupled R\"ossler systems. Notably, we find that R\'enyi
transfer entropy not only allowed us to detect a threshold of synchronization
but it also provided a non-trivial insight into the structure of a transient
regime that exists between region of chaotic correlations and synchronization
threshold. In addition, from R\'enyi transfer entropy we could reliably infer
the direction of coupling - and hence causality, only for coupling strengths
smaller that the onset value of transient regime, i.e. when two R\"ossler
systems were coupled, but have not yet entered a synchronization.
