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Uncovering causal interdependencies from observational data is one of thegreat challenges of nonlinear time series analysis. In this paper, we discussthis topic with the help of information-theoretic concept known as R\'enyiinformation measure. In particular, we tackle the directional information flowbetween bivariate time series in terms of R\'enyi transfer entropy. We showthat by choosing R\'enyi $\alpha$ parameter appropriately we can controlinformation that is transferred only between selected parts of underlyingdistributions. This, in turn, provides particularly potent tool for quantifyingcausal interdependencies in time series, where the knowledge of "black swan"events such as spikes or sudden jumps are of a key importance. In thisconnection, we first prove that for Gaussian variables, Granger causality andR\'enyi transfer entropy are entirely equivalent. Moreover, we also partiallyextend this results to heavy-tailed $\alpha$-Gaussian variables. These resultsallow to establish connection between autoregressive and R\'enyi entropy basedinformation-theoretic approaches to data-driven causal inference. To aid ourintuition we employ Leonenko et al. entropy estimator and analyze R\'enyiinformation flow between bivariate time series generated from twounidirectionally coupled R\"ossler systems. Notably, we find that R\'enyitransfer entropy not only allowed us to detect a threshold of synchronizationbut it also provided a non-trivial insight into the structure of a transientregime that exists between region of chaotic correlations and synchronizationthreshold. In addition, from R\'enyi transfer entropy we could reliably inferthe direction of coupling - and hence causality, only for coupling strengthssmaller that the onset value of transient regime, i.e. when two R\"osslersystems were coupled, but have not yet entered a synchronization.

Causal inference in time series in terms of R\'enyi transfer entropy