  In this paper, we quantify the statistical coherence between financial time
series by means of the Renyi entropy. With the help of Campbell's coding
theorem we show that the Renyi entropy selectively emphasizes only certain
sectors of the underlying empirical distribution while strongly suppressing
others. This accentuation is controlled with Renyi's parameter q. To tackle the
issue of the information flow between time series we formulate the concept of
Renyi's transfer entropy as a measure of information that is transferred only
between certain parts of underlying distributions. This is particularly
pertinent in financial time series where the knowledge of marginal events such
as spikes or sudden jumps is of a crucial importance. We apply the Renyian
information flow to stock market time series from 11 world stock indices as
sampled at a daily rate in the time period 02.01.1990 - 31.12.2009.
Corresponding heat maps and net information flows are represented graphically.
A detailed discussion of the transfer entropy between the DAX and S&P500
indices based on minute tick data gathered in the period from 02.04.2008 to
11.09.2009 is also provided. Our analysis shows that the bivariate information
flow between world markets is strongly asymmetric with a distinct information
surplus flowing from the Asia-Pacific region to both European and US markets.
An important yet less dramatic excess of information also flows from Europe to
the US. This is particularly clearly seen from a careful analysis of Renyi
information flow between the DAX and S&P500 indices.
