Using transfer entropy to measure information flows between financial markets

TLDR: In this paper, transfer entropy is used to quantify information flows between financial markets and propose a suitable bootstrap procedure for statistical inference, which allows to determine, measure and test for information transfer without being restricted to linear dynamics.

Abstract: We use transfer entropy to quantify information flows between financial markets and propose a suitable bootstrap procedure for statistical inference. Transfer entropy is a model-free measure designed as the Kullback-Leibler distance of transition probabilities. Our approach allows to determine, measure and test for information transfer without being restricted to linear dynamics. In our empirical application, we examine the importance of the credit default swap market relative to the corporate bond market for the pricing of credit risk. We also analyze the dynamic relation between market risk and credit risk proxied by the VIX and the iTraxx Europe, respectively. We conduct the analyses for pre-crisis, crisis and post-crisis periods.

Figures

Figure 5: VIX and iTraxx.

Table 1: Reference entities

Figure 1: Illustration of small sample bias.

Figure 4: Kernel Density plots of iTraxx and credit VIX first differences. The figure shows kernel density plots of the iTraxx and VIX first differences (solid line) together with the normal density (dashed line).

Table 4: Effective transfer entropy for VIX and iTraxx

Table 2: Effective transfer entropy estimates

Full text

SFB 649 Discussion Paper 2012-051
Using transfer entropy
to measure information
flows between financial
markets
Thomas Dimpfl *
Franziska J. Peter *
* Universität Tübingen, Germany
This research was supported by the Deutsche
Forschungsgemeinschaft through the SFB 649 "Economic Risk".
http://sfb649.wiwi.hu-berlin.de
ISSN 1860-5664
SFB 649, Humboldt-Universität zu Berlin
Spandauer Straße 1, D-10178 Berlin
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Using transfer entropy to measure information
flows between financial markets
Thomas Dimpfl
∗
Franziska J. Peter
∗
August 23, 2012
Abstract
We use transfer entropy to quantify information flows between financial mar-
kets and propose a suitable bootstrap procedure for statistical inference. Trans-
fer entropy is a model-free measure designed as the Kullback-Leibler distance
of transition probabilities. Our approach allows to determine, measure and test
for information transfer without being restricted to linear dynamics. In our
empirical application, we examine the importance of the credit default swap
market relative to the corporate bond market for the pricing of credit risk. We
also analyze the dynamic relation between market risk and credit risk proxied
by the VIX and the iTraxx Europe, respectively. We conduct the analyses for
pre-crisis, crisis and post-crisis periods.
Keywords: entropy; information flow; non-linear dynamics; price discovery;
credit risk; CDS.
JEL classifications: C14, G15
∗
University of T
¨
ubingen
Corresponding author: Franziska Julia Peter, University of T
¨
ubingen, Department of Statistics,
Econometrics and Empirical Economics, Mohlstraße 36, 72074 T
¨
ubingen, Germany, Phone: +49
7071 29 78165. Fax: +49 7071 29 5546. Email: franziska-julia.peter@uni-tuebingen.de.
We thank participants of the annual meeting of the Society of Nonlinear Dynamics and Economet-
rics, Washington 2011, of the interdisciplinary workshop on Econometric and Statistical Modelling
of Multivariate Time Series, UCL Louvain-la-Neuve, 2011, and of the 2012 Midwest Financial As-
sociation meeting, New Orleans, and two anonymous referees for helpful comments. This research
project is funded by the German Research Foundation (DFG) through grant GR 2288, data are pro-
vided through the DFG SFB 649 “Economic Risk”.
1

1 Introduction
Detecting and measuring interactions between different time series is the subject of
research in various areas. In finance the informational link between different mar-
kets is of particular interest. Yet, there is only a small range of methods to empiri-
cally examine these linkages. The predominant concept is that of Granger causality
(see Granger, 1969) which is widely applied to detect lead-lag relationships be-
tween time series. However, the insights that can be gained from this method are
limited to the mere existence of information flows rather than their quantification.
An actual measure for information transfer between financial markets exists only
for a particular setting of empirical applications: if prices in different markets re-
fer to the same underlying asset, price discovery measures such as the Hasbrouck
(1995) information shares can be used to determine the informational dominant
market. This approach requires a cointegration relationship between the different
time series and only provides a sensible interpretation of the results if cointegration
is implied by theory and supported by the data. Furthermore, Granger causality
and price discovery measures are based on a Vector Autoregressive (VAR) or Vec-
tor Error Correction Model (VECM) framework which imposes rather restrictive
assumptions concerning the underlying (linear) dynamics.
As an alternative we propose to apply the concept of transfer entropy to
measure information flows between different financial time series. Transfer en-
tropy is a model-free measure which is designed as the Kullback-Leibler distance
of transition probabilities. Under weak assumptions this approach allows to quan-
tify information transfer without being restricted to linear dynamics. It is therefore
an appealing tool if the assumptions required by the standard models are not sup-
ported by the data. There exist only few studies that apply transfer entropy within
the context of financial markets. Kwon and Yang (2008a), for example, analyze
the information flow between the S&P 500, the Dow Jones index and selected indi-
vidual companies on a daily basis. Baek, Jung, Kwon, and Moon (2005) examine
the information transfer between groups of NYSE listed stocks to determine market
sensitive and market leading companies, and Kwon and Yang (2008b) investigate
2

the strength and direction of information transfer between various stock indices us-
ing transfer entropy. The measurement of interactions between the Indian stock and
commodity market is the subject of a study by Reddy and Sebastin (2009).
In all these finance studies, statistical inference has not been accounted for.
We address this issue and propose to bootstrap the underlying Markov process (see
Horowitz, 2003) in order to derive the distribution of the estimates and assess the
statistical significance of the estimated information flows. Furthermore, we address
the issue of correcting for small sample bias of the transfer entropy estimator in a
novel way: we suggest to use the bootstrapped distribution under the null hypothesis
of no information flow to correct for finite sample bias rather than using shuffled
data which has been the standard procedure in the literature on information theory
so far (see, inter alia, Papana, Kugiumtzis, and Larsson, 2001).
This paper includes two empirical applications of the transfer entropy method-
ology to measure information flows between financial markets. First, we examine
the information flows between the credit default swap (CDS) and the bond market,
analyzing data on 27 iTraxx Europe companies. Both markets reflect the price of
credit risk for the same reference entity. Therefore, as outlined in Blanco, Brennan,
and Marsh (2005), assuming a cointegration relationship between the time series
is plausible. Our dataset consists of a sample of CDS and bond market data of 27
European companies from January 2004 to December 2011. Due to market imper-
fections cointegration is not always supported for each and every company (as is
also the case in the study of Blanco et al., 2005) so that a VECM cannot be reason-
ably applied. Using transfer entropy, we find in general significant bi-directional
information flow between the CDS and the bond market. Our results point towards
a slight dominance of the CDS market, in particular during the financial crisis, and
thus are in line with previous findings in the literature like Blanco et al. (2005) and
Coudert and Gex (2010).
In our second application we analyze factors that might influence the CDS
market by examining the link between market risk and credit risk. Thereby, we
follow Figuerola-Ferrett and Paraskevopoulos (2009) and consider the dynamic re-
lation between iTraxx and VIX. Again, we draw on a long sample of iTraxx and
3

VIX data ranging from March 2005 to November 2011. We find that in general the
transfer entropy estimates for the flow of information from the VIX to the iTraxx
are statistically significant and exceed the information flow from the iTraxx to the
VIX.
The remainder of the paper is organized as follows. Section 2 introduces the
concept of transfer entropy. Diagnostics and inference are discussed in Section 3.
Section 4 contains the empirical applications and Section 5 concludes.
2 Measuring information flows by transfer entropy
The concept of transfer entropy is best understood within the context of information
theory. In the era of early telecommunications, when communication was based on
Morse code, Hartley (1928) introduced a measure for information relying on the
logarithm of the number of all possible symbol sequences that can occur. The
general aim was to optimally encode messages such that they can be transmitted
more quickly. For that purpose it was necessary to quantify the information that
can be derived from a specific sequence of transmitted symbols.
Consider the following example: when flipping a fair coin, there are two
equally likely outcomes, head or tail. According to Hartley (1928) the information
(denoted by H) that can be gained from flipping a coin once is given by H = log(2).
If the base of the logarithm is 2, H = log
2
(2) = 1 and the measurement unit will be
bits. Consequently, n flips of the coin yield n bits of information (H = log(2
n
) =
n×log
2
(2) = n) and we would need n binary digits to specify the resulting sequence
(such as 1 for heads and 0 for tails).
In the case of symbols that are not equally likely, but occur with different
probabilities p
j
, the amount of information gained from a specific symbol j is given
by log(1/p
j
). The average amount (per symbol) of information one can get from
such a sequence is defined as H =
∑
n
j=1
p
j
log

1
p
j

, where n is the number of dis-
tinct symbols. This leads to the general formula of Shannon (1948): assume that J
is a discrete variable with probability distribution p( j), where j labels the different
4

Frequently asked questions

Q1. What are the contributions in "Using transfer entropy to measure information flows between financial markets" ?

The authors use transfer entropy to quantify information flows between financial markets and propose a suitable bootstrap procedure for statistical inference. Their approach allows to determine, measure and test for information transfer without being restricted to linear dynamics. In their empirical application, the authors examine the importance of the credit default swap market relative to the corporate bond market for the pricing of credit risk. The authors also analyze the dynamic relation between market risk and credit risk proxied by the VIX and the iTraxx Europe, respectively. The authors conduct the analyses for pre-crisis, crisis and post-crisis periods. 

Q2. What is the subject of a study by Reddy and Sebastin (2009)?

The measurement of interactions between the Indian stock and commodity market is the subject of a study by Reddy and Sebastin (2009). 

Q3. How do the authors derive the distribution of the transfer entropy estimates?

In order to derive the distribution of the transfer entropy estimates them-selves, the authors need to alter the proposed bootstrap procedure, since the authors have to preserve the dependencies between both time series. 

Q4. How do the authors calculate the spreads of bonds?

In order to calculate the corresponding bond spreads, the authors have to construct a time series of bond yields that matches the constant 5-years maturity of the CDS time series. 

Q5. Why is the intermediate bin kept large?

Since the time series are likely to contain a considerable amount of noise due to the illiquidity of the bond market and the OTC trading of both assets, the intermediate bin is kept rather large. 

Q6. What is the standard method to measure information transmission between financial markets?

The standard methodology to measure information transmission betweenfinancial markets is the information share technique of Hasbrouck (1995) which relies on the existence of a cointegration relationship. 

Q7. What is the way to correct for the small sample bias of the transfer entropy?

the authors address the issue of correcting for small sample bias of the transfer entropy estimator in a novel way: the authors suggest to use the bootstrapped distribution under the null hypothesis of no information flow to correct for finite sample bias rather than using shuffled data which has been the standard procedure in the literature on information theory so far (see, inter alia, Papana, Kugiumtzis, and Larsson, 2001). 

Q8. Why do empirical studies use a VECM framework?

these studies also note that due to market imperfections cointegration is not always supported by the data and a VECM might not suit the observed time series. 

Q9. What is the effect of the iTraxx on the transfer entropy?

In three cases, the estimate of the bias became larger than the estimate of transfer entropy, resulting in negative values of the effective transfer entropy (irrespective of using shuffling or bootstrapping for bias correction). 

Q10. What is the main advantage of transfer entropy?

Even if cointegration is supported, but the information share bounds are very large, transfer entropy may be used as a further back-check to determine the informationally dominant market. 

Q11. How do the authors calculate the yield of a bond with five years to maturity?

The authors then linearly interpolate these bond yields obtaining a yield curve for different maturities which is in turn used to predict the yield of an artificial bond with five years to maturity. 

Q12. What is the probability of observing The authorat time t +1?

Let The authorbe a stationary Markov process of order k, then it holds for the probability to observe The authorat time t +1 in state i conditional on the k previous observations that p(it+1|it , ..., it−k+1) = p(it+1|it , ..., it−k). 

Q13. What is the average amount of information that one can get from a sequence?

The average amount (per symbol) of information one can get fromsuch a sequence is defined as H = ∑nj=1 p jlog ( 1 p j ) , where n is the number of distinct symbols.