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Renyi's information transfer between financial time series
Petr Jizba, Hagen Kleinert, Mohammad Shefaat
TL;DR
The paper asks how to quantify statistical coherence and directional information flow in financial time series while retaining information about selected distribution sectors. It uses Rényi entropy and Campbell’s coding theorem to formulate Rényi transfer entropy, then applies it to world stock indices and DAX–S&P500 data. The analysis finds strongly asymmetric flows, especially from Asia-Pacific to Europe and the United States, with a weaker surplus from Europe to the United States.
Problem
The paper addresses the need to quantify information flow between financial time series while distinguishing information carried by different parts of their underlying distributions.
Method
The paper generalizes Shannonian transfer entropy to Rényi transfer entropy and uses Campbell’s coding theorem to characterize Rényi entropy’s distributional emphasis.
Results
The analysis finds strongly asymmetric bivariate information flow, with a distinct surplus from Asia-Pacific markets to European and US markets and a weaker surplus from Europe to the US.
Takeaways & Limitations
Rényi information flow provides a way to examine information transfer in selected distribution sectors, including marginal financial events such as spikes and sudden jumps.
Takeaways & Limitations
Rényi transfer entropy cannot affirmatively establish causation and is complementary to correlation and causality tests.
Abstract
from arXiv · showhide
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.
I. INTRODUCTION
The paper extends transfer entropy from Shannon’s framework to Rényi entropy to quantify information flow in financial time series, emphasizing selected distribution sectors. It motivates this approach through information-theoretic measures and their coding interpretations.
- Motivation and contribution: Transfer entropy quantifies statistical coherence between evolving systems and has applications in multivariate time-series and financial-market analysis.Its computational advantages are useful for analyzing large datasets.
- Motivation and contribution: The paper generalizes Shannonian transfer entropy to a Rényi setting for studying multivariate information flow between stock-index time series.The proposed measure is intended to provide more detailed information about excess or lack of information in parts of the underlying distribution.
- Motivation and contribution: The Rényi-based approach is particularly relevant to financial risk analysis because it can focus on marginal events such as spikes or sudden jumps.The paper connects such events to risk-reducing formulas in portfolio theory.
- Information-theoretic background: Shannon entropy measures the average number of bits required to optimally encode a discrete source, linking information quantification to source coding.The coding interpretation treats entropy as the minimal average number of binary questions needed to determine the source state.
- Information-theoretic background: Joint and conditional entropies extend self-information to dependent variables and support the chain rule for decomposing shared information.The conditional entropy averages the information in X when Y is known.
B. R´enyi’s entropy
R´enyi entropy is a one-parameter information measure whose q-dependent coding interpretation emphasizes different sectors of a distribution. Its conditional, mutual, and related forms support distribution-sensitive comparisons, but equality at a single q does not generally establish independence.
- Coding interpretation: R´enyi entropy assigns an exponentially weighted coding cost, so varying q selectively emphasizes different probability sectors.For q > 1, escort distributions emphasize more probable events; for 0 < q < 1, they accentuate rarer marginal events.
- Conditional entropy: The conditional R´enyi entropy is defined as an escort-distribution-weighted entropy and has boundedness, determinism, and independence-related properties that differ from Shannon entropy.Unlike Shannon’s case, extra knowledge can increase the nonlinear coding price, and some familiar implications do not hold generally.
- Distribution identification: All q > 1, or equivalently all 0 < q < 1, are generally required to uniquely identify the underlying probability distribution.A single order q probes only limited information about the distribution.
- Conditional entropy: Equality between conditional and marginal R´enyi entropies at one q or within a neighborhood does not generally imply independence.Independence requires equality across all q in the relevant range so the conditional and marginal distributions coincide throughout.
- Mutual information: R´enyi mutual information can be negative because nonlinear coding costs may increase after conditioning, especially when probability mass shifts across emphasized sectors.For q > 1, suppression of large probabilities can outweigh enhancement of marginal events; at q = 1, the Shannonian nonnegative behavior is recovered.
- Mutual information: R´enyi mutual information serves as a q-dependent rating of gain or loss in risk from learning one variable about another.This interpretation connects distribution-sensitive information changes with risk considerations in financial settings.
A. Shannonian transfer entropy
Shannonian transfer entropy addresses the lack of directionality in mutual information by measuring information about X supplied by Y’s history after accounting for X’s history. It excludes effects attributable to common historical factors.
- Shannonian mutual information is symmetric, so it cannot by itself identify the direction of information flow between time series.
- Schreiber’s formulation removes information shared through common histories, such as a common external driving force.
- Transfer entropy measures the gain in information about X at the next time step from Y’s history, conditional on X’s history.
- For finite-history processes, transfer entropy is nonnegative and vanishes when Y’s history has no influence on X’s next value.
- The measure is directional because it quantifies dependence of X on Y rather than dependence of Y on X.
B. Effective transfer entropy
Effective transfer entropy addresses finite-sample effects and the difficulty of estimating transfer entropy when relevant histories are long or non-Markovian. It uses surrogate data to remove residual correlations caused by finite data.
- Effective transfer entropy accounts for the finite size of real data sets, unlike standard transfer entropy.
- Finite data hinder the infinite-history limit required for non-Markovian systems, motivating effective transfer entropy to reduce finite-size effects.
- Surrogate sequences preserve mean, variance, autocorrelation, and power spectrum while destroying nonlinear phase relations.
IV. R´ENYIAN TRANSFER ENTROPIES
The paper generalizes transfer entropy from Shannon’s information measure to Rényi’s framework, with q controlling which distributional regions receive emphasis. For q below one, the measure highlights information flow involving tail events.
- Rényi transfer entropy generalizes Shannonian transfer entropy using a q-dependent information measure and recovers the Shannonian form as q →1.
- The Rényi measure subtracts the compound historical effect of X while retaining the effect of Y’s history on X’s future value.
- Rényi transfer entropy can be interpreted as a gain or loss in risk concerning X’s future behavior after incorporating Y’s historical values.
- Unlike Shannonian transfer entropy, Rényi transfer entropy can equal zero without implying independence and can become negative under nonlinear pricing.
- For q ∈(0, 1), Rényi transfer entropy accentuates information flow between the tail parts of distributions, including marginal events in Y influencing marginal events in X.
- For q ∈(0, 1), the analysis confirms that surrogate data are not needed for the effective Rényi transfer entropy.
V. PRESENTATION OF THE ANALYZED DATA
The analysis uses daily data from 11 stock indices to study global coherence and net flows, complemented by minute-level DAX and S&P500 data for a detailed bilateral analysis.
- The first data set contains 11 stock-exchange indices sampled daily and is used to construct heat maps and net information flows.
- The second data set contains 183,308 simultaneous minute-tick observations from the DAX and S&P500 indices.
A. Numerical calculation of transfer entropies
The calculations estimate empirical distributions with amplitude bins, use alphabet length N = 3, and replace block-length-dependent transfer entropy with effective transfer entropy to manage finite-sample effects.
- Relative-frequency estimates construct empirical PDFs by binning the stock-index amplitude axis and dividing each bin count by the time-series length.
- N = 3 fixes the alphabet length because larger alphabets are considered incompatible with the available data in these time series.
- Effective transfer entropy balances the desire for large, stable block lengths against unwanted finite-sample effects.Surrogate data are used for effective Shannon and Rényi transfer entropies, which are calculated and visualized in R.
B. Analyzing the daily data — heat maps vs. net information flows
Daily-data heat maps show strong exchanges involving Asia–Pacific, the US, and Europe, while net flows expose directional asymmetries and distribution-sector differences more clearly.
- The Asia–Pacific region exchanges substantial information with the US and Europe, whereas information flow among European markets is comparatively weaker.
- Within Asia–Pacific markets, information transfer is more imbalanced across distribution wings than central parts, suggesting low liquidity risks.A subordinate wing imbalance also appears between the US and Asia–Pacific markets.
- Net flows show substantially more information moving from Asia–Pacific to the US and Europe than in the reverse direction.The net flow is defined as FY↔X = TY→X − TX→Y, making directional disparities explicit.
- Asia–Pacific-to-Europe net information flow is evenly distributed between central and tail parts of the asset distributions.
- Europe sends an important but weaker surplus of information toward the US than the reverse flow.
C. Minute-price information flows
Minute-level analysis finds long DAX–S&P500 memory and asymmetric Rényi information flow, with Europe-to-US transfer stronger across all q values and slightly tail-biased.
- Rényi transfer entropy is generally nonmonotonic in q because it is the difference between two Rényi entropies with the same q.
- The q-dependent results confirm asymmetric information flow between US and European markets despite the US trading-volume advantage.The observed direction is stronger from European and Asia–Pacific markets toward the US.
- DAX-to-S&P500 information flow is stronger than the reverse flow at minute time scales.
- Europe-to-US flow remains positive for every q value and shows a small bias toward tail parts of the distribution.
VI. CONCLUDING REMARKS
The paper extends transfer entropy into a Rényi framework that isolates selected distribution sectors, then finds asymmetric global-market information flows with stronger Asia–Pacific-to-Western transfers.
- Rényi transfer entropy measures information flow between selected parts of price distributions, with the chosen sectors controlled by Rényi’s parameter q.
- More information flows from Asia–Pacific toward US and European markets than in the reverse direction across both peak and wing distribution parts.The results indicate greater exposure of US and European markets to Asia–Pacific price shocks.
- The US receives more information from Europe than Europe receives from the US, with DAX–S&P500 analysis attributing much of this influx to tail-part transfer.Peak-part transfer is less pronounced.
- Rényi transfer entropy captures high-order correlations and compares distribution sectors, but it cannot affirmatively establish causation.Other correlation tests provide complementary information.
Appendix A
Appendix A provides a glossary of the stock indices used in the analysis, while Appendix B passages specify effective transfer-entropy data for alphabet size N = 3.
- Index glossary: The S&P 500 tracks 500 actively traded US stocks, while the DAX represents 30 major German companies.
- Index glossary: The Swiss Market Index represents about 85% of Switzerland’s free-float equity-market capitalization through 20 large, liquid stocks.
- Index glossary: The Nikkei 225 is a yen-denominated, price-weighted average of Tokyo Stock Exchange equities whose components are reviewed annually.
- Effective transfer-entropy data: Appendix B tabulates effective transfer-entropy values calculated for alphabet size N = 3, including data associated with ETE and ERTE figures.The listed ERTE datasets use q = 1.5 and q = 0.8 in the corresponding figure captions.