IDEAS home Printed from https://ideas.repec.org/a/eee/phsmap/v456y2016icp1-9.html
   My bibliography  Save this article

Monitoring the informational efficiency of European corporate bond markets with dynamical permutation min-entropy

Author

Listed:
  • Zunino, Luciano
  • Bariviera, Aurelio F.
  • Guercio, M. Belén
  • Martinez, Lisana B.
  • Rosso, Osvaldo A.

Abstract

In this paper the permutation min-entropy has been implemented to unveil the presence of temporal structures in the daily values of European corporate bond indices from April 2001 to August 2015. More precisely, the informational efficiency evolution of the prices of fifteen sectorial indices has been carefully studied by estimating this information-theory-derived symbolic tool over a sliding time window. Such a dynamical analysis makes possible to obtain relevant conclusions about the effect that the 2008 credit crisis has had on the different European corporate bond sectors. It is found that the informational efficiency of some sectors, namely banks, financial services, insurance, and basic resources, has been strongly reduced due to the financial crisis whereas another set of sectors, integrated by chemicals, automobiles, media, energy, construction, industrial goods & services, technology, and telecommunications has only suffered a transitory loss of efficiency. Last but not least, the food & beverage, healthcare, and utilities sectors show a behavior close to a random walk practically along all the period of analysis, confirming a remarkable immunity against the 2008 financial crisis.

Suggested Citation

  • Zunino, Luciano & Bariviera, Aurelio F. & Guercio, M. Belén & Martinez, Lisana B. & Rosso, Osvaldo A., 2016. "Monitoring the informational efficiency of European corporate bond markets with dynamical permutation min-entropy," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 456(C), pages 1-9.
    • Handle: RePEc:eee:phsmap:v:456:y:2016:i:c:p:1-9
    DOI: 10.1016/j.physa.2016.03.007
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0378437116002788
    Download Restriction: Full text for ScienceDirect subscribers only. Journal offers the option of making the article available online on Science direct for a fee of $3,000
    File URL: https://libkey.io/10.1016/j.physa.2016.03.007?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Ladislav Kristoufek & Miloslav Vosvrda, 2014. "Measuring capital market efficiency: long-term memory, fractal dimension and approximate entropy," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 87(7), pages 1-9, July.
    2. Pawe{l} Sieczka & Didier Sornette & Janusz A. Ho{l}yst, 2010. "The Lehman Brothers Effect and Bankruptcy Cascades," Papers 1002.1070, arXiv.org, revised Sep 2011.
    3. Grech, D & Mazur, Z, 2004. "Can one make any crash prediction in finance using the local Hurst exponent idea?," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 336(1), pages 133-145.
    4. Bentes, Sónia R. & Menezes, Rui & Mendes, Diana A., 2008. "Long memory and volatility clustering: Is the empirical evidence consistent across stock markets?," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 387(15), pages 3826-3830.
    5. Kristoufek, Ladislav & Vosvrda, Miloslav, 2014. "Commodity futures and market efficiency," Energy Economics, Elsevier, vol. 42(C), pages 50-57.
    6. Barunik, Jozef & Kristoufek, Ladislav, 2010. "On Hurst exponent estimation under heavy-tailed distributions," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 389(18), pages 3844-3855.
    7. Aurelio Fernández Bariviera & Luciano Zunino & María Belén Guercio & Lisana Martinez & Osvaldo Rosso, 2013. "Revisiting the European sovereign bonds with a permutation-information-theory approach," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 86(12), pages 1-10, December.
    8. Bassler, Kevin E. & Gunaratne, Gemunu H. & McCauley, Joseph L., 2006. "Markov processes, Hurst exponents, and nonlinear diffusion equations: With application to finance," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 369(2), pages 343-353.
    9. Alvarez-Ramirez, Jose & Rodriguez, Eduardo & Alvarez, Jesus, 2012. "A multiscale entropy approach for market efficiency," International Review of Financial Analysis, Elsevier, vol. 21(C), pages 64-69.
    10. Di Matteo, T. & Aste, T. & Dacorogna, M.M., 2003. "Scaling behaviors in differently developed markets," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 324(1), pages 183-188.
    11. L. Zunino & B. M. Tabak & D. G. Pérez & M. Garavaglia & O. A. Rosso, 2007. "Inefficiency in Latin-American market indices," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 60(1), pages 111-121, November.
    12. Kristoufek, Ladislav & Vosvrda, Miloslav, 2013. "Measuring capital market efficiency: Global and local correlations structure," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 392(1), pages 184-193.
    13. Plerou, Vasiliki & Gopikrishnan, Parameswaran & Rosenow, Bernd & Amaral, Luis A.N. & Stanley, H.Eugene, 2000. "Econophysics: financial time series from a statistical physics point of view," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 279(1), pages 443-456.
    14. Serinaldi, Francesco, 2010. "Use and misuse of some Hurst parameter estimators applied to stationary and non-stationary financial time series," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 389(14), pages 2770-2781.
    15. Sensoy, A., 2013. "Generalized Hurst exponent approach to efficiency in MENA markets," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 392(20), pages 5019-5026.
    16. Fama, Eugene F, 1991. "Efficient Capital Markets: II," Journal of Finance, American Finance Association, vol. 46(5), pages 1575-1617, December.
    17. Risso, Wiston Adrián, 2008. "The informational efficiency and the financial crashes," Research in International Business and Finance, Elsevier, vol. 22(3), pages 396-408, September.
    18. Weron, Rafał, 2002. "Estimating long-range dependence: finite sample properties and confidence intervals," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 312(1), pages 285-299.
    19. Simonsen, Ingve & Sneppen, Kim, 2002. "Profit profiles in correlated markets," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 316(1), pages 561-567.
    20. Yi-Cheng Zhang, 1999. "Toward a Theory of Marginally Efficient Markets," Papers cond-mat/9901243, arXiv.org.
    21. Grech, Dariusz & Pamuła, Grzegorz, 2008. "The local Hurst exponent of the financial time series in the vicinity of crashes on the Polish stock exchange market," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 387(16), pages 4299-4308.
    22. Ortiz-Cruz, Alejandro & Rodriguez, Eduardo & Ibarra-Valdez, Carlos & Alvarez-Ramirez, Jose, 2012. "Efficiency of crude oil markets: Evidences from informational entropy analysis," Energy Policy, Elsevier, vol. 41(C), pages 365-373.
    23. Fama, Eugene F, 1970. "Efficient Capital Markets: A Review of Theory and Empirical Work," Journal of Finance, American Finance Association, vol. 25(2), pages 383-417, May.
    24. Aurelio Fernandez Bariviera & María Belén Guercio & Lisana B. Martinez & Osvaldo A. Rosso, 2015. "The (in)visible hand in the Libor market: an information theory approach," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 88(8), pages 1-9, August.
    25. Aurelio F. Bariviera & Luciano Zunino & M. Belen Guercio & Lisana B. Martinez & Osvaldo A. Rosso, 2015. "Efficiency and credit ratings: a permutation-information-theory analysis," Papers 1509.01839, arXiv.org.
    26. Zunino, Luciano & Tabak, Benjamin M. & Serinaldi, Francesco & Zanin, Massimiliano & Pérez, Darío G. & Rosso, Osvaldo A., 2011. "Commodity predictability analysis with a permutation information theory approach," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 390(5), pages 876-890.
    27. Bariviera, A.F. & Guercio, M. Belén & Martinez, Lisana B., 2012. "A comparative analysis of the informational efficiency of the fixed income market in seven European countries," Economics Letters, Elsevier, vol. 116(3), pages 426-428.
    28. Alvarez-Ramirez, Jose & Rodriguez, Eduardo & Espinosa-Paredes, Gilberto, 2012. "Is the US stock market becoming weakly efficient over time? Evidence from 80-year-long data," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(22), pages 5643-5647.
    29. Lo, Andrew W, 1991. "Long-Term Memory in Stock Market Prices," Econometrica, Econometric Society, vol. 59(5), pages 1279-1313, September.
    30. Zunino, Luciano & Zanin, Massimiliano & Tabak, Benjamin M. & Pérez, Darío G. & Rosso, Osvaldo A., 2009. "Forbidden patterns, permutation entropy and stock market inefficiency," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 388(14), pages 2854-2864.
    31. Oh, Gabjin & Kim, Seunghwan & Eom, Cheoljun, 2007. "Market efficiency in foreign exchange markets," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 382(1), pages 209-212.
    32. Aurelio F. Bariviera & M. Belen Guercio & Lisana B. Martinez & Osvaldo A. Rosso, 2015. "A permutation Information Theory tour through different interest rate maturities: the Libor case," Papers 1509.00217, arXiv.org.
    33. Grammatikos, Theoharry & Vermeulen, Robert, 2012. "Transmission of the financial and sovereign debt crises to the EMU: Stock prices, CDS spreads and exchange rates," Journal of International Money and Finance, Elsevier, vol. 31(3), pages 517-533.
    34. Zunino, Luciano & Zanin, Massimiliano & Tabak, Benjamin M. & Pérez, Darío G. & Rosso, Osvaldo A., 2010. "Complexity-entropy causality plane: A useful approach to quantify the stock market inefficiency," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 389(9), pages 1891-1901.
    35. Sensoy, Ahmet & Tabak, Benjamin M., 2015. "Time-varying long term memory in the European Union stock markets," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 436(C), pages 147-158.
    36. Bariviera, Aurelio Fernández, 2011. "The influence of liquidity on informational efficiency: The case of the Thai Stock Market," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 390(23), pages 4426-4432.
    37. Cajueiro, Daniel O & Tabak, Benjamin M, 2004. "The Hurst exponent over time: testing the assertion that emerging markets are becoming more efficient," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 336(3), pages 521-537.
    38. Alvarez-Ramirez, Jose & Alvarez, Jesus & Rodriguez, Eduardo & Fernandez-Anaya, Guillermo, 2008. "Time-varying Hurst exponent for US stock markets," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 387(24), pages 6159-6169.
    39. Zhang, Yi-Cheng, 1999. "Toward a theory of marginally efficient markets," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 269(1), pages 30-44.
    40. Darbellay, Georges A & Wuertz, Diethelm, 2000. "The entropy as a tool for analysing statistical dependences in financial time series," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 287(3), pages 429-439.
    41. Wiston Adrian Risso, 2009. "The informational efficiency: the emerging markets versus the developed markets," Applied Economics Letters, Taylor & Francis Journals, vol. 16(5), pages 485-487.
    42. Kristoufek, Ladislav, 2012. "How are rescaled range analyses affected by different memory and distributional properties? A Monte Carlo study," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(17), pages 4252-4260.
    43. Les Gulko, 1999. "The Entropic Market Hypothesis," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 2(03), pages 293-329.
    44. Morales, Raffaello & Di Matteo, T. & Gramatica, Ruggero & Aste, Tomaso, 2012. "Dynamical generalized Hurst exponent as a tool to monitor unstable periods in financial time series," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(11), pages 3180-3189.
    45. Aurelio Fernández Bariviera & M. Belén Guercio & Lisana B. Martinez, 2014. "Informational Efficiency in Distressed Markets: The Case of European Corporate Bonds," The Economic and Social Review, Economic and Social Studies, vol. 45(3), pages 349-369.
    46. Martina, Esteban & Rodriguez, Eduardo & Escarela-Perez, Rafael & Alvarez-Ramirez, Jose, 2011. "Multiscale entropy analysis of crude oil price dynamics," Energy Economics, Elsevier, vol. 33(5), pages 936-947, September.
    47. Couillard, Michel & Davison, Matt, 2005. "A comment on measuring the Hurst exponent of financial time series," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 348(C), pages 404-418.
    48. P. Sieczka & D. Sornette & J. Holyst, 2011. "The Lehman Brothers effect and bankruptcy cascades," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 82(3), pages 257-269, August.
    49. Grau-Carles, Pilar, 2000. "Empirical evidence of long-range correlations in stock returns," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 287(3), pages 396-404.
    50. R. Cont, 2001. "Empirical properties of asset returns: stylized facts and statistical issues," Quantitative Finance, Taylor & Francis Journals, vol. 1(2), pages 223-236.
    51. Carbone, A. & Castelli, G. & Stanley, H.E., 2004. "Time-dependent Hurst exponent in financial time series," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 344(1), pages 267-271.
    52. Martinez, Lisana B. & Terceño, Antonio & Teruel, Mercedes, 2013. "Sovereign bond spreads determinants in Latin American countries: Before and during the XXI financial crisis," Emerging Markets Review, Elsevier, vol. 17(C), pages 60-75.
    53. Zunino, Luciano & Fernández Bariviera, Aurelio & Guercio, M. Belén & Martinez, Lisana B. & Rosso, Osvaldo A., 2012. "On the efficiency of sovereign bond markets," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(18), pages 4342-4349.
    54. Cajueiro, Daniel O. & Tabak, Benjamin M., 2004. "Evidence of long range dependence in Asian equity markets: the role of liquidity and market restrictions," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 342(3), pages 656-664.
    55. Kevin E. Bassler & Gemunu H. Gunaratne & Joseph L. McCauley, 2006. "Markov Processes, Hurst Exponents, and Nonlinear Diffusion Equations with application to finance," Papers cond-mat/0602316, arXiv.org.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Christoph Bandt, 2020. "Order patterns, their variation and change points in financial time series and Brownian motion," Statistical Papers, Springer, vol. 61(4), pages 1565-1588, August.
    2. Albarracín E., Eva Susana & Gamboa, Juan C. Rodríguez & Marques, Elaine C.M. & Stosic, Tatijana, 2019. "Complexity analysis of Brazilian agriculture and energy market," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 523(C), pages 933-941.
    3. Trindade, Marco A.S. & Floquet, Sergio & Filho, Lourival M. Silva, 2020. "Portfolio theory, information theory and Tsallis statistics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 541(C).
    4. Fernandes, Leonardo H.S. & de Araujo, Fernando H.A. & Tabak, Benjamin M., 2021. "Insights from the (in)efficiency of Chinese sectoral indices during COVID-19," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 578(C).
    5. Charfeddine, Lanouar & Khediri, Karim Ben & Aye, Goodness C. & Gupta, Rangan, 2018. "Time-varying efficiency of developed and emerging bond markets: Evidence from long-spans of historical data," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 505(C), pages 632-647.
    6. Polanco-Martínez, J.M. & Fernández-Macho, J. & Neumann, M.B. & Faria, S.H., 2018. "A pre-crisis vs. crisis analysis of peripheral EU stock markets by means of wavelet transform and a nonlinear causality test," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 490(C), pages 1211-1227.
    7. Argyroudis, George S. & Siokis, Fotios M., 2019. "Spillover effects of Great Recession on Hong-Kong’s Real Estate Market: An analysis based on Causality Plane and Tsallis Curves of Complexity–Entropy," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 524(C), pages 576-586.
    8. Parker, Edgar, 2017. "The Entropic Linkage between Equity and Bond Market Dynamics," MPRA Paper 80036, University Library of Munich, Germany.
    9. Christoph Bandt, 2019. "Order patterns, their variation and change points in financial time series and Brownian motion," Papers 1910.09978, arXiv.org.
    10. Liu, Zhengli & Shang, Pengjian & Wang, Yuanyuan, 2020. "Characterization of time series through information quantifiers," Chaos, Solitons & Fractals, Elsevier, vol. 132(C).
    11. Siokis, Fotios M., 2018. "Credit market Jitters in the course of the financial crisis: A permutation entropy approach in measuring informational efficiency in financial assets," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 499(C), pages 266-275.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Lisana B. Martinez & M. Belén Guercio & Aurelio Fernandez Bariviera & Antonio Terceño, 2018. "The impact of the financial crisis on the long-range memory of European corporate bond and stock markets," Empirica, Springer;Austrian Institute for Economic Research;Austrian Economic Association, vol. 45(1), pages 1-15, February.
    2. Sukpitak, Jessada & Hengpunya, Varagorn, 2016. "The influence of trading volume on market efficiency: The DCCA approach," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 458(C), pages 259-265.
    3. Zunino, Luciano & Zanin, Massimiliano & Tabak, Benjamin M. & Pérez, Darío G. & Rosso, Osvaldo A., 2009. "Forbidden patterns, permutation entropy and stock market inefficiency," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 388(14), pages 2854-2864.
    4. Bariviera, Aurelio F. & Basgall, María José & Hasperué, Waldo & Naiouf, Marcelo, 2017. "Some stylized facts of the Bitcoin market," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 484(C), pages 82-90.
    5. Zunino, Luciano & Zanin, Massimiliano & Tabak, Benjamin M. & Pérez, Darío G. & Rosso, Osvaldo A., 2010. "Complexity-entropy causality plane: A useful approach to quantify the stock market inefficiency," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 389(9), pages 1891-1901.
    6. Argyroudis, George S. & Siokis, Fotios M., 2019. "Spillover effects of Great Recession on Hong-Kong’s Real Estate Market: An analysis based on Causality Plane and Tsallis Curves of Complexity–Entropy," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 524(C), pages 576-586.
    7. Zunino, Luciano & Tabak, Benjamin M. & Serinaldi, Francesco & Zanin, Massimiliano & Pérez, Darío G. & Rosso, Osvaldo A., 2011. "Commodity predictability analysis with a permutation information theory approach," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 390(5), pages 876-890.
    8. Ladislav Kristoufek & Miloslav Vosvrda, 2014. "Measuring capital market efficiency: long-term memory, fractal dimension and approximate entropy," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 87(7), pages 1-9, July.
    9. Kristoufek, Ladislav & Vosvrda, Miloslav, 2014. "Commodity futures and market efficiency," Energy Economics, Elsevier, vol. 42(C), pages 50-57.
    10. Anagnostidis, P. & Varsakelis, C. & Emmanouilides, C.J., 2016. "Has the 2008 financial crisis affected stock market efficiency? The case of Eurozone," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 447(C), pages 116-128.
    11. Sensoy, Ahmet & Tabak, Benjamin M., 2016. "Dynamic efficiency of stock markets and exchange rates," International Review of Financial Analysis, Elsevier, vol. 47(C), pages 353-371.
    12. Sukpitak, Jessada & Hengpunya, Varagorn, 2016. "Efficiency of Thai stock markets: Detrended fluctuation analysis," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 458(C), pages 204-209.
    13. Siokis, Fotios M., 2018. "Credit market Jitters in the course of the financial crisis: A permutation entropy approach in measuring informational efficiency in financial assets," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 499(C), pages 266-275.
    14. Sensoy, Ahmet & Tabak, Benjamin M., 2015. "Time-varying long term memory in the European Union stock markets," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 436(C), pages 147-158.
    15. Auer, Benjamin R., 2016. "On time-varying predictability of emerging stock market returns," Emerging Markets Review, Elsevier, vol. 27(C), pages 1-13.
    16. Kristoufek, Ladislav & Vosvrda, Miloslav, 2016. "Gold, currencies and market efficiency," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 449(C), pages 27-34.
    17. Gu, Rongbao & Xiong, Wei & Li, Xinjie, 2015. "Does the singular value decomposition entropy have predictive power for stock market? — Evidence from the Shenzhen stock market," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 439(C), pages 103-113.
    18. Li, Daye & Nishimura, Yusaku & Men, Ming, 2016. "The long memory and the transaction cost in financial markets," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 442(C), pages 312-320.
    19. Aurelio F. Bariviera & Luciano Zunino & Osvaldo A. Rosso, 2018. "An analysis of high-frequency cryptocurrencies prices dynamics using permutation-information-theory quantifiers," Papers 1808.01926, arXiv.org.
    20. Fernandes, Leonardo H.S. & Araújo, Fernando H.A., 2020. "Taxonomy of commodities assets via complexity-entropy causality plane," Chaos, Solitons & Fractals, Elsevier, vol. 137(C).

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:phsmap:v:456:y:2016:i:c:p:1-9. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.journals.elsevier.com/physica-a-statistical-mechpplications/ .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.