Entropy and Efficiency of the ETF Market

  • Lucio Maria CalcagnileEmail author
  • Fulvio Corsi
  • Stefano Marmi


We investigate the relative information efficiency of financial markets by measuring the entropy of the time series of high frequency data. Our tool to measure efficiency is the Shannon entropy, applied to 2-symbol and 3-symbol discretisations of the data. Analysing 1-min and 5-min price time series of 55 Exchange Traded Funds traded at the New York Stock Exchange, we develop a methodology to isolate residual inefficiencies from other sources of regularities, such as the intraday pattern, the volatility clustering and the microstructure effects. The first two are modelled as multiplicative factors, while the microstructure is modelled as an ARMA noise process. Following an analytical and empirical combined approach, we find a strong relationship between low entropy and high relative tick size and that volatility is responsible for the largest amount of regularity, averaging 62% of the total regularity against 18% of the intraday pattern regularity and 20% of the microstructure.


Market efficiency Shannon entropy Information theory ARMA processes 


Supplementary material


  1. Aït-Sahalia, Y., Mykland, P. A., & Zhang, L. (2011). Ultra high frequency volatility estimation with dependent microstructure noise. Journal of Econometrics, 160(1), 160–175.CrossRefGoogle Scholar
  2. Andersen, T. G., & Bollerslev, T. (1997). Intraday periodicity and volatility persistence in financial markets. Journal of Empirical Finance, 4, 115–158.CrossRefGoogle Scholar
  3. Barndorff-Nielsen, O. E., & Shephard, N. (2004). Power and bipower variation with stochastic volatility and jumps. Journal of Financial Econometrics, 2, 1–48.CrossRefGoogle Scholar
  4. Brownlees, C. T., & Gallo, G. M. (2006). Financial econometric analysis at ultra-high frequency: Data handling concerns. Computational Statistics & Data Analysis, 51, 2232–2245.CrossRefGoogle Scholar
  5. Cajueiro, D. O., & Tabak, B. M. (2004). Ranking efficiency for emerging markets. Chaos, Solitons and Fractals, 22, 349–352.CrossRefGoogle Scholar
  6. Giglio, R., Matsushita, R., Figueiredo, A., Gleria, I., & Da Silva, S. (2008). Algorithmic complexity theory and the relative efficiency of financial markets. EPL 84, 48,005Google Scholar
  7. Grassberger, P. (2008). Entropy estimates from insufficient samplings ArXiv:physics/0307138v2
  8. Oh, G., Kim, S., & Eom, C. (2007). Market efficiency in foreign exchange markets. Physica A, 382, 209–212.CrossRefGoogle Scholar
  9. Risso, W. A. (2009). The informational efficiency: The emerging markets versus the developed markets. Applied Economics Letters, 16, 485–487.CrossRefGoogle Scholar
  10. Shmilovici, A., Alon-Brimer, Y., & Hauser, S. (2003). Using a stochastic complexity measure to check the efficient market hypothesis. Computational Economics, 22, 273–284.CrossRefGoogle Scholar
  11. Shmilovici, A., Kahiri, Y., Ben-Gal, I., & Hauser, S. (2009). Measuring the efficiency of the intraday forex market with a universal data compression algorithm. Computational Economics, 33(2), 131–154.CrossRefGoogle Scholar
  12. Taylor, S. J. (2011). Asset price dynamics, volatility, and prediction. Princeton: Princeton University Press.Google Scholar
  13. Weiss, G. (1975). Time-reversibility of linear stochastic processes. Journal of Applied Probability, pp. 831–836.Google Scholar

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Authors and Affiliations

  1. 1.Scuola Normale SuperiorePisaItaly
  2. 2.LIST S.p.A.PisaItaly
  3. 3.Università di PisaPisaItaly
  4. 4.City, University of LondonLondonUK

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