Noncausal Autoregressive Model in Application to Bitcoin/USD Exchange Rates

  • Andrew Hencic
  • Christian GouriérouxEmail author
Part of the Studies in Computational Intelligence book series (SCI, volume 583)


This paper introduces a noncausal autoregressive process with Cauchy errors in application to the exchange rates of the Bitcoin electronic currency against the US Dollar. The dynamics of the daily Bitcoin/USD exchange rate series displays episodes of local trends, which can be modelled and interpreted as speculative bubbles. The bubbles may result from the speculative component in the on-line trading. The Bitcoin/USD exchange rates are modelled and predicted.

JEL number:

C14 G32 G23 


  1. 1.
    Andrews, B., Calder, M., Davis, R.: Maximum likelihood estimation for \(\alpha \)-stable autoregressive processes. Ann. Stat. 37, 1946–1982 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Andrews, B., Davis, R.: Model identification for infinite variance autoregressive processes. J. Econom. 172, 222–234 (2013)CrossRefMathSciNetGoogle Scholar
  3. 3.
    Balkema, G., Embrechts, P., Nolde, N.: The shape of asymptotic dependence. In: Shirayev, A., Varadhan, S., Presman, E. (eds.) Springer Proceedings in Mathematics and Statistics, special volume. Prokhorov and Contemporary Probability Theory, vol. 33, pp. 43–67 (2013)Google Scholar
  4. 4.
    Blanchard, O.: Speculative bubbles: crashes and rational expectations. Econ. Lett. 3, 387–389 (1979)CrossRefGoogle Scholar
  5. 5.
    Blanchard, O., Watson, M.: Bubbles, rational expectations and financial markets. In: Wachtel, P. (ed.) Crisis in the Economic and Financial Structure, pp. 295–315, Lexington (1982)Google Scholar
  6. 6.
    Bitcoin: Introduction. Retrieved 4 December 2013, (2013)
  7. 7.
    Bitcoin: Controlled Supply. Retrieved 4 December 2013, (2013)
  8. 8.
    Bitcoin: Trade. Retrieved 4 December 2013, (2013)
  9. 9.
    Blockchain. Bitcoin Market Capitalization. Retrieved from Bitcoin Block Explorer,
  10. 10.
    Breidt, F., Davis, R.: Time reversibility, identifiability and independence of innovations for stationary time series. J. Time Ser. Anal. 13, 273–390 (1992)CrossRefMathSciNetGoogle Scholar
  11. 11.
    Breidt, F., Davis, R., Lii, K.: Maximum likelihood estimation for noncausal autoregressive processes. J. Multivar. Anal. 36, 175–198 (1991)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Brunnermeier, M.: Asset Pricing under Asymmetric Information: Bubbles, Crashes, Technical Analysis and Herding. Oxford University Press, Oxford (2001)CrossRefGoogle Scholar
  13. 13.
    Cheng, Q.: On the unique representation of non-Gaussian linear processes. Ann. Stat. 20, 1143–1145 (1992)CrossRefzbMATHGoogle Scholar
  14. 14.
    Davis, J.: The Crypto-Currency, The New Yorker. Retrieved from (2011)
  15. 15.
    Davis, R., Resnick, S.: Limit theory for moving averages of random variables with regularly varying tail probabilities. Ann. Probab. 13, 179–195 (1985)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Davis, R., Resnick, S.: Limit theory for the sample covariance and correlation functions of moving averages. Ann. Stat. 14, 533–558 (1986)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Davis, R., Song, L.: Noncausal Vector AR Processes with Application to Economic Time Series, DP Columbia University (2012)Google Scholar
  18. 18.
    Evans, G.: Pitfalls in testing for explosive bubbles in asset prices. Am. Econ. Rev. 81, 922–930 (1991)Google Scholar
  19. 19.
    Flitter, E.: FBI shuts alleged online drug marketplace, Silk Road, Reuters. Retrieved from (2013)
  20. 20.
    Gourieroux, C., Jasiak, J.: Filtering, Prediction and Estimation of Noncausal Processes. CREST (2014)Google Scholar
  21. 21.
    Gourieroux, C., Zakoian, J.M.: Explosive Bubble Modelling by Noncausal Cauchy Autoregressive Process. CREST (2013)Google Scholar
  22. 22.
    Gourieroux, C., Zakoian, J.M.: On Uniqueness of Moving Average Representation of Heavy Tailed Stationary Processes. CREST (2013)Google Scholar
  23. 23.
    Lanne, M., Saikkonen, P.: Noncausal autoregressions for economic time series. J. Time Ser. Econom. 3(3), Article 2 (2011)Google Scholar
  24. 24.
    Lanne, M., Luoto, J., Saikkonen, P.: Optimal forecasting of nonlinear autoregressive time series. Int. J. Forecast. 28, 623–631 (2010)CrossRefGoogle Scholar
  25. 25.
    Lanne, M., Saikkonen, P.: Noncausal vector autoregression. Econom. Theory 29, 447–481 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  26. 26.
    Li, S.: Bitcoin now accepted as tuition payment at a Cyprus University, Los Angeles Times. Retrieved from,0,3194094.story#axzz2mXKIff7E (2013)
  27. 27.
    Litecoin Block Explorer: Litecoin Block Explorer Charts. Retrieved from 4 December 2013, (2013)
  28. 28.
    Liu, J.: BTC China the world’s largest Bitcoin trading platform, ZD Net. Retrieved from (2013)
  29. 29.
    Muth, J.: Rational expectations and the theory of price movements. Econometrica 29, 315–335 (1961)CrossRefGoogle Scholar
  30. 30.
    Newbold, P.: The exact likelihood function for a mixed autoregressive-moving average process. Biometrika. 61, 423– 426 (1974)Google Scholar
  31. 31.
    Phillips, P., Shi, S., Yu, J.: Testing for Multiple Bubbles, DP Cowles Foundation, 1843 (2012)Google Scholar
  32. 32.
    Phillips, P., Wu, Y., Yu, J.: Explosive behavior in the 1990s Nasdaq: when did exuberance escalate asset values? Int. Econ. Rev. 52, 201–226 (2011)CrossRefGoogle Scholar
  33. 33.
    Rosenblatt, M.: Gaussian and Non-Gaussian Linear Time Series and Random Fields. Springer, New York (2000)CrossRefzbMATHGoogle Scholar
  34. 34.
    Sparshott, J.: Web Money Gets Laundering Rule, The Wall Street Journal. Retrieved  from  (2013)
  35. 35.
    Tagaris, K.: Cyprus details heavy losses for major bank customers, Reuters. Retrieved from (2013)
  36. 36.
    Velde, F.R.: Bitcoin: A primer, Chicago Fed Letter. Retrieved from (2013)

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.York University, Department of EconomicsTorontoCanada
  2. 2.University of Toronto and CRESTTorontoCanada

Personalised recommendations