Long-Range Dependence of Financial Time Series Data
This paper examines long-range dependence or longmemory
of financial time series on the exchange rate data by the
fractional Brownian motion (fBm). The principle of spectral density
function in Section 2 is used to find the range of Hurst parameter (H)
of the fBm. If 0< H <1/2, then it has a short-range dependence
(SRD). It simulates long-memory or long-range dependence (LRD) if
1/2< H <1. The curve of exchange rate data is fBm because of the
specific appearance of the Hurst parameter (H). Furthermore, some
of the definitions of the fBm, long-range dependence and selfsimilarity
are reviewed in Section II as well. Our results indicate that
there exists a long-memory or a long-range dependence (LRD) for
the exchange rate data in section III. Long-range dependence of the
exchange rate data and estimation of the Hurst parameter (H) are
discussed in Section IV, while a conclusion is discussed in Section V.
[1] E. Alos, O. Mazet, and D. Nualart, "Stochastic calculus with respect to
fractional Brownian motion with hurst parameter less than ›",
Stochastic Processes and their Applications, 2000, vol. 86, pp.121-139.
[2] A. Assaf and J. Cavalcante, "Long range dependence in the returns and
volatility of the Brazilian stock market", European Review of Economic
and Finance, vol 4, 2005, pp. 1-19.
[3] R.T. Baillie, "Long memory processes and fractional integration in
econometrics", Journal of Econometrics, 1996, vol. 73, pp.5-59.
[4] Y.W. Cheung, "Long memory in foreign-exchange rates", Business and
Economic Statistics, vol. 11, 1993, pp. 93-101.
[5] F. Comte and E. Renault, "Long memory continuous-time models",
Journal of Econometrics, 1996, vol.73, pp.101-149.
[6] F. Comte and E. Renault, "Long memory in continuous time stochastic
volatility models", Mathematical Finance, 1998, vol.8, pp.291-323.
[7] W. Dai and C.C. Heyde, "Ito-s formula with respect to fractional
Brownian motion and its application", J. Appl. Math. Stoch. Anal., 1996,
vol.9, pp.439-448.
[8] L.Decreusefond and A.S. Ustunel, " Fractional brownian motion:theory
and applications", Fractional Differential Systems: Models, Methods and
Applications, 1998, vol. 5, pp.75-86.
[9] C.W.J. Granger and Z.Ding, "Varieties of long-memory models",
Journal of Econometrics, 1996, vol. 73, pp.61-77.
[10] C.W.J. Granger and R.Joyeux, "An introduction to long-memory times
series models and fractional differencing", Journal of Time Series
Analysis, 1980, vol. 1, no 1, pp.15-29.
[11] C.C. Heyde, "A risky asset model with strong dependence", Journal of
Applied Probability, 1999, vol. 36, pp.1234-1239.
[12] C.C. Heyde and S.Liu, "Empirical realities for a minimum description
risky asset model. The need for fractal features. Principal invited paper,
Mathematics in the New Millenium Conference, Seoul, Korea, October
2000", J. Korean Math. Soc., 2001, vol. 38, pp.1047-1059.
[13] J.R.M. Hosking, " Fractional differencing", Biometrika, 1981,vol. 68,
no 1, pp.165-176.
[14] A. Lo, "Long-term memory in the market prices", Econometrica, 1991,
vol. 59, pp.1279-1313.
[15] B.B. Mandelbrot and J.W. Van Ness, "Fractional Brownian motions,
fractional noises and applications", SIAM Review, 1968, vol.10, no 4,
pp.422-437.
[16] I. Norros, E. Valkeila, and J. Virtamo, " An elementary approach to a
Girsanov formula and other analytical results on fractional Brownian
motion", Bernoulli, 1999, vol. 5, pp.571-587.
[17] W.Willinger, M.S. Taqqu, and V.Teverovsky, " Stock market prices
and long- range dependence", Finance and Stochastics, 1999, vol. 3, pp.
1-13.
[1] E. Alos, O. Mazet, and D. Nualart, "Stochastic calculus with respect to
fractional Brownian motion with hurst parameter less than ›",
Stochastic Processes and their Applications, 2000, vol. 86, pp.121-139.
[2] A. Assaf and J. Cavalcante, "Long range dependence in the returns and
volatility of the Brazilian stock market", European Review of Economic
and Finance, vol 4, 2005, pp. 1-19.
[3] R.T. Baillie, "Long memory processes and fractional integration in
econometrics", Journal of Econometrics, 1996, vol. 73, pp.5-59.
[4] Y.W. Cheung, "Long memory in foreign-exchange rates", Business and
Economic Statistics, vol. 11, 1993, pp. 93-101.
[5] F. Comte and E. Renault, "Long memory continuous-time models",
Journal of Econometrics, 1996, vol.73, pp.101-149.
[6] F. Comte and E. Renault, "Long memory in continuous time stochastic
volatility models", Mathematical Finance, 1998, vol.8, pp.291-323.
[7] W. Dai and C.C. Heyde, "Ito-s formula with respect to fractional
Brownian motion and its application", J. Appl. Math. Stoch. Anal., 1996,
vol.9, pp.439-448.
[8] L.Decreusefond and A.S. Ustunel, " Fractional brownian motion:theory
and applications", Fractional Differential Systems: Models, Methods and
Applications, 1998, vol. 5, pp.75-86.
[9] C.W.J. Granger and Z.Ding, "Varieties of long-memory models",
Journal of Econometrics, 1996, vol. 73, pp.61-77.
[10] C.W.J. Granger and R.Joyeux, "An introduction to long-memory times
series models and fractional differencing", Journal of Time Series
Analysis, 1980, vol. 1, no 1, pp.15-29.
[11] C.C. Heyde, "A risky asset model with strong dependence", Journal of
Applied Probability, 1999, vol. 36, pp.1234-1239.
[12] C.C. Heyde and S.Liu, "Empirical realities for a minimum description
risky asset model. The need for fractal features. Principal invited paper,
Mathematics in the New Millenium Conference, Seoul, Korea, October
2000", J. Korean Math. Soc., 2001, vol. 38, pp.1047-1059.
[13] J.R.M. Hosking, " Fractional differencing", Biometrika, 1981,vol. 68,
no 1, pp.165-176.
[14] A. Lo, "Long-term memory in the market prices", Econometrica, 1991,
vol. 59, pp.1279-1313.
[15] B.B. Mandelbrot and J.W. Van Ness, "Fractional Brownian motions,
fractional noises and applications", SIAM Review, 1968, vol.10, no 4,
pp.422-437.
[16] I. Norros, E. Valkeila, and J. Virtamo, " An elementary approach to a
Girsanov formula and other analytical results on fractional Brownian
motion", Bernoulli, 1999, vol. 5, pp.571-587.
[17] W.Willinger, M.S. Taqqu, and V.Teverovsky, " Stock market prices
and long- range dependence", Finance and Stochastics, 1999, vol. 3, pp.
1-13.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:61454", author = "Chatchai Pesee", title = "Long-Range Dependence of Financial Time Series Data", abstract = "This paper examines long-range dependence or longmemory
of financial time series on the exchange rate data by the
fractional Brownian motion (fBm). The principle of spectral density
function in Section 2 is used to find the range of Hurst parameter (H)
of the fBm. If 0< H ", keywords = "Fractional Brownian motion, long-rangedependence, memory, short-range dependence.", volume = "2", number = "8", pages = "587-5", }