Detecting long-memory: Monte Carlo simulations and application to daily streamflow processes

Abstract. The Lo's modified rescaled adjusted range test (R/S test) (Lo, 1991), GPH test (Geweke and Porter-Hudak, 1983) and two approximate maximum likelihood estimation methods, i.e., Whittle's estimator (W-MLE) and another one implemented in S-Plus (S-MLE) based on the algorithm of Haslett and Raftery (1989) are evaluated through intensive Monte Carlo simulations for detecting the existence of long-memory. It is shown that it is difficult to find an appropriate lag q for Lo's test for different short-memory autoregressive (AR) and fractionally integrated autoregressive and moving average (ARFIMA) processes, which makes the use of Lo's test very tricky. In general, the GPH test outperforms the Lo's test, but for cases where a strong short-range dependence exists (e.g., AR(1) processes with φ=0.95 or even 0.99), the GPH test gets useless, even for time series of large data size. On the other hand, the estimates of d given by S-MLE and W-MLE seem to give a good indication of whether or not the long-memory is present. The simulation results show that data size has a significant impact on the power of all the four methods because the availability of larger samples allows one to inspect the asymptotical properties better. Generally, the power of Lo's test and GPH test increases with increasing data size, and the estimates of d with GPH method, S-MLE method and W-MLE method converge with increasing data size. If no large enough data set is available, we should be aware of the possible bias of the estimates. The four methods are applied to daily average discharge series recorded at 31 gauging stations with different drainage areas in eight river basins in Europe, Canada and USA to detect the existence of long-memory. The results show that the presence of long-memory in 29 daily series is confirmed by at least three methods, whereas the other two series are indicated to be long-memory processes with two methods. The intensity of long-memory in daily streamflow processes has only a very weak positive relationship with the scale of watershed.

[1]  C. Hurvich,et al.  Plug‐in Selection of the Number of Frequencies in Regression Estimates of the Memory Parameter of a Long‐memory Time Series , 1998 .

[2]  Nigel Meade,et al.  Evidence of long memory in short-term interest rates , 2003 .

[3]  Liudas Giraitis,et al.  RATE OPTIMAL SEMIPARAMETRIC ESTIMATION OF THE MEMORY PARAMETER OF THE GAUSSIAN TIME SERIES WITH LONG‐RANGE DEPENDENCE , 1997 .

[4]  C. Hurvich,et al.  ASYMPTOTICS FOR THE LOW‐FREQUENCY ORDINATES OF THE PERIODOGRAM OF A LONG‐MEMORY TIME SERIES , 1993 .

[5]  H. Akaike,et al.  Information Theory and an Extension of the Maximum Likelihood Principle , 1973 .

[6]  Philip Hans Franses,et al.  A seasonal periodic long memory model for monthly river flows , 1998, Environ. Model. Softw..

[7]  M. Taqqu,et al.  Fractionally differenced ARIMA models applied to hydrologic time series: Identification, estimation, and simulation , 1997 .

[8]  A. Ramachandra Rao,et al.  Hypothesis testing for long-term memory in hydrologic series , 1999 .

[9]  Michalis Faloutsos,et al.  Long-range dependence ten years of Internet traffic modeling , 2004, IEEE Internet Computing.

[10]  Fallaw Sowell.  Modeling long-run behavior with the fractional ARIMA model , 1992 .

[11]  Wen Wang,et al.  Testing and modelling autoregressive conditional heteroskedasticity of streamflow processes , 2005 .

[12]  Kuldeep Kumar,et al.  Modelling Financial Time Series with S‐PLUS , 2007 .

[13]  P. Gelder,et al.  Forecasting daily streamflow using hybrid ANN models , 2006 .

[14]  W. Willinger,et al.  A critical look at Lo's modified R/S statistic , 1999 .

[15]  Jan Beran,et al.  Statistics for long-memory processes , 1994 .

[16]  P. Newbold,et al.  BIAS IN AN ESTIMATOR OF THE FRACTIONAL DIFFERENCE PARAMETER , 1993 .

[17]  W. Newey,et al.  A Simple, Positive Semi-Definite, Heteroskedasticity and Autocorrelationconsistent Covariance Matrix , 1986 .

[18]  D. Andrews.  Heteroskedasticity and Autocorrelation Consistent Covariance Matrix Estimation , 1991 .

[19]  Jiahui Wang,et al.  Modeling Financial Time Series with S-PLUS® , 2003 .

[20]  Demetris Koutsoyiannis,et al.  Uncertainty, entropy, scaling and hydrological stochastics. 2. Time dependence of hydrological processes and time scaling / Incertitude, entropie, effet d'échelle et propriétés stochastiques hydrologiques. 2. Dépendance temporelle des processus hydrologiques et échelle temporelle , 2005 .

[21]  C. Granger,et al.  AN INTRODUCTION TO LONG‐MEMORY TIME SERIES MODELS AND FRACTIONAL DIFFERENCING , 1980 .

[22]  B. McCarl,et al.  Economics , 1870, The Indian medical gazette.

[23]  Murad S. Taqqu,et al.  A seasonal fractional ARIMA Model applied to the Nile River monthly flows at Aswan , 2000 .

[24]  J. R. Wallis,et al.  Computer Experiments with Fractional Gaussian Noises: Part 2, Rescaled Ranges and Spectra , 1969 .

[25]  S. Hussain,et al.  Fractional order estimation and testing, application to Swedish temperature data , 1999 .

[26]  P. Tonellato,et al.  Evaluating maximum likelihood estimation methods to determine the Hurst coeficient. , 1999, Physica A.

[27]  Thomas L. Burr,et al.  Modeling Financial Time Series With S—Plus , 2007, Technometrics.

[28]  J. R. Wallis,et al.  Computer Experiments with Fractional Gaussian Noises: Part 3, Mathematical Appendix , 1969 .

[29]  H. E. Hurst,et al.  Long-Term Storage Capacity of Reservoirs , 1951 .

[30]  J. R. Wallis,et al.  Computer Experiments With Fractional Gaussian Noises: Part 1, Averages and Variances , 1969 .

[31]  J. Geweke,et al.  THE ESTIMATION AND APPLICATION OF LONG MEMORY TIME SERIES MODELS , 1983 .

[32]  Zeng Zhen-yu.  The Empirical Analysis of Long-term Memory in Stock Market , 2003 .

[33]  Bovas Abraham,et al.  ESTIMATION OF PARAMETERS IN ARFIMA PROCESSES: A SIMULATION STUDY , 2001 .

[34]  Hans von Storch,et al.  Long‐term persistence in climate and the detection problem , 2006 .

[35]  A. Lo.  Long-Term Memory in Stock Market Prices , 1989 .

[36]  Marc Henry,et al.  Robust Automatic Bandwidth for Long Memory , 2001 .

[37]  Jussi Tolvi,et al.  Modeling Financial Time Series with S‐Plus , 2003 .

[38]  P. Bloomfield.  Trends in global temperature , 1992 .

[39]  A. Raftery,et al.  Space-time modeling with long-memory dependence: assessing Ireland's wind-power resource. Technical report , 1987 .

[40]  Demetris Koutsoyiannis.  The Hurst phenomenon and fractional Gaussian noise made easy , 2002 .

[41]  V. Klemeš.  The Hurst Phenomenon: A puzzle? , 1974 .