Abstract
A class of regression type estimators of the parameterd in a fractionally differencedARMA (p, q) process is introduced. This class is an extension of the estimator considered by Geweke and Porter-Hudak. In a simulation study, we compared three estimators from this class together with two approximate maximum likelihood estimators which are based on two separate approximations to the likelihood. One approximation ignores the determinant term in the likelihood and the other includes a compensating factor for the determinant. When the determinant term is included, the estimate tends to be much less biased and is in general superior to the other estimate. The approximate maximum likelihood estimator out performed, by a large margin, the regression type estimators for pureARIMA (0,d,0) processes. However, forARIMA (1,d,1) processes, a regression type estimator turned out to be the best for realizations of length 400 in 3 out of the 5 cases we tried.
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References
Brockwell, P.J.; Davis, R.A. 1987: Time Series: Theory and Methods. Springer-Verlag, New York
Fox, R.; Taqqu, M.S. 1986: Large sample properties of parameter estimates for strongly dependent stationary Gaussian time series. Ann. Stat., 14, 517–532
Geweke, J.; Porter-Hudak, S. 1983: The estimation and application of long memory time series models. J. Time Series Anal., 4, 221–238
Granger, C.W.J.; Joyeux, R. 1980: An introduction to long memory time series models and fractional differencing. J. Time Series Anal., 1, 15–30
Gupta, S.N. 1987: Parameter estimation in fractionally differenced ARMA processes, Ph.D. dissertation, Colorado State University, Fort Collins
Hosking, J.R.M. 1981: Fractional differencing. Biometrika, 68, 165–176.
Hosking, J.R.M. 1984: Modeling persistence in hydrological time series using fractional differencing. Water Resour Res., 20, 1898–1908
Hurst, H.E. 1951: Long-term storage capacity of reservoirs: Trans. Am. Soc. Civ. Eng. 116, 770–808
Li, W.K.; McLeod, A.I. 1980: Fractional time series modeling. Biometrika, 73, 217–221
Lomnicki, Z.A.; Zaremba, S.K. 1957: On estimating the spectral density function of a stochastic process. J. Joy. Stat. Soc. Ser. B. 19, 13–37
Rice, J. 1979: On the estimation of the parameters of a power spectrum. J. Multivariate Analysis, 9, 378–392
Todini, E.; O'Connell, P.E. (eds.) 1979: Hydrological simulation of Lake Nasser, vol. 1. BM Italica Scientific Centers, Italy, and Institute of Hydrology, Wallingford, Oxfordshire, United Kingdom
Yajima, Y. 1985: On estimation of long-memory time series models. The Australian J. Statistics, 27, 303–320
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Boes, D.C., Davis, R.A. & Gupta, S.N. Parameter estimation in low order fractionally differenced ARMA processes. Stochastic Hydrol Hydraul 3, 97–110 (1989). https://doi.org/10.1007/BF01544075
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DOI: https://doi.org/10.1007/BF01544075