Type I and type II fractional Brownian motions: a reconsideration. (English) Zbl 1453.62077
Summary: The so-called type I and type II fractional Brownian motions are limit distributions associated with the fractional integration model in which pre-sample shocks are either included in the lag structure, or suppressed. There can be substantial differences between the distributions of these two processes and of functionals derived from them, so that it becomes an important issue to decide which model to use as a basis for inference. Alternative methods for simulating the type I case are contrasted, and for models close to the nonstationarity boundary, truncating infinite sums is shown to result in a significant distortion of the distribution. A simple simulation method that overcomes this problem is described and implemented. The approach also has implications for the estimation of type I ARFIMA models, and a new conditional ML estimator is proposed, using the annual Nile minima series for illustration.
MSC:
| 62-08 | Computational methods for problems pertaining to statistics |
| 62M10 | Time series, auto-correlation, regression, etc. in statistics (GARCH) |
| 60G22 | Fractional processes, including fractional Brownian motion |
| 62P20 | Applications of statistics to economics |
References:
| [1] | Abry, P.; Sellan, F., The wavelet-based synthesis for fractional Brownian motion proposed by F. Sellan and Y. Meyer: Remarks and Fast Implementation, Applied and Computational Harmonic Analysis, 3, 377-383 (1996) · Zbl 0862.60036 |
| [2] | Beran, J., Statistics for Long Memory Processes (1994), Chapman and Hall: Chapman and Hall New York · Zbl 0869.60045 |
| [3] | Billingsley, P., Convergence of Probability Measures (1968), John Wiley and Sons · Zbl 0172.21201 |
| [4] | Byers, D.; Davidson, J.; Peel, D., Modelling political popularity: an analysis of long range dependence in opinion poll series, Journal of the Royal Statistical Society Series A, 160, 3, 471-490 (1997) |
| [5] | Byers, D.; Davidson, J.; Peel, D., Modelling political popularity: a correction, Journal of the Royal Statistical Society Series A, 165, 1, 187-189 (2002) |
| [6] | Caporale, G.-M.; Gil-Alana, L., Modelling the US, UK and Japanese unemployment rates: Fractional integration and structural breaks, Computational Statistics & Data Analysis, 52, 11, 4998-5013 (2008) · Zbl 1452.62889 |
| [7] | Cheung, Y.-W.; Diebold, F. X., On maximum likelihood estimation of the differencing parameter of fractionally integrated noise with unknown mean, Journal of Econometrics, 62, 301-316 (1994) |
| [8] | Coakley, J.; Dollery, J.; Kellard, N., The role of long memory in hedging effectiveness, Computational Statistics & Data Analysis, 52, 6, 3075-3082 (2008) · Zbl 1452.62755 |
| [9] | Davidson, J.; de Jong, R. M., The functional central limit theorem and convergence to stochastic integrals II: fractionally integrated processes, Econometric Theory, 16, 5, 643-666 (2000) · Zbl 0981.60028 |
| [10] | Davidson, J., 2008. Time Series Modelling 4.27. at http://www.timeseriesmodelling.com; Davidson, J., 2008. Time Series Modelling 4.27. at http://www.timeseriesmodelling.com |
| [11] | Davidson, J., Alternative bootstrap procedures for testing cointegration in fractionally integrated processes, Journal of Econometrics, 133, 2, 741-777 (2006) · Zbl 1345.62057 |
| [12] | Davidson, J.; Sibbertsen, P., Generating schemes for long memory processes: regimes, aggregation and linearity, Journal of Econometrics, 128, 253-282 (2005) · Zbl 1335.62129 |
| [13] | Davidson, J.; Hashimzade, N., Alternative frequency and time domain versions of fractional Brownian motion, Econometric Theory, 24, 1, 256-293 (2008) · Zbl 1281.60036 |
| [14] | Davidson, J., Hashimzade, N., 2009. Representation and weak convergence of stochastic integrals with fractional integrator processes. Econometric Theory, (forthcoming); Davidson, J., Hashimzade, N., 2009. Representation and weak convergence of stochastic integrals with fractional integrator processes. Econometric Theory, (forthcoming) · Zbl 1189.60109 |
| [15] | Davies, R. B.; Harte, D. S., Tests for Hurst effect, Biometrika, 74, 95-102 (1987) · Zbl 0612.62123 |
| [16] | Dolado, J.; Gonzalo, J.; Mayoral, L., A fractional Dickey Fuller test for unit roots, Econometrica, 70, 1963-2006 (2002) · Zbl 1132.91590 |
| [17] | Doornik, J. A., Ox: An Object-Oriented Matrix Programming Language (2006), Timberlake Consultants Ltd |
| [18] | Doornik, L.A., Ooms,, 2006. A Package for Estimating, Forecasting and Simulating Arfima Models: Arfima package 1.04 for Ox, at http://www.doornik.com; Doornik, L.A., Ooms,, 2006. A Package for Estimating, Forecasting and Simulating Arfima Models: Arfima package 1.04 for Ox, at http://www.doornik.com |
| [19] | Enriquez, N., A simple construction of the fractional Brownian motion, Stochastic Processes and their Applications, 109, 203-223 (2004) · Zbl 1075.60019 |
| [20] | Gradshteyn; Ryzhik, (Jeffrey, A.; Zwillinger, D., Tables of Integrals, Series, and Products (2000), Academic Press) · Zbl 0981.65001 |
| [21] | Granger, C. W.J., Long memory relationships and the aggregation of dynamic models, Journal of Econometrics, 14, 227-238 (1980) · Zbl 0466.62108 |
| [22] | Granger, C. W.J.; Joyeux, Roselyne, An introduction to long memory time series models and fractional differencing, Journal of Time Series Analysis, 1, 1, 15-29 (1980) · Zbl 0503.62079 |
| [23] | Haldrup, N.; Nielsen, M.Ø., Estimation of fractional integration in the presence of data noise, Computational Statistics & Data Analysis, 51, 6, 3100-3114 (2007) · Zbl 1161.62325 |
| [24] | Hurst, H., Long term storage capacity of reservoirs, Transactions of the American Society of Civil Engineers, 116, 770-799 (1951) |
| [25] | Johansen, S., Nielsen, M.Ø., 2008. Likelihood inference for a nonstationary fractional autoregressive model. CREATES working paper. University of Aarhus; Johansen, S., Nielsen, M.Ø., 2008. Likelihood inference for a nonstationary fractional autoregressive model. CREATES working paper. University of Aarhus |
| [26] | Mandelbrot, B. B.; van Ness, J. W., Fractional Brownian motions, fractional noises and applications, SIAM Review, 10, 4, 422-437 (1968) · Zbl 0179.47801 |
| [27] | Marinucci, D.; Robinson, P. M., Alternative forms of fractional Brownian motion, Journal of Statistical Inference and Planning, 80, 111-122 (1999) · Zbl 0934.60071 |
| [28] | Marinucci, D.; Robinson, P. M., Weak convergence of multivariate fractional processes, Stochastic Processes and their Applications, 8, 6, 103-120 (2000) · Zbl 1028.60030 |
| [29] | Meyer, Y.; Sellan, F.; Taqqu, M. S., Wavelets, generalized white noise and fractional integration: the synthesis of fractional Brownian motion, Journal of Fourier Analysis and Applications, 5, 5, 465-494 (1999) · Zbl 0948.60026 |
| [30] | Pipiras, V., 2004. On the usefulness of wavelet-based simulation of fractional Brownian motion, Working paper, University of North Carolina at Chapel Hill; Pipiras, V., 2004. On the usefulness of wavelet-based simulation of fractional Brownian motion, Working paper, University of North Carolina at Chapel Hill |
| [31] | Pipiras, V., Wavelet-based simulation of fractional Brownian motion revisited, Applied Computational and Harmonic Analysis, 19, 4, 9-60 (2005) · Zbl 1074.60048 |
| [32] | Sowell, F., Maximum likelihood estimation of stationary fractionally integrated time series models, Journal of Econometrics, 53, 165-188 (1992) |
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