Abstract
The autocorrelation of long memory processes decays much slower than that of short memory processes, e.g. autoregressive processes. Fractional integration and Gegenbauer are basic models that show long memory behavior. Many tests for long memory of the fractional integration type (at f = 0) have been developed based on test statistics (e.g. the R/S-type statistics) or parameter estimators (e.g. the GPH estimator). But, so far, no work has shown reasonable power on testing for cyclic long memory. The authors investigate a parametric bootstrap procedure with an R-squared statistic as an assessment of cyclic long memory behavior. According to simulation results, the R-squared-bootstrapping method performs excellently in detecting Gegenbauer-type processes (e.g. with long memory behavior associated with frequencies \(f\in (0,0.5]\)) while at the same time controlling observed significance levels. The R-squared-bootstrapping test is also applied to the Lynx data and the result suggests the presence of long range dependence.
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References
Bao, W. (2014). Analysis of time series with time-varying frequency behavior and long memory. Ph.D. thesis, Southern Methodist University.
Bhansali, R. (1979). A mixed spectrum analysis of the lynx data. Journal of the Royal Statistical Society. Series A (General), 142(2), 199–209.
Boubaker, H. (2015). Wavelet estimation of Gegenbauer processes: Simulation and empirical application. Computational Economics, 46(4), 551–574.
Boutahar, M., Marimoutou, V., & Nouira, L. (2007). Estimation methods of the long memory parameter: Monte Carlo analysis and application. Journal of Applied Statistics, 34(3), 261–301.
Campbell, M., & Walker, A. (1977). A survey of statistical work on the mackenzie river series of annual Canadian lynx trappings for the years 1821–1934 and a new analysis. Journal of the Royal Statistical Society. Series A (general), 140(4), 411–431.
Davidson, J., & Rambaccussing, D. (2015). A test of the long memory hypothesis based on self-similarity. Journal of Time Series Econometrics, 7(2), 115–141.
Ford, C. R. (1991). The Gegenbauer and Gegenbauer autoregressive moving average long-memory time series models. Ph.D. thesis, Southern Methodist University.
Franco, G. C., & Reisen, V. A. (2007). Bootstrap approaches and confidence intervals for stationary and non-stationary long-range dependence processes. Physica A: Statistical Mechanics and its Applications, 375(2), 546–562.
Geweke, J., & Porter-Hudak, S. (1983). The estimation and application of long memory time series models. Journal of Time Series Analysis, 4(4), 221–238.
Giraitis, L., Kokoszka, P., Leipus, R., & Teyssiere, G. (2003a). On the power of R/S-type tests under contiguous and semi-long memory alternatives. Acta Applicandae Mathematica, 78(1), 285–299.
Giraitis, L., Kokoszka, P., Leipus, R., & Teyssiere, G. (2003b). Rescaled variance and related tests for long memory in volatility and levels. Journal of Econometrics, 112(2), 265–294.
Gray, H. L., Zhang, N.-F., & Woodward, W. A. (1989). On generalized fractional processes. Journal of Time Series Analysis, 10(3), 233–257.
Hurvich, C. M., Deo, R., & Brodsky, J. (1998). The mean squared error of geweke and Porter-Hudak’s estimator of the memory parameter of a long-memory time series. Journal of Time Series Analysis, 19(1), 19–46.
Hurvich, C. M., & Deo, R. S. (1999). Plug-in selection of the number of frequencies in regression estimates of the memory parameter of a long-memory time series. Journal of Time Series Analysis, 20(3), 331–341.
Hurvich, C. M., & Ray, B. K. (1995). Estimation of the memory parameter for nonstationary or noninvertible fractionally integrated processes. Journal of time Series Analysis, 16(1), 17–41.
Lo, A. W. (1991). Long-term memory in stock market prices. Econometrica, 59(5), 1279–1313.
Murphy, A., & Izzeldin, M. (2009). Bootstrapping long memory tests: Some Monte Carlo results. Computational Statistics and Data Analysis, 53(6), 2325–2334.
Reisen, V. A. (1994). Estimation of the fractional difference parameter in the arima (p, d, q) model using the smoothed periodogram. Journal of Time Series Analysis, 15(3), 335–350.
Reisen, V., Abraham, B., & Lopes, S. (2001). Estimation of parameters in ARFIMA processes: A simulation study. Communications in Statistics-Simulation and Computation, 30(4), 787–803.
Tong, H. (1977). Some comments on the Canadian lynx data. Journal of the Royal Statistical Society. Series A (General), 140(4), 432–436.
Whitcher, B. (2004). Wavelet-based estimation for seasonal long-memory processes. Technometrics, 46(2), 225–238.
Woodward, W. A., Bottone, S., & Gray, H. (1997). Improved tests for trend in time series data. Journal of Agricultural, Biological, and Environmental Statistics, 2(4), 403–416.
Woodward, W. A., & Gray, H. (1983). A comparison of autoregressive and harmonic component models for the lynx data. Journal of the Royal Statistical Society. Series A (General), 146(1), 71–73.
Woodward, W. A., Gray, H. L., & Elliott, A. C. (2017). Applied time series analysis with R (2nd ed.). Boca Raton: CRC Press.
Xing, Y. (2015). Analysis and model identification of long memory time series. Ph.D. thesis, Southern Methodist University.
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Xing, Y., Woodward, W.A. R-Squared-Bootstrapping for Gegenbauer-Type Long Memory. Comput Econ 57, 773–790 (2021). https://doi.org/10.1007/s10614-020-09977-1
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DOI: https://doi.org/10.1007/s10614-020-09977-1