Abstract
Many time series in diverse fields of application may exhibit long-memory.The class of fractionally integrated (FI) processes can be used to try to model this strong data dependence. Asymptotic tests for FI include the re-scaled range statistic test and its modified form, the frequency-domain regression-based procedure, the modified Higuchi's test and Jensen's test. De Peretti and Marimoutou (2002) finds that proper finite-sample inferences are not possible using these techniques without correcting for size distortions. Some attempt this correction through `bootstrapping', but this method is not perfect and needs more study and improvements. In this paper, I examine and compare the finite-sample properties of parametric andnonparametric bootstrap tests by using graphical techniques of Davidson and MacKinnon (1998a) for showing whether they properly correct the distortions while retaining their power relative to the corresponding asymptotic tests.One of the tests uses a double bootstrap that provide better true power and size properties. I use a bilateral P value that permits the true power of the tests to grow when the size distortions are asymmetric. We then apply these procedures to a realtime series to investigate its long memory properties.
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Andersson, M.K. and Gredenhoff, M.P. (1998). Robust testing for fractional integration using the bootstrap. Working Paper Series in Economics and Finance No. 218.
Bardet, J.M., Lang, G., Moulines, E. and Soulier, P. (2000). Wavelet estimator of long-range dependence processes. Statistical Inference for Stochastic Processes, 3(1—2), 85–99.
Beran, J. (1994). Statistics for Long-Memory Processes. Chapman and Hall. London.
Booth, G. and Kaen, F. (1979). Gold and silver spot prices and market information efficiency. Financial Review, 14, 21–26.
Booth, G., Kaen, F. and Koveos, P. (1982). R/S analysis of foreign exchange rates under two international monetary regimes. Journal of Monetary Economics, 10, 407–415.
Davidson, R. and MacKinnon, J. (1993). Estimation and Inference in Economics. Oxford University Press, New York.
Davidson, R. and MacKinnon, J. (1998a). Graphical methods for investigating the size and the power of hypothesis tests. The Manchester School, 66, 1–22.
Davidson, R. and MacKinnon, J. (1998b). The size distortion of bootstrap tests. Working Paper, GREQAM.
Davidson, R. and MacKinnon, J.G. (1996). The power of bootstrap tests. Queen's University Institute for Economic Research, Discussion Paper 937.
de Peretti, C. and Marimoutou, V. (2002). Are the long memory tests really effective? GREQAM Working Paper No. 02A14.
Ding, Z. and Granger, C.W.J. (1996). Modelling volatility persistence of speculative returns: A new approach. Journal of Econometrics, 73, 185–215.
Ding, Z., Granger, C.W.J. and Engle, R.F. (1993). A long memory property of stock market returns and a new model. Journal of Empirical Finance, 1, 83–106.
Efron, B. (1979). Bootstrap methods: Another look at the Jacknife. Annals of Statistics, 7, 1–26.
Fama, E. and Frensh, K. (1988). Permanent and temporary component of stock prices. Journal of Political Economy, 96, 246–273.
Geweke, J. and Porter-Hudak, S. (1983). The estimation and the application of long memory time series models. Journal of Time Series Analysis, 4, 221–238.
Granger, C.W.J. and Ding, Z. (1996). Varieties of long memory models. Journal of Econometrics, 73, 61–77.
Granger, C.W.J. and Joyeux, R. (1980). An introduction to long-memory time series models and fractional integration. Journal of Time Series Analysis, 1(1), 15–29.
Greene, M. and Fielitz, B. (1977). Long-term dependence in common stock returns. Journal of Financial Economics, 4, 339–349.
Helm, B., Kaen, F. and Rosenman, R. (1984). Memory in commodity futures contracts. Journal of Futures Markets, 4, 559–567.
Higuchi, T. (1988). Approach to an irregular time series on the basis of the fractal theory. Physica, D 31, 277–283.
Horowitz, J.L. (1994a). Bootstap-based critical values for the information matrix test. Journal of Econometrics, 61, 395–411.
Horowitz, J.L. (1994b). Bootstrap methods in econometrics: Theory and numarical performance. Paper Presented at the 7th World Congress of the Econometric Society, Tokyo.
Hosking, J.R.M. (1981). Fractional differencing. Biometrika, 68, 165–176.
Hurst, H.E. (1951). Long-term storage capacity of reservoirs. Transac. Am. Soc. Civil Eng., 116, 770–808.
Jensen, M.J. (1994). Wavelet analysis of fractionally integrated processes. Dept. of Economics, Washington University, St. Louis, MO 63130.
Lo, A.W. (1991). Long-term memory in stock market price. Econometrica, 59, 1279–1313.
Lo, A.W. and MacKinlay, C. (1988). Stock market prices do not follow random walks: Evidence from a simple specification test. Review of Financial Studies, 1, 41–66.
Mandelbrot, B. (1971).When can price be arbitraged efficiently? A limit to the validity of the random walk and martingale models. Review of Economics and Statistics, 53, 225–236.
Mandelbrot, B. (1972). Statistical methodology for non-periodic cycles: From the covariance to R/S analysis. Annals of Economic and Social Measurement, 1, 259–290.
Mandelbrot, B. and Wallis, J.R. (1969). Some long-run properties of geophysical records. Water Resource Research, 5, 321–330.
Newey, W.K. and West, K.D. (1987). A simple, positive semi-definite, heteroscedasticity and autocorrelation consistent covariance matrix. Econometrica, 55, 703–708.
Poterba, J. and Summers, L. (1995). Mean reversion in stock returns: Evidence and implications. Journal of Financial Economics, 22, 27–60.
Robinson, P.M. (1995). Log-periodogram regression of time series with long range dependence. Annals of Statistics, 23, 1048–1072.
Schwarz, G. (1978). Estimating the dimension of a model. Annals of statistics, 6, 461–464.
Weber, N.C. (1984). On resampling techniques for regression models. Statistics and Probability Letters, 2, 275–278.
White, H. (1982). Maximum likelihood estimation of misspecified models. Econometrica, 50, 1–26.
Wornell, G. and Oppenheim, A. (1992). Estimation of fractal signals from noisy measurements using wavelets. IEEE Transaction on Signal Processing, 40, 611–623.
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de Peretti, C. Bilateral Bootstrap Tests for Long Memory: An Application to the Silver Market. Computational Economics 22, 187–212 (2003). https://doi.org/10.1023/A:1026129729224
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DOI: https://doi.org/10.1023/A:1026129729224