On a Class of Estimation and Test for Long Memory

Fu, Hui (2012): On a Class of Estimation and Test for Long Memory.

Abstract

This paper advances a new analysis technology path of estimation and test for long memory time series. I propose the definitions of time scale series, strong variance scale exponent and weak variance scale exponent, and prove the strict mathematical equations that strong and weak variance scale exponent can accurately identify the time series of white noise, short memory and long memory, especially derive the equation relationships between weak variance scale exponent and long memory parameters. I also construct two statistics which SLmemory statistic tests for long memory properties. The paper further displays Monte Carlo performance for MSE of weak variance scale exponent estimator and the empirical size and power of SLmemory statistic, giving practical recommendations of finite-sample, and also provides brief empirical examples of logarithmic return rate series data for Sino-US stock markets.

• Item Type: MPRA Paper
• Language: English
• Keywords: Long Memory, Weak Variance Scale Exponent, SLmemory Statistic, Time Scale Series.
• Subjects: C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C12 - Hypothesis Testing: General; C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C13 - Estimation: General; C - Mathematical and Quantitative Methods > C2 - Single Equation Models ; Single Variables > C22 - Time-Series Models ; Dynamic Quantile Regressions ; Dynamic Treatment Effect Models ; Diffusion Processes

References

Abadir, K.M., Distaso, W., Giraitis, L., 2009. Two estimator of the long-run variance: Beyond short memory. Journal of Econometrics 150, 56-70.

Anderson, T.W., 1971. The Statistical Analysis of Time Series, New York: Wiley.

Baillie, R.T., 1996. Long memory processes and fractional integration in econometrics. Journal of Econometrics 73, 5-59.

Beran, J., 1994. On a class of M-estimators for Gaussian long-memory models. Biometrika 81(4), 755-766.

Boutahar, M., Marimoutou, V., Nouira, L., 2007. Estimation methods of the long memory parameter: Monte Carlo analysis and application. Journal of Applied Statistics 34, 261-301.

Brockwell, P., Davis, R., 1991. Time Series: Theory and Methods(2nd ed.), Berlin: Springer- Verlag.

Fox, R., Taqqu, M.S., 1986. Large-Sample Properties of Parameter Estimates for Strongly Dependent Stationary Gaussian Time Series. Annals of Statistics 14(2), 517-532.

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., 2003. Rescaled variance and related tests for long memory in volatility and levels. Journal of Econometrics 112(2), 265-294.

Giraitis, L., Kokoszka, P., Leipus, R., Teyssiere, G., 2005. Corrigendum to ‘Rescaled variance and related tests for long memory in volatility and levels’. Journal of Econometrics 126(2), 571-572.

Granger, Clive W.J., Joyeux, R., 1980. An Introduction to Long-memory Time Series Models and Fractional Differencing. Journal of Time Series Analysis 1(1), 15-29.

Gradshteyn, I.S., Ryzhik, I.M., 2007. Table of Integrals, Series, and Products(Seventh Edition). Academic Press, Elsevier Inc.

Haslett, J., Raftery, A.E., 1989. Space-Time Modelling with Long-Memory Dependence: Assessing Ireland’s Wind Power Resource. Journal of the Royal Statistical Society. Series C (Applied Statistics) 38(1), 1-50.

Hosking, J.R.M., 1981. Fractional differencing. Biometrika 68(1), 165-176.

Hurst, H.E., 1951. Long-term storage capacity of reservoirs. Transactions of the American Society of Civil Engineers 116, 770-779.

Kwiatkowski, D., Phillips, P.C.B., Schmidt, P., Shin, Y., 1992. Testing the null hypothesis of stationarity against the alternative of a unit root: how sure are we that economic time series have a unit root? Journal of Econometrics 54, 159-178.

Lo, A.W., 1991. Long-term memory in stock market prices. Econometrica 59 (5), 1279-1313.

Mandelbrot, B.B., Wallis, J.R., 1968. Noah, Joseph, and Operational Hydrology. Water Resources Research 4(5), 909-918.

Mandelbrot, B.B., Wallis, J.R., 1969. Robustness of the Rescaled Range R/S in the measurement of Noncyclic Long-run Statistical Dependence. Water resources Research 5(5), 967-988.

Mandelbrot, B.B., 1975. Limit theorems of the self-normalized range for weakly and strongly dependent processes. Probability Theory and Related Fields 31(4), 271-285.

Mandelbrot, B.B., Taqqu, M.S., 1979. Robust R/S analysis of long run serial correlation. 42nd Session of the International Statistical Institute, Manila, Book 2, 69-99.

Palma, W., 2007. Long-Memory Time Series (Theory and Methods). New Jersey: John Wiley \& Sons, Inc.

Robinson, P.M., 1995. Gaussian Semiparametric Estimation of Long Range Dependence. The Annals of Statistics 23(5), 1630-1661.

Robinson, P.M., 2003. Time Series with Long Memory. New York: Oxford University Press Inc.

Robinson, P.M., 2005. Robust covariance matrix estimation: HAC estimates with long memory/ anti-persistence correction. Econometric Theory 21, 171-180.