Rescaled variance and related tests for long memory in volatility and levels. (English) Zbl 1027.62064

J. Econom. 112, No. 2, 265-294 (2003); corrigendum ibid. 126, No. 2, 571-572 (2005).
Summary: This paper studies properties of tests for long memory for general fourth order stationary sequences. We propose a rescaled variance test based on V/S statistics which is shown to have a simpler asymptotic distribution and to achieve a somewhat better balance of size and power than A. W. Lo’s [Econometrica 59, 1279-1313 (1991; Zbl 0781.90023)] modified R/S test and the KPSS test of D. Kwiatkowski et al. [J. Econom. 54, 159-178 (1992; Zbl 0871.62100)]. We investigate the theoretical performance of R/S, KPSS and V/S tests under short memory hypotheses and long memory alternatives, providing a Monte Carlo study and a brief empirical example. Assumptions of the same type are used in both short and long memory cases, covering all persistent dependence scenarios. We show that the results naturally apply and the assumptions are well adjusted to linear sequences (levels) and to squares of linear ARCH sequences (volatility).


62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62P20 Applications of statistics to economics
62E20 Asymptotic distribution theory in statistics
62F05 Asymptotic properties of parametric tests


Full Text: DOI


[1] Anderson, T. W., The Statistical Analysis of Time Series (1971), Wiley: Wiley New York · Zbl 0225.62108
[2] Andrews, D. W.K., Heteroscedasticity and autocorrelation consistent covariance estimation, Econometrica, 59, 817-858 (1991) · Zbl 0732.62052
[3] Baillie, R. T.; Bollerslev, T.; Mikkelsen, H. O., Fractionally integrated generalized autoregressive conditional heteroskedasticity, Journal of Econometrics, 74, 3-30 (1996) · Zbl 0865.62085
[4] Billingsley, P., Convergence of Probability Measures (1968), Wiley: Wiley New York · Zbl 0172.21201
[5] Breidt, F. J.; Crato, N.; de Lima, P., On the detection and estimation of long memory in stochastic volatility, Journal of Econometrics, 83, 325-348 (1998) · Zbl 0905.62116
[6] Brillinger, D. R., Time Series: Data Analysis and Theory (1981), Holt, Rinehart and Winston: Holt, Rinehart and Winston New York · Zbl 0486.62095
[7] Campbell, J. Y.; Lo, A. W.; MacKinlay, A. C., The Econometrics of Financial Markets (1997), Princeton University Press: Princeton University Press Princeton · Zbl 0927.62113
[8] Cheung, Y. W., Long memory in foreign-exchange rates, Journal of Business and Economic Statistics, 11, 93-101 (1993)
[9] Cheung, Y. W., Tests for fractional integrationA Monte Carlo investigation, Journal of Time Series Analysis, 14, 331-345 (1993) · Zbl 0800.62546
[10] Cheung, Y. W.; Lai, K. S., Do gold market returns have long memory?, The Financial Review, 28, 181-202 (1993)
[11] Crato, N.; de Lima, P. J.F., Long-range dependence in the conditional variance of stock returns, Economics Letters, 45, 281-285 (1994) · Zbl 0800.62791
[12] Davidson, J., Establishing conditions for the functional central limit theorem in nonlinear and semiparametric time series processes, Journal of Econometrics, 106, 243-269 (2002) · Zbl 1041.60032
[13] Davidson, J.; de Jong, R., Consistency of kernel estimators of heteroscedastic and autocorrelated covariance matrices, Econometrica, 68, 407-423 (2000) · Zbl 1016.62030
[14] Davidson, R.; MacKinnon, J. G., Graphical methods for investigating the size and power of hypothesis tests, The Manchester School, 66, 1-26 (1998)
[15] Davydov, Y. A., The invariance principle for stationary processes, Theory of Probability and its Applications, 15, 487-498 (1970) · Zbl 0219.60030
[16] Ding, Z.; Granger, C. W.J., Modeling volatility persistence of speculative returnsa new approach, Journal of Econometrics, 73, 185-215 (1996) · Zbl 1075.91626
[17] Durbin, J., Distribution Theory for Tests Based on the Sample Distribution Function (1973), Society for Industrial and Applied Mathematics: Society for Industrial and Applied Mathematics Philadelphia · Zbl 0267.62002
[18] Feller, W., The asymptotic distribution of the range of sums of independent random variables, Annals of Mathematical Statistics, 22, 427-432 (1951) · Zbl 0043.34201
[19] Geweke, J.; Porter-Hudak, S., The estimation and application of long memory time models, Journal of Time Series Analysis, 4, 221-238 (1983) · Zbl 0534.62062
[20] Giraitis, L.; Kokoszka, P.; Leipus, R., Stationary ARCH modelsdependence structure and central limit theorem, Econometric Theory, 16, 3-22 (2000) · Zbl 0986.60030
[21] Giraitis, L.; Kokoszka, P.; Leipus, R.; Teyssière, G., Semiparametric estimation of the intensity of long memory in conditional heteroskedasticity, Statistical Inference for Stochastic Processes, 3, 113-128 (2000) · Zbl 1054.62104
[22] Giraitis, L.; Robinson, P.; Surgailis, D., A model for long memory conditional heteroskedasticity, Annals of Applied Probability, 10, 1002-1024 (2000) · Zbl 1084.62516
[23] Giraitis, L.; Kokoszka, P.; Leipus, R., Testing for long memory in the presence of a general trend, Journal of Applied Probability, 38, 1033-1054 (2001) · Zbl 1140.62341
[24] Goetzmann, W. N., Patterns in three centuries of stock market prices, Journal of Business, 66, 249-270 (1993)
[25] Gourieroux, C.; Monfort, A., Time Series and Dynamic Models (1997), Cambridge University Press: Cambridge University Press Cambridge · Zbl 1420.91461
[26] Hauser, M. A., Semiparametric and nonparametric testing for long memorya Monte Carlo study, Empirical Economics, 22, 247-271 (1997)
[27] Hosking, J. R.M., Asymptotic distributions of the sample mean, autocovariances, and autocorrelations of long-memory time series, Journal of Econometrics, 73, 261-284 (1996) · Zbl 0854.62084
[28] Hurst, H., Long term storage capacity of reservoirs, Transactions of the American Society of Civil Engineers, 116, 770-799 (1951)
[29] Kiefer, J., \(K\)-sample analogues of the Kolmogorov-Smirnov and Cramér-v. Mises tests, Annals of Mathematical Statistics, 30, 420-447 (1959) · Zbl 0134.36707
[30] Kirman, A.; Teyssière, G., Microeconomic models for long-memory in the volatility of financial time series, Studies in Nonlinear Dynamics and Econometrics, 5, 281-302 (2002) · Zbl 1079.91565
[31] Kuiper, N. H., Tests concerning random points on a circle, Proceedings of the Koninklijke Nederlandse Akademie Van Wettenschappen, Series A, 63, 38-47 (1960) · Zbl 0096.12504
[32] Kwiatkowski, D.; Phillips, P. C.B.; Schmidt, P.; Shin, Y., Testing the null hypothesis of stationarity against the alternative of a unit roothow sure are we that economic time series have a unit root?, Journal of Econometrics, 54, 159-178 (1992) · Zbl 0871.62100
[33] Lee, H. S.; Amsler, C., Consistency of the KPSS unit root test against fractionally integrated alternative, Economics Letters, 55, 151-160 (1997)
[34] Lee, D.; Schmidt, P., On the power of the KPSS test of stationarity against fractionally-integrated alternatives, Journal of Econometrics, 73, 285-302 (1996) · Zbl 0856.62075
[35] Liu, Y. A.; Pan, M. S.; Hsueh, L. P., A modified R/S analysis of long-term dependence in currency futures prices, Journal of International Financial Markets, Institutions and Money, 3, 97-113 (1993)
[36] Lo, A., Long-term memory in stock market prices, Econometrica, 59, 1279-1313 (1991) · Zbl 0781.90023
[37] Lobato, I.; Robinson, P. M., A nonparametric test for I(0), Review of Economic Studies, 68, 475-495 (1998) · Zbl 0910.90070
[38] Lobato, I.; Savin, N. E., Real and spurious long-memory properties of stock-market data (with comments), Journal of Business & Economic Statistics, 16, 261-283 (1998)
[39] Mandelbrot, B. B., Statistical methodology for non-periodic cyclesfrom the covariance to R/S analysis, Annals of Economic and Social Measurement, 1, 259-290 (1972)
[40] Mandelbrot, B. B., Limit theorems of the self-normalized range for weakly and strongly dependent processes, Zeitschrift für Wahrschein lichkeitstheorie und Verwandte Gebiete, 31, 271-285 (1975) · Zbl 0288.60033
[42] Mandelbrot, B. B.; Wallis, J. M., Robustness of the rescaled range R/S in the measurement of noncyclic long run statistical dependence, Water Resources Research, 5, 967-988 (1969)
[45] Robinson, P. M., Testing for strong serial correlation and dynamic conditional heteroskedasticity in multiple regression, Journal of Econometrics, 47, 67-84 (1991) · Zbl 0734.62070
[46] Robinson, P. M., Gaussian semiparametric estimation of long range dependence, The Annals of Statistics, 23, 1630-1661 (1995) · Zbl 0843.62092
[47] Robinson, P. M.; Henry, M., Long and short memory conditional heteroskedasticity in estimating the memory parameter of levels, Econometric Theory, 15, 299-336 (1999) · Zbl 1054.62584
[48] Robinson, P. M.; Zaffaroni, P., Nonlinear time series with long memorya model for stochastic volatility, Journal of Statistical Planning and Inference, 68, 359-371 (1998) · Zbl 0937.62109
[49] Rosenblatt, M., Limit theorems associated with variants of the von Mises statistic, Annals of Mathematical Statistics, 23, 617-623 (1952) · Zbl 0048.36003
[50] Shin, Y.; Schmidt, P., The KPSS stationary test as a unit root test, Economics Letters, 38, 387-505 (1992) · Zbl 0800.62531
[51] Teverovsky, V.; Taqqu, M. S.; Willinger, W., A critical look at Lo’s modified R/S statisitic, Journal of Statistical Planning and Inference, 80, 211-227 (1999) · Zbl 1044.60508
[52] Teverovsky, V.; Taqqu, M. S.; Willinger, W., Stock market prices and long-range dependence, Finance & Stochastics, 3, 1-13 (1999) · Zbl 0924.90029
[54] Tsay, V.-S., On the power of Durbin-Watson statistic against fractionally integrated processes, Econometric Reviews, 17, 361-386 (1998) · Zbl 0913.62087
[55] Vogelsang, T. J., Sources of nonmonotonic power when testing for a shift in mean of a dynamic time series, Journal of Econometrics, 88, 283-299 (1999) · Zbl 0933.62092
[56] Watson, G. S., Goodness-of-fit tests on a circle, Biometrika, 48, 109-114 (1961) · Zbl 0212.21905
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