Sample autocorrelations of nonstationary fractionally integrated series. (English) Zbl 0883.62022
Summary: We derive the asymptotic distribution of the sample autocorrelations of nonstationary fractionally integrated processes of order \(d\). If \(d\geq 1\), the sample autocorrelations approach their probability limit one with a rate equal to the sample size. If \(d<1\), the rate is slower and depends on \(d\). These findings carry over to the case of detrended series. Monte Carlo evidence and an empirical example illustrate the theoretical results.
MSC:
| 62E20 | Asymptotic distribution theory in statistics |
| 62M10 | Time series, auto-correlation, regression, etc. in statistics (GARCH) |
References:
| [1] | Anderson, O. D., Partial Autocorrelation Properties for Non-Stationary Autoregressive Moving-Average Models, Journal of Time Series Analysis, 13, 486-500 (1992) · Zbl 0755.62063 · doi:10.1111/j.1467-9892.1992.tb00122.x |
| [2] | Bierens, H. J., Higher-Order Sample Autocorrelations and the Unit Root Hypothesis, Journal of Econometrics, 57, 137-160 (1993) · Zbl 0776.62065 · doi:10.1016/0304-4076(93)90062-A |
| [3] | Cheung, Y.-W., Long Memory in Foreign-Exchange Rates, Journal of Business & Economic Statistics, 11, 93-101 (1993) · doi:10.2307/1391309 |
| [4] | Davydov, Y. A., The Invariance Principle for Stationary Processes, Theory of Probability and Its Applications, 15, 487-489 (1970) · Zbl 0219.60030 · doi:10.1137/1115050 |
| [5] | Dickey, D. A.; Fuller, W. A., Distribution of the Estimators for Autoregressive Time Series with a Unit Root, Journal of the American Statistical Association, 74, 427-431 (1979) · Zbl 0413.62075 · doi:10.2307/2286348 |
| [6] | Diebold, F. X.; Rudebusch, G. D., Long Memory and Persistence in Aggregate Output, Journal of Monetary Economics, 24, 189-209 (1989) · doi:10.1016/0304-3932(89)90003-2 |
| [7] | Diebold, F. X.; Husted, S.; Rush, M., Real Exchange Rates under the Gold Standard, Journal of Political Economy, 99, 1252-1271 (1991) · doi:10.1086/261799 |
| [8] | Durlauf, S. N.; Phillips, P. B.C., Trends versus Random Walks in Time Series Analysis, Econometrica, 56, 1333-1354 (1988) · Zbl 0653.62068 · doi:10.2307/1913101 |
| [9] | Granger, C. W.J.; Joyeux, R., An Introduction to Long-Memory Time Series Models and Fractional Differencing, Journal of Time Series Analysis, 1, 15-29 (1980) · Zbl 0503.62079 · doi:10.1111/j.1467-9892.1980.tb00297.x |
| [10] | Hamilton, J.D. (1994): Time Series Analysis; Princeton University Press. · Zbl 0831.62061 |
| [11] | Hassler, U., The Sample Autocorrelation Function of I(1) processes, Statistical Papers, 35, 1-16 (1994) · Zbl 0803.62081 · doi:10.1007/BF02926395 |
| [12] | Hassler, U.; Wolters, J., On the Power of Unit Root Tests Against Fractional Alternatives, Economics Letters, 45, 1-5 (1994) · Zbl 0800.62532 · doi:10.1016/0165-1765(94)90049-3 |
| [13] | Hassler, U.; Wolters, J., Long Memory in Inflation Rates: International Evidence, Journal of Business & Economic Statistics, 13, 37-45 (1995) · doi:10.2307/1392519 |
| [14] | Hasza, D. P., The Asymptotic Distribution of the Sample Autocorrelations for an Integrated ARMA process, Journal of the American Statistical Association, 75, 349-352 (1980) · Zbl 0462.62019 · doi:10.2307/2287457 |
| [15] | Hosking, J. R.M., Fractional Differencing, Biometrika, 68, 165-176 (1981) · Zbl 0464.62088 · doi:10.1093/biomet/68.1.165 |
| [16] | Hosking, J.R.M. (1984): Asymptotic Distributions of the Sample Mean, Autocovariances and Autocorrelations of Long-Memory Time Series; Mathematics Research Center Technical Summary Report #2752, University of Wisconsin-Madison. |
| [17] | Hosking, J. R.M., Modelling Persistence in Hydrologic Time Series using Fractional Differencing, Water Resources Research, 20, 1898-1908 (1984) · doi:10.1029/WR020i012p01898 |
| [18] | Lo, A. W., Long-Term Memory in Stock Market Prices, Econometrica, 59, 1279-1313 (1991) · Zbl 0781.90023 · doi:10.2307/2938368 |
| [19] | Mandelbrot, B. B.; Van Ness, J. W., Fractional Brownian Motions, Fractional Brownian Noises and Applications, SIAM Review, 10, 422-437 (1968) · Zbl 0179.47801 · doi:10.1137/1010093 |
| [20] | Newbold, P.; Agiakloglou, Ch., Bias in the Sample Autocorrelations of Fractional Noise, Biometrika, 80, 698-702 (1993) · Zbl 0800.62519 · doi:10.1093/biomet/80.3.698 |
| [21] | Park, J. Y.; Phillips, P. C.B., Statistical Inference in Regressions with Integrated Processes: Part I, Econometric Theory, 4, 468-497 (1998) |
| [22] | Phillips, P. C.B., Time Series Regression with a Unit Root, Econometrica, 55, 277-301 (1987) · Zbl 0613.62109 · doi:10.2307/1913237 |
| [23] | Sowell, F., The Fractional Unit Root Distribution, Econometrica, 58, 495-505 (1990) · Zbl 0727.62025 · doi:10.2307/2938213 |
| [24] | Taqqu, M., Weak Convergence to Fractional Brownian Motion and to the Rosenblatt Process, Zeitschrift für Wahrscheinlichkeit und verwandte Gebiete, 31, 287-302 (1975) · Zbl 0303.60033 · doi:10.1007/BF00532868 |
| [25] | Wichern, D. W., The Behaviour of the Sample Autocorrelation Function for an Integrated Moving Average Process, Biometrika, 60, 235-239 (1973) · Zbl 0261.62070 · doi:10.1093/biomet/60.2.235 |
| [26] | Yajima, Y., Asymptotic Properties of the Sample Autocorrelations and Partial Autocorrelations of a Multiplicative ARIMA Process, Journal of Time Series Analysis, 6, 187-201 (1985) · Zbl 0573.62085 · doi:10.1111/j.1467-9892.1985.tb00409.x |
| [27] | Yajima, Y., On Estimation of a Regression Model with Long-Memory Stationary Errors, Annals of Statistics, 16, 791-807 (1988) · Zbl 0661.62090 · doi:10.1214/aos/1176350837 |
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