Averaged periodogram estimation of long memory. (English) Zbl 0854.62088
Summary: This paper discusses estimates of the parameter \(H \in (1/2,1)\) which governs the shape of the spectral density near zero frequency of a long memory time series. The estimates are semiparametric in the sense that the spectral density is parametrized only within a neighborhood of zero frequency. The estimates are based on averages of the periodogram over a band consisting of \(m\) equally-spaced frequencies which decays slowly to zero as sample size increases.
The second author [Ann. Stat. 22, No. 1, 515-539 (1994; Zbl 0795.62082)] proposed such an estimate of \(H\) which is consistent under very mild conditions. We describe the limiting distributional behavior of the estimate and also provide Monte Carlo information on its finite-sample distribution. We also give an expression for the asymptotic mean squared error of the estimate. In addition to depending on the bandwidth number \(m\), the estimate depends on an additional user-chosen number \(q\), but we show that for \(H \in (1/2, 3/4)\) there exists an optimal \(q\) for each \(H\), and we tabulate this.
The second author [Ann. Stat. 22, No. 1, 515-539 (1994; Zbl 0795.62082)] proposed such an estimate of \(H\) which is consistent under very mild conditions. We describe the limiting distributional behavior of the estimate and also provide Monte Carlo information on its finite-sample distribution. We also give an expression for the asymptotic mean squared error of the estimate. In addition to depending on the bandwidth number \(m\), the estimate depends on an additional user-chosen number \(q\), but we show that for \(H \in (1/2, 3/4)\) there exists an optimal \(q\) for each \(H\), and we tabulate this.
MSC:
| 62M15 | Inference from stochastic processes and spectral analysis |
| 62E20 | Asymptotic distribution theory in statistics |
Keywords:
averaged periodogram estimation; spectral density near zero frequency; long memory time series; Monte Carlo; finite-sample distribution; asymptotic mean squared errorReferences:
| [1] | Davies, R. B.; Harte, D. S.: Tests for Hurst effect. Biometrika 74, 95-101 (1987) · Zbl 0612.62123 |
| [2] | Delgado, M.; Robinson, P. M.: Optimal spectral bandwidth for long memory time series. (1993) |
| [3] | Delgado, M.; Robinson, P. M.: New methods for the analysis of long memory time series: application to Spanish inflation. Journal of forecasting 13, 97-107 (1994) |
| [4] | Fox, R.; Taqqu, M. S.: Noncentral limit theorems for quadratic forms in random variables having long-range dependence. Annals of probability 13, 428-446 (1985) · Zbl 0569.60016 |
| [5] | Fox, R.; Taqqu, M. S.: Large-sample properties of parameter estimates for strongly dependent stationary Gaussian time series. Annals of statistics 14, 517-532 (1986) · Zbl 0606.62096 |
| [6] | Geweke, J.; Porter-Hudak, S.: The estimation and application of long memory time series models. Journal of time series analysis 4, 221-238 (1983) · Zbl 0534.62062 |
| [7] | Janacek, G. J.: Determining the degree of differencing for time series via the log spectrum. Journal of time series analysis 3, 177-183 (1982) |
| [8] | Lee, D. K. C.; Robinson, P. M.: Semiparametric exploration of long memory in stock prices. Journal of statistical planning and inference special issue in econometric methodology (1993) · Zbl 0848.62061 |
| [9] | Robinson, P. M.: Gaussian semiparametric estimation of long range dependence. (1993) · Zbl 0843.62092 |
| [10] | Robinson, P. M.: Semiparametric analysis of long-memory time series. Annals of statistics 22, 515-539 (1994) · Zbl 0795.62082 |
| [11] | Robinson, P. M.: Rates of convergence and optimal spectral bandwidth for long range dependence. Probability theory and related fields 99, 443-473 (1994) · Zbl 0801.60030 |
| [12] | Robinson, P. M.: Time series with strong dependence. Advances in econometrics: sixth world congress 1, 47-95 (1994) |
| [13] | Robinson, P. M.: Log-periodogram regression of time series with long range dependence. Annals of statistics 23, 1048-1072 (1995) · Zbl 0838.62085 |
| [14] | Sinai, Ya.G.: Self-similar probability distributions. Theory of probability and its applications 21, 64-80 (1976) · Zbl 0358.60031 |
| [15] | Taqqu, M. S.: Weak convergence to fractional Brownian motion and to the rosenblatt process. Zeitschrift für wahrscheinlichkeitstheorie und verwandte gebiete 31, 287-302 (1975) · Zbl 0303.60033 |
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