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**Exact local Whittle estimation of fractional integration.**
*(English)*
Zbl 1081.62069

From the introduction: Semiparametric estimation of the memory parameter \((d)\) in fractionally integrated \((I(d))\) time series is appealing in empirical work because of the general treatment of the short-memory component that it affords. Two common statistical procedures in this class are log-periodogram (LP) regression [see P. M. Robinson, Ann. Stat. 23, No. 3, 1048–1072 (1995; Zbl 0838.62085)] and local Whittle (LW) estimation [see P. M. Robinson, ibid. No. 5, 1630–1661 (1995; Zbl 0843.62092)]. LW estimation is known to be more efficient than LP regression in the stationary \((|d|<1/2)\) case, although numerical optimization methods are needed in the calculations. Outside the stationary region, it is known that the asymptotic theory for the LW estimator is discontinuous at \(d=3/4\) and again at \(d=1\), is awkward to use because of nonnormal limit theory and, worst of all, the estimator is inconsistent when \(d>1\). Thus, the LW estimator is not a good general-purpose estimator when the value of \(d\) may take on values in the nonstationary zone beyond \(3/4\). Similar comments apply in the case of LP estimation.

To extend the range of applications of these semiparametric methods, data differencing and data tapering have been suggested. These methods have the advantage that they are easy to implement and they make use of existing algorithms once the data filtering has been carried out. Differencing has the disadvantage that prior information is needed on the appropriate order of differencing. Tapering has the disadvantage that the filter distorts the trajectory of the data and inflates the asymptotic variance. As a consequence, there is presently no general-purpose efficient estimation procedure when the value of \(d\) may take on values in the nonstationary zone beyond \(3/4\).

The present paper studies an exact form of the local Whittle estimator which does not rely on differencing or tapering and which seems to offer a good general-purpose estimation procedure for the memory parameter that applies throughout the stationary and nonstationary regions of \(d\). The estimator, which we call the exact LW estimator, is shown to be consistent and to have \(N(0, 1/4)\) limit distribution when the optimization covers an interval of width less than \(9/2\). The exact LW estimator therefore has the same limit theory as the LW estimator has for stationary values of \(d\). The approach seems to offer a useful alternative for applied researchers who are looking for a general-purpose estimator and want to allow for a substantial range of stationary and nonstationary possibilities for \(d\). The method has the further advantage that it provides a basis for constructing asymptotic confidence intervals for \(d\) that are valid irrespective of the true value of the memory parameter.

The exact LW estimator given here assumes the initial value of the data to be known. This restriction can be removed by estimating it along with \(d\). Also, computation of the estimator involves a numerical optimization that is more demanding than conventional LW estimation. Our experience from simulations indicates that the computation time required is about ten times that of the LW estimator and is well within the capabilities of a small notebook computer.

To extend the range of applications of these semiparametric methods, data differencing and data tapering have been suggested. These methods have the advantage that they are easy to implement and they make use of existing algorithms once the data filtering has been carried out. Differencing has the disadvantage that prior information is needed on the appropriate order of differencing. Tapering has the disadvantage that the filter distorts the trajectory of the data and inflates the asymptotic variance. As a consequence, there is presently no general-purpose efficient estimation procedure when the value of \(d\) may take on values in the nonstationary zone beyond \(3/4\).

The present paper studies an exact form of the local Whittle estimator which does not rely on differencing or tapering and which seems to offer a good general-purpose estimation procedure for the memory parameter that applies throughout the stationary and nonstationary regions of \(d\). The estimator, which we call the exact LW estimator, is shown to be consistent and to have \(N(0, 1/4)\) limit distribution when the optimization covers an interval of width less than \(9/2\). The exact LW estimator therefore has the same limit theory as the LW estimator has for stationary values of \(d\). The approach seems to offer a useful alternative for applied researchers who are looking for a general-purpose estimator and want to allow for a substantial range of stationary and nonstationary possibilities for \(d\). The method has the further advantage that it provides a basis for constructing asymptotic confidence intervals for \(d\) that are valid irrespective of the true value of the memory parameter.

The exact LW estimator given here assumes the initial value of the data to be known. This restriction can be removed by estimating it along with \(d\). Also, computation of the estimator involves a numerical optimization that is more demanding than conventional LW estimation. Our experience from simulations indicates that the computation time required is about ten times that of the LW estimator and is well within the capabilities of a small notebook computer.

### MSC:

62M10 | Time series, auto-correlation, regression, etc. in statistics (GARCH) |

62E20 | Asymptotic distribution theory in statistics |

65C60 | Computational problems in statistics (MSC2010) |

### Keywords:

discrete Fourier transform; long memory; nonstationarity; semiparametric estimation; Whittle likelihood
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\textit{K. Shimotsu} and \textit{P. C. B. Phillips}, Ann. Stat. 33, No. 4, 1890--1933 (2005; Zbl 1081.62069)

### References:

[1] | Geweke, J. and Porter-Hudak, S. (1983). The estimation and application of long memory time series models. J. Time Ser. Anal. 4 221–238. · Zbl 0534.62062 |

[2] | Henry, M. and Robinson, P. M. (1996). Bandwidth choice in Gaussian semiparametric estimation of long range dependence. Athens Conference on Applied Probability and Time Series. Lecture Notes in Statist. 115 220–232. Springer, New York. |

[3] | Hurvich, C. M. and Chen, W. W. (2000). An efficient taper for potentially overdifferenced long-memory time series. J. Time Ser. Anal. 21 155–180. · Zbl 0958.62085 |

[4] | Kim, C. S. and Phillips, P. C. B. (1999). Log periodogram regression: The nonstationary case. Mimeographed, Cowles Foundation, Yale Univ. |

[5] | Künsch, H. (1987). Statistical aspects of self-similar processes. In Proc. First World Congress of the Bernoulli Society (Yu. Prokhorov and V. V. Sazanov, eds.) 1 67–74. VNU Science Press, Utrecht. · Zbl 0673.62073 |

[6] | Marinucci, D. and Robinson, P. M. (1999). Alternative forms of fractional Brownian motion. J. Statist. Plann. Inference 80 111–122. · Zbl 0934.60071 |

[7] | Phillips, P. C. B. (1999). Discrete Fourier transforms of fractional processes. Cowles Foundation Discussion Paper #1243, Yale Univ. Available at cowles.econ.yale.edu. |

[8] | Phillips, P. C. B. and Shimotsu, K. (2004). Local Whittle estimation in nonstationary and unit root cases. Ann. Statist. 32 656–692. · Zbl 1091.62084 |

[9] | Phillips, P. C. B. and Solo, V. (1992). Asymptotics for linear processes. Ann. Statist. 20 971–1001. JSTOR: · Zbl 0759.60021 |

[10] | Robinson, P. M. (1995). Log-periodogram regression of time series with long-range dependence. Ann. Statist. 23 1048–1072. JSTOR: · Zbl 0838.62085 |

[11] | Robinson, P. M. (1995). Gaussian semiparametric estimation of long-range dependence. Ann. Statist. 23 1630–1661. JSTOR: · Zbl 0843.62092 |

[12] | Robinson, P. M. (2005). The distance between rival nonstationary fractional processes. J. Econometrics . · Zbl 1335.62143 |

[13] | Robinson, P. M. and Marinucci, D. (2001). Narrow-band analysis of nonstationary processes. Ann. Statist. 29 947–986. · Zbl 1012.62100 |

[14] | Shimotsu, K. (2004). Exact local Whittle estimation of fractional integration with unknown mean and time trend. Mimeographed, Queen’s Univ., Kingston. |

[15] | Velasco, C. (1999). Gaussian semiparametric estimation of non-stationary time series. J. Time Ser. Anal. 20 87–127. · Zbl 0922.62093 |

[16] | Zygmund, A. (1977). Trigonometric Series. Cambridge Univ. Press. · Zbl 0367.42001 |

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