## Time Series with Long MemoryLong memory processes constitute a broad class of models for stationary and nonstationary time series data in economics, finance, and other fields. Their key feature is persistence, with high correlation between events that are remote in time. A single 'memory' parameter economically indexesthis persistence, as part of a rich parametric or nonparametric structure for the process. Unit root processes can be covered, along with processes that are stationary but with stronger persistence than autoregressive moving averages, these latter being included in a broader class which describesboth short memory and negative memory. Long memory processes have in recent years attracted considerable interest from both theoretical and empirical researchers in time series and econometrics.This book of readings collects articles on a variety of topics in long memory time series including modelling and statistical inference for stationary processes, stochastic volatility models, nonstationary processes, and regression and fractional cointegration models. Some of the articles are highlytheoretical, others contain a mix of theory and methods, and an effort has been made to include empirical applications of the main approaches covered. A review article introduces the other articles but also attempts a broader survey, traces the history of the subject, and includes a bibliography. |

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### Contents

Introduction | 1 |

References | 25 |

On Largesample Estimation for the Mean | 33 |

An Introduction to Longmemory Time Series Models | 49 |

1 The d 0 Case | 63 |

Longterm Memory in Stock Market Prices | 82 |

Notes | 111 |

The Estimation and Application of Longmemory Time | 119 |

1 Proof of Strong Consistency for Spectrallikelihood | 209 |

1 Derivation of Score Statistic R | 238 |

4 Proof of Theorem 3 | 245 |

Estimation of the Memory Parameter for Nonstationary | 251 |

1 | 269 |

References | 276 |

References | 304 |

References | 332 |

References | 136 |

References | 174 |

Testing for Strong Serial Correlation and Dynamic Conditional | 175 |

References | 189 |

1 | 361 |

References | 371 |

### Common terms and phrases

alternatives Annals of Statistics applied approximation ARMA Assumption asymptotic normality asymptotic theory autocorrelation autocovariance autoregressive behaviour bias bounded central limit theorem coefficients cointegration computed covariance defined denote Econometrica Economic efficiency Eicker empirical example finite finite-sample follows forecasts Fourier frequencies Fox and Taqqu fractional differencing fractional Gaussian noise fractionally integrated function GARCH Geweke GPH estimator GPHT Hannan implies integrated series Journal of Econometrics least squares Lemma limit distribution linear log periodogram long-memory long-memory models long-memory time series long-range dependence Mandelbrot matrix mean square error Monte Carlo nonlinear nonstationary null hypothesis periodogram polynomial Porter-Hudak Proof of Theorem properties random variables re/ect regression rescaled range Robinson sample satisfies Section semiparametric sequence Series Analysis series models short-memory short-range dependence simulation spectral density standard stationary stochastic volatility Table tapered unit root values variance vector white noise Whittle estimates zero