## Time Series: Theory and MethodsThis edition contains a large number of additions and corrections scattered throughout the text, including the incorporation of a new chapter on state-space models. The companion diskette for the IBM PC has expanded into the software package ITSM: An Interactive Time Series Modelling Package for the PC, which includes a manual and can be ordered from Springer-Verlag. * We are indebted to many readers who have used the book and programs and made suggestions for improvements. Unfortunately there is not enough space to acknowledge all who have contributed in this way; however, special mention must be made of our prize-winning fault-finders, Sid Resnick and F. Pukelsheim. Special mention should also be made of Anthony Brockwell, whose advice and support on computing matters was invaluable in the preparation of the new diskettes. We have been fortunate to work on the new edition in the excellent environments provided by the University of Melbourne and Colorado State University. We thank Duane Boes particularly for his support and encouragement throughout, and the Australian Research Council and National Science Foundation for their support of research related to the new material. We are also indebted to Springer-Verlag for their constant support and assistance in preparing the second edition. Fort Collins, Colorado P. J. BROCKWELL November, 1990 R. A. DAVIS * /TSM: An Interactive Time Series Modelling Package for the PC by P. J. Brockwell and R. A. Davis. ISBN: 0-387-97482-2; 1991. |

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### Contents

Estimation of the Mean and the Autocovariance | |

Problems | |

2 5Projection | |

2 8Fourier Series 2 9 Hilbert Space Isomorphisms | |

Problems Chapter | |

Problems | |

8 6Recursive Calculationof theLikelihood of an Arbitrary | |

Estimators | |

ARMA Processes | |

9 2Identification Techniques | |

9 3Order Selection | |

9 4Diagnostic Checking | |

3 2Moving Average Processes of Infinite Order | |

3 3Computing the Autocovariance Functionof an ARMApq Process | |

The Spectral Representation of a Stationary Process | |

Process | |

Problems | |

Chapter 5Prediction of Stationary Processes | |

5 3Recursive Prediction of an ARMAp q Process | |

Problems | |

Multivariate Time Series | |

Problems | |

Problems | |

Chapter 13Further Topics | |

Problems | |

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### Common terms and phrases

absolutely summable approximation AR(p ARMA process ARMA(p asymptotic autocorrelation function autocovariance function autoregressive canbe causal coefficients complexvalued compute converges Corollary covariance matrix defined by image differencing Example Figure filter function F Gaussian Hence Hilbert space Ifimage image and image image image image Show image where image independent inner product innerproduct space integers inthe invertible isomorphism isthe Kolmogorov’s MA(q maximum likelihood mean squared error moving average multivariate normal multivariate normal distribution nonnegative definite nonzero numbers observations obtain ofthe onestep orthogonal orthogonalincrement process orthonormal partial autocorrelation polynomial Problem process defined process image projection theorem PROOF Proposition random variables random vector realvalued recursively Remark ROOF satisfying Section sequence show that image spectral density spectral distribution function spectral representation stationary process stationary solution stochastic uncorrelated unique values variance white noise