## Predictions in Time Series Using Regression ModelsBooks on time series models deal mainly with models based on Box-Jenkins methodology which is generally represented by autoregressive integrated moving average models or some nonlinear extensions of these models, such as generalized autoregressive conditional heteroscedasticity models. Statistical inference for these models is well developed and commonly used in practical applications, due also to statistical packages containing time series analysis parts. The present book is based on regression models used for time series. These models are used not only for modeling mean values of observed time se ries, but also for modeling their covariance functions which are often given parametrically. Thus for a given finite length observation of a time series we can write the regression model in which the mean value vectors depend on regression parameters and the covariance matrices of the observation depend on variance-covariance parameters. Both these dependences can be linear or nonlinear. The aim of this book is to give an unified approach to the solution of statistical problems for such time series models, and mainly to problems of the estimation of unknown parameters of models and to problems of the prediction of time series modeled by regression models. |

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### Contents

Hilbert Spaces and Statistics | 1 |

12 Preliminaries from Statistics | 6 |

13 Estimation of Parameters | 12 |

14 Double Least Squares Estimators | 24 |

15 Invariant Quadratic Estimators | 30 |

16 Unbiased Invariant Estimators | 37 |

Random Processes and Time Series | 51 |

22 Models for Random Processes | 53 |

34 Maximum Likelihood Estimation | 118 |

Predictions of Time Series | 147 |

42 Predictions in Linear Models | 149 |

43 Model Choice and Predictions | 165 |

44 Predictions in Multivariate Models | 179 |

45 Predictions in Nonlinear Models | 189 |

Empirical Predictors | 197 |

52 Properties of Empirical Predictors | 198 |

23 Spectral Theory | 61 |

24 Models for Time Series | 65 |

Estimation of Time Series Parameter | 73 |

32 Estimation of Mean Value Parameters | 74 |

33 Estimation of a Covariance Function | 99 |

53 Numerical Examples | 208 |

223 | |

229 | |

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### Common terms and phrases

approximation AR(p arg min arithmetic mean assumption asymptotic BLUP X*(n Box-Jenkins called computing the MLE Cov(X covariance function covariance matrix covariance stationary defined denotes depend derived design matrix DOOLSE DOWELSE empirical predictor estimated mean value Example expression fc=l finite observation following theorem Fourier frequencies function g Gaussian given Hilbert space invariant quadratic estimator invariant quadratic form iterations known least squares residuals Let us assume Let us consider lim R(t linear trend matrix F mean value function method mixed LRM MLRM n x n matrix nonlinear nonlinear regression normal distribution notation observed time series OLSE ß ordinary least squares orthogonal periodogram polynomial predictor U*(X problem properties random process random variables random vector regression parameters sin Ai spectral densities stationary time series statistical symmetric unbiased estimator unbiased invariant quadratic uniformly best unknown parameter vector with components weighted residuals WELSE white noise write