## Periodic Time Series ModelsThis book considers periodic time series models for seasonal data, characterized by parameters that differ across the seasons, and focuses on their usefulness for out-of-sample forecasting. Providing an up-to-date survey of the recent developments in periodic time series, the book presents a large number of empirical results. The first part of the book deals with model selection, diagnostic checking and forecasting of univariate periodic autoregressive models. Tests for periodic integration, are discussed, and an extensive discussion of the role of deterministic regressors in testing for periodic integration and in forecasting is provided. The second part discusses multivariate periodic autoregressive models. It provides an overview of periodic cointegration models, as these are the most relevant. This overview contains single-equation type tests and a full-system approach based on generalized method of moments. All methods are illustrated with extensive examples, and the book will be of interest to advanced graduate students and researchers in econometrics, as well as practitioners looking for an understanding of how to approach seasonal data. |

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### Contents

Introduction | 1 |

12 Readership | 3 |

13 Why periodic models? | 5 |

14 Outline of this book | 8 |

Properties of seasonal time series | 11 |

22 Typical features of seasonal time series | 17 |

23 Summary and outlook | 25 |

Univariate periodic time series models | 27 |

43 Testing for unit roots | 77 |

44 Forecasting trending time series | 92 |

45 Effects of neglecting periodicity | 97 |

46 Conclusion | 99 |

Multivariate periodic time series models | 103 |

51 Notation and representation | 104 |

52 Useful representations in practice | 108 |

53 Cointegration testing single equation approach | 111 |

31 Representation | 28 |

32 Stationarity in periodic autoregressions | 34 |

33 Model selection and parameter estimation | 39 |

34 Forecasting | 46 |

35 Effects of neglecting periodicity | 48 |

36 Periodic conditional heteroskedasticity | 54 |

37 Discussion | 58 |

Periodic models for trending data | 61 |

41 Representation of unit roots | 64 |

42 Intercepts and deterministic trends | 72 |

54 Cointegration testing fullsystem approach | 117 |

55 Discussion | 122 |

Critical values of the Dickey and Fuller statistics | 125 |

Critical values of the Johansen trace statistics | 126 |

Critical values of the Boswijk and Franses statistic | 129 |

References | 131 |

Author Index | 141 |

145 | |

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### Common terms and phrases

ahead forecasts analysis analyze autocorrelation autoregressive parameters Boswijk and Franses chapter characteristic equation cointegration relations consider corresponds covariance matrix critical values denoted discuss Econometrics economic time series equal error correction model EViews example F-test first-order Food and tobacco fourteen series Ghysels given Hence Hylleberg implies Johansen trace statistics lag order likelihood ratio test linear deterministic trends LM-test matrix method non-periodic models non-seasonal nonlinear notation null hypothesis number of unit Osborn PAR(l PAR(p parameter estimates periodic autoregression periodic data periodic differencing filter periodic integration periodic models periodic time series periodically-integrated quarterly data quarters rank regression relevant Schwarz criterion seasonal adjustment seasonal dummies seasonal heteroskedasticity seasonal time series seasonal unit roots seasonal variation Section serial correlation series models series yt simulated stationary stochastic trends Table test for periodic test statistic Testing for unit univariate variance vector vector autoregressive vector process VQ models VQ representation Wald test Yt process yt series