### Abstract

Let [X_{t}] be a stationary process with spectral density g(λ.).It is often that the true structure g(λ) is not completely specified. This paper discusses the problem of misspecified prediction when a conjectured spectral density f_{θ}(λ), θ ε Θ, is fitted to g(λ). Then, constructing the best linear predictor based on f_{θ}(λ), we can evaluate the prediction error M(θ) Since θ is unknown we estimate it by a quasi-MLE θ_{Q}. The second-order asymptotic approximation of M(θ̂_{Q})is given. This result is extended to the case when X_{t} contains some trend, i.e. a time series regression model. These results are very general. Furthermore we evaluate the second-order asymptotic approximation of M(θ̂_{Q}) for a time series regression model having a long-memory residual process with the true spectral density g(λ). Since the general formulae of the approximated prediction error are complicated, we provide some numerical examples. Then we illuminate unexpected effects from the misspecification of spectra.

Original language | English |
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Pages (from-to) | 543-564 |

Number of pages | 22 |

Journal | Journal of Forecasting |

Volume | 20 |

Issue number | 8 |

DOIs | |

Publication status | Published - 2001 Dec 1 |

Externally published | Yes |

### Keywords

- Best linear predictor
- Conjectured spectral density
- Long-memory process
- Misspecified prediction
- Multistep prediction
- Quasi-MLE
- Spectral density
- Stationary process
- Time series regression model

### ASJC Scopus subject areas

- Modelling and Simulation
- Computer Science Applications
- Strategy and Management
- Statistics, Probability and Uncertainty
- Management Science and Operations Research

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## Cite this

*Journal of Forecasting*,

*20*(8), 543-564. https://doi.org/10.1002/for.807