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Predictive Inference for Locally Stationary Time Series

  • Chapter
Model-Free Prediction and Regression

Part of the book series: Frontiers in Probability and the Statistical Sciences ((FROPROSTAS))

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Abstract

The Model-free Prediction Principle of Politis (Test 22(2):183–250, 2013) has been successfully applied to both regression problems, as well as problems involving stationary time series. However, with long time series, e.g., annual temperature measurements spanning over 100 years or daily financial returns spanning several years, it may be unrealistic to assume stationarity throughout the span of the dataset. In the paper at hand, we show how Model-free Prediction can be applied to handle time series that are only locally stationary, i.e., they can be modeled as stationary only over short time-windows. Both one-step-ahead point predictors and prediction intervals are constructed and compared.

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Notes

  1. 1.

    An alternative approach to prediction that does not rely on models such as (9.2) is given using wavelet representations of locally stationary processes; see, e.g., Fryzlewicz et al. (2003), and Antoniadis et al. (2006).

  2. 2.

    For simplicity, we assume that the bootstrap estimators \(\check{\mu }^{{\ast}}(\cdot )\) and \(\check{\sigma }^{{\ast}}(\cdot )\) are based on the same window width b used in the real world.

  3. 3.

    If the spectral density is equal to zero over an interval—however small—then the time series {Z t } is perfectly predictable based on its infinite past, and the same would be true for the time series {Y t }; see Brockwell and Davis (1991, Theorem 5.8.1) on Kolmogorov’s formula.

  4. 4.

    The joint normality of \(Z_{1},\ldots,Z_{n}\) follows immediately if one assumes that the stationary process {W t } appearing in Eq. (9.24) is Gaussian; but even without this additional assumption, it is difficult to construct examples where the joint normality of the Z t s may break down.

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Acknowledgements

Many thanks are due to Srinjoy Das and Stathis Paparoditis for helpful discussions.

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  1. Department of Mathematics, University of California, San Diego, La Jolla, CA, USA

    Dimitris N. Politis

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Politis, D.N. (2015). Predictive Inference for Locally Stationary Time Series. In: Model-Free Prediction and Regression. Frontiers in Probability and the Statistical Sciences. Springer, Cham. https://doi.org/10.1007/978-3-319-21347-7_9

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