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Wasserstein distance bounds on the normal approximation of empirical autocovariances and cross‐covariances under non‐stationarity and stationarity

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  • Andreas Anastasiou
  • Tobias Kley

Abstract

The autocovariance and cross‐covariance functions naturally appear in many time series procedures (e.g. autoregression or prediction). Under assumptions, empirical versions of the autocovariance and cross‐covariance are asymptotically normal with covariance structure depending on the second‐ and fourth‐order spectra. Under non‐restrictive assumptions, we derive a bound for the Wasserstein distance of the finite‐sample distribution of the estimator of the autocovariance and cross‐covariance to the Gaussian limit. An error of approximation to the second‐order moments of the estimator and an m‐dependent approximation are the key ingredients to obtain the bound. As a worked example, we discuss how to compute the bound for causal autoregressive processes of order 1 with different distributions for the innovations. To assess our result, we compare our bound to Wasserstein distances obtained via simulation.

Suggested Citation

  • Andreas Anastasiou & Tobias Kley, 2024. "Wasserstein distance bounds on the normal approximation of empirical autocovariances and cross‐covariances under non‐stationarity and stationarity," Journal of Time Series Analysis, Wiley Blackwell, vol. 45(3), pages 361-375, May.
  • Handle: RePEc:bla:jtsera:v:45:y:2024:i:3:p:361-375
    DOI: 10.1111/jtsa.12716
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    References listed on IDEAS

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    1. Jirak, Moritz, 2014. "Simultaneous confidence bands for sequential autoregressive fitting," Journal of Multivariate Analysis, Elsevier, vol. 124(C), pages 130-149.
    2. Anastasiou, Andreas, 2017. "Bounds for the normal approximation of the maximum likelihood estimator from m-dependent random variables," Statistics & Probability Letters, Elsevier, vol. 129(C), pages 171-181.
    3. Kley, Tobias & Preuss, Philip & Fryzlewicz, Piotr, 2019. "Predictive, finite-sample model choice for time series under stationarity and non-stationarity," LSE Research Online Documents on Economics 101748, London School of Economics and Political Science, LSE Library.
    4. Anastasiou, Andreas, 2017. "Bounds for the normal approximation of the maximum likelihood estimator from m -dependent random variables," LSE Research Online Documents on Economics 83635, London School of Economics and Political Science, LSE Library.
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