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Empirical spectral processes for stationary state space models

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  • Fasen-Hartmann, Vicky
  • Mayer, Celeste

Abstract

In this paper, we consider function-indexed normalized weighted integrated periodograms for equidistantly sampled multivariate continuous-time state space models which are multivariate continuous-time ARMA processes. Thereby, the sampling distance is fixed and the driving Lévy process has at least a finite fourth moment. Under different assumptions on the function space and the moments of the driving Lévy process we derive a central limit theorem for the function-indexed normalized weighted integrated periodogram. Either the assumption on the function space or the assumption on the existence of moments of the Lévy process is weaker. Furthermore, we show the weak convergence in both the space of continuous functions and in the dual space to a Gaussian process and give an explicit representation of the covariance function. The results can be used to derive the asymptotic behavior of the Whittle estimator and to construct goodness-of-fit test statistics as the Grenander–Rosenblatt statistic and the Cramér–von Mises statistic. We present the exact limit distributions of both statistics and show their performance through a simulation study.

Suggested Citation

  • Fasen-Hartmann, Vicky & Mayer, Celeste, 2023. "Empirical spectral processes for stationary state space models," Stochastic Processes and their Applications, Elsevier, vol. 155(C), pages 319-354.
    • Handle: RePEc:eee:spapps:v:155:y:2023:i:c:p:319-354
    DOI: 10.1016/j.spa.2022.10.008
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    References listed on IDEAS

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    1. Vicky Fasen-Hartmann & Celeste Mayer, 2022. "Whittle estimation for continuous-time stationary state space models with finite second moments," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 74(2), pages 233-270, April.
    2. Dahlhaus, Rainer, 1988. "Empirical spectral processes and their applications to time series analysis," Stochastic Processes and their Applications, Elsevier, vol. 30(1), pages 69-83, November.
    3. Thornton, Michael A. & Chambers, Marcus J., 2017. "Continuous time ARMA processes: Discrete time representation and likelihood evaluation," Journal of Economic Dynamics and Control, Elsevier, vol. 79(C), pages 48-65.
    4. Marquardt, Tina & Stelzer, Robert, 2007. "Multivariate CARMA processes," Stochastic Processes and their Applications, Elsevier, vol. 117(1), pages 96-120, January.
    5. Mikosch, T. & Norvaisa, R., 1997. "Uniform convergence of the empirical spectral distribution function," Stochastic Processes and their Applications, Elsevier, vol. 70(1), pages 85-114, October.
    6. Can, S.U. & Mikosch, T. & Samorodnitsky, G., 2010. "Weak Convergence of the function-indexed integrated periodogram for infinite variance processes," Other publications TiSEM 3be90f1b-2f53-4987-b46e-c, Tilburg University, School of Economics and Management.
    7. Marquardt, Tina, 2007. "Multivariate fractionally integrated CARMA processes," Journal of Multivariate Analysis, Elsevier, vol. 98(9), pages 1705-1725, October.
    8. Kokoszka, Piotr & Mikosch, Thomas, 2000. "The periodogram at the Fourier frequencies," Stochastic Processes and their Applications, Elsevier, vol. 86(1), pages 49-79, March.
    9. Jean‐Marc Bardet & Paul Doukhan & José Rafael León, 2008. "Uniform limit theorems for the integrated periodogram of weakly dependent time series and their applications to Whittle's estimate," Journal of Time Series Analysis, Wiley Blackwell, vol. 29(5), pages 906-945, September.
    10. Kokoszka, P. & Mikosch, T., 1997. "The integrated periodogram for long-memory processes with finite or infinite variance," Stochastic Processes and their Applications, Elsevier, vol. 66(1), pages 55-78, February.
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