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Parametric estimation of the driving Lévy process of multivariate CARMA processes from discrete observations

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  • Brockwell, Peter J.
  • Schlemm, Eckhard

Abstract

We consider the parametric estimation of the driving Lévy process of a multivariate continuous-time autoregressive moving average (MCARMA) process, which is observed on the discrete time grid (0,h,2h,…). Beginning with a new state space representation, we develop a method to recover the driving Lévy process exactly from a continuous record of the observed MCARMA process. We use tools from numerical analysis and the theory of infinitely divisible distributions to extend this result to allow for the approximate recovery of unit increments of the driving Lévy process from discrete-time observations of the MCARMA process. We show that, if the sampling interval h=hN is chosen dependent on N, the length of the observation horizon, such that NhN converges to zero as N tends to infinity, then any suitable generalized method of moments estimator based on this reconstructed sample of unit increments has the same asymptotic distribution as the one based on the true increments, and is, in particular, asymptotically normally distributed.

Suggested Citation

  • Brockwell, Peter J. & Schlemm, Eckhard, 2013. "Parametric estimation of the driving Lévy process of multivariate CARMA processes from discrete observations," Journal of Multivariate Analysis, Elsevier, vol. 115(C), pages 217-251.
    • Handle: RePEc:eee:jmvana:v:115:y:2013:i:c:p:217-251
    DOI: 10.1016/j.jmva.2012.09.004
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    Cited by:

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    2. P. Brockwell, 2014. "Recent results in the theory and applications of CARMA processes," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 66(4), pages 647-685, August.
    3. Claudia Klüppelberg & Viet Son Pham, 2021. "Estimation of causal continuous‐time autoregressive moving average random fields," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 48(1), pages 132-163, March.
    4. Wang, Fangfang & Ma, Chunsheng, 2019. "ℓ1-symmetric vector random fields," Stochastic Processes and their Applications, Elsevier, vol. 129(7), pages 2466-2484.
    5. Robin Merkle & Andrea Barth, 2023. "On Properties and Applications of Gaussian Subordinated Lévy Fields," Methodology and Computing in Applied Probability, Springer, vol. 25(2), pages 1-33, June.
    6. Stefano Iacus & Lorenzo Mercuri, 2015. "Implementation of Lévy CARMA model in Yuima package," Computational Statistics, Springer, vol. 30(4), pages 1111-1141, December.
    7. Basse-O’Connor, Andreas & Nielsen, Mikkel Slot & Pedersen, Jan & Rohde, Victor, 2019. "Multivariate stochastic delay differential equations and CAR representations of CARMA processes," Stochastic Processes and their Applications, Elsevier, vol. 129(10), pages 4119-4143.
    8. Jorge Ignacio Gonz'alez C'azares & Aleksandar Mijatovi'c & Ger'onimo Uribe Bravo, 2018. "Geometrically Convergent Simulation of the Extrema of L\'{e}vy Processes," Papers 1810.11039, arXiv.org, revised Jun 2021.
    9. Fred Espen Benth & Marco Piccirilli & Tiziano Vargiolu, 2017. "Additive energy forward curves in a Heath-Jarrow-Morton framework," Papers 1709.03310, arXiv.org, revised Jun 2018.
    10. Benth, Fred Espen & Karbach, Sven, 2023. "Multivariate continuous-time autoregressive moving-average processes on cones," Stochastic Processes and their Applications, Elsevier, vol. 162(C), pages 299-337.
    11. Vicky Fasen & Florian Fuchs, 2013. "Spectral estimates for high-frequency sampled continuous-time autoregressive moving average processes," Journal of Time Series Analysis, Wiley Blackwell, vol. 34(5), pages 532-551, September.
    12. Péter Kevei, 2018. "Asymptotic moving average representation of high-frequency sampled multivariate CARMA processes," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 70(2), pages 467-487, April.
    13. Mayerhofer, Eberhard & Stelzer, Robert & Vestweber, Johanna, 2020. "Geometric ergodicity of affine processes on cones," Stochastic Processes and their Applications, Elsevier, vol. 130(7), pages 4141-4173.

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