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Asymptotic moving average representation of high-frequency sampled multivariate CARMA processes

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  • Péter Kevei

    (Technische Universität München)

Abstract

High-frequency sampled multivariate continuous time autoregressive moving average processes are investigated. We obtain asymptotic expansion for the spectral density of the sampled MCARMA process $$(Y_{n\varDelta })_{n \in {\mathbb {Z}}}$$ ( Y n Δ ) n ∈ Z as $$\varDelta \downarrow 0$$ Δ ↓ 0 , where $$(Y_t)_{t \in {\mathbb {R}}}$$ ( Y t ) t ∈ R is an MCARMA process. We show that the properly filtered process is a vector moving average process, and determine the asymptotic moving average representation of it, thus generalizing the univariate results to the multivariate model. The determination of the moving average representation of the filtered process, important for the analysis of high-frequency data, is difficult for any fixed positive $$\varDelta $$ Δ . However, the results established here provide a useful and insightful approximation when $$\varDelta $$ Δ is very small.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:aistmt:v:70:y:2018:i:2:d:10.1007_s10463-017-0601-5
    DOI: 10.1007/s10463-017-0601-5
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    1. Marquardt, Tina, 2007. "Multivariate fractionally integrated CARMA processes," Journal of Multivariate Analysis, Elsevier, vol. 98(9), pages 1705-1725, October.
    2. Todorov, Viktor & Tauchen, George, 2006. "Simulation Methods for Levy-Driven Continuous-Time Autoregressive Moving Average (CARMA) Stochastic Volatility Models," Journal of Business & Economic Statistics, American Statistical Association, vol. 24, pages 455-469, October.
    3. Benth, Fred Espen & Klüppelberg, Claudia & Müller, Gernot & Vos, Linda, 2014. "Futures pricing in electricity markets based on stable CARMA spot models," Energy Economics, Elsevier, vol. 44(C), pages 392-406.
    4. Robinson, Peter M., 1977. "The construction and estimation of continuous time models and discrete approximations in econometrics," Journal of Econometrics, Elsevier, vol. 6(2), pages 173-197, September.
    5. Peter J. Brockwell & Vincenzo Ferrazzano & Claudia Klüppelberg, 2012. "High‐frequency sampling of a continuous‐time ARMA process," Journal of Time Series Analysis, Wiley Blackwell, vol. 33(1), pages 152-160, January.
    6. P. Brockwell, 2001. "Lévy-Driven Carma Processes," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 53(1), pages 113-124, March.
    7. Arne Andresen & Fred Espen Benth & Steen Koekebakker & Valeriy Zakamulin, 2014. "The Carma Interest Rate Model," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 17(02), pages 1-27.
    8. Marquardt, Tina & Stelzer, Robert, 2007. "Multivariate CARMA processes," Stochastic Processes and their Applications, Elsevier, vol. 117(1), pages 96-120, January.
    9. Bergstrom, A. R., 1988. "The History of Continuous-Time Econometric Models," Econometric Theory, Cambridge University Press, vol. 4(3), pages 365-383, December.
    10. 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.
    11. Peter J. Brockwell & Vincenzo Ferrazzano & Claudia Klüppelberg, 2013. "High-frequency sampling and kernel estimation for continuous-time moving average processes," Journal of Time Series Analysis, Wiley Blackwell, vol. 34(3), pages 385-404, May.
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    Cited by:

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