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Aspects of non‐causal and non‐invertible CARMA processes

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  • Peter J. Brockwell
  • Alexander Lindner

Abstract

A CARMA(p, q) process Y is a strictly stationary solution Y of the pth‐order formal stochastic differential equation a(D)Yt = b(D)DLt, where L is a two‐sided Lévy process, a(z) and b(z) are polynomials of degrees p and q respectively, with p > q, and D denotes differentiation with respect to t. Using a state‐space formulation of the defining equation, Brockwell and Lindner (2009, Stochastic Processes and their Applications 119, 2660–2681) gave necessary and sufficient conditions on L, a(z) and b(z) for the existence and uniqueness of such a stationary solution and specified the kernel g in the representation of the solution as Yt=∫−∞∞g(t−u)dLu. If the zeros of a(z) all have strictly negative real parts, Y is said to be a causal function of L (or simply causal) since then Yt can be expressed in terms of the increments of Ls, s ≤ t, and if the zeros of b(z) all have strictly negative real parts the process is said to be invertible since then the increments of Ls, s ≤ t, can be expressed in terms of Ys, s ≤ t. In this article we are concerned with properties of CARMA processes for which these conditions on a and b do not necessarily hold.

Suggested Citation

  • Peter J. Brockwell & Alexander Lindner, 2021. "Aspects of non‐causal and non‐invertible CARMA processes," Journal of Time Series Analysis, Wiley Blackwell, vol. 42(5-6), pages 777-790, September.
    • Handle: RePEc:bla:jtsera:v:42:y:2021:i:5-6:p:777-790
    DOI: 10.1111/jtsa.12589
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    1. Wyłomańska, Agnieszka & Gajda, Janusz, 2016. "Stable continuous-time autoregressive process driven by stable subordinator," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 444(C), pages 1012-1026.
    2. Brouste, Alexandre & Fukasawa, Masaaki & Hino, Hideitsu & Iacus, Stefano & Kamatani, Kengo & Koike, Yuta & Masuda, Hiroki & Nomura, Ryosuke & Ogihara, Teppei & Shimuzu, Yasutaka & Uchida, Masayuki & Y, 2014. "The YUIMA Project: A Computational Framework for Simulation and Inference of Stochastic Differential Equations," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 57(i04).
    3. Peter J. Brockwell & Alexander Lindner, 2010. "Strictly stationary solutions of autoregressive moving average equations," Biometrika, Biometrika Trust, vol. 97(3), pages 765-772.
    4. Brockwell, Peter J. & Lindner, Alexander, 2015. "Prediction of Lévy-driven CARMA processes," Journal of Econometrics, Elsevier, vol. 189(2), pages 263-271.
    5. Stefano Iacus & Lorenzo Mercuri, 2015. "Implementation of Lévy CARMA model in Yuima package," Computational Statistics, Springer, vol. 30(4), pages 1111-1141, December.
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    Cited by:

    1. 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.

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