IDEAS home Printed from https://ideas.repec.org/a/bla/jtsera/v42y2021i5-6p777-790.html
   My bibliography  Save this article

Aspects of non‐causal and non‐invertible CARMA processes

Author

Listed:
  • 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
    as

    Download full text from publisher

    File URL: https://doi.org/10.1111/jtsa.12589
    Download Restriction: no
    File URL: https://libkey.io/10.1111/jtsa.12589?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    References listed on IDEAS

    as
    1. Stefano Iacus & Lorenzo Mercuri, 2015. "Implementation of Lévy CARMA model in Yuima package," Computational Statistics, Springer, vol. 30(4), pages 1111-1141, December.
    2. 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.
    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. 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).
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    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.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Iacus, Stefano M. & Mercuri, Lorenzo & Rroji, Edit, 2017. "COGARCH(p, q): Simulation and Inference with the yuima Package," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 80(i04).
    2. Long, Hongwei & Ma, Chunhua & Shimizu, Yasutaka, 2017. "Least squares estimators for stochastic differential equations driven by small Lévy noises," Stochastic Processes and their Applications, Elsevier, vol. 127(5), pages 1475-1495.
    3. Asmerilda Hitaj & Lorenzo Mercuri & Edit Rroji, 2019. "Lévy CARMA models for shocks in mortality," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 42(1), pages 205-227, June.
    4. George Karabatsos, 2023. "Approximate Bayesian computation using asymptotically normal point estimates," Computational Statistics, Springer, vol. 38(2), pages 531-568, June.
    5. Sikora, Grzegorz & Michalak, Anna & Bielak, Łukasz & Miśta, Paweł & Wyłomańska, Agnieszka, 2019. "Stochastic modeling of currency exchange rates with novel validation techniques," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 523(C), pages 1202-1215.
    6. Ling, S. & McAleer, M.J. & Tong, H., 2015. "Frontiers in Time Series and Financial Econometrics," Econometric Institute Research Papers EI 2015-07, Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute.
    7. Yining Chen, 2015. "Semiparametric Time Series Models with Log-concave Innovations: Maximum Likelihood Estimation and its Consistency," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 42(1), pages 1-31, March.
    8. Shu, Yin & Feng, Qianmei & Liu, Hao, 2019. "Using degradation-with-jump measures to estimate life characteristics of lithium-ion battery," Reliability Engineering and System Safety, Elsevier, vol. 191(C).
    9. Martin Drapatz, 2016. "Strictly stationary solutions of spatial ARMA equations," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 68(2), pages 385-412, April.
    10. Pavel Kříž & Leszek Szała, 2020. "Least-Squares Estimators of Drift Parameter for Discretely Observed Fractional Ornstein–Uhlenbeck Processes," Mathematics, MDPI, vol. 8(5), pages 1-20, May.
    11. Oscar García, 2019. "Estimating reducible stochastic differential equations by conversion to a least-squares problem," Computational Statistics, Springer, vol. 34(1), pages 23-46, March.
    12. Ling, Shiqing & McAleer, Michael & Tong, Howell, 2015. "Frontiers in Time Series and Financial Econometrics: An overview," Journal of Econometrics, Elsevier, vol. 189(2), pages 245-250.
    13. Tanujit Chakraborty & Ashis Kumar Chakraborty & Munmun Biswas & Sayak Banerjee & Shramana Bhattacharya, 2021. "Unemployment Rate Forecasting: A Hybrid Approach," Computational Economics, Springer;Society for Computational Economics, vol. 57(1), pages 183-201, January.
    14. Jan Gairing & Peter Imkeller & Radomyra Shevchenko & Ciprian Tudor, 2020. "Hurst Index Estimation in Stochastic Differential Equations Driven by Fractional Brownian Motion," Journal of Theoretical Probability, Springer, vol. 33(3), pages 1691-1714, September.
    15. Driver, Charles C. & Oud, Johan H. L. & Voelkle, Manuel C., 2017. "Continuous Time Structural Equation Modeling with R Package ctsem," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 77(i05).
    16. Brandes, Dirk-Philip & Lindner, Alexander, 2014. "Non-causal strictly stationary solutions of random recurrence equations," Statistics & Probability Letters, Elsevier, vol. 94(C), pages 113-118.
    17. Peter Brockwell & Alexander Lindner & Bernd Vollenbröker, 2012. "Strictly stationary solutions of multivariate ARMA equations with i.i.d. noise," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 64(6), pages 1089-1119, December.
    18. Spangenberg, Felix, 2013. "Strictly stationary solutions of ARMA equations in Banach spaces," Journal of Multivariate Analysis, Elsevier, vol. 121(C), pages 127-138.
    19. Masahiro Kurisaki, 2023. "Parameter estimation for ergodic linear SDEs from partial and discrete observations," Statistical Inference for Stochastic Processes, Springer, vol. 26(2), pages 279-330, July.
    20. Cebrián, Ana C. & Abaurrea, Jesús & Asín, Jesús, 2015. "NHPoisson: An R Package for Fitting and Validating Nonhomogeneous Poisson Processes," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 64(i06).

    More about this item

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:bla:jtsera:v:42:y:2021:i:5-6:p:777-790. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Wiley Content Delivery (email available below). General contact details of provider: http://www.blackwellpublishing.com/journal.asp?ref=0143-9782 .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.