 Aspects of non‐causal and non‐invertible CARMA processesPeter J. Brockwell and Alexander Lindner Journal of Time Series Analysis, 2021, vol. 42, issue 5-6, 777-790 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. Date: 2021 Downloads: (external link) Related works: Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text Persistent link: https://EconPapers.repec.org/RePEc:bla:jtsera:v:42:y:2021:i:5-6:p:777-790 Ordering information: This journal article can be ordered from Access Statistics for this article Journal of Time Series Analysis is currently edited by M.B. Priestley More articles in Journal of Time Series Analysis from Wiley Blackwell |
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