A construction of continuous-time ARMA models by iterations of Ornstein-Uhlenbeck processes

• Argimiro Arratia ^[1] ; Alejandra Cabaña ^[2] ; Enrique M. Cabaña ^[3]
  1. [1] Universitat Politècnica de Catalunya

     Universitat Politècnica de Catalunya

     Barcelona, España

     
  2. [2] Universitat Autònoma de Barcelona

     Universitat Autònoma de Barcelona

     Barcelona, España

     
  3. [3] Universidad de la República

     Universidad de la República

     Uruguay

     

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• Localización: Sort: Statistics and Operations Research Transactions, ISSN 1696-2281, Vol. 40, Nº. 2, 2016, págs. 267-302
• Idioma: inglés
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• Resumen

  • We present a construction of a family of continuous-time ARMA processes based on p iterations of the linear operator that maps a Lévy process onto an Ornstein-Uhlenbeck process. The construction resembles the procedure to build an AR(p) from an AR(1). We show that this family is in fact a subfamily of the well-known CARMA(p,q) processes, with several interesting advantages, including a smaller number of parameters. The resulting processes are linear combinations of Ornstein-Uhlenbeck processes all driven by the same Lévy process. This provides a straightforward computation of covariances, a state-space model representation and methods for estimating parameters. Furthermore, the discrete and equally spaced sampling of the process turns to be an ARMA(p, p−1) process. We propose methods for estimating the parameters of the iterated Ornstein-Uhlenbeck process when the noise is either driven by a Wiener or a more general Lévy process, and show simulations and applications to real data.

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