A Construction of Continuous Time ARMA Models by Iterations of Ornstein-Uhlenbeck Processes

SORT-Statistics and Operations Research Transactions, 40 (2), 267-302, 2016

32 Pages Posted: 31 May 2018

See all articles by Argimiro Arratia

Argimiro Arratia

Polytechnic University of Catalonia (UPC)

Alejandra Cabaña

Autonomous University of Barcelona

Enrique Cabaña

affiliation not provided to SSRN

Date Written: July 22, 2016

Abstract

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.

Keywords: Ornstein-Uhlenbeck Process, Levy Process, Continuous ARMA, Stationary Process

JEL Classification: C32, C02

Suggested Citation

Arratia, Argimiro and Cabaña, Alejandra and Cabaña, Enrique, A Construction of Continuous Time ARMA Models by Iterations of Ornstein-Uhlenbeck Processes (July 22, 2016). SORT-Statistics and Operations Research Transactions, 40 (2), 267-302, 2016, Available at SSRN: https://ssrn.com/abstract=3180348

Argimiro Arratia (Contact Author)

Polytechnic University of Catalonia (UPC) ( email )

C. Jordi Girona, 31
Barcelona, 08034
Spain

Alejandra Cabaña

Autonomous University of Barcelona ( email )

Plaça Cívica
Cerdañola del Valles
Barcelona, Barcelona 08193
Spain

Enrique Cabaña

affiliation not provided to SSRN

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