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

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