  An Ornstein-Uhlenbeck (OU) process can be considered as a continuous time
interpolation of the discrete time AR$(1)$ process. Departing from this fact,
we analyse in this work the effect of iterating OU treated as a linear operator
that maps a Wiener process onto Ornstein-Uhlenbeck process, so as to build a
family of higher order Ornstein-Uhlenbeck processes, OU$(p)$, in a similar
spirit as the higher order autoregressive processes AR$(p)$. We show that for
$p \ge 2$ we obtain in general a process with covariances different than those
of an AR$(p)$, and that for various continuous time processes, sampled from
real data at equally spaced time instants, the OU$(p)$ model outperforms the
appropriate AR$(p)$ model.
  Technically our composition of the OU operator is easy to manipulate and its
parameters can be computed efficiently because, as we show, the iteration of OU
operators leads to a process that can be expressed as a linear combination of
basic OU processes. Using this expression we obtain a closed formula for the
covariance of the iterated OU process, and consequently estimate the parameters
of an OU$(p)$ process by maximum likelihood or, as an alternative, by matching
correlations, the latter being a procedure resembling the method of moments.
