  In system identification, it is often difficult to find a physical intuition
to choose a noise model structure. The importance of this choice is that, for
the prediction error method (PEM) to provide asymptotically efficient
estimates, the model orders must be chosen according to the true system.
However, if only the plant estimates are of interest and the experiment is
performed in open loop, the noise model may be over-parameterized without
affecting the asymptotic properties of the plant. The limitation is that, as
PEM suffers in general from non-convexity, estimating an unnecessarily large
number of parameters will increase the chances of getting trapped in local
minima. To avoid this, a high order ARX model can first be estimated by least
squares, providing non-parametric estimates of the plant and noise model. Then,
model order reduction can be used to obtain a parametric model of the plant
only. We review existing methods to perform this, pointing out limitations and
connections between them. Then, we propose a method that connects favorable
properties from the previously reviewed approaches. We show that the proposed
method provides asymptotically efficient estimates of the plant with open loop
data. Finally, we perform a simulation study, which suggests that the proposed
method is competitive with PEM and other similar methods.
