  Standard system identification methods often provide inconsistent estimates
with closed-loop data. With the prediction error method (PEM), this issue is
solved by using a noise model that is flexible enough to capture the noise
spectrum. However, a too flexible noise model (i.e., too many parameters)
increases the model complexity, which can cause additional numerical problems
for PEM. In this paper, we consider the weighted null-space fitting (WNSF)
method. With this method, the system is first modeled using a non-parametric
ARX model, which is then reduced to a parametric model of interest using
weighted least squares. In the reduction step, a parametric noise model does
not need to be estimated if it is not of interest. Because the flexibility of
the noise model is increased with the sample size, this will still provide
consistent estimates in closed loop and asymptotically efficient estimates in
open loop. In this paper, we prove these results, and we derive the asymptotic
covariance for the estimation error obtained in closed loop, which is optimal
for an infinite-order noise model. For this purpose, we also derive a new
technical result for geometric variance analysis, instrumental to our end.
Finally, we perform a simulation study to illustrate the benefits of the method
when the noise model cannot be parametrized by a low-order model.
