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.