  Subspace identification methods (SIMs) have proven very powerful for
estimating linear state-space models. To overcome the deficiencies of classical
SIMs, a significant number of algorithms has appeared over the last two
decades, where most of them involve a common intermediate step, that is to
estimate the range space of the extended observability matrix. In this
contribution, an optimized version of the parallel and parsimonious SIM
(PARSIM), PARSIM\textsubscript{opt}, is proposed by using weighted
least-squares. It not only inherits all the benefits of PARSIM but also attains
the best linear unbiased estimator for the above intermediate step.
Furthermore, inspired by SIMs based on the predictor form, consistent estimates
of the optimal weighting matrix for weighted least-squares are derived.
Essential similarities, differences and simulated comparisons of some key SIMs
related to our method are also presented.
