This paper is concerned with the frequency domain quantification of
noise induced errors in dynamic system estimates. Preceding
seminal work on this problem provides general expressions that are
approximations whose accuracy increases with observed data length
and model order. In the interests of improved accuracy, this
paper provides new expressions whose accuracy depends only on
data length. They are therefore `exact' for arbitrarily small
true model order. Other authors have recognised the importance of
such expressions and have derived them for the case of FIR-like
model structures in which denominators are fixed at
true values and only numerator terms are estimated.
This paper progresses beyond this situation to address the much more
general Output-Error
and Box--Jenkins structures in which full dynamics models (both
numerator and denominator terms) and noise models
may be estimated. A key aspect of the work here is that it
establishes that the variance quantification problem is equivalent
to that of deriving the reproducing kernel for a subspace that
depends on the model structure being employed.