Heat-Jarrow-Morton interest rate dynamics and approximately consistent forward rate curves

We study a finite{dimensional approach to the Heath Jarrow Morton model
for interest rate and introduce a notion of approximate consistency for a family of functions
in a deterministic and stochastic framework. This amounts to asking the decrease
of the minimum distance in least squares sense. We start from a general linearly parameterized
set of functions and extend the theory to a nonlinear Nelson Siegel family.
Necessary and sufficient condition to have approximately consistency are given as well
as a criterion of stability for the approximation.