Exogenous equations approximation

Dear all,

I have a question about perturbation approximation of nonlinear exogenous stochastic processes like - say - stochastic volatility.

z_t = \rho_z z_{t-1} + \sigma_t \eta_t
ln(\sigma_t) = \rho_s ln(\sigma_{t-1}) + \epsilon_t

In this case, perturbations will approximate the z process taking derivatives and constructing polynomial according to the selected order.

My question is:
Since the process does not involve expectations and is known, why don’t we just leave it as it is (I.e. no approximations)?

Is this because in a perturbation we want the same order of approximation for all model equations (even exogenous ones)? In this case would it be possible/beneficial to substitute the true law of motion ex post (I.e. after approximation)?

This is just curiosity and I might be missing something. Every comment would be very much appreciated.

One reason is consistency in approximation orders. This also has the advantage that regardless of the form of the exogenous process, the solution to the model always takes the same form. This is a big advantage for software packages, because they do not need to be tailored to arbitrary nonlinear processes. Put differently, you always get the same state space representation for your solution.

Yes, I was expecting something like that. It would be nice to quantify the cost of such inconsistency in model solution.

Many thanks for your answer.