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
and
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.