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.