I am simulating some models with stochastic volatility in the fashion of Fernandez-Villaverde et al. (2011). I find that they compute comments with the 3rd order approximation. I wonder if a higher-order appromation than the 2nd one is necessary.
Yes, you need third order. At first order, there is certainty equivalence. At second order, there uncertainty affects only the constant term in the decision rules. Only at third order does time-varying uncertainty matter.
OK, thank you first. But I then have a follow-up question: I want to compute theoretical moments, yet I know Dynare does not directly provide them at the 3rd order. What way do you suggest? Should I replicate different paths and then aggregate them?
Theoretical moments at order=3 are not yet supported. For now, you could only use simulated ones.