Second-order Perturbation in DSGE model with Stochastic Volatility

Dear Forum,

I am working with a DSGE asset pricing model with stochastic volatility. Somebody suggests me a third-order perturbation should be the minimum requirement. The reason is, in the second-order perturbation, only constant volatility affects the decision rule.

However, in Caldara et al (2012RED)'s case, the second-order and third-order perturbation shares almost the same result in the benchmark calibration. In my own research, the second-order perturbation with stochastic volatility can match the equity premium. Once I turn off the stochastic volatility or second-order perturbation to be first-order, the high equity premia and volatility disappear.

My question is, if, in a second-order perturbation, merely the constant volatility drives the equity premia. Why does the mere original TFP shock fail to produce the sizable equity premia? Why Caldara et al (2012RED) suggests that second-order perturbation is acceptable?

Or, the second and third-order perturbation, who is the minimum requirement? why?

If I adopt the second-order perturbation, is there any negative effects in terms of asset pricing research?

All comments are welcome.

If you want to have uncertainty shocks you need a third order approximation. At second order, volatility only has effects via a constant term. But the equity premium is the mean difference between the return on stocks and a risk-free bond. In this case, getting the effect of uncertainty on the mean is usually sufficient.

Dear Professor Pfeifer,

Nice to hear from you and thanks for your comments! I understand the reasons after a long discussion.

The second-order perturbation can quantitatively match the unconditional asset pricing moments and business cycle fluctuations as well as a third-order. However, if we would like to discuss the time-varying risk premium generated from the innovation term of the stochastic volatility, we need third-order perturbation. Some paper (Benigno et al 2013JEDC) may argue the first or second perturbation can generate time-varying risk premium, but these effects might not come from the innovation term of SV. Asset Pricing is all about co-variance risks, thus this implication would be important.