Recently my mentor advised me to model risk shocks in DSGE models.

Though I’ve found some literature for reference, I’m still confused…How could changes of volatility affect the behavior of the model in a first-order linear approximation? By affecting the steady states?

Christiano/Motto/Rostagno (2014) is a leading study. Did they use log-linear procedure? How did their sigma affect play a role?

I had the same question. Checking the code for CMR (AER,2014) they use a 1st order pertrubation. So I decided to send an email to Christiano to ask him why is that the case.
His answer is that even though hocks to variances have zero effect, to first order, the risk shock has an impact when they use first order perturbation because it is a very different type of shock than a shock to aggregate volatility.
It is a shock to cross sectional volatility. Different values of that shock change the terms of debt contracts, and those changes are first order.

Yes, have a look at that paper. But the reason is quite simple. They model a shock to the idiosyncratic volatility. Because of the underlying log-normal distribution, a volatility shock will increase the mean of the level. As they do the aggregation manually, you get a first order condition, where volatility shows up as a first order term, requiring only first order perturbation techniques.