Hi all,
I am modelling stochastic volatility as in Born and Pfeifer (2014), using their functional form for the stochastic driving process, of the type:
log(z_t) = (1-\rho^z)\bar{z} + \rho^z z_{t-1} + e^{\sigma_t^z} \varepsilon_t^z,
\sigma_t^z = (1-\rho^\sigma)\bar{\sigma} + \rho^{\sigma}\sigma^z_{t-1} + \eta^z \varepsilon_t^\sigma.
Given that we are working with the log-normal distribution we observe that the mean of z increases when simulating the stochastic process with higher steady state volatility \bar{\sigma}. Does it make sense to compare the impact of an uncertainty shock under different levels of steady-state volatility, given that we hold the level process fixed?
Thanks!
If I would have to write that paper again, I would use a level specification in both the level and volatility equations rather than a log-log specification to avoid the problem of non-existent moments implied by the latter (Andreasen, 2010, EL):
z_t = (1-\rho^z)\bar{z} + \rho^z z_{t-1} + \sigma_t^z \varepsilon_t^z,
\sigma_t^z = (1-\rho^\sigma)\bar{\sigma} + \rho^{\sigma}\sigma^z_{t-1} + \eta^z \varepsilon_t^\sigma.
Also note that your question is ill-poised. In steady state, there is zero volatility of shocks. I presume you had a concept like the ergodic mean or the stochastic steady state in mind.
Ah, thanks for pointing out my mistake, and for the pointer to Andreasen (2010)!
What I had in mind was whether it is meaningful to compare a two standard deviation shock to the second moment process for different values of mean log-volatility \bar{\sigma} with the log-log specification (I guess not given your answer). I presume that the stochastic steady state would change, but should we also expect an additional effect from the fact that the conditional distribution of z will be increasingly right-skewed?
The answer depends on the exact research question you have in mind. But you are correct that the log-specification will have additional terms related to curvature that will affect the IRFs when you change the mean log volatility.