Additive vs multiplicative shocks

Just a basic question. I intend to put an additive shock (say z) in a quadratic adjustment cost function. There are two options. I can specify this as z(t) =(1-rho) *zbar + rho *z(t-1) + eps(t) or
z(t)=zbar^(1-rho)*z(t-1)^rho * exp(eps) where eps is a Gaussian white noise term and zbar is the steady state value of z. I understand that the latter loglinear option precludes negative z(t) but I am not concerned about it because the shock appears within the quadratic term and this negative shock won’t cause any trouble.

Both specifications yield similar results but the estimated variance of eps is implausible large in the loglinear form compared to other multiplicative shocks in the model (1.85 vs 0.02). My hunch is that dynare loglinearizes the second specification around steady state of zbar. The variance of eps actually involves zbar^2 multiplicatively which makes it it so large. My inclination is to stick to the first additive form which gives plausible estimates.
Any comments?

I am not sure I understand the problem, but in the additive specification, the shock is measured in units of z_t, while it is in percent for the multiplicative specification.

Thanks for this comment. To be on par with all other multiplicative shocks measured in percent, is it ok to have this particular shock with a high estimated standard error? If a particular shock has a very high estimated variance, is it a problem?

If you are so worried about this, why don’t you consistently define your processes? In the above discussion, z in the additive process corresponds to log(z) in the multiplicative one.