Log linearization of the technology dynamic

Question

I am still dealing with DSGE models and with log-linearization of characteristic equations of the model. The one that creates me more doubts is the technology shock, that is usually modeled as an AR(1).

Usually, we are given technology processes like \begin{gather} \ln(z_t)=(1-\rho) \ln(\bar z)+\rho \ln(z_{t-1})+\epsilon_t \end{gather} so that it is already linear and we can re-write it in deviations from the steady state just moving $\ln(\bar z)$ on both sides of equation, obtaining \begin{gather} \hat z_t=\rho \hat z_{t-1}+\epsilon_t \end{gather} where hat variables are log deviations from the steady state.

But when the process is something like \begin{gather} \ln(z_t)=(1-\rho) (\bar z)+\rho \ln(z_{t-1})+\epsilon_t \end{gather} how should I proceed? What I cannot understand is how the log linearization should be used, taking into account that $\epsilon_t$ is a white noise process. I mean applying logs and differentiating gives me something I do not understand.

Answer 1

The shock process is already linear. any log linearisation will result in identical expression. The underlying non linear shock process can be something like this:$

$$ Z_t = Z^{(1-\rho)} Z_{t-1} e^{\epsilon_{t}}$$

This when you linearise, you obtain your linear shock processes.