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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.

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  • $\begingroup$ I don't think this makes any sense, as $\overline{z}$ is the expected value of the process in the first equation, only. Do you have a source for the second one? $\endgroup$ – Chris tie Oct 10 '16 at 15:46
  • $\begingroup$ @Christie we were actually wondering if it were a typo (having forgotten to put logs in $\bar z$, the problem is that the source is a problem set by my Adv. Macro TA $\endgroup$ – PhDing Oct 10 '16 at 16:03
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    $\begingroup$ Ok, I have a strong feeling that it is a typo, then $\endgroup$ – Chris tie Oct 11 '16 at 8:07
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    $\begingroup$ As a teacher I find it curious that students first contact messageboards and only if that fails do they maybe contact the TA. Has anyone ever lost points by contacting their TA? (Or is the TA a werewolf or something similarly dangerous?) $\endgroup$ – Giskard Oct 11 '16 at 14:19
  • $\begingroup$ It depends on the TA's availability to answer ;) $\endgroup$ – PhDing Oct 11 '16 at 15:24
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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.

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