I have two simple questions that I really want to ask.
I tried looking for the answers online but I have trouble finding the answers to my questions.
Suppose I have a set of log linearized equations.
Within the set, a particular log linearized equation is
C_hat = sigma*(i_hat-Zeta_hat+Zeta_hat(+1))
where C_hat is a % deviation from the steady state consumption,
sigma is a constant, i_hat is a % deviation from the steady state interest rate,
Zeta_hat is a % deviation from the steady state Zeta.
If Zeta runs in AR(1) process and persistence parameter (rho) is 0.9, what is a proper way of writing Zeta in the model block? I am a bit confused because I have a set of log linearized equations in the model block.
exp(Zeta_hat) = 0.1+ rho*exp(Zeta_hat(-1)) - eps_zeta?
where eps_zeta is an error term. I added 0.1 because Zeta_hat has to equal to 0 in the steady state. I also placed -eps_zeta instead of +eps_zeta, because I want to examine IRFs to a negative shock.
What is the difference between Zeta_hat and Zeta_hat_eps_zeta? If there is only one exogeneous variable, which is eps_zeta, why are Zeta_hat_eps_zeta and Zeta_hat not the same value?