Questions about adding AR(1) process in a set of log linearized equations


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.

Question 1:

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.

Is it

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.

Question 2:

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?



  1. Usually, you would have
Zeta_hat = rho*Zeta_hat(-1) - eps_zeta
  1. What do you mean with Zeta_hat_eps_zeta. You did not define it.
1 Like

Dear Professor Pfeifer,

Thank you for the reply. I really appreciate it. I figured out the answers to my questions.

For Question 1, the authors did not provide the process for Zeta nor steady state value for Zeta. So I assumed that Zeta follows log AR(1) process. That is,

Log(Zeta) = rho*Log(Zeta(-1)) - eps_zeta

I hope making this assumption is okay. Then Zeta in steady state would be 1. Hence,

Zeta_hat = rho*Zeta_hat(-1) - eps_zeta

For Question 2, Zeta_hat_eps_zeta is the Zeta_hat impulse response function values to eps_zeta. Zeta_hat is the simulation values. Question 2 was silly. I was confused between simulation and impulse response function.



I see. The nonlinear process

Log(Zeta) = rho*Log(Zeta(-1)) - eps_zeta

approximated at first order will result in

Zeta_hat = rho*Zeta_hat(-1) - eps_zeta

Thus, it makes no difference.