Technology process, Taylor rule and Output gap

Hello everyone,

I’m trying to replicate a model and have a few questions since I’m not experienced at it yet.

  1. Technology in this paper follows an AR(1) process which is:

ln(A_t) = rho_A ln(A_(t-1)) + s_A e_(A,t)

My question is how to log-linearize it. Should I just subtract the steady state which is ln(A) = rho_A*ln(A) ?

Another question is just for a better understanding: why do we multiply the technology shock e_(A,t) with its standard deviation s_A?

And why do we write this process in logs and not just like this: A_t = rho_A*A_(t-1)+ e_(A,t)

  1. The policy rate is set according to a Taylor rule which is:

ln(R_t) = (1-rho_r)lnR + (1-rho_r)[phi_pi*(ln(Pi_t)-ln(Pi))+phi_x*(ln(Y_t)-ln(Y*t))] + s_r e(r,t) (1) where Y*_t is the steady state of Y_t.

I think there’s a term +rho_r*ln(R_(t-1)) missing in this interest rate rule, as the log-linearized version of this rule is the following:

r_t = rho_r r_(t-1) + (1-rho_r)[phi_pi pi_t + phi_x x_t] + s_r*e_(r,t)

Would you say this term +rho_r*ln(R_(t-1)) is missing in the (1) equation?

Another question would be also how to log-lin it. Just subtract the from (1) which is ln® = (1-rho_r)lnR + (1-rho_r)[phi_pi*(ln(Pi)-ln(Pi))+phi_x*(ln(Y)-ln(Y*))] (probably + rho_r*lnR for that missing term) ?

  1. My last question is about the steady state value of the output Y_t. Apparently, they define it as Y*_t. But I don’t understand why the time subscript is used in the variable which describes the steady state value (Y*_t) of the output (Y_t). As I understand the meaning of the steady state it’s a constant value of a variable which doesn’t depend on time.
    Could you help me with the understanding of this?

Thank you all.

  1. That equation is already linear. Simply define ln(A) as the variable.
  2. You can either specify a standard deviation for e_A in the shocks-block or use a unit standard deviation there and premultiply the standard normal shock by its standard deviation. Both ways are equivalent.
  3. Sometimes the process is written this way to show how it is related to the level A. The other way defines \tilde A_t=ln(A_t)
  4. Yes, it seems the lagged interest rate is missing.
  5. Again, the equation is already linear in logs.
  6. Without knowing the paper, it is impossible to tell what Y_t^* is. But often it is not steady state output, but rather natural output, which is time-varying.