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