Non-linear model: Why some variables are taken to log (and exp) and some are not

[Daria]

Hello all,

I am trying to replicate the model by Sims and Wu (2021) “Evaluating Central Banks’ Toolkit” in Dynare and was wondering why in this non-linear model some variables are taken to the exponential, but others are not? Is there any specific rule behind it? (see screenshots of the code attached)

The second thing that confuses me is that in equation 34 output Y is not taken to the exponential but in equation 37 it is (see attachment 2).

I would be grateful for any hints.

Kind regards,
Daria

[jpfeifer]

1. The typical rule is that everything that has a positive steady state and is not yet in percent or logged gets put in exp(). If something is already in percent or in logs, you do not again put it in exp(). There is another exception: if there is a variable you are not interested in at all, you often do not bother to do a substitution.
2. The multiplicative version of the Taylor rule would include \frac{R_t}{R}=\left(\frac{Y_t}{Y}\right)^{(1-\rho)\phi_\pi}. Then you take logs to get \log R_t-\log R=(1-\rho)\phi_\pi(\log Y_t-\log Y). However, due to the exp substitution, what is called Y_t in the code is already \log Y_t. So you would need to enter \log e^{Y_t}=Y_t, which is what you have there: no exp() and no log.