Partial Model Log-Linearization


#1

Dear All,

I have a question that I hope you can help me with. I am new to this literature, so this may be a dumb question.

Is it acceptable to study a model in which only some equations are expressed in log-deviations, while others are not?
In my case this would imply that some variables will be expressed in their levels in some equations and in their log-deviations in other equations (all the equations will be linear though).

To my, very limited, experience I have never encountered a paper doing this but, for what I want to do, it seems to me to be the only solution.

In case the answer is yes, it is possible to implement this in Dynare?

Thank you very much.


#2

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#3

Dear Johannes,

thank you for your quick and helpful reply.

Would you be so kind to direct me towards some studies in the literature doing this? For some reasons I could not find any (all the models I have encountered are entirely in levels or in log-deviations).

Specifically, I am working on a NK model (where inflation is linearized around a zero ss), in which I don’t want to loglin the monetary rule:

R_t=A(Pi_t-Pi*) + B(Y_gap)

The reason is that I want to study the model for different values on the CB’s inflation target (Pi*). As this is constant, I’d loose it in the log-lin version of the monetary rule and I won’t be able to study how the system changes by changing Pi*.
The model also contains equations with multiplicative terms that I think is better to log-lin in order to have a linear system of equations.

Thanks.


#4

Hi,
I don’t understand your problem here. A loglinearized model equation can still contain constant terms. That is not a contradiction. If you don’t linearize by hand, Dynare will do so (of course keeping the constant).
The reason papers don’t talk about this is that in the end, everything is in log-deviations from a steady state. There is no point in stating the approach to get there.


#5

Andrea,

I think the “non linear” monetary rule you have mentioned is actually already linear. The non-linear version of that monetary rule is

R_t = (Pi_t / Pi*)^A times (Y_t/Y*)^B

Take the log of this and you have
ln (R_t) = Aln(Pi_t - Pi) + Bln(Y_t - Y)

If R = 1 + r and Pi = 1 + pi are gross terms with r and pi sufficiently small (less than or equal to 10%), then ln R = r, and ln Pi = pi so that the above is actually

r_t = A*(pi - pi*) + B*yhat.

My guess is that some of the other “non-linear” terms you mention are actually already linear if you look really closely.