IRFs of ratios in levels

General DSGE Modeling
1

Dear Prof. Pfeifer,

thanks a lot for your helpful guidance in this forum :slight_smile: Its truely great that you answer all these questions! I hope that mine can be answered quickly.

I have a model in non-linear form without any exp() substitution. Some of my endogenous variables are ratios of two variables as for instance

  1. Ratio=A/B

Am I right to assume that, when entering the model in non-linear form without the exp() substiution and doing a first order approximation, Dynare produces IRFs that can be interpreted as level deviations from steady state?

  1. IRF_of_A = New_Level_A - Steadystate_A;
  2. IRF_of_B = New_Level_B - Steadystate_B;

I simply have to divide these IRFs by the steady state in order for them to be interpretable as deviations from steady state am I right?

Lets say I would like to back out the IRF of the “Ratio (A/B)” from the two IRFs of A and B in 2) and 3)

Am I right to assume that I could simply do the following

IRF_of_Ratio=(IRF_of_A+Steadystate_A) /. (IRF_of_B+Steadystate_B) - Steadystate_Ratio;

And would this give the same result that I would get from Dynare if I were to directly extract the IRF of the ratio after running the mod-file? I guess not because dynare also linearizes equation 1)?

Best,
Ben

2

You cannot reconstruct the original IRF of a nonlinear object by recombining the IRFs from linearized objects.

IRF_of_Ratio=(IRF_of_A+Steadystate_A) /. (IRF_of_B+Steadystate_B) - Steadystate_Ratio;

is what you would theoretically get for the ratio without approximations (if the IRFs were not based on an approximation as well). But that would be inconsistent with the fact that the IRFs are already from a linearized model. Thus, you should define

Ratio=A/B

in the model and consider its IRF. Of course Dynare will approximate this relationship, but that’s consistent with the rest of the model.

3

Alright thanks a lot for the fast response :slight_smile: